Theory of Defects in Solids: The Electronic Structure of Defects in Insulators and Semiconductors (Monographs on the Physics and Chemistry of Materials) 019851378X, 9780198513780

This book surveys the theory of defects in solids, concentrating on the electronic structure of point defects in insulat

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Theory of Defects in Solids: The Electronic Structure of Defects in Insulators and Semiconductors (Monographs on the Physics and Chemistry of Materials)
 019851378X, 9780198513780

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CONTENTS I. THE

PERFECT

SOLID

1. CRYSTALS AND LATTICE GEOMETRY Holle Introduction: types of crystal 1.2. Structure of simple lattices 1.2.1. NaCl structure 1.2.2. CsCl structure

1.2.3. CaF, structure 1.2.4. Diamond and zincblende structures 1.2.5. Wurtzite structure 1.3. Geometric properties: lattice potentials

—"N W HW LS OOD —

2. ELECTRONIC STRUCTURE OF THE PERFECT LATTICE Zale Basic equations and approximations 2.1.1. Static lattice approximation 2.1.2. The one-electron approximation 2.1.3. Koopmans’ approximation 2.1.4. Approximations for exchange 2.1.5. Coulomb correlation DD Band theory 2.2.1. Basic assumptions 2.2.2. Band structures: general features 2.2.3. Band structures: wavefunctions 2.2.4. Band structure: examples 2.2.5. Correspondence between electrons and holes

14 14 14 15 16 17 18 22 4p 22 26 29 33

3. LATTICE DYNAMICS Syke Adiabatic approximations 3.1.1. The Born—Oppenheimer approximation 3.1.2. Adiabatic approximations for degenerate systems 3.1.3. Accuracy of adiabatic approximations 3.2. The harmonic approximation 3.2.1. Normal modes 3.2.2. General results in the harmonic approximation 3.2.3. Limits of the harmonic approximation 3.3. The dipole approximation 3.3.1. Introduction 3.3.2. Lattice dynamics

36 36 36 37 38 40 41 42 44 45 45 46

CONTENTS

3.4. Models for interatomic forces 3.4.1. Introduction 3.4.2. Ionic crystals 3.4.3. Valence crystals 3.4.4. Partly-ionic crystals 3.4.5. Rare-gas crystals SiS) Lattice dynamics 3.5.1. Dispersion curves 3.5.2. Density of states 3.6. The electron-phonon interaction 3.6.1. The Hellman—Feynman theorem 3.6.2. Electron—phonon interaction: delocalized case 3.6.3. Local models for electron-lattice coupling

IPELECRRONIGISTRUGIUWRE OF ISOLATED, DEFECTS 4. EFFECTIVE-MASS THEORY 4.1. Introduction 4.2. Simple effective-mass theory 4.2.1. Fundamental equations for the defect lattice 4.2.2. Approximations of effective-mass theory 4.2.3. The function A(r) 4.2.4. The effective-mass equation 4.2.5. The wavefunction 4.2.6. More complicated band structures 4.2.7. Matrix elements in effective-mass theory 4.3. The quantum-defect method 4.4. Effective-mass theory for a many-body system 4.4.1. Many-body eigenfunctions for the perfect lattice

4.4.2. Fundamental equations for the defect lattice 4.4.3. Approach using exact many-body states as a basis 4.4.4. Approach using Hartree-Fock states as a basis 4.5. The accuracy of the effective-mass approximation 5. GREEN’S FUNCTION METHODS Dale Introduction 5.1.1. Choice of basis 5.1.2. The Green’s function 5.1.3. Matrix elements of the Green’s function 522: Bound states: the Koster-Slater model 5.2.1. The one-band, one-site model 5.2.2. The eigenvalue equation aya} Resonant states 5.3.1. The Green’s function 5.3.2. The scattering problem

48 48 48 50 51 Sv 52 52 53 54 54 5) 59

65 65 65 65 67 68 69 70 71 75 76 77 78 79 80 83 84 87 87 87 87 87 90 90 91 Se) 97 98

CONTENTS

xi

5.4. Band structures with subsidiary minima 5.4.1. General theory: minima at different points of the zone 5.4.2. Resonant states

5.4.3. Expressions for AE, 5.5. Other approaches using Green’s function methods 5.5.1. The Bassani, Iadonisi, and Preziosi method

5.5.2. KKR and T-matrix methods 5.6. Choice of potential: validity of Green’s function methods

. VARIATIONAL

METHODS

103 104 105 107 108 108 109 2 115 115 115 116 117 Uy 118 119 121 126 126 130

6.1. Introduction 6.1.1. Basic theorems 6.1.2. The few-electron approximation

6.2. The orthogonality constraint 6.2.1. Modulated band functions 6.2.2. Atomic orbital methods 6.2.3. Explicit orthogonalization 6.2.4. The pseudopotential method 6.2.5. Defects with several electrons 6.3. The accuracy of a variational calculation 6.4. Choice of a variational wavefunction

. MOLECULAR METHODS AND MODEL CALCULATIONS 7.1. Introduction 7.2. General methods 7.2.1. Molecular-orbital and valence-bond methods 7.2.2. Separation of singlet and triplet states

sa

7.2.3. Localized bonds and hybridization 7.3. Approximate methods 7.3.1. Approximations for matrix elements 7.3.2. Semi-empirical approaches 7.3.3. Model calculations 7.4. Weak covalency 7.5. Method of localized orbitals 7.5.1. The density operator 7.5.2. Self-consistent solutions 7.5.3. Conditions on the orbitals 7.5.4. Application of localized orbitals methods 7.5.5. Approximate methods and local orbitals

SLATTICE DISTORTION 8.1. Introduction 8.2. Linear coupling

FOR)

DEFECT

SYSTEMS

18S 133 133 133 135 136 138 138 141 144 146 149 150 152 153 155 157 159 159 161

Xil

CONTENTS

8.3. Static distortion near defects 8.3.1. Linear response formalism 8.3.2. Methods for calculating static distortion 8.3.3. Long-range distortion from point defects 8.4. The Jahn-Teller instability 8.4.1. Introduction: the quasi-molecular hypothesis 8.4.2. Potential energy surfaces 8.4.3. Static and dynamic Jahn-Teller effects 8.4.4. Matrix elements of electronic operators : the Ham effect 8.5. Other asymmetric systems 8.5.1. Introduction 8.5.2. Fast- and slow-rotation limits 8.5.3. Other energy levels for rotating systems 8.5.4. Choice of potential for rotating systems 8.6. The bound polaron 8.6.1. Introduction 8.6.2. Weak electron-lattice coupling 8.6.3. Intermediate coupling: more advanced models 8.6.4. Strong coupling 8.6.5. The Toyozawa—Haken-Schottky model 8.6.6. Local phonon modes associated with bound polarons 8.6.7. The bound polaron: final comments

9. GENERAL RESULTS ole Introduction O72; Number of bound states Os! Nature of the ground state 9.4. Order of energy levels 3) Scaling relations and the virial theorem OIG: Symmetry

164 164 166 181 186 186 187 197 218 223 223 224 229 230 231 231 235 237 248 250 JEsy) 253

256 256 256 259 262 264 266

III. CALCULATION OF OBSERVABLE PROPERTIES OF DEFECTS 10. OPTICAL PROPERTIES 10.1. Optical absorption by atoms 10.2. Optical properties of centres in a solid 10.3. The Condon approximation 10.4. The line-shape function 10.5. Smakula’s formula and the effective-field correction 10.6. Einstein coefficients and the principle of detailed balance 10.7. The line-shape function with electron-lattice interaction 10.7.1. Introduction 10.7.2. Linear electron-lattice coupling 10.7.3. Beyond linear coupling

271 271 276 2H 280 282 289 291 291 294 309

CONTENTS

10.8.

Method of moments 10.8.1. Introduction 10.8.2. Non-degenerate states: linear and quadratic coupling 10.8.3. The configuration-coordinate diagram 10.9. Forbidden transitions: non-degenerate states 10.10. Electronic degeneracy 10.10.1. Introduction 10.10.2. Splitting of optical bands due to degeneracy 10.10.3. The Huang-Rhys factor with orbital degeneracy 10.10.4. Antiresonances 10.11. Photoionization of defects 10.11.1. Introduction 10.11.2. Dependence on the conduction band 10.11.3. Dependence on the defect wavefunction 10.11.4. Photoionization with phonon assistance

Ml, DYNAMICS OF IMPERFECT LATTICES Wb. Introduction UL Comparison with electronic defect systems Pikesh Defects in a linear chain 11.4. The method of classical Green’s functions U3): The method of thermodynamic Green’s functions 11.5.1. Introduction 11.5.2. Equations of motion 11.5.3. Correlation functions 11.6. Response functions 11.6.1. Introduction 11.6.2. Relation to classical Green’s functions 11.6.3. Absorption of energy from external forces SF. The isotopic impurity 11.8. Asymptotic expansions 11.8.1. Local modes 11.8.2. The existence of local modes 11.8.3. Bounds on local modes 11.8.4. Low-frequency resonance modes 11.8.5. Anharmonicity of local modes and resonances iS). Infrared absorption 11.9.1. The dipole moment 11.9.2. Charged defect in a homopolar lattice 11.9.3. Charged defect in a polar lattice 11.10. The T-matrix

xiii

310 310

312 ei) 317 320 320 B28 328 328 331 331 333 334 340

342 342 345 345 348 350 350 Spill 555 354 354 355 356 359 361 363 365 365 368 370 375 375 376 377 382

CONTENTS

XIV

11.11. Thermal 11.11.1. 11.11.2. 11.11.3. 11.11.4.

conductivity Introduction: the relaxation time The long-wavelength limit: Rayleigh scattering The peak theorem Summary and comparison with infrared absorption

. EXTERNAL FIELDS AND THEIR EFFECTS ale Introduction 1 Electric and optic fields: the Stark effect 12.2.1. 12.2.2. 12.2.3. 12.2.4. 12.2.5. 12.2.6.

Introduction: linear and quadratic Stark effects Linear Stark effect Effects of electric fields on wavefunctions Reorientation of defect under electric fields External optical fields Secondary radiation: the Raman effect and hot luminescence a3 Magnetic fields: the Zeeman effect 12.3.1. Introduction 12.3.2. Zeeman effect in atoms and effective-mass systems 12.3.3. Linear Zeeman effect :general defects 12.3.4. Faraday rotation and related phenomena 12.4. Stress fields 12.4.1. Effects of stress on energy levels 12.4.2. Effects of stress on wavefunctions 2.5). Strong electron-lattice coupling: moment methods 12.5.1. Introduction 12.5.2. Zero-phonon lines 12.5.3. Broad-band transitions 12.5.4. Raman effect 12.5.5. The rigid-shift hypothesis

3k ELECTRON-SPIN RESONANCE 1320 Introduction 132% The spin Hamiltonian 1ESES! The Zeeman effect 13.3.1. Orbital angular momentum 13.3.2. Examples of g-factors 13.4. The zero-field splitting 13h3) Electron—nucleus interaction :hyperfine structure 13.5.1. Isotropic hyperfine interaction 13.5.2. Anisotropic hyperfine interaction 13.5.3. Coupling of nuclear moment to the electronic orbital moment

383 383 385 386 389 390 390 390 390 392 395 397 397 400 402 402 403 413 416 422 423 426 428 428 428 429 434 437

438 438 439 442 444 447 450 455 457 462 465

CONTENTS

XV

13.6. Quadrupole interaction 13.6.1. The electric field gradient 13.7. Measurement of lattice distortion by spin resonance

14. NON-RADIATIVE PROCESSES AND INTERACTION OF FREE CARRIERS DEFECTS 14.1. Introduction

466 467 473

WITH

14.2. Non-radiative transitions 14.2.1. General theory 14.2.2. Paramagnetic relaxation 14.2.3. Resonant and non-resonant absorption 14.2.4. Temperature dependence of the zero-phonon line 14.3. Scatter of conduction electrons 14.3.1. Introduction 14.3.2. Limiting cases 14.3.3. Scatter by neutral impurities 14.3.4. Scatter by ionized impurities 14.3.5. Other approaches 14.4. Capture of electrons in solids 14.4.1. Introduction 14.4.2. Capture by phonon emission 14.4.3. The Auger effect and related phenomena 14.5. Kinetics

477 477 477 477 490 501 505 512 ol 514 515 ol y/ 519 520 520 Sy) 539 547

IV. COMPARISON OF THEORY AND EXPERIMENT 15. THE F-CENTRE AND RELATED ONE-CARRIER SYSTEMS 15.1. Introduction 15.1.1. The hydrogen atom model of the F-centre 15.1.2. Empirical rules for optical absorption 15.1.3. Optical emission and the relaxed excited state 15.1.4. Empirical rules for optical emission 15.1.5. Other one-carrier centres 15.2. The ground-state configuration of the F-centre 15.2.1. Introduction 15.2.2. Optical absorption 15.2.3. Electron spin resonance 15.2.4. Spin—orbit coupling 15.2.5. Response to stress 15.2.6. Response to electric fields 15.2.7. Optical absorption line-shapes

555 S25) 5D») 556 558 561 561 562 562 562 571 574 iil 581 583

Xvi

CONTENTS

15.3. Excited-state configuration of the F-centre 15.3.1. Introduction 15.3.2. Calculations of the relaxed excited state 15.3.3. Model calculations of the F-centre excited states

15.4. Perturbed F-centres 15.4.1. Introduction 15.4.2. The F,-centre 15.4.3. Photochromic centres in CaF, 15.4.4. The M *-centre

595 395 596 602

608 608 608 613 613

15.4.5. The Z,-centre

614

15.5. Antimorphs of the F-centre :trapped-hole centres 15.5.1. Trapped-hole centres in oxides 15.5.2. The zinc vacancy in ZnSe 16. CENTRES WITH TWO CARRIERS 16.1. Introduction 16.2. The F’-centre 16.2.1. Bound states in the alkali halides 16.2.2. Bound states in the alkaline earth oxides 16.3. Two electrons at close anion vacancies

615 615 618 620 620 620 620 623 625 625 627 629

16.3.1. The M-centre 16.3.2. The F,-centre in alkaline earth oxides 16.4. Two-hole centres in the alkaline earth oxides

17. THE R-CENTRE 17.1. Introduction 17.2. Electronic structure of the R-centre 17.2.1. The Kern—Bartram model 17.3. The R-centre and the Jahn-Teller effect 17.3.1. Introduction 17.3.2. Wavefunctions 17.3.3. Optical matrix elements and transition probabilities 17.3.4. Response to external fields 17.3.5. Comparison of LHOPS theory with the R-centre: summary

631 631 632 632 638 638 638 640 640

18. THE 18.1. 18.2. 18.3. 18.4. 18.5. 18.6. 18.7.

653 653 655 658 661 663 664 667

V,-CENTRE AND THE RELAXED Introduction Atomic configuration of the V,-centre Optical transitions Spin resonance The V,-centre as a molecule in a crystal Motion of the V,-centre The relaxed exciton in ionic crystals: [V,e]

EXCITON

651

CONTENTS

XVil

. THE H-CENTRE AND OTHER INTRINSIC DNGGE RSTn vATes 19.1. Introduction 19.2. Anion interstitials in ionic crystals 19.2.1. Defect structures 19.2.2. Theory of the H-centre lattice configuration 19.2.3. Interactions with other defects 19.2.4. Electronic structure of the H-centre 19.3. Interstitials in valence crystals 19.3.1. Experiment and the interstitial 19.3.2. Theories of interstitial migration 19.3.3. Theories of interstitial electronic structure 20. HYDROGEN IN IONIC U,-CENTRES 20.1. Introduction

CRYSTALS:

U-, U,-, AND

20.2. The interstitial atom H? (U,-centre) 20.2.1. Nature of the ground state: spin resonance

20.3. The substitutional hydrogen ion H- (U-centre) 20.3.1. Optical properties 20.3.2. Infrared absorption: the local mode oe THE REORIENTATION 21.1. Introduction 21.2. Electric dipole centres

OF

670 670 670 670 672 674 675 677 677 678 680

DEFECTS

IN SOLIDS

681 681 682 682 687 688 688 692 699 699 699

21.2.1. CN” and OH 7 in alkali halides 21.2.2. Hydrogen halides in rare-gas hosts 21.3. Elastic dipole centres: the O; molecular ion 21.4. Off-centre ions 21.4.1. The KCI: Li system 21.4.2. Occurrence of off-centre ions 21.4.3. Oxygen interstitial in silicon

699 703 704 710 710 le 714

2. TRANSITION-METAL IONS 22.1. Introduction 22.2. Classes of theory and coupling schemes 22.3. The iron-group in cubic symmetry 22.3.1. The cubic field splitting 22.3.2. Theories of the cubic field splitting 22.3.3. Distribution of electrons over orbitals 22.3.4. Optical properties 22.3.5. Spin resonance parameters

WG 717 718 720 720 721 WS Wey) 733

XVili

CONTENTS

22.4. Rare-earth ions 22.4.1. Free rare-earth ions

742 742 745

22.4.2. Crystal fields for rare-earth ions

23. SHALLOW DONORS SEMICONDUCTORS

AND

ACCEPTORS

IN

23.1. Introduction

751

751

23.2. The application of effective-mass theory :non-degenerate

bands

751

23.2.1. Energy levels

752

23.2.2. Wavefunctions 23.2.3. Response to perturbations 23.3. Applications of effective-mass theory :degenerate bands

755 TSU 758

23.3.1. Energy levels and wavefunctions 23.4. Corrections to effective-mass theory 23.4.1. Terms which depend only on the host 23.4.2. Corrections which depend in detail on the electronic structure of the defect 23.4.3. Lattice deformation

758 762 764

766 Ws

24. ISOELECTRONIC IMPURITIES 24.1. Introduction 24.2. Binding mechanisms 24.2.1. Mechanisms independent of lattice distortion 24.2.2. Mechanisms which involve lattice distortion

781 781 782 782 787

24.3. The one-band, one-site model 24.3.1. Band structure 24.3.2. Energy levels: bound states and resonances

790 790 792

24.3.3. 24.4. Beyond 24.4.1. 24.4.2. 24.4.3.

794 800 800 801

Optical line-shapes the one-band, one-site model The single impurity bound state The nitrogen pair bound state Phenomenology: use of an effective range and depth potential 24.5. Phonon structure in optical spectra

25. DIPOLAR SYSTEMS AND DONOR-ACCEPTOR PAIRS 25.1. Introduction 25.1.1. Lattice geometry 25.2. Transition energies 25.2.1. One carrier bound to a finite dipole 25.2.2. Pair recombination spectra 25.3. Transition probabilities 25.3.1. Basic theory

802

803 806 806 807 808 808 813 822 822

CONTENTS

25.3.2. Time dependence of recombination 25.3.3. Other transitions

26. BOUND EXCITONS 26.1. Introduction 26.2. Quantum chemistry of the bound exciton 26.2.1. Introduction 26.2.2. Basic transition energies 26.2.3. Electron bound to a neutral donor, and its antimorph 26.2.4. Exciton bound to ionized donor, and its antimorph 26.2.5. Exciton bound to neutral donor, and its antimorph 26.2.6. Other systems 26.3. Other properties of bound-exciton systems 26.3.1. The Jahn-Teller effect 26.3.2. Optical line-shapes 26.3.3. Undulation spectra 26.4. Response to external fields 26.4.1. Zeeman effect 26.4.2. Response to stress

Dip VACANCIES IN VALENCE CRYSTALS Pbifetle Introduction PHP. Vacancy centres in silicon PieSs Other observations of vacancy centres 274. Electronic structure in the undistorted lattice 27.4.1. Defect-molecule models 27.4.2. Cluster models 27.4.3. Other methods 27.4.4. The one-electron approximation 27.4.5. Summary Poles Electronic structure with lattice distortion 27.5.1. Basic assumptions 27.5.2. Forces causing distortion 27.5.3. Effective frequencies :the lattice response 27.5.4. Consequences of lattice relaxation 27.6. Summary of problems APPENDIX

xix

826 827

829 829 829 829 830 831 832 838 842 843 843 844 846 848 848 850 851 851 852 857 860 860 865 869 871 872 873 873 874 875 878 882

I

884

Sum rules

APPENDIX II The factorization of secular equations

887

REFERENCES

889

AUTHOR

INDEX

925

SUBJECT

INDEX

945

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1 CRYSTALS

AND

LATTICE

GEOMETRY

1.1. Introduction: types of crystal ParRT I of this book discusses those aspects of perfect crystals which are relevant to the defect systems treated later. There will be no attempt to review all the properties of perfect crystals. Instead, the opening chapters will be confined to defining notation, to stating essential assumptions, and to asserting without proof results which are needed later. The present chapter is primarily concerned with the geometric properties of the important crystal types. But, since crystal structure is intimately related to other microscopic properties (see, e.g. Phillips 1970), it is useful to start by classifying crystals into a few basic types. Seitz (1940) has made a detailed comparison of the various cases. Some of his categories will not be discussed at all here, notably those in which behaviour is dominated by free carriers. Examples are metals, semi-metals, and heavily-doped semiconductors. Nor is there any appreciable discussion of organic solids, which have their own range of defect phenomena. The four categories considered in most detail are: 1. Ionic crystals, notably the alkali halides, alkaline earth oxides, and alkaline earth fluorides. These crystallize in the cubic NaCl, CsCl, or CaF, structures.

2. Semiconductors, which are (by any definition) more covalent and less ionic. The most interesting are the II-VI and III-V compounds, which form zincblende and wurtzite structures.

3. Valence crystals, including diamond, silicon, and germanium; most involve only a single atomic species, but there are cases,

like silicon carbide, with several types of atom. The strong covalent bonds in these systems favour relatively open structures, in contrast to the relatively close-packed ionic crystals. 4. Rare gas solids and solids whose basic units are saturated molecules are mainly bound by weak forces like the van der Waals

4

CRYSTALS

AND

LATTICE

GEOMETRY

interaction. The short range of the interaction favours close packing, and the weakness ofthe interaction leads to a low melting point.

1.2. Structure of simple lattices We now consider some of the commonly-met structures, and tabulate properties of interest in discussing defects. General symmetry properties are not given. Apart from general books (e.g. Slater 1965; Tinkham 1964), three reviews are particularly relevant. Koster (1957) discusses Brillouin zones and the symmetries of points in them. His discussion can also be used to discuss the Wigner— Seitz cells and their symmetries. Bell (1954) describes a method of obtaining the angular parts of wavefunctions centred on a point in a lattice. She also gives tables of these functions for cubic and close-packed hexagonal lattices. Loudon (1964a) relates the crystal space group and the point group ofa lattice site in a form convenient for discussion of the selection rules for processes involving defects 1.2.1. NaCl structure

Nearest-neighbour distance Lattice parameter

a 2a

Volume per unit cell

2a?

These are cubic crystals, with two atoms per unit cell. The sublattices are both face-centred cubic. A. Substitutional sites. The first few shells of neighbours are these.

Shell I D. 3 4 D

Typical member 100 110 tat 200 210

Number ofsites 6 he? 8 6 24

Sublattice O S O S O

(S = some sublattice as substitutional site, O = other sublattice.) The various shells are at points (A,B,C) where (A even for S and odd for O sublattices. The degeneracies are 48 for (A, B,C), 24 for (A, A, B) and (A, B,0), 12 8 for (A, A, A), and 6 for (A,0,0). Here A # B # C ¥ also some accidental degeneracies, i.e. shells at the same not related by symmetry.

+ B + C) is of the shells for (A, A, 0), O. There are distance but

CRYSTALS

AND

LATTICE

GEOMETRY

WN

B. Interstitial sites. The two most obvious sites are:

(a) the body-centred site: this has tetrahedral symmetry, with four cation and four anion neighbours. There are two distinct sites related by inversion; (b) the face-centred site: this has two cation and two anion neighbours. It is readily distorted into the . 16/3,/3 = 3-0792 a’.

These

are cubic crystals. The

anion

cations lie on a face-centred cubic interstices in the anion sublattice.

lattice is simple cubic;

lattice occupying

the

alternate

A. Substitutional sites. The first few shells of neighbours are:

Shell 1 2 3 4 5

Typical member | 200 220 311 222

Anion site Degen-

eracy 4 6 12 12 8

Sublattice + — + _

Typical member 111 220 311 400 331

Cation site Degen-

eracy 8 12 24 6 24

Sublattice — + _ + -

The neighbours to the anion site are trivially related to those for the CsCl structure. Cation neighbours of the cation site occur with coordinates (A, B, C) a/./3 and at distances a,/(8N/3), with A, B, and C even. Anion neighbours have A, B, and C odd and occur at

distances a,/{(8N — 5)/3}. B. Interstitial sites. The two most obvious sites are these: (a) The body-centred site. This is the body-centred site of both sublattices. The nearest neighbours are a cube of eight anions. (b) The ‘face-centred’ cubic site (equivalently, the edge centre of

CRYSTALS

O

AND

LATTICE

GEOMETRY

Anion

@® Cation Ml

Edge-and

face-centred

interstitial sites

Fic. 1.2. The CsCl structure.

the fluorine lattice). This is at the centre of a (110) face; it has two anion neighbours at distance al,/3 and two cation neighbours at a ,/2/3. There are six equivalent sites, ie. three pairs corresponding to the different cube axes joining the anion neighbours with two possible directions for the join of the cation neighbour. This interstitial position readily becomes the ¢110> ‘crowdion’ configuration. The sites are shown in Fig. 1.3.

© Anions ® Cations (MM Body-and edge-centred interstitial sites Fic. 1.3. The CaF, structure.

7

8

CRYSTALS

AND

LATTICE

GEOMETRY

1.2.4. Diamond and zincblende structures

Nearest-neighbour distance Lattice parameter

a 4a/./3

Volume per unit cell

am, 16/3,/3 ~ 3-0792 a.

These are cubic crystals with two atoms per cell. The diamond structure is often best considered as zincblende in which both components are chemically identical. The structure can be derived from face-centred cubic by placing two atoms on each site, one from each sublattice, and separating them along a )°

DY

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QO; oo pktt Yim(;, 0;)

EL ee emia i4 LM\Wi> 3) Yi): Y OYiuG;:, time

(1.3.2)

(3) cos

The infinite sum in g,y is evaluated by the method of de Wette and Nijboer (1958). The potential is in atomic units if the lengths are in atomic units; to convert to angstrom units and electron volts one

multiplies by 14-4. Note that the expansions may be about any origin, not necessarily a lattice site. The results may be illustrated by a number of examples: 1. The spherically-symmetric part of the potential (L = M = 0) about an anion site is needed in a number of calculations of Fcentre energies. Fig 1.6 shows results for a number of different crystal structures. It is remarkable how similar the results are in all cases, apart from a simple scaling.

12

CRYSTALS

AND

LATTICE

GEOMETRY

Point-ion potential

L r/(nearest-neighbour distance)

Fic. 1.6. The point-ion potential. The spherically symmetric part of the point-ion potential near an anion vacancy is shown. The resulting potential wells have been normalized to the same width and depth for the different structures (NaCl ——, CsCl or CaF, ----- , and ZnS ——-).

2. The Madelung constant « enters in a number of defect energies. We use the definition in which the potential at a site is written:

V = a(Ze?/a),

(1.3.4)

with a the nearest-neighbour separation and Z an assigned charge; this is different from the definition common in discussions of cohesion (e.g. Born and Huang 1954) in which the energy per unit cell is written in terms of the lattice parameter. Values of a for systems of interst are 1-74756 for the NaCl structure, 1-76267 for CsCl, and 1-63805 for ZnS. In the fluorite structure «

is 1-76267 at the anion site and ~3-2761 at the cation site; the bodycentred interstitial site is attractive to positive charges and has « equal to —0:2492. Other values are given by Waddington (1959), Johnson & Templeton (1961), and Tosi (1964). Birman (1955) gives results using Gaussian charge distributions instead of point charges. Potentials at sites other than lattice sites are given by Slater (1967, p. 310), Hajj (1972), Curie & Barthou (1971), and Barthou (1970). 3. Discussions of the properties of magnetic ions often invoke the effect of the L = 4and L = 6 components of the point-ion potential of near-neighbours. Expressions for four-, six-, and eight-fold coordination are given by Griffith (1961, p. 204) and Hutchings (1964).

CRYSTALS

AND

LATTICE

GEOMETRY

In these cases it is always assumed that (r/r,) is negligible, wavefunction does not overlap the lattice ions. Similar available for corrections to the potential when there lattice distortions (e.g. Van Vleck 1940; Tucker 1965). a measure of the spin-lattice interaction.

13

so that the results are are small These give

2 ELECTRONIGASTRUCRURESO PERFECT LATTICE

Ts hit

2.1. Basic equations and approximations

WE begin from the non-relativistic Hamiltonian for a system of electrons and nuclei. Omitting small spin-dependent terms, the Hamiltonian has the form:

(2.1.1)

KH = HKH.+ Ant An, where the purely-electronic term is

#,=5. eas

fmt?

2

is

(2.1.2)

Sa Ine}

the purely-nuclear term is 2

Hy = 2 =a

2

sD oon

(2.1.3)

and the electron—nuclear interaction is e2

Hy = — A

red

(2.1.4)

Solution of the Schrédinger equation using (2.1.1) is difficult for atoms, and essentially impossible for solids. We now outline the set of approximations used to resolve the difficulties. 2.1.1. Static lattice approximation

This approximation is a combination of two steps. First, an adiabatic approximation is used to separate the electronic and nuclear motion. These approximations are discussed in §3.1: in

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

15

essence, the electronic Schrédinger equation is solved for fixed nuclear positions, and the electronic eigenvalue contributes to the potential determining the nuclear motion. Secondly, it is recognized that the nuclei are not usually displaced very much from their mean positions, either by thermal vibrations or near defects. Thus the electronic Schrédinger equation is only solved in detail for the nuclei at their mean positions. We discuss in § 3.6 the electronlattice coupling, which determines the changes when the nuclei are moved from their mean positions. 2.1.2. The one-electron approximation The electronic wavefunction in the static lattice approximation is a function of the coordinates of all the electrons and satisfies (44+ Ay)O(r,,..., ty; R) = €.(R)O®.

(2.1.5)

Exact solution is very rarely possible, and even numerical solutions become intractable for systems with more than a few electrons. The one-electron approximation factorizes (2.1.5) into equations involving only one coordinate r; explicitly. Thus some of the correlations in the motion of the different electrons are lost. There are two main versions of the approximation when the orbital wavefunctions do not depend on the electron spin. First, there is the Hartree form (Hartree 1928)

OY = Pi(r,)... ON(ty).

(2.1.6)

Secondly, there is the Hartree-Fock form (Fock 1930; Slater 1930):

DHF — S{d,(r,)... by( ty)} = Det||¢,(r,)

...

y(t)

Oren)

prj)

ilty) ... x(t) I where . is an antisymmetrizing operator. We shall only consider the Hartree-Fock approximation, since the Hartree form does not satisfy the exclusion principle, nor are its solutions orthogonal.

16

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

We seek an optimum solution of(2.1.5) using (2.1.7) as a variational trial function. This leads to a set of equations for the one-electron functions:

-f 94,ree 1ferrioien i — jFi

et oH

- feeaa lr—r| Be » Zi TeaRra

meee)

(2.1.8)

The potential which determines the one-electron function is a sum of three terms: a consistently-chosen Coulomb potential from the electron-electron interaction, an exchange term, and the nuclear potential. The exchange term correlates the motion of electrons with parallel spins, and is a consequence of the exclusion principle. No such term appears in the Hartree approximation. Correlation of the motion of electrons with parallel spins is not included. 2.1.3. Koopmans’ approximation

The one-electron eigenvalues ¢; can have physical significance. Koopmans (1933) showed that ¢; is Just the energy required to remove an electron in state ¢; from the system, provided that the other one-electron states @j # i) are not affected by the removal. Thus the energy of a transition which takes an electron from state ; to state ¢, is given by

AE = &—&.

(2.1.9)

The relation fails for small systems, including molecules and localized defects, where the transition leads to a substantial change in the other wavefunctions. The relation also fails if the exchange term in (2.1.8) 1s approximated. The success of(2.1.9) in practice stems from a cancellation of the terms resulting from changes in the ¢, in a transition and from those associated with the neglect of correlation. Fowler (1966a) has discussed these problems for the alkali halides. The sum of the one-electron eigenvalues )';¢; has no special physical significance. It is not the total electronic energy, since the electron—electron interaction is counted twice. The sum is easily calculated and is, in consequence, often misused.

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

17

2.1.4. Approximations for exchange

The potential for the exact exchange contribution to (2.1.8) is a nonlocal operator, i.e.

V.xb(0) = |dr! Vale, eb)

(2.1.10)

For computational convenience V,, is sometimes replaced in the equations for one-electron orbitals by an approximate local operator. The form usually adopted is

VX" I= —6a(3p,/4z2)* Ry

—3-7221ap; Ry

(2.1.11)

(Slater & Johnson 1972). Here p, is the density of electrons with parallel spins in ag *, and « is determined by some other criterion. There is no variational principle for a. Slater (1951) suggested a = | is appropriate for an average over occupied states in a band. Kohn & Sham (1965) argued « = 2 was suitable for the highest occupied states in a partly-filled band. Both these results are based on estimates of the interaction in a nearly-homogeneous electron gas. However, the p;-dependence can be argued much more generally. Slater & Johnson’s (1972) Xa method exploits this. They suggest that, within each atomic sphere in a polyatomic system, « be replaced by the value (Schwarz 1972) which gives the correct total energy for the corresponding free atom. Since the virial theorem (§ 9.5) still holds, the kinetic and potential energies will also be correct. Exchange potentials are usually used to give an approximate form of Hartree-Fock theory. Thus one-electron energies are found from the differences in energy with and without an electron in a particular orbital. However, Koopmans’ result (2.1.9) breaks down when approximations are used for exchange. One-electron energies given by differences in total energy for two occupancies n; = 0, 1

Ej = Eur(n; = 1)— Eur(n; = 0)

(2.1.12)

are not accurate, and special corrections are needed (Lindgren 1965; Herman, van Dyke, & Ortenburger 1969). Experience with (2.1.12) has suggested that local exchange operators are only useful for orders of magnitude, and that the correct exchange operator is worth the extra effort. However, Slater & Johnson argue that

18

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

(2.1.11) is merely being used in the wrong way. For one-electron energies, one should use

=

oR eri On;

(255213)

where n; is fixed to minimize the total energy. Instead of using Koopmans’ theorem to predict transition energies, one uses energies ¢/(n;) where nz = (Ninitias + Ninas)/2 iS the occupancy of the transition state. This value allows for some of the changes in other one-electron orbitals between the initial and final states. The way in which occupation numbers of states are used means that, for multiplets, the energy corresponds to a weighted mean; the multiplet structure is not given. In this mode, local exchange gives very good one-electron energies. It is not clear yet how good the orbitals obtained are. The nonlocal nature, and hence the momentum-dependence of the operator, is missing. Overhauser (1970, 1971) has commented on a number of special problems with local exchange, and shows that they become especially acute when correlation is not included at the same time. The forms given here only involve p,, and clearly do not lead to any correlation of electrons with antiparallel spins. Some approximate treatments replace p, by half the total electron density, although this may be very poor. 2.1.5. Coulomb correlation The correlation energy is defined here as the difference between the exact energy and the energy calculated from Hartree-Fock theory |he

7

ect

2 OAR

(2.1.14)

Other definitions are possible. Thus Goddard (1967) has argued convincingly that we should not use Eyr, but use instead the energy of the best independent-particle model Eg,. One then refers to (Egi—Eyr) as the ‘static’ correlation and to (E,,,.,— Eg ) as the ‘dynamic’ correlation. Only this second dynamic term depends on the detailed relative motion of the electrons. We shall use (2.1.14), since this is the standard convention, although static correlation dominates in many systems. It is a matter of convention too whether relativistic corrections are included in Ey. The distinction is of

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

19

importance in accurate atomic calculations, but rarely significant in solid state theory. For many purposes E,,,, may be neglected. About 99 per cent of the total energy is given for molecules by Hartree-Fock theory (Lowdin 1959a; Moshinsky 1968). But other properties may be sensitive to correlation, such as the small energy differences between Hartree-Fock states. We now describe some of the methods for estimating the electron correlation, most of which were first developed for molecules. The one general rule is that quantitative estimates need not be too difficult if the qualitative nature of the correlation is understood. A. Improved independent-particle models. By an independentparticle model, we mean that the total wavefunction is constructed from one-electron orbitals determined by a mean field from the other electrons. Only the static correlation contribution is obtained here, although this may be the major part. Various forms exist, including the method of ‘split orbitals’, ‘unrestricted Hartree— Fock theory’, and Goddard’s (1967) GI method. In Hartree-Fock theory, the many-electron wavefunction is a single determinant in which the one-electron orbitals have the same spatial parts for opposite spins. The first generalization is to allow different orbitals for different spins. This is especially useful in simple two-particle systems like the H™ ion, or in magneticallyordered systems where the internal field gives an important spindependent term in the Hamiltonian. The second generalization is to replace the Slater determinant method of antisymmetrizing by the use of a projection operator which ensures the resulting wavefunction is an eigenfunction of total spin (Goddard 1967, 1968; Lowdin & Goscinski 1969). The spin eigenfunctions are generated from the properties of Young tableaux (McWeeny & Sutcliffe 1969, § 3.6; Rutherford 1948). The projection-operator method does not seem to have been used in solid state problems, although it seems eminently suitable in some cases. B. Explicit dependence on interelectron separation. This method adopts the most direct picture of correlation, namely that detailed electron motions adjust to avoid close encounters. The effect may be described by the ‘correlation hole’ for electrons with antiparallel spins. The exchange interaction gives an analogous effect for parallel spins (e.g. Kittel 1964, pp. 94-5). Hylleraas (1964) and

20

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

Sinanoglu (1961a,b) have discussed the approach. It proves very difficult to evaluate matrix elements of r;; = r;—r;, the interelectron separation, and the approach is only really useful for two-electron systems. The dependence onr;, gives some insight into the slow convergence of other calculations of the correlation energy. The Schrodinger equation has singularities when r; and r, are equal. The singularities give cusps in the wavefunction which are hard to reproduce in other forms. Slater (1960, p. 39) demonstrates this clearly, and shows that the leading term in the wavefunction has WY a

exp (r;2/2ao)

a

1+(r,2/2a9)+

eee

(PANES)

near r, = r,. The higher terms are not so simple as our example suggests. Nevertheless, the cusp behaviour at small r,, and the correlation hole are both shown. Gilbert (1963) has argued that rather good convergence to the correlation energy can be achieved by including dependence on r,,, primarily because the cusp can be so simply represented by an expansion in rj. C. Configuration interaction. For a given Hartree-Fock wavefunction ®p, there will always be other Hartree-Fock wavefunctions ®,, with the same symmetry and spin. The one-electron orbitals of ®, and @®,,; will, of course, have different occupancies, and the energies of these states will differ. In the configuration interaction method, one includes the admixture of states ®,; into ®) by the electron-electron interaction. The new many-electron wavefunction is a sum of appropriately-weighted determinants @

=

ApPo+

> a,®,i,

(2.1.16)

and the energy reduction gives the correlation energy. In principle, configuration interaction can give exact manyelectron eigenstates and eigenvalues. In practice only a few determinants are included, and good results are obtained only if the nature of the correlation is understood. It may be radial correlation, where electrons tend to occupy regions at different distances from the defect. It may be angular correlation, where the electrons tend to move in different solid angles. Or the electrons may tend to associate with different lattice sites. Banyard & Baker (1969) observe that angular correlation tends to replace radial correlation as the nuclear attraction increases. The configuration interaction

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

21

method is especially attractive when there are two nearly degenerate configurations, since they will give most of the correlation. In all cases it is convenient to ensure that the higher states ®,, are orthogonal to the one of interest ®). This question, and the choice of basis, are discussed by Gilbert (1965) and Adams (1967).

D. Correlation potentials. The aim here is to represent the effects of correlation by a one-electron potential. The important result (Hohenberg and Kohn 1964) is that the ground state of a manyelectron system is uniquely given by the electron density. Further, the many-electron problem can be replaced by an exactly-equivalent set of one-electron equations. The potential in these equations is a functional of the electron density, but is unknown in general. Various approximations are possible (March & Stoddart 1969). The commonest are based on the exact results for the electron gas at high densities (Gell-Mann & Brueckner 1957) and low densities (Wigner 1934, 1938). Thus Wigner’s suggested interpolation is corr

=e

ere aes

(2.1.17)

- 7:8 a (8 Popp)*

where p,,, is the density of electrons of opposite spin. This form becomes inaccurate at high densities. Herman et al. (1969) discuss a number of such potentials. Overhauser (1971) gives an expression for the sum of the exchange and correlation contributions VG

Views a

—2-07(p

| + Dopp)? Ry

(2.1.18)

which appears to be accurate over a wide range of electron densities. E. Use of empirical data. Measured properties, notably atomic energy levels, dielectric constants, and band gaps, can be built into calculations. These already include correlation, and the resulting calculation may include most, if not all, of the correlation energy. This is true of effective-mass theory, for example, where manyelectron effects are included in the experimental masses and dieelectric constants. F. Other methods. In molecular physics, both the theory and its application are developed further than for solids. The theory has been extended in two ways: the qualitative nature of the correlation has been explored in more detail, and the formal theory has been rewritten in various more general and more convenient ways. This

22

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

work is reviewed by Loéwdin (1959), Yoshizumi (1959), McWeeny (1960), Sinanoglu (1964), Nesbet (1965), and Pauncz (1969). Sutton, Beracino, Das, Gilbert, Wahl, & Sinanoglu (1970) give a useful tabulated comparison of the various methods. A very penetrating analysis of the related problem of pair correlations in nuclear matter is given by Gomes, Walecka, & Weisskopf (1958).

2.2. Band theory 2.2.1. Basic assumptions Band theory is an approximate theory of the perfect lattice in a form which emphasizes the translational symmetry of the system. It uses the approximations outlined in § 2.1, namely the static lattice, one-electron, and Koopmans’ approximations. Corrections to these are needed before comparison with real crystals, especially because of the neglect of correlation and the fact that Koopmans’ result is only valid when the effective one-electron potential is not sensitive to the actual occupancies of the one-electron orbitals. In addition to these general approximations, there are several other common assumptions. First, an exchange potential like (2.1.11) may be used. This invalidates the Koopmans’ result, and further corrections are needed if it is to be used. Secondly, a correlation potential may be used to improve a little on the one-electron approximation. And thirdly, a ‘muffin-tin’ potential may be used. In this approximation the volume is divided up into non-overlapping spheres centred on each atom, and the potential assumed constant between the spheres. The muffin-tin approximation is not a good assumption for any of the insulators or semiconductors we shall be considering, and the truncation of the potential outside the spheres has a profound effect on the resulting band structure (Kunz, Fowler, & Schneider 1969; Painter, Ellis, & Lubinsky 1971). 2.2.2. Band structures: general features

The one-electron eigenstates of band theory are classified by a band index n and a wavevector k. We now consider the eigenvalues E,(k) for typical insulators and semiconductors. The eigenstates which are occupied at absolute zero temperature constitute the valence bands; the unoccupied states belong to the various conduction bands. Collective modes, like the plasmon modes, are usually ignored in band calculations.

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

23

Here we consider the qualitative features of E,(k). Some of these concern the extrema ofthe bands—the lowest states of the conduction band and the highest valence band states. Thus we are interested in the values of k for which E,(k) is extremal, whether the energy surface is degenerate near the extremum, and the energy surface near the extremum. Other properties, such as the density of states, are also clearly important. The band extrema occur at points k,,, where V,E,(k) vanishes. As a rule, the lowest conduction band has just one minimum at the centre of the zone, k = 0. This is so for most alkali halides and II-VI and III-V compounds. In other important cases the conduction

band has several equivalent extrema away from k = 0. Examples are given in Table 2.1. The maxima of the valence bands are almost TABLE 2.1 Positions of conduction band minima

in the Brillouin zone

Number of minima

Position in zone

Silicon

6

(K00) and equivalent

Germanium

4

(LLL) and equivalent

AISb

3

(K00) and equivalent

Most others

1

(000)

Crystal Diamond

GaP

invariably at the centre of the zone, k = 0. For direct-band-gap materials the extrema of the lowest conduction and highest valence bands lie at the same point of the zone; others, such as diamond,

silicon and germanium, have an indirect band gap (Fig. 2.1). For the materials

of interest to us, the conduction

bands

are

non-degenerate near their extrema. There is just a single energy surface E,(k), although it may exhibit a different type of degeneracy through having several equivalent minima. The lowest conduction band is often overlapped by higher conduction bands, but this complication will rarely be important. In most cases, however, the valence bands are degenerate. This occurs when the valence band is formed primarily from a full shell of atomic p-electrons, for example. The valence band degeneracy can be lifted by spin-orbit coupling (e.g. Elliott 1954) or by low-symmetry crystal fields. Such splittings

24

aa an

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

(b)

Fic. 2.1. Band structures with direct (a) and indirect (b) gaps.

are shown for the wurtzite and zincblende structures in Fig. 2.2. The spin-orbit splitting Aso is largest when the corresponding atomic spin-orbit coupling is large, as for anions of high atomic number. Thus Ago is larger for germanium than for silicon or diamond, and larger for iodides than for the other alkali halides. Cubic,

|

no spin-orbit

|

coupling

Cubic, spin-orbit

splitting

k

Axial, no spin —orbit

coupling

|

vain

YOO NaS

Axial, with spin — orbit

splitting

Fic. 2.2. Effects of spin-orbit coupling and low-symmetry fields on band structure.

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

25

The energy surface near the extrema of a non-degenerate band can be obtained by expansion

E(k)

d7E,(k) Ee g dyk— ah ak dk, | |

220)

If the expansion is truncated at this point we have the effective-mass approximation, with an effective-mass tensor

:

idea ened diene.

O22)

My];;

Different combinations of the components of the mass tensor enter in the: various predictions. For electrical conduction one

uses );-,,3m;

', and for the density of states ([];-,.3 m,)*, where

both expressions use a principal-axes representation. The energy surfaces are anisotropic near the extremum for non-cubic crystals and also for cubic crystals where the extrema lies away from the centre of the Brillouin zone. The expansion (2.2.1) cannot be used near the extremum of a degenerate band. For such degenerate cases the matrix elements of the Hamiltonian among the various states can be expanded in powers of (k—k,); the energy surfaces are complicated and usually anisotropic. The density of states N(E)dE, that is the number of states between E and (E+dE), is also important. Given E(k), there are general numerical methods for evaluating the density of states. The most important feature is that N(E) shows singular behaviour. We write the density of states for electrons of one spin as

NAE) =

{d?k5{E—E,(k)}

= @, |,ASV En.

(2.2.3)

where Q, is the volume of the Brillouin zone and the second integral is over an energy surface of constant energy E. Clearly there is a singularity at the critical points (van Hove 1953; Phillips 1956) where V,E,(k) is zero. Critical points are involved in defect problems in several ways. Thus the corresponding band wavefunctions are important in

26,

ELECTRONIG

STRUGTUREVORF

THE

PERE E GIs EAI iiGE

constructing wavefunctions of weakly bound defects. Secondly, transitions such as photoionization have a probability which depends on the density of states and which may show structure corresponding to the critical points. Thirdly, the band-structure data used in defect calculations is often best obtained by modulation methods which exploit the singularities. Exactly analogous critical points occur in phonon densities-of-states. 2.2.3. Band structures: wavefunctions A. Band functions and Bloch functions. The one-electron functions of band theory are found from the equation How

Ak, r) =

E,(k)W,(k,

r).

wave-

(2.2.4)

The band functions w,(k, r) can be written to make their translational symmetry explicit

w,(k,r) = e™Tu,(k, r). We adopt as normalizations functions

of the Bloch

(2:2'5) functions

and band

{ derw*(k, r)W,(k’, r) = 64, (kK —k’),

(2.2.6)

d?r u*(k, r)u,(K,r) = 6,,/Q, = cell

=|

@rw*(k, nW,(k,r).

(2.2.7)

cell

The Bloch functions are periodic in the lattice, that is u,(k, r) = u,(k, r+) for any translation t which leaves the lattice invariant. These functions are not orthogonal for different values of k; indeed

I d?r u*(k, r)u,(k’, r) = cell

= Onnq/Q,+ terms linear and higher in (k—k’).

(2.2.8)

But the u,(k,r) for a given k and for all bands n form a complete

orthonormal set of states satisfying the equation h

Ho+—k.n+

Mo

Zip

2 Mo

u,(k,r) = E,(k)u,(k, r).

(2.2.9)

ELECTRONIC

STRUCTURE_OF

THE

PERFECT

LATTICE

27

In this equation m is the momentum p. plus a correction term (6 A VV)/4moc? which appears when there is spin-orbit coupling.

B. Wannier functions. The theory of localized defects can be developed in terms of the band functions or Bloch functions of the host lattice. But it is convenient to adopt different basis functions, related

to the Bloch functions, which are localized in space. One set of functions commonly used are the Wannier functions (Wannier 1937) 1 a,(r —Ro) = Tx » exp (—ik. Ro)y,(k, r)= k

=e

|e exp (—ik .Ro)W,(k, r).

(2.2.10)

For the Wannier functions is just the mean band energy ), E,(R)/>\, 1. They are defined so as to be orthogonal for different bands and for different lattice sites Rg. Other choices are discussed by Ferreira & Parada (1970). Formal properties of Wannier functions are discussed by many authors. The localization properties and those of related functions are discussed by Blount (1962, p. 327) and Lix (1971). Extremal properties of the expectation value of the Hamiltonian are analysed by Koster (1953), Parzen (1953), and Blount (1962, p. 324). One property important in defect calculations concerns the choice of phase. Whilst any phase can be used if chosen consistently, certain choices have advantages. One might choose the phases to localize the Wannier functions as much as possible, by minimizing the expectation value of r* for example (Blount 1962, p. 327). Another possibility is to ensure continuity of symmetry even when bands cross at special points in the Brillouin zone (e.g. Callaway & Hughes 1967; Ferreira & Parada 1970). Or the Wannier functions may be defined by imposing the condition E,,, ,(k) > E,(k) throughout the zone. Special problems are discussed by Turner & Goodings (1965) and Goodings & Harris (1969). The Wannier functions reduce to atomic functions in the limits of extreme tight binding. If there is one atom in the unit cell, and if the overlaps of the atomic functions ¢(r) centred on sites separated

by R, are S,,, then (Landshoff 1936; Wannier 1937; Lowdin 1950)

a(r—Ro) = He —Ro)—5 Y S,G(r—Ro—R,)+ u#0

28

ELECTRONIC

STRUCTURE

(OF) THE

PERFECT

LATIICE

The small admixtures of wavefunctions at other sites ensure the orthogonality properties of the Wannier functions, and enhance the values of a(r) at these sites over the unorthogonalized values. When there is more than one atom per cell, special precautions must be taken if Wannier functions are to be associated with individual crystal atoms (Hughes 1968). C. Point symmetry and band functions. Defect problems involve the point symmetry of a state for a particular defect. We may need to know the point symmetry of band functions for defects on different lattice sites. No problems arise if the band is accurately described by the tight-binding approximation in terms of atomic orbitals. But other cases can be more subtle, such as the conduction band extrema in GaP (Morgan 1968). A simpler system, the diatomic linear chain of Fig. 2.3, shows the same effects if its conduction band ‘is assumed, like GaP, to have

Fic. 2.3. Band functions for a linear diatomic chain of atoms.

minima at the edge of the Brillouin zone. If the atoms A and B are identical, then there are two degenerate states at the edge of the zone. They are Wa(ko,r) = Au)(k, r) sin (k yx), which has a node at the B sites and a maximum at the A sites, and W,(ko,r) = Aud(k, r) cos (kgx), whose nodes are at the A sites. If the A and B sites are made inequivalent then w, and wz are split. When the A sites are the more attractive, W, is the lower in energy. We may then ask: What is the parity of the lowest conduction band function appropriate to a defect on the two sites? The results given in Table 2.2 show that the wavefunctions at the band extrema have different parities for defects on the two different sites. Since the matrix elements for the transitions of shallow donors depend on the symmetry of these band functions, the difference in parity can have observable consequences.

ELECTRONIC

STRUCTURE

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LATTICE

29

TABLE 252 Parity of zone-boundary wavefunctions for the diatomic chain Parity of lowest conduction band state More attractive

Lowest conduction

site

band state

At A site

At B site

A

Vs

+

B

We



-

+

The theory for GaP is exactly analogous. The conduction band extrema lie at the edge of the zone in the [100] and equivalent directions. The P sites are attractive relative to the Ga sites. The lowest conduction band for defects on the P sites (group IV acceptors, group V isoelectronic traps, or group VI donors) and the lowest conduction band for defects on the Ga site (group II acceptors, group III isoelectronic traps, or group IV donors) have different point symmetries. The selection rules, which are consistent with observation, suggest that strong emission without phonon cooperation should be observed from neutral centres on the P site, from donor-acceptor pairs with the donor on the P site, but only from deep donors on the Ga site. 2.2.4. Band structures: examples A. Relation to constituent atoms. A prototype band structure can be constructed to show features typical of insulators and semiconductors. The constituent atoms are first arranged on a lattice of the correct symmetry and large spacing. Secondly, the electronic configurations of the atoms are changed to those of the final structure. Thus the electron transfers needed to create ionic crystals, or the hybridization for covalent crystals, are made at this stage. For our model system we assume that the highest-energy orbitals which are occupied are p orbitals and that the lowest unoccupied orbitals (not necessarily on the same species of ion) are s orbitals. Thirdly, the lattice parameter is reduced to the correct crystal volume. The discrete levels of the ions are broadened out by the interactions between the ions to give the energy bands. The widths of the bands are a measure of the interaction between ions. Clearly,

the ratio of the widths to the splittings of the ionic levels depend on the nature of the band. The ratio is small in the tight-binding

30

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

limit, where intra-ionic terms are dominant, but is large in the nearly-

free-electron approximation. B. Typical band structure. The highest valence band in the model system is almost entirely derived from the p orbitals, and the lowest conduction band from s orbitals. The band structure near the zone centre is shown in Fig. 2.4. The effective mass can be calculated using (2.2.9) and assuming a matrix element P = /m. Perturbation theory gives the band energies near the extrema as Conduction band:

lenick

E = E, +5 a

(electron energies)

Me 1

Valence bands:

1

e

for

2)-2

E,= Eyo+5 i

(hole energies)

2

— imp? atesalt

:

us Z 2 fal +mP 2 = Bots4+=1 h?k? |1+—mpP?|—

Bo = (Eo

1 h?k?

2

E; 3 = Ey+= ots

1

1+=mP? sgt? al

~ A (22711)

Conduction band

Eg

dei ele on

hay ay Valence band

Fic. 2.4. ‘Typical’ band structure.

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

31

For small k the valence band is split by the spin-orbit coupling, which separates E; from E, and E, by A, and by the terms in P. E, and E, form a ‘heavy-hole band’ E, and a ‘light-hole band’ IOP.

We now compare this model with important classes of crystal.

Alkali halides. Here the p orbitals are anion orbitals and the S orbitals the cation ones. The valence bands are reasonably well represented by a tight-binding approximation. For the iodides the spin-orbit splitting A is much larger than the width of the lighthole band, but the opposite is true for the fluorides, for example. The conduction band has more free-electron character, and involves s orbitals mainly on the cations, but with a sizeable admixture of anion s orbitals. The conduction band structure is complicated by other nearby bands involving d-states which have minima away from the zone centre. Recent calculations include those of Kunz & Lipari (1971a,b). Valence crystals: diamond, silicon, germanium.

Both the conduc-

tion and valence bands are complicated by the closeness of s and p orbitals, which leads to hybridization. For diamond sp* hybrid orbitals are formed, and a good description of the valence band can be obtained as linear combinations of bonding orbitals formed from the hybrids. Clearly the promotion energy (which is E(2s 2p*)—

— E(2s? 2p’) for carbon) is an important parameter. For silicon and germanium admixtures of d orbitals are important. For all these crystals the conduction band has minima away from the centre of the Brillouin zone. Further, the widths of the conduction

and valence bands are large compared with the band gaps. This is typical of systems where the nearly-free-electron model, rather than the tight-binding model, is appropriate. One can get reasonable descriptions of the band structures starting from either extreme.

C. Free-ion and crystal ion orbitals. onic wavefunctions in crystals are often approximated by orbitals of free ions. We review here the accuracy of this approximation for crystals reasonably described in the tight-binding approximation. The main differences between free-ion and crystal-ion wavefunctions come from (i) the effect of the crystal potential, such as the

32

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

Madelung potential in ionic crystals, (ii) the effects of orthogonalizing orbitals on one ion to those on neighbouring ions, and (iii) covalency. We concentrate on ionic crystals and ignore the covalent contribution. Evidence comes from four sources. First, there are Hartree-Fock

calculations of wavefunctions for a free ion in a potential well; the well represents a crystal field. Thus Watson (1958) discussed O?~ in a square-well potential. The O*~ ion is not stable as a free ion; once stabilized by the extra potential, the wavefunction is not very sensitive to the depth of the well. The second observation comes from Compton scatter, which samples the electron momentum distribution in the crystal. Measurements on O77 (Togawa, Inkinen, & Manninen 1971) do not agree well with values based on Watson’s calculations. However orthogonalization should certainly eliminate some of the discrepancies. The third set of measurements are of X-ray structure factors, which measure the charge distribution. These are reviewed by Weiss (1966). Maslen (1967) finds that uncorrected free-ion wavefunctions are best for CaF,; Aikala & Mansikka (1970) find symmetricallyorthogonalized free-ion functions best for LiF, and Kim & Friauf (1971) find the best functions for KCl have Madelung corrections as well as orthogonalization corrections. The relative importance of the corrections in the different crystals is also confirmed from the fourth set of measurements, the isotropic hyperfine constants of the F-centre. Stoneham, Hayes, Smith, & Stott (1968) find the orthogonalization correction small for CaF,, whereas Wood (1970) found a significant correction needed in KCl. One other simple modification to atomic wavefunctions which is used frequently is scaling (Yamashita 1952, Yamashita & Kojima 1952; Mansikka & Bystrand 1966; Petersson, Vallin, Calais, & Mansikka 1968; Calais, Makulé, Mansikka, Petersson, & Vallin 1971). The wavefunctions of the outer electron shells are scaled so

that W(r) > W(nr). The value of 7 is fixed by some simple variational procedure, or a fit made to some observable such as X-ray scattering. Scaling factors 7 usually differ from 1 by a few per cent. Clearly this is a rather primitive modification, and it would be surprising if it lead to valid quantitative results for other properties. But qualitative information may be of wider value. Thus it is useful to know that, in LiF, it is the fluorine orbitals alone which are

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

33

changed significantly under compression (Mansikka & Bystrand 1966). More complicated corrections are necessary for the wavefunctions of impurity ions in crystals ;we discuss these systems later. 2.2.5. Correspondence between electrons and holes The correspondence between electrons and holes is used in several forms in solid state physics. The best-known example is of transport in semiconductors. Suppose there are no free carriers when a given semiconductor contains N electrons. Then the same sample with (N—n) electrons may be treated as to an n-particle problem in which the n carriers are supposed to have a positive effective mass and a positive charge. Another example occurs in crystal-field theory (Griffith 1961, p. 245). Suppose we are interested in the states of systems in which the occupancies of one shell are varied. For iron-group transition-metal ions in cubic symmetry we might consider systems with different occupancies of the t, levels derived from the 3d shell. Griffith shows that the states with n electrons in this shell can be related to the states in which n electrons have been removed from a full shell. Thus, in our example, the states

of t5, are related to those of t$~”. In this case the phases defining the various states must be chosen with care for the correspondence to be useful. In this section we have a very limited aim. We write down the matrix elements of a Hamiltonian between various Slater determinants, and rewrite them to correspond to contributions from a full band of N electrons, together with corrections from the n holes or electrons additional to the band. Thus the situation is closer to the example of transport theory rather than the crystalfield case. The total Hamiltonian is the sum of #,, which consists of oneelectron operators, and #,, the electron-electron interaction,

I H =H, +H, = HAZYVy. i

(2.2.12)

i,j

We consider the matrix elements of # among determinantal wavefunctions |J> and |J> describing a system with (N-n) electrons.

34

ELECTRONIC

STRUCTURE

THE

OF

PERFECT

LATTICE

First we consider the expectation value NE

nN-n

CSET) = SsCENId+5 y dL Ki

-—. Suppose t is the order of the permutation needed to get the identical one-electron functions in |J) and |J> in the same order. Then, if |) and |J> differ by just a single

ELECTRONIC

STRUCTURE

OF

THE

PERFECT

LATTICE

35

one-electron wavefunction |k> N

CII \J> = Ck NKYD+(— DY (Ck iRtiD — Cik iy) +

+(-1)'*! y ( — (ik! ki), Fk

(2.2.15)

and what is essentially the electron-hole interaction changes sign. If |J> and |J> differ by two functions |k> and |/> we find

CSET = (— INCRE — CREEP KD). There is no one-electron term here, and the results the same for n extra electrons and n holes. Clearly cases of |I) # |J> the phase relations between the portant, and hence the importance of Griffith’s work field problem.

(2.2.16)

are essentially in both these states are imin the crystal-

3 LATTICE

DYNAMICS

3.1. Adiabatic approximations THERE are three major approximations in conventional lattice dynamics. They are an adiabatic approximation to separate the electronic and nuclear motion, a harmonic approximation to relate the theory to an exactly soluble model, and a dipole approximation to simplify both microscopic models and the parameterization of experiments. We review these in turn. The term adiabatic refers to the evolution of a system when some external parameter is changed such that no transitions are induced. Thus, for a system of electrons and nuclei, the electronic system often responds adiabatically when the nuclear configuration is changed. The total Hamiltonian (2.1.1) can be written

H = T+ Tat Vit, R),

(3.1.1)

where T, and 7, represent the electronic and nuclear kinetic energies and V the various contributions to the potential energy. The electron and nuclear coordinates are r,R and the corresponding momenta p, P.

3.1.1. The Born—Oppenheimer approximation

If there is no degeneracy the adiabatic approximation indicates a total wavefunction of the form

M(r, R) = (r; R)x(R).

(3.1.2)

The electronic part yw depends parametrically on R. We shall treat ® as a variational trial function and seek equations for y and yw. The electronic wavefunction w satisfies

{T.+V(r, R)}W(r; R) = ¢.(R)Y(r; R).

(3.1.3)

It is determined by solving the electronic problem when the nuclear coordinates are fixed at R. The direct interaction between nuclei is included in ¢,(R) for convenience. Our assumption that

LATTICE

DYNAMICS

37

® is non-degenerate means that y/(r; R) is non-degenerate. Thus we can choose to be real and normalized over r, and we may assume w and e, vary smoothly with R. The equation for nuclear motion is found variationally (Longuet—Higgins 1962) and is:

b pure1

+ eel)+w®)) in(R) = exin(R),

3.1.4)

where

(R)

W’'(R)

=

1

Slee d3 ry *___p2 aM, ow q

al (3.1.5)

The effective potential energy in this equation for the nuclear motion is dominated by the electronic eigenvalue ¢,(R). This term also incorporates the direct interaction between nuclei. The other term W’(R) is small, in part because of the nuclear (rather than electronic) mass in the denominator and in part because the y(r; R) are often insensitive to nuclear displacements. The effective potential for nuclear motion is independent of the nuclear masses only if W’'(R) is neglected. This independence is well verified for molecules. The eigenvalue éy obtained when W’(R) is neglected is itself of interest, since it can be shown that the exact eigenvalue without adiabatic approximations ¢,,,,, Satisfies the bounds (Bratsev 1965; Epstein 1966)

ENTE Tee

(3.1.6)

Ene

A further approximation is sometimes used, where the wavefunctions y(r; R) are assumed to be independent of R both in eliminating W’(R) and in calculating ¢,(R). The electronic expectation value used instead of ¢,(R) is then d*r y*(r; Ro) {T. + V(r, R)} Wr; Ro). No

In this limit the electrons follow translations and rotations of the

nuclear system, but not vibrations.

3.1.2. Adiabatic approximations for degenerate We consider degenerate states in the same determine the electronic wavefunction at a guration. The simple factorization (3.1.2) is no

systems spirit as before, and fixed nuclear confilonger adequate for

38

LATTICE

DYNAMICS

the total wavefunction, ®(r, R). Instead we postulate

@(r;R) = Db)war; R)xAR)

(3.1.7)

where the Wr; R) are a set of real orthonormal functions, degenerate when R = Ro. The equations which replace (3.1.4) are

1 152M Py + UAR)} n+ v v

s » v j#i

Qi;

Pix; aa ENXi>

(3.1.8)

in which

UR) = ¢.(R)+ > v

d*r w¥(r; R) “No

P2wr; R), 2M,

OP Oe i d?r W#(r; R)(P,/M, Wr; R). Noo

Small coupling terms of the form )’, fy. d°r w(P?/2M,)y; have been dropped. The set of coupled equations (3.1.8) is more complex than (3.1.4) in several respects. First, there may be heavy mixing of the y{R). Thus U(R) may not be a smooth function which can be expanded in powers of (R—R,). Secondly, the configuration Ry for which the Wr; R) are exactly degenerate does not usually minimize the total energy. This is a consequence of the Jahn-Teller theorem, discussed in Chapter 8.

3.1.3. Accuracy of adiabatic approximations

The accuracy of the approximations can be discussed formally or by explicit calculations for model systems. We consider formal estimates first. When egn (3.1.2) is substituted into the appropriate Schrédinger equation, there are extra terms which would not appear if ® were an exact eigenstate. These terms mix different adiabatic states ®; and, in essence, cause transitions during the nuclear motion. The off-diagonal elements are found by taking expectation values of the Hamiltonian (3.1.1)

1 |IA|D > ==Daag {| (O)9010)) for itr)||eryrrty + +2

{d°R xt Pa | {dr vr)

(3.1.9)

LATTICE

DYNAMICS

39

This has been simplified by noting the electronic functions Wr; R) and w(r; R) are orthogonal. For the adiabatic approximation to be good, the admixture of other electronic states must be small. The energy separation of these states E;; must be large compared with (O|#|®;>. This condition can be made more transparent by approximations in the matrix element. We assume (i) that the electronic matrix elements are insensitive to R, (ii) that y; and y, have opposite parity so that only the second term contributes, and (iii) that the lattice modes all have frequency «, so that we may use a sum rule (Appendix

I) to write f d°R y*P,y; as (iM,/h)(hw)AR,, where AR, is a typical atomic displacement. With these simplifications approximation is valid if (Herring 1956)

E,,> Dro, AR, fPrytay;

the adiabatic

(3.1.10)

i.e. if the electronic wavefunction w(r;R) does not vary much for typical nuclear displacements. Clearly AR, is smaller for larger nuclear masses M,. Indeed, Born & Huang (1954) show that the adiabatic approximation is obtained by neglecting higher terms in an expansion in powers of(mo/M,)*. A result essentially equivalent to (3.1.10) is found if y; and ,; have the same parity, so that only the first term in ¢®| #|®,> remains. We emphasize that we have only shown to be ‘small’ when calculating eigenvalues, where Q is the Hamiltonian. In some problems these off-diagonal matrix elements are important. Thus non-radiative transitions are induced by just the terms omitted in the adiabatic approximations. Also the approximation may be inadequate for the estimating electron—phonon interaction (Ziman 1960, §V.2). Moshinsky & Kittel (1968) have discussed the adiabatic approximation for a simple model system where exact results are also available. The model system consists of three particles A, B, and C.

Two represent nuclei and are of mass My and coupled harmonically with a force constant Ky. The third represents an electron; it has mass m, and is coupled harmonically to both the nuclei with force constants K,. At small values of m = m/My, independent of the force constants, the fractional error in the ground-state energy (E-E,ygiapatic)/E is always less than m/4. In the same limit the overlap of the exact and adiabatic ground states |nic wavefunctions

overlap from cell to cell: this overlap effect is best

introduced as part of a short-range interaction. Different charges again would result if Z is chosen to modify the displacement dipoles so as to include the effects of deformation dipoles. In terms of e* of§ 3.3, this definition would give Z* = Zle*|/|e|. This deformation effect is best included by way of a model like the shell model. The remaining interactions are predominantly short-range interactions representing a complicated and inter-related set of terms. These terms include (a) corrections to the monopole monopole Coulomb interaction, (b) exchange terms, (c) terms which appear from the constraint that the one-electron functions remain orthogonal in a distorted lattice, (d) correlation corrections such as the van der Waals interaction, and (e) local field correction terms. It is hardly surprising that empirical expressions are adopted, and the few parameters in them fitted to experimental data. The empirical expressions belong to three main classes. First, there are two-body central forces of the form ¢,,(r), where J and J refer to the species of ion, and r is their separation. Secondly, there are non-central two-body forces. These are much less important, and it is often difficult to determine anything but an upper bound on their magnitude. Strictly, the dipole-dipole interaction belongs to this category, but it is always treated separately. Thirdly, there are many-body forces involving more than two ions. To lowest order perturbation theory, these terms come from the deviations from additivity of charge distributions of free ions caused by the exclusion principle (Tosi 1964). Strictly, many-body terms appear whenever the electrons of an ion do not move rigidly with the nucleus (e.g. Johnson 1969). These interactions are not usually important for ionic crystals, although they may help to determine the relative stabilities of different crystal structures (Jansen & Lombardi 1964) and account for deviations from the Cauchy relations (LO6wdin 1956; Lundquist 1952, 1955, 1957; Cochran 1971). The questions of importance in providing a model for the central two-body forces are these. First, what analytic function is used for

$,,(r)? The standard assumptions are exponential

gd ~ bexp(—r/p)

(3.4.1)

50

LATTICE

DYNAMICS

and power law

(3.4.2)

p ~ (p/r)”,

or sums of terms like these. Both expressions individually can give respectable fits to alkali-halide data. The van der Waals interaction, if included explicitly, is always chosen to have an r-° dependence, even though the interaction does not have this asymptotic form at the close distances at which it is important. Indeed, it is hard to justify the separate inclusion of this lowest-order correlation correction except by the modest improvements in cohesive energy which result (Tosi 1964). The second question is how do we decide the dependence of the ¢,,(r) on the species J and J? The standard assumption (e.g. Barr & Lidiard 1970) is to take {b,,(r)}? = O,(r)hjs(7),

which

relates the interaction

between

(3.4.3)

different species to those

between identical ions, and to assume, consistent with this, that

$(r) = bexp jee

(3.4.4)

Alternatives to (3.4.3) and (3.4.4) are discussed by Smith (1972). Various criteria are used in selecting ionic radii r, and r,. The choice affects the relative importance of first- and second-neighbour interactions (Barr & Lidiard 1970). The third question applies specifically to the shell model, and concerns the division of the short-range interactions into core—core, core-shell, and shell-shell contributions. For ionic crystals the usual assumption is that only the shell-shell interactions are important, since the major interactions are those between the outer electrons on each ion.

3.4.3.

Valence crystals

Three models are commonly used for the inter-atomic forces in valence crystals, such as diamond, silicon, and germanium. One is the shell model, where the most important difference from other

systems is that the ionic charge is zero: the displacement dipoles vanish. The second model, which is only acceptable in special cases,

LATTICE

DYNAMICS

51]

assumes pairwise central forces described by the Morse potential :+ a V(r) = o| -om exp EE)

a ee = ae exp {—axr—ra

|

(3.4.5) Here D is the binding energy and ry the equilibrium separation. Usually a, is assumed to be 2«,. The third model uses a valenceforce potential (Musgrave & Pople 1962; McMurry, Solbrig, Boyter, & Noble 1967; Singh & Dayal 1970). This model adopts the usual picture in which valence crystals have strongly-directional covalent bonding between nearest neighbours with much weaker interactions between more distant neighbours. Nearest-neighbour central interactions do not give an adequate description of the forces (Smith 1948; Herman 1959). The inter-atomic forces are described by an expression quadratic in the bond lengths and bond angles, such as

V = yar. A,,. Ar+3{Ar. Ajp. (roA9)+(roA@). Ag, . Ar} +

+4(roA). Ago«(PoA9).

(3.4.6)

It is thus a harmonic potential. It is also an equilibrium potential ; if a bond is broken when the lattice is undistorted there are no forces on the neighbouring atoms unless we allow rebonding of the electrons.

Since terms

occur

which

depend

on bond

angles,

three- and four-body interactions appear. 3.4.4. Partly ionic crystals

This category includes most III-V and II-VI semiconductors. The models used are a mixture of models used for ionic crystals and valence crystals. Thus the shell model, valence-force potentials, and simpler forms like the rigid-ion model are used. One important question is what is the effective ionic charge needed? + For reference we cite the Sato potential used for an antibonding state rather than a bonding one a>

exp {—a,(r—ro)} + V(r) = o| a, +4, Je } +

ay

exp H2

{—a,(r—r 1 es

:

This is a slight generalization of the usual exponential form, and clearly D has a different significance (Sato 1955).

a2

LATTICE

DYNAMICS

Ionic charges are usually defined with reference to the rigid-ion model. Simply because the model omits important features, fits of the change to different observables will give different values: Z from the dispersion curves. Z’ from the Szigeti relations, or Z” from the difference in frequency of long-wavelength optic modes. Thus for GaAs, Z is 0-04, Z’ is 0-51, and Z” is 2:20. When more complicated models are used, the values can be reconciled, since the differences

represent the effects of electronic and deformation dipole terms. It is also possible to generalize the model so that the ionic charges change in the lattice motion, a change which represents changes in covalent bonding during the displacements. Whilst physically plausible, the results (e.g. Korol & Tolpygo 1971) do not give any significant improvement over the shell model. The

ionic charges

Z, Z’, and Z” are determined

from

lattice

dynamics and dielectric properties. There are many other definitions (Cochran 1961) which are associated with the electron distribution, such as the charge within the atomic volume or the chemical shifts of X-ray lines, or with descriptions of covalent bonding. Needless to say, these definitions give different values of ionic charges. 3.4.5. Rare-gas crystals The attractive interaction which dominates in rare gases is the van der Waals interaction. The weakness of the interaction means that three-body forces are relatively important and that the zeropoint motion may be sufficiently large for quantum effects to be appreciable.

3.5. Lattice dynamics

We summarize here some of the features of observed phonon spectra: the frequency spectra, the density of states, and the trends in properties from crystal to crystal. 3.5.1. Dispersion curves Many of the general properties have been mentioned already. Briefly: 1. There are 3n phonon branches for a crystal with n atoms per unit cell: 3 acoustic branches and (3n—3) branches for optic phonons. 2. The acoustic modes reduce to the usual (anisotropic) continuum elastic waves in the limit of long wavelengths. The elastic

LATTICE

DYNAMICS

53

waves have a frequency w(k) linear in the wavevector k. This linearity holds for acoustic modes through a substantial part of the Brillouin zone, and the success of the Debye model stems from this feature. 3. The optic modes primarily involve relative motion of atoms within each cell. The optic branches have frequencies roughly constant over the zone, and which usually (but not invariably) decrease with increasing wavevector. The Einstein model, with w {k) independent of k, is usually an adequate description. At long wavelengths the longitudinal and transverse optic modes of ionic crystals are split by the electric field associated with the longitudinal modes. The longitudinal and transverse optic modes are degenerate in cubic valence crystals or cubic rare-gas crystals. 4. The optic-mode frequencies and acoustic-mode frequencies may overlap, giving a quasi-continuous spectrum. However, there may be a gap between the highest acoustic and lowest optic-mode frequencies. This 1s most likely when there are atoms in the unit cell with widely-different masses. Thus for a linear chain with nearest-neighbour interactions only, @,,, 1s related to Mmmax> the maximum frequency, by

= Wmax

ay

|—— eae

max

G51)

ut)

where yp is the ratio of the two masses. Gaps are observed for Nal and KBr, but not for KCl. 3.5.2. Density ofstates

The main features can be analysed in the same way as the electronic density of states in §2.2. But it proves useful to define two related densities, associated with the density of states as functions of m? and w. For n atoms per unit cell these are

eye

p(w?) d(w?) =

wt+dw

;

ys kK

w* is an exact eigenstate ofW, that W is the Hartree-Fock Hamiltonian

for which |y> is a single-determinant closed-shell function, and that the Born—Oppenheimer approximation is valid. Some relaxation of these assumptions is possible. Coulson (1971) has extended the theorem to treat open-shell cases; the theorem holds for any solution of the full Hartree-Fock equations. Further, Stanton (1962) has argued the result still holds when correlation is included. Wannier & Meissner (1971) give a derivation which does not involve the Born—Oppenheimer approximation. The only change in the result is that an average over the nuclear motion appears. The theorem has had a history of misuse and criticism. Most of the objections are unnecessary in the problems we discuss, but difficulties do occur and should be mentioned (Benston & Kirtman 1966; Kleinman

1970; DeCicco

1971).

3.6.2. Electron—phonon interaction: delocalized case Three interactions are commonly considered in this class. In all crystals there is the deformation potential interaction. In polar crystals there is the Frohlich interaction with longitudinal optic modes. The third interaction is piezoelectric coupling. A. The deformation potential interaction. The band extrema of a crystal shift in energy under a homogeneous strain. Under pure hydrostatic stress, band extremum n of the perfect crystal shifts linearly with the dilatation A

(LITE TEHIN The energy &, is the It can be generalized 1956) and to discuss (Brooks 1955). Thus vector a;

(3.6.2)

simplest example of a deformation potential. to treat more general strains (Herring & Vogt bands which have several degenerate minima for a valley i whose axis is given by the unit

Ei = E,o+(Zanl + Zyndiaj) - @,

(3.6.3)

where e is the strain tensor. The interaction takes a more complicated form for degenerate bands. For the three-fold-degenerate valence bands of most cubic crystals the interaction is written in terms of angular momentum matrices L (spin-orbit coupling is

56

LATTICE

DYNAMICS

ignored here, but is readily included). The effective interaction 1s

He, = a(Tr e)1 —3b{(L2 —4L7)e,,. +permutations} — —2,/3d{(L,L, + L,L,)e,,+ permutations}.

(3.6.4)

Shockley & Bardeen (1949) have shown that =,A, or its generalization, can be treated as an effective potential for the motion of an electron in a state near the band edge. The result requires that the strain vary slowly in space. The deformation potential coupling can thus be used to describe the interaction of an electron with long-wavelength lattice vibrations. The essential assumption is that interaction with a long-wavelength mode is equivalent to the effect of a locally-homogeneous strain. Thus the deformation potential parameters can be obtained from transport measurements or uniaxial stress data on the perfect crystal. The assumption of long wavelengths is essential. Thus the deformation potential works badly in estimating inter-valley scatter, where phonons of short wavelength are involved. One other assumption is implicit, namely that the interaction which is responsible for the electron—lattice coupling is short-ranged. Quadrupole forces have long range and give a significant contribution to the coupling, even for nonpolar semiconductors. Indeed, this contribution may be finite even when the usual deformation potential vanishes by symmetry. However, the quadrupole term is usually important only for the optical phonon contribution to the deformation potential (Lawaetz (1969) and references therein). Accurate deformation potentials are slowly becoming available, but detailed values are altered too frequently to be usefully quoted. However, two trends are apparent. First, the change in band gap

under hydrostatic pressure is roughly constant for a given class of crystals; it is about 12 eV for the IJ-V compounds, for example. Secondly, the ratio d/b of(3.6.4) is roughly constant for a given class, and increases with ionicity (d/b ~ (1-1-35) for Group IV valence crystals, ~(1-4-1-9) for III-Vs, ~(2-5-4-5) for II-VIs, and =7 for alkali halides). B. The Frohlich interaction. The electron—lattice coupling in ionic crystals and the electric screening of a static charge by ionic displacement are different aspects of the same phenomenon. When the ions are displaced the potential at any point r in the crystal is altered, and the terms linear in the displacements give the electron-—lattice

LATTICE

DYNAMICS

57

coupling. In lattice screening the static charge interacts with the ions, displacing them. The displacements are determined selfconsistently by both this interaction and by the interactions of the displaced ions with each other. The potential at any point is then the sum of the unscreened potential due to the static charge and the screening potential from the consistently-chosen ionic displacements. The Frohlich interaction is derived by exploiting the relation between lattice screening and electron—lattice coupling (see, e.g., Frohlich 1962). Several simplifying assumptions are used. The lattice is considered as an isotropic dielectric continuum, assuming that the detailed lattice structure is not important. The longitudinal optic modes are assumed to dominate in the interaction (there is no electric field associated with transverse modes in an isotropic continuum), and their energy ha, is taken to be independent of wavelength. The interaction is then chosen so that the potential of a point charge q is modified from q/é,,r, with just electronic screening, to q/éor, when lattice screening is included. Clearly, (1/e,, —1/é9) is an important parameter. The coupling is characterized by a dimensionless coefficient

=

Viele!

Qe,

-

e?

&/(h/2m*a_)?

:

1

hwy’

(3.6.5)

6.

where the characteristic length (h/2m*w,)? is the ‘polaron radius’. Other discussions of the electric fields associated with lattice modes are given by Born & Huang (1954) and Dow & Redfield (1970) for perfect lattices and by Page (1969) for imperfect lattices. Markham & Ritter (1969) have extended the methods described here to use the actual lattice modes rather than a continuum limit. C. Piezoelectric coupling. In piezoelectric crystals there is a coupling analogous to the Froéhlich interaction, but which involves the acoustic modes rather than optic modes. The possible screening of this interaction by conduction electrons is of great importance (Duke & Mahan 1965). It is usual to make simplifying assumptions, namely, that the distinction between transverse and longitudinal mode velocities may be ignored and that all anisotropy may be ignored by taking simple averages. The appropriate coupling constant is proportional to é, a piezoelectric constant, which relates the longitudinal electric field E to the strain e (Hutson 1961) EE; =

— Ci €ij-

(3.6.6)

58

LATTICE

DYNAMICS

A convenient dimensionless constant describing the coupling is 2

ap

GSS)

=

hvEg

-

2

PvE

(3.6.7)

;

where v is the velocity of sound, p the crystal density, and Whar y deel) Qtpihame ab ound Alah,

i

“Wik Ag heyWye’ii he Mimi

6

» VWire@al pat, bet) yey

elhigener

gh tucaphe etfs \ ne wae ‘ip ali iam os he Pritts) ural byirrelve67 daphne lye hind ne Pista ye agg bh a ari oryents 1) rmbaity Lenoe's dor

alitrelaha BAW? hen

GftShhe

ths

ke

wails, ow

nat

sat) DOr

iniP ats gat

ety Tleh

litte

sont

J ttyl

nit 2A)

PHY, _

(PinBcrat:

aerate (

eu;

1 jo——

heuicderwind

Vel

rer ara.) ms

Ralnieechate "

thet, :

a- vibe

te) OG pred

poiieliy

;

ag —— * 1

7

q

:

a Uiaeo: i

beiegtise

djineheal

filed beagri met

>

\ Aaa

i” eo eyete|

es Tat >

; Ris

call ™

iclat ealiiy diFicaattah; auiny Neen jgire

pte

, :

ied: it

aay) goaitdsam

eel!

;

bel

cling sate triage|

Fey

Jey, om» elleso pub ian

ales a) pet

f.Gpri eee

i

Pepi

tale u PA Sait raha yb “ne thentee eT pan

ATRIA) SAG)

(pe

hag),

bal Mpe®

he

wheel

stag, maples aan Wap joie

OY. See

TT “er nl

ABAD)

cl) >,

Gri seras mee mall Agha GVM

cyte gute oc: slenlinvetiedin ni Ital were,

‘4

* ely tie

& a

jin

Ly

om ce

ta a

‘9

iy

ae nye

2

-

nea li ehh

ira)

ee a

;

A

II

ELECTRONIC STRUCTURE [ISOLATED DEFECTS

OF

Hf ,

4 EFFECTIVE-MASS

THEORY

4.1. Introduction

THE methods known as effective-mass theory are probably the commonest and most useful in the theory of defects in solids. Their value comes in part from the ease of applying simple versions of the theory and in part from the fact that the approximations in the method are well defined. A consequence of the ease of application is that the method is often applied well outside its region of validity. Even then it is often possible to obtain useful qualitative predictions. The central result of effective-mass theory is that, in many cases, the eigenvalues of a system consisting of one electron plus filled bands can be found by solving an equation for a single particle moving in the field of the difference between the actual potential and the perfect-lattice potential. An (N+ 1)-body problem is reduced to a one-body problem, and the complexity of the perfect-lattice potential eliminated, entering only through an effective mass and a dielectric constant. Even many-body effects can be included through these parameters. 4.2. Simple effective-mass theory We begin by deriving the usual effective mass equations in the one-electron approximation (Kittel & Mitchell 1954). The manyelectron extension will be treated separately later. For clarity we shall first concentrate on non-degenerate bands with just a single extremum in the Brillouin zone.

4.2.1. Fundamental equations for the defect lattice The unperturbed band functions satisfy the equation

How, (k, r) = E,(k)Y,(k, r).

(4.2.1)

When a perturbation U(r) is applied, related to its Fourier transform

U(k) by U(k) = | Geren" Uin);

(4.2.2)

66

EFFECTIVE-MASS

THEORY

1 3 { d?k e*'*U(k),

U(r) =

(4.2.3)

we must now solve an equation

(HEE U)Y = EY.

(4.2.4)

The wavefunction of the defect electron (r) is normalized, and may be written as a linear combination of the band functions of the perfect crystal

Yr) =D {d?k’ (kW y(k’, 1).

(4.2.5)

Clearly we must determine the amplitudes ¢,,(k’). To do this we substitute (4.2.5) in (4.2.4), multiply from the left by w*(k,r), and integrate over all space

(Esk) E}o,00)+ F | drvztk, Ue) |dk’ by (k Welk’ 2) = 0 or, since the band functions can be written

Walk, 1) = exp (ik. r)u,(k, 1), {E,(k) — E} lk) + + d Idk’ ¢,,(k’) [.der u*(k, r)U(r)u,(k’, r) exp {i(k’—k). r}

= 0.

(4.2.6)

This change allows us to exploit the periodic properties of the Bloch functions u,(k, r)

u*(k, ru

= Lar en ik

(4.2.7)

This expression depends only on the fact that the Bloch functions have the lattice periodicity ;the x,, are the reciprocal lattice vectors

EFFECTIVE-MASS

THEORY

67

of the perfect lattice. With (4.2.7), the innermost integral in (4.2.6) becomes

| d*r u¥(k, r)U(r)u,(k’, 1) exp {i(k’—k) . r} fe.9)

=> | d?ra”" U(r) exp {i(k’-k—x,,)

.r}

= Vian U(k—k’ +x,,).

(4.2.8)

Eqns (4.2.6) and (4.2.8) are the basic results. They have assumed the one-electron approximation but are otherwise quite general. 4.2.2. Approximations of effective-mass theory

There are just two fundamental approximations: Approximation ]. That the potential U(r) is varying so slowly that its Fourier components outside the first Brillouin zone may be neglected. The effect of this assumption is that only terms with m = 0 need be considered in (4.2.8); k’ and k are confined to the first Brillouin zone, and we shall later require (k —k’) to be small so the term with K,, = 0 should suffice. With this approximation (4.2.6) and (4.2.8) become

{E,(k)— E}¢,(k) + | d3k’ ¢,,(k’)ag" U(k—k’) = 0,

(4.2.9)

in which

1

frye 0

d3ru*(k, r)u,(k’, r).

(4.2.10)

cell ¢ cell

Approximation 2. That the eigenfunction of (4.2.4) (strictly an envelope function, the Fourier transform of ¢,(k)) is so diffuse that we need only consider very close values of k and k’. Here ‘very close’ means that 22

=

ey?

« band gap Eg.

2m* With this approximation we may write nn’

ag

__

=

3 | Onn’ + terms of order {/(6/Eg)}7

(4.2.11)

(4.2.12)

Strictly, the last equation is only valid ifk’ and k are near a minimum at k, in k-space; there are corrections if they are not near an

68

EFFECTIVE-MASS

THEORY

extremum. In essence (4.2.12) is the assumption that u,(k,r) and u,(k’,r) are adequately given by u,(Ko, r). The effect of the second approximation is to eliminate the interband terms with n 4 n’. The weakness of the potential U(r) is not involved directly. This is important, since it means that effectivemass theory is still valid asymptotically for tightly-bound defects, at large distances from the defect. Some problems of matching a discrete region near a deep defect to an asymptotic effective-mass region are discussed by Opik & Wood (1969). Note also that the results can be valid for unbound, positive-energy states too, if their energy is close to a band extremum. Eqn (4.2.9) becomes

{Ex() —E} 4,9) + |PRO AK

U(k—-k’)=0.

= (4.2.13)

In § 4.2.4 we shall show that, apart from some manipulation, (4.2.13)

is equivalent to the usual effective-mass equation. 4.2.3.

The function A(r)

Let us define a function f(r) by

f(r) = { der’ A(r—r’) f(r),

(4.2.14)

where 1

33

A(r—r’) = (=| [ox een)

(4.2.15)

We show that f(r) differs from f(r) only by the removal of its Fourier components which lie outside the first Brillouin zone. To show this we note

f(k) = { d?re~*-*7(r) |

hoe ar’ | ee] ex(5| elk

= D(k) i dries = D(k) f(k),

8)

ik-r 6p)

f(r) (4.2.16)

where D(k) is unity ifk is in the first Brillouin zone and zero otherwise.

PAC IW ESIMUNSS,

IWISUS ORG

69

Approximation | gives an example of the use of (4.2.16): by ignoring the higher Fourier components of U(r) we are essentially replacing U(r) by

U(r) = { der’ A(r—r’)U(r).

(4.2.17)

Approximation 2 involves a stronger constraint: k and k’ must be close; it is not sufficient that (k —k’) lie in the first Brillouin zone. Clearly (4.2.17) would be exact if A(r—r’) were the usual Dirac delta function o(r—r’). In fact A(r) is essentially the Wannier function

for free electrons in the crystal structure of interest. For a spherical band with a zone of radius k oe

r sin (Kr) Dig

Malena

(4.2.18)

r

In general, the convolution with A(r), as in (4.2.14) and (4.2.17), amounts to averaging over a unit cell. The assumption that f(r) can be replaced by f(r) is only acceptable if f(r) varies slowly over the unit cell.

4.2.4. The effective-mass equation We now make a transformation to reduce (4.2.13) to a real-space form. Instead of the amplitudes ¢,(k) we introduce an ‘envelope function’

Fi(r)= |d°k e*",(k).

(4.2.19)

Note that if E,(k) may be written as a power series in k then we may write

E,(k)F,(r) = { d?k E,(—iV) e*"¢,(k).

(4.2.20)

Z

When zone

(4.2.13) is multiplied by exp (ik.r) and integrated over the

{E,( iV) — E} F(t) +| ak

leesek

|ax

(k')U(k

—k’)

= 0.

70

EFFECTIVE-MASS

THEORY

If U(r) is reintroduced instead of its transform the second term is 3

( Ue

| ak |ark’ | dr'd (k’ye7 ek) Zz

UF)

fee)

Zz

= i ar | dek’A(r—r')U(r') e*'-"' ,(k’)

= | drA(r—r')U(r')F(r’). Thus (4.2.13) becomes

(E,(—iV) - E}F,tr)+ | dr’A(r—r')U(r')F(r’) = 0.

(4.2.21)

But our discussion of A(r) showed that rewriting the second term as U(r)F,(r) was equivalent to using Approximation | a second time. Thus a simpler equation of equivalent accuracy is

{E,(—iV)+ U(r) — E}F,(r) = 0.

(4.2.22)

This is the basic equation and gives energy levels relative to band extrema. Note that it is still valid, although possibly difficult to solve, when the band extremum is anisotropic. In its commonest

form, E,(—iV) becomes —(h?/2m*)V?. It is sometimes argued that (4.2.21) is more accurate than (4.2.22) and that (4.2.21) may be used to estimate corrections in the basic equation arising from the rapid variation of U(r) (e.g. see Schechter 1969). This is not correct, for the approximation relating these two equations has already been used in deriving the first of them. At best corrections estimated from (4.2.21) will be order-of-magnitude values only. 4.2.5. The wavefunction

The original expression for is (4.2.5). It is simplified by the loss of the interband terms through Approximation 2, giving

Yr) = |dK’ blk Walk’, v) =

|d3k’ $,(k’) elk'-ry (k’, r). Zz

If the state is sufficiently compact in k-space (i.e. if F(r) is diffuse

EFFECTIVE-MASS

THEORY

7)

in real space) the k dependence of u,(k’,r) near the extremum may be ignored

ko

P(r) ~ ;{d?k’ ¢,(k’) oil. r) = F(r)u,(Ko, ¥).

(4.2.23)

The wavefunction is the product of the envelope function F,(r) and the Bloch function at the extremum. 4.2.6. More complicated band structures The results given so far have been for simple nondegenerate bands with just a single minimum at the centre of the Brillouin zone. Two other cases are of practical importance: the case when the band is degenerate and the case when there are several equivalent minima at different points of the Brillouin zone.

A. Degenerate bands. The theory can be extended, with caution, to cases where the band is degenerate. Such cases occur for acceptors in silicon and germanium, where the valence band is degenerate; these systems are discussed in § 23.2. Some changes prove necessary, however. One change results from the fact that E,(k) for the energy surfaces at the point of degeneracy may not be expandable as a Taylor series in k. Thus the change from E,(k) to E,(—iV) in (4.2.20) cannot be made. This problem can be avoided by the use of different basis functions which give matrix elements analytic in k. A convenient choice of orthonormal basis functions is that of Kohn & Luttinger (1954)

z,(K,r) = e*-"u,(0,r) = e*-"y,(0, r) for a degenerate band extremum

(4.2.24)

at k = 0. For extrema at other

points in the zone we use ;,(k, r) = exp {i(k—k,) .r}W,(k,, Pr). Matrix elements of the perfect crystal Hamiltonian among the x(k, r) are (Callaway 19645, p. 219)

{ dr yf(k, 1)otelk’, 8) h hk2m tae 21-2

= 0(k—k’) |? Ex~

(4.2.25)

m

in which

ee

3

| dr u*(0, r)nu,(0, r) cell

(4.2.26)

(2

EFFECTIVE-MASS

THEORY

and mn = p+(1/2mc’)S A VV is an operator which reduces to p when there is no spin-orbit coupling. In most cases we can choose the degenerate states u,(0,r) so that there are no off-diagonal elements of m between them. This is clearly true when there is a centre of inversion, since one can take appropriate linear combinations to construct u,(0, r) of definite parity. The states from the degenerate set will all have the same parity unless there is accidental degeneracy. But x has odd parity, and so n,,, will vanish for these states at the point of degeneracy. The matrix elements of m which connect the degenerate states to other bands produce several effects. These interband terms cause the deviations

of the effective mass

from

the free-electron

mass,

and they give the coupling which prevents E,(k) being expanded as a Taylor series near the point of degeneracy. The perturbation HK, = (h/m)k.m can be taken into account by introducing a new effective Hamiltonian % valid only in the basis of the degenerate states (Pryce 1950; Appendix II). Thus .% operating in the degenerate states |d> alone gives the same eigenvalues for these states as % operating among both the degenerate states and the other states |e. The effects of these other states are included implicitly in %

5

P,H,P,H,P. OEE

aS

we

(4.2.27) ae

P, and P, are projection operators for the degenerate and excited states respectively. #%, is linear in k, so that we have truncated H#, by dropping terms of order (k3). This is roughly equivalent to using Approximation 2 and assuming that the envelope function

is diffuse. The matrix elements of U(r) can be calculated as before. Again

it is possible to eliminate the interband terms if U has no Fourier components outside the first Brillouin zone and the envelope function is diffuse. In outline, the theory goes through as before. We postulate a wavefunction

Wit) = DY |ak’ bad K Vad 2)

(4.2.28)

nd

Instead of (4.2.13) we obtain a matrix equation, with matrices of

EFFECTIVE-MASS

the order of the degeneracy from (4.2.27)

THEORY

73

and with off-diagonal terms derived

; h2 »:oar [Eo Home} t+ |d?k vik—kieak'} te ae Aanarkakytat

= ().

(4.2.29)

xB

In appropriate cases U will have off-diagonal elements too. The envelope function is now defined by a vector with components

= idk ep ,(k)

(4.2.30)

Whilst the E,(k) cannot be expanded in powers of k, each of the matrix elements in (4.2.29) deriving from # can be expanded in this way. We finally obtain

» » Dii(—iV.)(—iV,)+ U(r) daa |Fat = (E—Ep,)F,r), a

(4.2.31)

a,B

in which

Dé, = h?

h?

amet 2m Ong + mi © Ek 2 0)— E(k = 0)

(4.2.32)

e

The D tensor for degenerate bands is precisely analogous to the (h?/2m*) tensor for a single band. The parameters in D can be determined from cyclotron resonance. B. Several equivalent minima.

The conduction

bands

in diamond,

silicon, germanium, GaP, and other indirect-gap materials have several equivalent minima at points k, in the Brillouin zone. The theory for such systems is very similar to our earlier discussion. Indeed, if we extend Approximation | to the assertion that the Fourier components of U(r) which connect the different extrema are very small

|U(k, —k,)| « |U(k ~ 0)|

(4.2.33)

then the energy levels can be found from an equation involving just one minimum. The effective-mass equation is only trivially altered (4.2.22) becoming

{E,(k; —iV) + U(r) —E\F,(r) = 0.

(4.2.34)

The wavefunction, however, involves all the equivalent extrema.

74

EFFECTIVE-MASS

THEORY

Approximation 2 may also be usefully extended so as to decouple the various minima. Thus, if we assume that the wavefunction is given by

w=

> {d?k o{k)yA(k, vr) minima

(4.2.35)

“z

t

then we make the approximation that ¢(k,—k,) is negligible: the envelope function is so diffuse that the wavefunction is just a sum of independent contributions from the different minima

V(r) =

oy a;F(r)u(k;, r).

(4.2.36)

The small terms U(k;—k,;) which were dropped in deriving the energy levels are important because they admix the various degenerate F(r)u(k;,r) and cause small ‘intervalley’ splittings. These splittings cannot be predicted quantitatively with any accuracy in effective-mass theory, since large Fourier components of U(r) are involved and comparable terms have been ignored at other points in the calculation. But the qualitative effects can be predicted, since the weights «; can be found by symmetry arguments alone. Thus for a cubic crystal with six minima, such as silicon, the six F(r)u(k;,, r) recombine to give levels of A, E, and T symmetry; for germanium,

with four minima, A and T levels result. We discuss these phenomena in more detail later.

C. Cases of near degeneracy. Two types of near degeneracy may be important. In the first two bands have extrema at the same point of the zone. In the second the extrema are at different points in the zone. When the extrema are at the same point in the zone we may use the theory developed for the degenerate case, with the minor change of diagonal terms in the D tensor corresponding to the energy splitting of the extrema. InSb presents special problems because of its band structure. Donors in InSb have been discussed by Larsen (1968) and acceptors by Sheka & Sheka (1969).

The problem of inequivalent minima at different points in the zone is more difficult. We discuss this system in § 5.5.

EFFECTIVE-MASS

THEORY

75

4.2.7. Matrix elements in effective-mass theory

Consider

two

states, both well represented

by effective-mass

theory, Vy

=

F,,(r)u,(Ko,,

Y

1

F (Um

and the matrix element of some suppose that O is a local operator that it is a function of r but not of If O varies only slowly in space

r)

(4.2.37)

(Kom » r)

operator O between them. We which commutes with the , so p. we may neglect.the variation of

F,,(r), F2,(r), and O over each unit cell. Then

CY IOP.) = { d?r F¥,(t)OF p_(t)U* (Koy, P)tg(Kom 5¥) =

y

Fcd0.Faltoh ;

cells

d*r ux(Kon> T)Un(Kom ) o|

cell

~ if ar FNOF. (0 | d°rufig. (oe)

(4.2.38)

cell

The unit operator, O = 1, clearly falls into this class, so the normalization can be calculated in this way. For diagonal elements (m =n, Ko, = ko,,) we find

CPOIPS/CP|P) = {d?r F*(r)OF(r)

(4.2.39)

if the envelope functions are normalized to unity over all space. Thus one need only know the envelope functions to calculate diagonal matrix elements of slowly-varying operators. For offdiagonal elements the matrix elements of the Bloch functions give additional selection rules. Thus if kp,, = ko, when n # m then

CP |O|W,> =

i‘PFI (NOF. «(0 }ae

(4.2.40)

Even if the envelope functions give a finite element, the Bloch function term causes the whole matrix element to vanish. Similar methods can be used when kp,, and Ko, are not equal.

76

EFFECTIVE-MASS

THEORY

The opposite extreme occurs when O varies rapidly in space, as in the isotropic hyperfine interaction of a host nucleus at r, with the defect electron, where

O = Ad(r—r,). Only

the envelope

functions

are

slowly

(4.2.41) The

varying.

diagonal

elements of O take the form

CPIO|P> ~ AIF(r.)|?lu*(Ko, F.)1”, so that the expectation value of O is

CHOR2=, a;AlF(r,)| sl) hye one (Ko,

“OPS

= (FOF

9)|

3

6” |u*(k, 0)|7.

2 (4.2.42)

cell

The expectation value calculated using the envelope function alone is multiplied by a factor depending on the Bloch functions. This factor may be very large, often of order 10%.

4.3. The quantum defect method The two fundamental approximations of §4.2.2 become valid asymptotically at large distances from a defect, where both the potential and the envelope function vary slowly in space. The quantum defect method (Ham 1955; Seaton 1958 ; Bebb & Chapman 1967; Bebb 1969) concentrates on the wavefunction in these distant

regions. It exploits the fact that the long-range properties of U(r) are often known rather well, even though the short-range potential may be both complicated and hard to determine. Thus the potential U(r) for an impurity of charge Z|e| tends to — Ze?/er at large distances r from the defect; ¢ is the appropriate dielectric constant, usually the static constant in the cases of interest. The quantum defect method for charged defects like this derives functions op(r) which are solutions of

WN Zee Se eta

0

(4.3.1)

They are chosen to have the experimentally-observed energy E,,..

In general, of course, Ey,, is not an eigenvalue of (— h?/2m* — Ze?/er). Thus, whilst we can ensure Yop(r) has the correct behaviour far from the defect, it may not be finite at r = 0. The misbehaviour at r ~ 0 is not important for some properties, particularly optical

EFFECTIVE-MASS

THEORY

77

matrix elements, and it is these properties which are studied by the quantum defect method. The method gives a prescription for calculating these matrix elements in terms of observed energy levels (e.g., see Bebb & Chapman 1971). We shall be concerned mostly with the ground state of a donor or acceptor. It is an s-state Vopr) = am

(4.3.2)

The general solution for P, is a Whittaker function, but this can be approximated by P(r), =

Pa

I

‘(| mantra

ee

P exp(—r/va*).

(4.3.3)

The effective principal quantum number v is given by

Zeon i =s4 eat

or y-

(4.3.4)

Its difference from the principal quantum number is (n—v), the quantum defect of atomic spectroscopy. The effective Bohr radius a* is (he/e?m*). The deviations of U(r) from the simple Coulombic potential — Zle|*/er are reflected in v rather than in a*. The quantum defect method also provides a convenient interpolation technique between the pure Coulomb potential, with no short-range corrections, and a purely short-range potential for which the Coulomb tail to the potential is not too important. The delta-function potential would be a typical short-range potential. In the Coulomb limit we have v = 1; P(r) varies as exp (—1r/a*) and E,,, is just the effective Rydberg. In the short-range limit v tends to zero. E,,, is much larger than the Rydberg if all other factors are kept constant, and P(r) has the expected asymptotic form r-‘ exp (—r/a*v). 4.4. Effective-mass theory for a many-body system The proof of effective-mass results in the one-electron approximation involved a number of intuitive steps. Thus the effective

mass entering through E,(—iV) = (—h*/2m*)V? was identified with the mass observed experimentally. The potential U(r) was identified

with the usual screened potential, and the screening described by

the static dielectric constant appropriate for the perfect insulator.

78

EFFECTIVE-MASS

THEORY

The potential is independent of the defect wavefunction in this form, which is remarkable and different from the usual Hartree— Fock results. So it is likely that electron correlation is important, and it is useful to rederive the results in a way which verifies the intuitive steps when many-body effects are included. 4.4.1. Many-body

eigenfunctions for the perfect lattice

We consider an (N + 1)-electron system, corresponding to the full band of N valence electrons plus the single defect electron considered earlier. When discussing the defect lattice we can use two different choices of basis. We may use the eigenstates in the Hartree-Fock approximation, and these can clearly be classified by the band indices and wavevectors of the one-electron wavefunction involved. Another choice of basis consists of the exact eigenstates of the (N+1)-electron system W(k;1ro,...,ry). Here ro is the defectelectron coordinate, and we have anticipated that these states can also be classified by band indices n and a wavevector k. The exact eigenstates satisfy the equation

How AK tw 1) = E,(kK)W,(K; tw +1),

(4.4.1)

where the abbreviations TN

a

¥,,%2,---,IFn

Tu+14

=

To,T,,QT2,.--,0n

(4.4.2)

are used. Their translation properties may be shown explicitly. If T is the translation operator such that TUK? TG tie ert)

= (kK; {ro +t}, {r,+t},..., {r;+t},..., {ry+0),

(4.4.3)

where t is a lattice vector. The w, may be written in terms of other functions, related to the Bloch functions,

WiAK; tTy+1) = exp ik: oe |u,(K; tw +1)s

j which are strictly periodic in the sense Tu,(K;¥9,...,8y) = u,(K319,..., Fy).

(4.4.4)

(4.4.5)

The u,(K;ty4,) are also periodic in the coordinates of one of the

electrons, so that a translation which merely changes r; to r;+t

leaves both u,(k; ty 44) and u,i(K; ty 41) = exp (ik. P54, 6))u,(K3 ty44) invariant.

EFFECTIVE-MASS

by

THEORY

W®)

It is convenient to introduce a one-particle density matrix defined

PPT aed) IL dy (Ks ty4s)a(’ s ty 1) BN 1) I ee (kOe jue (eee ke No

(4.4.6)

The diagonal elements of p give the charge distribution of the extra electron in the (N + 1)-electron state. 4.4.2. Fundamental equations for the defect lattice The unscreened impurity perturbation is N+1

Ux(tv41) =

sy U(r;)+ Uy, i=1

(4.4.7)

the sum of one-electron terms plus a term from interaction with the nuclei. The Schrodinger equation for the perturbed system has the general form

(Ao +U7)¥ = EW.

(4.4.8)

Since the w(K; t, ,,) form a complete set of states, the eigenfunction

Y can be written

Wess) = ¥ |dk d(bplk ty)

(4.49)

Without approximation, the Schrédinger equation becomes

{E,(k) — E},(k) + ol d3k’¢,,(k’) = 0 (4.4.10) in which the matrix element is V(k—K’) Sjn’-

(4.4.12)

Once the effective potential is established, the standard effectivemass result follows directly. The Schrédinger equation becomes

{E,(k) — E},(k) + {d3k'V(k—k’)¢,(k’) = 0.

(4.4.13)

This is exactly like (4.2.13), and the proof of the standard result goes through as before. The result looks like a one-particle Schrodinger equation, but this does not imply that a one-electron approximation has been used: the coordinate R in the envelope function corresponding to ¢,(k) need not be simply related to ry. The step (4.4.12) is the one which ultimately shows the observed dielectric constant that should be used. To show that the effective mass used is the one derived from cyclotron resonance is simpler in principle; one merely has to verify that an equation of the form (4.4.13) can be derived both when U;, is the potential due to a defect and when U;, is an electromagnetic field appropriate to cyclotron resonance experiments. 4.4.3. Approach using exact many-body states as a basis We derive the effective potential of (4.4.12) in three stages. First we show that the interband terms (n 4 n’) may be dropped by an analogue of Assumption 2. Secondly, we show that the matrix elements of U; can indeed be rewritten as in (4.4.12) if the potential varies sufficiently slowly in space, as required in Assumption 1. Thirdly, we relate the rewritten form ofthe perturbation to the usual screened potential. Parts of the derivation are lengthy and will merely be sketched; full proofs are given by Kohn (1958) and Klein (1959). The first stage uses the fact that the N-electron system, without the defect electron, is an insulator. When U; is zero, this requirement can

EFFECTIVE-MASS

THEORY

81

be expressed by the assumption that, for the (N + 1)-electron system,

Ew 4nAk)—E,(k)

> AE > 0

(4.4.14)

for any k within the first Brillouin zone. This relation is not valid for metals, where two electrons at the Fermi surface can be excited with

conservation of total wavevector and only an infinitesimal change in energy. For an insulator AE is of the order of the separations of the bands. The impurity potential admixes higher states by terms proportional to (U;k?/AE) and, exactly as in the discussion of Approximation 2, these give a negligible contribution to the eigenvalues of a very diffuse state. Thus the terms in with n #n’' may be dropped, as in the one-electron case. The second stage involves rewriting of ¢nk|U;|nk’> in the form V(k—k’). The matrix element can be rewritten in terms of the oneelectron density matrix (4.4.6), giving E+io,

(53377)

which is allowed to tend to zero later. Both Koster (1954) and Callaway (1964a) give explicit calculations for the asymptotic form of the Green’s function at large distances. Thus, for a simple spherical band h2

EXk)= lk) =5

5.3.8 (5.3.8)

k?

we may define q by the energy E h2

E= eat

(5.3.9)

and obtain the Green’s function

GuReE

2

(h2/2m*)Q,)

exp (igR)

RO

(5.3.10)

+ In some cases, as in the scatter of holes or where the band structure is complicated, the contour of integration appears to correspond to incoming waves, even though the usual convention of outgoing waves is being used (Liu & Brust 1967).

)

98

GREEN’S

FUNCTION

METHODS

For a general band the results are more complicated. Koster and Callaway give results for an energy surface defined by E = E,(q) which has a number of points of stationary phase for each R/R. These results differ greatly from atomic scattering results just because of the loss of spherical symmetry.

5.3.2. The scattering problem

We now return to the scattering problem, and put some of the equations in a more familiar form. From the two fundamental equations (5.3.3) and (5.3.4) we may write

B= Bo+G.Us(1-GoU)s! 48°.

(5.3.11)

This shows a scattered wave superimposed on the incoming wave.

The factor

U.(1—G.U)~'

is known

as the T-matrix.

Exact

solutions are rarely possible, so we discuss three special cases. A. The Born approximation (Goertzel & Tralli 1960, p. 190). Suppose G . U is in some sense small, so that we may expand (1—G. U)7’. The scattered wave becomes

G .U{1+(G.U)+(G.U)*+...}B°. If we retain only the first term we have the first Born approximation, the first two terms yield the second Born approximation, and so on. The approximation is particularly bad for energies E close to the band extrema, but becomes increasingly good as the energy moves towards the middle of the band. In the effective-mass approximation

the expansion is good when h?q?/2m* is large compared with U. B. The one-band, one-site model.

Although

this model

is a poor

description of most real systems, it does provide a useful illustration of both the scattering problem and the bound states. The simplification appears because G.U is now a scalar. The perturbation matrix is = d

E—E(k)—io

(5.4.6 )

is reasonably represented by AE,,, which replaces E by E,,

ey

Abs = 2

_lenlUID?

= Fk) =i

rE: SiG.

Substituting into (5.4.3), and ignoring off-diagonal elements to any other bound states |n’>

is an eigenvalue of the Hermitian operator #. The variation is subject to the constraint that |~> be orthogonal to all eigenstates of # with lower eigenvalues than the state of interest. In practice one chooses a trial function |y(A)> and varies the parameters / to satisfy

0 WAIAWA) _ 4 dA WANA)

(6.1.1)

subject to the orthogonality constraint. A second theorem due to Hylleraas & Undheim (1930) and MacDonald (1933), is useful for excited states. If the trial functions are linear combinations of N constant orbitals {¢;} whose weights alone are varied, then the theorem states that the nth solution for

any given set {¢;} is an upper bound to the exact energy of the nth state of the system. The important point is that one has an upper bound for excited-state energies without needing to know exact solutions for the (n—1) states lower in energy. Perkins (1965) has

extended the theorem in several respects. In calculations of excited states it is, of course, essential to include correctly the off-diagonal elements of # between different ¢;.

116

6.1.2.

VARIATIONAL

METHODS

The few-electron approximation

All the methods described in this chapter begin by dividing the electrons into two groups. The first group contains the ‘defect’ electrons, the electrons associated with the defect and whose one-

electron wavefunctions are particularly sensitive to the perturbing potential. Thus in the F-centre (essentially an electron bound to an anion vacancy; see chapter 15) or in the simple effective-mass donors (chapter 23) there is just one defect electron. In other cases, such as aggregates of F-centres like the R-centre (chapter 17), there are several defect electrons. The second group of the electrons, the ‘core’ electrons, includes all the remaining electrons. It is usual to make two assumptions about the core electrons. First, it is assumed that their wavefunctions are not significantly altered from the perfect crystal form. This assumption, or some stronger form, is used in the orthogonality constraint. Second, it is assumed that the response of the core electrons to the presence of the defect can be represented by simple polarization effects and that the polarization can be calculated from the response of the perfect lattice. Thus the matrix elements of the perturbation due to the defect among core states must be appreciably less than the band gap of the perfect crystal. If the matrix elements are larger than this then more electrons must be classed as defect electrons. Indeed, in some cases it is not obvious how many defect electrons there are; this is particularly true for covalent crystals, where a defect may cause substantial local rebonding. Ina variational calculation, only the defect-electron wavefunctions are varied (the ‘few-electron approximation’). The core electrons contribute to the potential in which the defect electrons move and provide a constraint, for the defect electron wavefunctions must be orthogonal to those of the occupied core states. As the polarization of the core orbitals depends on the defectelectron wavefunctions, the potential in which the defect electrons move depends on their own wavefunctions, and hence on the varia-

tional parameters. When the Hamiltonian depends on the variational parameters we cannot use (6.1.1) to optimize the wavefunction. Instead we must use

0 WAIAAIWA)> =0) OA WAIWA)D lana where we use consistency as well as the variational principle.

(6.1.2)

VARIATIONAL

METHODS

117

Most variational calculations assume that W is a Hartree-Fock one-electron Hamiltonian. In these cases the one-electron eigenvalues of |y/(A)> which result are referred to the vacuum level. The energy zero is different from that for the effective-mass and Green’s function methods, where energies are referred to a nearby energy band. It may be difficult to refer energies obtained variationally to band edges without detailed estimates of relatively small terms such as Coulomb-correlation corrections and self-energies (e-g. Wood & Opik 1968). 6.2. The orthogonality constraint The major differences between variational calculations arise from differences in the way in which the orthogonality constraint is treated. First we consider just one defect electron outside full bands of core electrons. Three of the methods described include orthogonality to the core orbitals by a special choice of trial wavefunction. The fourth, the pseudopotential method, builds the constraint into an effective Hamiltonian.

6.2.1. Modulated band functions We may express the trial wavefunction in terms of the band functions y,(k, r) or Wannier functions a,(r—R,) of the unoccupied bands of the perfect crystal

WA) = dd, balk, AW, (Kk,1) n (unocc)

=

k

Y B,C, a)a,(r—R,).

(6.2.1)

(unocc)

This trial wavefunction is automatically orthogonal to the occupied band functions of the perfect crystal. The expansion in terms of band functions w,(k, r) is particularly useful when effective mass theory is valid, but where the equations are too complicated to solve exactly. Examples include excited states when there are several band minima or cases where there is an anisotropic effective mass. Expansion in terms of Wannier functions is most useful for well-localized defects,

118

VARIATIONAL

METHODS

when the variational condition becomes

0 (WA)Ao + UlWA) Gh GWANWA)> 3 YY BW, DBali, ALOn en Ry = Ry) Uy Pe a=

eed nym

V,u

=

24

;

:

» LIB A“

(6.2.2)

The prime on the sum over bands n, m means that only unoccupied bands are to be included, and =

2aE,(k

eik. (Ry

By

Un V, LL) ae (a,(F - R,)| U(r)la,,(r ia R,)>.

Kilby (1967) has suggested a simple analytic form for the B,(y, A). If the perturbation U is centred on site Ro, then his suggestion is that the band and space dependences be separated

B,(v, A) = B(A, |R, — Rol)B(n). In particular, he finds that several predictions of exact Koster— Slater solutions could be reproduced. The accuracy of his approximation appears to increase with the strength of the perturbation. 6.2.2. Atomic orbital methods

The core orbitals are often described quite well by atomic or atomic-like (§2.2) functions. The trial function is the standard L.C.A.O. form

>

B,C, A)a,(r —R,),

(6.2.3)

(unocc)

where «,(r —R,) is an atomic orbital centred on site R,. Our notation is chosen to resemble that for the expansion in Wannier functions since the methods become identical in the limit of small overlaps of atomic functions. Although the orbitals a,(r—R,) are orthogonal to all other orbitals in site v, they are not, in general, orthogonal to the orbitals on other sites which appear in y(A). We are using a nonorthogonal set of basis functions. The «,(r—R,) may also have a finite overlap with the occupied core orbitals on other sites, but this can usually be treated straightforwardly, since the overlaps are

VARIATIONAL

METHODS

119

small, and are often simply ignored. The variational calculation with a non-orthogonal basis can be set up without difficulty: In particular, the estimate of the lowest eigenvalue is given by the lowest root of

det || — E|| = 0

(6.2.4) (Goertzel & Tralli, § 15.4; Roothaan 1951). The matrix elements of Hy can be related to atomic energy levels in many cases. If there is only one defect electron then the matrix elements involve twocentre integrals such as , in which Q = 1 or U, the defect potential. Where there are several defect electrons,

three- and four-centre integrals occur ae (r,—

aedAA Oe R,)\e7/|r; —r,la(r,

—R,,)a,(r,

—R,,)>.

Three- and four-centre integrals are difficult to evaluate accurately and rapidly. They present a serious barrier to accurate calculations for several-electron systems using the atomic orbital basis. 6.2.3. Explicit orthogonalization A third choice of variational function is one in which some arbitrary trial function ¢(r, 4) of r and 4 is orthogonalized to the occupied core orbitals by the Schmidt process

\W(A)> = {96 Ayy— Yd In, v >}N (occ) n

(6.2.5)

v

The normalization N is given by

N? = ¢o|6>—-2) |Kn, vib? + + VY . mv

(6.2.6)

nyu

Two major differences from the earlier functions are apparent. First, we vary the form of @(r, 4), rather than just the weight of the various basis functions. Thus we use the Rayleigh, rather than the Ritz, variational principle. Secondly, the sums over atomic orbitals |n, v> are over occupied, rather than empty, orbitals. When the expectation value of an operator is taken between explicitly-orthogonalized states, three classes of term appear. This is

120

VARIATIONAL

METHODS

shown by the expression for N’. First, there are terms in the trial function ¢ alone. Second, there are cross-terms’ between the trial function and core functions, and, third, terms just involving the core

functions. It is clearly an advantage to use atomic functions for the core functions, both because free-atom energy levels may be used, and because free-ion wave functions are often known quite accurately. In simplifying the integrals which arise, two assumptions are particularly useful. The first is the assumption that @(r, A) varies much more slowly in space than the core orbitals. Consequently only the first few terms in an expansion of ¢(r, 4) about an atomic site v can be used in evaluating two-centre integrals. The second assumption is that one may use a continuum approximation of some sort for describing the host lattice at some distance from the defect. This means only a few terms appear explicitly in the orthogonalization. However, the variational principle has to be rewritten if the electron has an effective mass m* different from the free electron mass m in the continuum region. Let the region treated directly be I and that treated in the continuum limit be II. In region I ¢, is an approximate solution of h2

Hig = [s+ Vijo1 = Edy m

(6.2.7)

and, in region II, ¢,, satisfies

h?

m*

Hyoy = [—S50 + Mu = m m

m* — Edy. m

(6.2.8)

Then the variational condition is

d LG q aS 5>» = 0,

(6.2.9)

where (Opik & Wood 1969)

(E> =

Pi

Ah dt+ Su Pi Ayu At

rll? dt+(m*/m) fy lpul? dt

(6.2.10)

and ¢, and ¢,, and their derivatives are continuous at the join of the two regions. The present form of W is Hermitian, and so the variational principle follows directly. The formal theory of regional wavefunctions of this sort has been given by McCavert & Rudge

(1972).

VARIATIONAL

METHODS

121

6.2.4. The pseudopotential method In the pseudopotential approach the orthogonality constraint is included by adding an extra term to the Hamiltonian. Instead of solving

6 WALA) OA WA om

(6.2.11)

subject to constraints, we solve

0 (AIA +h P(A)> _ OA

(6.2.12)

without constraints. In general the pseudopotential h is neither local nor unique. In writing the variational condition we have assumed h to be Hermitian; where h is not Hermitian special methods must be used. We begin by outlining the inclusion of constraints by h, following the general approach of Weeks & Rice (1968). We shall then show that there is substantial cancellation between h and some terms in HA and that this can greatly simplify subsequent parts of the calculations. In particular, the negative potential energy of the electron when it is close to a nucleus and inside the screening core electrons largely cancels the positive kinetic energy associated with the rapid oscillations of the wavefunction within the core, arising from the requirement of orthogonality. However, the pseudopotential need not be small, and we shall not treat it as a perturbation. We define a projection operator P for the core functions |¢,> by

P=)

P, = Ylbe>, where

ly>= (1—P)|'P>.

(6.2.16)

VARIATIONAL

122

METHODS

The constraint = 0 is satisfied for arbitrary |‘¥>, and hence for arbitrary variations in |’). The essence of the pseudopotential method is that, if we minimize the functional «P|7H +h|P>/ = )a,|¢.>, we have

CpelVl¥> = Vide >.

(6.2.25)

124

VARIATIONAL

METHODS

On substituting V+ PQ for V, and recalling COP

ON

(PlQIY> = . This leads immediately to (6.2.22). The Cohen—Heine pseudopotential need not be Hermitian.

B. The point-ion potential and ion-size corrections. We give an example in which the Cohen—Heine pseudopotential is approximated by a Hermitian pseudopotential. We shall assume that the core wavefunctions are orthogonal and that they are much more strongly localized than the trial function |‘¥>. The long-range Coulomb terms in the potential will be included explicitly. We assume that the potential due to atom i is V,, and that the potential due to atom i treated asa point charge is Vf" The sum over core electron projection operators on site iis written P, = &, P.;. The total potential, including the Cohen—Heine pseudopotential, is

V, = V+ P(V—V) = Vayt+(V—Ve)+

P(V—V).

(6.2.27)

Vy = Vat D{U-P)Y-VE)+P(V-U)—PiV"},

(6.2.28)

After a certain amount of rearrangement

where U; is given by

U;= VVj~ ¥ ve. j#i

(6.2.29)

j#i

We now assume that |) varies much more slowly in space than the core functions. Each of the terms in the sum over i is well localized, so that we may write

CPVAIWD =

+

+ )+Y Or, EP.

(6.2.33)

l

Here the P, project out the wavefunction components with given angular momentum. This form has two main advantages (Abarenkov 1972). The potential is real, rather than an integral operator like the pseudopotential. Further, the model potential tends to be weaker than the conventional pseudopotential for highly-excited states. Model potentials for isolated ions can be generalized to those for solids by including the displacement of the origin of energy by the point-ion potential of the other ions. Thus for ion i, Q,(\r— Ril, E) becomes Q,(\r—R,|, E—U,) if the potential varies only slowly over the ion. Abarenkov

(1972) reviews

the various

choices

of Q,(r, E). Ob-

viously the point-ion potential itself and the Bartram—Stoneham— Gash

form (6.2.32) are special cases.

Another

special case, used

126

VARIATIONAL

METHODS

widely for metals and semiconductors, is to have a constant potential A, within a sphere of radius Ry and merely the point-ion potential outside. Matthew & Green (1971) use a special case of this for ionic crystals Or,Ar, E) = ((A—V"6,0, E) = O10 =

U

r

(6.2.34)

Ro

where A is related to the coefficients of (6.2.32). Abarenkov & Antonova (1967) discuss analytic forms designed to give good energy levels and wavefunctions for free atoms. Another convenient choice is that of Hellmann (1936) and Abarenkov & Bratzev (1963), who use Qr, E) = A, exp(—k,r)/r.

(6.2.35)

6.2.5. Defects with several electrons The orthogonality constraint 1s modified, and usually takes the form of ‘strong orthogonality’ (e.g. Parr, Ellison, & Lykos 1956; Szasz 1968; McWeeny & Sutcliffe 1969, p. 176). Instead of merely requiring that the total defect wavefunction is orthogonal to the core functions when integrated over all electron coordinates, here

one requires orthogonality when integrating over each coordinate separately 0 =

|d°R

Waerect(R,

oq;

Co")

rnd)Wcore(R,

Toc

COE)

Ic):

(6.2.36)

Pseudopotential methods can be developed (Szasz 1968, Huzinaga & Cantu

1971), but this has only been done

within

the Hartree

approximation. Thus for two electrons one uses a pseudowavefunction ‘W(1,2) related to the real wavefunction (1,2) by two projection operators, one for each electron

WL,2)= (1—P,)(1—

P,) (1,2).

(6.2.37)

Simple expressions can be derived only if wyand can be factorized into two parts, one for each electron. The problem becomes very difficult if correlation cannot be ignored. 6.3. The accuracy of a variational calculation

Variational methods are approximate by nature. We now discuss the three main methods by which their accuracy may be assessed.

e1]39q

uonoun)

[Jam-arenbs 10) 944](I1[

ueissnen

otusso1pAy

suonouny 10) ed4{ ({

ssajdsnp

19}e|5





a

0

uuu)

Ge

aG

[RIUDIOg OBZ

p>41°%—

+4) m/[

d

D

4‘4/0°4 < D

LOY) DS

[Jam-aIeNbs Se D in the explicitly orthogonalized case, or for the pseudowavefunction |‘¥>? The answer is based on the answers to three questions. Firstly, what symmetry has the wavefunction? Can we approximate the wavefunction by one of the form

eee » Rim

YP, >),

(6.4.1)

in which the sum keeps only the first terms of the correct symmetry? Secondly, if we are interested in the energy alone, is there a problem which we can solve exactly which has a similar Hamiltonian to the one of interest? Finally, if we want detailed properties of the wavefunction—for example its amplitude for use in hyperfine structure calculations—is the trial function sufficiently flexible? Table 6.1 shows a number of trial functions in common use. The effective potential V(r) for which they give exact solutions is defined by

2m

>

V=

{ed

Ragets Ren) ae

IR} dP? rR) dr

(6.4.2)

for a single term of the form (6.4.1). Additive constants in V(r) are

VARIATIONAL

METHODS

131

dropped, and normalizations chosen to make f dr r?R?(r) equal to unity. In practice the symmetry of the wavefunction is simplified as much as possible consistent with the predictions which are desired, and high angular-momentum terms are omitted. The reason that this works well is partly that there are cancellations between terms which vary rapidly in angle, and partly because there is a ‘centrifugal’ term in the radial Schrédinger equation; the anisotropic parts of the wavefunction tend to be largest in regions of higher potential energy and hence they have relatively small amplitudes. The ‘exact-soluble’ problems give either continuum functions— that is, wavefunctions for certain special wells—or else atomic functions. In fact one of the most useful features of continuum treatments is the provision of suitable variational trial functions. Some examples of functions are given in Table 6.1. To satisfy asymptotic forms one generally simply adds a correction term which increases the flexibility of the trial function slightly. The asymptotic forms can be expressed in terms of the binding energy E, of the

level. We define k* = 2mE,/h’. If the binding potential satisfies rV(r) > 0 asr — oo, then the radial function falls off as exp (—kr)/r at large distances. For a Coulomb potential one finds instead (some finite polynomial in kr) x exp(—kr). Newton (1960) discusses analytic properties in more detail. If only long-range properties of the wavefunction are needed, it may be simpler to use the quantum-defect method (§ 4.3), recognizing the asymptotic validity of effective-mass theory. In the opposite extreme the asymptotic part of the wavefunction can usually be ignored if only an approximate eigenvalue is needed for a deep state. We also have to decide whether to use a one-centre form of trial wavefunction (usually possible when we use explicit orthogonalization or a pseudopotential method) or a many-centre form, such as a linear combination of atomic orbitals. Both may involve two- or more-centre integrals, but usually the integrals are fewer and easier in the one-centre expansion. The choice depends on which functions or set of functions gives the best representation of the wavefunction with the simplest form. It appears that only rarely is the many-centre form the best choice. The reason can be seen from Fig. 6.1. We give the radial dependence of the spherically symmetric part of a wavefunction consisting of six hydrogen atoms at (1, 0, 0) and equivalent sites. It is clear that the radial dependence will be well described by a

VARIATIONAL

132

METHODS

1-4

) Ro,o(r

fr) Ro

SS) &

2:0

) Ro,olt

) ol RoH

aa a

3-* oO

Roof)

) oT Ro,

Fic. 6.1. One-centre versus few-centre functions. The radial dependence of the spherically-symmetric part of an L.C.A.O. function is shown. The function is constructed from hydrogenic 1s-orbitals with effective Bohr radius a* at a distance R from a common centre. Values of R/a* are given on each plot.

simple one-centre function unless the radial function of the individual atomic orbitals are very compact. However the result is less clear if a high degree of anisotropy is expected.

7 MOLECULAR METHODS AND CALCULATIONS

MODEL

7.1. Introduction

MOLECULES, like solids, are many-electron, many-atom systems, and the methods of calculating their electronic structure can often be carried over into the theory of defects. We now discuss some of the methods and ideas which are common to both fields. The fundamental methods in molecular theory are the valence-bond (Heitler-London) method and molecular-orbital theory, and we outline these initially. The concepts of localized bonds and of hybridization will also be described. Secondly, we discuss some of the approximations associated with the molecular-orbital approach. These include semi-empirical approximations for the integrals which arise in these treatments. Model calculations related to molecular-orbital theory are also discussed. Finally, we outline approaches which are particularly appropriate to defect systems in solids. Clearly the chapter is in no sense a review of the theory of molecules; it surveys instead molecular methods in the context of defect theory. Examples of the use of molecular methods are given for self-trapped holes in ionic crystals (chapter 18) and for vacancies in valence crystals (chapter 27). 7.2. General methods 7.2.1. Molecular-orbital and valence-bond methods

The two fundamental methods are the valence-bond approach and molecular-orbital theory. In their usual forms they give approximations to the total wavefunction. Whilst these usual forms are very different, both methods converge to the same result if properly >xtended (Longuet-Higgins 1948; Coulson 1961, p. 155). Molecular-orbital theory is very similar to atomic theory. Oneelectron orbitals w,(r,o), including spin, are constructed for an slectron moving in the field of all the nuclei and in a mean field from

134

MOLECULAR

METHODS

the other electrons. These orbitals are filled successively in the ground state, starting with the ones of lowest energy. The electronic waveNx N function for a system of N electrons is then given by the n ing t one-electro functions. determinan of the correspond By contrast, the valence-bond method concentrates on bond formation by pairs of electrons with opposite spins, rather than seeking a description in terms of a single determinant of oneelectron orbitals. At its simplest, each valence electron is assumed to participate in at least one pair-bond, and it is assumed that both electrons in a given bond are associated with different sites. Thus one ignores the ‘ionic’ configurations, such as A°B™~ as opposed to A~B™ in a molecule (AB)’~. This restriction means that fewer two-electron states can be derived from given atomic orbitals in this approach than with the molecular-orbital method. The molecular-orbital and valence-bond methods are conveniently contrasted by reference to the H, molecule (see especially Coulson 1961). Both predict a a ground state near to the equilibrium separation of the atoms in a free molecule. In both cases too, the electronic charge distribution is concentrated between the two nuclei. The differences are most apparent at large separations, where the valence-bond approach goes over correctly to free atom states (albeit the incorrect spin state, cf. Bingel et al. 1961 and § 9.3), whereas the molecular orbital form fails. It is straightforward to correct the error by configuration admixture (Coulson and Fischer 1949 ;Coulson 1961, p. 156). Such calculations indicate that, for H,, the molecular-orbital method is best at small separations, the valance-bond method at large separations, but that in many cases both methods are comparable. The valence-bond method becomes difficult to apply in practice unless all the overlaps of atomic orbitals are very small. The difficulty appears because the many-electron wavefunction in the valencebond method is a sum of determinants whose elements are nonorthogonal atomic orbitals. It is thus the orthogonality problem which is inconvenient (Coulson 1970). The molecular-orbital method uses instead a single determinant whose elements are ortho-

normal one-electron molecular orbitals, and the corresponding difficulty does not arise. It is possible to use orthogonalized atomic orbitals as a basis for the valance-bond method (McWeeny 1954, 1955) or to work with a non-orthogonal basis (Newman 1971), but other difficulties may arise. Thus, Slater (1951b) showed

that

AND

MODEL

CALCULATIONS

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orthogonalized orbitals do not give a bound state for H, unless onic configurations are included. }.2.2. Separation of singlet and triplet states Conventional methods of calculating energy differences are often inaccurate in calculating the small energy separations of spin singlet and triplet states. The problems for H, were mentioned in }7.2.1. The special methods used in such cases are discussed here. The Hamiltonian for a system of two electrons has the form

H =7T,+T,+V,+V,+e7/r,,

(7.2.2)

when there are no spin-dependent interactions. The subscripts | and 2 label the electrons. The lowest singlet and triplet spatial wavefunctions are, respectively, ¥,(r,,r.) = Y,(r2,1r,)and P,(r,, rz) = —Y(r.,1r,) with energies E, and E,. The energy separation can de written in two forms. One, the most familiar, is

E,-E, = { dr, Gr, {P2V,—V,H¥,}.

(7.2.3)

[he second is obtained by multiplying the Schrédinger equation for ¥, by ¥, and integrating over a region Q, and then subtracting he corresponding equation with Y, and ’,, in opposite roles

Ey

=e

Da

ee d°r, datctal T, at Na

Pe YT;

fled, o'r, ¥,¥,

ay Ts

.

Wee Cis

This equation is meaningless if the denominator vanishes, so that Q

cannot be taken as all of the configuration space. Note that (7.2.4) Joes not involve the potential energy explicitly. Both expressions sive the same value for exact eigenfunctions VP,,. However, the econd expression can be made much more precise when only ipproximate functions are available (Herring 1962; Gorkov & Pitaevskii 1963: Herring & Flicker 1964; Berezin 1968, 1972; Norgett 1970). We rewrite the (7.2.4) using linear combinations

od, =—-(¥,+¥,).

(7.2.5)

na system like H,, for example, ®, is largest when electron | is

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near one proton and electron 2 is near the other. The splitting is

en

(7.2.6)

ee (agar, Soe tee est + T,)®, J ffq d?r, d?r,(®4 — ®-)

The choice of region Q depends on the system being considered. If we consider a two-centre system like H, lying along the x-axis and centred at the origin of coordinates, then Q can be described in terms of the coordinates x, and x, of the electrons. Thus, at very large nuclear separations R, one defines Q by (x, < X,) to obtain (Herring & Flicker 1964)

E,—E, = (3-282 Ry)R# exp (—2R/ay){1+O(R~*)}.

(7.2.7)

For intermediate values of R, appropriate for the colour centres discussed later in § 16.3, Q is defined by x, < 0 and x, > 0. In this

region (1—ff,d°r, d’r,{) and (ff,d°r, d°r,®3) are both very small. The denominator of (7.2.17) is close to unity, and we have

re (EGE)

=|: ck



Ie dy, dy,

dz, dz, dx,(0e

o® 0,0)

(7.2.8) This can be evaluated analytically in one case of interest, namely when one has Heitler-London functions

oe

Tal

N,)

1 '

Ne Wp (A(F (rz) + br; )a(r)}

(7.2.9)

constructed from wavefunctions proportional to exp (—a|r—R,,|) and with negligible overlap. The splitting is given in terms of « and the separation R by Berezin (1972)

(E,—E,) = PR{L aR +

aR

|exp(—2uR)

(7.2.10)

7.2.3. Localized bonds and hybridization

The molecular-orbital method is usually used in solid state problems, and can be made reasonably precise quantitatively. Often, however, simpler approximations still are useful. They may simplify complex problems so that the important factors and

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general trends may be isolated without the need for a full calculation. The basic elements of these approximations are the concepts of localized bonds, directed bonds, and hybridization.

The molecular orbitals in solids may spread over many sites. Thus the Bloch orbitals are the molecular orbitals of the perfect crystal, and they spread through the whole crystal. Despite this feature, one can often analyse electronic properties in terms of twocentre bonds. We shall do this at two levels; the qualitative picture

here and the quantitative method of localized orbitals in § 7.4. Qualitatively, an analysis in terms of localized bonds is particularly useful if the bonds are reasonably reproducible from system to system. Bond energies and lengths are often sensibly reproducible. Compton profiles for hydrocarbons (Eisenberger & Marra 1971) indicate

that

C—H,

C—C,

and

C=C

bonds

are

localized

and

consistent in behaviour. The reason that bond properties can be additive in this sense is that the charge distributions which correspond to different bonds are well separated. Coulson (1949 and 1961, p. 173) has discussed the rather close equivalence between the molecular-orbital and bond pictures. The distribution of charge in a bond also appears in the two basic ideas of stereochemistry :the concept of localized-bond orbitals and the concept of bonds directed so as to maximize the overlap between the atomic orbitals involved. These ideas are mainly used to predict bond angles, and so they have less application in solids apart from questions of crystal structure (Born & Huang (1954) and Phillips (1970) give interestingly distinct views on this point). However, the

concept of hybridization appears quite often. The localized directed bonds need not be formed from equivalent atomic orbitals; they may be linear combinations of inequivalent orbitals on each of the atoms involved. Thus a hybridized orbital may be a linear combination of distinct orbitals |a> and |a’> on site A and distinct orbitals |b) and |b’) on site B. The energy reduction from bonding more than compensates the energy needed to admix the higher energy orbitals |a’> and |b’>. The classic case of hybridization is, of course, the |2s> and |2p> hybridization in carbon and its compounds. Similar s—p hybridization is probably found in the four-fold-coordinated H-VI and III-V compounds (Coulson, Redei, & Stocker 1962). Pauling (1931) suggested the possible formation of hybrids from the |3d), 4s), and |4p) states of transition-metal ions in ionic crystals. Owen & Thornley (1966) argue that this is not a good picture

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because the energies of the |4s> and |4p) states are usually so much higher than the |3d) energy. However s—p-d hybridization may be important in perfect crystals of silicon and germanium, although the d part is often ignored. 7.3. Approximate methods Molecular orbital methods lead to secular equations of the type shown in § 6.2.2. The matrix elements in such equations can be too complicated to evaluate accurately. There may be too many matrix elements, as in systems which involve many orbitals on many atoms. Or the matrix elements may be too difficult, as in the multicentre

integrals mentioned before. A range of approximate methods has been derived to deal with these problems. Some approximate the Hamiltonian, e.g. the Xa method of §2.1.4. Others approximate specific terms in the matrix elements, or construct matrix elements by schemes which use some experimental data. All these approximations should be used with caution; their failures pack the literature of theoretical chemistry. The greatest temptation is to believe that an approximation is justified if an end result agrees with experiment. This is misleading. An approximation is acceptable if an exact theory and the approximate one agree when all other assumptions are the same. Agreement with experiment is irrelevant in judging an approximation unless all other approximations are known to have been eliminated. 7.3.1. Approximations for matrix elements

We now consider a variety of simplifications which are commonly used. A detailed review of these approaches has been given by Nicholson (1970). We shall adopt the notation that Ppalt,) 1S an atomic orbital for electron 1 on site A,and bya(t,) isan orthogonalized atomic orbital on that site. The orthogonalized orbitals are closely related to the Wannier functions (§ 2.2.3), and are usually defined in terms ofthe overlap matrix S by Lo6wdin’s symmetric orthogonalization procedure $= @. S~*. Matrix elements (abcd) follow the convention of(2.2. 13a).

A. Mulliken approximation. This method (Mulliken 1949) simplifies matrix elements by using the relation Ppa(1)oyn(1)

az TS pg P pal)

pa(1) + Pap

1)>ga(1)} ‘

(7.3.1)

It is often used for reducing three- and four-centre integrals to

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combinations of two-centre integrals (e.g. Coulson & Kearsley 1957). The approximation is best for matrix elements of operators which are fairly constant over ¢, and #,, such as the potential from a distant charge. By contrast, kinetic energy integrals are poorly treated. There are three main sources of error. First, the charge distribution is altered by (7.3.1). Two-centre integrals such as are particularly affected. Secondly, certain finite integrals may vanish because the overlap S,, vanishes. Thirdly, the Hartree-Fock equations which result from use of the Mulliken approximation are not invariant under orthogonal transformations among the basic orbitals. A choice of basis orbitals can be made which optimizes the approximation. It is usually best to use hybridized orbitals when appropriate. B. Ruedenberg approximation. Ruedenberg (1951) showed that a series of approximations like (7.3.1) could be constructed systematically using expansions of molecular orbitals in terms of atomic orbitals. In an integral the Mulliken approximation results if centres A and B are not too far apart. However, if Aand A or B and B are the centres which are close, another form is favoured

pall) Ppal2) > 2S ppl Ppa(l)bpal2)+Ppa(l)bpa2)}.

(7.3.2)

Clearly the forms (7.3.1) and (7.3.2) are suitable in different circumstances and should not be used in the same integral. The main sources of error in (7.3.2) are similar to those of the Mulliken approximation. The specific result (7.3.2) is sometimes called the Ruedenberg approximation.

C. Zero differential overlap. This approximation is to drop certain

matrix elements involving different orthogonalized basis orbitals. It is usually used with a second approximation in which the remaining matrix elements are evaluated over non-orthogonalized basis orbitals. The two approximations can be summarized in the equa-

tions

>

CGlQ\Q> > ?-

(7.3.3)

CAQID — 0 The algebra of solving secular equations is greatly simplified when

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the off-diagonal terms in ES, involving the eigenvalue and overlap, are eliminated. Zero differential overlap approximations for orthogonalized orbitals are closely related to the Mulliken approximation for the non-orthogonal atomic orbitals. There are several degrees of zero differential overlap approximation: Complete neglect of differential overlap (CNDO) = 0 for all potential energy integrals.

—_(7.3.4a)

Intermediate neglect of differential overlap (INDO) = 0 for all potential energy integrals other than one-centre exchange integrals.

(7.3.4b)

Neglect of diatomic differential overlap (NDDO) = 0 for two-centre integrals.

potential

energy (7.3.4c)

Zero differential overlap approximations are usually used in conjunction with semi-empirical estimates of the remaining matrix elements. These are discussed later. D. Neglect of inner shells. The assumption here is that the effects of core orbitals can be completely represented by using valence orbitals orthogonalized to free-atom core orbitals and adjusting the nuclear changes to include the effects of screening by the cores. It appears to be a good approximation (Manne 1966, 1967 ; Nicholson 1970, p. 277), and is related to the pseudopotential method described in § 6.2. One must orthogonalize valence and core orbitals to get an acceptable charge distribution. Clearly, for an atom with s core shells only, the ratio of s- and p-like character of the valence electrons will be strongly affected by orthogonalization. E. Approximation of Goeppert-Mayer and Sklar. This approximation is normally confined to compounds containing carbon or other s—p hybridized atoms. It is designed to simplify the Coulomb, exchange, core, and nuclear terms from a given atom. The assumption is that the potential due to one carbon atom (including core and valence electrons) can be represented by V(r) a

Veonerigat) =

VeemalX);

(7.3.5)

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where

Vepherical = POtential due to a neutral carbon atom in the valence state sp,

Voona = potential due to the hybridized bond-orbital singled out in the integral and where all exchange integrals are ignored. The approximation has a history of expedience rather than accuracy (Goeppert-Mayer & Sklar 1938 ;Daudel, Lebfevre, & Moser 1959; Parr 1964: Coulson &

Kearsley 1957). 7.3.2. Semi-empirical approaches

The semi-empirical methods use atomic spectroscopic data to simplify some of the matrix elements in the Hartree-Fock equations. The data are usually ionization potentials, since these can be related to free-atom Hartree-Fock energy levels by Koopmans’ approximation. Koopmans’ approximation (§2.1.3) can only be used for ionization potentials, and not for other transition energies, since it depends on the cancellation of the errors associated with neglecting correlation and with neglecting the changes in orbitals of other electrons in a transition. Many of the justifications of semi-empirical methods are invalid (Nicholson 1970). Two arguments do seem to be correct. The first is that these methods can successfully correlate results. The methods should be regarded as mean of relating observables, rather than first-principles calculations (Streitweisser 1961; Murrell, Kettle, & Tedder 1965). The second is that the methods can make one’s intuition more quantitative. Thus the methods might give one zuidance as to how many orbitals should be considered in a more accurate treatment. There have been many recent applications of semi-empirical methods to solid state problems, and Baetzold (1972) has given an optimistic summary. A comprehensive review of the methods is ziven by Pople & Beveridge (1970).

A. Extended Hiickel theory. The Hartree-Fock equations predict sigenvalues E which are the roots of the secular determinant

det ||94,— ES;,|| = 0. in extended

Hiickel

theory

(Hoffmann

1963)

(7.3.6) the one-electron

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METHODS

matrix elements %; are approximated. If J; is the one-electron ionization energy for orbital |i), then the approximated diagonal elements are

Heme

(7.3.7)

This may be corrected by the Madelung energy in ionic crystals (Birman 1966). Two approximations for the off-diagonal elements are common. Both are proportional to the overlap S;; (Mulliken 1949). One (e.g. Helmholtz & Wolfsberg 1952; Hoffmann 1963) assumes

Ki = =

K

Ui FT )Si;>

(7.3.8a)

in which K is an empirical constant, usually 1-75. The second form (e.g. Ballhausen & Gray 1962, Birman 1966) uses a geometric, rather than arithmetic, mean

Hy = —k (IL )S,;-

(7.3.8b)

Again K is an empirical parameter, usually between 1 and 2. Unfortunately, different choices of k and K are best for different properties (Parr 1964) and the results are sensitive to the choice of energy unless K is unity. These difficulties are not too important if one is interpolating results for a sequence of similar systems, but absolute predictions become hard to trust. The off-diagonal elements are less important in ionic crystals, where the overlaps are small and Madelung terms large. The early Hilsch—-Pohl approach corresponds to neglect of the overlaps, and is sometimes useful (e.g. Serban 1971). The two forms (7.3.8a) and (7.3.8b) are so similar in spirit and accuracy that we discuss them together. The original Hiickel theory applied to the z orbitals of planar conjugated and aromatic systems. Hoffmann’s extension was to apply it to o orbitals too. He found that the relative importance ofoand z terms was well predicted, that three-dimensional shapes were predicted adequately, but that

its spectral predictions were ‘miserable’. Blyholder & Coulson (1968) found that the L.C.A.O. molecular-orbital Hartree-Fock equations could be reduced to extended Hiickel theory by using the Mulliken approximation (7.3.1) and the assumption that the charge distribution was reasonably uniform. The kinetic energy terms lead to special difficulties in the choice of K, since they do not scale with

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the overlaps S as desired. Blyholder & Coulson confirm that the approximations have rather low accuracy, being particularly good for ionization energies and particularly poor for total energies. The total energy is needed if the distortion of a molecule or a crystalline defect is needed. It is customary to make a further gross assumption, that the sum of one-electron energies 2, may be used instead of the total energy E;. This sum £, counts the electron— electron interaction twice and omits the nucleus—nucleus interaction. This approximation is often grossly inaccurate (Larkins 1971d) and should not be used. In some cases the approximations may predict bond angles correctly, as Hoffmann found, simply because much less information than the total energy is needed (Allen & Russell 1967). If @is the angle ofinterest, then 2, may be used instead of Ey provided

(62 ,/06) and |{0(E;—42,)/00}| have the same sign and

(7.3.9)

|(CX,/A8)| > | {Ey —5z,)/08} |. It is the qualitative shape of the energy surface near the minimum, rather than its details or absolute energy, which matters. The numerical checks by Allen & Russell show that these conditions are satisfied when electronegativity differences are slight. Thus the bond angle for F,O is predicted well but for Li,O rather badly. But, even in the best cases, there are big differences between 2, and the true total energy. Semi-empirical methods have been used in quite a few defect problems (Jorgensen 1962; Moore & Carlson 1965, 1971; Moore 1966; Birman 1966: Messmer & Watkins 1970; Watkins & Messmer 1970, 1971; Dunn 1971) with varying success. Jorgensen finds, for instance, that the relative equilibrium charges on transition metals and their nearest neighbours are rather poorly given. This is a consequence of the general rule (e.g. Allen 1970) that extended Hiickel theory is only successful when electronegativity differences between constituent atoms are small. Other problems, too, can arise. Messmer & Watkins have discussed donor and acceptor states in diamond with some success. However, some features which come

from the long-range Coulomb field in these cases are omitted. The only long-range terms included are implicit: the defect atom alters one of the ionization potentials (and some overlaps, although

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MOLECULAR

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these are a secondary effect since conventional effective-mass theory would count these as central-cell corrections) which affects the various matrix elements. The number of bound states predicted is finite, not infinite as in the full problem (see chapter 9). Further, it does not seem easy to reduce effective-mass theory to extended Hiickel theory, or vice versa. B. CNDO methods. The CNDO methods are one stage more advanced than extended Hiickel theory, and have been used in several solid state problems (e.g. Hayns 1972a, b). We concentrate on the CNDO/2 form, since this is most commonly used (Pople & Segal 1966; Pople & Beveridge 1970). In essence, CNDO/2 is a self-consistent calculation which involves zero differential overlap,

the neglect of some terms which cause problems under change of basis, and the use of some empirical data. There are no three- or four-centre integrals, and no overlaps between different orbitals are retained in the secular equation or in the normalization. In the notation of § 7.3.2, the main approximations are:

one centre, one electron:

= = 2p A;) (Za Savane two centre, one electron: (®4|Vgl@,a> ~ OngZpYap> two centre, one electron: COPMe

two centre, two electron:

Via ar VelPop>

=

abe

+

Ba) ~ OprOgs)aB-

Here the average of the ionization potential J and electron affinity A is more accurate than the adoption of J alone in extended Hiickel theory. Z is the effective nuclear charge (atomic number minus number of core electrons). The ‘bonding parameters’ B% are chosen to give agreement with detailed calculations of simple systems, and the average electron repulsion integral y,, is found in terms of s-like valence orbitals appropriate to the atoms in question. 7.3.3. Model calculations

The semi-empirical schemes are most successful when used to relate properties of a series of similar defects. Another class of approaches also adopts this strategy. However, the model calculations use simple exactly-soluble models, rather than attempting to approximate the matrix elements of an accurate Hamiltonian.

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Strictly, the use of either effective-mass theory of the Koster—Slater treatment outside their ranges of validity are examples of model approaches discussed earlier. A. The Kronig—Penney model. In the original Kronig—Penney treatment, the one-electron potential of a perfect crystal was represented by an array of attractive or repulsive square-wells, later taken to be delta functions. Saxon & Hutner (1949) and Stéslicka & Sengupta (1971) have both considered cases where the strength of one of the delta functions is altered to represent the defect. Both bound and unbound states can be treated.

B. Thefree-electron network. This approach (Ruedenberg & Scherr 1953;

Coulson

1954,

1955) also represent

the atoms

by delta-

function singularities. The atoms are at the vertices of a network, usually chosen to consist of the lines joining nearest neighbours. The Schrédinger equation is solved for an electron confined to the network. Thus there is a set of one-dimensional Schrédinger equations to solve. The potential is assumed constant along any line joining two atoms, so the main problem is satisfying the boundary conditions at each vertex. A set of linear homogeneous equations for the wavefunction at each vertex is found, and from this one may obtain eigenvalues. Coulson has stressed that the method resembles a simple form of molecular-orbital theory. The method has been used for defect calculations. Fukuda (1965) combined it with the Koster-Slater method to discuss both bound states and scattering by an impurity in graphite. The impurity was represented by a change in one of the delta function strengths. Goosens & Phariseau (1966) discuss the vacancy in diamond using two models. In one the delta-function at the vacancy site is removed. In the other the network is reconstructed by joining the nearest neighbours to the vacancy in pairs; this represents the rebonding assumed to occur near the vacancy. C. Delta-function potential approximation. This method (Berezin & Kirii 1970; Chkartishvili & Papava 1971) explicitly attempts to relate properties of similar isoelectronic defects to one another. For these defects there is no long-range binding, and it is reasonable to expect only the one s-like bound state as for a 6-function potential. If one can obtain the strength of the potential from results for a single centre (e.g. an F’-centre in alkali halides) then one can predict

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MOLECULAR

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results for cases where N similar defects are present (e.g. the M’and R’-centres in alkali halides). The method appears to be quite accurate if the systems considered are chosen with care.

7.4. Weak covalency The simplest example of weak covalency is a system in which we may concentrate almost entirely on a single electron predominantly in a one-electron orbital ¢,. The weak covalency leads to corrections largely described by another one-electron orbital $y. An example of such a system is a cluster containing a transitionmetal ion in an ionic crystal, where ¢, is an occupied orbital on the neighbouring host ions and @¢, an orbital on the transitionmetal ion. The orbitals need not be associated with a single ion; indeed $, would usually be a symmetrized combination of the orbitals on the neighbouring ions. We shall compare the molecularorbital description with that of the related ‘configuration-interaction’ method, which offers advantage in this case (Rimmer 1965 ; Hubbard,

Rimmer, & Hopgood 1966). The simplest description is the molecular-orbital bonding and antibonding orbitals are then

version. The

dp = (1+2BS + B*)-3(¢,+ Boy),

(7.4.1)

Pa = (1—2AS+ A’) 4(bu — Ady),

(7.4.2)

in which S is the overlap (|¢,, and A, B, and S are related by the orthogonality of dg and od,

_ B+Ss BeoBS

< BS

(7.4.3)

The admixture coefficients A and B can be defined in terms of the one-electron Hamiltonian h and the one-electron energies € and &,, Rae

ya Chlhloo — ES = (CA Gu> - (Ego — Exo) — (|AEg| —|AE,|).

(7.4.15)

The covalency correction can be a large fraction of the total splitting, as shown in chapter 22. The other observable is the spin-resonance spectrum, which monitors the spin density. If @, corresponds to a (usually occupied) ligand orbital and ¢y to a (usually empty) transition-metal orbital, then it is usual to observe the transitionmetal resonance and to look for admixtures of the ligand orbital. In essence, one measures the spin density of the hole in the antibonding orbital. In the molecular-orbital approach the fraction of spin density on the orbitals of the neighbours is approximately A2

ome

aa

from (7.4.2) and (7.4.3). 7.5. Method of localized orbitals

We now discuss one very powerful class of methods for treating a polyatomic system in the Hartree-Fock approximation. The methods can be extended beyond this, but, as always, at the expense of greater complexity. In the Hartree-Fock approach the manyelectron wavefunction is a single determinant of one electron functions. Choosing the optimum determinant by using the variational principle, the one-electron wavefunctions satisfy a set of eigenvalue equations which must be solved self-consistently. In the method of local orbitals the problems of determining the eigenvalues and of obtaining self-consistency are separated, with considerable practical advantages. The method also makes contact with the idea of ‘atoms in molecules’. This useful concept implies that the wavefunctions of chemically-constant parts of a sequence of polyatomic systems need not be recalculated from first principles for each system. The molecules can be thought of as constructed from standard subunits. It

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MOLECULAR

METHODS

also suggests that overlaps between the basic units may be small, giving approximate methods like Hiickel theory through expansion in the overlaps, and simplifying the correlation problem, since the Coulomb correlation within units is likely to be much greater than that between units. The methods in the form in which they are needed here have been developed, in particular, by Adams (1961, 1962), Gilbert (1964, 1969),

and Kunz (1969). Weinstein, Pauncz, & Cohen (1971) review molecular applications. We shall confine our attention to a Hartree— Fock treatment of systems with full electron shells. The equation which is satisfied by the one-electron wavefunctions is a slight generalization of the standard Hartree-Fock equation. It can be written

Fo; = 2" ojfu-

(7.5.1)

The @; are not necessarily eigenstates of the Hartree-Fock operator F, and in general are linear combinations of the eigenstates. Consequently f,; is not just the diagonal matrix of the eigenvalues. Further, the ¢; need not be orthogonal, and we define an overlap matrix S whose elements are

Sij = {d°r b¥(r)d (1).

(7.572)

The Hartree-Fock operator is the one-electron operator

F=T+ht+hik

(7.5.3)

in which T is the kinetic energy and the three potential energy terms give the interaction with the nuclei, the Coulomb interaction, and the exchange terms. The superscript H on the summation indicates a sum over the occupied orbitals which constitute the Hartree-Fock manifold. Other orbitals will be referred to as ‘virtual’ orbitals. The matrix elements f;, are Lagrangian parameters which arise from the constraint that the eigenfunctions of F be orthonormal. Formally, we may write F = S.f, or

f=S"'.F.

(7.5.4)

7.5.1. The density operator

In discussing the one-electron eigenfunctions, we shall often make use of the properties of the Fock—Dirac one-electron density operator defined by

Perr) = pelts’) = YY" bin)Si5' Ar).

(7.5.5)

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It is idempotent (pf = p,) and is a projection operator; if | Daas some linear combination of the occupied orbitals

Pre

(7.5.6)

whereas if | > is one of the unoccupied (virtual) orbitals

Pr

= 0.

(7.5.7)

An important result follows from this, namely that we may define the projection A* of an arbitrary one-electron operator A

AP = peApe.

(7.5.8)

whose effect on orbitals within the Hartree-Fock manifold is given by

i

The projected operator has zero matrix elements outside the manifold. When acting on a virtual orbital it gives zero

Atl

t=O)

(7, 5;10)

The density operator is also an invariant of the Hartree-Fock equation. If T is any non-singular matrix then the transformation

¢-Te=¢'

7.5.11)

changes the orbitals and also changes the matrix elements of the Hartree-Fock operator. But the density matrix is invariant pr > T 'prT = pr,

(7.3.12)

and the eigenvalues of the Hartree-Fock operator are similarly unchanged. Finally, we note that the Hartree-Fock operator is itself a linear functional of pp. For example, we may write the Coulomb and exchange terms as follows: e2

Vefolr, r')} = ate—r) |a

p(r’, r” ror’

Velole,r')}= — p(t,r)}——.

(7.5513)

(7.5.14)

If all the spatial orbitals are doubly occupied p(r, Vr’) > dss(r, vr’)

(72555)

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and (noting sums are over spatial orbitals only) 2)

Vo an

2 { dr”

pr’,

r’

re. Vy >,



at

(7.5.16)

2

dss P(F, r ears

(7.517)

7.5.2. Self-consistent solutions We now return to the problem of solving the self-consistent equations for the orbitals ¢;. There prove to be far more convenient sets of equations for self-consistent solution than the usual eigenfunction form of the standard Hartree-Fock equations. The ¢; satisfy not only the equations

Fo, = 0" oifi

(7.5.18)

but also, adding eqn (7.5.9), the modified Hartree-Fock equations

(F+ A"); =)" o,fM.

(7.5.19)

j In the second form AF is given by (7.5.8) in terms of arbitrary operator A. The ¢ satisfy a whole class of equations with different choices of A. In both cases (7.5.18) and (7.5.19) we may take different linear combinations of the ¢; by transforming the equations

o> ¢' = Td,

(7.5.20)

and this transformation will affect the matrix elements of both F and F + AF. It is clear that the orbitals ¢, and the matrix elements are only given uniquely when some extra condition is imposed. The advantages of the local orbitals method stem from this extra condition. The extra condition may be imposed on the orbitals @¢; or, alternatively, on the matrix elements f;; or f}. The usual condition is that the Hartree-Fock operator be diagonal, so that

Fy, = &y;.

(7.5.21)

With this choice the &; are just one-electron energies in the sense of Koopmans’ approximation. The similar condition for the modified Hartree-Fock equations

(F+AF)yM = EMyM

(7.5.22)

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has eigenvalues which are of no physical significance. Both these conditions were imposed on matrix elements. It proves to be advantageous to impose instead conditions on the wavefunctions g; by some variational condition. The self-consistency problem can then be solved with comparative ease, giving wavefunctions which are linear combinations of the one-electron eigenfunctions. The eigenfunctions and eigenvalues themselves are found separately by diagonalizing the Hartree-Fock operator using the ¢; as a basis. 7.5.3. Conditions on the orbitals

The condition on the wavefunctions is imposed by requiring some functional &(@,) of the orbitals to be extremal. We shall also want the ¢; to be normalized, but they need not be orthogonal, in general. The condition that G(,) be extremal is conveniently expressed in terms of an operator G

CMD

OG

ma

OL rer Se jk

(7.5.23)

4)

The variational condition which is imposed on ¥ can be expressed in the form Go; = 7 bg,i.

(7.5.24)

One set of modified Hartree-Fock equations which incorporates the variational condition will then be the diagonalized form (F+ G— pF pd; = Pi;

(75:25)

obtained by taking A = G—F. A variety of conditions have been imposed at various times in the past. These are summarized by Gilbert (1964), whose discussion we follow. A. Well-localized orbitals. Suppose we wish to localize orbitals on a particular site A. Then an appropriate condition on these orbitals is obtained by first defining a localizing potential W,,,(r, A) and by minimizing the expectation value of the localizing potential, summed over all orbitals to be localized at that site. The localizing potential might be an attractive square-well potential of radius of the order of the tabulated atomic radii. The depth of the well is chosen so that the number of eigenvalues of W,,, which lie below the lowest

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eigenstate of F is equal to the number of electrons we wish to localize at site A. The modified Hartree-Fock equations are thus (7.5.26)

(F + ppWrocPr)Pai = ExiPai-

The depth of the well is chosen so that the electrons localized at A have eigenfunctions which are approximate eigenfunctions of pe W,,-Pr- The constraint on these eigenfunctions is just that

(T5208)

ye < bail Mocl Pair siteA

is minimal.

Whilst we give here a condition

for localization in

atoms, localization in bonds can also be described in a similar, but

more complex, way. B. Distorted orbitals. We may want the localized orbitals ¢; to resemble free-atom orbitals as closely as possible. The ¢; are then free-atom orbitals distorted by the environment. It is convenient to express the resemblance by requiring that the overlap between the free ion and local orbitals be as large as possible, so one minimizes CPV

AD

De

(7.5.28)

siteA

in which p§ is the density matrix for free-ion orbitals. The constraint is already similar to the ‘well-localized’ constraint with a non-local localizing potential

Wis t= Date xs):

(7.5.29)

Again the depth parameter D is chosen so that the appropriate number of eigenstates of W{'s' lie below the lowest eigenstate of F.

C. Energy-localized orbitals. Here the Coulomb self-repulsion part of the energy is maximized so that some corrections, particularly exchange and interorbital correlation, are minimized. Edmiston & Ruedenberg (1963) maximize the functional

6 =

Kili),

(7.5.30)

subject to the constraints of normalization and orthogonality. The localizing operator is then 2

Be =D" oe}fe lr—r'| - bien? boc"

(7.5.31)

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155

and the stationary condition gives E,,, = E,. An equivalent description can be derived without the orthogonality condition, but has the disadvantage that E,,, is non-Hermitian in general.

D. Screened orbitals. Here the localized orbitals are chosen to minimize the Hartree-Fock energy of a part of the whole system, usually a single atom or ion. The Hartree-Fock energy of the subsystem is extremal, and the modified Hartree-Fock becomes F—pFp+pFip

= F,+U,— pup,

C7523)

where U, = F—F, is the potential produced by the rest of the polyatomic system. Adams (1962) calls pU,p the ‘screening potential’ since it completely cancels U, within the Hartree-Fock manifold. Its role as a screening potential is exactly analogous to that of the localizing potential. E. Exclusive orbitals. Boys & Foster (1960) suggested the minimization of the functional

G= II” ( b: rd; | = fb,r0,0%| lee (7.5.33) subject to orthonormality constraints. This choice maximizes the differences in the centroids of electronic charge. It is often used to generate and analyse electron densities after a Hartree-Fock calculation has been completed. 7.5.4. Application of localized orbitals methods In the constraints discussed the orbitals have been separated into those associated with a particular site and those associated with the environment. We now indicate how the Hartree-Fock operator may be divided in the same way. The density matrix may be divided into local and environment terms, exactly like the localized orbitals PATST) = pyar)

Pt (rer):

(7.5.34)

The local term contains just orbitals on site A Prat, r)=

oy bald Pad’). i siteA

(7.9.39)

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The environment term t, contains four terms, one independent of overlaps CO =

(7.5.36)

yy >, baihsiB#A

i

The others contain intra-atomic overlaps Ciiz

(75,31)

2 Pad) (Saiaj— Ll)pair’),

which are an indirect consequence of the finite overlap integrals, inter-atomic overlaps

Taz = YY (GaSaibsbajt OBjSpj aiPai> BHA i,j

(7.5.38)

and extra-atomic overlaps, which do not involve orbitals on site A at all

OS p (ba(SoahVon j— + at

y

(beck, bi+

uSaphbed

|

(7.3:39)

C#B#A

The modified Hartree-Fock equation contains the Fock operator and a localizing potential. The Fock operator is readily divided into local and environment terms since it is a linear functional of the density matrix. Thus we write

F=F,+V,,

(7.5.40)

where the local part F, includes the kinetic energy terms, nuclear attraction from site A and the terms in p,,. The environment term V, is usually dominated by Madelung energies, and includes all the terms in ty. The localizing potential can also be simplified without difficulty Pr WoocPrF =

(Paa WrocPaa) ak (Ts WrocPAaa st PAA Wrocla ateTa WiocTa)

= Wi)+Weer),

(7.5.41)

and splits into local and environment terms. The self-consistent solutions for the local orbitals are then found

by first iterating, keeping the environment terms Vi a wer) loc

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157

fixed, and then, if necessary, allowing the environment orbitals to change. In fact, iteration on the environment terms will only give a small correction if the orbitals chosen at the start of the calculation are reasonably good. In most cases use of free-ion or free-atom wavefunctions as a starting point will be adequate. 7.5.5. Approximate methods and local orbitals

In addition to the full local orbitals method there are several schemes which exploit the method but which are technically much simpler. Both exploit expansions in the overlaps between local orbitals on different sites. A. Hiickel theory (Gilbert 1969). The prescription in Hiickel theory is to assume that the matrix elements of the Hartree-Fock operator can be written in the form

oak SHE

(7.5.42)

Gilbert showed that if the matrix elements of the Fock operator are expanded in powers of the overlap integrals, keeping only the first-order terms, then the approximate matrix elements can be factorized as above. If we choose local orbitals as a basis, rather than Hartree-Fock eigenfunctions, the expansion is more satis-

factory since the overlaps are smaller. Moreover, Kunz has shown (1969, unpublished) that the local orbitals often resemble the free-ion Hartree-Fock orbitals closely, and this further facilitates calculation.

B. Development in powers of overlap. perturbative solutions. The modified Hartree-Fock equations can be written (Fa t+V,—Ex) Gai =

— PrWocPrPai-

(7.5.43)

The right-hand side of this equation can be simplified greatly if the overlaps between different sites

and the corresponding matrix elements of W,,. =

Waipj

(7.5.45)

are both small compared with the one-site values. We shall assume S,;,3; and W,;,3; are small quantities of the same order, so that expansions in them may be truncated after the same

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number of terms. To lowest order in these terms Kunz shows that (F

+

Va

Ex)

Pai

=

0.

(7.5.46)

The localizing potential then causes no distortion of the Hartree— Fock orbitals. To the next order (FeV

= Ca@Oag ea mas(u) =

(P(r; u)|Up(r; u)|‘P(r; u)> CHE

(EDD)

The potential energy for the lattice motion is just V,(u)+ Vp(u). Often V,(u) can be expanded as a power series in the displacements.

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There is a term independent of u, which has no dynamic effects but contributes to the defect-formation energy. The coefficient of the linear term is, by the Hellmann—Feynman theorem, just the effective force for the particular displacement deOVo(u) cu

i

«P(r: u)|(r:u)>

;

(8.1.3)

Higher-order terms involve the dependence of ¥ on u directly, and will be discussed later. Clinton & Rice (1959) and Sturge (1967) discuss the extension of (8.1.3) to degenerate levels. Higher-order terms appear in both VY, and V,. Those from V, are just the anharmonic part of the perfect-lattice potential energy. Our knowledge of V,(u) for real crystals is often so poor that it may be misleading to include the anharmonic terms, and in defect problems the harmonic part is usually sufficient. The effects of anharmonicity can often be predicted qualitatively (e.g. Flynn 1971a), but quantitative estimates are not usually possible. The most common technique is to construct model two-body atom-atom interactions expressed in a convenient analytic form and parametrized. Such models do not provide a complete prescription for treating anharmonicity.

8.2. Linear coupling

For many defect systems one can assume that the host lattice is harmonic and that the potential energy due to the defect is linear in the atomic displacements

V,(u) =4u.A.u Vu) = Voo=Foo u:

(8.2.1) (8.2.2)

We now show that the effect of the defect term is to displace the lattice modes without mixing the modes or changing their frequencies. This result is of enormous importance. Before deriving the general result, we illustrate it for a system with a single mode Q. The Hamiltonian is

ices

fg

errs dQ?

+4KQ?—FQ.

(8.2.3)

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At the minimum of potential energy the coordinate Q has the value

ONE OM

OS

(8.2.4)

We transform from coordinate Q to @ =(Q—Q,). The kinetic energy term is unaltered since 0/6Q and 6/60 are equal, so the Hamiltonian becomes 5 KH Bors = eh30 5+7KQ 1KO7)=4F7/K

:

ll (8.2.5)

The first term is a harmonic Hamiltonian, exactly equivalent to the perfect-lattice Hamiltonian apart from the displacement of the mode. The frequency of the mode is unaltered. The second term is equal to the change in the value of the minimum potential energy. We derive the general result in second-quantized form. The Hamiltonian has the general form

H =) hofaja,+fa,t+S

i

af),

(8.2.6)

q

in which the terms in f and f* give the defect contribution to H The transformation which simplifies this equation is

It leaves the commutation relations unaltered, since the f and f* are not phonon operators. The Hamiltonian becomes

H => hw,fija,— > ho,| fl’. q

(8.2.7)

q

The linear terms have been eliminated

by displacing the lattice

oscillators, without affecting the frequencies, occupancies, or relative

energies of the various states. The result is exact, and valid for any phonon spectrum. Again there is a term corresponding to the change in the value of the minimum potential energy; it represents a decrease in the defect-electron—lattice energy Vp and an increase in V_, the lattice strain energy. This is the ‘lattice relaxation energy’. One consequence is that in many defect problems we may concentrate on finding the configuration which minimizes the potential energy. The relaxation energy can be found, and so can the displacements of the atoms themselves if the normal modes of the perfect lattice (or equivalent information) are known.

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163

Another important result is that the lattice relaxation energy is independent of the isotopic constitution of the host (Hughes 1966). The Hamiltonian for the perfect lattice can be written

Ho =50.M,.a+4u.A.u,

(8.2.8)

in which I indicates dependence on isotope. The defect introduces an extra term given by (8.2.2) in which neither Vj, nor F, depend on the host’s isotope content. We want to find the dependence on isotope of the terms in the energy which derive from the defect term Vp(u). The change in isotopic composition affects the perfect lattice: the normal modes are altered, there may be displacements u, of the mean position of the host atoms, and there will be an energy change independent of the defect. This energy change we ignore. However, the displacements from the change in isotopes give an energy contribution associated with the unrelaxed lattice near the defect. We call this the static shift AE,

=

Fy e U;.

(8.2.9)

The lattice relaxation energy due to the defect is associated with displacement measured from the configuration of a perfect lattice having the appropriate isotopic configuration. This term is called the dynamic shift. The effect of Vj is simply to displace the lattice modes;

it does not mix them, nor cause

any further changes of

frequency. The relaxation energy is most easily found by transforming from atomic displacements u to normal modes Q = %;.u. The unitary transformation is, of course, isotope dependent, and was discussed in § 3.

H = T(a)t+5u.A.u—Fy.u T(a)+3(u Wh

A- AW. W—(Fo - A(X

BD:

The potential energy is now a sum of terms, each involving just one mode. The relaxation energy is the sum of the contributions of these modes; the transformed force-constant matrix is now diagonal, and

the relaxation energy can be formally written AEp =

—3(Fo WW

Ax)

—7(Fo Wh

Avl. X)(X1- Fo)

Bet Oey Vitec UA

10% - Fo)

(8.2.10)

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But both Fy and A are factors in potential energies, and so are

independent of nuclear masses. Thus AEp is independent of the isotopic composition of the host. This assertion, that the potential energy is independent of the nuclear masses, is a slightly stronger assumption than the Born—Oppenheimer approximation, although the difference (§ 3.1) is not usually important.

The result that AE, is independent of host-lattice masses is not valid if higher-order terms are included in Vj. Some results in this case are given by Hughes (1966). In essence, the failure comes from two sources: the changes in energy (including zero-point energy) which result from the change in phonon frequencies and a massdependent admixture of the modes by the higher-order terms. 8.3. Static distortion near defects

8.3.1. Linear response formalism We now examine the methods of calculating the static distortion for a given system. It is legitimate to neglect the nuclear kinetic energies in the harmonic approximation, provided that it is realized that the displacements derived are mean displacements and that the relaxation energy is a thermodynamic internal energy. It is convenient to write the defect terms in a different form. Instead of V,(u) we write

Vo(d, u) = T(A)+Vo,(A, u) + Vp,(u).

(8.3.1)

The parameter 4 is a variational parameter describing the defectelectron wavefunction; for simplicity we show just a single parameter, but the generalization is obvious. The new form V,(A, u) shows that the defect energy depends on both the displacements and on the wavefunction. The kinetic energy of the defect electrons T(A) is independent of the displacements. The defect-electron-lattice interaction is given by V,,(A, u), and initially we assume there is no electronic degeneracy to complicate this term. Finally, V,(u) includes those terms which are independent of the defect-electron wavefunction. It includes, for instance, the changes in energy from shortrange repulsive forces when an anion is removed in forming an F-centre in an alkali halide. Changes in force constants and in ionic polarizability near the defect may fall into this category.

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We may usually expand V,(A, u) in the displacements u and in the change in A from J, its undistorted lattice (u = 0) value. The leading term is just the one discussed in § 8.2, Vo(4, w) = Vo(4o, 0) —Fo(Ao)

.u+ (higher-order terms).

(8.3.2)

The linear terms do not depend on (A— A) at all, and depend only on the defect wavefunction for u = 0. The higher terms can usually be expanded in the convenient form

higher-order terms ~3u. B(A,).u+(A—A))A. ut lA’ +5A'(A—Ap)*

2

+...

(8.3.3)

As a rule, both F, and the higher-order terms are of short range, 1.e. components involving displacements of atoms far from the defect are small. This is not true when there are long-range Coulomb terms because of a net charge on the defect, although these terms can

be treated too. In seeking the static distortions, two minimum principles are used. The first is the variational principle for the defect-electron wavefunction = 0. aya Vol’ w) =

is

3 (8.3.4)

Thus V,(4,0) has an absolute minimum at 4 = Aj, which is the reason that there is no term linear in (A—A,) and independent of u in (8.3.2) or (8.3.3). If the expansion (8.3.3) is valid then we are using a perturbation—variation expansion to simplify the problem. The second minimum principle is simply the statement of the static

equilibrium of the lattice yeni ou

V,(A, u)} = 0,

(8.3.5)

and it is to the solution of this equation that we now turn. We shall describe three main methods: the Kanzaki method (or the method of lattice statics), various computer approaches, and the Mott— Littleton method ; we shall relate these methods to each other and to continuum elasticity theory. Lidiard & Norgett (1972) have reviewed the approaches recently, and Johnson (1972) has given a quantitative comparison for one specific system.

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8.3.2. Methods for calculating static distortion

A. The Kanzaki method (Kanzaki 1957; Hardy 1960; Stoneham & Bartram 1970). The method solves the static problem in two stages: first the solution of the linear problem, which gives the linear response of the lattice to the ‘defect forces’ F (4p), and secondly the treatment of the higher-order terms. The linear problem seeks solutions of

7)

—(—F,).ut+ju.A.u) = 0. ou

(8.3.6)

This represents a set of coupled equations, since the displacements of different atoms are connected by the force-constant matrix A. Their formal solution 1s:

WAG ER:

(8.3.7)

But direct inversion of A may be intractable in practice, especially when there are long-range Coulomb interactions. Kanzaki resolved this difficulty by Fourier-transforming the equilibrium equations. The coupled equations for the displacements u become separate equations for the different wavevectors, exactly as in the dynamics of the perfect lattice. The transforms of the displacements are

u(q) = )iu,e

1 F,

u, = =5” u(q) et,

(8.3.8)

(8.3.9)

q

The U are the Fourier transforms of the displacements. They are not normal modes, but combinations of normal modes with the same wavevector ;the normal modes are defined by a more complicated transform ofthe sort (3.2.5)

Qiq) = Y x(q)ur. l

(8.3.10)

In this section we use (8.3.8) rather than (8.3.10), since the manipulation of the equations is easier. The Fourier transform of the forces

F, is Fo,

Fo(q) = > Fo(l) et ®, l

(8.3.11)

LATTICE

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F,(/) = ~ DFo@e- co

SYSTEMS

167

(8.3.12)

and the transform ofthe force-constant matrix A gives the dynamical matrix

D(q) = & Aw el (RR),

(8.3.13)

[t is through D, the dynamical matrix, that the contact with models of the lattice dynamics of the host is made. As in lattice dynamics, the contribution of long-range Coulomb forces to D can be treated readily (Kellermann 1940). The separated equilibrium equations have the form

D(q) . i(q) = Fo(q).

(8.3.14)

For a crystal of N cells each with n rigid ions the (3Nn x 3Nn) matrix equation (8.3.6) has reduced to N(3n x 3n) matrix equations. Thus the matrix equations are (6 x 6) for the alkali halides in the rigid-ion limit, since there are two ions per unit cell. However, in the shell model each ion has core and shell coordinates, giving

12 x12) matrix equations for the alkali halides. As discussed in ) 3.3.2, shell coordinates are kept explicit in defect calculations. The decoupled equations can be solved by matrix multiplication

u(q) = D~ ‘(q). F,(q),

(8.3.15)

which presents no problems since the matrices are of low rank. The lisplacements u follow immediately after the Fourier transform 8.3.9). The final result gives the linear response of the lattice to he forces F,, and can be written

u=R.Fo,

(8.3.16)

where R = A !. Since this equation is linear, the displacements from jarious contributions to the force are simply added. The forces *, do not have to have any particular point symmetry, and asymnetric distortions can be calculated without significant modificaions. The actual matrix multiplication and subsequent Fourier transorm may be performed in two ways. If the displacements of atoms lose to the defects are wanted, as in calculations of transition energies and other electronic properties, the operations are carried ut numerically at a mesh of Ny, points q in the Brillouin zone. This

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corresponds to treating a superlattice of defects with one defect every Ny sites. The accuracy can be improved by raising Ny as necessary. One convenient choice of points q for alkali halides is the Kellermann (1940) sett of 1000 points in the zone. It is probably adequate for many calculations, and is readily extended in other cases. A second approach is used when the long-range distortion is needed, as in estimates of volume changes due to defects. Here the displacements are dominated by the small wavevector contributions. The manipulations can be carried out analytically by expansion in |q|. In this limit the Kanzaki method is essentially equivalent to elasticity theory. The relaxation energy may also be calculated conveniently, starting from (8.2.10),

= —1F,.R.Fo.

(8.3.17)

This equation can be simplified by taking symmetrized linear combinations of forces F,, rather than using forces on specific neighbours of the defect. The advantage is that cross-terms between forces of different symmetry in (8.3.17) vanish for reasons of symmetry, and can be ignored completely. Eqns (8.3.16) and (8.3.17) are the solutions of the distortion problem in linear coupling. The higher-order terms in (8.3.2) are treated by a calculation in real space rather than reciprocal space. We work with displacements u rather than their Fourier transforms u, and exploit the fact that

very few of the displacements are large. For simplicity, we begin by using the expansion (8.3.3) of the higher-order terms in u and in (A—A,). The variational condition (8.3.4), which presumes the adiabatic approximation for the defect electrons, relates 2 to the displacements

Au) = Ap—A.u/A’,

(8.3.18)

so that Vp can be written as a function ofu alone

Vp(A, u) = Vp -Fo.u+ju.B.u—3(u.A)?/A’,

(8.3.19)

+ Kellermann’s set, as printed, includes (10 5 0) in error, and omits (77 1) in error. This is easily corrected.

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169

This can be written more simply in terms of a distortion-dependent

force,

Vp(A, u) = Vy9—F(u).u,

(8.3.20)

av, F(u) = Sean =F,+B.u.

(8.3.21)

Even if the expansion (8.3.3) is not valid, the variational parameter A can be eliminated in terms of u, and a distortion-dependent force F(u) found. However, F(u) will be more complicated than the linear xpression in (8.3.21). The expansion (8.3.3) tends to break down when the defect has close energy levels which are mixed by small distortions. In all cases we may find the solutions of the equations for static equilibrium by solving (8.3.16) simultaneously with (8.3.21) or its generalized equivalent

u = R. Flu).

(8.3.22)

Thus if the F(u) is given by the simple analytic form (8.3.21) the distortions are

u=(1—R.B)'.R.F, including the higher-order terms. The variational hese distortions is given by

(8.3.23) parameter

1 =1,—A.(1—-R.B)"!.R.F,/A’.

for

(8.3.24)

As a rule, the second-order terms can be managed without diffi-

culty. Few displacements are large enough to give significant 1igher-order terms, so the B matrix of (8.3.21) will have low rank. Further, the algebra of the set of simultaneous equations (8.3.22) -an be simplified by the use of symmetry-adapted coordinates. The static equilibrium configuration of the crystal can still be ound even when the harmonic approximation breaks down. One egards the crystal as a Region I near the defect and a Region II urther away. The boundary is chosen so that the harmonic approxination

holds

in Region

II, where

the methods

outlined

earlier

ipply with straightforward modifications. The displacements in Region I must be treated another way. However, since only a small 1umber of these displacements are involved the problem can be olved by direct minimization of the total energy. This leads to the irst major problem: the anharmonic energy is not known accurately

170

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even for perfect lattice. Model potentials, such as the Born—Mayer form, do not solve this problem of anharmonicity; at best they introduce plausible additional approximations. The second major problem arises when the defect-electron wavefunction is very sensitive to the lattice configuration. Whilst the static equilibrium configuration can be found, it is not sufficient to predict the observable properties of the defect. Because of the anharmonicity which

results, the relaxation energy will no longer be the same as the change in mean energy averaged over thermal oscillations. Nor can optical transition energies be found directly from the parameter A appropriate to the minimum, since the Condon approximation (§ 10.3) and Franck—Condon

principle need not hold.

B. Computer calculations of distortion. There are several strategies by which the sum {V,(u)+ V,(0)} of the defect and lattice potential energies may be minimized. Two broad classes of approach are molecular dynamics methods (Gibson, Goland, Milgram, & Vineyard 1960; Bullough & Perrin 1968) and static methods which alter the positions of atoms near the defect to try to minimize the potential energy or to cause the forces on these atoms to vanish (Huntington, Dickey, & Thomson 1955; Johnson 1964; Doyama & Cotterill 1965; Norgett & Fletcher 1970). The static methods differ in the way in which they seek the minimum of V. = V>+ \ as a function of the displacements. Broadly, there are three levels of sophistication. First, there are search methods. These minimize V,(u) with respect to each variable in turn. No derivatives of V.(u) are needed, although the most efficient approaches (e.g. Powell 1964) do, in effect, make use of numerical derivatives during the calculation. Secondly, there are gradient methods, such as the method

of steepest descents, which need both

V,(u) and its

first derivatives 0V(u)/du. And, thirdly, there are matrix methods, which need the second derivatives 07 V./du, du, as well. For a general function of N variables, these methods have comparable efficiency. The search methods need O(N’) iterations, the gradient methods O(N) iterations, and the matrix methods O(1) iterations, but the calculation of derivatives in the gradient and matrix methods largely compensates for the smaller number of iterations. However, V.(u) is not an arbitrary function. The derivatives of V.(u) are particularly simple and easy to calculate, and the matrix method, with its small number of iterations, is the best approach. This advantage comes

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primarily from the fact that V,(u) is usually a sum of two-body potentials.+ Thus the potential energy itself is a double sum over the displacements, its first derivative is a single sum, and the second derivative consists of just a single term. As a result, the higher derivatives can be evaluated more quickly. If there are just shortrange forces then the calculation of V.(u) is not very much longer than that of its derivatives, and the other methods (particularly the gradient method) may become comparably efficient. Suppose that the first and second derivatives of V,(u) are f,, and W,,,, respectively, at the mth iteration. The earlier calculations using static methods (Huntington et al. 1955; Johnson 1964; Doyama & Cotterill 1965) were based on the Newton—Raphson method. Thus at each iteration an improved value of the displacement is found by

Uns 1= Uy —(W,,)~! « (Ey)

(8.3.25)

At each iteration f,,, W,,, and (W,,) ' are recalculated. The calculation of (W,,) ’ is particularly time-consuming and, in cases where there are long-range forces, especially difficult. A much more satisfactory method has been given by Norgett & Fletcher (1970), using a variable-metric algorithm (Davidon 1959; Fletcher & Powell 1963; Fletcher 1970). In this method only a single calculation of Ry = (W,)7! is needed; R is improved in subsequent iterations without recalculating W first and then its inverse. Clearly this feature speeds the calculation considerably. Further, since the distortions from the perfect crystal are usually fairly small, Rp, can be calculated for the perfect crystal configuration. This is a great advantage if there are long-range forces. The prescription is this. At each iteration, starting from m = 0, an improved

value of the

displacement is found by

Unt

= Un—R,, - fi,

(8.3.26)

exactly as before. The f,, are then recalculated, again as before, so that we know the differences

Ou,,

= Un+1—Un

(8.3.27)

Of, = fnoi fn

(8.3.28)

+ Strictly we want to minimize a total energy asa

function of configuration. There is

no need to introduce the concept ofa force-constant at all, although this is customary. Other formulations have been given by Ho (1971) and Stoneham (1970).

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The response matrix R is then improved by using the expression . Fe T (8.3.29) nae OU n (Un) _R,,- fn (Oho) cs a Rete (ou) .(of,,) (ot,)) oR, (ol,,) so that the iteration can be continued. Implicit here is the assumption that Ro is already quite a good approximation to the final value. The molecular-dynamics methods numerically integrate the equations of motion ofthe defect lattice. As the lattice configuration changes in time, the total kinetic energy builds up from its initial zero value. After a suitable number of steps, the kinetic energy is set to zero once more, and the process restarted. As the equilibrium configuration is approached the rise in kinetic energy gets smaller, and the configuration which minimizes V,(u) is assumed to be found. This method is akin to the gradient methods, since the first deriva-

tives of V,(u) are essentially forces. However, the dynamical method has useful extra flexibility in the choice of timestep and the choice of the stage at which the kinetic energy is quenched. Several criteria may be used to compare these methods. One is the speed with which the distortions are found. A second is the ease of treating long-range forces, particularly Coulomb forces. Thirdly, does the method find the real minimum of V,(u), or is it easily diverted into metastable minima or to regions where V,(u) is merely slowly varying with u? The matrix method of Norgett & Fletcher and the Kanzaki method (where the computer merely inverts matrices and Fourier transforms displacements) are particularly good by the first two criteria: they are rapid and can handle longrange forces. It is less clear that they treat subsidiary minima properly, although they are probably adequate in most cases met in practice. The molecular-dynamics method is particularly good by the first and third criteria: it can be rapid if the timesteps are properly chosen and, shallow metastable minima are usually avoided, because the system has kinetic energy. However, long-range forces cannot

be managed easily. The computer methods just described are intended for defects without defect electrons, such as vacancies in ionic crystals. They are inefficient for systems with defect electrons, since small changes

in the approximations for the electronic structure will change Vp(u), and it is clearly tedious to have to do a full-scale minimization of V. = V)+\, after each revision. A strategy for these systems has

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been given by Larkins & Stoneham (1971a, b). Instead of seeking the minimum of V., where

V(u) = V,(u)+ Vp(u),

(8.3.30)

one minimizes instead functions of the form

V(u) =

V(u)—F,.u.

(8.3.31)

One finds the linear response of the lattice to certain external forces exactly as in the Kanzaki method

u=R.F,.

(8.3.32)

The forces F, are generally symmetrized combinations of the forces on near neighbours to the defect site. Their magnitude depends on the model of the defect, and is given by an expression of the form

F° = —0V,(u)/du.

(8.3.33)

The main differences between this method and the Kanzaki method are these. First, the response function is found by a ‘computer experiment’, in which the distortion is related to the applied force, as opposed to a Fourier transform and matrix inversion method. Secondly, with the computer method it is particularly easy to treat terms which will be common to all models for the defect energy V>(u). For instance, if the defect has a net charge, then the Coulomb contribution to the forces will be present in all models for V,(u), and only the short-range terms will be model-dependent. It is then clearly an advantage to solve a slightly different problem, the minimization of

Vu) = V,(u)+ Y%,(u)— Fo. u,

(8.3.34)

where Vp, does not depend on the model. It is possible, but rather more difficult, to treat terms of this sort in the Kanzaki method.

Thirdly, one occasionally meets another class of distortion problem. If external uniaxial stress is applied to a crystal, how do the neighbours of the defect move, and how does this motion compare with that in a perfect crystal? Again, problems of this sort are most easily handled by direct computer methods. Siems (1970) has discussed the related elastic analogue of local field corrections. The computer can only handle finite systems directly. When the distortions in a large system are wanted, the defect lattice is divided into a Region I, close to the defect, and a Region II beyond it.

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As in the Kanzaki method, the boundary is chosen so that Region II can be treated as harmonic. If the atoms in Region II follow the motion of those in Region I adiabatically, then the displacements in Region II can be found as functions of the displacements in Region I, and the total energy can be written in terms of the displacements in Region I only. This rewritten form of Vu), depending on the displacements in Region I explicitly, is minimized in the methods described here. Two special cases should be mentioned. First, when there are only short-range forces, it may be possible to make Region I sufficiently large that the displacements in Region II can be set to zero without inaccuracy. However, this should only be done if the results have been shown to be insensitive to the choice of boundary between Regions J and II. Secondly, some approximate calculations keep all lattice atoms fixed except for the nearest neighbours of the defect. This is clearly a poor approximation for estimating the distortion, but can be useful for some simple semiquantitative calculations (e.g. Lidiard & Stoneham 1967). This class of calculation leads to a ‘sudden’ response function, that is to the lattice distortion under a force applied in a time much less than the shortest vibration period. Such forces might occur in atomic collisions. For all defect calculations of interest we need the ‘adiabatic’ response of the lattice to the forces Fy. Displacements should not be set to zero arbitrarily. Special caution is necessary in two cases. The first occurs when the lattice distortion lowers the symmetry of the defect lattice. Examples are the Jahn-Teller effect, and the off-centre substitutional ions. The force which drives the asymmetric distortion is usually zero in the symmetric configuration from which one starts: the symmetric configuration is an extremum of V,(u) between different equivalent asymmetric distortions. It is then essential to check for asymmetric distortions by starting from an asymmetric initial configuration. In the Jahn-Teller effect, the symmetric configuration is not an extremum; it is a point at which several energy surfaces intersect. One of these surfaces may be chosen arbitrarily, and this selects one of the possible equivalent distortions. There is no need to start from an asymmetric configuration, but it helps to know the symmetry of the expected distortion. The second case is that configurations reached at intermediate stages of the minimization process may differ sufficiently from the final configuration to cause an instability. The danger arises because the

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175

inter-atomic potentials are usually expressed in forms appropriate at large separations. If an intermediate configuration pushes ions too close together, attractive terms, extrapolated to these small separations, may overwhelm the repulsive ones. This error can be avoided by modifying either the potential (Quigley & Das 1967; Faux 1971) or the minimization procedure. C. The Mott-Littleton method for defects in ionic crystals. Mott & Littleton’s (1938) classic method finds those displacements which cause the forces on ions to vanish in a Region I near the defect. Region I is surrounded by a harmonic Region II. The fundamental assumption is that the moments in Region II can be obtained from the macroscopic polarization P induced by the effective charge Zple| of the defect. The method is iterative in that the size of Region I is progressively increased until the results are insensitive to the boundary. The forces on the ions in Region I are sums of two terms. One term includes the interaction of each ion in Region I with the defect charge Zple| and with the other ions in Region I. It is a function of the displacements and moments of the ions in Region I only. The other term gives the coupling between the displacements in Region I and the total moments in Region II. The total energy is reduced to a function of the Region I variables alone by an assumption relating displacements and moments in Region II to the macroscopic

polarization. This relation may be derived as follows. The macroscopic polarization induced by the defect charge is

P= | p_E 2 e (1-2). 4n

4n

figs honed Eo An

Cole

ae

Eo

cent

(8.3.35)

This polarization is the sum of the electronic polarization

1

1

P.. =—|1-—]D, 4n oe

(8.3.36)

and the ionic polarization from relative displacement of the ions in each cell



+-2|p.

(8.3.37)

176

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We have assumed a system of cubic symmetry, so that the dielectric constants are scalars. For simplicity we also assume two atoms per unit cell, so that the cell volume is 2Q for ionic volume 2. The electronic dipole moment of a unit cell which can be written in terms of the electronic polarizabilities of the ions, «, and o_, or

simply as 2QP,,. By analogy, we define a displacement polarizability Qgisp by the relation 2Qdisp (cee ateoa)

=

Jeftes: lee

(8.3.38)

or, in our simple case, A) =

2hdisp = (%4 +O_)

Bes

Eg(Eq—

(8.3.39)

1)

The total dipole moment per cell is now divided into two parts defined in terms of their sum and their ratio by

M,+M_

= 2QP,

(8.3.40)

MM sae ua ian

(8.3.41)

Mi

OSes

It is easily found from these definitions that

Grape2 Cie ese ACI 2oae.

1 Eo

(8.3.42)

Thus M, and M_ are uniquely defined in terms of D and a factor which depends on «1, «_,

&,,, €), and the atomic volume.

So far the results derived are quite general. The Mott-Littleton assumption is that Region II may be represented by placing point dipoles M_., on positive ions and M_ on negative ions. Each dipole is placed at the perfect lattice position of the ion, and the moment is calculated using the value of D at this position. There are thus several approximations. Near the defect there is no way of constructing unit cells so that M, and M_ are parallel and given by the ratio (8.3.41). There is also the simplification of point dipoles, and the elastic distortions are ignored. In addition to the special assumptions used in estimating M, some other special assumptions are made. One is implicit in that Region II is described by macroscopic parameters (e.g. &9 and ¢,,), whereas microscopic constants, such as force constants, are given in Region I. Problems of consistency arise if the electronic moment

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177

of an ion in Region | is taken as the electric field multiplied by the ionic polarizability. The point polarizable-ion models of the perfect crystal, which make the same approximation, do not give the vv. ect value of the dielectric constant: &é) is overestimated because the deformation dipoles are ignored. If correct answers are to be obtained as the size of Region I is increased, one must use a model giving the correct €,) used, such as the shell model (Faux & Lidiard 1971). Thus treatments are suspect if they define the displacement polarizability in terms of interatomic forces (8.3.39) instead (e.g. Boswarva & Lidiard 1967). The other assumptions are ones commonly made in defect calculations: the assumption of a Lorentz—Lorenz local field correction for all sites, and the neglect of non-linear effects of the large fields (~ 10° V cm ~') at ions close to charged defects. Several natural extensions of the Mott—Littleton method have been tried. The important ones are reviewed by Boswarva & Lidiard (1967) and Barr & Lidiard (1970). One is to include the elastic distortions in Region II. The problem here is matching at the boundary ; all methods assume the (r/r’) fall-off of the displacements predicted by elasticity theory. Brauer (1952) suggested finding the coefficient of (r/r?) from matching the elastic displacements in Region II to the displacements in Region I; a better approach is to match the total displacements, optic and elastic, in Region II to those in Region I (Boswarva & Lidiard 1967). Even simpler models than the Mott—Littleton approach can be useful. These include the Jost model (Jost 1933), in which the vacancy in an ionic crystal is treated as a spherical hole in a dielectric continuum. Crystal structure only enters in choosing the volume of the hole to be an atomic volume. The model is useful both for the vacancy (e.g. Flynn 1970) and, when an electron is trapped, for the F-centre (Simpson 1949). Its value comes from its simplicity and from the inclusion of two important elements: the polarization and a suitable length parameter. Quantitatively, however, there are almost always better ways of solving the problems of interest, and we shall not consider the Jost model further. D. Relations between methods. Calculations of the distortion near defects all involve response functions, which relate specific displacements to the forces giving rise to them. Relations between methods are conveniently discussed in terms of these response functions. In particular, one can relate the response function to the

178

LATTICE

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lattice Green’s functions of continuum elasticity theory and to the dynamic Green’s functions of chapter 11. In the theory of differential equations Green’s functions are defined by (Goertzel & Tralli 1960)

Gu, w) = d(u—w) for the inhomogeneous

differential

(8.3.43)

equation

in the differential

operator Y S f(u) = s(u).

(8.3.44)

In the present case the differential equation consists of the equations of motion of the lattice dynamics—either the classical equations, or the Heisenberg equations. The static-lattice limit is found by dropping the momentum terms in these equations, in which case the Green’s function is formally the same as the response function R of the Kanzaki method (eqn (8.3.16)) and physically the same as the computer-generated response functions. Although the static-lattice theory we have described can be regarded as a special case of Green’s function theory, some differences in emphasis are obscured. The Green’s function approaches, described later, are primarily concerned with lattice dynamics. Thus they concentrate on the effects of isotopic substitution and of changes in force-constant, i.e. of terms in Vp(u) which are second-order in u. Terms in V,(u) linear in the displacements u do not affect the lattice dynamics: the frequencies are unaffected and the modes merely displaced, which has no effect unless there is anharmonicity. By contrast, the methods described in this chapter concentrate entirely on the displacements caused by the linear terms in V,(u); the changes in force-constant are higher-order terms, which are often small in their effect on

the distortion.

Further,

the lattice relaxation

energy is independent of the isotopic content of the host crystal. The response function, like the Green’s function, may be found

for either a perfect second-order terms then there are two (Tewary 1970). We

lattice or an imperfect lattice. If only first- and in the defect potential energy V,(u) are important, equivalent ways of calculating displacements write the crystal potential energy

V(u) = (—F,.u+3u.B.u)+3u.A.u,

(8.3.45)

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179

in which the last term is the perfect-lattice term V,(u). The effective force driving the distortion is — 6V,(u)/du = Fp -—B.u = F(u),

(8.3.46)

and the equation for equilibrium is oV./ou = 0 = —Fo+B.u+A.u.

(8.3.47)

The distortion can be written either in terms of the perfect lattice response function A™' and the effective force at the final lattice position

u=A~!. Flu)

(8.3.48)

or in terms of the imperfect lattice response function (A+B)7' and the effective force for the undistorted lattice

u = (A+B)~!. F(u = 0).

(8.3.49)

Only the first result (8.3.48) remains valid when terms of higher order in V)(u) than quadratic are needed ; (8.3.49) breaks down if there are anharmonic terms of any sort. We now discuss the Green’s function approaches of continuum elasticity theory. The Green’s function G;,r, r’) of an infinite elastic medium is defined as the displacement at point r in direction i due to a unit force at r’ in thej direction. It is precisely equivalent to the earlier response functions, except that it is defined in terms of the elastic equations, rather than the atomic equations of motion. There is a boundary condition lim

G;,,r,r’) = 0,

(8.3.50)

dim Si Aloo}

and translational invariance allows one to write

G,fr,r) = Gifr—r).

(8.3.51)

It is convenient to introduce the Fourier transform defined by

He Gilt) =5I | nad4°aG,0l0< ae

=

= cos 6, 6, = |E,> = sin 6). The difficulties are compounded by a number of trivial misprints. The eigenfunctions are described by two quantum numbers. One, j, is related to the angular momentum around the potential trough; the other, p, describes the radial excitations, which depend on p.

In the absence of Jahn-Teller coupling the solutions reduce to those of a two-dimensional harmonic oscillator whose eigenfunctions are Xnm(P)

=

Rajmi(P) Cue.

(8.4.23)

Here |m| is an integer less than n. Longuet-Higgins et al. show that the exact solutions can be expressed in terms of the oscillator functions as fo@)

Di

ip/2 cad oy Eee tyCarey mereey ae

+e 10?

ys Q2N+2,pXm+2N+2,m+1°

(8.4.24)

N=0

Here m = j—4 and the a; are coefficients satisfying known relations. For each j the eigenvalues increase with p. Simple trends can be seen in both the strong and weak coupling limits. In both cases, the energies of the low-lying states can be expressed as the sum of three terms. The first is independent of the Jahn-Teller coupling; it is essentially a harmonic oscillator term and can be written phw,. The second is the minimum value of the potential energy due to the Jahn-Teller effect, E,,. The third term gives the dependence on j. For weak coupling, when Ej; is much less than the phonon energy, perturbation theory holds and gives a term {hwp—(—1)?Ejy}j linear inj. In the opposite extreme ofstrong coupling the third term is quadratic inj and of order (hw,j)*/Ey;. As the coupling increases, the splitting in this free-rotor limit tends to zero. For the strongest coupling, E,; will be much larger than hw,, and the eigenstates will be concentrated near the potential energy minimum. Exact eigenstates and eigenvalues of (8.4.22) are needed for the optical properties of defects discussed later; less information is usually needed for spin-resonance work. So far our only approximations have been the use of an adiabatic approximation and the neglect of higher-order @-dependent terms

LATTICE

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201

in the potential. It is convenient to make further approximations. The most useful of these approximations decouples the two solutions f, by dropping the right-hand side of (8.4.22). This should be valid for the lower energy levels with strong Jahn-Teller coupling. In practice the condition for validity Jahn-Teller energy > zero-point energy holds in almost all cases where the Jahn-Teller effect has observable consequences. Zgierski (1970) has given a more precise condition and discussed the dependence on j. The equations for f, reduce to

hea(Co*astt

0s)

G74)

Fulton (1967) discusses general properties of the solution of equations of this type.

The second approximation is to drop the term h?(j? +4)/2Mp?. This term, essentially a centrifugal term, is important in determining the low-lying states associated with the upper energy surface (Sloncewski 1963; Miiller 1968a). However the centrifugal term is negligible for the low-lying states associated with the lower energy surface. As can be seen from Fig. 8.4, the energy surface near the

Fic. 8.4. Centrifugal terms for an E-state coupled to E modes. The dotted lines ignore centrifugal terms, plotting +Ggpo+}Kgp’.

tribution (h?/2M)(j* +4)/p?.

The full lines add the centrifugal con-

202

LATTICE

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lowest minimum is not greatly affected. For many properties we may concentrate on the lower surface and use

Y = O,(p, dr = fre,

(8.4.26)

where L is the sign of Gpg. To this approximation fi = Ef, ép? +5K,p* —Gigp 2Mbrags

(8.4.27)

where f, is independent of j.

Effect of cubic perturbations. We now examine the effect of small cubic terms on the lowest state of the system (O’Brien 1964). This state is assumed to derive entirely from the lowest energy surface, and has a wavefunction ofthe form ®,(p, )W,. If the cubic term is a small perturbation, @, may still be factorized into f,(p) g(¢). But g(#) is no longer simply exp(ij@), since jis no longer a good quantum number. We shall treat the lowest-order perturbation of the correct

symmetry

V3() = A3(p)p* cos 3

(8.4.28)

in a manner reminiscent of the hindered-rotor problem. A,(p) is finite at p = 0, but must tend to zero at large p faster than 1/p, to avoid overwhelming the harmonic strain energy (Dixon & Smith 1972). The function g(@) now satisfies 62 “ag?

g+f cos (3o).g+eg = 0,

(8.4.29)

where ¢ is the corresponding energy. Here « is given by

= Siolara WP slo,

(8.4.30)

and is typically 10cm~'. B is a measure of the barrier between potential wells produced by V; and has the form p=

—< frol A(p)p7| fro»:

(8.4.31)

Terms from quadratic electron—lattice coupling may be incorporated by suitable modifications of « and £. The solutions of (8.4.29) are conveniently written in the form

= > yp, eV? i

(8.4.32)

LATTICE

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203

The boundary condition g(@ + 2) = —g(@) requires (j+4) to be an integer. The coefficients y, satisfy

ya?—8)= SB(yj-3 + yj43)-

(8.4.33)

Thusj’ = j modulo 3 is a good quantum number, irrespective of the detailed form of the cubic term. The cubic term only couples states withj values separated by 3. The various states g(¢) may be classified in terms of the C,, symmetry group, since this gives the symmetry of the potential in (p, P) space. Summarizing

j= +3,+3,+49,... transform as A, ify; = +)_, and as A, ify, = —y_,,

j= 4, —3,3, j=

—++,43,... transform asE (labelasE,),

$.

(8.4.34)

—3, +3, -3, +44, —43,... transform as E (label as £,))

The E states have two components, E, and E,. The lowest eigenstates of these symmetries are |A,> = Wrfroty; cos (56)+ 72 cos (3) +p

A> = Wrfol; sin 3h) +7ssin (3h) +)

cos (h)+...}

sin 436)+

IE,> = Wrfrotys Cos (3d) +Y3 cos (3¢)+73 cos(5) +..

|E2> = Wi frotys sin 2d) —y, sin Gh) +p; sin (Fd)—a where the electronic functions

W,(@) were

(8.4.35)

given in (8.4.18) and

8.4.19), and f,0(p) satisfies (8.4.22) or (8.4.27). Eigenvalues for various values of A, are shown in Fig. 8.5. Note hat only the j = 3 term is split to first order by the perturbation; his occurs because V;(¢) mixes j= +3 states. For the two extreme imits of the barrier, measured

by f, we have (f = 0) a potential

surface independent of ¢ and (f — oo) a static slow-rotation limit.

I. T-state equally coupled to E and T modes. This model provides a ‘emarkably good description of the excited state of the F-centre in CaO (§ 15.3.3). O’Brien (1969, 1971) has discussed the vibronic sroperties by exploiting the extra degeneracy resulting from equal -oupling. When the tetragonal relaxation energy (8.4.8) and trigonal

204

LATTICE

(ee

DISTORTION

#0

FOR

DEFECT

SYSTEMS

Static

Dynamic

oat

limit

—|

40 Blaa Fic. 8.5. Effects of anharmonicity on an E-state coupled to E modes. The energies are given as a function ofthe barrier parameter f for the case Gggf positive. A, and A, should be interchanged if GggB is negative. Values of j= 4,3,... are given in the dynamic limit. Results are those of O’Brien.

relaxation energy (8.4.10) are equal, the potential energy is a minimum over a surface defined by

where

3K (0? +67) + 9K (S* +07 +0*) = Eg, (8.4.36) Ey; = 2G2,/K, = 2G?,/3Ky. The various displacements

can be defined by two polar coordinates (0,, @,)

(0, 2) = (Grp/Kz){(3 cos? 6,— 1), (./3 sin? 0, cos 2¢,)} (¢,n, 6) = (Grz/Ky) {(sin 26, sin @,), (sin 20, COS ,),

(sin? 6, sin 2¢,)}.

(8.4.37)

The boundary conditions are such that (0,,¢,) and (—0,, = Oe) are identical. The usual tetragonal extrema (8.4.9) occur at (8,> Pp) values of (0,0), (2/2, 0), and (2/2, 2/2) or (x, 0), (x/2, 2), and (x/2, 3n/2). If n is integral and 0, = cos ‘(3/,/3), the trigonal extrema (8.4.11) occur at (0,, nn/4) and (x—0,, n7/4).

LATTICE

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205

Ei

ee qT,

A,

B

=

5

0

a4



if

SS

SS Ts

E

3 T,

2

E

eee

of l

—10

4

ae ay

A

|

Eee

Z

Lb

a,

!

2

oD

ea

|

|

eI

cig

4-1

T,; T,

fe

, weak

S

T,

He Sails

10 A 8

A i

==)

6

;

Ts +—3

of Tetragonal minima Fic. 8.6. Coupling of a T,-state to

hs Trigonal minima

E and T, modes. The vibronic levels are shown

as a function of A, the difference in the E and T, Jahn-Teller energies. The units are (hw)*/|A|. The dotted lines mark the lowest potential energy minima. Degeneracies of groups of levels at large |A| are shown; at zero |A| the degeneracies of the groups are 3,7, 11,15,...., in order of increasing energy. Results after O’Brien.

The simplest solutions occur when the distortions are confined to the minimum energy surface. If the phonon energies hw, and ha, have the same value hw, then the energy levels are in groups of degeneracy (4N — 1) with energy N(N + 1)(hw)?/4E,;. The spectrum resembles that of a free rotor in this limit. In practice, one wants to understand the effects of deviations from precise equality of E and T coupling. Rather than solve a complicated vibronic problem, it is simpler to put in a potential V(8,, @,) of the correct symmetry

V(0,,Pp) = A[Y3(6,,,)+

(4) {Y4(0,,o,)+ Ya *@,, $,)}I- (8.4.38)

When A > 0, trigonal distortions are favoured; A < 0 favours tetragonal distortions. There is no simple relation between A and

2G2,/K,—2G?_/3Ky. The effect on the energy levels of V(6,, ¢,) is shown in Fig. 8.6 as a function of A.

LATTICE

206

III. A,- and discussed by excited state Hamiltonian § 8.4.2, and a

DISTORTION

T,-states coupled by a Ham (1972, 1973) as of the F-centre in contains the potential kinetic energy term i

DEFECT

FOR

SYSTEMS

T, mode. This model has been a possible description of the alkali halides (§ 15.3.3). The energy contribution given in

5(P2+P3 + P2). m

(8.4.39)

The major simplification here occurs because the potential energy

surface depends only on p = (x?+y?+z’)*

and not on the T,

modes (x, y, z) separately. The eigenstates can be classified conveniently after introducing several operators. Three are angular momenta. One is the vibrational angular momentum L, with components such as Le

1 prey

YEx)

(8.4.10)

Similarly, the electronic angular momentum is l, and we may define a total angular momentum

J=L+l

(8.4.41)

as the sum of the lattice and electronic contributions. In addition to angular momenta, we define inversion operators. In configuration coordinate space we use I,, where I, f(x, y,z) = f(—x, —y, —2z). The analogous electronic operator I, is conveniently written in terms of projection operators

I, = |a> Cyl +12) Cz). The product of these two operators is

Hance)fee

(8.4.42)

Both A and J commute with the total Hamiltonian. The states can be classified by their eigenvalues. Specifically, the quantum numbers are (a)

J = 0,1,2,..., where J(J +1) is the eigenvalue of J?;

(b) M = 0,1, 2,...,J, where M is the eigenvalue of J,; (c) A’ = +1, where A’ is the parity of the state, and the eigenvalue of A. The eigenstates are, of course, combinations of the states derived from the electronic A,- and T,-states. The electronic A ,-states give

LATTICE

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207

vibronic states with

(J, M) = (L, m) iE eenyee a's

(8.4.43)

and the electronic T,-states give vibronic states with

rar iande eae

only for: D.= 0)s.and,

iN === (1) A’ =(—1)"** _

>.

(8.4.44)

(es {yA 1

Comparing these quantum numbers, we see there are two main slasses of states. The first, Type I, derive from both electronic states. A typical eigenstate can be written

fi(J, p)lJ,M; J,Ay> +f, p)lJ,M; J—1,T,>+ Tes

p)iJ,M3J+1,

fips

The second term is omitted when J = 0. The second class, Type II, comes solely from the T, electronic states. A typical eigenstate has the ‘orm 6PCr) ) PAY fol kgsEe

[Inspection shows the Type II states can be written as products of a hree-dimensional harmonic oscillator wavefunction and an elecronic T,-state; they are independent ofthe electron-lattice coupling. The functionsf,, f,, and f; for the Type I states are the solutions »f coupled equations. They have a simple solution in one limit, when the stabilization energy E, = (Gz,/2K-) is much larger than he A,—T, splitting A and the phonon energies fw. In this limit the ow-lying states of Type I have energies

(ha)? E(J) = a (Ie):

(8.4.45 )

(0)

ind the levels resemble those of a free rotor. The ground state has J = 0, A = 1), irrespective of the sign of the A,—T, splitting for ero distortion. The next excited state (also Type I) has (J = 1, \’ = —1), and lies higher in energy by (hw)?/2E,. Since we have

issumed Ep > hwo, this is a small fraction of the phonon energy. In his limit the Type II states all lie higher in energy by about Eo.

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IV. T-states coupled by T modes. Caner & Englman (1966) have analysed numerically the coupling of T modes to T-states. This system is more complicated than the others, although one can see from Fig. 8.7 that similar behaviour is found. In weak coupling, there is a dynamic Jahn-Teller effect. The static regime is achieved as the coupling, measured by G,,/,/Ky, becomes larger. There is no need to introduce any ‘barrier’ terms like (8.4.28). The shape of the potential energy surfaces with linear Jahn-Teller coupling automatically tends to localize the system in one of the four equivalent wells corresponding to trigonal distortions.

—_ @ En)

Weak coupling dynamic limit

Strong coupling static limit

ef)

ho

pe IE

Fic. 8.7. Vibronic levels for a T-state coupled to T modes. The labels are for a A state, the subscripts are interchanged for T,. The results, after Caner & Englman,

give energies in units of hw as a function of G;//Ky. A term Eyr = 3(G,/,/K,)? has been added to make clear the strong-coupling limit.

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Jahn-Teller coupling to many modes. Almost all work on the Jahn— Teller effect has assumed that phonons with only a single frequency are involved. This is an oversimplification for any real system, and several authors have given more general results (Sloncewski 1963; O’Brien 1972; Fletcher 1972a, b; Englman & Halperin 1973; Gauthier & Walker 1973). If the lattice modes have just one frequency, there is always a transformation to new modes which removes the linear coupling to all but one of the modes. If there are many frequencies, there will always be residual coupling to the other modes. O’Brien has shown that, despite this, there is a transformation which gives a useful simplification. The steps are these. First, one singles out a principal mode such that the reduction in potential energy associated with the mode is as large as possible. A static displacement is then made so that this mode is measured from the position of its extremum. Secondly, one transforms the remaining modes such that the only cross-terms in the kinetic energy involve the principal mode and one of the others. The Hamiltonian is now the sum of three terms: one involving just the principal mode, one independent of the principal mode, and a third containing cross-terms with no diagonal elements among the eigenstates of the term involving the principal mode. Thus the effects of the cross-terms can be included by secondorder perturbation theory. Englman & Halperin have used an analogous method. Their approach differs from O’Brien’s in choosing the initial displacement of the modes to minimize the ground-state energy, rather than the adiabatic potential energy. The results show the order of magnitude of the corrections arising from phonon dispersion. If phonon modes with energy hw, have coupling coefficients G;, then the leading corrections depend on

—?, where ,G;. A number of results are unaltered, like certain moments of optical bands and the relations between orbital reduction factors considered later. However, it is not yet possible to understand the empirical observation that a single effective frequency can be chosen to describe a range of phenomena, including their dependence on temperature. We return to this point in § 10.8. Fletcher considers in detail the coupling of an E-state to many E modes, and derives an accurate variational description of the lowlying energy levels. The trial function is obtained as follows. We first arbitrarily constrain the electronic wavefunction to contain

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specific proportions of the |e> and |@> states measured by 6. With this electronic wavefunction we can find the static displacements of symmetrized combinations of the lattice modes of each frequency. Even though the Jahn-Teller energy may be large, it is assumed that no individual mode is displaced greatly; local modes are specifically excluded. Analogous assumptions are made for nondegenerate systems in § 10.7.2. Now the static displacements are unlikely to be accurate when there are significant dynamical effects. One constructs a vibrational wavefunction with modes whose mean displacements are f, times larger than the static values. The product of the constrained electronic function and the vibrational function is a vibronic function which depends on 6 and the f;. The vibrational trial function is obtained by projecting out that part of the vibronic function with a given j, that is with a specified total angular momentum in (p, ¢) space. The results obtained give remarkably good agreement with known test cases for all coupling strengths. Both the energy levels and eigenfunctions (as indicated by reduction factors, for example) are accurately given. Significant discrepancies are found for highlyexcited states, since the trial functions do not form a complete set,

and there are hints that the wavefunction does not fully describe the breakdown of the adiabatic approximation for strong coupling. Comparison with the cluster model shows that the correct values

of E(j = 5) and E(j = 3)— E(j = 4) are obtained if one defines

Gi

[» oi]{(o7')P?/w~?}

hog = h{w '/o~ 7},

(8.4.46) (8.4.47)

in which wY = )’; G?w’/S’; G?. These results are consistent with those of O’Brien. B. The static Jahn-Teller effect : the slow-rotation limit. The barriers between equivalent distorted states may become so large that transitions between these different configurations become negligibly rare. A defect prepared in one distorted configuration will remain there for a mean time t, long compared with the characteristic time of the experiment of interest and the periods of relevant lattice vibrations: the defect is sensibly localized in one configuration. Examples occur for some transition metal ions and for nitrogen in diamond.

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The ‘displaced-oscillator hypothesis’. For a strong Jahn-Teller effect the nuclear displacements are almost always close to the values o which minimize the potential energy. Fluctuations which take

the displacements Q! well away from Q/, are rare. For small values of (Qi—Q!,), the potential energy surface can be approximated by the harmonic potential

VQ) = + Y (Q}—Qio)At,(Q). — Qh)

(8.4.48)

yy

near extremum J. A great simplification can be achieved using the ‘displaced-oscillator hypothesis’ that we may treat these harmonic potentials as exact and ignore any regions where the true potential energy surfaces are anharmonic. The assumption is clearly best when the lattice relaxation energy E,; is much larger than the relevant phonon energies. It is exact only when the matrix of the electron-lattice coupling is diagonal, that is for non-degenerate states and the exceptional case of T-states coupled to E modes. In other cases there is an important difference, quite apart from the neglect of anharmonic terms: the lattice modes and their frequencies are altered by the transformation to modes centred on the minima Q,9. This behaviour is different from that in § 8.2, and is illustrated now for an E-state coupled to E modes. When there is no Jahn—Teller coupling, the E modes can be chosen in the form é = cet+s0,

6 = —se+c8,

(8.4.49)

where c?+s? = 1 and c is arbitrary. But when the Jahn-Teller terms and appropriate higher-order terms give potential energy minima at (€,, 9), the value of c which diagonalizes the harmonic expression (8.4.36) is fully determined : c—>cosdo,

s—singo,

bo = tan~! (&/9o).

(8.4.50)

The values of $9, and hence of é and 6, differ for each extremum. Further, the frequencies of the modes are altered. One of the modes

6 corresponds to oscillations in p = \/(e? +0”). Its frequency is the

same as in the zero coupling limit. However, the other mode é corresponds to motion along the potential trough p = Po shown in

AA

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Fig. 8.1. This mode has zero frequency in the absence of higherorder terms. Even with higher-order terms, the 0-mode frequency is likely to be appreciably higher than that of the é mode. A related assumption concerns the electronic wavefunction. Since the nuclear displacements are always close to Q.,..,, which minimizes the potential energy, it is usually adequate to write the electronic part of the wavefunction as p(r; Q,) ~ OF; Q,o))-

(8.4.51)

The implicit dependence on lattice configuration is assumed so small that any dependence on thermal motion may be ignored. This approximation is related to the Condon approximation of § 10.3. The two approximations (8.4.48), (8.4.51) allow one to write the vibronic functions associated with potential well I as es

Pr; Q,o)W({n,}, OF - On

(8.4.52)

where yw is a product of harmonic oscillator functions with occupation number n,,.

Transitions between potential wells. In the static limit the defect is sensibly localized in one distorted configuration. The localization is not exact, and it is convenient to begin by defining localized vibronic wavefunctions which are orthogonal to those localized near other potential minima. The orthogonal wavefunctions are linear combinations of the approximately-orthogonal wavefunctions obtained in the same way as one constructs Wannier functions from atomic orbitals. In special cases, such as T-states coupled to E modes, the vibronic functions for different distorted configurations are automatically orthogonal because of the orthogonal electronic functions. But it is clear in other cases that the electronic functions o(r;Q,o) are not orthogonal to corresponding functions in other wells. For suppose there are Ny potential minima. Electronic orthogonality would require at least Ny linearly-independent electronic functions. If the electronic degeneracy Np is less than Ny, this is not possible. Thus for E-states coupled to E modes Ny = 3 and Np = 2; for T-states and T modes Ny = 4, Np = 3. There are still matrix elements of the Hamiltonian between these orthogonal states. The orthogonalization has not localized the wavefunctions, but merely provided more convenient basis states.

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213

There are two ways of discussing the delocalization produced by the finite matrix elements. On the one hand, we may seek the stationary states of the system. On the other, we may imagine the system to be localized in one particular potential well, and examine the evolution in time of the system. Both methods have their virtues. We begin by seeking the stationary states (cf. Bersuker 1963), and concentrate on the set of the lowest vibronic states in each well. These are given by expressions like (8.4.52) with zero occupation numbers n,,;. The matrix elements of the total Hamiltonian among these states have the general form

|dQ. {Pr PH; 0, )H(t: Q,,)

~ |dO, W*({0}, 0,1 2,0H({0}, 0,9 Qyou) x x {dr HF(t; Qj) Hrb,(0; Q,0,)

(8.4.53)

omitting diagonal terms, which are absent for complete delocalization. The matrix element reduces to the product of a lattice overlap integral and an electronic matrix element. The general form of the matrix for a Jahn-Teller system with three equivalent distortions is

x=. B

ee all

(8.4.54)

Bp B « The stationary states of a Hamiltonian with these matrix elements are

(Eee ENCG/ (te yD

vd

rere

E’ = a—2f, E”

=a+bB.

he

(8.4.55)

These solutions are appropriate for E-states coupled to E modes. — E’) is 3B. The matrix is very similar in The ‘tunnel splitting’ (E”

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form for four minima, as appropriate for a T-state coupled to T modes. The states and their energies are

P+

P,+ P34 P4),

E'= a—3f8,

peeps v, =i)

eet)

E" = a+f.

(8.4.56)

yy, —P, +P; —P,) The tunnel splitting is 48. Note that the off-diagonal matrix element B, of the form (8.4.53), determines the splitting. As the Jahn-Teller effect becomes stronger, the asymmetric lattice distortions increase and the lattice overlap factor in the matrix element f decreases. The tunnel splitting, proportional to B, tends to zero in the limit of the strong Jahn-Teller effect. The sign of B is important, since it determines whether the lowest state is degenerate or non-degenerate. For a T-state coupled to E modes f is identically zero, and there is no tunnel splitting. For E-states coupled to E modes and for T-states coupled to T modes B is negative; the degenerate state is lowest. This may be seen from the electronic matrix elements, which are of the form COP; Q,0:)|Qoulr; Q,on)>, where Q is a totally symmetric operator. For the E-states we might have

lop ~ 18>,

di poN ee3 \e> gs

(84.57)

ignoring the configuration dependence of the basic electronic states. Since Q is totally symmetric, only the ¢6|Q\|@> term contributes, and this is negative because of the —4 coefficient in |¢,,>. In the same way for the T-states we might have

ld> ~ WE>4+In> H1O>//3, lou> ~ (E>-In>-1O)//3.

(8.4.58)

Again there is a negative factor, here —5, which makes f negative. It is often asserted that the ground state of any non-pathological system should be non-degenerate, so that the sign ofBwould appear surprising. Strictly, the general result is that the ground state wavefunction has fewest nodes (§ 9.3). The negative sign for B

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215

appears because of the novel boundary conditions. Thus for the E-state coupled to E modes the wavefunction changes sign when g > (b¢+2z). This may be contrasted with the usual hindered rotation, where the wavefunction is unaltered when ¢ > (f +2rn). For the Jahn-Teller case, however, the doubly-degenerate vibronic state has fewer nodes than the non-degenerate state. That the degenerate state lies lower than the non-degenerate one also fits in with the fast-rotation results: the ground state has the same symmetry from the static limit to the dynamic limit. The size of 6 can be estimated. For an E-state coupled to E modes the results depend on the detailed form of the higher-order terms chosen. A reasonable assumption is that the potential energy surface for p = po has the form

V(b) = 4Vo(1 —cos 3¢),

(8.4.59)

giving an effective frequency w’ for oscillations in @and a barrier Vo between minima. Sturge (1967, p. 130) gives the tunnel splitting

3 3B ee = ay w’'exp (1 —2V)/ha’),

(8.4.60)

where the | in the exponent comes from the zero-point energy. Caner & Englman (1966) evaluated the tunnel splitting for a Tstate coupled to T modes, and found it to be

4Bry X 1-32Ey, exp(—1-24E,q/ho,),

(8.4.61)

where Ej; is given in (8.4.10). We now discuss the time-dependent behaviour of the system if it is initially localized in one of the distorted configurations. The electron—lattice coupling greatly complicates the behaviour, and no really adequate calculations for Jahn-Teller systems are available. There are three major ways in which the system may progress from one configuration to another. We now discuss the qualitative features. Which process dominates depends on the details of the

system. The first process corresponds exactly to the processes effective in our discussion of the stationary states. The vibrational states y can be described by a distortion label and by oscillator occupation numbers for the modes in the displaced-oscillator approximation. At the lowest temperatures, these occupation numbers will be zero,

and there will be zero-point ‘motion

only in these modes.

The

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dominant transitions are those in which none of these occupation numbers change; the probability of a transition to an excited state is very low. Thus the initial and final vibronic states in the transition are related by asymmetry operation. The motion from one distortion to another is described as in tunnelling theory. The probability that the system is in one particular distorted configuration varies periodically in time (e.g. Landau & Lifshitz 1959a, pp. 143-4 and 176-7), and the rate is given by

1/t ~ |B\/h.

(8.4.62)

Since the penetration of a barrier is involved, rather than its surmounting, the process is conventionally called tunnelling. Tunnelling rates are readily calculated in the semiclassical (W.K.B.) approximation (e.g. Avamukov 1960) as the product of an ‘incidence frequency’ and a ‘transmission factor’ for the potential V(x) along the trajectory which crosses the lowest barrier between the configurations

1

B

voon

1

hw exp

aeTh |

K

eee

ee dx oe

(V(x)(x) on2

2

: oy |

(8.4.63 )

Here w is the effective frequency of the mode taking the system from one distortion to another and K is the corresponding force constant. As the temperature increases, higher vibrational states become involved. The overlaps of these lattice states are smaller, and the transition probability from the first process decreases in favour of transitions in which the occupation numbers of the lattice modes change. The lattice energy is conserved over all, but there is a redistribution of the energy over lattice modes. The initial and final states of the transition are no longer simply related by a symmetry operation, and the probability that the system is in a particular well is random, not periodic. The appropriate description of the motion is hopping, rather than propagating, from site to site. The precise temperature dependence of the hopping transition probability depends on the dependence of the electronic matrix element on the lattice distortion. The simplest approximation is the Condon approximation (§ 10.3) that the matrix element is independent of the Q,; it is implicit in (8.4.53), for example. In this case the processes which dominate at low temperatures are two-phonon processes, in

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217

which one phonon is emitted and another is absorbed, conserving energy over all. The temperature dependence is usually T’, although there is still some dependence on the model. At higher temperatures the transition probability becomes thermally activated ;the hopping rate varies as exp(—E,/kT). The activation energy has a simple explanation in the displaced-oscillator model, when the lattice potential-energy surface can be represented by a set of harmonic wells. E, is a lattice activation energy, namely the energy barrier for the lattice to change from one distorted configuration to another. There is no reason to believe that the result holds accurately when the energy surface differs a lot from the harmonic form. Unfortunately, no quantum treatments are available for cases where the energy surface is highly anharmonic. Further, it is an oversimplification to assume that the transition matrix element is independent of atomic configuration. Discussion ofthese points for the analogous case of light interstitial diffusion in metals is given by Stoneham (1972). The third mechanism of reorientation of Jahn-Teller defects appears when rapid tunnelling between excited states in each potential well can occur. Thus, whilst the lowest states in each well

are best described by a static Jahn—Teller effect (slow-rotation limit), the excited states are best described by a dynamic effect (the fastrotation limit). Again, we expect thermally-activated behaviour, with an activation energy of the order of the energy barrier in the lattice potential energy. The activated behaviour appears because of the thermal population of these activated states, and the magnitude of the energy will be of the order of the barrier because states with rapid tunnelling are only expected to occur with energies near the top of the barrier or higher. Random strains and internal fields can have a profound effect on the reorientation of Jahn-Teller systems, since they make the different orientations inequivalent. The major effects are these. First, if the perturbation causes one configuration to lie lower in energy by more than the tunnel splitting, of order B, the system will be effectively localized in this configuration. Dynamic effects will not be observed. Secondly, the transition probabilities between configurations are altered. The effects are seen most easily for the hopping mechanisms. At low temperatures one-phonon processes become possible, in which a single phonon is absorbed or emitted, with energy 6 equal to the difference in energy of the two configurations.

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At high temperatures the activation energy is changed from E, to

(E, + 6/2+ 67/16E,). We shall consider many of these phenomena in more detail later in discussing non-radiative transitions. Other, particularly useful, discussions of the processes are given by Sussmann (1967), Silsbee (1967), and Flynn & Stoneham (1970). 8.4.4. Matrix elements of electronic operators: the Ham effect

A. Reduction factors: the Ham effect. We now examine the relationships between the matrix elements of an electronic operator Q(r) among purely electronic wavefunctions and among vibronic wavefunctions. In practice Q might be the spin-orbit coupling or the perturbation from an external field. Suppose the electronic wavefunctions are ¢,(r;Q) and their vibrational counterparts are W(Q). The total wavefunction in the adiabafic approximation is a linear combination of vibronic functions of the form

P(r, Q) = .(r; Q)W.(Q). Matrix elements of Q between vibronic functions are just

Uae)

“= {ao {cls

AOE

= |€QuHQ(@ [Protr: QOMH(:Q.

(8.4.64)

If the dependence of the electronic matrix element or Q is ignored (essentially the Condon approximation of § 10.3) then Oe

=

TOS

PhO)

(8.4.65)

in which

Ry = |AQ YNQW AQ) (8.4.66)

ageron) = [ar p4(r; QOH: Q| Q=Q

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AMS)

The advantage of (8.4.65) is that both AYE") and R,, are often easily calculated separately. Further, the reduction factor Ryg 1s independent of the detailed form of Q(r), and need be calculated once only for a given system. The reduction factor has some simple properties. When a = f the two vibrational functions y, and we, are identical and Ryg = 1. But for off-diagonal elements (~ # f) the reduction factor may be much less than unity. Off-diagonal elements among vibronic states may be much smaller than the elements between the corresponding electronic states. This reduction of off-diagonal elements is known as the Ham effect. It is easily seen that the reduction will be greatest when there is a strong Jahn-Teller effect. In the displaced oscillator approximation, if w, and W, correspond to the ground states of harmonic oscillators centred on Q,, and Q,,, then

Mw?

Ri, = exp }=F HO. Op"

(8.4.67)

As the Jahn-Teller effect becomes stronger, Q,,,Q,,, and (Q,.— —Q,,) all increase. R,, approaches zero as exp (—dE,;/ha@), where d is of order unity. We can also see that a Jahn-Teller interaction reduces operators of different symmetry. For even if we diagonalize the Jahn-Teller coupling, off-diagonal elements will remain from operators of different symmetry (excluding A,,). Thus interaction with trigonal modes quenches coupling to tetragonal strains, and vice versa. So far we have concentrated on one set of vibronic functions related by symmetry. If an operator has only off-diagonal electronic matrix elements among these states, one must consider the admixture of other vibronic states. In essence, the second-order coupling through these states may be more important than the reduced direct matrix elements. Operators like orbital angular momenta are especially affected, since they have only off-diagonal elements among real orbital states. Ham (1965) gives details, and shows that the effects of states lying closer in energy than E,; or hw may be particularly strong. But most distant states do not give rise to substantial off-diagonal

vibronic

elements,

since

their

contribution

also

is

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reduced vibronic

OE

second order

ax

CHAO?



D

~

Ds te

CF

JQP

56>

jean

Q lbp

y

~ {#062 CE) cu, 1y4> ee

leer orderttag :

(8.4.68)

B. Symmetry-adapted reduction factors for standard systems. We now consider the reduction factors for the standard case in cubic symmetry, exploiting the fact that the symmetry of the ground state in these cases is the same for all strengths of Jahn-Teller effect. It is convenient to introduce a number of standard matrices M: four for the (2x2) matrices of E-states and nine for the (3 x 3) matrices of T-states. The matrix elements of any electronic operator may be written

ORE

= SHON by

(8.4.69)

L

so that the coefficients a; are sufficient to specify Q within these states. This formulation makes the symmetry properties particularly clear, and is used in many other contexts, such as the spin Hamiltonian (§ 13.2) and isotopic spin. The matrices are defined in Table 8.1. Whilst they can be related to spin matrices (e.g. for E-states %, = —2S,, U, = —2S,, U, = +28,), such a description obscures the symmetry used. We may now make a similar construction for the vibronic matrix elements in vibronic states of the same symmetry

Qvibronic) _ » Ra {Q)M,.

(8.4.70)

There is one reduction factor for each distinct matrix, and we may

evaluate these once and for all for a given system. Symmetry arguments may be used to reduce the number of independent R;, and

also to show that the R; only depend on the operator Q through its symmetry.

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TABLE 8.1 Matrices for E-states i

U, =

1

0

0)

1

—1

oa

-i

1

0

0

0

Cae

+1

0

(ie

|

1

0

Matrices for T-states De a

es

:

1

W2. Gy =

:

= (3/2

1/2

1

.

Ca

,

—1 Je=

al

—1

J, =

J,=|-1

—1

—1

: i

L=

Nie)2

—-l

aa

—1

LZ =

£,=)|+1

The various R; are readily calculated when the vibrational wavefunctions are known. In the case of an E-state coupled to E modes they are

R(.%#,) = 1, R(.%,) = p, R(O.)

RO)

;

(8.4.71)

a.

where for linear coupling with no anharmonic terms

2g = IEFp.

(8.4.72)

Ham (1965) has given estimates of p and q. For weak Jahn-Teller coupling

at ee

(8.4.73)

q ~ 1-2S

with S = (E,,/hw); for strong coupling p tends to zero and q to 3. Intermediate cases are given by Child & Longuet—Higgins (1961),

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but may be approximated by

p ~ exp (—1-974 S®7°")

(8.4.74)

q ~~ (i+p)/2. Sloncewski (1969) has shown that anharmonicity affects q rather little in the strong coupling limit, when 0-484 < q < 0-5. He conjectures that 0-484 is always a lower bound on gq. Halperin & Englman and Gauthier & Walker have examined the effects of phonon dispersion on eqn (8.4.72). Their results show that 3(1 +p) exceeds q and that qg may fall below 4+ when there is coupling to phonons with a range of energies. As examples of electronic operators reduced by these factors, the Zeeman and hyperfine interactions contain terms proportional to ,,U,, and U,; tetragonal lattice strains give terms in U, and U,, and trigonal lattice strains plus spin-orbit coupling or Zeeman interactions give a term in .%,. Some of these cases will be discussed in more detail later. The case ofa T-state is more complex, since it is coupled to both E modes and to T modes. The coupling to E modes alone can be solved exactly, since the interaction is diagonal. The Jahn-Teller interaction gives three harmonic energy surfaces centred on different configurations, and the vibrational wavefunctions are just the harmonic oscillator wavefunctions. As a result the reduction factors can be obtained exactly. When there is coupling to T modes alone, the calculation is not simple. Numerical calculations have been made by Caner & Englman. The reduction factors for separate T and E couplings are given in Table 8.2. Results for strong coupling to

TABLES Sez Orbital reduction factors for systems of T symmetry Reduction factors for modes

Operator symmetry and

matrices reduced

A, R(&%,) E R(&) = R(&,) T, RJ) = RU,) = R(J,) DeeRCAy = REA RD)

E

T.t

1 1 exp(—3S/2) exp (— 38/2)

1 exp (—9S/4) 4{2+exp(—9S/4)} exp (—9S/4)

t Approximate forms valid when S = E,,/hw is small.

Equal E+T

1 0:4 0:4 0

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223

both E and T modes when they both produce equal Jahn-Teller

energies, E\t) and E}7), are also given in the last column of the table (O’Brien 1969). Romestain & Merle d’Aubigne (1971) have shown that an exact relation analogous to (8.4.72) holds in this case

R(T,) = R(E) = 3R(T,)+2)/5.

(8.4.75)

Operators for which the Ham effect is important include the orbital g-factor and the spin-orbit coupling, which are reduced by R(T,), and the perturbations from external strains, which are reduced by R(E) (tetragonal strains) and R(T,) (trigonal strains). We discuss specific examples of these reductions later in § 12.5, § 15.2, and § 17.3. 8.5. Other asymmetric systems 8.5.1. Introduction

Earlier we mentioned defects other than Jahn-Teller systems which could take up different equivalent orientations. In this section we contrast the two situations. Electronic degeneracy plays no role in non-Jahn—Teller systems. This has two main consequences. First, the nuclear potential energy surface is single-valued, and one may use the Born—Oppenheimer approximation instead of a more general adiabatic approximation. The energy surface may still be strongly anharmonic. Secondly, the analogue of the Ham effect is not usually needed. Of course the physical content of the Ham effect—that vibrational overlap integrals are essential in the discussion of matrix elements of electronic operators—is still valid. The superficial similarities of asymmetric defects and Jahn-Teller systems are most apparent when we discuss reorientation. In both cases it is easiest to discuss two extreme limits—the fast-rotation limit, when the reorientation

is rapid, and the siow-rotation limit

when transitions between equivalent configurations are improbable. The theory of asymmetric defects for the fast rotation limit can be constructed as before, but starting from the free rotor (generally threedimensional, but two-dimensional in special cases, e.g. the oxygen interstitial in silicon (§21.5)). One major difference concerns the boundary conditions on the wavefunction. For an asymmetric rotor, the wavefunction is unaltered by rotations through 2z (z for systems with inversion symmetry), whereas the LHOPS wavefunction is

224

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SYSTEMS

invariant under changes of 4x in ¢. Further, the nuclear spin may affect the rotational levels of trapped molecules through the exclusion principle, as for free molecules. This does not occur for Jahn-Teller systems, where reorientation merely alters distortions rather than interchanging atoms. In the slow rotation limit the same types of transition occur between equivalent states for both classes of system. There are the direct transitions between the lowest localized states and transitions involving excited localized states. Brot (1969; also Lassier & Brot 1969; Brot & Darmon 1971) discuss the reorientation of rotors in a

classical limit appropriate for heavy molecules in deep wells at high temperatures. The main differences between Jahn-Teller systems and others in the slow-rotation limit occurs because the number of potential minima for non-Jahn—Teller systems can be large. In cubic symmetry [110] orientations are common, and systems may lack inversion symmetry, so that [100] and [100] are not equivalent. Such complicated situations only occur rarely for Jahn-Teller systems, as for cases of accidental degeneracy. The large number of minima means that more independent transition probabilities must be treated. In principle the low-lying levels need not be describable by a single tunnel splitting, but in practice a single transition usually dominates. Finally, we emphasize that in all reorienting systems the internal strains have a profound effect on all properties—so much so that it is not always easy to identify the intrinsic phenomena. These problems are discussed by Ham (1972) and Narayanamurti & Pohl (1970). 8.5.2. The fast- and slow-rotation limits A quantitative comparison of the fast- and slow-rotation limits can be made which shows the way the two extremes approach each other. Although we derive the results for systems with single-valued potential energy surfaces, the general features are probably valid for Jahn-Teller systems too. The fast-rotation limit starts with free-rotor energy levels and the effects of low-symmetry potentials are considered. This gives the ‘hindered-rotor’ picture (Devonshire 1936; Sauer 1966: Jain & Tewary

1972; Beyeler 1972). The slow-rotation limit assumes

the

defect is strongly localized in one of several equivalent orientations by a deep potential well. Transitions from well to well occur because

LATTICE

DISTORTION

FOR

DEFECT

SYSTEMS

225

the defect wavefunction is not completely localized. This is the ‘tunnelling’ picture (e.g. Gomez, Bowen, & Krumhansl 1967; Estle 1968). Both these pictures can be appropriate in the limit of low temperatures. At high temperatures the hopping processes, discussed in § 8.4, must be treated too. We shall compare the two limits for the simplest cases of defects in cubic symmetry which have either [100] or [111] orientations. Whilst [110] orientations can, and do, occur in practice, they only appear in the hindered-rotor model when more complicated nonspherical potentials are used. In comparing the two methods, we must observe that the qualitative shapes of the potential energy surfaces are different. The hindered-rotor model constrains the defect to a sphere on which the potential energy varies as Vir)=

Si

lxtayt+20-3

oa

a

3)

(8.5.1)

Beyeler has given results for a hindered rotor with sixth-order terms too, describing the simplest system which can have [110] minima. When K is positive the surface has [100] minima with Vir00) =

—IKI;

(8.5.2)

when K is negative the minima are at [111] with potential energy een

—3|K|.

(8.5.3)

One can go from a system with [100] minima to one with [111] minima merely by varying K. The wells in the surface are separated by a saddle point. By contrast, the tunnelling model is the exact analogue of the displaced-oscillator model for Jahn—Teller systems. There are a set of harmonic potential wells centred on different distorted configurations. No simple transformation relates the [100] and [111] cases uniquely by a simple change of parameter, and the most probable transition path between the wells passes over a cusp rather than a saddle point. The two models are contrasted in Fig. 8.8. Our comparison of the two limits, following Dreyfus (1969) and Jain & Tewary, will concentrate on the levels which can be derived by considering only the lowest state in each well in the tunnelling model. In order of increasing energy, these states are

Avealaen Es 1g?

226

LATTICE

FOR

DISTORTION

DEFECT

SYSTEMS

(a)

(b)

Fic. 8.8. Contrast of hindered-rotor and tunnelling-model potentials. For the tunnelling model (a) the system moves between parabolic wells. For the hindered rotor, the

system moves along a path at constant radius; the path I

SII in (b) has no cusp

like that in (a).

for [100] minima, and oes Ata

Tos; Adu

for [111] minima. We shall compare the relative energy splittings and their absolute values as a function of

k = K/(h?/21),

(8.5.4)

a dimensionless parameter relating the well-depth and J, the moment of inertia of the rotor. A. Relative energy splittings for (100) minima (k > 0). We denote by A, the tunnelling-model matrix element for two wells related by a rotation of °. Thus Ago refers to a transition in which the system rotates from the (100) well to the (010) well. As in our discussion of the Jahn-Teller effect, we assume that the basis states of the tunnelling model have been orthogonalized. This simplifies the notation, without physical consequences. Results which involve overlaps explicitly are given by Gomez et al. The tunnelling model gives as the ratio of the splittings

Ely

Eig

E(B) EAP

ind

al akan

AS ans

(8.5.5)

LATTICE

DISTORTION

FOR

DEFECT

SYSTEMS

227

®

0

20

40

60

80

,

@&

Fic. 8.9. Energies in the fast- and slow-rotation limits: 100 case. The ratio of the T,,-A,, splitting to the E,-A,, splitting for various values of the barrier parameter k (Devonshire @ ; Sauer @; Dreyfus). The value q for slow rotation should be reached

at large k.

This ratio is shown in Fig. 8.9. Clearly it agrees well with the large-k limit of the hindered-rotor model. Agreement is not significantly

improved by the small model-dependent term A, g9/3Ago. B. Relative energy splittings for [111] minima (k < 0). The tunnelling model predicts these ratios of splittings

B(Tiy)—E(Aig) _ Aco+2Ar20+Arso _ | E(A2,)— E(A,,) ne

3A60+ A180

3

E(T2,)— E(A1,) 2 2A60+2A120 Rs 2 E(Ay)— E(Ai,)

When These model terms

3A6o0+ A180

(8.5.6) (8.5.7)

3

the 60° elements dominate the levels are all equally spaced. ratios agree well with the large-k limit of the hindered-rotor shown in Fig. 8.10. Again inclusion of the A,y9 and Ajgo does not affect the agreement significantly.

C. Absolute values of splittings. Tunnel splittings can be calculated throughout the whole range of barrier heights from fast to slow rotation. But the methods of calculation are different in these two extremes, for the usual hiadered-rotor

picture becomes

inefficient

228

LATTICE

DISTORTION

FOR

DEFECT

SYSTEMS

E(Ax,)—E(Aig)} { (Ay,)} {E(P)—£

Fic. 8.10. Energies in the fast- and slow-rotation limits :(111) case. The ratios of the T,,-A,, and T,,-A,, splittings to the E,-A,, splitting for various values of the barrier parameter k (Devonshire @; Sauer

x ; Dreyfus). At large k the slow-rotation theory

predicts + and 2, respectively.

when the wavefunction is very strongly localized in the wells. This is clear from lack of a smooth trend in Sauer’s results at large k. Here we compare results for intermediate k obtained from extrapolation from both extremes. The earliest calculations (Dreyfus; see also Jain & Tewary) used the tunnelling model for large k and the hindered-rotor model at small k. The disadvantage of this approach is that the potential energy surfaces are qualitatively different, and so there is some arbitrariness in matching parameters. Suffice it to say, agreement is about as good as one might expect. Jain & Tewary give a discussion using Devonshire’s hindered-rotor model throughout. We concentrate on their results, since a single parameter k defines the problem in all regimes. Jain & Tewary show that a linear combination of functions like

exp (—a sin’ @) become increasingly accurate ask = 4a2/5 increases. This function is localized near 6 = 0. The tunnel splitting, defined

by |E(E,)— E(A,,)|/3, tends to

4a,/(na)(1 +a/8) exp (—a)

(8.5.8)

for [100] minima. Table 8.3 compares Sauer’s numerical results with (8.5.8). Agreement is quite good, being best near k ~ 20.

LATTICE

DISTORTION

FOR

TABLE

DEFECT

SYSTEMS

229

8.3

Tunnel splittings: comparison of fast- and slow-rotation methods. Sauer’s values start from the fast limit (k small) and those of Jain & Tewary from the large k limit. Values quoted are

|E(E,) — E(A,,)|/(h?/21) k

0

10

20

30

40

50

Sauer Jain-Tewary

2 —

1-1273 1-4214

0-558 0-547

0-270 0-240

0-136 0-115

0-072 0-059

8.5.3. Other energy levels for rotating systems

It is often necessary to consider higher excited states of rotators. Results for the Devonshire model are summarized in Fig. 8.11. Naturally there is much in common with the Jahn-Teller systems (Figs. 8.5 and 8.6) and a fairly direct relationship can be made. The main features for rapid rotation (|k| small) are those of a free rotor; groups of levels appear whose degeneracies and mean energies can be found from the free-rotor problem. At large |k|, libration appears: in essence, there are excited states of the rotor within each potential well. The groupings of the levels change too, but can be easily understood by regarding each well as a two-dimensional (librational) oscillator. If the degeneracy of the nth level in each well is D(n) and there are Ny wells, then the levels will form groups of D(n)Ny levels. Note that the librational splittings increase with |k| as the effective force constants increase. The tunnel splittings decrease at the same time because overlap between wells decreases. The level scheme of Fig. 8.11 is only an incomplete picture for several reasons. First, higher-order terms in the potential have been ignored and may be important. Secondly, the defect may have other degrees of freedom. Thus Jungst & Sauer (1967) considered the interaction of internal vibrations of a molecule with its rotation. Other authors (Friedmann & Kimel 1965a, b; Pandey 1965; Kirby, Hughes, & Sievers 1970; Keller & Kneubiihl 1970) both find centreof-mass motion as well as rotation essential in understanding data for HCl in rare gases and OH in alkali halides. Finally, there may be interactions between various molecular motions (e.g. Hougen 1970) which are especially important if modes occur whose frequencies are in a simple ratio.

230

LATTICE

DISTORTION

r

Lae Peres

20-

Ev

&

vee

DEFECT

FOR Mig

z

Ald

SYSTEMS

2

ia

aoe eh

ve

~

4/21 units in Energy

(6) Trigonal

Tetragonal

minima

minima

— 307-

—100

—80

| —60

i -—40

n —20

0

[eseae | oaa 20 40

= 60

80

k Fic. 8.11. The Devonshire model: energy levels of a hindered rotor. The lowest levels are shown as functions of the barrier parameter k in units h?/2]. Typical values for tetragonal minima are 16 (KCI:CN) and 33 (RbCl: CN); values for trigonal minima

are —83 (KCI:Li®) and —96 (KCI:Li’) (Narayanamurti & Pohl). The bracketed numbers are the degeneracies of the groups of levels in the slow-rotation limit.

8.5.4. Choice of potential for rotating systems

In eqn (8.5.1) the potential is defined phenomenologically in terms of a single constant K. This constant represents the defect interaction with the host lattice. But the lattice atoms are not held rigidly at their lattice sites ; they move both from thermal motion and in response to the rotation of the defect. In general a single constant K will not give a good description of the rotational levels.

LATTICE

DISTORTION

FOR

DEFECT

SYSTEMS

231

Two extreme limits can be given clearly. In one the rotational (or librational) motion of the defect is slow, with a frequency much less

than typical lattice frequencies. The lattice follows the defect motion adiabatically: the lattice-strain energy can be calculated for each defect configuration, and this strain energy is part of the potential energy for rotation or libration. In the other extreme limit the rotational (or librational) motion of the defect is rapid, with a frequency well above typical lattice frequencies. The defect follows the lattice adiabatically: the rotational or librational motion of the defect can be found from the potential of a static lattice with displacements determined, in part, by the dependence of the defect energy on lattice configuration. Thus there are two limits in which (opposite) adiabatic approximations are used. The isotope effect is different in the two limits. If the host atoms follow the motion of the defect, then the effective mass of the defect is some average of the defect and host atom masses; the isotope effect should be relatively small. 8.6. The bound polaron 8.6.1. Introduction

We now discuss a generalization of the effective-mass theory of chapter 4 to include electron-lattice coupling. The simplest example is the bound polaron, an electron trapped by a point charge Ze and coupled to the longitudinal optic modes of the host. This system is both a good description of defects such as donors in III-V semiconductors and also a limiting case with which other, more complex, models may be compared. Detailed lattice structure is ignored in the Hamiltonian, which consists of an electronic term, a lattice term, and electron—lattice coupling. The electronic term contains the kinetic energy and the Coulomb potential with electronic screening. If m* is the effective mass for the rigid lattice

#9 =

h?

> VV),

Hee

2

(8.6.1) (8.6.2)

RAL

Only longitudinal optic modes are included in the lattice and electron—lattice terms, and they are further simplified by assuming

232

LATTICE

DISTORTION

FOR

SYSTEMS

DEFECT

a single frequency wo for all these modes. The lattice term is thus

HH? = hod, (Gj ag +2):

(8.6.3)

q

This will sometimes be generalized to include phonon dispersion. Finally, the Frohlich Hamiltonian (§ 3.6.2) is used for the electron— lattice coupling

Hy, = hoo Y {Vaa(e's*— 1)+ oar Claie ialeossh)

(8.6.4)

q

in which

V= -i| ie Ne UZ. 2mMWo

%

=

Iq|

lie leet -

{=

e?

“ata

1 hwo

,

(8.6.5) 8.6.7 ( )

This interaction is linear in the displacements and does not mix the normal modes with each other, nor does it affect their frequencies.

The Hamiltonian can be simplified by eliminating the interaction of the lattice with the (fixed) defect potential. This is done by a canonical transformation which displaces the lattice modes. Its effects are to change the form V(r) by introducing lattice screening, and to introduce an infinite self-energy (that of a static point charge) which may be subtracted. The three terms become

h? Ze? 1 aay See 2m Eq |r|

(8.6.8)

Han = hao Y (ag ag +3)

(8.6.9)

i=

q

Hy = ¥ (Vaig elt’+Vea* eoi9},

(8.6.10)

q

Solutions of the bound-polaron problem are given in terms of two dimensionless parameters which characterize the relative magnitude of the three terms in the Hamiltonian. If Ro is the effective Rydberg (m*e*Z*/2esh7), then the parameters are a and R = Ro/hwo. Clearly special care is needed for those values of R which put an

LATTICE

DISTORTION

FOR

DEFECT

SYSTEMS

233

electronic splitting close to a phonon energy. In comparing

the

results of various workers it is convenient to note the identities

and

a*./R =a,

(8.6.11)

where a* is the effective Bohr radius (h7¢9/m*e*) and a, the polaron

radius {(h?/2m*)/hag}?.

Two different approaches have been used. The most direct approach (Buimistrov & Pekar 1957; Buimistrov 1964; Brandt & Brown 1969; Larsen 1969b) is a variational method. A trial solution is chosen which describes both the electronic and vibrational degrees of freedom. The energy functional is then minimized, first with respect to the lattice variables to give an effective electronic energy functional, then with respect to the electronic variables. Since the lattice modes are dynamically independent, the lattice minimization can be done separately for each mode. Of course, the success of the variational approach depends on the accuracy of the trial function. The second approach (Platzman 1962; Schultz 1963; Perlin & Gifeisman 1968) is based on the method of Feynman (1955). This approach rewrites the time-dependent Schrédinger equation in terms of transformation functions, related to the electronic Green’s functions, and concentrates on the Lagrangian rather than the Hamiltonian. It is a perturbation method in the following sense. Suppose the phonon variables can be eliminated somehow, giving a complicated exact Lagrangian Y,,,.,. Normally one cannot solve the equations of motion exactly with this Lagrangian; but one may know the solution for a related zero-order Lagrangian %. Then (Loxact Lo) is treated as a perturbation on the motion. Unfortunately, very few convenient % are known, and the lowest-order results may be rather poor. In the present case Yo replaces the Coulombic V(r) by a parabolic potential well, and the interaction with the phonons by a harmonic interaction with a massive particle which represents the inertia of the lattice coupling. 8.6.2. Weak electron—lattice coupling For very weak coupling, « « 1, the lowest-order corrections to the ground-state energy may be found by second-order perturbation

234

LATTICE

DISTORTION

DEFECT

FOR

SYSTEMS

in #,,,- In practice such calculations are acceptable for « »

(8.6.15)

and have energies

En = Eh)

et3),

(8.6.16)

q

in which |,» is an electronic state and E,, is its eigenvalue. On substitution, taking all occupation numbers n, as zero,

rs de The sum

awd)

Val71 denotes an average over the electronic wavefunction. The transformation which eliminates the linear electron—lattice terms when averaged over the electronic motion is aq a

a,—,

(8.6.28)

aq > 4, —< f(r). corresponding to a lattice displacement

Q, = 0. But (O,,W) is just a rotation of W, so it too is never negative. Thus each term in Wo is nodeless, and so Wo is itself nodeless. Further, since the symmetry operations O;; commute with the Hamiltonian by definition, wo corresponds to the same eigenvalue as w. 3. Hund’s rules. These are a set of empirical rules for atoms. They state that for a given configuration, and consistent with the exclusion principle, the ground state of an atom has (a) the highest allowed value of the total spin and (b), consistent with this spin, the highest allowed value of total orbital angular momentum. The rule works well for atoms, although prediction (b) tends to break down for atoms with atomic orbitals other than s- and p-type. Also the rule may break down for excited configurations of a given atom. The rules can be derived by direct calculation of the individual energies (cf. Griffith 1961, p. 79). Basically the high-spin states are favoured because of the signs of the important electron-electron interaction integrals. Loosely one might argue that electrons with parallel spins must have orthogonal spatial one-electron wavefunctions, and that the orthogonality implies they tend to keep apart in space with a corresponding reduction in Coulomb repulsion energy. At first sight Hund’s rules appear to contradict 1(a), which argued that the ground state of a two-electron system must have spin zero. There is, of course, no real contradiction. For the systems described by 1(a) the ground state has no orbital degeneracy. There

are two electrons in this orbital which is a singlet because of the exclusion principle. The systems to which Hund’s rules are usually applied include transition-metal ions in crystals where degenerate levels are only partly filled. Other cases are the two-hole centres found in oxides. Whereas two-electron centres have singlet ground states, two-hole centres have triplet states lowest. It seems (§ 16.4)

262

GENERAL

RESULTS

that the triplets occur through the admixture of configurations like

[O?~ O°] instead of [O~ O7]; it is the intra-atomic exchange interaction which gives the triplet ground state. These systems also demonstrate that configuration mixing may be important in determining the ground-state spin. 9.4. Order of energy levels Energy levels may be classified according to the symmetry of the state to which they correspond. The states of a particular symmetry, labelled y, can be placed in order of increasing energy BG

Pie oh) < EaAV)ce

eee

(9.4.1)

What general results exist which order the levels of one symmetry relative to one another, such as

E,(71) < E,(?2)

(9.4.2)

E,(?15V2) < En(v3ȴ4)?

(9.4.3)

or such as

Here E,(y;,, 72) is the nth level of the set including both states with y, symmetry and those with y, symmetry. Results have been derived by Saffren (1968) for square, cubic, and tetrahedral symmetries. They may be considered as generalizations of the results for spherical symmetry (Sachs 1953)

E,() < E,(/+ 1), in which / is the orbital angular momentum.

(9.4.4) Indeed, a number of

results are identical with this apart from the removal of certain degeneracies in ways which are model-dependent. The relations are derived by using two theorems due to Titchmarsh (1958) and generalized by Saffren. They are: 1. If, in an eigenvalue problem, the eigenfunctions are constrained to vanish on ever-increasing portions of the boundary, the eigenvalues increase; 2. If, in an eigenvalue problem defined in a region R, the portion of the boundary on which the eigenfunctions are constrained moves inwards into R, then the eigenfunctions increase.

We summarize the results without proof in Table 9.1.

GENERAL

RESULTS

TA BLE

9.1

Ordering of energy levels in different symmetries Square symmetry

(a)

xy (x2 — y?)

(Db) [xy(x?— y2) + (x?— y2)

[cy

xy]

xy

xe+y

[ G2+y?)+(x?-y?)

(a) tee

xy(e—y2) +xy]

Cubic

(x2+y2)+xy J

symmetry

xy2(x? — y?)(y? — 2?)(z? — x?)

El

7 6

(x? — y?) Q? = z?) (2=—

fib,

xyz (322—?)

Ne

; 4

reste y?)

3

2(x?-—y

;

xyz

ee

2

te Ser ey

1=x?+y?+2?

b (b)

TY

T

(iad (c)

Mm]

xy

[ips r ]

(d)

[Pis+T, ]

ies

ey

[13 +Ms}

r,

ls

r,)

}

Vou)

Tetrahedral symmetry

Ge

ie

(b)

1m

[Tas ]

[ties

sae ie,

i

[any

T+

T, J

[Ms]

| lhigel In these tables [y;] means the nth level with symmetry y;,, [y;+y,] means the nth level of the class of y,- and y,-states taken together. Two classes placed alongside in energy without a specific relation being shown are not ordered by symmetry arguments alone. For cubic symmetry we give examples of functions with the appropriate symmetry and the lowest angular-momentum states from which they can be derived. The lowest energy states are printed lo west, the highest at the top.

264

GENERAL

RESWETS

9.5. Scaling relations and the virial theorem

The virial theorem is one of the standard checks of wavefunctions in atoms. It relates the total kinetic energy of a system to the total potential energy ar

(9.5.1a)

R > oR.

(9.5.1b)

Under this scaling the total energy changes

W(a) > «? + a U(aR, 8)»

(9522)

where the interactions vary inversely as the Nth power of the separations. The most common case is N = 1, corresponding to Coulomb interactions. The variational principle indicates that W(a) is extremal when « = 1, corresponding to the exact eigenstate. The virial theorem simply expresses the vanishing of GW/da at the extremum

ow a

2a + NaX~* +

CI

+ o *R . 4+ aR

d¢U>

aR. 4 +a"R aR

(9.5.3)

When « tends to unity,

OW epe e

ee

ow

odSoe

(9.5.4)

GENERAL

RESULTS

265

The last term is essentially a pressure term. For atoms it is zero, and, with N = 1, gives the well-known form ¢ JQ*(O|D.

The integral is a simple example of a correlation function.

(10.1.8)

OPTICAL

PROPERTIES

23

We now define Q explicitly. Strictly, we should start from the Dirac equation, but for simplicity we start from the Schrédinger equation and add the appropriate corrections separately. The interaction with radiation changes the kinetic energy term in the electron Hamiltonian

=p?

a

1

CaN

>—I|p+-A] “(pe

1

e

= ~p*+—A.

|

as

Pre

e

Pup

.A+-A.Al. Boo

i

(10.1.9) The Coulomb gauge is used, so the scalar potential is that obtained from a static charge distribution. There are no free charges for a pure radiation field, so there is no potential energy change here. The perturbation theory we use is gauge invariant (Griffith 1961). The termin A. Acan be dropped unless we want non-linear effects. Then, for atom I, the operator Q is

Q=-—

JY

0

cos (o(t—215/¢)} =,

(10.1.10)

Vs

electrons s

and the cross-section becomes

o(w) =

4n*he? m?>cw

Koy

etude

jp? oe)

(10.1.11)

electrons Ss

This expression can now be simplified by two changes. First, the optical wavelengths of interest are much greater than the dimensions of the atoms. This is still true for extended centres in solids. Thus

we may expand the exponential in powers of (@z,,/c),

i

ee co mnenroh perc (ko a) 4(:24,3)1 Vieyamey MDE) aay oz te oye Oz @-

@&

A yee

wl

@

0

onaae

10.1.12

in which L, is the x component of orbital angular momentum. Had we started with the Dirac equation, L, would be replaced by L, + +2S,; we simply add in the spin term here. The second change

)

274

OPTICAL

PROPERTIES

affects the first and third terms. Using the sum rules of Appendix I, the matrix elements of p can be transformed into those of r,

MO;

cain j> and |j> have the same parity, as mentioned in § 9.6. In most cases we need only consider electric dipole transitions. The cross terms in o(w) cancel out for unpolarized light and an isotropic medium such as the vacuum. Another way of writing o(@) is to introduce the oscillator strength. We consider electric-dipole transitions alone, and average the

OPTICAL

PROPERTIES

275

corresponding square matrix element over all orientations of the electric field vector, givingt

Avi|? = |R,A?.

(10.1.17)

We define the oscillator strength by 2m

fig = = Odd’,

(10.1.18)

so that the averaged cross-section is

DRec. mc The oscillator strengths satisfy the important ‘f sum rule’

BES ri

(10.1.20)

derived and discussed in Appendix I. We finally consider spontaneous emission, which dominates over stimulated emission in all but special systems like lasers. The rate follows quite directly from the quantum formulation, but can also be found from detailed balance. The transition rate must be summed over all final states (hence a factor (1/227)k? dk per mode per unit

volume, identically equal to (1/2?)(w?/c?)(dqw/c) summed over the two polarizations) and averaged over directions of the field vector. After averaging over directions, the absorption cross-section is a(w) = Bd(w—a,;,),

(10.1.21)

where (10.1.19) gives B = (227e7/mc) f,;. The emission probability per unit time is thus

are

|f

+ Several authors use instead the maximum

(10.1.22)

value of the matrix elements, giving

[Ri*|? = 3|R,,|?. This leads to an extra + in their papers (Dexter 1958; Markham

1959).

276

OPTICAL

PROPERTIES

10.2. Optical properties of centres in a solid Changes occur when atoms or other centres are incorporated into a solid, because their environment is neither inert nor isotropic. We ignore for the present resonant coupling between the photon field and collective modes of the lattice (cf. the polariton) and concentrate on the changes which result from the polarizability and anisotropy of the lattice and from the existence of phonons. The high-frequency polarizability of the lattice changes the refractive index n from unity. This changes the photon flux, since the velocity of propogation is altered, c—> c/n.

(10.2.1)

Similarly, the photon density of states, used in the emission problem, changes, although by a different factor. Further, the magnetic dipole and quadrupole cases are affected in a different way, since they come from a higher term in an expansion in terms of wz/c. In all cases we should use the real part of the dielectric constant if there is absorption and n is complex. In cases where there is dispersion, 1.e. n = n(w), the energy density, and hence the flux, must be defined with more care; these cases will not normally be important (Stern 1963, p. 309; Landau & Lifshitz 1961, p. 253). A further

change which occurs because the lattice is polarizable is that the effective field which induces transitions is not simply the applied field, but contains a contribution proportional to the polarization. We shall discuss the local ‘effective’ field later in more detail. The effect of these changes on our previous formula for the absorption cross-section 1s 24

a(w) > (F 6(w),

(10.2.2)

1)

and the effect on the reciprocal lifetime is E ete ¢ + W — n|—) wr.

(10.2.3)

In these expressions we write é and W, rather than o and W, since there are also changes in the transition matrix elements.

OPTICAL

PROPERTIES

277

These occur partly because the lattice is anisotropic. The operator (10.1.16) becomes Tl .r+

hn 2mc

KAT. M+—"(x. r)(m.r), OKs

where 7 is the unit polarization vector and x is the unit wavevector. The selection rules are modified both by the reduction in point symmetry of the centre and by the coupling of the centre to the lattice, since phonons may be emitted and absorbed during the transition. The coupling to the lattice is of major importance, and causes a shift and a broadening ofthe resonance line. The absorption and emission energies need not be the same, and the Einstein relations between absorption and emission probabilities must be changed. The question of the line-shape in the presence of interaction with collective modes of the crystal, such as phonons, will be treated later ; it is a problem which is sufficiently complex and important to need special attention. However, we can make some useful extensions here. We denote states of the defect plus lattice by |i, {n}>, where i indicates the defect state and n the occupation numbers ofthe various lattice modes. The matrix elements for dipole transitions become

Ci, {n}|m lj, {n'}> and the transition probability includes an average over the initial lattice states and a sum over possible final lattice states. The crosssection becomes

1 Eere\? 4n7e? o(@) = | | Fe AY Dds (alle

Ar}? 50 — Om (10.2.4)

in which AO; jay = E;,—E,; ,. Similar changes occur in the reciprocal lifetime and the oscillator strength. 10.3. The Condon approximation We shall often need to calculate matrix elements of electronic operators between different electron—lattice states. With the adiabatic approximation the matrix elements have the form

Vee idQ |dry QW#(r; Q)A(r; QW (tr;Q)z,(Q),

(10.3.1)

278

OPTICAL

PROPERTIES

where Q denotes all the normal coordinates of the lattice. The Condon approximation asserts that the integral over electronic coordinates

m, AO} = |ruse; QOr; Qw(r;Q)

(10.3.2)

is essentially independent of Q. The total matrix element may then be factorized

Moraceae |dQy*(Q)z,(Q).

(10.3.3)

The factorization is particularly useful because the integrals over y(Q) can often be found simply, or sometimes eliminated by completeness relations. The accuracy of the approximation depends on the sensitivity of both the (r; Q) and Q(r; Q) to Q. If, as we have presumed, the adiabatic approximation is valid, then /(r; Q) is relatively insensitive to Q. We expect states with a large electron—lattice interaction to show deviations from the Condon approximation. In optical transitions the electron—lattice interaction is responsible for the line-width in absorption AE, andso we may use(AE,/E,) as a measure of the error involved. More detailed general rules are hard to establish, even for optical transitions. One reason is that Q(r;Q) in this case contains an effective field factor which may be sensitive to Q. The accuracy also depends on the operator 2. Thus it makes a difference if momentum rather than coordinate matrix elements are used in optical absorption. As can be seen from (10.1.13) the matrix elements are related by a factor involving the transition energy and this extra factor affects the line-shape of the broad band. Further, as Klein (1970) observes, Raman scatter is strongest when polarizabilities are sensitive to atomic displacements, in just the circumstances in which the Condon approximation breaks down. In principle there is no difficulty in going beyond the Condon approximation and treating the terms in m;,(Q) which are linear in Q. In practice this is rarely necessary except when considering forbidden transitions (§ 10.9) or when a particularly accurate analysis is attempted (see e.g., Brothers & Lynch 1967). Detailed estimates of the accuracy of the approximation confirm that it is usually acceptable, as in Gilbert & Wood’s (1970) work on the F-centre in

KCl. Most other theoretical checks have been made on the related

OPTICAL

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279

‘r-centroid’ approximation of molecular physics, which asserts that if

ge idO x*(2)0x,(0)

(10.3.4)

then Q” = (Q)" and that one may approximate the matrix elements using

Mi, = m,(0) idO y#(Q)x,(0).

(10.3.5)

Such results are examined by Drake & Nicholls (1969). Some experimental checks of the Condon approximation are also possible. Schnatterly (1965) examined the optical absorption of the F-centre under stress and, in particular, examined the ratio of the fractional

change in the total integrated absorption to the fractional change in mean absorption energy. This is a check ofthe ‘law of constant area’ (10.4.8) and should be unity if the Condon approximation is valid. Schnatterly finds the ratio is one to the experimental accuracy, provided that the absorption in both the F and K bands is included; both bands appear to derive from the F-centre. His results are given in Table 10.1. Other estimates of the accuracy of the Condon approximation can also be estimated from the stress-dependence of parameters, e.g. the photoelastic constants (Aggarwal & Szigeti 1970), which measure strain-induced changes in refractive index.

TABLE 10.1 Check of the Condon approximation (AA/A)(AE/E) Crystal

(F + K bands)

(F band alone)

KCl KBr KI RbCl NaCl

10+0:3 1-4+0:3 1-2+0-6 1-2+0:3

2:4+0-2 5-140:3 3-4+0-4 3-3+0:3 7-1+0-7

Results

taken

from

Schnatterly

(1965).

A

is the

integrated absorption and E the mean energy of the F-centre transitions.

280

OPTICAL

PROPERTIES

10.4. The line-shape function

The cross-section is usually written in terms of a line-shape function G(q). It is this function which is actually calculated when there is electron-phonon interaction. For simplicity we shall relate G(@) to o(w) using the Born—Oppenheimer approximation and the Condon approximation. The first of these lets us write

li, {2}>= WG; Q)xn(Q).

(10.4.1)

The resulting matrix element is

Ci{nn-rilmt}> = |€Q 22QO/Qin(Q.

(10.4.2

The second approximation drops the configuration dependence of

2,{Q):

2,{Q) = Id?q W#(q; Q)n. rdf; Q).

(10.4.3)

Its use is subject to the qualifications mentioned in § 10.3. Note that we may not be able to assume that the electronic matrix element is the same for absorption and emission. The matrix element (10.4.2) becomes

Kinin nlifw}>? = 93||dQ Qe

(10.4.4)

With these approximations the expression for o(w) may be factorized a

1 Eee

ae?

o(w) = ‘at i

Glo) =

2

QF,0G(a),

[40 x8Q)r(Q))

2

(10.4.5)

= j,m) (104.6)

G(q) is the shape function and is normalized to unity over w. Thus

. dw Glo) = AVY, {dQ x¥(Q)rn(Q)

| 2

= AVY. |dQ 2hQven(Q) |dQ (Q)xKQ)

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PROPERTIES

281

By closure, this becomes

[ao Gta) = av fag [40 121Q),(Q) 4Q—Q)

Av [dQ Ix(Q)? =,

(10.4.7)

from the normalization of the y. This result leads to the important ‘law of constant area’. The cross-section (10.4.5) varies like wG(). The total area of a band is proportional to the integral: area oc [do @G(w@) ~ wr |do G(@) = @r.

(10.4.8)

The integrated absorption is independent of the electron—lattice coupling, apart from any effects on the transition energy wr. Further, if the transition energy is altered by any perturbation, the fractional change in area should be the same as the fractional change in frequency. This result is the one used by Schnatterly to verify the Condon approximation, since (10.4.7) only holds when the approximation is valid. Deviations from the approximation bring in an extra Q-dependence to the integrands which prevent the simple use of closure and normalization. In principle (10.4.5) defines a function G(w) even when the adiabatic and Condon approximations break down. We may also rewrite the oscillator strength using these approxima-

tions pol>

(10.4.9)

where the ¢ >,,,; denote an average over all the directions of the polarization vector nm. Similarly, the reciprocal lifetime is

Ee\?4e?

=.

W= || 03

2. (10.7.12)

The effective phonon energy hw,,,, chosen so that S,fim,,, is the lattice relaxation energy, is (7/8)hv,/a*,v, being the velocity of sound. The result for Frohlich coupling is in acceptable agreement with observation for a variety of defects.

B. The configuration coordinate diagram and the Franck—Condon principle. The configuration coordinate diagram gives a simple description which shows the significance of the Huang—Rhys factors. The diagram is very widely used since it helps one to keep track of terms even in complicated processes. An example is shown in Fig. 10.2. As in several models of chapter 8, we consider only the electron-phonon interaction with a single ‘effective’ mode Q. The figure shows the total potential energy (total energy apart from nuclear kinetic energy) as a function of Q for two different electronic states. The different vibrational states are also shown. Optical transitions are best understood in terms of the Franck—Condon principle (Franck 1925, Condon 1928): Electronic transitions occur with no change in nuclear configuration. Thus a cycle involving

296

OPTICAL

PROPERTIES

/3,

Eas

Fic. 10.2. The configuration—coordinate diagram. Absorption and emission energies in the Franck—Condon approximation are shown.

absorption and emission might be the following. At low temperatures, the defect system will be in the lowest state A. Absorption occurs primarily to B, the excited state with the same configuration. Nonradiative processes occur rapidly, and the system relaxes to C. The energy released is S,)hw,, and the mean number of phonons emitted is the Huang—Rhys factor. Emission of light occurs primarily in a transition to D at constant configuration. Finally, non-radiative processes take the system back to its initial state A with emission of phonon energy 5S,ohw,. Note that the difference between absorption and emission energies (the ‘Stokes shift’) is 2S,.>hw,, and that the absorption energy is the larger

EE abs

POhGSie

(10.7.13)

The justification of the Franck—Condon approximation depends on the form of the vibrational wavefunctions. As seen from (10.4.6), the transition probability depends on the overlap of the vibrational wavefunctions in the initial and final states. In the ground state A,

the vibrational function is concentrated near Q = Q,,. All the excited states, apart from a few of the lower-lying ones like C, have their wavefunctions concentrated near the classical turning points which lie on the upper energy curve. The ones which overlap most strongly with the initial state A are those with turning points at B. The

OPTIGAL

PROPERTIES

297

essence of the Franck—Condon principle is that only these states need be considered. Clearly, the principle should predict reasonable ‘most probable’ transitions, even when the energy curves are more complicated than harmonic. But exceptions will occur. First, when the electron— lattice coupling is weak, the most probable transition will be from A to C, the so-called zero-phonon transition. Secondly, many authors have attempted to derive line-shapes by weighting Franck-Condon transition energies by the probabilities of different Q in the initial state. This semiclassical method works fortuitously for linear coupling in nondegenerate states, but fails in detail when there is electronic degeneracy. This is particularly clear for E-states coupled to E modes (O’Brien 1965, 1971, Longuet—Higgins et al. 1958) where predicted line-shapes are too symmetric and have a spurious

zero. Other

slightly-different

versions

of the

principle

are

found.

Rebane (1970, in his Appendix I) contrasts these classical and semi-

classical versions with the quantum form used here. C. The characteristic function. The significance of the S, can also be made clear in another way. We recall that the lattice eigenfunctions y(Q) can be written as a product of factors, one for each mode

Xum(Q) = [] h(n, ;Qia).

(10.7.14)

0.= O08 and {n} is an abbreviation for the set of occupation numbers (n,,",,...,My) of the modes. The line-shape function G(q@) is related to the overlap of the initial and final vibrational states,

G(w) =

mera)

(10.7.15)

{n}v2

The overlap between the initial and final states also factorizes into a product over modes. For one lattice oscillator we define

ole,Mi:diz0)=| dO, ht(n,,:O,)h(n,: Qj.)

(10.7.16)

and rewrite the line-shape function as G(@)

=

Nd, [TotNy» Ny 3 Gj, ae (w =

WF nn’)

(O17)

298

OPTICAL

PROPERTIES

The importance of the Huang-Rhys factors S, arises because these lattice oscillator overlaps can be expressed simply in terms of the S,. In general

lo(n, m;q)2 = S"-™e7S.(m!/n!) {Lt ™(S)}2,

(10.7.18)

where L"~™ is an associated Laguerre polynomial. In most cases the coupling to any one mode is small, and an expansion in S, is possible. Explicitly, the important overlaps are

lo(n,n;q)|? = 1-S+O(S?), |o(n, n+15;q)? = 3(S£S_)+ O(S?),

lo(n,n+2;q)? = O(S?).

(10.7.19)

The expansion in S, is particularly useful when there are no local modes, since the S, are then of order (N~'). This is still true for systems where the electron-—lattice coupling, summed over all modes, is substantial. Our expression for G(w) may be rewritten to give an expression for g(t) by using the spectral representation of the delta function. If we define the change in occupation number of mode a as A, = (n,—n,), then the characteristic function g(t) is

g(t) = exp (iw,;t) Av)[] lo(n,,n+ Ag; dij)? exp (iA,@,). UD dN

The transition energy is hw;; for zero electron-phonon coupling. The sum of products can be rewritten as the product of the partial sums

g(t) = exp (iw, ;t)Av [| x {n}

a

x ($lottesnet Aas dal? exp (id.cr!). (10.7.20) Aa

Since this is a simple product over modes, it follows from the convolution theorem that G(w) is just the convolution of the lineshapes which would be obtained by coupling to each mode separately. This result is useful if the lattice modes fall into well-defined groups.

OPTICAL

PROPERTIES

299

If there are no local modes, only the terms A, = 0, +1 need be

retained in the sum over A,. Thus

g(t) = exp (ia;Hav [T{1—S. (1 —cos @,t) +189, Sin w,t + O(N ~7)}.

(10,721)

Each item in the product is linear in the occupancies of the modes, since S, is just (2n, + 1)S,>. The thermal average is trivial, and simply means that n, should be replaced by its thermal equilibrium value. The expression for g(t) can now be written ina more convenient form, accurate to O(N ')

g(t) = exp (ia,,t) exp E

Sse}

ne

—isin oat ;

(10.7.22)

The spectrum of lattice frequencies is continuous for practical purposes. Sums over lattice modes can be rewritten as integrals over phonon wavevectors or, more conveniently, as an integral over phonon frequency. We introduce a smooth function which describes the relative contributions of different frequency modes to the coupling

A(w) dw =

>

Soa/>,Sos:

OfOg. Soa alla

S=Y¥

S,.

(10.7.24)

alla

We term S, the Huang—Rhys factor. This one factor is often used as a measure of the electron-phonon coupling. We now find

g(t) = exp (iw; ;t)e~* exp E ida’ A(w’) x x J 008 (w’t) coth fa +isin wo},

and we proceed to examine this expression in special cases.

(10 7a20)

300

OPTICAL

PROPERTIES

D. The weak coupling limit. When the electron-lattice coupling is very weak the factors S, )are small. We shall assume

So =tys,p ,

(10.8.13) which can be expanded into terms up to fourth order in the Q. The odd powers of Q give a zero average, and may be dropped. We find h’M

,(0) =

(hd; ;)”mp S {2S, o(h@,)? fa: LOM

a

2 »(Age +3A

ga

pp) (Ne aI 3)(ng ar 3) =F

+ DARA 3{(n, +4)? +4}. aa8

(n, ay) at

(10.8.14)

The last term is negligible unless one mode dominates in the coupling.

314

OPTICAL

PROPERTIES

Now the square of hM,(0) is {nM ,(0)}? = (he; ;)° — YA gah®; (ny +4) +

+ YY GA Agel, +3)(Ng +4)a

(10.8.15)

p

The second moment referred to the mean transition energy is thus

h?{M2(0)—[M,(0)]}?} = Y Syolhe,)*(2n,+1) + +

ys (Ags Se aA ua\pp) (Na a 3) (ng a 4).

a,B

(10.8.16)

Other useful results can be obtained from the Huang—Rhys model (§ 10.7.2E) which considers linear coupling to modes of a single frequency. The first few moments are easily found hM

,(@;)) = 0

h?M(@;;) = S(hao)?(2n+ 1) (10.8.17) h° M 3(@;;) a S(hao)° h*M

4(@;;) =>

3[h? M

»(0;,)}°[1 +

{3S9(2n+

1)} oe |

or, alternatively, M,/M3’?

ees

M,/M;

=

{S(2n+1)}~#

= 3+1/S

;

(10.8.18)

The moments are measured from the mean energy. As the Huang— Rhys factor S increases, the skewness (measured by M3) and the deviations from M,/M3 = 3 decrease. The line becomes increasingly Gaussian, and its full width at half intensity is related to the second

moment by

A? = 8In2.h2M).

(10.8.19)

We may use (10.8.16), (10.8.19) and the asymptotic forms of coth x

OPTICAL

PROPERTIES

315

to obtain approximate line-widths for linear coupling A=

‘8In 2 L (hw,)? S49 coth (hos 2KT

> 'In2>) (ho)S.o}

~ 'In2 } hw,S.0

4

4

kT

4

(low temperature)

(high temperatures). (10.8.20)

10.8.3. The configuration-coordinate diagram One of the few strong-coupling models which can be solved analytically for the line-shape function is the Huang—Rhys model. The qualitative features of this model—e.g. the temperature dependence of the line-width—are very similar to those observed for centres where the Huang—Rhys theory does not strictly apply. A natural question (Lax 1952) is this: when can we describe the electron—phonon interaction in terms of interaction with a single mode (or, equivalently, with modes of a single frequency)? The advantages of such a description are these. First, we may use the configuration-coordinate model of (§ 10.7.2B) to describe the various processes in a way which is conceptually clear. Secondly, the model suggests a Gaussian band, which is usually observed. Thirdly, we may obtain an ‘effective Huang—Rhys factor’, which is a measure of the electron-phonon coupling. This may be found from a variety of expressions, all equivalent for this simple model

Cio/Izpr=0 = XP (So),

(10.8.21)

E.ts—E abs

(10.8.22)

emiss

= 2S oho,

A? = 8In2.So(hwo)? coth (hw/2kT).

(10.8.23)

The deviations from the simple model are useful pointers to unusual effects—particularly as indicators of electronic degeneracy or of changes in force constants. As a measure of the accuracy of the configuration-coordinate picture, we attempt to choose the coordinate frequency and coupling

316

OPRTICALSP

ROPE REIES

to fit the line width at all temperatures. It is straightforward to fit the expression for A at high and low temperatures,

So = (A(0)/ha,)2/8 In 2, lim {A(@)/A(T)}?2KT, hwo but, of course, it is by no means

(10.8.24) (10.8.25)

obvious that the other relations

involving Sy and hay will be satisfied, since these involve different averages over the phonon spectrum. In practice, careful analysis of line-shapes and other properties have tended to verify the model (Henderson, King, & Stoneham 1968; Henry & Slichter 1968). The reason that the model is so successful can be seen from some results due to Housely & Hess (1966) and to Johnson & Kassman (1969). They show that bounds can be put on the mean square displacement and the mean square velocity of any particular atom in a harmonic solid. If the mean square displacement < X) is known at a temperature Ty then there are strong restrictions on displacement Mean exp {i%t/h}|F>.

(10.10.4)

Secondly, we shall use ey |f> 7E transition of KMgF;,:V**. The full line gives their theoretical results.

Sturge et al. give a detailed analysis of transitions in KMgF, Vacate where again sharp transitions are superimposed on a broad band. We concentrate on one ofthe clearest cases, whose interference term is shown in Fig. 10.10. The broad band is a spin-allowed magnetic

330

OPTICAL

PROPERTIES

dipole transition *A, +*T,. The Huang-Rhys factor is large because the two states derive from different 3d° configurations,

t3(*A,) and t3 e(*T,), so there is a substantial difference in charge

distribution. The broad band is well-described by the Pekarian form (10.7.41), apart from interference terms. The sharp band is the spin-forbidden transition *A, > 7E, and is sharp because both states derive from f3; there is no appreciable change in charge distribution. Since the transition is forbidden there is no resonant line. The transition is detected through the antiresonance which results from admixture of spin-allowed states associated with *T, by the spin—orbit coupling. When the spin-orbit coupling is zero, only the broad band occurs. The final states of the transition are |b>|{n,}, q,>, where |b> is the electronic state (b for broad), {n,} the phonon occupation numbers in the final state, and q, the change in mode displacement. Effects associated with the orbital degeneracy of the *T, will be ignored. The line-shape can be found by our earlier methods, or equivalently by writing down the time-dependent Schrédinger equation for the state amplitudes. The Schrédinger equation proves easier to generalize to include the E-state when spin-orbit coupling is added. In either case, one calculates the susceptibility y giving the response to a weak oscillating field. The absorption is proportional to x”, the imaginary part of x. The susceptibility is proportional (omitting some small terms unimportant in the antiresonance region) to the sum of the broad band contribution alone G(w) and a term from coupling of the broad band to the zero-phonon line of the sharp transition

A?G(o) 3 fo) w—ow,— A?G(o)

~ G(w)+

as

(10.10.11)

Here G = G’+iG” is complex, «, is the transition energy of the sharp line and A? is the spin-orbit coupling matrix element. The absorption profile for the broad band alone is proportional to G"(w); G'(q@) is related by the Hilbert transform 1

(o.0)

Go) = -| dx

G"(x)

O- xX

(10.10.12)

The absorption profile with interference is proportional to x", and

OPTICAL

PROPERTIES

331

is given by

x" ~ wy} +p pects ‘} lene Here pisa

(10.10.13)

factor, less than 1, introduced because not all the electronic

states are coupled by the spin-orbit frequency separation from resonance

oO

coupling.

— 0.0),

€ measures

the

(10.10.14)

and q is just G'(a,)/G"(@,). Note that the resonance is shifted from w, to w,+q. In other cases the shift may differ from the g parameter of (10.10.13). The particular form of y” is chosen to match Fano’s expressions for rare-gas atoms. The antiresonance term changes character as q varies. For q = © a Lorentzian resonance occurs, and for qg= 0 a negative Lorentzian appears. But for intermediate values, dispersionlike behaviour is found. For the transition of Fig. 10.10 q ~ 1-3, and the intensity of the structure is very close to that predicted from the known spin-orbit coupling. 10.11. Photoionization of defects

10.11.1. Introduction We now turn to processes in which either the initial or the final state is not localized near the defect. These processes include photoionization and the inverse process of free-to-bound transitions or radiative capture. The theory is very similar to that developed earlier in this chapter, apart from two modifications. First, there is a continuum of states when there is a free carrier. The optical lineshapes must include contributions from all relevant continuum states : one cannot separate the line-shape into discrete components. Secondly, since the charged localized at the defect alters by one electronic charge, the wavefunctions which remain localized will be significantly altered by the transition. These modifications must be taken into account when calculating absolute transition probabilities. Relations between photoionization and radiative capture crosssections have been derived by several authors. The simplest is to use the principle of detailed balance, equating the recombination rate in thermal equilibrium with the generation rate of free carriers from optical absorption (e.g. Ascarelli & Rodriguez 1961b). Other approaches are possible (Bebb 1972), giving results which are

332

OPTIGALSRPRORE

RITES

largely equivalent. These formulae are not always of great practical value, because the absorption and capture transitions do not always correspond. Photoionization is normally from the ground state, whereas capture often proceeds first to an excited electronic state. Further, there may be competition between different capture modes. We shall derive photoionization probabilities in this section, and begin by considering a very special case. There are three simplifications beyond the usual assumptions of an adiabatic approximation, the one-electron approximation, and neglect of the variation of the photon phase over the defect. These simplifications are: 1. That the lattice state does not change in the transition, so we may suppress the phonon variables {n}; 2. That the free-carrier state is one of the unperturbed states of the perfect crystal’s conduction band, |k> = w(k, r) = exp (ik. r)u,(k, r). This is equivalent to the first Born approximation ; 3. That the conduction band is isotropic, non-degenerate, and parabolic,

E(k) = h?k?/2m*.

(10.11.1)

These approximations will be relaxed in later sections. The absorption is proportional to a line-shape function

. Glo) =—1 Av YD [Kitm}ip ea f(a}? Oey tam) inl lal final States statesf

(10.11.2)

which, using | and 2, reduces to

we 1 G(w) = i [d?k|m . A(k)|? 6{@ —(E(k)+E,)/h}.

(10.11.3)

Here ris the polarization vector, E, the defect ionization energy, and A(k) is the electronic matrix element . Slightly different results would be obtained using a coordinate matrix element (cf. §10.1). The integral is readily evaluated using the result (Callaway 19646, eqn (4.253)):

|dx a(x) 0(/(K) = ¥ glxo)l dx/df|z-x,,

(10.114)

OPTICAL

PROPERTIES

333

in which Xo is the root of f(x). For (10.11.3), the roots kg satisfy f (Ko) = [w—{E(Ko)+ E;}]/h = 0, so that we have

ko = (2m*/h) (hw—E,a afidx\,. = m*/hke

(10.11.5)

There are no roots of f(k) for hw < E,, so there is no absorption below this energy. The other factor in (10.11.4) is g(x) =

k*f,,dQ\n. A(k)|’. If the average over directions (k/ko) of |m . A(k)|? is written |z . Aol’, then (10.11.3) becomes

G(w) = 4nm*|n. inal? Aol (>|= Bi)

+

(10.11.6)

for hw > E;. Eqn (10.11.6) has a simple interpretation. There is no photoionization below the threshold energy E,. Above the threshold, the crosssection is simply proportional to the density of states in the conduction band. There may be an additional energy dependence from the behaviour of the momentum element A(k). The main modifications in later sections come from phonon cooperation, from specific dependence of A(k) on energy, and from the more complex electronic density of states of real systems. 10.11.2 Dependence on the conduction band When the conduction band is more complicated than (10.11.1), both the electronic density of states and the averaged matrix elements are altered. The changes are especially simple for spheroidal bands 3

2

E(k) = p> ake om,

(LO P17 )

where the transformation from (k,,k,,k3) to (/(m/m,)k,,

/(m/m)k,

/(m/m3)k3)

=

(K,,K2,K3)

reduces the energy surface to the earlier form h2

E(k) = —x?’. 2m

(10.11.8)

There is a slight change in the definition of the average of |r. A(k)|?, since it is over a constant-energy surface, rather than a sphere

334

OPTICAL

PROPERTIES

in k-space. The only other change from (10.11.6) is the appearance of the Jacobian of the transformation from k to k, namely an extra

factor (m?/m,m,m3)*. Other features of (10.11.6) are unaltered (e.g. Haga 1967). G(w) can also be calculated for complicated non-parabolic bands. The main alteration is that the electron density of states will have structure corresponding to critical points (§ 2.2.2). This structure will be especially important when several conduction bands overlap (Bassani, Iadonisi, & Preziosi 1969). It should be observed in photoionization unless there is phonon-broadening, unless the matrix elements A(k) vanish for symmetry reasons, or unless there are competing processes which obscure it. G(w) is not directly proportional to the density of states, however. There is an additional dependence on energy through A(k) which is often important. Allen (1969) has stressed this point for defects in valence crystals. He observes that for sp>-hybridized systems the energy dependence of the Bloch functions may alter the line-shape significantly.

10.11.3. Dependence on the defect wavefunction We now consider the energy dependence of G(w), which depends on the details of the envelope functions for the initial and final states |i> and |k>. This arises from the matrix element A(k) = , and is conveniently discussed separately for the two states.

A. Dependence on initial state. To make the trends clear, we shall continue to use simplifications 2 and 3 for the band states |k>, and merely introduce other approximations in |i>. The initial state is assumed to be described by effective-mass theory and associated with band i. Band i is distinct from the conduction band c since it is not consistent to assume that a ground-state |i) is principally derived from the conduction band and yet that the conduction-band states |k> near threshold are unaffected by the defect. Thus we distinguish between the defect binding energy E,, measured from band i, and the ionization energy E,; measured from band c. There are also distinct effective masses. The wavefunction we assume has the form i> = or)u,(k,,, ©) exp (ik,, .r),

(10.11.9)

where we shall assume the band extremum k,, to be at the centre of

OPTICAL

PROPERTIES

335

the zone. Written in full, the matrix element becomes

A(k) = { dr ,(r)u{0, rp exp (ik . r)u,(k, r)

- | dr o,(r) e**{u(0, r)(p+ hk)u,(k, 1)} ~ (2n)*${—k)p;.(k).

(10.11.10)

In the last step, we have assumed @(r) varies slowly over the unit cell and used the results of §4.2.7. To much the same accuracy the momentum matrix element is constant in the region of the Brillouin zone for which @(—k) is appreciable (e.g. Kane 1957), so that we may use

A(k) = (22)°6,(—k)p;,(0).

(10.11.11)

The matrix element is roughly proportional to the Fourier transform of the envelope function, with a proportionality constant depending only on the host. One approach which shows the variation of $k) with system clearly is the quantum-defect method (§ 4.3). This, and its analogues (e.g. Adelman 1972), also exploit the feature that the long-range tail of ¢,r) is important because this part overlaps most strongly with the continuum states. Here ¢,(r) is a function of an effective

quantum

number

v = (R,/E,)?, where Ro = m*¥ e*/2he” is the

effective Rydberg. The states change with v from Coulombic (v = 1) to those for a delta-function potential (v > 0). The transforms can

be written in terms of a* = h7¢/e?m*, the effective Bohr radius

jn. Al? ~ |w. p;(0)|? . 47

2?*(va*)> sin? (v+ 1) tan~ '(vka*) (vka*)?{1+(vka*)?}"*? (10.11.12)

In studying the line-shape, it is useful to introduce X, defined by

X = (vka*)? = 2m*(hw— E,)/mF Eg.

(10.11.13)

The line-shape function can be written in a form which separates out all dependence on the potential S(v, X) *

G(w) = 256n?h? In - pie(0)|?

S(v, X) =

I

ho

0

2?y? sin?{( 1)v+ tan”? JX}

TG Kena em taz

10.11.14

(

(10.11.15)

)

336

OPTICAL

PROPERTIES

When the defect is hydrogenic, v tends to 1 and S(v, X) becomes

S(1,X) = X#/(1+.X)*.

(10.11.16)

This result is appropriate for the ionization of a simple acceptor, and is equivalent to forms given by Eagles (1960), Dumke (1963), and others. The opposite extreme, v close to zero, is appropriate for a potential for which there is no long-range Coulomb tail. The result in this limit

ay

St, X)20

Ve Ee Bw 1X eStas

(JOR)

exp

is equivalent to one given earlier by Lucovsky (1965). The main trend with v can be seen from Fig. 10.11. As the potential becomes less Coulombic, the shape G(w) spreads out and the peak photoionization rate moves to higher energies. The quantumdefect method interpolates between the two extremes. Remarkably, the delta-function limit (10.11.17) often appears to give a better description of the behaviour of simple acceptors (Messenger & Blakemore 1971). The reasons for this are not clear, although Dow, Smith, & Lederman (1973) have observed that microfields from

phonons or random impurities can affect the threshold behaviour.

Delta-function limit

o nN

Hydrogenic limit

S(v,X)/S absorption Normalized max aeie 0-0

O72

ere LL Teeeal 2)04 80:6 F0-8OnTl

Ee

18 Omi

1-4

Fic. 10.11. Dependence of the photoionization cross-section on initial state. Results from the quantum-defect calculations of Bebb show trends as a function of y.

OPTICAL

PROPERTIES

337

Several other forms of initial state have also been tried. Berezin (1972) has discussed a state which is the ground state of the Hiilthen potential. This potential varies as exp (—ar)/{1—exp(—br)}, which is Coulombic (~—1/ar) at small r but falls off exponentially (~exp (—ar)) at larger distances ; it can be considered an extended delta function. It is clearly very good for systems like H~ solids, although it is slightly less good for the F’-centre (two electrons at an anion vacancy) for which Berezin makes detailed application. For molecules (e.g. Sharma 1967; Hernandez 1968; Johnson & Rice 1969; Kaplan & Markin 1969), the effective-mass approximation (10.11.9) is avoided and a linear combination of atomic orbitals used for |i}. One remarkable effect is that interference may occur in A(k) because of the different photon phases at the different sites (Johnson & Rice; Kaplan & Markin) to produce novel variations with angle and energy. B. Dependence on the excited state. Two important changes can be made in the envelope function defining the final state, previously taken to have the plane-wave form exp(ik.r). The first is most important when the ionized defect has no net charge. One then uses envelope functions appropriate to some short-range potential. The second change includes the effects of long-range Coulomb fields when the ionized defect has a net charge. In transitions (X~ + X°+e), the interaction of the released electron with the ionized defect X° can often be represented by a square-well potential of depth V and radius a. The remaining longrange terms usually vary as r * from the different polarizability of X° to that of the host, but are generally negligible. A square-well perturbation is reasonable, for example, for the F’-centre in alkali

halides, transition-metal ions in ionic crystals, and trapped electrons in metal-ammonia solutions. The potential has resonances, and these affect the line-shape. The resonances can be seen in several ways. One may examine detailed calculations (e.g. McVoy, Heller, & Bolsterli 1967 ;Kajiwara, Funabashi, & Naleway 1972), or simply examine the phase shifts for the p-states, which dominate in the optical absorption from an s-like ground state. Schiff (1949, eqn (19.27)) shows that this phase shift passes through an odd multiple of m/2 when the electron kinetic energy (hw — E)) satisfies

hw —E, = (nn)?

2

2m*a?

A

(10.11.18)

Cross-section

0-2 0-4 0-6 0-8 Transition energy (eV)

1-0

(b)

Cross-section

0-0

IS Transition

oa

T re

oH

energy (eV)

Fic. 10.12. Dependence of the photoionization cross-section on final state. Calculations of Kajiwara et al. contrast results for plane-wave final states (I) with those (II)

including the eflect of a square-well potential of radius 5 A and depths (a) 0-564 eV, (b) 0-939 eV.

OPTICAL

PROPERTIES

339

Clearly the resonances are associated with situations where the electron wavelength bears a simple relation to the well radius. However, the effects of the resonances can be seen at much lower

energies. This is shown in Fig. 10.12, which compares photoionization cross-sections using plane-wave final states with those using correct eigenfunctions of the effective-mass equation. There is little difference between the two cases for a shallow well. But, for a

deeper well, the exact functions lead to a narrower more-symmetric peak at slightly lower energies. The differences are considerable, and would affect values of derived parameters. Jaros (1972) has verified modifications of this sort in pseudopotential calculations for Si:Zn~. He notes that the relation between the energy above threshold of the peak in the photoionization cross-section (hw,,—E,) and E, depends on the details of the model. In the quantum-defect model, (hw,, — E;) is proportional to E,, the constant depending on v. But, in other cases, hw,, may be near threshold whilst E, is small; the modifications depend on the relative sizes of the short- and longrange parts of the potential. More often, the photoionization leaves the defect in a charged final state (e.g. X° + X* +e). The Coulomb potential in the final state modifies the eigenstates considerably. However, for a purelyCoulomb potential there is no extra structure produced. Thus it may be difficult in practice to detect the characteristic features. Most striking is the fact that there is a sharp discontinuity in G(q) at threshold, and not merely the discontinuity in gradient predicted by (10.11.6). Bethe & Salpeter (their eqn (71.7)) show that the absorption of a hydrogen atom is proportional to se -1 Y) fa4 exp(—4Ycot™' ho} 1—exp(—2zxY) ’

(10.11.19)

where Y? = 2/X is the ratio of the ionization energy to the electron kinetic energy. Immediately above threshold, this factor varies as

(hw/E,)*, and well above threshold as (hw/E,)*. When there are both short-range and Coulomb terms, structure appears because of resonances, as before. This is clearly shown by Page, Strozier, & Hygh’s (1968) treatment of the L bands of the F-centre. They find that the peaks just above threshold arise from the resonant behaviour of the matrix element A, whereas structure in the electronic density of states is responsible for the peaks at higher energies (see

340

OPTICAL

PROPERTIES

Fig. 15.5). Their results give satisfactory agreement with experimental L bands in KCl. 10.11.4. Photoionization with phonon assistance There is no adequate theory in this case, and we shall be content with describing the main features. We begin by describing transitions well above threshold. Transitions to any individual continuum state occur with the absorption and emission of phonons. Instead of a sharp delta-function absorption to this state, a distribution proportional to G(w)/a is obtained. If there is no interference between the contributions of the different states, the final line-shape from all transitions is given by a convolution

a(w) ~ {da’ G(w')G(w — a’)/a’.

(10.11.20)

The structure in G(q@) from critical points will probably be smeared out and hard to observe. Phonon assistance also modifies processes near threshold. There are two main classes of process. In one, there is a single transition which involves both photon absorption and phonon absorption or emission. This corresponds to (10.11.20). Alternatively, there may be a two-step process: first the absorption of a photon and then a separate transition involving phonons. Both transitions are met in practice. Single-step transitions occur when the no-phonon transition is forbidden. The recombination of an indirect bound exciton weakly localized near a defect is an example. If the conduction band and valence band extrema are at points of the zone differing by k,,, then the crystal momentum selection rule can be satisfied by the absorption or emission of a single ‘momentum-conserving’ phonon with wavevector k,,. The energy of the phonon hw,, can be found from the phonon dispersion curves of the host. Kogan & Sedunov (1967) have argued that two-step transitions dominate in the photothermal ionization of simple defects with weak phonon coupling. In photothermal ionization, the photon energy is too low to produce ionization by itself; phonons are also absorbed to give ionization. Clearly, the process vanishes at zero temperature. One expects the probability to be greatest when the photon energy coincides with one of the bound—bound transitions

OPTICAL

PROPERTIES

341

of the defect. The reason why the two-step process dominates when the phonon coupling (measured by a Huang—Rhys factor S) is small can be seen as follows. For the one step process, photothermal ionization occurs for a fraction of order S of the transitions induced by photon coupling to a given excited bound state. This is the content of our earlier (10.7.30). Thus the cross-section a; is of order 7,5: But for the two-step process, the defect makes a direct transition with cross-section o,,,. The subsequent non-radiative transition may be to a higher or lower state, but the fractional probabilities are independent of S. Thus oj is of order a,,, f,, where f; is the fraction of processes which give ionization. For sufficiently weak coupling o, will always exceed o,. Analyses of experiments on photothermal ionization are given by Tasch & Sah (1970) and Sah, Ning, Rosier, & Forbes (1971). Herman & Sah (1973) appear to have observed the two-step process in Si:Zn. Problems may occur in practice because of impurity banding among the higher excited states, and can lead to discrepancies between purely optical and thermal results, for reasons quite distinct from those associated with the Franck—Condon principle (see also § 8.4; Stillman, Wolfe, & Dimmock 1971).

1] DYNAMICS

OF

IMPERFECT

LATTICES

11.1. Introduction

THE presence of a defect in a lattice affects the phonon modes and their frequencies. In this chapter we discuss those phenomena which result from the modification of the lattice dynamics by the imperfection. The two most important types of modification lead to the appearance of local modes and in-band resonances. In both cases the amplitude of the mode near the defect is strongly perturbed. Local modes occur most commonly for light imperfections like the Ucentre (§ 20.3), giving mode frequencies higher than those for perfect crystals. Genuinely local modes, in which the amplitude of vibration falls rapidly to zero away from the defect, can also occur at frequencies within gaps in the perfect crystal spectrum. Such gaps appear when there are several atoms per unit cell, and the masses of the different atomic species are widely different. In-band resonances occur at frequencies within the perfect crystal spectrum. They occur for heavy atom impurities and for defects which are only weakly coupled to the host. Defects which can reorient (see chapter 21) have low-lying energy levels and often produce resonances. Indeed, resonances provide a useful tool for studying such centres, especially when spin resonance is not appropriate. Since the resonances occur within the perfect crystal spectrum, the most profound observed effects occur for very low-lying frequency resonances, where the phonon density of states of the perfect crystal is low. In this limit the perfect lattice modes are most like those of an elastic continuum. Fig. 11.1 shows the various forms of local mode and

resonance. The modification of the modes shows up particularly clearly in infrared absorption, in optical absorption, which we discussed earlier, and in thermal conductivity. Other phenomena, such as observations of the heat capacity, X-ray or neutron scattering, the Mossbauer

effect,

or

spin-lattice

relaxation

can

be useful

but

DYNAMICS

OF

IMPERFECT

LATTICES

343

energy

Gap local

mode

Resonance

Light mass, high forceconstant impurity

Perfect crystal

Heavy mass, low forceconstant impurity

Fic. 11.1. Local and resonant modes. The schematic diagram shows various possibilities for different impurity masses and force constants. The resonance is shown broadened because a range of modes is affected.

usually are of more limited value. We shall discuss later the connection between lattice dynamics and observed properties. Two of the several methods for obtaining defect lattice properties have become widely used (Krumhansl (1963) and Pryce (1969) describe some others). The first is the method of classical Green’s functions, which gives the perturbed modes and their frequencies directly. It is particularly useful in giving the orthodox insights into the frequencies and the behaviour of the amplitude of a mode near the defect. The second method is that of double-time thermal Green’s functions, as pioneered by Zubarev (1960). In this approach thermally-averaged correlation functions are calculated. For example, if u(l, t) and p(l, t) are the displacement and momentum of ion I, then we may calculate , , and . The brackets < > represent a thermal average. The

advantages of this second approach, apart from certain purely technical ones, are, first, that the temperature dependence of these correlation

functions

is automatically

treated

and, second,

that

the correlation functions are very closely related to observed quantities. This second advantage may be decisive. It is sometimes difficult to derive observables from the perturbed modes, quite apart from any loss of elegance in doing so.

344

DYNAMICS

OF

IMPERFECT

LATTICES

Throughout this chapter we shall represent the defect by changes of atomic mass and interatomic force constant. Special treatments are needed in the rare cases that other models are necessary (e.g. § 8.6.6). The harmonic approximation will be used almost exclusively ; cases where more complicated approaches are needed, as for the Jahn-Teller affect, are not discussed. The changes in mass are given

by M,()) > Mil) = MD

{1—e()}

(11.1.1)

and in force constant by

ALD) > Abt) = Ald Aare):

(itt)

following the notation of chapter 3 for the perfect lattice. The Heisenberg equation of motion for the imperfect lattice becomes

Mii) +d) Aggll, 'ug() = AM ,(Dii,(l)+ Bl

+ YAMA(LMug(t).

(11.1.3)

BL

In this equation the u,(l) are the displacements from the mean position of a particular ion. The equations do not contain the static distortion near a defect. Such distortions may be implicit in the changes in force constant, but it is not necessary to include them explicitly unless we go beyond the harmonic approximation. We shall only consider dilute solutions of defects, and obtain properties of interest from solution of the isolated impurity problem. The observable changes in macroscopic properties are then superpositions of the contributions of the individual defects and are linear in the defect concentration. To discuss terms of higher order in the concentration we need to estimate the way in which one defect affects the contribution of another to some observable property. One must solve the defect pair problem to obtain terms quadratic in the concentration, representing interference of the defects with each other. The terms cubic in the concentration are found by considering the three-defect problem, and so on. Local modes also occur in magnetic systems, where one is concerned with a defect with different magnetic properties from the host atoms. This case can be simpler than the defect lattice dynamics we consider here, since the magnetic exchange interaction is of very

DYNAMICS

OF

IMPERFECT

LATTICES

345

short range. Cowley & Buyers (1972) have reviewed the theory for magnetic systems with defects. 11.2. Comparison with electronic defect systems

The Green’s function methods which we shall use resemble those used in the calculation of electronic properties (§§ 5.2 and 5.3). There is some similarity in the mathematics and in the simpler manifestations, such as the existence of resonances and local states.

The source of this similarity can be seen from Table 11.1, where we compare the equations in the two cases using both a local basis and a running-wave basis. Despite these similarities, there are significant differences. For lattice dynamics we constrain the atoms to remain centred on specific sites, and omit the diffusive modes. In effect this gives a finite number

of bands. For electrons there is no such constraint,

and the admixtures of higher bands by strong perturbations cause real problems. We may also examine the limits of validity of the methods. In lattice dynamics the harmonic approximation sets the limit. In electron dynamics the limit is that the electron—electron interaction terms are not changed appreciably, so that the effective potential can be independent of the defect-electron wavefunction. The conditions for validity of the harmonic approximation for lattice dynamics are much less restrictive than those in the electronic case. This is especially clear in the simplest cases: the isotropic defect, with large AM and no force constant change, is a soluble problem of great practical value; the Koster—Slater model, whilst pedagogically admirable, is rarely of practical use. 11.3. Defects in a linear chain This, the simplest of defect lattice problems, is exactly soluble.

It gives qualitative insight into the defects of various sorts of defects but, regrettably, cannot be generalized to two or three dimensions. For real systems, a Green’s function method is necessary. Derivations of the one-dimensional results are given elsewhere (e.g. Kittel 1966; Pryce 1969) and will not be reproduced here. But the results

are of interest and will be quoted. Consider a linear chain of masses M coupled by nearest-neighbour springs with force constant A and equilibrium length a. The frequency spectrum

of the perfect lattice varies as sin (qa/2), with a

:b b ae

pueq :u

=p (b)"g }dxa — br

(4 ‘b)“f

Ls

JOWSAIARM

I

(1 “bi — )dxo

‘1)"2(1

{(1—D*

tb

1

(b)0 ik

jeo1ueudxljew p

(Peet

juguoduI0d

UeISa}IvO

7m = (b)40(b)*"q

SSC O

I

JOJOOAZARM yourlq

IMPERFECT

A

:())’n= =

Gg =

0:

d

suena Kak

:D :f

(b);o(b)

uonenby ‘paysnes [290] siseq

— 1

1516 (11.5.16)

Using the identity for the positive infinitesimal o

1 ae x+io

1 |= — 271 0(x) x-—10

Cheeta)

it is readily verified that

J 4—(@) = port

(As Bs), — «A; B;wy,). (11.5.18)

Having expressed the spectral density in terms of the Green’s functions, we now express the correlation functions in terms of the spectral density. It is seen easily from the definition (11.5.14) that

354

DYNAMICS

OF

IMPERFECT

LATTICES

the correlation functions are basically Fourier transforms of J4,(),

the spectral density

(A(t), Bt) = lp doo Jya(co) exp {—iat — ¢)} ). Ted

a

(11.11.16)

This expression may be simplified by using the relations between the displacements at individual sites |L> and in normal modes |\q>. The T matrix has only one non-vanishing matrix in the site representation and G is diagonal in the mode representation. After simplification, the first term in 1/t, becomes

AM 2 caoyr (Se) M

(11.11.17)

Wg

which vanishes with 0. The second term is more complicated, and becomes Q (AM\?

Im eI

KalODI? 21" x

4 2 7) oe (“8 — 07 — 240i

Wa — Wa +i0

\

(11.11.18)

We may transform the sum into an integral, introducing a phonon density of states. The term which remains when 0 is zero is just

Q/AM

al || 23a a\Mo

|dx?) pol? 1G |O>|? OF x

x d(we— x’).

(11.11.19)

Since ||? = |o2(q)|?/N, and since at low frequencies p,(w?) is A w/w?,,,, where A is a constant and w,,,, is the maximum fre-

quency, the result becomes

Liven T

.

aon 4(q)|7|o(@,)|? , AM\2|[35 PAGAC lal>4) ~

Na

M

(11.11.20)

(60) max

where the second factor in the polarization vectors a is the average oc modes with frequency w,. The result shows the well-known wa and (AM)* dependence of Rayleigh scatter. 11.11.3. The peak theorem Thermal conductivity experiments depend on an average of the cn We now calculate this average, and derive a theorem due to Klein

DYNAMICS

OF IMPERFECT

LATTICES

387

(1969). The relaxation rate when there are N, defects per unit volume is

N,Q

— = Tq

— $m (q\T(w2 +io}|q)

(11.11.21)

Dq

for mode q. The average relaxation rate for all phonons of frequency w is then

ad = » Hw—ogych /{5Hoo} (2) tT

q

(11.11.22)

q

It is this relaxation time which is measured (albeit indirectly) in thermal conductivity experiments. We begin by simplifying (11.11.18) using the definition of G, which we express as

G(w? +io) = >q a y, the probability of pure phonon transitions is much higher than that of radiative emission. Apart possibly from special behaviour very soon after excitation, one expects the equilibrium luminescence of § 10.7.2. The transition probabilities W(E’, E”) approximately factorize into one factor G,(E’). for absorption, and another G,(E”), for emission. More interesting cases occur for large emission energies E” > E, at low temperatures, since the hot luminescence and Raman contributions will dominate. Detailed calculations have been done for different classes of system and show interesting differences between those with small a and B (Hizhnyakov & Tehver) and those with large values (Rebane et al.). For small « and f, as for KCl: NO;, one expects hot luminescence and conventional luminescence to dominate over the Raman contribution, provided the exciting light has a bandwidth broader than I. Hot luminescence has been reported for NO; and OH ions in KCl (Rebane, Saari, & Mauring 1972). For large « and f, as for the F-centre in alkali halides, the Raman terms dominate for excitation by highly monochromatic laser light. Fitchen & Buchenauer (1972) have observed resonant Raman scattering, including spectra up to fourth order for NaBr. Their results verify several of the predictions. Thus the integrated intensity of the first-order spectra agrees with the expected dependence on the position of the laser line relative to the F absorption band. The relative intensities ofthe different orders of Raman spectra show the anticipated dependence on order for NaBr. Fitchen & Buchenauer also compare their spectra with the projected phonon density of states. For NaBr, many of the details

402

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

agree well if allowance is made for a reduction in the local force constants. There are, however, some discrepancies in the longitudinal optic region which become more pronounced at higher excitation energies where the K band is more important. These differences are probably associated with the nature of the electron—lattice coupling, which becomes closer to the Frohlich form for the more spread-out states corresponding to the K band. We shall consider moment analyses of Raman spectra in § 12.5.3, and discuss in chapter 14 the non-radiative processes preceding conventional luminescence. 12.3. Magnetic fields: the Zeeman effect 12.3.1. Introduction The perturbation from an applied magnetic field can be derived from the Dirac equation fairly readily. The method is given in detail elsewhere (Foldy & Wouthuysen 1950; Frosch & Foley 1952; Griffith 1961) and will not be repeated here. The main effect is to change the momentum operator in the Schrédinger equation po 1 =p+eA/c,

(12.3.1)

affecting especially the kinetic energy and spin-orbit coupling terms. The vector potential A is related to the magnetic field H

by A =4H A r. We adopt the Coulomb gauge, with

V. A = 0, so

the scalar potential in the Schrodinger equation is determined from the static charge distribution. The Zeeman terms for a single electron are 2

#, = BA.(L+g.S)+ teal A rl?+5-Etr)(S A r).(HA of. (12.372) In the cases of interest the contributions of additional electrons simply superimpose. The first term usually dominates, giving paramagnetism. The spin and orbital angular momenta are S and L, and g, = 2+4a/z ~ 2.0023 is the electronic g factor; « is the finestructure constant. The second pair of terms is usually negligible and comprises a diamagnetic term and the field-dependent spin—orbit coupling. They may often be ignored.

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

403

Clearly the Zeeman terms present more problems than the Stark terms. There are three main reasons. First, results must be gaugeinvariant. Consequently some small higher-order terms must be retained in perturbation theory. Secondly, the orbital angular momentum L plays a prominent role. It is not a good quantum number in a solid, where the potential V(r) does not have spherical (or even axial) symmetry. This leads to ‘quenching’ of orbital angular momentum. Thirdly, the spin-orbit coupling is important. Moreover

€(r), which describes this, is dominated

by the regions

close to nuclei, and it is common to have a many-centre problem. This many-centre feature can make it hard to retain the desired gauge-invariance. We discuss these three difficulties later in connection with spin-resonance. The Zeeman effect provides a wide range of physically interesting phenomena, ranging from the splitting of optical transitions between orbitally degenerate states (typically AL, = +1, AS, = 0) and of spin states, observed in spin resonance (typically AL, = 0, AS, = +1) to the various magneto-optic phenomena such as Faraday rotation. 12.3.2. Zeeman effect in atoms and effective-mass systems We begin by discussing the Zeeman effect for atoms. This simplifies some of the important features and can be taken over directly to certain defect systems. These systems are ones which are adequately described by effective-mass theory, where we can concentrate on the defect potential which binds the electron rather than on the total potential of all crystal atoms. If the defect potential has spherical symmetry (e.g. — Ze*/er), the eigenstates may be classified by angular momenta S, L, J, and J,. The orbital parts of the angular momenta

strictly refer to the envelope wavefunction, rather than the total wavefunction, of such a system. A. Low-and high-field limits. Twoextreme cases may be distinguished, depending on the relative sizes of the spin-orbit coupling and the Zeeman energy. When the Zeeman energy is large it mixes the states of different J derived from particular L and S ; when the Zeeman energy is small we may concentrate on a single value of J. We begin by discussing the weak-field limit, and anticipate later developments by writing the paramagnetic term of (12.3.2) as

#', = BH. (yL+g.S).

(12.3.3)

404

EXTERNAL

FIELDS

AND

THEIR

The analogue of the Landé formula S, L, J, M,, splits up into components

holds:

EFFECTS

the multiplet with

AE = gL Ss, J)pHd-,

(12.3.4)

where the spectroscopic splitting factor is

a(S)

=) (85-7)



Tce

—_—*Y

sisee bt)

IIT +N)

(

125325

)

For a hydrogenic defect the ground state has parameters

L=0,S

=4,J =4

Geencee

(12.3.6)

The excited states derived from the 2p-state are

Quartet (L = 1,5 =4,J = 3)

g = 47-42,

Doublet (L = 1,8 =3,J =41

g = 4yt+4e,.

~—«(12.3.7)

(12.3.8)

The simplest way in which a reduction factor can appear is for the effective mass of the electron to differ from the free mass. Only the orbital term is affected, since it involves the momentum

directly.

In this case y = mo/m* and y may exceed unity. Reduction factors appear in other ways, discussed later. In the strong-field limit we must decide which J states will be admixed. For the hydrogenic centre described above the two multiplets derived from the 2p-state (J = 3 and J = 4) are separated by the spin—orbit coupling and would be mixed by a magnetic field. When the field is strong it is more convenient to work in the L, S,

L,, S, representation, when it is readily shown that the transitions AS, = +1 have a splitting factor g, and the transitions AL, = +1 a splitting factor y. For intermediate values, when the Zeeman energy and spin-orbit coupling are comparable, one must solve the full secular equation for both the interactions in the secular equation of order (2L+ 1)(2S+1). The results, shown in Fig. 12.3, are analogous to the Paschen—Back effect in atomic spectra. The small-field limit is observed for a number of rare-earth ions, and the high-field limit for certain excited states of defects like donors in GaAs.

B. Ground states of donors and acceptors. We now consider the Zeeman effect in the ground (1s) states of donors. The wavefunctions in these cases may be so extended in space that the splitting parameters are mainly determined by the band structure of the host

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

405

Strong field

Weak field

| |

| ! i}

j=)

| 1

NEN NeN/V)

1 !

i}

| | ! 1

|

i | 1

i i} | | |

( | | i} i}

{ | | i | | | i

;

| '

alee [02>

|—1,—5>

Fic. 12.3. High- and low-field behaviour of a p-state. The figure is drawn for a spin— orbit coupling A > 0 and for y < g,/2. In the high-field limit the states |L,,S,> are given; the corresponding energies are BH(g.S,+yL,).

lattice. For an s-state (L = 0) there is no direct orbital contribution to the magnetic moment, but the spin-orbit coupling causes deviations from the free-spin g-value. Calculations of g-factors for band states are given by Roth (1960), Liu (1961), and reviewed by Yafet (1963). The most useful result for present purposes is derived by Kittel (1964, p. 282) who treats the simple system shown in Fig. 2.4. The g-factor for an electron in the conduction band is given

in terms of the band gap E, = (E,o—£E,), and spin-orbit splitting of the valence band A by ex

2A El

e-|

os i). m*

(12.3.9)

The correction to the free-electron value may be large, since m* can be small; in InSb, for instance, g* ~

—50.

This result is valid for a crystal with at least tetrahedral symmetry and with the conduction-band minimum at the centre of the Brillouin zone. Many important systems have minima at other points in the zone (e.g. Si and Ge) or are anisotropic (e.g. wurtzite CdS). Consequently, both the effective mass m* and g-factor g* are tensors. We discuss the simplest situations, where g* can be specified by two components g, and g,. The Zeeman interaction within one valley

406

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

of a multivalleyed system (e.g. Si or Ge) or within the single conduction band of an anisotropic system can be written

Ho! =6.9.HB68,

(12.3.10)

where o is a Pauli spin vector. Choosing principal axes

8|

= #g) +2g,)1+(g) —g.)(nn—41),

(12.3.11)

where n is a unit vector along the valley’s axis of symmetry. When transitions in just one valley are observed, the spectroscopic splitting factor is

g* = ,/(g] cos” 6+ sin’ 6),

(12.3.12)

where H.n = H cos@. This expression is appropriate for donors in anisotropic crystals. A second case is appropriate for a multivalleyed system with cubic symmetry over all. The defect wavefunction in this case is a linear combination of products of envelope functions and the band functions at the minima

w=

>

a,Fi(r)u({r).

(12.3.13)

minima i

The expectation value of #°'! p z is easily W shown to be o.

P a? {3g +22 ,)+(g; ~8,)(an—4)} |. BH,

using the slow space-variation of the envelope functions F, and the orthogonality of the u;. In cubic symmetry the second term sums to zero over the minima, so that

g* = (g, +22,)/3 is the isotropic g-factor in this case. Expressions for g|, and g, can be derived analogous to (12.3.9), such as

&\ = Be

2 Alm 3 z(t}

2 Ajmo 81 == &e 2g.+~>—|—-1 3 ae .

(12.3.14) (12532)

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

407

for silicon, where E, is the indirect gap. In practice, there are always slight deviations of g* from the values predicted from unbound electron data. These are associated with the finite radius of the envelope function and other small corrections. Corresponding results can be obtained for acceptors. The complication here is that the valence band is degenerate, although there will be spin-orbit splittings and possibly splittings from non-cubic terms in the potential. In the absence of these splittings there is an orbital angular momentumL = |associated with the band functions, as opposed to the envelope function. Luttinger (1956) has shown how Zeeman terms can be treated for unbound holes in a valence band. The extension to acceptor states is straightforward in principle, if not in practice, and is given by Bir, Butikov, & Pikus (1963), Suzuki, Okazaki, & Hasegawa (1964), Butikov (1969), and by

Lin-Ching & Wallis (1969). C. Excited p-states of donors. The Zeeman effect in the ground state of a simple donor is usually observed in spin resonance, when AS, = +1. There is no orbital angular momentum, apart from the small corrections coming from the spin-orbit coupling. The Zeeman effect in the first excited state (2p) of a donor is usually observed optically (AL, = 0, +1; AS, = 0), so that it is the orbital Zeeman energy which is important and the spin terms which may be ignored. Thus, if the orbital Zeeman energy is strongly quenched (y ~ 0) no Zeeman splitting will be observed. We shall ignore spin-orbit coupling in our discussion. At sufficiently high magnetic fields, the optical Zeeman effect can be related to cyclotron resonance. In discussing the Zeeman effect in the excited 2p-state of a donor, we introduce the cyclotron resonance frequency @,.=eH/m*c

(12.3.16)

in addition to the effective Rydberg

Ro = m*e*/e7h?.

(12.3.17)

The dimensionless parameter I’, defined by

T = ho,/2Ro,

(12.3.18)

408

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

is small at low fields and large at high fields. The effective-mass Hamiltonian has the form Zz

=

sat V(r) +4ho(xp, — yp,) + 3moz(x? + y’) m

(12.3.19)

for a field along the z-axis of a cubic crystal. To simplify the treatment at high fields we put W in a dimensionless form, with the unit of length

(cyclotron radius)/,/2 = \/(h/2m*a,)

(12.3.20)

and unit of energy hw,. The Hamiltonian becomes

KH =ho(Ho+%)).

12325)

The terms independent of V(r) are taken into %

Hy Ho

Cran CAO =|-—+-=

ae 20m

| Sp Oban ptm

1

Lc

p

tae

a)12322 )

where p = A (x? + y’). This is the Hamiltonian of a two-dimensional harmonic oscillator. The attractive potential V(r) gives 7, ,

2

|

At low fields the term linear in ha, in (12.3.19) can be treated by perturbation theory. We have assumed no spin-orbit coupling, so the high-field theory discussed earlier applies. The splitting is then described by a g-factor g* = y, in which y is the reduction factor of(12.3.3). At the highest fields, #% dominates and #, is a relatively small perturbation, whose main effect is to localize the defect electron near z = 0. To the lowest order we need only recognize the presence of V(r) by ignoring the translational motion of the electron along

the z-axis, so that the term h*k?/2m* for an unbound electron should be dropped. The eigenvalues of % are essentially those of a twodimensional harmonic oscillator. The lowest states may be writtent (N, L,), where

HN, L.) = (N +3)A(N, L,).

(12.3.24)

+ Various notations appear in the literature. Hasegawa & Howard (1962) use the same as Larsen (1968). Wallis & Bowlden use (I,m), where N = 1+4(\m|+m) and L, = m; their notation is more useful in the low-field limit.

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

409

This classification is not affected by the +L, term in %. Thus the lowest states with L, = 0 or +1 are

(0, 0) = exp (—3p”) ¢(0, —1) = pexp(—gp*/—id) (1, 1) = pexp(—sp’ +id)

(E = ha,/2) (E = hw,/2)

>.

(12.3.25)

(E = 3hw,/2)

These states have a Gaussian dependence on p, unlike the hydrogenic forms appropriate at low fields. In the large-field limit the solutions of (#) +#,) are separable, in the sense

(p, z) = $(p)G(z).

(12.3.26)

Here #(p) is one of the ¢(N, L,), and G(z) is a solution of a onedimensional Schrédinger equation with a potential (Loudon 1959; Haines & Roberts 1969) V(z) = [as idy |f(x, y)17 V(x, y, z).

@2327))

The eigenvalue of % differs from (N +4)hm, by the eigenvalue of this one-dimensional equation involving the small term V(r). At intermediate fields a variational calculation is needed; exact solutions are not available. Larsen (1968) showed that convenient variational functions are

ESR

exp |— p?/8a? — K,/(p? + az?)}

L. 2 = +1:

ee

v4

ee

|

ek

y

(CAB 2QE

| (

)

in which the four parameters a, K, «, and f may be varied independently. These trial functions have the correct behaviour at both small and large fields. They give a very accurate description of donors in GaAs (Stillman, Wolfe, & Dimmock 1969) and CdTe (Cohn, Larsen, & Lax

1970), as shown

in Fig. 12.4. The actual value of

T = haw,/R at which the hydrogenic low-field wavefunctions go over into Gaussian high-field functions depends on the system. Horii & Nisida (1970, 1971) cite F ~ 0-15 for donors in Ge. The intermediate-field results can be expressed conveniently in a number of ways. The simplest is to calculate an ionization energy EF, defined as the difference between the eigenvalues with and without V(r). The differences in ionization energy can be related, if desired, to the splitting of lines in optical absorption. Table 12.1 gives values

EXTERNAL

410

FIELDS

EFFECTS

THEIR

AND

80

E 6

oe

2

8

(a)

g i

(1s-=2p, m= +1) 50

40

(ls 2p,

=)

LIS

aL

m=-—1)

ei

eS ek

Magnetic field (kG)

=

240

eh

200}

E sot Saad Oe

5

ee

3 AD a

(b)

ea (Is--2p,m

LO phonon _ =+

1)

140

(ls—e2p, m= —1)

0

50

100 150 Magnetic field (kG)

200

250

Fic. 12.4. Zeeman effect for donors. In (a) Larsen’s theory for zero electron-phonon coupling is compared with Stillman, Wolfe, & Dimmock’s results for GaAs. In (b) Larsen’s calculations for « = 0-4 are compared with data of Cohn for CdTe.

EXTERNAL

FIELDS

AND

TABLE

THEIR

EFFECTS

411

12.1

Ionization energies for a simple donor in a magnetic field, after Larsen (1968); the energies are in units of the Rydberg R for zero magnetic field, and the field is determined by T = ha,/2R

r

E\L, = 0)

E(L, = 1)

0-0 03 0-5 0-7 1-0 15 2.0 25 3-0 5-0 25.0 100-0

1.000 1-26 139 1-51 1-661 1-87 2:04 2.19 2:32 2-75 4.733 7-457

0.2500 EN; = = 0-9118 = = ae = L717 3-183 5-261

of ET) for L, = 0 and 1. A second description of the results is to concentrate on the energy

ACH) SEL =O) Hy Efi

A),

(12.3.29)

We return to the relation between cyclotron resonance and the Zeeman effect in optical absorption later. Larsen (1968) has also discussed the effects of a non-parabolic conduction band, as needed for InSb where the band gap is particularly narrow (Kane 1957; Bowers & Yafet 1959). The main effect is to reduce A(H) from the parabolic-band value at large fields. Further, Larsen has discussed the effects of the electron-phonon interaction. In the simplest cases the effective mass m* is changed to the polaron mass in both the kinetic energy and the cyclotron frequency. However, in some systems (e.g. CdTe) the energy difference between a Zeeman-split 2p level and the 1s level of the donor may be very close to the longitudinal-optic phonon energy (Cohn et al. 1970, 1972). In these cases the resonance must be treated in detail (Larsen 1970).

D. Relation to cyclotron resonance. Both cyclotron resonance and optical absorption may be induced by an electric field normal to the

412

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

magnetic field, and both involve transitions with AL, = +1 and AS, = 0. The difference is that cyclotron resonance corresponds to the extreme when V(r) is negligible. Since V(r) becomes less important as the field H and I become large, the optical absorption by a donor should tend to the cyclotron resonance spectrum of unbound electrons. We demonstrate this now. Hasegawa & Howard (1962) have examined the optical transition probabilities in the high-field limit T > 1, using wavefunctions factorized in the form (12.3.26). They show that the opticallyallowed transitions fall into three classes, whose intensities vary with I as follows

Class A: 1 ~ T° Class B: J ~ (InT)~>

Glass

F

Gul’ ~ tr(In) eerie

(12.3.30)

10

Only one transition occurs in Class A. As I becomes larger, this transition tends to hw,, and other lines fade out of existence. In this limit the energy levels form a Balmer series. The various classes of transition, their energies and intensities, are shown in Fig. 12.5. E. Donors in anisotropic crystals. There is no detailed theory for donors in anisotropic crystals other than in the low-field limit. However, it is useful to have results for a donor with small axial

Continuum

G

iSat eT

ee

ae ---4-

"Continuum

A|B|B|B

c

Seed | SA

Left-hand

Right-hand

circularly

circularly

polarized

polarized

Intensity

Fic. 12.5. Donor transitions and cyclotron resonance. Schematic diagram of the optical transitions of a hydrogenic defect in a magnetic field. As the field increases, transitions B and C become negligible, and transition A tends to energy ha,.

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

413

anisotropy (Henry & Nassau (1970), whose results are an extension of those of Wheeler & Dimmock (1962)). We define Eo = (m¥/éo 1£0) Ry,

(12.3.31)

and «, a measure of the anistropy, by Eo, mi

i

a

leer

(12:33:32)

Zon)

together with the auxiliary variables C=

omit ks,

(123333)

ag, = mito, /mF,

(12.3.34)

A = E,(2a+ 3a7)/40.

(1273335)

The energy levels with H along the symmetry axis are shifted by

(1s)

+ 39),BH+o,H?

(2s)

+ 4g)BH+140,H?

(2p,,2p,) (2p.)

+ 28 BH +PH(mo/m,)+ 120)H late

;

(12.3.36)

4g) BH +60,H?

where g, is the appropriate unbound-electron g-factor. When H is normal to the symmetry axis the shifts are

(1s) +3¢,BH+0,H?

(2s) +49, BH+140,H? BH +./(A?+20,H?)+120,H?| L2pe 2D, ) +4g,

(12.3.37)

(2p, ) +3¢,BH+60,H? 12.3.3. Linear Zeeman effect. general defects

In § 12.2.2 we saw that the nature of the splitting of an optical transition by an external field permitted a determination of the defect symmetry. We now discuss the analogous results for applied magnetic fields. Again the theory can be developed quite generally by symmetry arguments alone. We shall concentrate on defects in cubic crystals, where full tabulations are given by Archangel’skaya & Feofilov (1958), Feofilov & Kaplyanskii (1962), Zakharchenya & Rusanov (1964), Johnston et al. (1968), and Runciman (1969).

414

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

As before, there are two types of degeneracy : electronic degeneracy, as in the degeneracy of the 2p,-, 2p,-, and 2p,-states of a simple donor with cubic symmetry, and the orientational degeneracy of an anisotropic defect, which can adopt several equivalent orientations in a cubic crystal. Of course, defects may exhibit both types of degeneracy at the same time. The various possible electric-dipole transitions in cubic symmetry are listed in Table 12.2, together with the number of components into which each line splits for unpolarized light and different field directions.

TABLE 12.2 Twenty-one classes of Zeeman spectra Magnetic field direction Type of centre Electronic degeneracy 1. A,>T,,A,>T,

2, E>T,,E>T, Sh Iya IA 4.T, >T,,T, -T, SE ig al Pag PC 6.1, 7Ts 7. Do => Ws 8 Tg oT, Electronic + orientational degeneracy 9. Tetragonal ¢, A> E 10. Trigonal E 11. Tetragonal , E> E 12. Trigonal ¢111), E > E§ 13. Tetragonal ¢100)

14. Trigonal

Ceartess

SEND Aa2 (O|pila>

2

rea (12.5.21)

where the incident polarization and final polarization are i and j respectively. The total cross-section is

8 d Cro = ul (sal

(12.5.22)

ij

and is independent of i. Henry (1966) has discussed the total cross-section of simplifying approximations. First, the admixture by the lattice vibrations is ignored. Secondly, E, and to be close, so that |E; — E,| is less than the maximum

with a number of other states E, are assumed phonon energy

EXTERNAL

FIELDS

AND

THEIR

EFFECTS

435

(say E,,4x). Thirdly, E, and E, lie sufficiently far from the mean

transition energy E that (E

—E,) and (E—E,) are both much larger

than E,,,,. Finally, the electron—lattice coupling is linear. This last assumption is largely justified for the F-centre where the onephonon spectrum has been seen (Worlock & Porto 1965). Henry uses an expansion in {(E,—E,—E)/(E#£,)} based on the relation (E,—E,+£,)' = ((EFE)+(E,—E,—E)}~ I tesE\)+ (b| He. + 4o0t(A.— Evollb>}~

(12.5.23) The leading term gives Rayleigh scatter; the phonons are not involved. The second term gives the first-order Raman effect and is the one we want. Other terms give higher-order Raman contributions,

e.g. two-phonon spectra and corrections of order {E,,,,/(E—,)} to the one-phonon spectrum. The term of direct interest reduces to Gio On

EME? +ER)? key

= OED

fe F2/p2. p24 E(E-—EF)

(12.5.24)

where o; is (82/3)(e?/mc?)? = 0-665 x 107 ®A’, the Thomson crosssection for free-electron scatter, and f is the oscillator strength, including local-field corrections. The values of the moment ,...,|n> by appropriate modifications of the matrix elements. Pryce (1950) has shown that these modified matrix elements can be obtained by a perturbation treatment. The proof is given in Appendix II. Within the basis |1>,...,|n> the modified matrix elements are the same as the matrix elements of the effective operator Q4¢¢,

PyQP,QP, erp

=

PoQPo —

eS e#0

mee e

(3.255)

0

in which Pp and P, are projection operators acting inside and outside the basis |1>,...,|n>, respectively. Expressions for the eigenstates can also be obtained. It is conventional, but not necessary, to take

the perturbation theory to second order only; this is adequate for the vast majority of systems met in practice. The second fundamental idea is that Q.;, can be written in terms of spin operators (see, e.g. Stevens 1963). We define a spin Hamiltonian operator Q,, equivalent to Q.4, by identifying each of the states \1) |2>,...,|n> with the eigenstates of the z component S_, of an effective spin S’ = 4(n—1). The n? constants giving the matrix elements of Q.,, can then be represented by the coefficients of the identity operator 1 and the various products of components of S’

440

ELECTRON

SPIN

RESONANCE

of degree less than n. Thus Q, is given by an expression of the form

042.01

OFSce 07 Sos a

ere

(13.2.2)

ap

We emphasize that S’ is a fictitious spin, determined entirely by the number of levels involved. It need bear no simple relationship to S, the real spin of the defect; instead, S’ depends on the value of n,

and hence on the properties of the total Hamiltonian. In many cases S and S’ are equal, or there is a simple correspondence. However, equality is not necessary. Thus for the 3d’ system MgO:Co?*, the real spin is S = 3 and the effective spin S’ = 3. Moreover, if there are energy levels a few wavenumbers above the ground state, then S’ may depend on temperature. A two-level system with an energy splitting A would be described by S’ = 0 for temperatures kT « A, when only the lowest state is populated, and by S’ = 3, when kT > A. The advantage of writing the matrix elements of Q.¢, in terms of matrix elements of spin operators is that the well-developed algebra of angular momenta can be exploited. The spin Hamiltonian formulation consists of constructing a spin operator Q,, whose matrix elements within the effective-spin basis are identical with those of Q..,¢ among the states |1>,...,|n>. The spin Hamiltonian comprises the spin operators equivalent to the total Hamiltonian within the set of low-lying states. This concept of an effective spin is exactly analogous to the isotopic spin of elementary-particle theory. There is no unique way of constructing a spin Hamiltonian. The states |1>,...,|n> can be identified in a variety of ways with the eigenstates of S’. There are three main criteria: simplicity, symmetry, and correspondence to free atoms. Wherever possible the correspondence is chosen to make the symmetry properties of the spin Hamiltonian and the real Hamiltonian as similar as possible. Thus it is usually arranged that the effective spins S’ transform under rotation of axes in the same way as the real spins S. Likewise, it is usually possible to ensure that the symmetry is exhibited explicitly. However,

other choices are legitimate, if unattractive.

Moreover,

there are cases where the simplest choice of spin Hamiltonian does not have these symmetry properties. An example is the case of a non-Kramers’ doublet in an electric field, where one must choose between the simplest form without the desirable symmetry properties (Bleaney & Scovil 1952) and other, less-simple forms, which

ELECTRON

SPIN

RESONANCE

44]

keep the symmetry manifest (Griffith 1963; Miiller 19685: Washimlya, Shinagawa, & Sugano 1970). Another criterion used is that the results correspond as closely as possible to free-atom results. Thus, for a free atom, the sign of the spectroscopic splitting factor g gives the sense of the precession of the magnetic moment in an external field. This may be used to fix the signs of the various components of g, the spectroscopic splitting factor in a solid (Pryce 1959, Blume, Geschwind, & Yafet 1969). The sign of det ||g|| (i.e. g,g,g, in the principal-axes representation) is well defined ; if it is negative the moment precesses in the opposite sense to a free spin. The signs of the individual components of g are not well defined, and it is conventional to take components as positive wherever possible. The criterion that the spin Hamiltonian should reflect the symmetry of the system has led to a number of general calculations of spin Hamiltonians. These consist of constructing the most general combinations of effective-spin operators and external-field components in symmetrized combinations and of determining the smallest number of independent parameters. Such treatments are given by Koster & Statz (1959) for the Zeeman interaction, by Kneubihl (1963) for polyatomic systems, and by Grant & Strandberg (1964) and Ray (1963), who discuss tensor decomposition methods. The forms of spin Hamiltonian found from symmetry arguments are not necessarily restricted to the forms obtained from low-order perturbation theory. We shall not discuss the results here, but instead indicate the simplest form of spin Hamiltonian which contains all the features of immediate interest. This is Ah ace+1P,1), eet iJest] H. = ¥ (BH g,8,+1,4,S,+5;D,8,

(13.2.3)

ij

in which S is an electronic effective-spin and I a nuclear spin. The terms are, respectively, the Zeeman energy, the hyperfine interaction, the zero-field splitting, and the quadrupole interaction. These terms are discussed in detail in the next sections. In practice, spin resonance is performed at a fixed frequency. Thus one wants to find the condition that energy levels be split by hy and also the relative intensities of transitions. Clearly, this can be done by calculating the eigenvalues and eigenfunctions of %. However Coope (1966) has shown that both the resonance condition and intensities can be calculated directly in terms of the spin Hamiltonian matrix elements.

442

ELECTRON

SPIN

RESONANCE

13.3. The Zeeman effect We now consider the terms in the spin Hamiltonian which depend on the magnetic field. They are usually written in the form

H, = BH.g.S,

(13.3.1)

where f is the Bohr magneton. The spectroscopic splitting factor g measures the energy splitting under a magnetic field. It is related, but not generally equal (in contrast to the free atom case) to the magneto-mechanical ratio g’. Blume et al. (1969) show that, in a wide range of cases,

g = g/(g—p),

(13.3.2)

in which p = /S, and is the expectation value of the real spin component S, in the state labelled S;. For the remainder of this chapter we shall work in terms of the real spin S, unless otherwise stated. The interaction with the magnetic field is given by

H, = BH.(L+28)+ 2

tall wrf2+ re E(r)(S Ar).(HA of =

H,+

H2,.

(15535)

For simplicity the result is given for just one defect electron. The generalization to several electrons is usually straightforward. The suffix i labels the various nuclei and r; is the defect-electron coordinate relative to nucleus i. In simplifying the last term the potential near a particular nucleus is assumed to be spherically symmetric in the region in which the term in appreciable. Thus the electric field is approximated by

2m? Saye » (rr ;-

(13.3.4)

A similar assumption is also used in writing the spin-orbit coupling

Heo = Y ElrL; . S,

(13.3.5)

which is important here. The spin-orbit Hamiltonian is shortranged, and is only significant close to each nucleus. We shall use

ELECTRON

SPIN

RESONANCE

443

Russell-Saunders coupling, so that the electron-electron interaction is assumed to have been treated already, and the spin— orbit coupling can be treated by perturbation theory, possibly treating a few levels more exactly. For jj coupling, where the spin— orbit coupling is more important than the crystal-field energies or electron-electron interaction, a rather different formulation is needed (Griffith 1961, p. 111; Condon & Shortley 1935). Our notation distinguishes between r; and L;, which are defined relative to a particular site, and r and L, which refer to an arbitrary origin. The distinction is essential if the final results are to be gaugeinvariant. Errors in choice of origin may lead to results which depend on the choice of the arbitrary origin. The expression (13.3.3) for #, is itself gauge-invariant. If we assume a Coulomb gauge, where V. A = 0 and the electrostatic potential is determined from the static charge distribution, then the form of 4% is invariant under the transformation

jeg ees ae Vy =0

13.3.6 (13.3.6)

However, the separate terms in #%, are not individually gaugeinvariant. Since #,, and ”,, are different in order of magnitude, problems may arise in taking them to different orders of perturbation theory. Indeed, .%,, is a factor of about 1/137* smaller than %,, (Bartram et al. 1967) and has been included simply to ensure gaugeinvariance. Briefly, we need a consistent choice of origin r to ensure gauge-invariance, and we want a local choice of r to simplify terms such as the last term of #,. A number of approaches have been devised to deal with these problems. Griffith (1961) has shown that the results of perturbation theory to second order in A are gauge-invariant, i.e. independent of x in (13.3.6). Slichter (1963) has shown that if the wavefunction can be written as a sum of non-overlapping terms centred on the different sites then one can choose a different vector potential for each site. These vector potentials must imply the same H, of course, so they differ at most by a gauge term. Stone (1963), in his particularly useful treatment, uses a perturbation expansion analogous to (13.2.1), and keeps track of the various terms in gauge-invariant combinations. He derives an expression for the g-factor of an orbitally non-degenerate system, starting from the Dirac equation. Casselmann & Markham (1965) extend his treatment so that their

444

ELECTRON

SPIN

RESONANCE

results do not depend on the validity of the perturbation expansion. They include spin-orbit coupling, unlike Griffith, and do not need to divide the wavefunction into non-overlapping components. The relative merits of these treatments depend on the system under discussion. For present purposes the result that perturbation theory to second order in H is gauge-invariant will suffice. The effective Zeeman operator calculated from the perturbation expansion (13.2.1) to second order in the magnetic field is

H, = Pot, Po + PoH2,Po— a

y Po(H12+ e#0

Aso) P(A 12+ Aso) Po Ee

(13.3:7)

eG

The projection operators are of the form Pp = |S,>, or |z> p-states) then the expectation value of L, in each is zero, If we choose eigenstates of L, then only the one with zero expectation value is real: the others are complex.

ELECTRON

SPIN

RESONANCE

445

states. It is useful to recall that L is the operator which induces infinitesimal rotations ; it is not a good quantum number in a solid, where only certain discrete rotations are symmetry operations. The quenching of angular momentum has been demonstrated, very directly, by Pryce (1957). The orbital angular momentum operator is, algebraically, purely imaginary Slit) 0 L,= ily ce

(13.3.8)

Thus its expectation value in a real eigenstate is an imaginary number. But this expectation value is a real observable, and so must be real. These statements are consistent only if the expectation value vanishes. The spin-orbit coupling can remove the quenching to give L a finite expectation value. The point is that off-diagonal elements of L may be finite. The spin-orbit coupling AL. S, can mix in excited states |e> into the original state |0>

|0> + |6> = |O- > le>

(13.3.9)

Eo

The cross-terms in , |>, |k>, so that the matrix elements of L are the same as the matrix elements of aL in the pstates |x, |y>, |z>. Thus, if ijk is a permutation of xyz, = 1OE; jx

(13-5201)

where e;;, is +1 for even permutations and —1 otherwise. The value of « depends on the specific angular dependence of the states involved. Some examples are given in Table 13.1.

446

ELECTRON

SPIN

TABLE Symmetry

Atomic states from which derived

RESONANCE

13.1 Example of corresponding polynomial form

a

T,

p

x

+1

A

f

x(5x? — 3r?)

—3

T,

d

yz

—1

we

f

x(y? — 2?)

+4

Another example of particular interest (Stevens 1953) is that a covalent admixture of orbitals on the neighbours to a defect changes the angular dependence of the wavefunction and hence «. This change is usually described by an ‘orbital reduction factor’ k, such that the true value a = ka , where «, ignores the covalency. Similar factors appear when an envelope function is orthogonalized to core orbitals. As a simple illustration, consider T, orbitals in octahedral coordination formed from an admixture of d orbitals on the central ion and p orbitals on the six ligands. One such orbital is

IXy> = N(ld, xy> +A{|p,(010)> —|p,(010)> + + |p,(100)> —|p,(100)>})

(13.3.12)

in which |p,(Imn)> is the p, orbital on the ligand at (/mn). The normalization N is given by

N~? = 1484S +4/?,

(13-3913)

where S is the d—p overlap and the p—p overlaps are ignored. The orbital reduction factor is easily calculated, and reduces the orbital elements from their values for pure d-states by

keel

Pyes TRIS ae

(13.3.14)

For any reasonable admixture, k < 1. Most experimental results lie in the range 0-85 to 1-00 in cases where this ‘covalency’ mechanism operates. The value of the ‘p’-isomorphism’ (13.3.11) is that operators involving L may be replaced by operators using an equivalent orbital angular momentum aL“, whose matrix elements are readily evaluated. Examples of such replacements are (Griffith, 1961, § 9.5.3) the spin-orbit coupling, which becomes éaL*4.S, and the orbital contribution to the Zeeman energy, which becomes BH . «L‘?.

ELECTRON

SPIN

RESONANCE

447

Our discussion has concentrated on matrix elements between electronic states. We often need elements among vibronic states. As discussed in § 8.4.4, there is then an extra reduction factor from the overlap of the vibrational parts of the vibronic wavefunctions, as well as the covalent contribution. The two reduction effects are distinct in origin, but it may be hard to identify their separate contributions. 13.3.2. Examples of g-factors

We discuss a number of cases to demonstrate aspects of these calculations. The first case shows the importance of the spin—orbit coupling. We assume that the state of interest |0) does not have orbital degeneracy and that just one electron need be considered. The electron is localized on one defect atom, so there are no gauge problems, and the spin-orbit coupling may be simplified to

aes:

(13.3.15)

Strictly, this simplification from é,(r)L;.S requires that all the states of interest, both |0> and the set of excited states, have the same radial functions. Finally we assume that the other states |e) lie sufficiently far from |O> that

(E,—Ep)| > |Al.

(13.3.16)

These assumptions are obeyed rather well for a number oftransitionmetal ions in ionic crystals, such as iron-group ions in MgO. The g-factor can be found directly from (13.3.7) and (13.3.3). We express the results in terms of a spin Hamiltonian

where, in this case, the

real spin and effective spin are identical. The effective Hamiltonian is the sum of paramagnetic and diamagnetic terms

=

para

a NT

(13.3.17)

where the diamagnetic term contains contributions from all electrons, including the core electrons, in which we are not directly interested, e2

Haia a

y electrons

8mc?

H?a is the lowest

state. %,,,, becomes yarn = Y, 2B(6;,—AAi)S;Hj;— ¥) 07 AS:Sj— Nef]

i,j

— Y BA HH; + tJ

+

a H.SCér2) = 2c

AS which dominate in A all have energies of the order of E, ~ Ey +A. Then we may write

A,, ~5 ¥ may involve orbitals centred on different sites, when the

result is non-trivial. Two points have emerged. One free-spin value is a measure of 4/A, and A some suitable excitation anisotropy of the wavefunction We shall show shortly how the reflected in the g-factor.

is that the deviation of g from the where J is the spin-orbit coupling energy. The second is that any enters through these deviations. symmetry of the defect may be

A. Electron and hole centres. The importance of the spin-orbit coupling has one direct application. The spin-orbit coupling of a single electron is always positive in a defect of the type we are considering. If the state |0> is the ground state, then A is positive. The g-shift is then negative: g;; < 2. However, if we consider a single hole instead of an electron, the spin—orbit coupling is reversed in sign:g;; > 2. Thus the sign of (g;; —2)can be used to identify electronand hole-centres. Like most simplifications, this rule is often incorrect, although it gives useful guidance. The cases where the rule breaks down most readily are those when g is close to the freespin value. Configuration interaction (§ 2.1) can lead to more important admixtures than the spin-orbit coupling (Kahn & Kittel 1953; Adrian 1957; Carrington, Dravnieks, & Symons 1959; Bartram et al. 1967). It is particularly important when the defect electron interacts with ions with unfilled shells, which can be polarized readily, or when two configurations are especially close in energy. The configuration admixtures are often just the ones expected from simple covalency arguments. The g-factor in these cases is the weighted sum of the various configurations, and it is clear that the simple rule relating (g—2) to hole- or electron-centres is no longer valid. B. Symmetry and the g-tensor. We now relate the anisotropy of the g-tensor to the symmetry of the defect. The anisotropy comes from the orbital contribution to the magnetic moment, and depends

on the tensorA of(13.3.19). Its origin depends on whether the asymmetry of the environment affects the states whose g-factor we are calculating more than the other states, which are mixed in by the spin-orbit coupling. For the F-centre, an electron trapped at an

450

ELECTRON

SPIN

RESONANCE

anion vacancy in cubic symmetry, the ground state is non-degenerate. Under uniaxial stress the g-factor becomes anisotropic because the degeneracy of the excited p-states is removed. The energy denominators in A, and to some extent the matrix elements, are altered. Thus it is the effects of asymmetry on the excited states which gives the anisotropy in the g-factor. By contrast, for Cu** in octahedral coordination the excited states are negligibly affected, and it is the effects on the ground state which dominate. In cubic symmetry Cu? * has a single hole in its 3d shell. The lowest states are orbitally

degenerate and transform like (x? —y*) and (3z*—r?). This degeneracy is removed by tetragonal distortions, which usually cause the (x* — y”) state to drop below the other in energy. The excited states are negligibly affected. The A are profoundly affected since we have now singled-out as the ground state one of a set of states which were equivalent in cubic symmetry. Here the effects of asymmetry on the states of direct interest is the important feature. So far we have just considered cases where the manifold of states of interest was either non-degenerate or degenerate but without finite matrix elements of L among them. The theory is more complicated, without being more difficult, when there are matrix elements

of L between the states of interest. Second-order perturbation theory is replaced by a combination of this perturbation approach with the solution of a finite secular equation. Examples of these calculations are given by Griffith (1961, p. 340 et seq.).

13.4. The zero-field splitting There are terms which cause small splittings even when the magnetic field vanishes and there is no hyperfine interaction. Two examples which are particularly common are written

#4 = D{S? —4S(S +1)} + E(S2 —S?)

(13.4.1)

and, for cubic symmetry,

Hg = Za{Sz +S} +S2—48(S + 1)(3S?+3S—1)}.

(13.4.2)

We now discuss the origin of these terms, and the relation of the parameters to properties of the defect. There are two major contributions to #%,,, apart from mechanisms involving exchange between neighbouring magnetic ions (e.g. Abragam & Bleaney 1970, chapter 9). The first was treated in

ELECTRON

SPIN

RESONANCE

451

§ 13.4, and came from the spin-orbit interaction taken to second

order. The appropriate term in (13.3.20) which can be written - » A*A,,S ipiS 2 =

ite

x Oe

Ay,y) {82-335(S+

1)} x

i XK) (S? ie S?)],

(1 3.4.3)

referred to principal axes ; a term proportional to S(S + 1) is omitted. The second contribution comes from the dipole-dipole interaction of the electrons involved. The term #,, only causes a splitting of the levels of systems with S >» 1, and these defects must involve at least two electrons. For simplicity, we concentrate on two-electron systems with S = 1. The dipolar interaction has the form Hye

oF \S..Su 351: r12)(S2 .1 42)

r 12

le

\

(13.4.4)

where S, and S, are the spins of the two electrons and r,, is their separation. If the defect wavefunction is (1, 2) then the parameters of (13.4.1) are Sule

D = gb) «wl, 2y —— 74 7 |W(1, 2)>

(13.4.5)

i2

and

E = gf), By sia Le 2M, 2)>.

(13.4.6)

Some cautionary comments are necessary. First, throughout, we shall be using real spins S and not effective spins. The g-factor in eqns (13.4.4}13.4.6) is strictly 2.0023, the free-spin value, and the wavefunction y(1, 2) is the one appropriate to the spin distribution in space. When there is a substantial orbital contribution it is customary to generalize the equations by changing the g-factor appropriately and to let S be the effective spin. The argument is that the g-factor measures the response of a spin to an external field, and so it can be used to represent the response both to externally applied fields and to the magnetic fields of other electrons. Some simplifications are involved here. This approach bypasses several of the steps of the full perturbation treatment strictly needed, and so presumes

452

ELECTRON

SPIN

RESONANCE

a number of relations between matrix elements which may not be valid. Further, this approach assumes that a single space-function describes the distribution of both spin and orbital angular momentum, a result which is not always satisfactory. The second general comment concerns electron correlation. In the lowest-order configuration interaction, two types of configuration can be admixed: those (|J>, say) differing from the dominant configuration |0> by a single one-electron state and those (\II>, say) differing in two such states. Other states are admixed only by higher-order perturbation in the Coulomb interaction between electrons. Since #4, of (13.4.4) is a two-electron operator, both the types of configuration |I> and \II> are important because and , |P2>, and |P3> lie symmetrically in the plane normal to the dissimilar bond

pay

PS

\P2) = 9+ (os

Bi7y : 3

~4x>-2 17

(13.6.16)

470

ELECTRON

SPIN

RESONANCE

The general form of these expressions is

Dasa ales

(13.6.17)

in which |i) is one of the bond orbitals and |«> an atomic function. The component of the field gradient of interest is q,,. Since only the p orbitals contribute, we may write q,, as a sum over the occupied orbitals 1 qzz =

225 naj,

— a, — aiy)

(13.6.18)

= Gif.

In this expression n; is the occupancy of orbital |i> and q2, is the field gradient from a single |pz» orbital. We first consider cases where there is no ionic character for any of the bonds. The n; are then integers. If n; = 1, so that there 1s one electron in each bond, then f = 0 identically. If there is an extra electron in the inequivalent bond then

at 1—3 cos? 0 : sin? 0

f=

(13.6.19)

which is unity if 9 = 47; this case would correspond to planar bonds plus a lone pair of p electrons normal to the plane. The sign of f is reversed when there is an extra hole instead of an extra electron. These results are changed somewhat also when there is electron transfer from the central atom to its neighbours. We measure the transfer by I, for the dissimilar bond and I, for the others. In the case n; = | the field gradient no longer vanishes and is given by

1—3 cos? fe le

—10)

(13.6.20)

These results show that the field gradient is sensitive to the bonding of the electrons associated with the particular nucleus. A similar, but more complicated, description has been given for nitrogen in diamond by Every & Schonland (1965). In cases where there is electron transfer, it is not always clear which ionic state should be considered in estimating +a,|p>) from a hybridized orbital on a given neighbour. Then the hyperfine constants associated with the nucleus of that neighbour contain contributions 16x

(13.7.1)

a = 4; ~8nBBls(0)’, ; 4/1 b = gnBbnBli = apSNPNB T= 73 )p?

(13.7.2)

using (13.5.16) and (13.5.28). The ratio of a, to a, is given by

(«,/a,)

Ui 2

=

amend

sO)? 202 B P

—_—_—_—-



(13.7.3)

If the atomic parameters are known, «, and a, can be obtained from the hyperfine constants. The «, and «, may then be interpreted in terms of bond angles, using relations like (13.6.14). This prescription is open to several sources of error. One must be sure that b, in particular, comes only from intra-atomic terms. The assumption ofonly s and p character, with no d or higher terms,

is also doubtful. And it is not clear that values of |s(0)|? and ¢1/r?>, for free atoms are a good description for atoms in solids. has been used by many authors. Some (e.g. Watkins & 1964, 1965) use the analysis as a means of describing data, and show commendable caution about relating to distortions. Others interpret results in terms of good example being Messmer & Watkins’s (1970)

The approach Corbett 1961, the hyperfine the «, and as distortions, a discussion of

ELECTRON

SPIN

RESONANCE

475

nitrogen in diamond; in this discussion wavefunctions for various distorted configurations were constructed from s and p orbitals using extended Hiickel theory. One plausible set of distortions gave good s—p characters for the nitrogen and its most important carbon neighbour, together with a good value for the nitrogen quadrupole interaction. Messmer & Watkins also noted that the sum of the one-electron Hiickel energies was minimized by this distortion, but this feature is deceptive since it is the total energy, not the sum, which should have a minimum; the minimum in the Hiickel sum here is partly fortuitous and partly a result of constraints put on the distortion.

C. Quadrupole interactions in ionic systems. In ionic materials such as alkali halides, a large contribution to the quadrupole interaction comes from the point-ion part of the inter-atomic terms ; Feuchtwang (1962), Hartland (1968), and Kersten (1970) are among those who have exploited this feature to derive distortions. However, the polarization contribution can be large also, and uncertainties make it difficult to get accurate distortions. The method is best for very simple centres. Examples are the substitutions of one alkali or halogen by another in an alkali halide. In systems like KCI: Na, distortion gives the dominant quadrupole terms (Hartland 1968). Off-centre ions, like Lit and Cu* in KCl, can be treated by natural extensions of this case (Wilson & Blume 1968). The F-centre is slightly more complicated since the charge cloud overlaps the near neighbours (Feuchtwang 1962), although the comparison of Fcentres and F,-centres, with an adjacent alkali impurity, can be treated reasonably (Kersten 1970). D. Scaling of parameters. In these methods one assumes the defect properties are determined by the nearest neighbours only. The distortions are deduced from comparison ofthe pressure dependence of some property with its variation along a series of similar crystals. Suppose the nearest-neighbour distance is a, and that it differs from the corresponding value @ in the perfect host lattice. The property F(a) is assumed to depend on the local spacing a through

F = Fa.

(13.7.4)

Experiments on one crystal at different pressures involve the mean

476

ELECTRON

SPIN

RESONANCE

spacing @, and give d(In F)/d(In @). If the defect does not affect the local elastic constants, d(In a) and d(In a) are equal, and we have

d(In F)

~~

d(In F)

d(na)—d(In a’

(13.7.5)

where the second form can be obtained from the variation from host to host. Now if the distortions in the different hosts i at zero pressure are 6, = (a,—4;) then Fy = F(a;+6,)% is independent of the host. Thus if we know the distortion in one case (or if we can guess it) to fix Fy, the 6; can be obtained from the observed F; and N, using

6; = (Fo/F,)** —G,.

(13.7.6)

The essential elements here are the assumption of a simple power-law relationship and the neglect of changes in force constant. The powerlaw fails if there are both long-range and short-range contributions to F,so that the dependence on a and acannot be separated properly. And, as shown in (12.4.2)}{12.4.4), changes in local force constant completely alter the analysis. We mention two examples of treatments largely equivalent to these arguments. Hurren, Nelson, Larson, & Gardner (1969) discussed crystal field parameters of rare-earth ions in alkaline-earth fluorides. Covalent terms may well dominate (cf. § 22.4), and so the power law (13.7.4) is reasonably plausible. It is also possible that the local force constants are not greatly changed, so their conclusions may be correct. By contrast, Bailey (1970) has chosen the less likely example of the F-centre hyperfine structure. Long-range terms contribute to the wavefunction, so (13.7.4) should involve @ too. But even if (13.7.4) is accepted, all evidence (cf. § 15.2) suggests that there is a substantial change in force constant and very small distortion 6,;, so that the analysis of § 12.4 should be used instead of the present treatment.

14 NON-RADIATIVE PROCESSES AND THE INTERACTION OF FREE CARRIERS WITH DEFECTS 14.1. Introduction

THE interaction of free carriers with defects gives rise to three main classes of process. The carrier may be scattered, or it may be captured with or without the emission of light. In this chapter we discuss some of the processes involved, together with the important non-radiative transitions between bound states. Free-carrier recombination is particularly important in two contexts. The free-carrier lifetime is an important parameter in the possible use of systems in electronic devices. Further, the relative importance of non-radiative and radiative processes affects both the efficiency of luminescent devices and the possible creation of defects by optical irradiation (e.g. Pooley 1966). Non-radiative transitions occur in many systems and occur as intermediate stages in the trapping of carriers. We shall try to relate the various treatments here. 14.2. Non-radiative transitions

14.2.1. General theory

A. Basic definitions. The theory developed in this section covers many cases of interest, including multiphonon transitions. We consider the transitions between two vibronic states which differ in electronic energy but have the same total energy. Usually we consider the states between which transitions occur to be eigenstates of the Born—Oppenheimer equations. The transitions are induced by the non-adiabatic terms in the Hamiltonian. These terms may be treated as a perturbation if the two energy surfaces do not intersect for lattice configurations of interest. We assume no such problenis occur. We consider the transition from a vibronic state such as

Is,n> = Wor; Q)x({n,}, Q)

(14.2.1)

478

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

to another state

[n> = Wir; Q)x({m}, Q—A).

(14.2.2)

The electronic parts of the wavefunctions y, and w, are assumed to have no orbital degeneracy. The vibrational wavefunctions y are specified by a set of occupation numbers {n}. There is a displacement of the modes in the transition, given by A. The adiabatic potential energy surfaces are thus

U(Q) = Eot+ ¥ 3M,ox Qi,

(14.2.3)

k

U(Q) = 3M.

(Q.—A,),

(14.2.4)

k

apart from anharmonic terms, which must be treated separately. As usual we assume that the normal modes in the initial and final states are the same and that their frequencies are unaltered. These assumptions are strictly unnecessary, but simplify the theory. Their effect seems to be small (Freed & Jortner 1970) except possibly in the reorientation of asymmetric defects. For later use we define some energies, as in the configurationcoordinate model for optical transitions. The energies are defined in Fig. 14.1, and are

Ey = ¥4M,2A?2

(14.2.5)

k

and

E, = (Eo — Ey)’/4Ey.

(14.2.6)

Energy

Fic. 14.1. Configuration-coordinate diagram for non-radiative processes.

OF

FREE

CARRIERS

WITH

DEFECTS

479

Thus Ey would be half the Stokes’ shift in the optical case, whereas

E, would be the activation energy for radiative decay in various classical models of the process. We shall usually assume that Eo is large compared to the phonon energies, a result often described as the statistical limit. As in the optical case, it is also convenient to define dimensionless parameters

So, = 4M, @2A2/ho,, Spee. oun

(14.2.7) (14.2.8)

k

The So, are of order (1/N) when there are no local modes. We define the strong coupling regime by Sy > 1 and the weak coupling regime by So « 1. The transition probability can be calculated in a number of ways. The simplest is an extension ofthe ‘golden rule’: we may use ordinary time-dependent perturbation theory provided that we use (W — E) as the perturbation and provided the initial and final states are strictly orthogonal (Kronig 1928; Huang & Rhys 1950). Here # is the total Hamiltonian and E the zeroth-order eigenvalue of the (degenerate) vibronic states between which the transition occurs. The expression for the transition probability can be found directly as in the optical case, or by use of generating functions (Englman & Jortner 1970; Freed & Jortner 1970). Another approach is to work directly from the time-dependent Schrédinger equation (Brailsford & Chang 1970; Holstein 1959). These cases are all roughly equivalent although there are many differences in detail. All these authors assume that the probability is given adequately by the lowest-order perturbation theory. Fischer (1970) avoids this assumption in his correlation function approach. Reviews ofthe various methods, including some not discussed here, are given by Perlin (1963) and by Henry & Kasha (1968). B. Matrix elements. The matrix element for the transition can be

written in the general form hj, = [ae Ji(Q)x({ns}:

)x({n},

Q — A).

(14.2.9)

If there is no change in total spin in the transition (‘internal conversion’ in molecular physics) then J comes from the non-adiabatic terms in the Hamiltonian or from the configuration dependence

480

NON-RADIATIVE

PROCESSES

AND

INTERACTION

THE

of the wavefunctions (cf. § 10.9). We shall give expressions for the non-adiabatic terms, since the other (Herzberg—Teller) case can be obtained readily by comparison. The non-adiabatic matrix element can be written (cf. § 3.1) y=

OXKs/OOr [ery 3p ot - Dye |e

k

Loy 4 ae Sihaasi +5]

(14.2.10)

The y,, are the eigenstates for single phonon modes

(14.2.11)

x(Q) = [] x(Qx). k=1

The modes k which are effective in breaking the adiabatic approximation are known in molecular physics as ‘promoting modes’ or ‘kinetic’ modes (e.g. Gummel & Lax 1957). The modes with finite displacements A, are often called ‘accepting modes’ or ‘lattice relaxing modes’. When there is a change in total spin (‘intersystem crossing’) it is the spin-orbit coupling, rather than the non-adiabatic terms, which causes the transitions,

= |dor Wi}How,.

(14.2.12)

Intersystem crossing is especially important in the decay of metastable excited states of colour centres. C. Transition probabilities. The transition probability involves the usual sum over final states and thermal average over initial states

2m =

aA Av Yalhaail?

ey

o(E,;—

E,;),

(14.2.13)

where i and j refer to the different vibrational states {n,} and {n,}. Usually the average is a simple thermal average ;one assumes that the intralevel transitions which establish equilibrium among the vibrational states for a given electronic state are much more rapid than the interlevel transitions. This assumption may be invalid in the decay of excited states. If these states are populated optically,

OF

FREE

CARRIERS

WITH

DEFECTS

481

the distribution over states will certainly differ from the equilibrium distribution for some time ; whether this period is significant depends on the system under study (see e.g. § 12.2.6). It is straightforward, however, to consider the decay of a single vibronic state (Brailsford & Chang 1970) so these points can be resolved in any special case. Dexter & Fowler (1967) have argued that, in general, the lower the ‘cooling rate’ at which the system in a given electronic state cools to the lowest vibrational state, the higher the decay rate to another electronic state. This result simply requires that the decay rate increases with increasing excitation of the vibrational state of the initial level. We now consider some of the calculations in detail. There are minor differences in the results, most of which come from the approximations used to simplify eqn (14.2.10) for the transition matrix element. Thus some authors assume that the promoting modes and accepting modes are distinct, so that the mean value of a mode which causes a breakdown of the adiabatic approximation is never displaced in the transition. The selection rules for the promoting modes are then just An, = +1. However, if their mean value changes in the transition then An, = 0, +2 are also allowed and are proportional to the corresponding S,. Two calculations which discuss this approximation (Huang & Rhys 1950, especially §6; Gummel & Lax 1957, p. 47), confirm that the neglect of the displacements of the promoting modes is a good approximation. With this simplification, the theory is very similar to that of forbidden optical transitions. The differences come solely from the replacement of a displacement which affects the symmetry by one which causes a breakdown of adiabaticity. A second difference is that most authors drop the second term in (14.2.10). This is usually arbitrary. However, in some cases (e.g. Huang & Rhys 1950) the matrix element is identically zero. Consider a transition between two states of opposite parity; irrespective of the parity of the promoting mode Q,, the matrix element vanishes. The third distinction between the methods concerns the use of the Condon approximation, which is the assumption that a matrix element is independent of Q, the lattice configuration. A relatively weak approximation (e.g. Freed & Jortner 1970) is to assume that

ad; Qla5. =o,|s,.Q> = mu ee Qla5. o Is;Q>= YM op

(14.2.14)

482

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

and is independent of Q. A slightly stronger approximation (Brailsford & Chang 1970) is to assume that J,,, of (14.2.10), is independent of Q. The matrix element (14.2.9) has then the simple form

cr |dQ x({n,}, Q)x({n,}, Q—A), which contains just the lattice overlap factor which

(14.2.15) appears

in

optical transitions. The notation C*’ and C¥° is close to that of the papers concerned; the two coefficients are related by

h

ICRP = Au tS 4 BICEP,

(14.2.16)

where the signs + refer to transitions in which An, = +1 in the Freed & Jortner model. Nitzan & Jortner (1972) show that going

beyond the Condon approximation affects the detailed quantitative results but does not greatly affect the form of the results. But in special cases the results may be profound. An example is the ‘latticeactivated’ process in the quantum theory of diffusion (Flynn & Stoneham 1970).

D. Weak coupling. In the weak-coupling limit So, of (14.2.8), is small. Brailsford & Chang (1970) find the transition probability from a single vibronic state to be

axl(nchis Qld» Q—A)>!* (EE)

2

W = TCH? (eae

=

oo)

exp (—G°9) | dt exp (—ia@pt+GBo +

+ GE + GE + O(S})).

(14.2.17)

They derive the last line in detail, using methods very similar to those in chapter 10, together with the neglect of terms of high order in S;. The various functions are

GPC = ¥ {(2n,+ 1)Soj+3n(n;+ 1)S3,+.0(S3,)},

(14.2.18)

J

C= dD(nj+7+2)So; exp (+i,t), J

GEE =

Ue ala

nn,+1)Soj{2 cos (co,t)—4 cos(2w,t)}, (14.220) J

OF

FREE

CARRIERS

WITH

DEFECTS

483

in which n, is the phonon occupation number in the initial state. In a solid, oe there are no local modes, the terms in S$; may also be ignored as O(1/N). The transition probability simplifies considerably at low temperatures, where n; = 0. Without dropping terms we have

Wn,

BC}2 =0) => ot exp (— S.) {_ _stexp} —ivot+ k

+ }¥ So, exp fio}

(14.2.21)

The integrals in W cannot be obtained analytically and exactly. Instead the method of steepest descents is used (Erdelyi 1956, p. 39; Jeffreys & Jeffreys 1950, chapter 17), with the assumption

“oh

Soule + (@«/@m)

Eo

Ql

LOD

where wy is the frequency of the highest frequency mode finite Soy. The transition probability becomes

w=5

(Ge k

h

x exp

2n E

hoy

_ gilz

2n,+1)So;} i

= at 1n Gacaane

i

tal

with

| Xx

) es

(14.2.23)

dropping all terms in Q,,. The appropriate averages of W over thermal distributions of phonons can be made readily, since the occupation numbers {n,} appears explicitly. One important feature is that the dominant terms in W vary as exp (—aE,), where a depends slowly on Eg. This is the ‘energy-gap law’. Riseberg & Moos (1968) have verified the dependence over a range of 10° in W in extensive work on rare-earth ions in ionic crystals. They have also given a simple derivation of the general form. Suppose the probabilities of nth- and (n—1)th-order phonon processes are related by

W, ~ eW,_4,

(14.2.24)

where ¢ is small. Then W, can be written as Ae”. Further, suppose the

484

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lowest energetically-allowed process dominates. Then n ~ Eo/h@y, so that W ~= As" ~ Agkolhom

~ Aexp {(Eo/hw,y) In ¢}.

(14.2.25)

In the cases analysed by Riseberg & Moos, ¢ was always less than 0.2 and n was typically 6. German & Kiel (1973) note that transitions with the smallest n always seem to dominate for rare earth ions, even though the phonons with maximum energy are not obviously favoured in the density of states. Another approach (Freed & Jortner 1970) involves a function related to W and defined by F(e)i= Av yy Asal? O(E,;- Ey +6),

(14.2.26)

i

so that

W=

2n GZ FO).

(14.2.27)

Instead of seeking F(e) directly, we seek its Fourier transform

fos { de F(e) exp {i(e/h)t}.

(14.2.28)

The useful feature is that f(t) factorizes into a product of separate factors for the various modes. The function takes the form

OS

DICK IT£0,

in which k labels the various promoting modes, which break the adiabatic approximation, and j labels the other (accepting) modes. The one-phonon factors are

i= thu}co (w,t) coth sc +isin wo}, 2kT

fj = exp |-3s,{1— cos (wt) coth Ke —isin wo.

(14.2.30)

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These results are again closely analogous to those for the optical case, and particularly to the forbidden transitions (where the analogues of promoting phonons merely need to destroy a parity restriction rather than the adiabatic approximation). They are also convenient for evaluating the moments of F(e). An expression for W is then obtained by writing the transition probability in terms of f(t), giving

ee FJ ) W =) —+ho,.exp(—G

coth =11

afdt exp {—i(@p —um,)t + GF2+ GE},

(14.2.31)

where

Fy

> Sof2n;+ 1)

; GE + Y Sofnj+2+%) exp (tia,t)

(14.2.32)

j

These expressions can be evaluated in the strong- and weakcoupling limits. The weak coupling results differ from those of Brailsford & Chang, partly because Freed & Jortner assume an equilibrium thermal distribution over the initial states and partly because Brailsford & Chang use a stronger form of the Condon approximation. In corresponding notation the Freed & Jortner result is

le arnEo, - hOy 7 STE

Fox 1 Fox eae ee \-honan (a i.

Denoye 14.2.33

)

in which Eo, = Eo —hw,. The changes are the replacement of Eo by Eo,, which is negligible in the statistical limit Ey > hay, the altered C, related by (14.2.16), and the replacement of (ny, +1) in the denominator of the logarithm by a degeneracy weighting dy. The effects of these changes is probably small, but I am not aware of any quantitative check. E. Strong coupling. In this other extreme, So is large compared with unity. Two different approaches have been given (Huang & Rhys 1950; Freed & Jortner 1970), which we consider in turn.

486

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Huang & Rhys made the simplification that all the phonon modes have a single energy hw,, corresponding to the longitudinal optic modes in their discussion of the F-centre. Their result is derived using exactly the methods of chapter 10, and is exceptionally complicated in its final form. However, in the statistical limit Ey > hw,, considerable simplification is possible, giving a transition probability

ee

4 ant

NCHRP"Rp,

14.2.34 (14.2.34)

in which h2

jon" = 5p [ferves wn] HR 2a

3

(14.2.35)

involves matrix elements analogous to |C*?|? and

R, = exp {—(2n+ 1)So}

n+1\?/2 - | 1,[2S,/ {n(n + 1)}].

(14.2.36)

The integer p is given by phw, = Ey, which gives energy conservation ; for most purposes p may be regarded as a continuous variable.

The factor R, is normalized so that )’, R, is unity; R, may thus be regarded as a type of shape function. The dependence on p at high temperatures (n >» 1) is mainly determined by the Bessel function J,. This dependence, given by

I,(z) ~ sa exp (z—p?/z),

(14.2.37)

shows that the transition probability decreases as p, the number of

phonons required for energy conservation, increases. This has a direct physical consequence. If the masses of the atoms involved are increased by isotope-substitution, as in deuteration, the lattice frequencies are reduced. The energy Eg is not significantly affected, so p is increased and W falls. The competing radiative contribution becomes more important. The Huang-Rhys theory has been applied to radiative and nonradiative recombination in alkali halides (Pooley 1966). The matrix elements for the non-radiative transition can be estimated fairly easily in a continuum approximation, if it is assumed that the important part of the Hamiltonian is proportional to the electric

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field of the longitudinal optic modes, and that this electric field is essentially constant over the wavefunctions of the initial and final states. The radiative and non-radiative transitions then involve the same matrix elements, and one can easily obtain their relative probabilities. Pooley’s results show that the Huang—Rhys theory gives a good qualitative description, but there is disagreement in detail. The radiative processes are correctly predicted to dominate at low temperatures and to fall off rapidly at temperatures of a few hundred K. However, a large radiative term is predicted at high temperatures. The discrepancy is presumably caused by the omission of important terms in the approximations described. The details are still speculative, although one possibility (Pooley) is that there is a local mode with large So;, corresponding to the (electron plus M-centre) configuration of the excited state. The large value of So, affects the non-radiative probability, but has no effect on the total radiative probability (§ 10.4, for example). Anharmonic effects can also be large (Makshantsev 1971; Sturge (1973) gives a particularly good example) and may be important here. The strong-coupling limit is found by Freed & Jortner by expanding (14.2.32) in powers of t. This is a moment method and closely parallels the treatments by Kubo & Toyozawa (1953) and Pryce (1966). Truncation of the series by retaining terms up to t? gives a Gaussian form for the F(e) of (14.2.26). To this order the transition probability is =



d

|CF |?

Tt es

NO,

ae

x exp {-(-==

cot

ra.

/2D

h

ho,

2

a \

+up

xX

(14.2.38)

in which D? = ¥ (ho,)?S(2n,;+ 1). j

(14.2.39)

The moments are given by Pryce, in whose notation h?y2 replaces

4\CF|?hw,. The expression for W becomes more transparent in the limit that the w, can be replaced by a mean frequency @, such that

(Eo —Ey) = 2/(EmEa) > 10, hor,

(14.2.40)

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and if we define an effective temperature T* by

Ene othe

(14.2.41)

T* and T become equal at high temperatures, whereas T* tends to ha/2k at lower temperatures. The transition probability at high temperatures takes a simple thermally-activated form

= SI hy 2) con (A) en [—4). oaaen only, é

2n

hw

log.

The result, with the dominant exponential dependence on 1/T*, has been derived without using the concept of a ‘saddlepoint configuration’ or an ‘activated complex’ at any stage. The results follow directly from the formalism through the overlaps of the lattice oscillator functions. The configuration at which the two energy surfaces intersect appears partly because of these overlaps and partly because the non-adiabaticity operator can be most effective in mixing the two sets of states where they cross. It is straightforward to verify the relative magnitudes of the upward and downward transitions. In accord with the principle of detailed balance, the probabilities are in the ratio exp {—(E) + E,)/kT} to exp {—(E,)/ kT}. Whilst (14.2.4) gives a thermally-activated transition probability, it must not be assumed that thermal activation always appears. In the weak-coupling case, for example, (14.2.4) becomes absurd because E, becomes large. In essence, different transitions dominate and the transition probability is governed by (14.2.23). The thermally-activated expression in strong coupling is not expected to be accurate at low temperatures. In this limit a different approach is needed (Holstein 1959; Flynn & Stoneham 1970). We start from (14.2.30), which contains integrals of the form iy dt exp (—iw4t) exp |BSont {(2n;+

1) cos (wt) +isin (@,9}

(14.2.43) in which @, = @)+a,, and we have used (14.2.32). This integral can be transformed from an integral along the real t axis to one

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which is along an altered contour but with no imaginary term in the second exponent. This is achieved by

h aires t>tt+in—

(14.2.44) 2

Then it can be shown that the integral is dominated by the contribution from along the real axis: we need only consider

i._drexp(+ ad exp (—i@,7t) exp

2 So,

x COs we)

hw

Gr (14.2.45)

At low temperatures the thermal occupation numbers are low;; it is the few-phonon processes which dominate. This is particularly true for small transition energies, when energy conservation can be achieved with just a few phonons (the opposite of the ‘statistical limit’ we have considered until now). The final exponent can be expanded, giving exp 52] faedt exp (—im@¥4T)x Z

x {1+| »T; cos oye)+4[3100s or] Bee j j

(14.2.46)

in which I’; = So, cosech (hw,/2kT). The integral then simplifies by use of the spectral representation of the 6-function

{

die

f= 2710(0):

(14.2.47)

The first term always vanishes even when w, = 0, when the perturbation theory used to derive these expressions breaks down. This point is discussed in detail by Holstein. The second term gives the one-phonon terms

yal{os +o)+5(@,—@)},

— (14.2.48)

490

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the third term gives two-phonon processes ha + = 5 exp (22)

ya

T

OST ,

=I I,

+ 5(@4 +0@;—0;)+5(@4

—0;+;)},

(14.2.49)

and so on. In the low-temperature limit, the sums )’; reduce to a power-law dependence on temperature. The approach (14.2.43)} (14.2.49) to the integrals in the transition probability is particularly appropriate for diffusion and reorientation problems. 14.2.2. Paramagnetic relaxation

The non-radiative transitions which have been studied in most detail quantitatively are those of paramagnetic ions in insulators. The transitions studied are basically spin-reversals, in which the ion goes from one Zeeman level to another. Typical energy separations are in the range 0-1-1-0 cm’, although they can be changed by varying the applied magnetic field. The transitions are monitored by magnetic resonance methods. There are two main classes of experiment :microwave acoustics and paramagnetic relaxation. In microwave acoustics the effects of externally-stimulated phonon pulses on a spin system are observed. This can be done by measuring the attenuation of the phonon pulse or by monitoring the populations of the defect levels by conventional spin resonance. Tucker (1965) and Altshuler, Kochelaev, & Leushin (1962) have reviewed these methods. In paramagnetic relaxation the most important method involves altering the populations of the defect levels by an applied electromagnetic field. The recovery of the populations to equilibrium through interaction with thermal phonons is measured by spin resonance. Manenkov & Orbach (1966), Stevens (1967), and Standley & Vaughan (1969) have surveyed this field. Fig. 14.2 shows the energy levels of a typical defect. The lowest states |a> and |b» are close in energy; their splitting 6 is adjusted by an applied magnetic field to a value convenient for spin resonance. Most experiments adopt a splitting of about 0-3cm~', corre-

sponding to a frequency 2x x 10'° rads~'. States |c> and |d> are appreciably higher in energy. Their separations A, and A, may arise from spin-orbit or crystal-field interactions, and are usually in the

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pote seee a i Tae

——————

|c>

AA:

ae

| 5 see eae (tain

ee AG |b> ehiget a Cs

Fic. 14.2. Energy levels involved in spin-lattice relaxation.

range 20-1000 cm’. These excited states may be assumed unoccupied at the temperatures at which experiments are carried out. A. Nature of the non-radiative transitions. We now consider the recovery of this defect system when the relative populations of |a> and |b» are disturbed. Two types of transition are particularly

important. In one-phonon processes, a single phonon of energy 6 is absorbed or emitted. In two-phonon processes, one phonon is emitted and one, of energy differing by 6, is absorbed. The analysis which we give resembles that in eqns (14.2.45)—(14.2.48), but there are many minor differences. The various calculations differ from each other on two basic questions: the precise definition of the states between which the transitions occur and the order to which some of the intermediatesized terms in the total Hamiltonian are taken. The states between which transitions occur are usually approximate eigenstates of the static-lattice Hamiltonian. It is perfectly possible to use the better approximate eigenstates of the Born—Oppenheimer Hamiltonian (Foglio 1962; Levy 1970), but this is rarely done and is usually unnecessary unless the spin-lattice interaction is exceptionally strong. The total Hamiltonian is written

KH =HA-+Ant+M,

(14.2.50)

where %, is the lattice Hamiltonian, #%, is the electron—lattice interaction, and .%, is the Hamiltonian of the defect ion. This electronic term is itself a sum of terms such as Hy

=

Hyt+Vot

Moot

Mai t+ Az.

(14.2.51)

Here % is the free-ion Hamiltonian, Vj represents the crystalline environment, “%pQ is the spin-orbit coupling, and #7, and #7; are the orbital and spin Zeeman energies. Occasionally other divisions

492

THE

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NON-RADIATIVE

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are more convenient. Thus, if an ion has an orbitally-degenerate ground state in the absence of spin-orbit coupling (e.g. MgO: Coz) then it is useful to separate the effects of the spin—orbit coupling among these degenerate states |g> from contributions involving other states. This can be done by writing

(14.2.52)

H 80); Heo = H$o+(Heowhere #8 operates only in the degenerate states KH so = P erent g?

(142.53)

P, = Yilg>,

(14.2.63)

df

where the denominator shows the electronic energy and the energy of the phonon emitted or absorbed in going from state |i) to |J). We distinguish three cases: 1. ha much smaller than the electronic energy splittings, so that the denominator never vanishes ; this leads to the Raman process ;

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larger than the electronic splittings, so that the never

vanishes;

this

gives

a

modified

Raman

process; 3. the denominator vanishing for interaction with some phonon modes ; this leads to the resonant-Raman or Orbach process. We discuss these in turn. Non-resonant two-phonon processes. When a general term in the matrix element is

there is no resonance,

FIO A1Qilt) | A121) AO) E,—E,;+ho, E,—E,;—hw, —

(14.2.64)

The operators Q, are the parts Of Myansition Teferring to phonon mode «. The first term represents emission of phonon | followed by absorption of phonon 2; in the second term phonon 2 is absorbed first. For non-Kramers’ ions it is now straightforward to calculate the transition probability in the limits (E;,—E, ~ A> how or (E,—E;)A « hw. The probability involves an integral over wm, and @, of the square of the matrix element together with a factor 6(@,—@2+q@) to conserve energy. We assume the temperature is well below the Debye temperature. It is easily shown that the relaxation rate is proportional to

for

(G*/A?)T’

(14.2.65)

GT?

(14.2.66)

A > hw and to

when ha > A. For Kramers’ ions there is a complication, since the equality

C fIQaIT> = — = Mx,r (14.2.67) (Orbach 1961a; van Vleck 1940) holds from time-reversal symmetry. There is a cancellation, only saved by the difference in the energy denominators.

Thus, when

A > hw, a general term in (14.2.64) is

given by

Mx,1

fest

any

At+hw,

mestliest

A-ha,

adh pen Ost Onda 71)

498

NON-RADIATIVE

The Raman

relaxation

PROCESSES

AND

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rate is then proportional

to (G*/A*)T’.

Strictly, other terms occur from Zeeman energies in the denominators; these are invariably small and will be ignored. When A « ha

the relaxation rate becomes proportional to (G*A’)T°. Resonant case: the Orbach process. The resonant case, when the energy denominator may vanish, can also be treated simply. The apparent divergence shows that the perturbation theory being used is inadequate. Heitler (1954) has shown that the correct modification is to change the electronic energies E to E(1 —io), where the spontaneous lifetime is 4/2cE. In practice the spontaneous emission of phonons by the excited state dominates in g, so that the lifetime is virtually independent of temperature (Orbach 19616). Note that the transitions caused by dipolar interaction between the various magnetic ions in the crystal #4, do not contribute to o. Strictly, one is interested in lattice-induced transitions between eigenstates of the sum of “%,, and the single-ion Hamiltonians, since these are the states studied experimentally. However, it is common to ignore HA, entirely when it is small compared with the Zeeman energy. The important term in the transition element has the form M/{A(1—ic)—how}. If o were zero this term would diverge when hw and A are equal. The transition probability is proportional to an integral over phonon modes whose integrand contains the square modulus

|M|?/{(A—ha)? + Aa7}.

(14.2.69)

The other factors in the integrand are the same resonant

case.

These

include

phonon

occupation

as in the nonnumbers,

the

energy-conserving delta function, and the density-of-states terms. If a, is small, the integral is dominated by the phonons with energies close to A. The relaxation rate can be evaluated without difficulty and is proportional to G*A° exp (—A/kT)/o for both Kramers’ and non-Kramers’ ions. Since o is determined by the orbit-—lattice coupling we finda ~ A’G*. The resonant Raman rate is proportional

to G*A? exp (— A/kT). Interference terms. In general there will be non-resonant contributions to the two-phonon the total rate is not the sum ofthe two; there is them (Stoneham 1967). The interference terms more than one electronic excited state giving

both resonant and processes. However, interference between arise when there is a significant part of

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the transition matrix element. In the simplest case of all, suppose one level J gives a predominantly resonant contribution and another level J gives only a non-resonant part. The transition matrix element is asum

M,/{A,(1 —io,)—ho} + M,/{A,—hot.

(14.2.70)

The square modulus of this sum enters in the transition probability and contains cross-terms in M,M,. These are predicted to show a characteristic dispersion-like behaviour: they tend to zero at very low or very high temperatures, and change sign at intermediate values. The interference terms are hard to identify experimentally because they are important in ranges where other approximations, such as the use of a Debye model, cause deviations from the simple theory. The most important effect of interference is that energy splittings A obtained by fitting resonant terms may be in error because of these extra contributions. D. The phonon bottleneck and the phonon avalanche. The calculations we have outlined give the rate at which electronic energy of the defect is converted

into energy

of lattice vibration. In two cases,

the direct process and the resonant Raman process, the electronic energy is transferred at first to the very small fraction of phonon modes whose energies are close to the electronic transition energies. In these processes the rate of recovery of the electronic system to equilibrium may be determined by the rate of energy transfer from these phonon modes to other degrees of freedom and not by the spin— lattice interaction just discussed. The two situations described now are ones in which the rate-determining process is the transfer of energy from the modes which couple directly to the electronic transition. They are the phonon bottleneck (van Vleck 1941) and the phonon avalanche (Brya & Wagner 1965, 1967). The rate of change of the populations of the electronic levels can be found by solving coupled equations for these populations and for the occupancies of the phonon modes. Different formulations are given by Faughnan & Strandberg (1961), Scott & Jeffries (1962), Stoneham (1965b), and Brya & Wagner (1967). The important features can be demonstrated using a simple model. We regard the experimental system as three separate parts: the defect (or ‘spin’) systems, the ‘lattice’ system, including just the phonon modes which

500

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are directly coupled to the defects, and the ‘bath’ system, consisting of all other degrees of freedom. Two simplifying assumptions are made. First, a temperature is assigned to each of the subsystems. Thus each subsystem is assumed to come to equilibrium internally much more rapidly than it does with the other subsystems. It is, of course, well known (e.g. Abragam 1961) that a unique spin temperature 7; cannot be defined in general. However, this affects the details, not the essence, of the argument. The lattice temperature J; is determined by the occupation number for phonons of energy 6 in the direct process. It is important that T, cannot be negative for the direct process, since this would require a negative occupation number. By contrast, Tj, for the Orbach process is defined by the relative occupation numbers for phonons of energy A and (A—0); T,, can take negative values (Stoneham 1965b), as can the spin temperature. The bath temperature Tj is just the macroscopicallymeasured temperature of the host crystal or its environment. The second simplifying assumption is that the transfer of energy between the subsystems can be described by two relaxation times. The first Ts is the parameter calculated earlier and is a measure of the energy transfer between the spin and lattice systems. The second is the phonon relaxation time t,,,. It measures the rate at which the lattice temperature 7, returns to the bath temperature T, after a disturbance. It should not be confused with the phonon relaxation time of thermal conductivity theory, which is concerned with modes of a given wavevector. In thermal conductivity, elastic scatter is important; in the phonon bottleneck elastic scatter is irrelevant. Instead, there is exchange of energy through the surface of the crystal containing the defect (‘spatial bottleneck’) or exchange of energy with other modes (‘spectral bottleneck’). The various mechanisms are reviewed by Stoneham (1965b). We now relate the observed relaxation time to Ts and T,,. The net transfer of energy from the spin system is determined primarily by two factors. One is the coupling to the lattice, measured by ts ‘(T,) = W(T;), where T, enters because the phonon occupation numbers occur in the transition probability. The second factor is the difference between the spin and lattice temperatures (T;— T;.). The phonon bottleneck reduces the observed relaxation rate. It is usually observed when both T; and Tj, are close to the bath temperature T,. The coupling of the spin and lattice systems is sufficient to reduce the temperature difference (T;— T,) appreciably. However,

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the increase of T, is not large enough to have a significant effect on W(T,). In the simplest model the observed relaxation time will be nee

an teen Cs Ce,

(14.2.71)

where c, and c, are the specific heats of the spin and lattice systems. For the direct process the second term usually varies as T~

(ton ~ T°,cs ~ T~?,c, ~ T°), giving one of the best known indications of a bottleneck. The phonon avalanche increases the observed relaxation rate. It is observed during the recovery of an inverted spin population, where Tg is negative. The temperature difference (T;— T,,) (or, more strictly, (1/T,—1/Ts) here) remains large for the direct process, since 7; must be positive. The main effect on the observed rate is that W(7,) is enhanced by the rise in T,. As the phonon occupation numbers increase, the energy transfer between the spins and the lattice rises, and the cumulative effect is the phonon avalanche. Clearly an inverted spin system may exhibit an avalanche initially, and when close to equilibrium may exhibit a bottleneck later in the same decay sequence. 14.2.3. Resonant and non-resonant absorption In the examples considered earlier the absorption of energy in optical absorption

(chapter

10) and

spin resonance

(chapter

13)

was a maximum at an energy equal to some defect energy separation. This is not always the case. In non-resonant absorption the maximum occurs at a frequency

wo ~ 1/t,

(14.2.72)

where t is a characteristic relaxation time. Different processes are known by different names according to the source from which energy is absorbed : electric field: dielectric loss magnetic field: magnetic relaxation or non-resonant absorption acoustic field: internal friction. Thus for dielectric loss one might have resonant absorption from purely electronic transitions and non-resonant absorption due to ionic motion, with characteristic time t determined by transitions between equilibrium positions separated by a barrier. Frohlich (1958) has given a comprehensive analysis of this situation.

502

NON-RADIATIVE

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Two cases will be discussed here. The first indicates the form of the absorption when the resonant contribution is ignored. The second discusses a model system in which the transition from resonant to non-resonant behaviour can be seen.

A. Non-resonant absorption. It is customary to characterize the absorption by a complex susceptibility

Xo) = x iY", which measures the response R of the system to F the applied field. In simple cases the response is the sum of two terms. There is an instantaneous response

R; = iF;

(14.2.73)

such as the electronic polarization under an applied electric field. There will also be a delayed response, Rj, which saturates to Rg,. For simplicity we assume there is just one characteristic relaxation time. Gevers (1946) gives results for a distribution of relaxation times. The response is:

(14.2.75)

Ree = "Kal

Suppose F oscillates with a frequency w. The response also oscillates with this frequency w, but may be out of phase, so that energy is absorbed. The response is given by

eked

R,+

Rg)

dpandae

LS = aLR (Ra= R Ray)

4.2.7 (14.2.76)

When expressed in terms of susceptibilities, this becomes :

NY See

fear (r= LimeXa),

(14.2.77)

so that after a final rearrangement ia

ae

all yo

Xa

aag +71? ROT

~ L402?

(14.2.78)

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The energy absorption is proportional to y’, and has a peak at

Dia lt

(14.2.79)

determined by the relaxation time rather than by an energy splitting. For dielectric relaxation, the results (14.2.78) are the familiar Debye theory (Debye 1945; Frohlich 1958; Daniel 1967). For a perfect ionic crystal at frequencies below the optic mode frequencies Xi = (€.—L/4n and xq = (€—«,,)/4n. For a crystal containing defects which may reorient, at low frequencies y; = (¢)— 1)/4n and Xa 18s determined by the defects. Absorption from a magnetic field is more complicated, since there are two relaxation mechanisms:

the rapid transfer of energy within the magnetic system (spin-spin relaxation) and slower transfer to the phonon system (spin—lattice relaxation; the important

relaxation time is characteristic of this

process). Casimir & Dupré (1938) have shown that y; = 7,Cy/Cy and Xa = X1/t

@ «il,

— Wo

Once

(14.2.82)

Resonant absorption corresponds to the weak-damping limit. A second approach is closer to methods used in magnetic resonance, and can be generalized considerably (Abragam 1961). For definiteness we consider an ensemble of spins with S = 3 interacting with a magnetic field and with some bath, causing relaxation. The density matrix for this system p can be related to the ensemble average & of the spin,

(1+2,)

(2,—i2,)

pee ;

(14.2.83) (2-12) (ae)

The time-dependence of p can be obtained from

ao

ih

—ih = (#2, p]-—(p— po)

(14.2.84)

in which po is the equilibrium value of p and #, describes the interaction with a magnetic field H

KH, = hayS,+4hw;,(S, e+ Se 1").

(14.2.85)

Thus there is a constant field in the z-direction and a smaller field of constant magnitude rotating in the x-y plane. We now seek the steady-state solutions of (14.2.85). The energy absorption from the magnetic field is proportional to &. (dH/dt) and is proportional to

Wi WyW@t 1+qaj{rt*+77(@—@)*

(14.2.86)

The maximum energy absorption occurs at a frequency w,,,

1

Om = —/{1 + (wi +9)0"} > I/t

t./(w6 + @7) « |

> J(@§+o7)

t./(@§ +.) > 1.

(14.2.87)

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One interesting feature is that this model shows the effects of saturation. When the exciting field is large, that is when a, is appreciable, both the frequency and magnitude of maximum absorption are affected. Other discussions of the transition between resonant and nonresonant behaviour are given by van Vleck & Weisskopf (1945), Daniel (1967), and Sturge (1967). 14.2.4.

Temperature dependence of the zero-phonon line

A. Line-widths and lifetimes. The zero-phonon line contains contributions from all transitions in which the total number of phonons is unaltered. The mean energy and line-width of the zero-phonon line change with temperature as the transitions which contribute to it vary in relative importance. This is still true if the individual transition energies are independent of temperature. The source of the temperature dependence is the electron—lattice interaction. In practice two temperature-independent contributions may be important. One comes from random internal fields, which broaden the zero-phonon line inhomogeneously by causing the transition energy to vary from one site to another. The other, the homogeneous part, is related to the finite lifetime of the states involved against the emission of photons or phonons. The simplest form of the relation is the Weisskopf—Wigner (1930) result

l= WwW,

(14.2.8)

relating the full width at half intensity of the transition I’,, to the reciprocal lifetimes of the two states involved. Their original result referred to photon emission alone;

we shall, of course, be concerned

with the effects of electron—lattice coupling. It is not obvious that the Weisskopf—Wigner result holds for the systems under discussion, and there has been controversy on this point (Brout 1957; Culvahouse & Richards 1969; Stedman 1970a, b, 1971). The most complete discussion is that of Stedman (1971), who obtains the line-shape by a diagram expansion. His result is valid for all orders of perturbation theory and for both linear and quadratic electron—lattice coupling. Stedman shows that the transition probabilities W,, W, may be obtained from the self-energy operator of the one-particle Green’s function. The line-width I is obtained from a two-particle Green’s function. There proves to be a certain class of terms in I’ which violate

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the Weisskopf-Wigner equation (14.2.88). This class consists of those transitions in which there is no net energy transfer between the defect and the lattice. Stedman & Cade (1973) have demonstrated the importance of these terms for Nd** in yttrium aluminium garnet. However, the Weisskopf—Wigner equation often gives results which are qualitatively correct, such as the temperature dependence of a line-width. A second general result concerns the shifts of zero-phonon lines. Given a set of N electronic levels, can a set of N shifts be defined,

one for each level, which transition energies due to has shown that such a set although in many systems of accuracy.

give correctly the shifts of the N(N —1)/2 electron—lattice coupling? Stedman (1972) of N shifts cannot be obtained in general, they can be defined to an adequate degree

B. Energy levels and electron—lattice coupling. Systems with a variety of different energy-level structures have been treated in the literature. Some are shown in Fig. 14.4, where the range of phonon energies is also indicated. Case (a) (Fig. 14.4) is fairly common in that the electronic transition energies are much larger than the maximum phonon energy. Krivoglaz (1962, 1964), Silsbee (1962), McCumber (1964), and Ganguly (1970) consider this system. Case (b) also occurs often : the R-lines in ruby come from such a system. The transition energy is much larger than the phonon energies, but there is a third state separated from one of the others by less than the maximum phonon energy. This situation has been treated by McCumber & Sturge (1963), Kushida & Kikuchi (1967), and Kushida (1969). The third class, discussed by Barrie and co-workers (Nishikawa & Barrie 1963;

Barrie & Nishikawa 1963; Barrie & Rystephanik 1966) includes donors in silicon. The transition energy is less than the maximum

ee (a)

(b)

N (c)

\\

Phonons

Fic. 14.4. Relative energies of electron and phonon systems for zero-phonon widths.

line-

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phonon energy. Related cases have been discussed by Rodriguez & Schultz (1969) and by Harris & Prohofsky (1968). The Hamiltonian consists of the usual three terms: an electronic term independent of lattice configuration, a harmonic lattice term independent of electron state, and the electron—lattice coupling. The coupling contains terms linear in the displacements “4 and quadratic in the displacements #%,. The linear terms operating within each electronic state are responsible for the Stokes shifts; they cause the mean lattice configuration to depend on the electronic state. The quadratic terms mix the normal modes so that the modes appropriate for the system in electronic state |i) are different from those for state |j>. If the linear terms are

H, = ¥.V(agtaz)

(14.2.89)

q

and the quadratic terms are

Hy = Y Vaqlag + 4g (Gq + 4¢'),

(14.2.90)

4.4’

then two assumptions are almost invariably made: 1. that the electronic operators V, depend on q in the same way in all electronic matrix elements

= C;,S(qQ);

(14.2.91)

2. that the second-order term can be factorized in the sense

Voge = OV Var

(14.2.92)

where « is independent of q and q’. The first approximation is probably acceptable. In the long-wavelength limit the coupling to acoustic phonons is roughly proportional to the strain, which varies

as |q|?. Thus all the factors S(q) in (14.2.91) satisfy

S(q) ~ lal.

(14.2.93)

Almost all workers use (14.2.93), or the essentially equivalent form in which S(q) varies as wz. The second approximation, which assumes that modes which dominate in quadratic coupling are just those which are important in linear coupling, is probably wrong. Modes which are forbidden by parity to contribute linearly can contribute in second order. Nevertheless, we use (14.2.92), faute de mieux.

C. Line-widths and -shifts. Two main classes of methods are used here; the first includes moment methods, and the second uses

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Green’s function approaches and various diagram techniques related to them. We shall not discuss here the simple but important shift coming from thermal expansion of the host lattice. Moment methods are less satisfactory, since there is a problem in truncating the electron-lattice coupling so that only the shape of the zero-phonon line is considered. Only the non-resonant case (Fig. 14.4(a)) is easily treated. Silsbee (1962) discusses this system for just quadratic coupling. The truncation is performed in two parts. First, one drops terms in ™, in which the net number of phonons is altered (e.g. ag aj: or a,a,,). Secondly, one divides the terms in the first four moments obtained with the truncated #, into two groups. The division is by direct inspection of which modes contribute, or by comparison with perturbation theory. The first group of terms is associated with the zero-phonon line. The second group gives a weak broad background which contributes to the moments, but which gives negligible intensity. This second group is dropped. The line-width is then deduced from these four truncated moments. This is only possible with a qualitative assumption about the shape (see e.g. Grant 1964). Silsbee argued that the centre of the zero-phonon line should be Lorentzian, but with a faster fall-off in the wings of the line. This shape can be represented by a cut-off Lorentzian (Kittel & Abrahams 1953) which is zero beyond a particular separation from the centre of the line. The cut-off is chosen to fit M, and M,, the second and fourth moments. The full width at half intensity is then

T

r= FN Ona)

(14.2.94)

The displacement of the line A is found from the odd moments. The results show that the width is given by

T = «hw, f(T),

(14.2.95)

where f(T) varies as T’ at low temperatures and T? at high temperatures, and that the shift variation is

A = ahwpg(T),

(14.2.96)

where g(T) is proportional to T* at low temperatures and T at high temperatures. The cut-off proves reasonably large in all limits.

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The values of I(T) and A(T) can be related to other physical

observables. If we introduce a weighted density of states PA@) = Y |VI? (@— W,)

(14.2.97)

q

then we find wo

A(T) > & { ae p(w) {n(co) +4}.

(14.2.98)

10)

If V, ~ wg then |V,|? ~ hw,, and the shift A(T) is proportional to the heat content of the crystal. It seems to be a general rule that line-shifts are related to some macroscopic crystal property; this is the case for inhomogeneous broadening too (Stoneham 1969). The form ofthe integrand in (14.2.98) can be understood from the configuration-coordinate model. If the effective frequencies in the ground and excited states are w,, w, respectively then the zerophonon transition with n phonons initially present has energy E, = Ey +nh(, —©,).

(14.2.99)

But the most probable value of n is n, the Bose-Einstein value. Thus

A(T) = nh(w, —@,),

(14.2.100)

which corresponds to(14.2.98) apart from a temperature-independent term. Further, we can rewrite (14.2.100) in terms of the fractional change in force constants (AK/K),

A(T) = fiho(AK/K) + kT(AK/K) asT— ©.

(14.2.101)

Comparing the last result with (10.8.12) gives a special case of a general result: for non-degenerate states, the centroid of the broad band and the zero-phonon line shift by the same amount with temperature. The line-width takes the form

I(T) — a?n

Om

dw {p,(w)\?n(w) {n(w)+1}.

(14.2.102)

0)

Ganguly (1970) has noted that the temperature dependence of the Raman cross-section can be related to the functions f(T) of (14.2.95) and g(T) of (14.2.96) Ramis

=

wl) —

f(T)+8n1n2.9(T) ORaman(O)

f(0)+8x

In 5 ; 2(0)

9

dd

4.2.103

)

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THE

This result is derived using approximations leading to (12.5.24) for the Raman cross-section. Both resonant and non-resonant cases (Fig. 14.4) are readily treated in the Green’s function approach, where one examines the effect of the electron-lattice coupling on the one-electron density of states. This function is, in effect, a susceptibility in which the response of the electron creation operator A(t) to an external field FA*(t) is measured. It is defined by

peE) = >) W,, O(E,— Em—E), —(14.2.104) where W,, is the thermal weighting factor (11.5.15) and is related to G,(E), the one-electron Green’s function, by

pE) = i{Go(E +io)— Go(E —io)}.

(14.2.105)

The Green’s function is found from its Heisenberg equations of motion, as in our discussion of lattice dynamics in chapter 11. However, the solution of the equations is more difficult here: a hierarchy of equations is formed which does not give a closed finite set of equations without approximation. The usual approximation is to factorize the Green’s functions into electronic and lattice parts at the lowest non-trivial order. Thus

KO) {AG

AMSBrae

CAR

AMEE,

(14.2.106)

where the definitions are analogous to those in § 11.5 and Q,,,, is some operator involving only lattice variables. The one-electron density of states takes the form

pAE) =

ie 1 2n (E—Ey—A)? +(1/2)2

(14.2.107)

instead of d(E—Ep), in the absence of coupling. For very weak coupling, the result resembles Brillouin—Wigner perturbation theory in that the perturbed energy appears in the denominator. In general, the shift, A and width I are given by complicated expressions and depend on energy. Usually the variations of A, I and the optical transition matrix element with E are ignored, although several authors (e.g. Nishikawa & Barrie and Stedman) have gone beyond this approximation. Then p,(E) is a simple Lorentzian shape which gives the shape of the zero-phonon line. The simplest contributions to Aand TI are the ‘direct process’ terms, which appear because there

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are other states |k> which can be reached from |i) by the emission or absorption of a single phonon. These give 1

A(E)= YY ICiel1SQl?5 (1+ 1 2! a Pat) a ne

sic

.

(14.2.108)

T(E) = 20 bY 1Cial1Sql?{(%q + 1) O(E — Ey,— Eq) +

+n, (E—E,,+E,)}.

(14.2.109)

There are also higher-order processes for which more complicated expressions are needed. They are similar to the earlier expressions. However, since E;, and E, can be equal, further resonant processes

are possible. These are analogous to the Orbach or resonant-Raman processes of spin-lattice relaxation, and can lead to an exponentiallyactivated dependence of the width on temperature.

D. Resonant phonon interaction. In special cases an electronic transition almost exactly coincides with a phonon energy. Examples are Si:Bi (Onton,

Fisher, & Ramdas

1967a), where the Ils > 2p,

donor transition is very close to the transverse optic phonon energy at the (100) zone boundary, and Si:Ga (Onton, Fisher, & Ramdas 1967b), where one of the transitions lies close to the silicon Raman

energy. In this case, the zero-phonon line-shape needs special treatment and has been discussed by Rodriguez & Schultz (1969) and Harris & Prohofsky (1968). Harris & Prohofsky’s treatment is appropriate when the electronic transition is coupled to one mode only, such as a local mode, when this mode is weakly coupled to others. Rodriguez & Schultz’s treatment includes a continuum of phonons and adopts selection rules appropriate to silicon. Their case should describe the Onton, Fisher, & Ramdas results, but no detailed comparison has been made to give estimates of parameters. Rodriguez & Schultz calculate the one-electron density of states in the presence of electron—lattice coupling. An isotropic hydrogenic defect wavefunction is assumed, and the natural radiative decay of

the centre included. The results are described in terms of a function T’.,(@), which describes the energy dependence of the electron— phonon coupling summed over all modes. I’,, is assumed to have a peak value I’, at m = Wo, usually close to a peak in the phonon density of states, and a half-width 6; wp lies at energy A below the

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energy of the electronic transition in the absence of coupling. The resulting line-shapes can be described as follows. Usually there are three peaks, although they may not all be resolved. One peak is close to the pure electronic transition at ¢ ~ @)+A, but is pushed further away from the phonon energy near @, as the coupling increases. A second peak is close to Wg, and a third peak appears near (W, — A). When A is large, far from resonance, the peak near ¢ dominates, as expected. As A is reduced, the other peaks at wy and

(@ — A) increase in intensity, so that the line is symmetric about wo for the exact resonance A = 0. The precise line-shape depends on 6, the width of the band of phonons to which coupling occurs, relative toits peak value I). As 6is reduced, the three-peak structure becomes more clearly resolved, and the two peaks at (@ )+A) become more important relative to the wo peak. Onton, Fisher, & Ramdas (1967a) managed to vary A for the Si: Bi system by applying uniaxial stress. Their results do not show a three-peaked structure but a single line, which was asymmetric except for A ~ 0. As predicted, the asymmetry of the shape corresponded to the transition being pushed away from the phonon energy by the coupling. The lack of resolved structure suggests that the width of the band of phonons coupled 6 is quite large relative to I), although inhomogeneous broadening may smear out some of the features. Regrettably, none of the papers attempts to estimate the parameters involved, and it is hard to do so from the published data. 14.3. Scatter of conduction electrons 14.3.1. Introduction

The scattering of free carriers by defects affects the thermal and electrical conductivity, the Hall effect, magnetoresistance, and the Thomson, Ettingshausen, and Nernst coefficients. All of these

quantities can be related to thermal averages of the sort

CE",

(14.3.1)

in which E is the carrier energy and t,(E£) a characteristic scattering time (e.g. Smith 1959). We now examine 1,(E) in various cases. The scattering time t, is related to a cross-section o, by

ty! = Now,

(14.3.2)

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where there are N defects per unit volume and v is the electron velocity. The effective cross-section for use in transport problems o, is related to the differential cross-section by 2n

C=

Tt

i ag | dé sin 6 (1 —cos A)a(8, ¢). 0 0

(14.3.3)

This expression may be derived (Ziman 1960, §7.4) as part of an exact solution of the Boltzmann equation, assuming that the energy surfaces are spherical (so that the electron velocity vector vy is always parallel to k) and that the differential cross-section depends only on the angle through which scatter occurs, not on the absolute orientation in space. The corresponding expression for the total cross-section is 2n

T

gwgele iL do i dO sin 0 o(6, d).

(14.3.4)

The mean time between collisions t is given by con

=

Nororv.

(14.3.5)

Both o, and o7o7 can be expressed in terms of phase shifts 4n

2

o.= Ta) bsin? (m—1—m)

(14.3.6)

l=1

4n & . Ae = y (21+ 1) sin? 1).

(14.3.7)

l=

The two cross-sections are identical for s-wave scattering, in which No is the only finite phase shift, 4n ay

=

OTono

=

.

psi’

No-

(14.3.8)

We shall now calculate cross-sections in a number of cases. All the systems studied will be assumed isotropic. This is, of course, a simplification. However, various factors make it a tolerable assumption. For example, polaron effects are especially important in ionic crystals at frequencies w less than the longitudinal optic frequency @,9, and the polaron effective mass is usually less anisotropic than the band mass (Pekar 1968, Kahn

Borders, & Brown

1968; Hodby,

1969). A further assumption, useful for showing

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trends, is to assume that all electrons have the mean kinetic energy 3kT. This assumption is acceptable in expressions which do not vary rapidly with electron energy. 14.3.2. Limiting cases

We begin by describing results for very energetic and for very slow electrons. For electrons whose kinetic energy is large compared with the scattering potential, the Born approximation is appropriate. It may be expressed concisely in terms of the phase shifts

ny =

otal {+ (kep\i2 Fe | dr 2 r°Vir)i(kr)},

(14.3.9)

(@)

where j,(kr) is a spherical Bessel function (Messiah 1962, p. 406). A related approximation, the distorted-wave Born approximation, may be used when the total potential is U(r)+ V(r) and when one knows the regular radial solutions kry(kr) for the potential U(r) alone

fle aoe {edr r?V(r){y(kr)}?. Jo h°

(14.3.10)

This expression is particularly useful for examining the effects of weak long-range parts of the perturbation when the short-range terms can be solved exactly. In the opposite extreme of slow electrons, effective-range theory (Newton 1966, p. 308 ; Messiah 1962, p. 406) is suitable. This amounts to an expansion of k cot in powers of the energy. Let us define u(r) and v(r) as the regular solutions of (14.3.11): h2

d2

h2

[ass gatM5 heeds

i

h?

[2 aes

i

and let uo(r) and vo(r) be the limits of these functions as ¢ tends to 0.

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The s-wave phase shift is given by k cotyo = ——+e| dr (vvg — uu) a 0 1

(oo)

= --+e| dr (vg —ud) 4

=

0

wee

(14.3.12)

The parameter a depends on the potential: a < 0 if there are no bound states of V(r), a > 0 if bound states exist, and a = oo if there is a bound state of energy zero. For a shallow bound state of binding energy Eg = h?|e,|/2m* the effective range is given by

2.~

2

agers ‘ key ake

(14.3.14)

in which |e,| = kg. At the lowest energies, when er,,, is negligible, the cross-section has the limit (Landau & Lifshitz 1959a, § 109)

h?

1

C-=

14.3.3 Scatter by neutral impurities We now discuss scatter by defects analogous to the neutral hydrogen atom with effective Rydberg R and Bohr radius a”*. At low energies pure s-wave scatter dominates, so that only one phase shift is needed. In this limit 0, = o,, and the scatter is isotropic. Erginsoy (1950) extrapolated the results of Massey & Moiseiwitsch (1950) on hydrogen to the scatter of electrons by donors. The essential result is that for low-energy electrons, typically those whose energy satisfies S16) ee Sik, (14.3.15)

the cross-section varies roughly inversely with the electron velocity,

ay

(14.3.16)

A ~ 20 is a constant determined by a calculation using Hulthen’s (1944) variational principle, including exchange and correlation.

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For a thermal electron at room temperature o, is about 2x 10° (a*/1 A) (A)?. The important feature of o, is that t, is independent of the incident electron velocity

1 * h b= No. act /[™ =N,.4.—,

(143.17)

and so is independent of temperature. This scattering mechanism becomes important below 100 K. These hydrogenic defects provide a simple example of spindependent scatter: the cross-section is different for carriers with spins parallel (o;) or anti-parallel (c,) to the trapped electron’s spin. If the carrier polarization is measured by P, and the defect spin polarization by P,, where

pe tae Nip tniy

(14.3.18)

then the appropriate average cross-section 1s

o = (1—P.P,)o,+4(3+ P.P,)o, + spin-independent terms. (14.3.19) The Erginsoy result A ~ 20 is the spin-averaged result

6 = 46,+-36,. However, Schwartz (1961) has obtained cross-sections separately

(14.3.20) the singlet and

k?a5 = 48-8 (ka*) k?o_ = 10-4 (ka*) }. k?é = 20-0 (ka*) A number been made.

of measurements For semiconductors

of spin-dependent (Maxwell

triplet

(14.3.21) transport have

& Honig

1966; Honig

1966 ;Ohyama, Murase, & Otsuka 1968) the ratio (a5/a,) is about 4; close to the theoretical 4.654. For scatter by F-centres in ionic crystals the ratio is much larger, and o,/o; 2 38 for KBr (Hodby, Borders, & Brown 1969). The large ratio presumably means that the system with anti-parallel spins has a resonance which is very nearly a bound state. If a bound state existed it would correspond to an excited state of the F’-centre. There is no evidence for such a bound state (Hodby et al. 1969 ;Crandall 1965).

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14.3.4. Scatter by ionized impurities

The scattering cross-section for a Coulomb potential can be found exactly, and is given by the Rutherford formula Daye o(0, d) = (S| nisin O12) ime) (14.3.22) which we write in a form suitable for comparison with the screened Coulomb case later. The long range of this potential causes sufficient small-angle scatter to lead to a divergence in o,. The divergence is unphysical for two reasons. First, in any real system there are many scattering centres, and the scattering potential should be the superposition of their contributions. Second, there are many charge carriers, and these screen the interactions with the scattering centres. In simple situations both features lead to similar changes in the cross-section. Conwell & Weisskopf (1950) assumed that the scatter of particles with impact parameter greater than r,, could be ignored, where r,, is related to the average defect separation

r, = 4iN;)+.

(14.3.23)

This is equivalent to ignoring scattering through angles less than 6... Where the equation

Ze m*v

E

6. = 2tan-? (|

= 2tan-! 2]

(14.3.24)

defines an energy E,,. The conductivity cross-section becomes Ze2

2

E

2

mae 2n| 5 ops 25| In 11+|2—] Z| >.

(14.3.25 )

Usually E is much greater than E,,. Thus, for ¢ = 10 and 101° scattering centres per cubic centimetre, we have E,, = kT,,, where T,, is 8 K. Clearly, at room temperature E ~ 3kT > E,,. The cross-

section at room temperature is about 2 x 10* (A)?. Brooks (1955) and Dingle (1955) assumed instead that the Coulomb

potential was screened, Z

Vinee

eh

(14.3.26)

ér

a

Br

4ne*n\ ~} | ekT | ‘

1327,

(

)

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where we have assumed a single type of carrier with concentration n; Brooks discusses other situations. In the Born approximation the scatter cross-section becomes 1

6, 6) =

Ze2

2

h2

4 —

in?(0/2)+———=,

=o

,

(14.3.28

which may be compared with the earlier result. At low energies the Born approximation becomes invalid; a phase-shift analysis by Blatt (1957) examines this low-energy case. Typically, the Born expression is good above 100K, but then falls off too fast with temperature. The conductivity cross-section is

Zen ){naan em*y? 14+X

o3P — 2n c

(14.3.29)

in which X can be written

x=

a

ie

-1

1

| | = E/E,. 4 2m*d? 4m*v?

(14.3.30)

defining E,. At room temperature E is much larger than the other energy E,. The general form is very similar to the earlier case, especially when the argument in the logarithm is large 2

Zee oo

—>

2n ( Em”

2 In (EJ ES)

v

Wa 2

(14.3.31)

2

ge, on|=| In (E/E,) EmMm”~vU

where E,, = 2e?/er,, and E, is h?/8m*d?. We may also derive mobilities using 1/t- = No,v to find the energy dependence (and hence temperature dependence),

1

N(Ze?2/e)?

(14.332)

ow = ae = ttn(1+ (2E/E,,)*}], 1

{BD

{aN(Ze?/e)?

;

1

}JQm* E>) i {1+(E/E,)} aE

(14.3.33)

In both expressions the appropriate (single-valley) effective mass

m* is an average {3(mft) *+4(m*)7 of the longitudinal and transverse components.

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14.3.5 Other approaches The methods we have described use effective-mass theory implicitly. Other approaches are possible, such as the Green’s function method of §5.3. Eqn (5.3.30) gives the behaviour near a resonance. We now write down the corresponding formula for S-wave scatter by an isoelectronic defect which can bind an electron. In terms of our earlier notation, at very low energies,

Oh Fe op, c

1

AP

1

m*

pa

2

1— AG(0)



(14.3.34)

This cross-section can be large if the state is weakly bound. In our Koster—Slater example the wavefunction of the bound state falls off as exp(—r/L) at large distances. Writing o, in terms of L we obtain a result which may be considered a special case of (14.2.14)

o, = 4n.4.L?.

(14.3.35)

Faulkner (1968) gives similar expressions, slightly modified because of the three-valleyed structure of GaP. He cites ¢ ~ 4x 10’ (A)? for GaP:N. This large value is of the order of the geometric area of

the orbit 27L?. At high energies we might expect the scatter by an isoelectronic defect to be similar to that from a spherical square well of appropriate depth V, and radius do. In the effective-mass approximation, which

TABLE

14.2

Summary of scattering mechanisms Mechanism

Energy dependencef of a,

Typical valuet of o, (A?)

Near resonance (J 4.3.14)

(E+E,) ' (low energy)

3-6 x 10?

Neutral defect (14.3.16)

E~? (low energy)

13x 10°

E~? In {1+(2E/E)?} E-7In {(1+E/E,)—EM(E+E,)}

1-5 x 10* DV NO

Coulomb defect:

Conwell—Weisskopf Brooks—Dingle + Energy and concentration

dependence

of 1I/t are E*N times those of o,; Ey

varies as N+ and E, as N*'. t Data used: N = 10'5cm73,m = mo, = 10,Z = 1, kT = 0-:0375 eV. The effective Rydberg R = 0-136 eV with these figures.

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is probably good enough for trends, the Born approximation gives a differential cross-section (Schiff 1949, p. 167)

2m | 4 a0, 6) = «| 2

sin x — X COS x x3

bs

(14.3.36)

where x is ka. 2sin (0/2). Typical capture cross-sections are shown in Table 14.2.

14.4. Capture of electrons in solids 14.4.1. Introduction

We now discuss the non-radiative capture of electrons in solids. This is important in transport phenomena. Minority carrier mobilities are affected if they are captured and subsequently emitted by the trap. Further, if the minority carrier is trapped for sufficiently long, recombination may occur when a majority carrier is captured. There is a qualitative difference between traps and recombination centres which depend on their relative cross-sections for minority and majority carrier capture and also on the relative carrier concentrations. Specifically, if the minority carriers are holes we expect defects X~ to be recombination centres; they are attractive to the holes, and the species X° formed on capture may have a substantial electron-capture cross-section. On the other hand, defects X?~ are attractive to holes but repulsive to electrons, so they may be less efficient in causing recombination. The general orders of magnitude are shown in Table 14.3. The measured quantity is the rate coefficient ; we have quoted values of /, which is an appropriate average. For the giant traps, the cross-section may exceed the geometric area of the

orbit 2(a*)?, typically 10? or 10° A?. TABLE 14.3 Typical capture cross-sections

Type of trap

Example

Typical cross-sections

(A?)

Giant trap Neutral centre

Electron capture by Ge:Sb*t Electron capture by Ge: Ni°

10' to 10* 107' to 10!

Repulsive centre

Electron capture by Ge:Mn~

10-* to 1075

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521

Electron capture requires that the electron loses kinetic energy in some way. There are several different processes: (a) radiative capture: the emission of a photon; (b) Auger processes: the transfer of energy to other carriers; (c) phonon emission. In this case there are several subdivisions, notably multiphonon emission and cascade processes. Similarly, we distinguish between cases where the transition energies between electronic states are less than the maximum phonon energy (so that one-phonon processes are possible, in principle, and other cases where multiphonon processes are essential (although not always rate-determining). The general results are summarized in Table 14.4. TABLE 14.4 Capture mechanisms Mechanism

Typical cross-

Radiative

10m o=10ma Ac

section Conditions for dominance

Auger

Wide variation;

Phonon

10m tOmAc

10~' A? typical Depends on states and energies

High carrier and defect concentration

Low defect concentration

(N > 10'7cm~? typical for Ge) Temperature dependence

Small deperdence

Small dependence

Power law or exponential

Concentration dependence

None

Linear with concentration until screening

None

is important

Bonch-Bruevich & Landsberg (1968) have reviewed the experimental evidence on capture, and make a number of important points. First, the charge of the centre plays a role, but is not always the decisive factor. An attractive Coulomb potential, of course, does

favour a high cross-section. But neutral centres and centres with a repulsive Coulomb interaction may have similar cross-sections; the repulsive centre may even be more efficient in capturing carriers. This observation shows that the detailed electronic structure of the centre must be considered if any but the crudest results are needed. Indeed, the capture mechanisms for neutral and repulsive centres

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do not seem to be understood. None of the calculations to be described discuss such details. The only two features in this category which are discussed are the question of intervalley transitions for a multivalleyed semiconductor, and the relative magnitudes of maximum phonon energies and of energy spacings of the defect. Secondly, the temperature dependence is usually of the power-law form T~", with 2 S n S 5. Capture does not seem to be a thermallyactivated process, except possibly for centres with a repulsive Coulomb interaction (e.g. Sklensky & Bube 1972). It is unlikely that direct capture into the defect ground state with emission of one phonon is significant. Thirdly, the most rapid capture is the non-radiative capture by deep centres whose energy is near the centre of the energy gap. Radiative capture is particularly favoured when the transition energy is large, of the order of the band gap. Finally, the capture rate can be significantly affected by applied fields and by deviations of the carrier energy-distribution function from the thermal-equilibrium value. One problem in comparing experiment and theory is establishing that the conduction mechanisms correspond. At low temperatures and concentrations, hopping conduction may occur (e.g. Miller & Abrahams 1960). In this process conduction is achieved by the hopping of trapped carriers from one trap to a nearby, empty trap. This may be contrasted with conduction by free carriers which is occasionally interrupted by absorption by, and subsequent emission from, a trap. Brown (1966) has discussed the conditions for hopping conduction and argues that the cross-section should vary as T~* in this regime, changing over to a more rapid dependence at higher temperatures. The data for acceptors in Ge of Besfamil’naya, Ostroborodova

& Shlita (1969) show

these features, and may

be

illustrating just these mechanisms. 14.4.2. Capture by phonon emission In treating phonon emission we shall assume that we may concentrate on the single-carrier—single-trap problem; the screening radius and the mean separation between traps are both very large. We also assume

the electron-phonon

interaction is weak, so that

the mean free path for phonon collisions with free electrons is long. The opposite extreme is appropriate for capture of self-trapped carriers like the V,-centre (chapter 18). In the present cases the length for comparison is an orbital radius, roughly fixed by

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523

|V(r)| = kT or by o?, whichever is the larger. These assumptions are more restrictive than might be apparent. Suppose there are N traps per unit volume. The most probable separation of one trap from the nearest one in a random distribution is rp = (21N>p)*. Clearly, excited states whose characteristic radii r, are greater than rp/2 cannot be treated in the single-trap approximation. For donor states in Ge at 10'? donors per cubic centimetre this restricts us to

states with principal quantum number n < 10; for 10!* donors per cubic centimetre in Si states the restriction is n < 7 (Brown 1966). These concentrations are typical of values used in measurements of capture cross-sections.

A. Cascade processes. The cascade hypothesis asserts that the electron is captured into some high-excited state and subsequently loses energy, mainly by a cascade of one-phonon transitions between the defect states. The alternative processes are multiphonon transitions, possibly including direct capture into the ground state. Multiphonon processes are essential in deep traps, when electronic states are separated by more than the maximum phonon energy (BonchBruevich & Glasko 1962). We shall be mainly concerned with donors and acceptors in silicon and germanium, where the optic and maximum acoustic phonon energies are greater than the donor ionization energies. Here multiphonon processes are too small by several orders of magnitude (Gummel & Lax 1957). It is important to decide into which state the electron will be captured. Clearly the most highly excited states, with ionization energy less than kT, will not be important; an electron captured into such a state will be lost rapidly. There are two different schools of thought on the dominant states for capture. One school (Lax 1960; Hamman

& McWhorter

1964; Smith

& Landsberg

1966)

assumes that it is the higher excited states of large radius which dominate.

The other school (Ascarelli & Rodriguez

1961a, 19615,

1962; Brown 1966; Brown & Rodriguez 1967) argue that capture is mainly into low-lying s-states. We discuss both, since it is likely that the correct picture contains elements from each of the two concepts. The cascade mechanism predicts the cross-section for the capture of a carrier into the ground state of the defect. The mechanism of capture consists ofthe initial capture of a carrier with kinetic energy

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E into an excited state |x), with cross-section o,(E), and subsequent transitions among the bound states. These latter transitions are characterized by a ‘sticking probability’ P,, the probability that an electron in state «> will enter the ground state before escaping from the trap. For P, to be well defined, this probability must depend only on the state the electron is in and not on how the state was reached. The cascade must be a Markov process in energy. Further, the observed decay time for recombination or capture is assumed to be much longer than the timescale for the transitions between the bound states; if this were not so, the recombination or

capture would not have a single characteristic time. The observed capture cross-section is then given by

o(E) = ¥0,{E)P,

(14.4.1)

a

for capture into the ground state. Beleznay & Pataki (1966) have argued that a different cross-section is measured and that capture into certain low-lying excited states is also effective.

The Ascarelli-Rodriguez model and its generalizations. We begin by outlining the Ascarelli & Rodriguez (196la, b) calculations of o,(E) and P,, together with later, related work.

The initial capture cross-section ¢,(E) is not calculated directly. Instead the average of o,(E) over a thermal distribution of carriers is found by relating it to the corresponding thermal ionization probability w,, using the principle of detailed balance (§ 10.6),

aii

=

—hw,. 8

exp (I,/kT) ET ie

(14.4.2)

In this equation g, is the degeneracy of |x, and I, is its ionization energy. The thermal ionization probability is found as in our treatment of photoionization (§ 10.10). It includes contributions from all accessible excited states, and is induced by the electron— phonon interaction. Only interaction with acoustic phonons is considered

here. We concentrate

on Coulombic

centres, such as

the capture of an electron by an ionized donor. The electronic states between which transitions occur are thus described by

OF

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52)

Coulomb wavefunctions. If |«> is an s-state of principal quantum number n, then

250

Re win Ui

m — p(a*)°(kT)* n? tes 3 exp Q, =2 ) dx. x°.

OF;

(14.4.3)

i eexa(2ney esas) Sree‘st RD? | (14.4.4)

Here a* is the effective Bohr radius of the lowest state, = is the

deformation-potential constant, and v, is the velocity of sound. The use of Coulomb wavefunctions is particularly important. If plane-wave

continuum

states

are

used,

the

predicted

thermal

ionization probability is changed by a factor R to R,w,. For the n=1 ground state R, = 32(h/m*a*)/, which is small and temperature-dependent. Using the parameters for Ge at 4K, Ascarelli & Rodriguez find R, = 00061, R, = 0-011, R, = 0-020,

and R, = 0-053 for the first few s-states. Plane-wave states given an underestimate of the capture rate. Specifically, R,w, is given by the right-hand side of (14.4.3) with Q, replaced by

=

kT\> 16/2) aor exp(—I,/kT).

(14.4.5)

The second part of the calculation estimates the sticking probability by setting up a system of coupled equations for the P,. We define the probability per unit time W,,, that the captured electron makes a transition between bound states |«> and |). The fractional probability of this transition is k,,, kup z= Wy

w+) We)

(14.4.6)

B

and the fractional probability that the trap will be ionized is i, = va | (v4 y Ws)

(14.4.7)

B

The sticking probability in state « is then P,, where

Ryall) in

(14.4.8)

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where i” is the probability of ionization after m transitions, without first reaching the ground state. Combining these equations (cf. Beleznay & Pataki) gives the so-called Kolmogorov equation Ae



2d KapP

(14.4.9)

The k,, are combinations of transition probabilities which can be calculated directly, like the w, which have been quoted already. For illustration we quote the W,, for hydrogenic s-orbitals with principal quantum numbers n and n’ (n> 17’). With the Born approximation and the assumption that a* is greater by a large factor than the wavelength {(J/,,—J,,)/hv,} of the phonons relevant for the transition

cm m

(nn’)°

(a*)? p(a*)° I} (n—n’)°(n+n')?{1—exp(—1,—]/kT)}

(14.4.10) The transition probabilities are appreciably smaller when / « 1. For non-s-states the transition probabilities are reduced by a factor

of order 47“*"), where | and I’ are the angular momenta involved. The set of equations for the P, is infinite, and the k,, which appear in them are too complicated for the set to be solved exactly. Ascarelli & Rodriguez made two assumptions to simplify their equations. The first assumption was that only a few of the states |x) need be considered, specifically the 1s-, 2s-, 3s-, and 4s-states. This assumption was based on the dependence of the w, and the W,,, on n and on the angular momenta / and |’. The second assumption was that the P, were given to sufficient accuracy by the ratios of the probabilities of a one-step transition to the ground state and direct ionization

P, =

Wat Wat

Ugy

=

k eee 5m

ew

k bake

14.4.11 (

)

Beleznay & Pataki have examined these assumptions in some detail, and conclude that neither assumption is valid. However, straight-

forward extensions make the theory much more satisfactory. Thus the equations for the P, are solved for a finite, but large, set of lowlying states, including those with finite angular momenta. The relatively small probabilities W,, for non-s-states are compensated for by the increasing number of these states as n increases. The general

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527

effect of including these states is to reduce the sticking probability in a given state. Brown & Rodriguez (1967) have generalized the original Ascarelli— Rodriguez work in some respects. They have treated the longitudinal and transverse acoustic modes separately, and find the distinction important. Also, they have corrected some numerical errors in the earlier work, and used more recent parameters. The effect of these changes is to alter the various transition probabilities. Brown & Rodriguez then argue that, in consequence, the correction suggested by Beleznay & Pataki for states of higher angular momentum is unnecessary. No numerical support of this statement is given. Later we shall compare the various models quantitatively and show that the Brown & Rodriguez values of the sticking probability are much larger than those of Beleznay & Pataki. Both Si and Ge have conduction bands with several minima. There has been controversy over the resulting corrections to the capture rate. Although the arguments (Brown & Rodriguez 1967; Hamann & McWhorter 1964) still conflict, the conclusions agree: the presence of several minima has very little effect on the capture rate. The Lax model and its generalizations. Lax assumed that capture into highly-excited states |«> is dominant. This assumption leads to two simplifications. The first is that we may use the correspondence principle: certain parts of the calculation of the o, may be made classically, rather than quantum-mechanically. The second is that the states of importance are sufficiently dense in energy for them to be treated as a continuum. This leads to an integral equation for P,, instead of a set of coupled equations. The observed capture cross-section becomes o(E) = il du o(E, u)P(u),

(14.2.12)

where (—u) is the ionization energy of the corresponding state. Both the assumptions are dangerous. At the concentrations met in practice, the wavefunctions of quite low excited states (n > 7 in Si) overlap neighbouring ions. Thus the single-trap approximation breaks down in just the regime where the two main assumptions of the Lax model become valid. Our calculation of o(E,u) follows the treatment of Smith & Landsberg (1966). The central part of the calculation gives the

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probability W.(E, q) that an electron with kinetic energy E emits a phonon of wavevector q, polarization s, and energy hw,,. The probability is calculated by first-order time-dependent perturbation theory, and treats the initial and final electronic states as if they were free-electron states. All the papers discussed make a simplification by omitting the modification of the states by the attractive potential of the trap. The potential enters because the transition probabilities have to besummed over the different final states, and the conservation

of energy and momentum conditions determine which states are relevant. The potential also enters into the average over initial states needed in calculating the capture cross-section. As the carrier follows its trajectory, its total energy Ey remains constant. However, its kinetic energy E = E,—V(r) changes, and so affects W,(E(r),q). The trajectory used is the one obtained from classical orbit theory. The W,(E,q) have been evaluated by Smith & Landsberg in several cases. They are conveniently written in the form

WE, q) =

Dye

WA

h? (2n)°

BY(E,p,4){n(q) +1} (AE),

(144.13)

in which V is the crystal volume, py is the cosine of the angle between the electron momentum

vector and q, and AE is the change in total

energy. The B,(E,y,q) depend

only on q and on the electron—

phonon interaction, because we have used free-electron states. The

examples in Table 14.5 depend on q = |q| rather than q. TABLE 14.5 Phonon factors Bq) for capture rates ; these assume a single isotropic band minimum. The transverse modes give additional intervalley contributions in other cases,

as for Ge System

Covalent

Longitudinal

Acoustic :

Optic lonic

Optic

hq

A

2pc

ae

h cos? 2PWo

2n

Transverse

0

e*h

fe 5 as | : q

h sin?

fe

Eo

sea

2PWo

: | £0

0

5

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529

We now calculate the probability that, somewhere in its classical trajectory, the carrier emits a phonon whose energy exceeds the instantaneous kinetic energy of the carrier. Both optic and acoustic phonons may contribute to this process. Classical concepts are used here and in the treatment of energy and momentum conservation. A quantum treatment is possible, although more complicated. For systems with complex band structure it is essential to use the quantum treatment for anything beyond order-of-magnitude calculations. It proves simpler at this stage to return to the calculation of a(E,), rather than o(E,,u). Eqn (14.4.12) can be rewritten, changing the variable of integration,

6(Eo) = [aq 6 (Eo, q)P(h®.q— Eo). The cross-section o,(E),q) depends assume the potential has the form

V(r) = —AR™ si

on

(14.4.14)

the trap potential.

Ol t-=R

eaflp 2

We

(14.4.15)

INS PE

Smith & Landsberg show that, when R is small,

# 6(Eo,

mhqB(q) {n(q) + 1} q) =

3h*E 4

x

M452

1

22 h*q

2

: [4es |cand

—Eo?

|

|.

14.4.16

aie

In such cases the potential can represent Coulombic centres with net charge Zle|,

A, = Zlel/é0,

game

(14.4.17)

and neutral centres with polarizability y»

A, = ylel?/2e0,

g= 4.

(14.4.18)

The polarizability for neutral donors or acceptors may be crudely estimated by scaling from the hydrogen-atom value

y~ 2 m

Ry)2.

(14.4.19)

530

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INTERACTION

It is usual to fix A from this equation and R from a very simple estimate of the bound-state energy. It remains to calculate the P(u). We follow the treatment of Hamann & McWhorter (1964), who describe the capture process by the classical Boltzmann equation for an electron interacting with both an equilibrium acoustic-phonon field and also the Coulomb potential. The Boltzmann equation describes the time dependence of the distribution function f(r,v) for electrons with coordinate r and velocity v. It can be solved to give the steady-state value of f(r, v) in the presence of applied fields. Here the fields are the Coulomb potential of the trap and the scattering terms which come from the phonon interactions. The important result shown by Hamann & McWhorter is that here the Boltzmann equation can be simplified to an equation in which the distribution function can be treated as a function of the total electron energy alone. They show that the transition rate predicted in this approximation is correct to order (bound-state radius/mean free path for phonon scatter).” In fact, this method can treat the capture and the cascade process simultaneously. If their equations are written to give the sticking probability directly, then 10)

P(u) = {

oo)

P(u')Kym(u, u’) aw

| du’ Kyu, u’),

(14.4.20)

with P(— oo) = | (essentially assuming the ground state is very remote in energy). For present purposes the important part of Kym(u, u’) is

(u’ —u)|(u' —u)

Kym(u, uv’) ~

{(u' —u)? — 4mce?(u' + u) +. 4m2c4}3_* x

Lax

makes

different

1 : exp {(u’—u)/kT\ —1

assumptions,

and

derives

(14.4.21) the P(u) using a

classical trajectory model throughout. Specifically, he neglects the dependence of P(u) on the angular momentum of the state involved, and uses an argument based on the virial theorem, assuming that fluctuations in the kinetic energy over the orbit may be ignored. The latter assumption, in effect, ignores some transitions between

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531

bound states, and this is the reason for the difference between the

Lax and the Hamann & McWhorter results. Lax derives an equation of the form

P(u) = { du” K,(u, u” —u)P(u"),

(14.4.22)

@)

in which K,(u, uv’) is related to the mean free path for emission of a phonon of energy u’ by a free electron of kinetic energy u,

Ps taanl (on)

Ku, uv’) = [du SiGe:

(14.4.23)

The similarity between (14.4.21) and (14.4.23) is clear. In the appropriate limits K,(u, u”) are equal. It is necessary to find the P(u) by numerical integration. Values are given by Lax (1960, p. 1511) and by Hamann & McWhorter (1964, p. 253); we compare these later. For neutral centres, as opposed to traps with a long-range attractive Coulomb field P(u) is not derived: it is asserted that P(u) is unity if the binding energy u exceeds kT and zero otherwise. The problem is, of course, that the potential of a neutral atom gives only a few bound states. Indeed, the cascade hypothesis may break down in this case. Eqn (9.2.2) can be applied to give the number of bound states for potential (14.4.15). For

g=4,

m* =m,

A=

1leV At, and

R=1A, we find at most

five s-states, one p-state, and one d-state. Screening of the potential by conduction electrons will make matters worse; the effect of reducing the number of bound states overwhelms the benefits of reducing the separations of the levels that remain. One major weakness of the continuum treatments is the assumption that it is possible to discuss states with very high quantum numbers and also to treat each trap as isolated. This is not true at practical concentrations. Another weakness is the assumption of a continuum of states which extends to energies below the correct ground state of the system. This leads to an overestimate of the capture rate. One check on this assumption is to use the finite ground-state energy as a cut-off when calculating the initial capture cross-section. The small errors found by this sort of test suggest that the model is satisfactory, but Beleznay & Pataki observe that the test is inappropriate. The expressions for K(u,u’) are strongly dependent on (u’ —u). Thus we need to check the effects of the change

532

NON-RADIATIVE

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in ground-state energy on both the direct capture of the electron and on the sticking probability. The first effect was checked by Lax,

and

found

satisfactory;

the second

effect,

on

the sticking

probability, is large and cannot be ignored. Direct calculation of P(u) for correct eigenstates gives a much lower value than the continuum approximation. Comparison of the models with each other and with experiment. (a) The stickling probability for Coulomb centres. The various calculations are compared for Ge in Fig. 14.5. The continuous curves, due to Lax and to Hamann & McWhorter, are universal curves for any host material. The discrete values, from Beleznay & Pataki and Brown & Rodriguez, are specific to Ge. Comparison is not

completely straightforward: different authors use different parameters, and values are not always given at the same temperature. However, the expected trends are common: the sticking probability increases rapidly with binding energy and decreases as the temperature is raised. The two discrete-level models should be best in the cases shown. Regrettably, their results differ more from one another

SF C7 = Sot ey Woy ee Se ory

ODiz probability Sticking

IS neo 5D)

2

SO

2y

50 100 10 Binding energy /(} mv,”)

50

100

500

1000

Fic. 14.5. Sticking probabilities. Results for donors in Ge from various theories. The binding energies of the levels are in units }mov2. The results are those of Beleznay & Pataki (BP), Brown & Rodriguez (BR), Lax (L), and Hamann & McWhorter (HM). Temperatures are shown with the curves.

OF FREE

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533

than from the other treatments. In part the discrepancies must come from the physical parameters used. It would, however, be useful to have a quantitative estimate of the effect of higher angular momentum states on the Brown—Rodriguez results. Experiments on Ge:In (Lifshitz, Lichtman, & Sidonov (1968)) suggest the Lax theory is adequate for the lowest states, but underestimates the sticking probabilities in some of the higher excited states. (b) Total capture cross-sections: Coulomb centres and acoustic phonon coupling. Fig. 14.6 compares predicted capture crosssections with the experiments on Ge:Sb by Koenig, Brown, & Schillinger (1962). All models give correct general trends and orders of magnitude. The errors of a factor of 5—10 are fully consistent with the coarse approximations made, together with the observed variations of a factor of 2 from sample to sample. But it is still disturbing that both the Lax and the Beleznay—Pataki calculations give smaller cross-sections than are observed. One check of 105

:

5 x 104F

S

a er.

(A?) Cross-section

3 ee

105

4

oe

el

(Se Ne NS

Same Se

Temperature

(K)

Fic. 14.6. Capture cross-sections for donors in Ge. Experimental results for Ge:Sb of Koenig, of Koenig, Brown, & Schillinger, and of Ascarelli & Brown are compared with theory. The predictions are labelled as in Fig. 14.5.

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the mechanisms discussed is to compare the capture cross-sections of a variety of donors or acceptors. If the long-range Coulomb field is indeed the important feature, the results should be independent of the species of donor or of acceptor, apart from minor differences arising because the energy levels differ somewhat. The results are sparse; Table

14.6 shows results mainly from Bonch—Bruevich

&

TABLE 14.6 Observed capture cross-sections

System

Electron | Ge:As traps | Ge:Sb

Cross-section

Temperature

Temperature

(Ome cms)

(K)

dependence

10*-10° 10*-10°

3-9 3-9

10*

Hole traps

4

Si:Zn7

IO

Ge:(?)

1-2-5x 103

GaP :(Zn, O)

Si:B

@p. the Debye temperature. It also requires a more complicated strong-coupling condition so that the saddle point can be found

538

NON-RADIATIVE

PROCESSES

readily. With a Huang-Rhys the condition is

AND

THE

INTERACTION

factor So, for the accepting modes,

a = 4 Sool) y Seon} hop 2

«K

Y Soq(@q/@p)*(2n,+Dp

(14.4.26)

The transition probability becomes

("| exp (—E,/kT) x M

is |ME Dag: We QD} «Tag

14.4.27) (144

where the Q; are the kinetic phonon modes. Thus the capture probability increases with temperature. The cross-section obtained from this expression can be used to show its dependence on the trap depth, the effective radius of the bound state a*, and temperature. Results are shown in Table 14.7,

using h@p = 0-03 eV (@p ~ 350K),

S = 10eV,

Ey = 3-8 hap ~

0-114 eV, E, = 0-35 eV or 0-24 eV, and m* = my. These are obsolete,

but serve for present purposes. These results differ by about two orders of magnitude from experimental data. It is not clear if this difference is significant, since the results are so sensitive to the data.

Gummel & Lax comment that in Si a change of m* from 0-38 my to

TABLE 14.7 Cross-sections (A?) for multiphonon processes

E, = 0:35 eV

T=10,

T = 10,

T=0)

a* = 3-4A a* = 415A

5-1 x 1074 3-4x 1075

2:8 x 10-3 2:8 x 107°

19x 107? 48x 1075

(b) Dependence on temperature and ionization energy

a* =3.4A

E, = 0-24eV E, = 0-35 eV

= 10,

T = 10,

T=,

345.107 4 Suilseil@r

42x 107! 2:8 x 1073

6:8x 107! 1-9x 10-2

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539

Mo produces a change of 10° in the cross-section. Further, no account has been taken of optic phonons, nor has adequate distinction been made between the various phonon modes. 14.4.3.

The Auger effect and related phenomena

In the Auger effect, the transition energy of the captured carrier is taken up as kinetic energy of other carriers. Four main classes of Auger processes occur, involving respectively 1. one bound carrier and free carriers; 2. two bound carriers and free carriers:

3. three bound carriers; 4. processes involving a free exciton. At least three particles (including the defect if necessary) are needed to satisfy conservation conditions. The Auger processes enter into a variety of physical problems, notably those involving recombination and exciton dissociation contributions to the photocurrent. They compete with radiative recombination, so that their rates can be technologically important. The capture mechanism can be regarded as a reciprocal process to impact ionization. A recent review has been given by Landsberg & Adams (1973). A. Basic theory Electron—electron interaction. The Auger transitions are caused by the electron—electron interaction. Strictly, it is the difference between the exact interaction and the mean-field (or Hartree-Fock) interaction which is important. The transitions occur between eigenstates of the Hartree-Fock

Hamiltonian,

which includes the mean-field

interaction as part of the one-electron potential. Several treatments use an interaction which includes screening by conduction electrons

(14.4.28)

exp (—r,2/A), Kee = aoe elfe

where A is the screening length. The effect of the screening is to reduce the transition probability as the concentration of conduction electrons increases.

However,

most

authors let 2 become

infinite

before deriving their final results, so this term is just a gesture. No authors appear to justify their choice of dielectric constant «. It is conventional

to use the static constant, so the transition rate will

be underestimated, particularly screening is less effective.

from

deep

traps, where

lattice

540

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THE

INTERACTION

Assumptions for wavefunctions. It can be seen from the list of processes that three types of state are involved : free-carrier states, bound-carrier states, and excitons. Further differences between the

approaches are found in the choice of wavefunctions. The free-carrier states are usually taken to be perfect-crystal band states, that is plane waves modulating Bloch functions. This is not always a good approximation, and Coulomb wavefunctions should be used when appropriate. Whilst the need for Coulomb functions is often mentioned, I am not aware of a treatment which uses them for Auger processes: the calculations become unmanageable. The bound-carrier states are usually those from the simplest form of effective-mass theory, that is hydrogenic wavefunctions modulating Bloch functions. When both holes and electrons are involved we shall use envelope functions with effective Bohr radius a, and a, respectively. Only in rare cases are states other than ls-states considered, and all treatments seem to assume an isotropic crystal with the band extrema at the zone centre. Strictly, the bound-state functions should be orthogonal to the free-carrier function. This constraint is not usually enforced, although the use of Coulomb continuum functions would ensure that it was satisfied in most cases. Landsberg, Rhys—Roberts, & Lal (1964a) have shown that the orthogonality can be included, with some increase in complexity. The main effect is to multiply the transition rate by overlap integrals of Bloch functions. In the same way, excitons are generally assumed to be Wannier excitons (§ 2.2.5) with a hydrogenic envelope function describing the relative motion of the electron and hole in terms of an effective Bohr radius a,. Matrix elements. In most cases the evaluation of the matrix elements for the transition is completely straightforward, and the usual Coulomb and exchange integrals are obtained from the electron—-electron interaction. The two contributions may be comparable in size. These integrals must be evaluated using the proper wavefunctions of the electrons or holes involved. It is not correct to use just the envelope function of effective-mass theory, nor the pseudo-wavefunction of pseudopotential theory. In the case of exciton states it is customary to assume the envelope function varies much more slowly in space than the Wannier functions. An expansion is then possible for the transition matrix element (Knox

OF

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DEFECTS

54]

1963, pp. 117-19). The leading term then gives the dipole transitions, the second term gives the quadrupole transitions, and so on. The matrix elements can then be related (if desired) to the oscillator strength for radiative decay of free excitons. This makes comparison of Auger processes with competing radiative transitions much easier. Finally, we comment that the phonon system is ignored in these various estimates. The emission or absorption of phonons is assumed to be improbable. This is clearly a gross oversimplification. However, if an analogue of the Condon approximation is used, and the transition matrix elements are considered independent of lattice configuration, then the total transition probability is not affected by the electron-lattice coupling. This follows from arguments equivalent to those used in § 10.4 for the line-shape function. There is no experimental evidence for a Condon-type approximation.

B. Detailed calculations Auger processes for one bound carrier and free carriers. The processes in this category contribute to the capture of free carriers by traps. Symbolically, the processes are shown in Table 14.8. The first two processes involve the capture of electrons, and the electron which is trapped transfers energy (E,+E;) to the other carriers. The other two processes are for hole capture, and the electron transfer is (E, + Eg— Ep). The results are characterized by a rate constant T, for the Ath process. The time dependence of the free-electron concentration n and the free-hole concentration p is then characterized by time constants t, and Tt,

=i| eoNn GaN 1/t, = 1/tpo + NnT;+NpTa, — a a 3

(14.4.29) (14.4.30)

in which N is the concentration of recombination centres, and tT, and t,o describe other competing processes. Of course, the T, (which are usually in cm®s~') can be converted into a crosssection per defect 6, =nT,/, but the results are inconvenient since they depend on the free-carrier concentration. Instead, one gives either the T, or the carrier concentration at which Auger processes and radiative processes become equally probable. Landsberg et al. (1964) have discussed all four Auger processes shown in Table 14.8. They use a low-temperature approximation,

542

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

TABLE 14.8 Auger processes: one bound carrier with free carriers Notation of different authors

Energy loss by carrier in being trapped

Process

t 1. Landsberg et al. T, ;

easear | => ear ©

te

E.+ Ey

e+h+

:+

E+E,

oe

Bess II 2. Landsberg et al. T,

—>je|+h

3. Landsberg et al. T;;

Bonch-Bruevich &

1

h+e+[e]>e+[

|

ess

E,, + (Eg — Ex)

h+h+[e]>h+{_

|

HA

Bet (ES= By

Gulyaev; Bess I

4 Landsberg et al. T,)

The energies are the binding energy of an electron to the trap E7, the free electron and hole kinetic energies E,, E,, and the band gap Eg. In the diagrams the changes in state of the electrons are shown; c and v label the band edges.

assuming that the mean free-carrier kinetic energy is negligible compared with the trap depth and band gap. The resulting expression is probably valid in most cases, for Auger processes do not have a strong temperature dependence. The rate constants can be written in the form

fe

ep

8n7e*h> Sn

Tee

y Pea an ie

+(small temperature-dependent terms).

(14.4.31)

The factor |F 4|? is a dimensionless overlap of Bloch functions, and

E, is the trap depth. In practical units (8m7e*h?/m?) is 1-44 x 10727 (eV)? cm® s~'. The y, depend on the process and are functions of T = E,/(Eg—E;) and of the mass ratio into a hydrogenic Is-state

¢o= m*/m*. For capture

y,=1

y2 = {20/(1+0)}* yy = TH QUL4

T)}4

y4 = T? {20/(1+0)}*

(14.4.32)

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543

The transition probability decreases rapidly with the energy transfer required to trap the carrier. For electron capture (T, and T>) this energy is of order Ey, and the rate falls off as E; >. For hole capture (T; and T,) the energy transfer is of order (Eg—E;), and the rate increases with E,. Estimates of the temperature dependence verify that it is not important. There is, however, an interesting dependence on the quantum numbers n, | of the state into which capture occurs. For hole capture, T; and T, decrease rapidly and monotonically with / at constant n. But for electron capture, T, and T) increase with ! at constant n with substantial oscillations. Only incomplete comparison with experiment is possible. Experimental data are available for T, and T; in n-type Ge. Koenig,

Brown,

& Schillinger (1962) find T, ~ 1-5x10~73cm®s~!

for

shallow donors Ge:Sb and Ge:P. The predicted value is 5 x 10°7? |F,|? cm®s~', using m* = 0-25m, and E; = 0-12eV. Karpova & Kalashnikov (1962) estimate T, ~ 10°?°cm°®s~! (Eg—E;) ~ 0-4eV.

for a trap with

Again this is close to the predicted value if

the overlap factor |F3|? is close to unity. Auger processes for two bound carriers and one free carrier. Discussions of these processes have been mainly qualitative. Thus Sheinkman (1964, 1965) considered the processes h+|ee|/7e+

3

(14.4.33)

together with the reciprocal process

e+([h h]>h+[_ |.

(14.4.34)

These processes may give observable effects on photocurrents and thermocurrents, in addition to their usual kinetic effects. Tolpygo, Tolpygo, & Sheinkman (1965) examine transitions involving two traps, of the general form

h+[e]tfe]>[

|+{_

+e.

(14.4.35)

They estimate the transition probabilities for various combinations of deep and shallow centres arranged on a lattice. The process is most effective when there are both shallow centres (at which electron-hole annihilation occurs) and deep centres. The results are, of course, sensitive to both the details of the traps and to their

544

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

separation. Such processes are only important at higher defect concentrations; Tolpygo et al. estimate N > 10'° per cubic centimetre for them to compete with radiative processes. Thus there is only a very restricted range of concentrations in which these processes are significant and yet impurity banding may be ignored. Auger processes for three bound carriers. These transitions are of interest for several systems. Dexter & Heller (1951) and Fuchs (1958) were concerned with the bound exciton recombination at an F-centre in alkali halides. The results are needed to interpret two observations: the removal of F-centres by excitons (Apker & Taft 1950, 1951a, b,c) and the width of the /> band, which may be determined by this transition. Nelson, Cuthbert, Dean, & Thomas (1966) were concerned with the competition between radiative and non-radiative recombination of an exciton bound to a donor in GaP:S, and Khas (1965) was concerned with a rather similar problem in Cu,O. These papers make different assumptions about the electron-hole correlation. All papers except that of Khas assume that the electron and hole may be considered as an exciton weakly bound to the defect. A typical process would be

[e (eh)|

>|

|+e

(14.4.36)

Khas, however, assumes that the electron and hole are independently bound to the same centre, so instead the process is

SUG) | ie

+e:

(14.4.37)

The F-centre discussions, mentioned above, both use the Frenkel

picture of the exciton. The trapped exciton is a hole in the outer p shell of an anion which is a next-nearest neighbour to the F-centre and an electron in an excited state of the vacancy. Since the centre is so compact,

an unscreened

electron—electron

interaction

is used.

Fuchs finds that the decay rate in NaCl contributes 5-10 per cent of the width ofthe # band. Dexter & Heller conclude that the Auger processes do indeed provide an adequate mechanism for the destruction of F-centres by excitons. Nelson et al. use a result derived for the internal conversion problem in nuclear physics (Blatt & Weisskopf 1952, p. 617) with

OF

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WITH

suitable scaling. Blatt & Weisskopf’s Auger and radiative processes to be

DEFECTS

545

result predicts a ratio the

Auger :radiative = 4(a*K)~3(ka*)~!.

(14.4.38)

An extra factor + has been included because the nuclear result included a sum over the two K-shell electrons which is not appropriate here. In the ratio, K is the wavenumber of the photon emitted in the corresponding radiative process (K = wn(w)/c, where nw~ Je), and k is the wavevector (k = ,/(2m*w/h)) associated with the Auger electron. The effective Bohr radius is a* = age/m*; for present purposes 4* is identical to a*, although Nelson et al. argue that slightly different values of ¢ and m* are appropriate. The ratio finally becomes S\N

Say)

PN

Auger :radiative ~ F |" = |(:|: ett hw he

(14.4.39)

With m* ~ 0-25mo, ¢ = 10-2, and w ~ 3-5x 10'° rads ', the predicted ratio is about 900, compared with the observed value of 500. The slightly different parameters of the original paper give 1200, an insignificant difference. The agreement is far better than should be expected. Scaled hydrogenic results are notoriously inaccurate (e.g. Sclar & Burstein 1955), and here the original internal-conversion

result does not agree well with a full treatment of the problem for which it was intended (Blatt & Weisskopf 1952). Dean, Faulkner, Kimura, & Ilegems (1971) have obtained Auger rates for a number of acceptors in GaP. They find that the Auger rate varies as the fourth power of the acceptor binding energy E,. This can be understood

as the product of two factors. One, pro-

portional to (exciton binding energy)? gives the normal transition probability for a diffuse exciton. The other, proportional to Ei (apart from slowly-varying terms), is associated with momentum conservation. Since the exciton binding energy is roughly proportional to E, (§ 26.2.5), an EX dependence results. Auger processes involving a bound carrier and a free exciton. Trlifaj (1964, 1965) has treated several examples of exciton dissociation mechanisms. These are important because they permit opticallyproduced excitons to give a photocurrent and because they compete with simple radiative decay. Two processes are of interest. One is

546

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

‘transfer of charge’, in which one ofthe carriers is trapped by a defect,

feh]+[

| > [e ]+h.

(14.4.40)

The other involves the annihilation of a trapped carrier,

[eh]+[e] > [_]+e.

(14.4.41)

The transition probability is most easily expressed by the defect concentration at which the radiative and non-radiative decay are equally probable, since some of the terms cancel. Trlifaj makes the standard simplifications. The anisotropy of the host is ignored, and average parameters ¢ and m* chosen. The boundcarrier states are taken as hydrogenic, with plane waves for free carriers and a hydrogenic function chosen to describe the internal motion of the Wannier exciton. The transition probabilities depend on the trap depth E; through a dimensionless parameter p? =

F,/(Ex—E,), where Ey is the exciton creation energy. Often p? is small; Trlifaj gives results for donors in CdS (p? ~ 0-013) and Cu,O (p? ~ 0-3) when the dependence on E; can be simplified. The process (14.4.40) should dominate in all cases. For shallow

traps the transition probability varies as E3 (or as Ez if quadrupole transitions are involved). Trlifaj gives as example donors in CdS (dipole case) and in Cu,O (quadrupole case). For CdS, assuming a trap depth of 0-027 eV, m* = 0-2 mo, m* = 1-16 mo, and ¢ = 9-3, he finds the non-radiative process should exceed the radiative for 4x 10'° centres per cubic centimetre. For Cu,O, with m* = m* = 0-5 mp, € = 6-25, and a trap depth of 0-6 eV, the radiative processes are overwhelmed at 3 x 10° defects per cubic centimetre. The associated cross-sections are large: for CdS, taking a mean exciton

velocity of 10’ cms~!, the cross-section is 4-5 x 10° A?. The alternative process (14.2.41), involving a full trap, is far less effective. Thus for donors in CdS a concentration of 5x 10'8 centres per cubic centimetre is needed for the non-radiative transitions to dominate, which is equivalent toa cross-section 3 x 10~* A?. For the Cu,O defect the critical concentration is 3 x 10'* defects per cubic centimetre. The transition probability contains comparable exchange and Coulomb terms, both of which vary as Ei for shallow traps.

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547

14.5. Kinetics

Each of the processes we have been considering—the capture process and the various transitions between bound states—is only a part of the complex of processes which occur in an experiment. We now examine the behaviour of a system when there are many competing processes. The fundamental equations used to analyse the behaviour are variously known as kinetic, rate, or master equations. Their general form gives the time-dependence of the occupancy n; of a state d

LWW), qtt = j#ti

(14.5.1)

where W,,; is the transition probability from |i> to |>, summed over all mechanisms,

We= 5, 2W,ij,p* ij

(14.5.2)

processes

These equations may be solved to predict the time-dependent response of a system or the effects of varying the relative importance of particular transitions. The two equations (14.5.1) and (14.5.2) are by no means trivial. The first equation shows irreversibility: the occupancies tend to equilibrium values, and do not show the same behaviour under time reversal as the Schrédinger equation. Demonstrations that microscopic reversibility and macroscopic irreversibility are consistent are summarized by Tolman (1938), Chester (1963), and Wu (1969). In essence they require that, if the system makes a transition |k> > |J) then the next transition will be to some other state |m), with a negligible probability of return to the initial state |k>. The system has then a negligible probability of simply oscillating reversibly between states |k> and |l>. The condition is usually valid, at least for transitions between vibronic states, because of the large number of different vibrational states. The second equation asserts that the different transition probabilities are independent. This assumption is essential if we are to talk of occupancies of different states and not of relative amplitudes in the wavefunction. The equation is invalid when there is interference between the processes. Interference can occur if two states are connected by more than one transition operator. The transition probability in simple cases is +V,)|/>|? and contains a cross-term then proportional to | and |/);it is not sufficient that the electronic parts of two states |k> and |l) are connected by both V, and V, if the two operators only

cause different changes in the vibrational states. Interference is fortunately rare in the cases where kinetic arguments are needed. Examples are given in §§ 10.10.4, 14.2.2, and 24.5. In solving rate equations there are two important initial steps. First, the nature of the equilibrium to which the system tends must be determined. In the absence of special perturbations, such as applied electromagnetic fields, it is usual to assume that the system tends to thermal equilibrium, defined by a single temperature T. The temperature is introduced by coupling the system, in some way, to an external heat bath; typically, it can be assumed that the phonon system is always in thermal equilibrium. However, it often happens that certain classes of transition take place much more rapidly than others. It is then instructive to assume that thermal equilibrium is achieved rapidly in different subsystems, which may be described by different effective temperatures. The temperatures of the subsystems become equal with some, much longer, time constant. This behaviour

is particularly common for magnetic systems (Abragam 1961 ; Gill & Vinall 1969) where different temperatures can be associated with the Zeeman energy and dipolar energy of a spin system. It may also occur for different frequency bands of phonon systems, as in the phonon bottleneck (§ 14.2.2). The second important step is to decide which of the many possible transitions are going to be important. Matters are simplified by the principle of detailed balance, since this relates W,; ,to the reverse probability W;;,,. We now consider some examples where radiative and non-radiative processes compete, and illustrate some of the processes which may be important. For simplicity we use the configuration-coordinate picture. The example is a two-level system. Three types of process occur when there are no external perturbations. There is radiative decay from the upper curve to the lower, and there is non-radiative decay between the different vibrational levels associated with one electron state (intra-level transitions). Finally, there are inter-level non-radiative transitions, in which the electronic state changes. For many common defects the orders-ofmagnitude for the lifetimes of states for the first two processes are

Allowed optical transitions:t ~ 1077-107* s Intra-level transitions:t ~ 107 11-107}? gs,

OF

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549

Whilst these estimates are very rough, it is clear that intra-level transitions are usually much more rapid unless the electron—lattice coupling is weak (see § 12.2.6). The inter-level transitions are much more sensitive to the details of the energy levels. We illustrate this for the system shown in Fig. 14.7, where the effective frequency is assumed to be the same for both states. Two classes of inter-level transition are particularly rapid. There are ‘vertical’ transitions like AB, which are favoured because of the large lattice overlap

Fic. 14.7. Inter-level transitions in a configuration-coordinate scheme.

factors, i.e. for the same reasons that the Franck—Condon transitions

often dominate in optical transitions, and there are ‘horizontal’ transitions like C< D, which occur because of near-degeneracy of levels associated with the two electronic states. This resonance effect has been demonstrated clearly by Sussmann (1967a). The behaviour of the inter-level transition probability as Q, and E, vary has been particularly well shown by Englman & Barnett (1970, especially p. 42), who also discuss the kinetics for a number of impurity systems. For several systems, including the F-centre and some bound exciton systems (Dean 1967; Bogardus & Bebb 1968), a particularly simple result is adequate for interpreting experiments. What is needed is a luminescent efficiency for the decay of the system shown in Fig. 14.1. This is defined by

1

aflPp oal Pa + Py

14.5.3 Wied)

where Pz is the radiative transition probability and Py the nonradiative one. Eqn (14.2.42) suggests Py ~ exp (— E,/kT), whereas

550

NON-RADIATIVE

PROCESSES

AND

THE

INTERACTION

Pz is independent of temperature. Thus 1 depends on temperature:

n = {1+cexp(—E,/kT)}~°.

(14.5.4)

This equation is adequate for phenomenological analysis of many experiments. The solution of the rate equations is straightforward when they are linear, when the transition probabilities are independent of the occupancies of the states. The linear rate equations then predict a decay to equilibrium with a number of time constants. Oscillatory recovery can occur (e.g. Zwanzig 1961, pp. 121-5; Heims 1965; Kutz, Davidson, & Konzak 1971), but is less common. The conditions for simple decay are discussed by Strandberg & Share (1973), with

emphasis on spin-lattice relaxation. The solution of non-linear rate equations is more difficult, and numerical solutions are usually essential. Such equations occur in cross-relaxation (Grant 1964) and in recombination (e.g. Nagae 1962, 1963), when special solutions are needed. Finally, we discuss the sort of predictions that can be made from kinetic arguments. The general aim in these approaches is to analyse a complex series of processes so that the individual contributions of different mechanisms can be obtained. One prediction is the time dependence of the intensity and spectrum of light emitted after the system has been excited. A good example is given by Thomas, Hopfield, & Colbow (1964), who considered the radiative recombination of electrons and holes as distant donor-acceptor pairs. We shall consider these systems later in § 25.3.1; for the present purposes it suffices to know that both the transition probability and the energy of the emitted photon vary monotonically with the donor-acceptor separation. Thus both the decay rate and the spectral distribution of the emitted light change during the decay, since the close-pair recombination proceeds faster. Analysis of the time dependence gives a measure of the dependence of the transition probability on the defect separation. A second prediction is the effect of varying the intensity of the initial excitation of a system on the subsequent decay. This is clearly shown by Dishman & DiDomenico’s (1970) analysis of three-level systems, in particular GaP (Zn, O). Their treatment is a generalization of the earlier work of Shockley & Read (1952) and Hall (1952). A third class of prediction is the dependence on the energy at which the system is initially excited. An

OF

FREE

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551

example occurs in photoionization. As the excitation energy changes it is sometimes found that both the photoconductivity and the luminescent intensity vary periodically. When the photoconductivity is strong, luminescence is weak. Here the origin of the behaviour is the coupling of the free carrier to optical phonons. When the initial carrier kinetic energy is nearly an integral multiple of the optic phonon energy, the carrier is slowed rapidly and captured, with the emission of a phonon. At other energies the photo-

conduction is relatively more effective. Thus in this case the periodicity in energy should occur at intervals of roughly the optic phonon energy. Another example is given by the spin-lattice relaxation of Fe?* in MgO (Lewis & Stoneham 1967). This is a three-level system, whose return to equilibrium is characterized by two relaxation times. The resonance lines are strain-broadened, and by saturating the line at different energies it proves possible to prepare the system so that these two relaxation times can be obtained separately.

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hg Ads M,. Here M, is the cation mass, which enters because we need phonon frequency rather than a phonon energy. The results are shown in Fig. 15.7; the trend is clearly right, but there is no detailed agreement.

125¢

oLil

Tie

50

0

25

it

5-0

75

(E,-/ahwy/M,hw

Fic. 15.7. Huang-Rhys factors for optical absorption by F-centres. Results from Dawson & Pooley are compared with the combination of F-band energy, effective frequency and cation mass discussed in the text.

ONE-CARRIER

CENTRES

585

Analysis of the line-shape data beyond (15.2.11) lead to two other features. The first is a shift in the peak position with temperature. This is partly a result oflattice expansion, and partly associated with changes in effective frequencies in the ground (,,) and excited («,,) states. These changes also enter in the second moment of the line. The results for (@,,/,,)” are usually less than unity; it is also found that w,, and w,,/w,, both tend to increase with the ratio M,/M _ of ionic masses at constant lattice spacing. The second feature is an asymmetry of the F band. The asymmetry follows from the simple theory (e.g. § 10.8.2), although it is common to analyse the data in terms of different Gaussian widths for the high- and low-energy sides of the line. It is hard to get any meaningful data from the asymmetry; the measurements are imprecise, and their interpretation depends on subtle points like the detailed use of the Condon approximation. B. Theories of the line-shape. The earliest detailed calculations of the F-centre bandshape were those of Pekar (1950) and of Huang & Rhys (1950). The method is outlined in § 10.7.2. Whilst they contain a number of simplifications, like the assumption of coupling only to dispersionless longitudinal-optic modes, the results are satisfactory both qualitatively and in general magnitude. Anomalies do occur, such as the incorrect prediction that the hw of(15.2.11) should

be haw, for the perfect crystal. The more recent calculations we describe (Benedek & Nardelli 1967; Mulazzi & Terzi 1967; Benedek & Mulazzi 1969; Ritter & Markham 1969; Kern & Bartram 1971) go beyond the earlier calculations in several ways. The methods are contrasted in Table 15.10. First, a microscopic calculation of the F-centre wavefunction is used, rather than a continuum

function. This should allow the

Jahn-Teller terms to be included, although at present a proper calculation of the line-shape (rather than moments) is not available. Secondly, the full spectrum of lattice modes is included. Thirdly, effects of changes in local force constants on the line-shape are included. Other calculations which have discussed some of these effects include McCombie, Matthew, & Murray’s (1962) calculation of the effect of perturbed lattice modes by considering the breathing modes of a large cluster and Ritter & Markham’s (1969) extension of the basic ideas of the Huang-Rhys model to treat coupling to a full spectrum of perfect-lattice modes. Lemos &

586

THE

F-CENTRE

AND

RELATED

ABER 1541.0 Theories of the F-centre line-shape Authors

Pekar Huang & Rhys Ritter & Markham Lemos & Markham McCombie, Matthew, & Murray Benedek & Mulazzi Loader & McCombie Kern & Bartram

Linear coupling

Lattice dynamics

Frohlich Frohlich Generalized

Dispersionless optic Dispersionless optic Deformation dipole

Frohlich Point-ion Point-ion

Cluster modes Cluster modes

Experimentt Experimentt

Deformation dipole Shell model

Point-ion plus ion size

Breathing shell model

Changes in local parameters None None None

Removal of repulsive interactions Theoryt Experimentt

Theoryt

+ Experiment here means direct use of observed data plus some simple general qualitative assumptions. Theory means a microscopic model was used.

Markham (1965) used a very different approach, developing a cluster model in which the relation to the configuration-coordinate model was especially clear. The two main features of interest are the models of the electron— lattice coupling and of the lattice dynamics. All recent theories agree that it is necessary to use a shell model or deformation—dipole model for the lattice dynamics. The Huang—Rhys factor S then depends mainly on the linear coupling coefficients. Consequently, it is no surprise that workers who use experimental data obtain values of the right general magnitude. Benedek & Mulazzi adopt this solution, using Gebhardt & Maier’s data. For the contribution

of the coupling to the change in local force constants, they use a mixture of a continuum model with a trial function orthogonalized explicitly to the ion cores and with one parameter (the effective radius of the potential well) chosen to make the transition energy agree with experiment. This is an unusual mixture of methods. Thus the energy contains a term amounting to a correction to the pointion potential, even though the point-ion model is not used for the zero-order Hamiltonian. Further, the authors do not seem to have

used the correct variational procedure for cases where different effective masses are appropriate in different regions (§ 6.2.3, especially

ONE-CARRIER

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587

(6.2.9}(6.2.10). The approximations will be more important in this line-shape problem than for transition energies, since high-order derivatives of the energy with respect to displacements (up to third order in the Benedek & Mulazzi paper) are involved. Benedek & Mulazzi discuss changes in central-force constants for the first (a, 0,0), second

(a,a,0), and fourth (2a,0,0) neighbours

and non-

central first-neighbour force constants. By contrast, Kern & Bartram concentrate on the central-force constants for first and second neighbours and calculate the linear coupling terms directly, rather than take them from experiment. Kern & Bartram use the exact pseudopotential method of Gash & Bartram for the nearest neighbours, and the point-ion potential for second-neighbours. With such a wide range of types of calculation, it is probably best to isolate the agreed features rather than compile lists of parameters. The main conclusions are the following. First, the Fcentre is ‘soft’. The local bulk modulus may be halved by the presence of the defect. This conclusion agrees with analyses of Raman

data

(Buchenauer,

Fitchen,

&

Page

1969) and

with

the

dependences of F-band energy on lattice parameter found from the Mollwo-Ivey law and from experiments under hydrostatic pressure (§ 15.2.2). Secondly, interactions of the defect electron with the nearest neighbours are by far the most important for the lineshape in absorption. This is clear when one considers the three contributions to the interaction. The overlap term, which appears because an ion has been replaced by a more extended defect charge distribution, falls off rapidly because the F-centre is compact. The ion-size terms, however calculated, fall off rapidly for the same reason. The short-range repulsive terms between the host ions are also very weak except for neighbouring ions (§ 3.4). Thirdly, the degree of ‘softness’ of the F-centre is sensitive to lattice relaxation near the defect. Thus Benedek & Mulazzi find the effective centralforce constant to nearest neighbours some 22 per cent lower in the relaxed configuration for NaCl than in the perfect-lattice configuration. Kern & Bartram find a 17 per cent reduction for KCl. These anharmonic corrections are most apparent in the one-phonon line-shape, although Warren (1967) finds evidence for them in the full line-shape. The one-phonon line-shape cannot be determined directly from F-centre spectra since the electron—lattice coupling is too strong (Mostoller, Ganguly, & Wood (1971) discuss cases where vibronic structure is resolved). The predictions of Kern & Bartram

588

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in Fig. 15.8 show how the one-phonon sideband moves to lower frequencies as the approximations improve. The conclusions agree with those of McCombie et al. The general features of the line-shape are predicted rather well, and reasonably accurate predictions made of the Huang-Rhys factor and the temperature dependence of the line-width. The lineshape is very nearly Gaussian, with a slight asymmetry. Both theory and experiment are conveniently compared with a ‘double-Gaussian’, in which different parameters are used on the high- and lowenergy sides of the peak. The logarithm of Kern & Bartram’s normalized line-shape I(E)/I,,,, proves very nearly linear in (E—E,,x)*, showing that the approximation works well. Calculations of the line-shape also imply that in most cases Jahn— Teller terms are significant, but not dominant. The caesium halides

and CaO are exceptions and will be treated separately. Estimates of the Jahn-Teller energies can be made from the contributions of

0-6

ft

0:54

r ! 1

S£:

Zero-phonon line

=> Ww

xS,(@) 10-4 (eV-) S tr

—-----y ¢.

0-1

0

5

10 1

15 1

20 25 wx 10-!2( §-1)

30

35

40

Fic. 15.8. One-phonon sidebands for the F-centre in KCl. Kern’s calculations are for three cases :(a) an unperturbed F-centre with neither force constant changes nor lattice distortion, (b) the case of altered force constants, and (c) altered force constants with ground-state distortion.

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asymmetric modes to the second moment of the line-shape. Thus, if a single effective frequency determined from the temperature dependence of the line-width is appropriate for all modes, the relaxation energies are

totally symmetric modes:



Sho

(15.2.14)

and that the effective phonon energies for A, E, and T modes are all equal to hw. S, is the Huang—Rhys factor for the A-mode coupling. The non-Gaussian line-shape can be understood from the lower moments. Hughes (1970) observes that a critical quantity is R = M,/(M,)?, which can be measured more accurately than either M, or M, separately. The ratio is 3 for a Gaussian; if R falls below 3, the line becomes squarer in shape, with less intensity in the wings for a given width. In the model system the moments at absolute zero are

M,/(hw)* = (S, +38), (15.2.15) M,/(ho)* = 3{M,/(h)?}? —455? +(S, +38). (15.2.16) When S, is large and S zero, we find the usual Gaussian value R = 3,

appropriate for coupling to a single non-degenerate mode. When S is large and S, is zero, R is 2:1. The observed value of 2-48 indicates a band substantially squarer than Gaussian. Detailed calculations by O’Brien (1971) agree rather well with the observations of Escribe & Hughes (1971), and are shown in Fig. 15.11. O’Brien notes that the adiabatic theory of Cho (1968), similar to that used for the CsF F-centre, gives rather poor line-shapes. The apparent conflict of a broad band and a well-defined zero-

phonon line can also be resolved. There are two features. First,

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(a)

CENTRES

593

Absorption

300 K ,:

if

Me Uh a

SS.

28 000

30 000

~~, —s

Energy (cm =)

(b)

Absorption

Emission

See

OE ee

ne

L 4

l 6

Energy/hw

Fic. 15.11. The CaO F-centre. (a) Optical absorption data (Escribe & Hughes) for the (oneelectron) F-centre ;(b) O’Brien’s predictions fora system with equal E-and T-mode coupling in the excited state. These theoretical curves use S = 5-45 for the Jahn-Teller coupling and A-mode coupling corresponding to a width 0-83 phonon energies.

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O’Brien has shown that the fractional intensity in the zero-phonon

line is not exp

{—(S,+5S)}, but the larger value S exp {—(S,+S)}.

The extra factor is a consequence of the equality of the E- and Tmode couplings. Second, the second moment contains a term

$S(hw)?, rather than the S(hw)* for non-degenerate modes. In the present case, for a given lattice relaxation energy, the line-width will be substantially larger. Of course, neither feature explains the smallness of S,. The real test of the ‘equa!l-coupling’ model of the CaO F-centre is quantitative. One can examine the various data to see if consistent values of S,S,, and hw can be deduced from them. Values of S and S, can be determined in five ways:

1. From fractional intensity in zero-phonon line: S,+S—In(S) = 3-7 (consistent with: Sx = 1-6.)

2. From R = M,/M3 and M,:

S, = 1-64,

3. From M, and the centroid—zerophonon line separation: Sc 4. From the observed stress splittings of the zero-phonon line:

5. From magnetic circular dichroism (Merle d’Aubigné & Roussel 1971):

S.=93-3):

S$ = 3-28,

Léon

ks = B23:

Sites Leys

Se—24:0!

Ser 2-055.

sSih= 73215:

Clearly the agreement is very satisfactory. These results assume a single phonon frequency. Estimates of hw are less consistent, although one should not expect a single value to be good for all

measurements : (a) Peaks in the one-phonon part of the broad band, identified by stress response of similar bonds in emission (Hughes, Pells, & Sonder):

emission: hw, = 205cm™', hag ~ hwy ~ (302+8)cm™! (absorption: ha, ~ 220cm™!, hw, ~ hw; ~ 310-50 cm *), The temperature dependence of the first and second moments in absorption suggests an effective phonon energy of about 0-9 ha in the excited state. This is consistent with the positions of the peaks and indicates that usually quadratic electron-—lattice coupling may be ignored.

(b) Limit of M,/M, at absolute zero: (c) Temperature dependence of M,:

hw = 290cm7!. ho =(264cem—1.

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52D

(d) Temperature dependence of the zero-phonon

line: There is no convincing explanation (Escribe & Hughes 1971).

hw = !140cm7/. of the last value at present

15.3. Excited state configuration of the F-centre 15.3.1. Introduction Calculations of the relaxed excited states fall into three main groups. There are a priori theories which attempt to calculate all features from first principles, and at the other extreme are model calculations which, for example, attempt to relate observations to

some specific vibronic model. In between are a number of approaches generally designed to show that some specific property can appear for plausible values of input parameters. These approaches cannot be regarded as quantitative theories because of their arbitrariness, and are best described as empirical or a posteriori. Two common assumptions in the empirical theories, which allow enough flexibility to ensure some kind of success independent of content, are the use of the Toyozawa—Haken-Schottky—Fowler potential (which contains a surfeit of arbitrary parameters) and the use of different Hamiltonians in absorption and emission. The response of the relaxed excited states to perturbations is harder to measure than that of the unrelaxed states. However, in some cases, results for stress, electric field, and magnetic field

perturbations are available. The results of the Stark effect show the influence of an s-like state lying just below the 2p*-state, as expected from bound-polaron theory, but opposite to the order of the levels in absorption. The results for response to stress are equally striking (Hetrick & Compton 1967) and show induced dichroism linear in the stress. However, the absence of temperature or wavelength dependence of the dichroism implies that the splitting of the relaxed excited state for a given stress is about two orders of magnitude less than that in the unrelaxed state. The stress coupling in the unrelaxed state is strong, and this appears to be true for at least part of the relaxation process, since it is presumably T, coupling causing transitions between the 2p-state which leads to unpolarized luminescence after excitation with polarized light. The change in stress coefficients is not yet understood, although explanations are possible in terms of both ‘extended-2p’ and ‘2s—2p’ models. Present experimental evidence is consistent with both these models. The response

596

THE

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AND

RELATED

to external fields, primarily for NaF, KF, and KCl, seems to require

an

s-like

contribution.

(Mollenaur & Baldacchini

Measurements

of hyperfine

constants

1972, Baldacchini & Mollenauer

1973)

suggest an extended 2p-state for KI and MBr. Conclusions from g-factors (Mollenauer, Pan, & Yngvesson 1969, Ruedin et al. 1971) are more ambiguous, but the simple trend of (g—2) with «e,, is also consistent with the extended state for KBr, KI, RbBr, and RbI. As

we shall see, theory does not resolve the. controversy. Results for CsF are different from those for other alkali halides (Fulton & Fitchen 1970). The Stokes shift is small, the emission line narrow,

and

the lifetime

short.

However,

it is found

that

the

characteristic time for the return to the (relaxed) 1s ground state of an optically-excited F-centre is about | ys longer than the luminescence lifetime. This suggests that the lattice relaxation 1s* — 1s following emission is very slow. There is no fully satisfactory explanation. 15.3.2. Calculations of the relaxed excited state

We now survey some of the calculations of the energy and wavefunction of the relaxed excited state. They divide themselves readily into calculations which include the Jahn-Teller effect and calculations which ignore it. Omission of the asymmetric distortions cannot easily be justified. Evidence from the Stark effect suggests that very close separations of energy levels occur, typically 0-010-02 eV. If these are to be estimated accurately, it cannot be reasonable to drop substantial Jahn-Teller energies, which Schnatterly’s results (Table 15.11) suggest are about 0-1 eV. A. Calculations without the Jahn-Teller effect. These calculations include the work of Fowler (1964), Bennett (1968), and Wood & Opik (1969). Fowler uses a continuum model throughout; Wood & Opik treat the interaction of the defect electron with extended ions in detail, whereas Bennett chooses an intermediate position. These theories are important for their qualitative results, rather than their quantitative predictions. The two main reasons for this are the omission of Jahn-Teller terms and the use of the Toyozawa— Haker-Schottky—Fowler theory for polarization and distortion (§ 8.6.5). All evidence available suggests this treatment of polarization and distortion is inaccurate quantitatively. There can be little justification at present for failing to use a proper microscopic description of the lattice and electron—lattice coupling in accurate calculations.

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Fowler’s (1964) thesis is that the lattice distortion in the relaxed

excited F-centre is so great that the wavefunction changes from being a compact function to being a very diffuse function. Thus his work concerns the ‘extended 2p-state’ model. The importance of his work is that it shows that such a change can happen for plausible values of the parameters involved. Within this range of parameters One can understand the long radiative lifetimes observed. Whilst the argument has been criticized in detail (e.g. there has been criticism on the arbitrariness of some parameters or the conflict between predictions of the model obtained variationally and by the numerical integration, discussed by Wood & Opik), its qualitative conclusion is borne out by other results, such as those of Bennett. Bennett (1968) observes that the spreading of the excited state is very sensitive to the interactions between the host ions. Wood & Opik’s calculation is more sophisticated in a number of ways. Their conclusion is two-fold: that the relaxed state is indeed more extended than the ground state, and that the 2s and 2p levels may cross, so that the 2p is lowest in absorption and 2s in emission. Some models considered by Bennett (1970) show the same features. The strengths of the Wood—Opik treatment are their detailed treatment of extended-ion effects and their use of very flexible variational functions. The weaknesses are consequences of the Toyozawa—Haken-Schottky—Fowler

model (§ 8.6.5). The most

important difficulties are the following. First, there is arbitrariness in the parameters of the polarization term. Wood & Opik make sensible choices, but find it necessary to select the values by the results they give rise to rather than other independent criteria. Thus the calculations are no longer strictly a priori. An example is the use of different parameters for absorption and emission: in an a priori theory such differences would be demonstrated by the theory and not assumed in advance. Secondly, there are technical difficulties which result from the odd nature of the polarization and distortion term. Wood & Opik comment on the difficulties involved in treating the nearest neighbours in an atomic model and the rest as a continuum, without counting some contributions twice. Also, it is hard to make predictions for Franck—Condon transitions, since the implicit lattice distortion automatically changes with electronic states. The qualitative results are illustrated at their best by Fig. 15.12, where energies are shown as a function of the totally-symmetric

598

THE

F-CENTRE

Conduction band

RELATED

AND

Absorption — Emission parameters | parameters

Emission

Is*

SAT aaen

Ow |

Outward

We 1

SAAS

distortion

wis

(percent)

Fic. 15.12. The KCI F-centre. Results of Wood & Opik show the effects of symmetric relaxation of the nearest neighbours, and demonstrate a possible crossing of the 2s and 2p levels.

motion of the nearest neighbours. Four screening parameters are needed in the Haken potential, two (p,, p,) for electronic polarization and two (v,,v,) for ionic polarization. The (p,,v,) refer to polarization by the defect electron, and the (p,, v,) refer to polarization by the anion vacancy. The parameters are chosen as follows. First, p, is predicted by comparing the vacancy contribution to the electronic polarization with that from a Mott—Littleton calculation (cf. § 8.3.2). Secondly, v, is chosen to be zero (no ionic polarization) in the ground state and to equal the polaron radius ,/(2m*a@,0/h) in the relaxed excited state. Thirdly, it is assumed that p, = p, and Vy, = v,. These assumptions are arbitrary, although one can argue that Fowler’s different choice (v, > v,) is inappropriate because it implies that the vacancy is much more effective than the electron in causing ion displacements. In turn, a strong correlation of ionic motion with the electron appears, and this is not consistent with the Franck—Condon principle. Fourthly, the radius at which one changes from a discrete-lattice to an effective-mass theory must be chosen.

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The wavefunctions are sensitive to the detailed values of parameters. The main features of interest concern the dipole matrix elements and the hyperfine constants discussed in § 15.2.3. The dipole matrix elements change significantly. Thus for the KCl case of Fig. 15.12 is 0-69 A in absorption but merely 0-24 A in emission, since the 1s—2p overlap is reduced. However, the ¢2s|z|2p> element is increased from 1-80 A in absorption to 5-44 A in emission, because of the increase in extent of the 2s and 2p functions. The conclusions of the Wood-Opik work are the following. First, the relaxed 2s- and 2p-states may be very close in energy. They predict separations of about 0-06 eV to 0-01 eV, although these cannot be trusted quantitatively for reasons given earlier concerning polarization. Second, the radial extent of the 2p function may be much larger in the relaxed state than in the unrelaxed state. Third, they demonstrate two technical points: the importance of including extended-ion effects and the importance of very flexible variational functions. Finally, we observe that Sak (1971b; see § 8.6) has shown that the 2s-state of a bound polaron with weak coupling lies just below the 2p-state. The coupling for the F-centre is too strong for this result to be used directly.

B. Calculations including the Jahn-Teller effect. Very few calculations have attempted a quantitative calculation of the Jahn-Teller terms (Wood & Joy 1964; Stoneham & Bartram 1970). We begin with that of Stoneham & Bartram, whose treatment was outlined in § 8.3.1 and § 8.3.2. The main strengths of the method are as follows. First, it uses the shell model to describe the lattice dynamic and dielectric properties of the host. Second, the lattice is treated as discrete throughout, without continuum approximations. There are no artificial restrictions on either the range or the symmetry of the lattice distortion. Third, the variation of the electronic wavefunction as the lattice distorts is included consistently. This can be done even when there are two close levels coupled to each other. Fourth, it is straightforward to incorporate a number of simple generalizations, such as the allowance for ion-size corrections to the defect electron— lattice coupling and the discussion of centres like the F-centre in alkaline earth oxides with a net charge. The main limitation of the method is that it concentrates on the static equilibrium configuration of the defect, assuming that the lattice responds to the average, but consistently chosen, charge distribution of the defect.

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In some cases a Static treatment is not sufficient, and then the method

may fail. Examples where this may occur include cases described by the model system of § 15.3.3, where the wavefunctions and energies are especially sensitive to the ion displacements. In such cases other common assumptions, e.g. the Condon approximation, may break down too. The

detailed

calculations

include

three

main

elements.

First,

a wavefunction for the defect electron is required. Stoneham & Bartram used the results of the pseudopotential calculation of Bartram, Stoneham, & Gash (1968). This calculation used trial functions with just a single variational parameter. Whilst the results show no evidence for the extended-2p model of the relaxed excited state, the restricted form of trial function prevents a firm conclusion

from being drawn. The second element involves the forces on the lattice due to the presence of the defect. This is the sum of three terms: a term from the changes in short-range forces between host ions from the removal of the anion, a term from the point-ion interaction of the electron and host ions, and a contribution from ion-size effects. The first two

contributions present no difficulties. The ion-size forces are more complicated, mainly because the derivatives with respect to displacements of the (V—U,) in (15.2.1) are needed. There is also a question of how the various forces are divided between the shells and cores of the host ions. The purely electrostatic point-ion forces are divided in proportion to the shell and core charges. The shortrange forces between ions are conventionally assumed to act only between shells. The position is less clear for the ion-size forces. But, since they are primarily associated with the outer valence electrons, it is natural to assume that they act on the shells alone. Finally, the third element in these calculations is a model of the lattice dynamics. The results are sensitive to the model, and the use of a very simple model (e.g. a rigid-ion model) is not good enough. Detailed results were given for the 2p excited state of the F-centre in KBr. The results confirm one conclusion of Fowler and Wood & Opik, namely that it is essential to allow the wavefunction to change as the lattice relaxes. The changes appear to affect the displacements more than the absolute energies. The relaxation energy in the excited state is typically 0-5 eV, depending on details which we shall discuss, and is also similar in order of magnitude to values obtained by other authors or deduced from the observed Stokes

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shift of 1-1 eV. However, there are qualitative conclusions of some

importance. First, the ion-size forces make a significant difference to the energies. Second, three different models for the lattice dynamics were used: a shell model fitted to neutron data, a shell model fitted to macroscopic dielectric and elastic data, and the rigid-ion model. One expects similar results, differing only in detail. But different symmetry distortions are found in the different cases. For example, with the neutron-data shell model a tetragonal distortion is favoured ; with the shell model fitted to long-wavelength data a trigonal distortion gives lower energy. The published calculations of Stoneham & Bartram concentrate on the lattice relaxation energy. It is also possible to calculate the Stokes shift directly, and this has several advantages. One advantage appears when a defect with a net charge is considered, for example an electron trapped at a divalent-anion vacancy. The relaxation energy then contains a large term associated solely with the forces F, from the net charge, a term solely from the short-range forces discussed above, and a cross-term involving both sets of forces. The term in F, alone is independent of the electronic state and is not important, but the cross-terms are difficult because they are so very model-sensitive. A second advantage concerns the adiabatic approximations used. Stoneham & Bartram’s assumption that the ion shell positions are determined by the average charge distribution of the defect electron differs from the usual assumption that the host-lattice electrons follow the detailed motion of the defect electron. There is some doubt about which assumption is best for any one state (see e.g. the discussion using the Buimistrov-Pekar model in § 8.6 and in Stoneham (1972)), but clearly one would like results for the Stokes shift which correspond to the usual Franck— Condon principle for optical transitions between two states. This can be done readily in the linear approximation. Let f, and f, be the differences in the forces on the shells and cores due to the defect between the ground and excited states (e.g.

f, = Fe&*—F®"), and let the equilibrium shell displacements be Y,, Yex and core displacements be x,, x,, in the two states. Then, if A, is the shell-shell part of the force-constant matrix for the perfect lattice, the Stokes shift is

AE. = =f, Ao! ff,

(y.,—Y,) +f. (Xx

X,)-

(i531)

Clearly, any constant terms in the forces cancel out, including all

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terms associated with the net charge of the defect. This expression assumes that the shell motion follows the defect-electron transition, but that the cores remain fixed. Wood & Joy discussed the excited states of 9 alkali halides, the fluorides, chlorides, and bromides of lithium, sodium, and potassium.

Their methods were simpler than those of Stoneham & Bartram in several respects. Thus electronic polarization was ignored, and only tetragonal and totally-symmetric distortions were included. Only the nearest neighbours to the F-centre were allowed to move. This assumption, that only a few atoms relax, may be contrasted with the Stoneham-—Bartram assumption that, whilst forces on only a few atoms are important, many atoms may move in consequence. Wood & Joy include extended-ion terms for the nearest neighbours only. Their results are in line with those obtained by Stoneham & Bartram and with the observed Stokes shifts. Thus they predict a large difference between absorption and emission energies; for KBr, for example, the absorption band is predicted at 2:2 eV and the emission band 1-05 eV lower. Relatively small changes in wavefunctions are found, although this may bea result ofa restricted variational function. The displacements are large (~ 10 per cent) and outward in the excited state, and are small and usually inward in the ground state. The sign of the tetragonal distortion in the excited state requires two neighbours to move out further than the other four. This agrees with the corresponding Stoneham-—Bartram result. 15.3.3. Model treatments of the F-centre excited state By a model treatment, we mean one which makes no attempt to estimate the values of parameters in the problem. Instead, empirical values are used when necessary. The examples include two treatments of relevant vibronic problems: Ham’s (1972, 1973) analysis of A,,- and T,,-states coupled by T, modes and O’Brien’s (1969, 1971) discussion of a T,,,-state coupled equally to E and T, modes. The third model system is the non-vibronic treatment of coupled s- and p-states of Bogan & Fitchen (1970) and Stiles, Fontana, & Fitchen (1970). A. Ham’s model in the strong-coupling limit. Ham’s analysis of A, and T, coupling by T, modes was outlined in § 8.4.3. He showed that the eigenstates could be characterized by two vibronic quantum numbers :a parity operator anda vibronic orbital angular momentum

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operator. The eigenvalues involve three energies: the A,—T, separation A in zero distortion, the Jahn-Teller energy, E,, which would occur if A were zero, and the phonon energy hw. When Eo > A+hqw, Ham showed that the ground state was s-like (J = 0, A’ = 1) and the first excited state p-like (J = 1, A’ = —1), irrespective of the sign of A. Moreover, the separation of the states is small,

of order (hw)?/2E,. We now consider whether the close s- and pStates observed experimentally could correspond to the s and p vibronic states of this model, rather than to the 2s and 2p electronic states of a hydrogen-atom model of the F-centre. The advantages of identifying the s- and p-states with those of Ham’s model are as follows. First, it removes the need for an acci-

dental near degeneracy of 2s- and 2p-states, with the 2s-state lower by a mere 0-01-0-02 eV. One can imagine near-degeneracies occurring in one or two alkali halides, but it is hard to believe them to be so universal. Second, the model explains some features of the response to applied fields which were hard to understand previously. Bogan & Fitchen’s (1970) results for the Stark effect are readily explained, although there is a modification: the induced linear polarization and the field-induced change in lifetime are no longer so directly related. Hetrick & Compton’s (1967) observation of stress-induced polarization in emission, independent of temperature at low temperatures, is also explained. The stress admixes the second excited vibronic state (J = 2, A’ = 1) into the ground state through the strain-induced splitting of the electronic p-states. Further, the very-small temperature-independent magnetic circular dichroism observed (Fontana & Fitchen 1969 ; Fontana 1970)

can be understood by noting that the admixed excited states are different from those of importance in the stress and electric-field cases. These earlier cases involved states of Type I lying higher in energy by terms of order (hw,)”/Eo. The magnetic field only admixes Type II states, which are higher in energy by the larger amount Ep. Against the identification of the F-centre states with the lowest states of the strong coupling Ham model are two pieces of evidence.

First, if Ey > how, then the s—p splitting (hw)?/2E) = ha(zhw/Eo) should be much less than hw. Experimental values from a variety of sources, including spin-lattice relaxation, give splittings which are of order hw (0-018 eV for KCI and 0-017 eV for NaF and KF). Further, in the strong-coupling limit, the radiative lifetime of the

lowest levels becomes independent of J. Thus there should not be

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field induced changes in the radiative lifetime, contrary to observation. Also, estimates of Ey from Zeeman data, or from the moment

analysis of the absorption spectrum, give values of order 0-1 eV, in the weak- to intermediate-coupling range. Ham & Grevsmiihl (1973) have recently shown that almost all observations can be understood in terms of weak coupling. The s*-state must be lower in energy. This requires that the Jahn—Teller energies for the p*-state be less than A. Further, weak coupling requires A exceed E,, and A > 4Eo, to avoid a static distortion. The calculation treats the coupling to T,,, modes as a perturbation, using harmonic-oscillator states as a basis. Since both energy levels and wavefunctions are found explicitly, it is straightforward to obtain expression for observables. There are three main stages in the analysis to see if consistent values of A, E,, and hw can be found, namely: 1. The temperature-independent part of the magnetic circular dichroism is —2g,BH/(A+hw,). If g,, the orbital g-factor, has the same value as in absorption (g, ~ 1) then (A+/ha ) is about 0-19 eV (KF) or 0:12 eV (KCl). Of course, g, may be larger: for a really extended orbit, g, tends to my/m*, which is closer to 2. 2. If the longitudinal-optic phonons dominate, hw, is known and A is found to be 0-15 eV for KF and 0-09 eV for KCl. As in the strong-coupling theory, there is no need to assume a particularly close 2s—2p degeneracy. 3. The Jahn-Teller energy from s—p coupling, E,, can be found

from the small energies 6E = hw,{1—2E,A/(A? —h?@?)} which characterize the temperature dependence of the radiative lifetime (cf. (15.3.7)). Ey proves to be 0-040 eV for KF and 0-014 eV for KCl, consistent with weak coupling. The ratio of the radiative lifetime to that of a pure p-state can be calculated in several ways, either from the parameters already obtained or from the use of Stark data. The conclusion is that no wavefunction spreading seems necessary in KF or KCl; the s—p mixing can explain the full increase in lifetime. This may not be true in all cases, however, because Mollenauer & Baldacchini’s (1972)

data imply an extended state for KI; there is some ambiguity as to the state’s s or p character. Some problems remain. The Stark shift with parallel electricfield and optic-field vectors proves to be opposite in direction and weaker than the predicted blue shift. However, this may be a result

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of electrostriction, for there is an induced

605

stress quadratic in the

applied field. If the hydrostatic component dominates, the effects should be similar to those observed. The other problems concern the magnitudes of the Jahn-Teller and spin-orbit couplings. Available data seem to suggest that the Jahn-Teller coupling is greatly reduced, and that coupling to trigonal modes is essentially non-existent. Further, the spin-orbit coupling parameters appear to be very small. This is not quenching of the orbital angular momentum, since the Zeeman results show no anomalies.

B. The

Bogan—Fitchen

contains the same

model.

Bogan

& Fitchen’s

(1970) model

basic elements as Ham’s treatment, namely, an

electronic 2p-state with a nearby s-state and coupling to lattice vibrations of appropriate symmetry. Whilst the s-state is naturally taken to be the electronic 2s-state (as, for instance, in the Wood-Opik

treatment), it could equally be an s vibronic state associated with an electronic state of different symmetry (Tomura 1967). The model assumes that the defect electron responds adiabatically to the fluctuating internal electric fields of phonon modes of T,,

symmetry. The Jahn-Teller effect for other modes is ignored. The |s> state is then mixed into the IP, state along the direction of the instantaneous field

Is> > |S = (L+a?)4Js> +01 Heat Ip) > > IBD = (1 +a?) 4]py> —a(1 +a?) 4I]s>) |

(15.3.2)

The other p states are not affected. The |§> state is assumed to lie

below the |p> state. Various predictions can be made assuming that applied electric fields merely admix the states without changing the |s> and |p» components. First, the radiative lifetime in the |§> state and zero applied field is given by

fle

aeee Gio),

(15.3.3)

The lifetime calculated using observed absorption data and emission energy is Tt, if the |s> and |p)> states are unaffected by lattice relaxation. When an external field is applied, the lifetime changes

ANG a

eaten

(Aa)?

(a*(1+a*)}

at

15.3.4)

(

606

THE

F-CENTRE

AND

RELATED

and the polarization parallel to the field is enhanced at the expense of that perpendicular to it, P

Lifer? iP Dieter

a y

2

(15.3.5)

04

Both changes are quadratic in the applied field. The relation

hl P

se 3° (140)

(15.3.6)

allows one to estimate «. This expression is not reproduced in the Ham model. Other relations involve the temperature dependence which results when the |p> and other |p» states are thermally populated. These states have shorter lifetimes against radiative decay. It appears that « is already large, and is not changed significantly by the temperature dependence of fluctuating internal fields. If the |p> and |p» states are all assumed to lie at OE above the |8> state, and if their lifetimes are a factor R larger than that of the |S> state on average

(R = (14 2a?/3)/a), then (T) ~ x(0) ‘+3R ant 1+3 exp (—6E/kT)

(15.3.7)

There is also a temperature dependence of the dichroism. The main test of the model is the consistency of parameters deduced from different experiments. Stiles, Fontana, & Fitchen (1970) give data for NaF, KF, and KCl (summarized in Table 15.12). There are some quite large experimental uncertainties, but it is clear that very low s—p splittings and large admixtures are found in all cases. Honda & Tomura (1972) have proposed a variant of the Bogan— Fitchen model. The main changes concern non-radiative transitions. Thus they argue that the temperature dependence of the lifetime does not involve repopulation of the |S», |p>, and |p> levels but comes from the temperature dependence of the admixture « because of changes in the mean-square amplitudes of T,,, modes. Once repopulation is important, thermal ionization dominates. Honda & Tomura

also note that, in four cases (NaF, NaCl, KF, RbF),

direct non-radiative transitions from |§> to the ground state appear to be important, since the quantum yield is less than at low temperatures.

ONE-CARRIER

CENTRES

607

TABLE 13.12 Admixture «? and s—p splittings for F-centre excited states in NaF, KF, and KCl; methods are (1) eqn (15.3.6), (II) eqn (15.3.7), and (III) Stark effect (Stiles, Fontana, & Fitchen 1970) Admixture «? Crystal

N

aF

KF KCl

Splitting (eV)

mM : +0-2 COR

026473 046402

: 0-2

05405 05

11 aS

027 0:38

Hl

1

: 0017

: 0.012

0.017 0.018

0.016 0.017

There are three main differences between the strong-coupling adiabatic model of Bogan & Fitchen and the weak-coupling vibronic analysis of Ham

& Grevsmiihl.

First, the adiabatic states do not

correspond to the exact energy levels. The only simple correspondence is with groups of exact states in the strong-coupling limit. Second, there proves to be no simple relation between values of a

derived from (15.3.3) and (15.3.6), except in the limit of very weak coupling. Third, 6E of (15.3.7) is not the energy difference A of the uncoupled s*- and p*-states. Instead, for Stark and lifetime effects OE is the separation of the J = 0 and J = 1 Type-I levels; the important excited state is basically s* with a single T,,, phonon excited. For magnetic phenomena, the dE of the adiabatic model corresponds closely to A. C. The O’Brien model. O’Brien’s model has been discussed in earlier sections. The model assumes equal coupling to E and T, modes, with weaker coupling to A, modes. It works well for the CaO F-centre, describing the observed line-shapes, including the intensity of the zero-phonon line, and the response to stress and to external magnetic fields. The model emphasizes coupling to asymmetric modes, just the features which are omitted in the Ham and Bogan-Fitchen models. Only one F-centre is known where the model clearly applies, but in other cases there is evidence that the E- and T-mode couplings are similar in strength. This is shown by the estimates in Table 15.11. Whilst the strong coupling to A modes

608

THE

F-CENTRE

AND

RELATED

hides the most conspicuous consequences of the model, it is possible that the theory has wider application. 15.4. Perturbed F-centres 15.4.1.

Introduction

Several defects occur which can be regarded as an F-centre in an alkali halide with an adjacent impurity or other defect. Examples are the following. 1. The F,-centre. Here one of the nearest-neighbour cations is replaced by a different alkali ion. In almost all observed cases the impurity is smaller than the replaced host ion. Thus the F-centre in KCl might have one Na* neighbour. Most of the work in this chapter will be concerned with the F,-centre, since it is the simplest and best understood of these defects. 2. Photochromic F-centre—Rare-earth complexes. Most of the theory treats centres consisting of an F-centre in the fluorite structure with an adjacent trivalent impurity ion (Alig 1970), although another charge state with an extra electron trapped is also observed. 3. F-centres with vacancies at the nearest anion or cation sites. The F-centre with an adjacent cation vacancy has only been identified in the alkaline earth oxides. It is a dipolar centre, with no net

charge to bind the trapped electron, and is discussed in § 25.1. The F-centre with an adjacent anion vacancy is the M*-centre, an

analogue of the H} molecular ion: the defect electron is trapped by two equivalent anion vacancies. 4. The Z,-centre. This probably consists of an F-centre with a nearest-neighbour cation vacancy at, say, (a, 0,0) and a divalent metal

ion at one

of the sites (a, +a, +a). The Z,-centre,

also

associated with a divalent cation, appears to be a two-electron centre of uncertain structure. It will not be discussed here.

15.4.2.

The F,-centre

A. Phenomenology. Liity (1968) has recently reviewed the many elegant experiments which have led to our understanding of these defects. Only a brief survey of the main experimental results will be given here. The analogue of the main F band now consists of two bands in absorption. The F,, band involves transitions to the p-like state along the axis of the centre which passes through the defect ion.

ONE-CARRIER

CENTRES

609

The F,, band includes transitions to the two p-like states normal to the axis of the centre. The transition probabilities are not strongly affected by the perturbation. To about 10 per cent, the F,, and F,, bands have the same integrated absorption as the F band to which they are related. The F,, band has the same oscillator strength as each of the two F,, transitions. The temperature dependence of the bandwidth is similar for the F,,, F,.,and corresponding F bands. However, the F, , band is slightly narrower than the F, ,band because of the reduced effect of the coupling to Jahn-Teller modes. The transition energies provide an important test of theories which go beyond the point-ion model. The F,, band lies close to the F band. As a rule the separation is less than 0-1 eV. The F,, band is shifted by a larger amount, but usually only about 10 per cent of the F-centre transition energy. No splitting is observed in the higher K and L bands associated with the F,-centre. However, dichroism can be observed after alignment of the centres. The K and L bands are somewhat stronger for the F,-centres than for the F-centre,

which tends to confirm the view that the lower L bands are a result of energy dependence of the transition matrix element, rather than of the final density of states. Other bands observed include the bound-exciton « and f bands. These show no splitting, suggesting that the exciton is not too strongly localized. Whilst the results in absorption are qualitatively those one might guess, the results in emission are novel. There are two distinct types

of F,-centre. Type I includes most F,-centres (Li centres in KBr, RbBr, and RbI; Na centres in KCl, KBr, and RbCI; K centres in

RbBr), and their line-widths and Stokes shifts are comparable to those of the F-centre. Type II is rarer (Li centres in KCI; other cases suggested by Liity, but disputed by Okhura (1970), are Li centres in RbCl and RbBr) and shows a very narrow, low-energy emission with a very short lifetime. Only a single emission is observed in both cases. There is no splitting of the emission band. Experiment suggests that Type-I centres can reorient easily in the relaxed excited state. Instead of an activation energy of about 1 eV, as is found for the ground

state,

reorientation

in the relaxed

excited

state

has

an

activation energy of order 0-1 eV. The electronic state profoundly affects the reorientation rate. More remarkable still, the reorientation

of the Type-II F,-centre KCI (Li) is rapid and independent of temperature. A full theory of these phenomena is still needed. Lity cites model calculations by Strozier for the Type-II centres. We shall concentrate on the absorption results.

610

THE

F-CENTRE

AND

RELATED

B. Theory of the F,-centre. Three mechanisms have been proposed to explain the difference in F,, and F,, transition energies. The first, which appears to dominate and which leads to results of the right general size, comes from the differences in electronic structure of the host and impurity ions. The second is present even in the point-ion approximation, and involves a different radial displacement of the impurity ion with a consequent change in Coulomb potential. Alig (1970) and Evarestov & Logachev (1971) have argued that this term is significant but small. Thirdly, differences in impurity and host-ion polarizabilities lead to a contribution which Alig has shown to be negligible. Most calculations have used pseudopotential approaches (Smith 1966, cited by Liity 1968; Bartram 1968, unpublished; Weber & Dick

1969; Alig 1970; Evarestov

1971). We concentrate

on these,

and merely mention the tight-binding calculation of Kojima, Nishimaka, & Kojima (1961). The differences among the pseudopotential approaches lie in the form of the pseudopotential and the form of the trial functions used. Smith and Alig used the Phillips—Kleinman prescription, Bartram and Weber & Dick used the Cohen—Heine prescription in the form given by Bartram, Stoneham, & Gash (1968), and Evarestov used the Abarenkov— Antonova form. The formulations of these pseudopotentials are distinct but largely equivalent, although the model potential form of Abarenkov and Antonova appears to be more complicated to use in practice. In both the papers for which most details are available (Weber & Dick and Alig) the pseudo-wavefunction is assumed to vary slowly over the cores. Thus one expects, as for the F-centre, that a reduction factor corresponding to the « = 0-53 of § 15.2.2 should be necessary in the pseudopotential coefficients. Both papers agree that the energy changes in going from the F-centre to the F,-centre

do not

need

the reduction

coefficient,

and

are

far too small if it is included. This is hard to understand, since the reduction factor appears just because the pseudo-wavefunction varies too rapidly in space. It is less surprising that « is closer to unity than for the F-centre, however, since all the observations of

F,-centres discussed involve impurity ions smaller than the host ion which they replace. The different trial functions may be described as follows. Weber & Dick use various vacancy-centred functions, which are linear combinations ofSlater functions, with radial dependencer” exp (— (7).

ONE-CARRIER

CENTRES

611

These functions are usually worse trial functions for the F-centre (invariably so in the point-ion model), but one could argue that we are interested in a specific term in the F, energy here, and the best function to use is one which gives this term most accurately. Evarestov (1969, 1970; see Petrashan et al. 1970) used only the very simplest Slater functions; Alig used the Gourary—Adrian functions which worked well for the F-centre. He found that the F—F, shifts were given well by lowest-order perturbation theory and that the results did not vary much if one used different trial functions. For example, perturbation results using point-ion wavefunctions and those using the Bartram,

Stoneham,

& Gash

wavefunctions

gave very nearly the same results. In his work on spin-resonance parameters only Evarestov (1971) followed Okhura’s (1969) suggestion, and argued that it was not adequate to use vacancycentred functions but that one should use ‘floating’ functions. Thus instead of a radial function f(r) one might use f(r—A), where A is a vector from the vacancy centre in the direction of the impurity ion. His results show an improvement of the order one would expect from the introduction of an extra variational parameter. Data for KCl:Na are shown in Table 15.13. The results for the transition energies are summarized in Table 15.14. The two energies quoted are: Och Op =tE(F

by E(x) O)— EPA)

(15.4.1)

PABLE 152133 Hyperfine constants for the Fx, centre in KCI (Na)

Theory:

Impurity ion

Host ions in plane normal to defect axis

Host ion opposite impurity

vacancy-centred

function Experiment

7?

Theory: floating function

0-036

0-119

0-119

0-045

0-103

0-094

0-051

0-112

0-085

The experimental values are compared with those for vacancy-centred functions and those for ‘floating’ functions centred on a point 0-185 A closer to the impurity. Values quoted are the square modulus of the envelope function.

S-

(8961) Ss ysTUIg “(9961) pert Ain

a

IGM

ee

AO}sIeAq : [YH

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AIG :

19q2M % AIC :

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sonjeA pojonb o1e JOJ JY}

PAM

se

A3I9U9

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aes

09 ea = me

(SIUSZOIPAY)

xq

99

58 £9 68 =

0-01 99 68 LL

-110d

jGeur 8°61 891 661 SLI

B8dM =

G

ST cel 91 cvl

{—uUoNIsueN (UoMIsuey pue“9 =

‘suoHOUNJaA

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-

Ov 96 70

€9 Gk cS aa:

a

‘SUOOUNJIARM

(P9}19X9) eM

AB19U9) Jo @Vy

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a

0

0

07 07 07

Of Of Of Oe

Vv

"SUOTIOUNJIARM

(P9}9x9) SJ

(TUDSOIPAY)

(Punos3) % [TWO

(punoys) ¥ [TVD

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[VD

ee a

$0:0—

Lae -; 60 1-0

06

S

*e

A8iou9) Jo °Vy

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Srgs ra: — a

ELE a =a =

qdA



0c = = =

67 = = a

S

sontea ,_OT)‘Me=

uontsuey — ‘V4

(4

el ST 81 vt

oT 07 (Z6 07

Vv

Z7LZ0-0 (A?

a

RELATED

‘(Gontsuey

L Cl L Lvl

6 SST 0°01 rs

dq

AND

70

4 07~ omg as

vl

qqm&

Z7LZ0:0 (A?

F-CENTRE

v0

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ION PN JE PN: 109dPN 1d9aPN:

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sanyea ,_QT)‘Ne=

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aTavy T VIS uo1pISUDL]SaiZ4aua 4of ay)J

612

ONE-CARRIER

CENTRES

613

If the impurity is on the z axis, then the F,, transitions are to x and y p-like states which have nodes at the impurity. Thus 6, represents the change in the energy of the ground s-like state. The splitting of the F, bands 6, is dominated by the effect of the impurity on the z p-like function which has a large amplitude at the defect site. There is general good agreement of experiment and theory, but no theory is completely satisfactory. Clearly, now that accurate ENDOR data are available (e.g. Kersten 1970), there is still room for further work on energy levels, and also on wavefunctions.

15.4.3. Photochromic centres in CaF ,

The photochromic centres in CaF, consist of an F-centre with a trivalent rare-earth ion on a nearest-neighbour cation site. The two major differences from an F,-centre are the difference between the host and impurity charges and the importance of covalent admixtures of rare-earth orbitals. The main interest in these centres stems from their photochromic properties (Faughnan, Staebler, & Kiss 1971). These, in turn, occur because optical transitions are observed at different energies in the different charge states of the defect. Alig (1971) has discussed the theory of these centres, and has shown his predictions to be consistent with optical and spinresonance data. His defect-electron wavefunction is a linear combination of orbitals on the F-centre site and on the rare-earth site. The most important admixtures for the excited p-state of the F-centre are with the 5d orbitals rather than the partly-filled 4f shell. Rather than use unperturbed F-centre wavefunctions, one must correct for the effects of the rare-earth ion. These effects include two terms obtained for the F,-centre, those from lattice distortion and from ion-size effects and an additional term arising from the difference between the host and impurity charges. These corrections give a rather small net splitting of the p-states of the F-centre, since the repulsive ion-size and attractive point-ion terms act against one another.

15.4.4. The M*-centre The defect electron is trapped at two adjacent anion vacancies. There is an analogy with the photochromic centres in that there are two attractive sites. However, the fact that the M*-centre sites are equivalent may be exploited to relate the energy levels to those of the

614

THE

F-CENTRE

AND

RELATED

H} ion (Herman, Wallis, & Wallis 1956). The eigenvalues at separa-

tion R ina medium ofdielectric constant ¢ can be obtained from the equation 7B (Re) (15.4.2) Eye =e where ¢ and R are fitted to give results closest to those observed for the M *-centre. Aegerter & Liity (1970) have shown a remarkably good fit to be obtained for KCI with ¢ = 2-33 (cf. ¢,, = 2:19) and R = 396A (cf. the anion-anion separation of 4-44 A). Whilst one knows that the model is not accurate, the agreement is very useful for identifying levels. The agreement is just as good for the relative oscillator strengths of the transitions. Obviously any features will be omitted which depend solely on the atomic nature of the lattice, like the small splitting of the 2p-states. This is discussed in S56. 3:1. Berezin (1966) has made estimates of the energy of the lowest transition using a point-ion model with electronic polarization only. The results are in general agreement with observation. 15.4.5. The Z,-centre The Z,-centre is one stage more complex than the others, since it probably consists of an F-centre with a [100] cation vacancy and [111] divalent metal ion near to it. There are still some doubts as to whether this is the only configuration found, but it seems reasonably established for alkaline-earth ions in a number of alkali halides. Theories of optical absorption have been given by Dick (1968) and by Weber & Dick (1969). Dick used a point-ion model only, and found very poor agreement with experiment: the p-states of the F-centre were split by a much larger energy than observed. Weber & Dick included ion-size corrections in the model of Bartram, Stoneham, & Gash and also allowed for some limited lattice relaxation. Agreement with experiment was greatly improved. The Z,-centre strongly resembles the F-centre in absorption. There is a Mollwo-Ivey law (Hingshammer & Jodl 1967), and both the transition energies and Faraday rotation (Paus & Liity 1968)

are almost independent of the impurity cation and are similar to F-centre values. There are also similarities in emission (Paus 1969), such as the large Stokes shift, but the results are more complicated. In particular there is a (110) dipole moment which is not easily understood. The dependence on impurity cation is larger in emission, but still small. For KCl, the F band is at 2-30 eV in absorption and

ONE-CARRIER

CENTRES

615

1-215 eV in emission. The Z, bands in absorption are at 2.106eV (Sa, Sr) and 2-107 eV (Ba); the emission 0-855 (Sr), and 0-701 eV (Ba).

values are 0-833 eV (Ca),

15.5. Antimorphs of the F-centre: trapped-hole centres The most striking feature about centres where a hole is trapped at a cation vacancy is that they differ in almost all respects from the F-centre, where an electron is trapped at an anion vacancy. This is not obvious from the theories we have described in § 15.2 and, indeed, these theories are not good approaches for the hole centres. In cubic crystals, the F-centre ground state has cubic symmetry. The hole centres, however, have lower symmetry. The hole is localized on one or two neighbours to the cation vacancy. Of course, it may hop between the various equivalent sites around the vacancy, but the localization is easily observed. Thus, for a hole trapped on an oxygen neighbour to a magnesium vacancy in MgO the time spent at a given site, at room temperature, can be shown from the absence of dichroism in optical bleaching, to be less than a few seconds, but, from the absence of motional averaging in spin resonance, is seen to be more than about 107s.

Trapped-hole centres have been seen in the alkali halides (Kanzig 1960a), various oxides (see the review of Hughes & Henderson (1972)) and in other II-VI compounds, such as ZnSe (Watkins 1971). The centre in the alkali halides can be regarded as an X, molecular ion next to the cation vacancy, X being the halogen. Thus it can be regarded as a V,-centre (cf. chapter 18) perturbed by the nearby vacancy. The main effect of the vacancy is to tilt the principal axes of the hyperfine interaction, with only small effects on other parameters. The centre in the oxides and other II-VI compounds differs from the alkali-halide centres in that the hole is localized on a single anion. Since fuller data are available we shall discuss these systems in more detail. 15.5.1. These

Trapped-hole centres in oxides centres

have

been

observed

in the cubic

oxides

MgO,

CaO, and SrO, in Al,O3 and in the wurtzite oxides BeO and ZnO. Optical data are available for MgO, CaO, and Al,O, and spinresonance data for all these systems. Hughes & Henderson review the experimental data. The properties can be understood reasonably well on the simple model (Bartram, Swenberg, & Fournier 1965,

616

THE

F-CENTRE

AND

RELATED

extended by Hughes & Henderson) of an O™ ion representing a hole trapped on O? -, perturbed by a point charge — 2\e| representing the vacancy. In this simple crystal-field model, the main effect of the point charge is to split the 2p levels of O-. The energy of the 2p,-state pointing towards the vacancy is raised by 3A, and the energies of the other two 2p,-states lowered by —3A. Here A is given by

Ger.» mar Wares

(15.51)

where R is the vacancy—O~ separation and

—Wii},

{Wo2, W2->},

Woon

—W,})

for the E-states. Their results give variational parameters ¢ for each basis function (or pair of functions for E-states). Linear combinations of these optimized basis functions are then used to give one-electron wavefunctions and energy levels ¢,,, where T = A,,E and k = 1, 2,3. As expected, the lowest one-electron level has A, symmetry.

B. Three-electron states. The nine one-electron wavefunctions lead to 53 distinct three-electron configurations by populating the levels subject to the exclusive principle. To simplify the problem, Kern & Bartram concentrated on the 10 configurations for which the sums of one-electron energies were lowest. These states are shown in Table 17.1. Note that the Silsbee (1965) and Wang & Chu (1966)

TABLES l/s Lowest configurations for R-centre in KCl One electron Configuration

S, W-Ct SWC

SWee

Ye; relative

promoted?

Terms

energy

2E 2A, 2E 7A,+7A,+°7E+7A, 27E+4E *E 21\ 5 7A,+7A,+7E +4A,+4A,+°%E *F +E 2E

— 0-0119 0:1662 0-2471 0:2590 0-2625 0-2709 0-4133

ae, AAs aqe> Aer Anke, aie; ade Michie,

aae Ba a gl AAG aee a AC agane! al al ey Ao

— Yes Yes Yes Yes Yes No Yes

Ae, ei

A Ale Gee

No No

0-4252 0-4942

+ Configurations included by Silsbee and Wang & Chu are marked S, W-C.

634

THE

R-CENTRE

calculations singled out just three configurations by treating only the lowest A,- and E-states; their selection is not adequate and omits a number of important states. New orbital parameters ¢ are needed for the neutral R-centre. Detailed calculations were carried out for the lowest energy levels of A,, Aj, and E symmetry,

with configurations

aaa’, aae, and aee

respectively. The resulting parameters ¢ were then used to calculate the energies of the other three-electron states of the same symmetry. Parameters for one-electron orbitals of E symmetry in the A,states were taken from the results for the A,-state, since these gave the best results after configuration admixture was included. The results, listed in Table

17.2, show

that the one-electron

functions

are more diffuse than for the R?*-centre. There is also a significant variation from state to state. TABMESIY 22 Orbital parameters ¢ for the KCI R-centre Wavefunction

R24

Lowest 7A,

Lowest *A,

Lowest *E

Woo

0-61

0-405

0-499

0.444

one-electron < W,

0-65

0-456

0-626

0-673

0-46

Not present

0-363

0-342

A,

orbitals

Wo

E

Uae

one-electron
7E,?A,, or *E) are well within the accuracy one would expect for a complex a priori calculation of this type. The only real discrepancy is that the Rp band is not predicted. The R, band is a very broad band whose existence was only established by stress-induced dichroism, thus there are still some doubts about its status. Silsbee’s results, scaled from those

for the H3; molecule, also agree reasonably well with experiment. This should probably be considered fortuitous, since the model omits some important configurations. The oscillator strengths are even more sensitive to the details of the calculation, and achievement of the correct trends (see Table 17.3) must be considered satisfactory. Results for higher energy

ABLE Mao Oscillator strengths for the KCI R-centre Relative oscillator strength

Transition

Band

Theory

Experiment

7E + 7E

Rn R, Re R, Rx Ry R*

0-25 3-91 0-44 {1-00} 0-01 0-09 0:26

0-37 + 0-07 1-20+0-12 oo [1-00] 0-18 + 0-04 0:29 +0-03 a

7E > 7A, 7E > 7A, *A, > *E

THE

R-CENTRE

637

transitions are not given, since they are hard to separate experimentally from transitions involving other defects. The most unsatisfactory result is the underestimate of the Ry, strength; other results are acceptable. The predicted optical bandshape confirms the general features of the model. The calculated bandwidths are generally larger than

Absorption

0

0-01

0-02

1

0-03

1

0:04

4

Energy (eV)

Fic. 17.3. R, band in KCl. The observed line-shape (broken line) is compared with the predicted shape (full line; Kern & Bartram).

observed, although the trends are right. For the important R, band the agreement is better, and the results very close to those indicated by experiment. For this band the predicted sideband structure (Fig. 17.3) suggests that the third one-phonon peak (0-02 eV above the zero-phonon line) is associated with A, modes; the other peaks are probably associated with modes of different symmetry.

638

THE

R-CENTRE

17.3. The R-centre and the Jahn-Teller effect 17.3.1.

Introduction

The electronic states of the R*,R~, and R’-centres have A,, A>,

or E symmetry. The E-states couple to lattice modes of the same symmetry, giving a Jahn-Teller effect. The Jahn-Teller effect proves to be well described by the LHOPS model of§ 8.4.3, and the response of these centres to external fields permits an accurate test of the model. The properties of the zero-phonon line and of the phononassisted sidebands have both been measured. The success of the LHOPS

model also casts doubt on analogies with the H,; molecule,

where detailed theory (Porter, Stevens, & Karplus 1968) shows that linear coupling and harmonic interactions give a very incomplete description. We may concentrate on transitions between an E-state and an A-state. The three transitions discussed in most detail are between these states: Ground state

Excited state

Host

A 2E BAG

'E 7A, (R, band) 213)

NaF KCl LiF

R* R R’

The host lattices given here are the ones for which most information is available. In using the data to check the LHOPS model, we shall use two main criteria. First, can one obtain consistent Jahn-Teller para-

meters from a variety of experiments? Secondly, is there any evidence that the two basic approximations—a single E mode frequency and the cylindrical symmetry of the potential—are violated? For simplicity we shall not discuss in any detail the coupling of the Rcentre to non-Jahn—Teller modes. 17.3.2.

Wavefunctions

We now give the vibronic wavefunctions associated with A,, A>, and E electronic states. When there are no anharmonic or trigonal terms in the potential to remove the cylindrical symmetry all levels are doubly degenerate. The vibronic states associated with the E electronic state were given in § 8.4.3. Since most discussions ofthe R-centre use a particular

THE

R-CENTRE

639

notation (different from § 8.4.3) and most papers contain misprints, we give the wavefunctions explicitly. The relation between the sets of notation is given in § 8.4.3,

De Ten Yum —1) cos (1) —Wy(— 1) 4u,(r) sin (16)}, (17.3.1) l

Pi =

Jn) {Wuup(— r) sin (ld) +W(—- 1)’ 4u,,(r) cos (Ip)}.

(17.3.2)

| =j describes the variation with @ of the vibrational factors and p = 1,2,... labels the states of given / in order ofincreasing energy. The electronic functions Wy, W, refer to the upper and lower potential-energy surfaces. The components in wy, dominate in the lowerlying states for a large Jahn—Teller effect. The electronic states W, and Wy can be related to |x> and |y>, with explicit symmetry properties, by a proper choice of phase. Thus

Wu = cos (f/2)|x> +sin ($/2)ly>

(17.3.4)

WL = —sin ($/2)|x> +cos ($/2)ly>

(17.3.5)

and

are the analogues of (8.4.18}(8.4.19). The |x> and |y> states are defined with reference to Fig. 17.1. With this choice the transformations properties of the vibronic functions (W£, PF) are (E(x), E(y)) for

1=43,344,43...and(A,,A.) for!= 3,3, ce , The vibronic cies associated with A, or A, electronic states are much simpler, and they can be written in the Born—Oppenheimer form

ee) re

= WaRnm(t) exp (img) ty = picts ay cairn

(17.3.6)

where the y,,, are two-dimensional harmonic-oscillator functions (8.4.23). The subscript a is 1 or 2 for A, or A, electronic states ,. The transformation properties of the vibronic states are given in Table 17.4. Note that A,, A,, and E vibronic states can be obtained from any of the possible electronic states. Further, we have ignored the configuration dependence of the purely electronic wavefunctions

640

THE

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TABLE 17.4 Transformation properties of vibronic states derived from A, and A, electronic states Vibrational quantum number m

Vibronic states

bee

ain

ioe

|m| a

Le2n4; 3; Upeiedc

|m| =

0, S; 6)

E(x)

Ay

E(y)

A;

|x), |y>, and w,. The Condon approximation is already built in. Of course, the proportions of wy and y, in the ‘YF do depend on the lattice configuration, and this must be taken into account. Matrix elements cannot be written as simple products of electronic and vibrational factors, but they can be written as a sum of a small number of terms of this sort. The u,,(r) can be expanded in terms of harmonic-oscillator functions, using (8.4.24). Thus lattice overlaps can be found in terms of standard harmonic-oscillator overlaps. When we treat the effect of external fields on the lowest vibronic states associated with an electronic E-state, we shall use the theory of§8.4.4. This involves the use of an effective Hamiltonian operating among the electronic states only. The vibronic properties are included in Ham reduction factors. In our earlier notation, the ones of interest are, together with the operators affected, p = R(A,) (orbital angular momentum about the R-centre z axis), q Ill R(E,¢) (electric fields or stress components in the plane of the R-centre),

1 = R(A,) (electric fields or stress components

normal to the R-

centre).

17.3.3. Optical matrix elements and transition probabilities A. Matrix elements. The optical matrix elements contain products of electronic and vibrational terms. The electronic terms are of the form Eand E — A, the initial state is taken to be the lowest vibrational state. The phonon-assisted transitions are then separated by energies characteristic of the final vibrational state. Thus for

E— A, the levels in the final state are those of a two-

dimensional harmonic oscillator and have equal spacing ho. However, for A > E the final state splittings are not quite constant, because of the Jahn-Teller coupling. Fig. 17.4 shows the two cases compared with R’-centre data for LiF. There is general agreement, especially for the two-humped structure (§ 10.10.2) in the A> E transition. However, there are anomalies. In emission (E > A), there is a sharp peak at 0-049 eV, corresponding to an E mode. It should be present, slightly displaced from vibronic effects, at about 0-054 eV in absorption (A > E). No peak is observed. Similar effects have been seen for the R *-centre in LiF (Toulouse & Lambert 1966) and NaF (Baumann et al. 1967). Possibly the discrepancy means that it is too simple to assume E modes with just a single frequency.

Absorption

[\—=— 8

——

Emission

E—+A

Fic. 17.4. R’ band in KCl. Comparison, after Davis, of the observed spectra with a line spectrum from the LHOPS model. Results use k? = 5 and hw = 0-049 eV.

THE

R-CENTRE

643

Fetterman estimated the ratio of the intensity in the zero-phonon line to that in the one-phonon sideband. This decreases with increasing Jahn-Teller coupling; it is shown in Fig. 17.5 for both E > AandA > E transitions. We measure the strength of the Jahn— Teller effect by the parameter ka



Ey7/shog

(Wesel)

used by Longuet—Higgins et al. It is dimensionless.

First peak Emode strength Zero-phonon strength line

Fic. 17.5. Relative intensities of the zero-phonon and one-phonon peaks in the LHOPS model. Results, after Fetterman, for A E and E— A as a function of coupling.

Of course, in any real solid, coupling to totally symmetric modes will occur. This does not cause any real complications in the electronic structure, but these modes will participate in phonon-assisted transitions. Fig. 17.6 compares the R’-centre emission in LiF with the phonon density of states, which involves phonons of both A and E symmetries. Coupling to phonons which do not have E symmetry is relatively weak, characterized by a Huang-Rhys factor of order unity.

17.3.4. Response to external fields The response to external fields of the R*-, R~- and R’-centres

allows quantitative estimates of the parameters in the Jahn-Teller effect. Comparison of the results from different experiments then indicates just how well the LHOPS model describes the defects. We

644

THE

R-CENTLRE

R’ emission

Phonon density of states

Fic. 17.6. R’ emission Fetterman.

and the phonon

density of states. Results for LiF, after

shall concentrate on the three best-documented systems: NaF(R*), KCI(R), and LiF(R’). It will appear that the LHOPS model provides a good basis for discussing their vibronic properties. Some general features are common to all the situations we discuss. We shall concentrate on A+ E transitions, although EoE transitions have been studied and give useful data. If the initial state of the optical transition is an E-state, then an external field will remove the double degeneracy. At low temperatures, only the component with lower energy will be populated. The dichroism which results, with its dependence on temperature and on the perturbation, are prime sources of information. However, the simple picture of an R-centre in an otherwise-perfect crystal proves inadequate. Random strains split the E-states by amounts compar-

able with the field-induced terms, and must be included in the analysis. The LHOPS model has cylindrical symmetry about the z axis normal to the R-centre; all terms in the potential which have just the three-fold symmetry of the real problem are ignored. Thus the electronic orbital angular momentum L, is unquenched, although the other components are completely quenched. Matrix elements of L, within the vibronic E-states are partly quenched by two effects. The first is the Ham effect (X|Lz1Y> vibronic

a

DOXA

tenets ,

Gide st l )

THE

R-CENTRE

645

where p is the reduction factor. Detailed expressions are marginally more complicated, because the vibronic wavefunctions do not factorize into products of electronic and vibrational parts. After simplification, the reduction factor appropriate to the lowest vibronic state is p= [array

=r

(4 2)

This has been tabulated by Child & Longuet—Higgins (1961). If a matrix element between two distinct E-states is involved, then we should need instead

£ |dr rful(—rutr) +u( =r) for | = 5. The second contribution to quenching of the orbital angular momentum comes from random strains. The general methods for treating the response to external fields were discussed in chapter 12. We shall only sketch the arguments here.

A. Response to applied stress. Uniaxial stress has been used in several ways. The earliest (Silsbee 1965; Hughes & Runciman 1965) involved the identification of the symmetry of the R-centre by stresssplitting of zero-phonon lines. This work confirmed the F, model and also established the symmetries of some of the states involved. Silsbee also examined the stress-induced dichroism in the broad phonon-assisted bands. Later, this technique was used for the R’centre, by Fetterman (1968), whose discussion we follow. The main ideas can be seen from Fig. 17.7, which shows transitions under an external stress which lowers the x-state relative to the ystate.t The allowed transitions for light polarized parallel (c) and perpendicular (z) to the stress are shown. At low temperatures only the x states are populated. The selection rules can be exploited as follows. First, we can estimate the effective frequency of the E-mode phonons by noting that o transitions are only allowed when accompanied by emission of an odd number of E phonons. Thus Fetterman found o-polarized peaks for the R’-centre in LiF at 0-049 eV, 0-076 eV, and 0-083 eV. These correspond to an E phonon + These are vibronic states referred to axes in which the normal to the R-centre is

the z direction, etc. It is straightforward to relate the axes to the axes of the cubic crystal, but unnecessary to do so here.

646

THE

R-CENTRE

Fic. 17.7. Splitting of the A, where L, is the sum of electronic angular momenta. The electronic spin-orbit coupling 4 and gp are both reduced by the same factor p. The parameters are such that either the spin Zeeman term or the internal strain term dominates. Further, the spin-orbit term is much larger than the orbital Zeeman contribution. When there is no internal strain, the levels are shown in Fig. 17.8(a). Positive spin-orbit coupling is shown. When the internal strain is large, the levels are shown in Fig. 17.8(b). Almost all the orbital angular

(x+iy)6B —— B———"

(x+iy) ¢——*

Oa

(x+iey) B

.

(x+ iey) x

XP

ue

(x—iy)

y,

me

i

Ne

:

fe)

a

xy / ——————

yi!

\

rene

\

\

¢

4

Va Spin Zeeman energy

Reduced

spin-orbit

coupling

Strain splitting

(ytiex)a

Spin-orbit and

Zeeman

energies

Orbital Zeeman

energy Zero strain

Large strain

Fic. 17.8. Zeeman effect for the R-centre: effects of internal strain. The modifications of energy levels and wavefunctions are shown, illustrating the profound effects of strain. If the strain splitting is A,, then ¢ is (ggBH + Ag)/A. The thermal average of S, enters here because the spin-orbit coupling is weak and can be regarded as providing an effective magnetic field A¢S_>/f. The g-tensor observed experimentally contains three contributions : the spin Zeeman term and the orbital Zeeman term, within the lowest vibronic state, and a term from the admixture of higher E-states by the spin-orbit coupling. Krupka & Silsbee give

Bar el a =g.+2

p?

ioe)

oamt

pe

n Set (17.3.14)

g. = g,//(1+p77/A’).

They have also given estimates of gy and 4 for KCl, using a Heitler— London model of the R-centre. These indicate fsa

2

tenia

gy = 0-48.

dena

The term (g, — g,) is positive: the spin-orbit interaction results from currents about the nuclei which have angular momenta opposite in sense to that of the centre as a whole. This is also the case for the isolated F-centre. Estimates of the Jahn-Teller coupling can be made from the stress-dependence of the g-tensor. The internal strains must be included here. The results suggest k? ~ 3+0-5 and lead to

SoA = —— O26 CMe

(17.3.16)

If the ratio of go and 4 is correctly predicted then this implies Zo

——

A=

1-1

—48 cm.

(L7Sithz)

Duval et al., Burke, and Shepherd all observed the magnetic circular dichroism of the R-centre in KCI. The important measurements are of changes in the total intensity and the first moment of the left- and right-hand polarizations for both the zero-phonon line and the broad phonon-assisted band. The changes in intensity simply involve the increase of one polarization at the expense of the other; there is no change in total intensity.

THE

R-CENTRE

649

There are three contributions to the changes in area of the components of the zero-phonon line: population effects, the admixture of higher vibronic levels of the electronic E-state by the spin-orbit coupling and magnetic field, and the admixture of higher vibronic states associated with the electronic A-state to which the transition occurs. The dominant effect is the population effect, with a smaller contribution (about 20 per cent) from the admixture of higher vibronic states derived from the electronic A-state. If A,, A_ give the area of the zero-phonon line for left- and right-hand circular polarizations respectively then the population effect gives OA4

42

gp

A

eauze KepeopHeth: jis Ser athes as +

g.pBH Ae \e Heda tanh as gob aa rat +04(27|

173.1 a,

averaged over spins. Note the cancellation of A factors between the populations and admixture parts. At the temperatures of interest the limit ¢ > 0 is adequate, and the result is very nearly linear in H. The contribution of the A-derived states is removed by estimating the term (17.3.18) from the change in first moment (AE,>

=

+p

H 8ohH — 3A tanh ue |.

(17.3.19)

These results give values of Rg o= 0-055 and RA = 0:24cm poe The ratio A/gy is about —4-4, similar to Krupka & Silsbee’s estimate. If Krupka & Silsbee’s values of A and go are used

p = 0.05,

(17.3.20)

which implies k? ~ 3-6. Similar calculations can be made for the broad band. The calculations are more complicated because of the Jahn-Teller effect, but can be carried through without difficulty. The area changes and first-moment changes are given by

bea A,

= (*As] 2® band

(17.3.21)

zpl

and CAE

hand

=

CAE s Dzpi®,

(173722)

650

THE

R-CENTRE

where only the population-change contribution to the zero-phonon line is included. The factor ® is close to 4. It can be written 1—U

(17.3.23)

> = —__, 1+U

where U involves overlaps of vibrational factors

a Oiy a 2al) hatin isle Dot g()}/ +ui )? (—

i dn lfdr. rppoltig

(17.3.24)

The sign of the moment changes depend on the final vibrational state, and is opposite for states corresponding to the excitation of an even or odd number of E modes. There are two contributions to the total angular momentum in the lowest states : one is vibrational, the other electronic. The electronic angular momentum has opposite signs in the two terms, and in the transition the vibrational angular momentum does not change. For transitions with the excitation of an odd number of E modes only one term contributes to the matrix element and for an even number the other, with its opposite electronic angular momentum. This selection rule allows one to assign phonon sidebands to A or E modes. C. Applied electric fields. Applied fields have two main effects. The first is a linear Stark effect arising from finite matrix elements within the E state. If the two independent electronic matrix elements allowed by symmetry are

M, = My =

(173.25)

then the effective Hamiltonian describing the interaction is Hors





qeE)M,(cU;,+sU;)+eE,

M,.

(17.3.26)

The component within the plane of the R-centre contains the Ham reduction factor q. For the strength of Jahn-Teller effect of interest

(k? = 2) q is very close to 4+. The values of M, and of qM \ can be measured ; (Johannson et al. 1969; Davis & Fitchen 1971); regrettably it does not seem possible to measure M | OF q separately. Results cited by Johannson et al. (1969) for R- and R’-centres in LiF show that the matrix elements do not vary much with the charge state of the centre.

THE

R-CENTRE

651

The second effect of an applied field is to admix the / = 3A,- and A,-States into the lowest vibronic E-state. In principle, such measurements can give an estimate of the splitting of the A,- and A,-states by the trigonal field. However, no effects have been observed, and it

is merely possible to put bounds on the separation of the A,- and A,-states from the lowest E-state. 17.3.5. Comparison of LHOPS theory with the R-centre: summary

The first test of the LHOPS model is that it should describe consistently a range of phenomena with the same parameters. This has been investigated in most detail by Davis (1970) for the R’centre in LiF. He was able to put bounds on k? in six different ways, summarized in Table 17.6. The results are all consistent, although the limits are not very close.

TABLE

17.6

Values of k? for the R’-centre in LiF (Davis 1970) Experiment Optical line-shape Stress-induced polarization of broad-band emission Ratio of intensities of emission with one E phonon to that with no E phonons Linearity of response to stress Linearity of response to electric field Temperature-independent magnetic circular dichroism (assume A = 0-5)

k? Bix [> 7,). The transition energies are interpolated from Gilbert & Wahl’s (1971) Hartree-Fock calculations. The cases for which experimental data are available are summarized in Table 18.1. The agreement

TABLE 18.1 V,. transition energies (eV)

ULTRAVIOLET

CaF,

LiF

NaF

LiCl

NaCl KCI

RbCl

404 3-70

3-48 3:30

3-38 3-30

3-15 2-69

3-28 2:65

3-40 2:20

1-72 1-87

1-65 1-60

BAND

Experiment Jette et al. theory INFRARED BAND Experiment Jette et al. theory

~

3-39 2:37 1-65 1-23

EPR BAND (MEAN OF AE, AND AE,)

Experiment (f, = 1) Jette et al. theory

2:57 2:55

2-63 19

2:75 18

for LiF and NaF is very good. However, the results for the chlorides appear to reflect the limited lattice relaxation allowed by Jette et al. The results for KCl can be made as accurate as the LiF and NaF ones by allowing about 40 per cent reduction in separation, instead of the calculated 33 per cent, this can be seen from Fig. 18.3. All these results for the chlorides suggest that the true equilibrium separation of the V, ions is closer to the free-ion separation than the calculations of § 18.2 predict. Preliminary results allowing more extensivelatticerelaxation(Norgett, Lumb, & Stoneham, unpublished

660

SELF-TRAPPED

HOLES:

© CaF,

o LiF

°oKC

Energy levels (eV)

5 4

3

Rese

:

E. E,

16, Fluorides

Chlorides

B32 3-4 3-6 3:8 4:0 4:2 4-4 46 48 5-0 5:2 5-4 5-6 5:8 6-0 R (atomic units)

‘ ae

ob

3

0 for the chlorides, but is inconclusive. Dreybrodt (1967) has measured the temperature dependence of the hyperfine constants of V,-centres. He finds a characteristic frequency for the centre which is similar to that in the free X; ion— 80 per cent of the free-ion value for KF and 95 per cent for KCI. The potential-energy curves of Jette et al. suggest that a modest increase in frequency should occur if the effective mass of the mode is the same as for the free ion. Presumably Dreybrodt’s result demonstrates the inertia associated with the motion of neighbouring ions in the

THE

V,-CENTRE

663

crystal as the V, ions move. However, for the alkaline earth fluorides, Assmus & Dreybrodt (1969) find effective frequencies which are not consistent with motion of the V, ions (see e.g. Norgett & Stoneham 1973) and are probably associated with cation motion. The hyperfine structure of neighbours to the V,-centre consists of two main parts. The first is the isotropic hyperfine interaction, which is a measure of the amplitude of the X; wavefunction at the site. The second is the anisotropic interaction, which is dominated by the dipole-dipole interaction. Ikenberry, Jette, & Das (1970) have shown that exchange polarization is important in the isotropic interaction. It is not sufficient to concentrate on the singly-occupied a, orbital. One must recognize that the electron in this orbital interacts differently with electrons in s orbitals on nearby atoms, according to their spin. The difference

in interaction is the exchange energy, and it leads to a modification in the spin density at the nuclei of neighbouring ions. There are three main terms. First, there is the direct polarization of the neighbouring s orbitals by the a, electron. Second, the a, electron affects the other paired orbitals of Xj, and these, in turn, affect the neighbouring s orbitals; this is the indirect term. Third, there is a crossterm. Ikenberry et al. show that in LiF the indirect term and crossterms cancel almost completely. The direct term gives good agreement with experiment for the large negative hyperfine constants of neighbours A and B in the equatorial plane of the V,-centre: Theory (MHz)

A site (Li*) B site (F~)

Experiment (MHz)

— 5.03 — 6:60

=—4:42 — 6-69

18.5. The V,-centre as a molecule in a crystal Qualitatively and quantitatively, the model of the V,-centre as an Xz ion is a considerable success. The separation of the component ions alone is significantly changed by the crystalline environment. We now examine the limitations of the model, and seek evidence that the X; wavefunctions are perturbed by the crystal field or that there is a significant admixture of crystal-ion wavefunctions. The crystal field shows directly in the differences between g,,, and E,| is only a small gy. Castner & Kanzig’s results indicate |E,.— KCl, 6 per cent and LiF for cent fraction of 4(E,+ E,}—about 4 per are modest differences These for KBr, and 14 per cent for NaCl. Likewise, wavefunction. the in and do not require large changes

664

SELF-TRAPPED

HOLES:

the hyperfine interaction with neighbouring ions indicates that the wavefunction is still close to that of the free molecular ion. A more sensitive test is found by seeking a separation of the X, ions which gives accurately the measured optical and spin-resonance properties. It proves possible to fit optical data and the g-factor (with f, ~ 1) well with a separation close to that found in the detailed distortion calculations. It is also possible to fit the hyperfine constants a and b, but only at a smaller separation. This discrepancy could be a consequence of crystal-field distortion or an admixture of valence-band functions. But it is also possible that better wavefunctions than the present Hartree-Fock ones are needed and that the X; model is still adequate.

18.6. Motion of the V, -centre So far we have considered a hole localized on a single X, molecular ion by the self-trapping distortion. The mean atomic displacements can be written in several ways, notably as Q,, where a labels the mode with wavevector q and polarization s, « = (q,s). The self-trapping energy has the form

Ey = -$¥M,w?0?

(18.6.1)

and is the sum of an increase in lattice-strain energy and decrease in the hole—lattice interaction. This result assumes a harmonic lattice. The motion of the hole through the lattice is by a hopping motion from site to site. It is different from the propagating motion of electrons in the conduction band. The motion is characterized by two parameters with the dimensions of energy (Holstein 1959; Song 1970, 1971; Flynn 1971; Norgett & Stoneham 1973). The first parameter J involves the overlap of the electronic wavefunctions of the hole for its initial and final sites. It is too difficult to calculate accurately at present, and is usually obtained phenomenologically. The second parameter is an activation energy E,, which is much easier to calculate using methods analogous to those in § 18.2. If the change in self-trapping distortion during the jump from one site to the next is AQ, = Q,-—Q,,, then

Ex = pyeM o|AO7215

(18.6.2)

Ex is the lattice activation energy; this is the energy required to distort the lattice to a type of saddle-point configuration such that the total energy is independent of the site to which the hole is attri-

THE

V,-CENTRE

665

buted. E, can be written in terms of lattice relaxation energies under suitable combinations of defect forces (Norgett & Stoneham) and can be found by methods described earlier. The hopping time t is then given by

Py

Sea |ooOheee a

c

“)

(18.6.3)

lJ| fae

which simplifies in many cases. At high temperatures, for practical purposes when all the hw,/kT are less than unity, the hopping rate is thermally-activated : 1 ae E Pe | ae salEET exp | al

6. ee)

The assumptions involved in (18.6.2)}(18.6.4) are primarily the following. First, the Born—Oppenheimer approximation is used, so there is a unique potential energy for the lattice with the hole present. Second, there is a linear approximation. In essence, it is assumed that the forces giving the self-trapping distortion do not change significantly as the relaxation proceeds, and in consequence the normal modes and their frequencies w, are not affected by the self-trapping distortion. Third, the Condon approximation is used: the electronic part of the transition matrix element is assumed to be independent of the Q, and to have the single value J. Finally, the harmonic approximation is used. It is possible to extend the theory to go beyond certain of these assumptions, as we see later. The simplest treatment of E, regards the hole as a point charge interacting only with longitudinal-optic modes of a single frequency 10 (Flynn 1971). The result for the alkali halides is that

0-838 eV A=

a

2

: (e, +29(4- *II,, with x polarization. These features are observed. Thus, in KCl the transition energies are H-centre o transition m transition

3-69 eV 2:38

Theory

V,-centre

4-53 2:16

3-39 1-65

The theory results come from interpolation of Gilbert & Wahl’s (1971) Hartree-Fock calculation to the 2.54 A separation predicted by Dienes et al. In CsBr effects of spin-orbit coupling are observed, and the z band is split (Chowdari et al. 1971). The g-factor gives a measure of the separation of the *Z, and *T1,, levels. In tetragonal symmetry one expects the energy difference AE to be related to the g-tensor by

&\| = 80> where ¢ is the spin-orbit coupling. For KCl, Kanzig & Woodruff (1958) give g, ~ 2:0023, as expected, and g, = 2-0227, g, = 2.0219. Using these values with € = 440 cm! (Slichter 1963) one obtains AE ~ 5-45 eV. This suggests a major qualitative difference from the V,-centre: the energy levels are inverted, as shown in Fig. 19.3. It also raises another problem, since the Gilbert-Wahl calculation shows no such crossing for any separation of Cl; that they consider. Of course, it may be that a smaller effective spin-orbit coupling is needed; a 37 per cent reduction would bring AE down to the value predicted by Gilbert & Wahl. Kanzig & Woodruff’s measurements of the hyperfine structure give isotropic constants on the neighbours less than those for the H

INTRINSIC

INTERSTITIALS

677

Fic. 19.3. Xz energy levels for H-centres. The order appears to be diflerent for H- and V,-centres, and the results are contrasted here. Full lines indicate optical transitions ’ broken lines the separation involved in g-factors.

central ions by factors 6-8 (LiF), 8-3 (KCl), and 8-4 (KBr). The KCl result is very similar to Hayns’ C.N.D.O. estimate of 8 for NaCl. No more detailed calculations are available. The theory of the electronic structure is incomplete in many respects. This is partly because of incomplete data; only in KCI are complete optical and spin-resonance results available.

19.3. Interstitials in valence crystals 19.3.1. Experiment and the interstitial

There are no confirmed direct observations of interstitials in diamond, silicon, and germanium; the only proposed identifications (e.g. Owen 1965; Lomer & Wild 1971; Lee, Kim, & Corbett 1972) are very tentative. Instead, the interstitials have been inferred from

their effects on other defects. The properties deduced are so novel that we summarize them here, although there is no confirmed explanation. The most tantalizing results are those of Watkins

1968) in his spin-resonance temperatures.

Under

electron

work

on

(1963, 1964,

silicon irradiated

irradiation,

lattice

at low

vacancies

are

formed and observed in spin resonance. The interstitial is not observed in spin resonance, either in silicon or in any diamond structure. Two explanations have been offered. Watkins has suggested, convincingly, that the interstitial is mobile even at 4-3 K. The interstitial moves through the lattice, until it is trapped in some way. Since substitutional group III acceptors are observed to become

678

THE

H-CENTRE

AND

OTHER

interstitial at very nearly the rate vacancies are produced, the sequence proposed is: 1. Production of vacancy and interstitial by radiation damage. 2. Motion of interstitial through the lattice. 3. Exchange of position by the interstitial and group III acceptor. Theoretically, the problem is to explain the high mobility, and we shall discuss models shortly. However, McKeighen & Koehler (1971) have suggested the possibility that the interstitial may only be mobile in special circumstances. Motion might occur under electron irradiation assisted by energy transfer from the electron beam, either directly (which is unlikely) or indirectly via secondary electrons, electron-hole recombination at the interstitial, or phonons

produced in the damage events. The experimental evidence is surveyed by Norris, Brower & Vook (1973). The second theory of the absence of observations of interstitials (e.g. Singhal 1971) is that interstitials are present but not observed; thus no localized states in the gap are predicted. This theory is unlikely to be correct. It does not explain the production of interstitial group III acceptors observed, and the approximate theory, which suggested the argument is in disagreement with other approximate calculations on the question of local states both for interstitials (Yamaguchi

1963; Watkins, Messmer, Weigel, Peak, &

Corbett 1971) and for other systems. Long-range motion of interstitials in Ge has been postulated (see the review of Mackay & Klontz (1971)), and there is some evidence for low-temperature motion of defects in diamond (Lomer & Wild

1971).

19.3.2. Theories of interstitial migration

Most theories have assumed that the stable interstitial site is the tetrahedral body-centred site and that the motion involves passing through the hexagonal site (§ 1.2). Bennemann (1965) estimated the difference in energy between these two interstitial configurations. He used the multiple-scattering theory reviewed in §5.4, and predicted an activation energy: Enex— Etetra = 0:85 eV (diamond) 0-51 eV (silicon)

0-44 eV (germanium).

INTRINSIC

INTERSTITIALS

679

These energies refer to neutral interstitials, and are far too large to explain the motion: energies of a few hundredths of an electron volt are needed if conventional diffusion theory is used. Hasiguti (1966) developed an observation of Weiser (1962) that, for charged interstitials, the polarization energy in the hexagonal site can compensate the increase in the short-range repulsive interaction. Indeed, his estimates suggest that the hexagonal site is stable for the positive interstitial: 0 = Enex—Etetra tetra

~

= — 0-25 (silicon, germanium).

Bourgoin & Corbett (1971) have suggested an extension of this idea, which is also in agreement with the comments of McKeignen & Koehler, that interstitital motion is detected under irradiation. The

argument is that the interstitial is stable at one site in one charge state and at another site in a second charge state. A thermal diffusion can then occur by alternate changes of charge state through capture of free carriers produced by the ionizing radiation. Other workers have considered different interstitial configurations. Watkins et al. (1971) have done an approximate molecularorbital treatment (extended Hiickel theory, § 7.3.3) calculation of four configurations in diamond. Two are the usual tetrahedral and hexagonal configurations. The third is a split —¢100> interstitial: two atoms separated in a (100) direction at a single site. The fourth configuration has the interstitial at a bond centre. The calculation gives the sum

of one-electron

eigenvalues for each configuration;

this is not the same as the total energy, and so the results must be treated cautiously. Further, only limited lattice distortion was allowed: the only two distortions included were the separation of the components of the split — interstitial and the linear motion of the atoms joined by the bond into which the bondcentred interstitial was inserted. Nevertheless, the results show the interesting feature that the split — and bond-centred interstitials are the most favoured in diamond. For the neutral interstitial the energies relative to that for the tetrahedral interstitial are: hexagonal

—9-1eV

split

—21-7eV

bond centre

—21-7eV.

680

THE

H-CENTRE

AND

OTHER

INTRINSIC

INTERSTITIALS

Results for the positive and negative interstitial are also given, but are more approximate because of limitations of the method. They are: ae

=

Hexagonal Split

— 10:8 —19-4

Bond-centred

— 20-9

—7-4 — 24-2 — 22:8

The important features are the observation that the tetrahedral site is unlikely to be stable, and the observation that two distinct configurations give much lower energies roughly equal to each other. 19.3.3. Theories of interstitial electronic structure

In view of the negligible experimental evidence, only a superficial description of the theories is appropriate. Yamaguchi (1963) and Watkins et al. (1971) have considered interstitials in diamond. Yamaguchi’s work on the tetrahedral configuration uses a molecular-orbital method similar to those used for the diamond vacancy (§ 27.4.1). The results are sensitive to the details of the calculation, but suggest a localized non-magnetic ground state, from which optical transitions to other local states should be observed. Watkins et al. give no details, but observe that local states within the bandgap were found in their extended Hiickel treatment of the various interstitial configurations. Singhal (1971) found no states in the gap in his Green’s function treatment of the tetrahedral interstitial in silicon. But his results on the bond-centre configuration suggest a bound state below the valence band, rather than in the bandgap (Singhal 1972). These methods have had rather poor success in the past in predicting bound states, and the result should be considered merely suggestive.

20

HYDROGEN IN IONIC CRYSTALS: Ue aUe AND UL CENTRES 20.1. Introduction

HYDROGEN in ionic crystals leads to three important classes of defect, depending on charge state and on site. These are as follows. 1. The U-centre, or the substitutional negative ion. We shall call

this H,. In the alkali halides and the caesium halides the H™ ion substitutes for the anion, and has an environment with cubic

symmetry. In the alkaline earth fluorides the site has just tetrahedral symmetry. 2. The U,-centre, or the interstitial negative ion. This will be labelled H; . The interstitial site in the NaCl-structure alkali halides has tetrahedral symmetry, in the CsCl-structure caesium halides has tetragonal symmetry, and in the alkaline earth fluorides has cubic symmetry.

3. The U,-centre, or the interstitial atom. We call this H?. Both H, and H? occupy the same body-centred interstitial sites, in contrast to the (§1.2) intrinsic interstitial atoms discussed in § 1922. The notation U, U,, and U, is a confusing remnant of the early work on these centres. It will not be used again in this chapter. There should be no confusion with the notation H-centre for the anion interstitial of chapter 19. The points we emphasize in this chapter are based on two observations. First, these centres contain a very simple impurity atom, with no core orbitals to complicate the theory. Thus one should hope for a rather detailed understanding of the nature of the states involved. We shall discuss the theory of the spin-resonance parameters of H? and the excited states to which optical absorption occurs. Secondly, hydrogen is an extremely light atom and gives rise to a local mode. The infrared absorption associated with this mode has been considered in detail, and we discuss the theory for

682

HYDROGEN

IN

IONIC

CRYSTALS:

H,. Local modes associated with hydrogen have been observed in a wide range of materials, including valence crystals and metals.

20.2. The interstitial atom H?

(U,-centre)

20.2.1. Nature of the ground state: spin resonance

The theory of the ground state of H? has been a series of extensions to the simple model (Mimura & Uemura 1959) in which the wavefunction of a free hydrogen atom is orthogonalized to the core orbitals on neighbouring ions. The first extension depends on the host-ion wavefunctions to which the hydrogen atom wavefunction is orthogonalized. Spaeth (1966) has shown that much better results are obtained when the core orbitals are orthogonalized to each other. Technically, this is done using Léwdin’s symmetric orthogonalization method; the hydrogen orbital is orthogonalized subsequently by the Schmidt procedure. Orthogonalization of the cores improves the relative values of the isotropic hyperfine constants on the nearest neighbour anions and cations (Table 20.1(a)). There is a similar improvement in the isotropic constant for the second cation shell. With orthogonalization, the theoretical value is two orders of magnitude less.

A second extension comes from configuration admixture. The main configuration is one with a single electron on the hydrogen and with full core shells on the neighbours. The admixed configuration is one in which a second electron is transferred to the hydrogen from the outer p shell of the halogen nearest neighbour. Thus the situation is very similar to the weak covalency discussion in §7.4. In the excited configuration there are two electrons with opposite TABLE 20.1 Ratios of predicted and experimental hyperfine constants for KCl: H? (a) Effect ofcore orthogonality (After Spaeth 1966) Ratio (theory/experiment)

Cores orthogonal Cores not orthogonal

Nearest cation

Nearest anion

1-68 4-63

0-732 0:796

(b) Effect of covalency (After Spaeth & Seidel 1971) Theory/experiment

Nearest cation

Nearest anion

Next-nearest cation

Covalency No covalency

0-55 1-44

0-595 0-825

1-26 0-51

U-, U,-, AND

U,-CENTRES

683

spins on the hydrogen site. Cho (1967) has argued the two 1s orbitals will have different orbital exponents, and uses values for the free H™ ion for this configuration. The effective Bohr radii for the two electrons are 0-961 ay and 3-54 ay respectively. The very widespread function for the second orbital leads to hyperfine constants on the second and third shells of anions which are much larger than observed. His calculation estimates the configuration mixing by fitting the central proton and nearest-cation hyperfine structures. Spaeth & Seidel (1971) observe that a different formulation, in terms of covalency, leads to a wavefunction which is equivalent in form. Consider, for simplicity, a three-electron system: one electron on the proton site and two electrons with opposite spin on the anion next to it. The ground configuration has a wavefunction

¥, = det du da Pa ||,

(20.2.1)

where +, — indicates spin. The excited configuration is

Pi = det Pada du |

(20.2.2)

and the admixed wavefunction a sum of the two:

Yo = No(Vit vy).

(20.2.3)

This can be rewritten using the properties of determinants as

Po = det li Ga(ba tubu)(i+y7)Fl

(20.2.4)

or, transforming again to put the orbitals into a common form,

Yo = det |l(u—uba) (ba +Hbu)* (Gat Hw) {I x

x(1+p)7?.

(20.2.5)

Thus, if the hydrogen orbital wavefunctions are the same in both configurations then the configuration admixture and the covalent change in the one-electron orbitals are equivalent. Of course, this is consistent with the discussion of §7.4; the two approaches are equivalent only because of the special assumptions. Spaeth & Seidel (1971) use the covalency picture, so that only one parameter need be fitted to experiment, and the problems of the spread-out second electron do not arise. They fixed the anisotropic hyperfine constant on the halogen nearest neighbours, since this is most sensitive to the covalency. Values of about 10 per cent were found for p. There is no significant improvement in the hyperfine constants. The agreement with experiment is generally good, and

684

HYDROGEN

IN

IONIC

CRYSTALS:

it is merely the details which are altered. The ratio of isotropic constants predicted to those observed is given in Table 20.1(b); there are similar results for other parameters. Sammel (1965) considered the modifications of the hydrogenatom wavefunction in the crystal. The usual approximation assumes

the free-atom wavefunction. Sammel considered the variation of the orbital exponent of the 1s function and the admixture of both 4f (since the site has tetrahedral symmetry) and 5g functions. The admixtures and the parameters in the different components were fixed by a variational calculation in the point-ion model. The improvement in agreement with experiment proves rather small. The percentage error in the proton hyperfine constant in KC] falls from 18 per cent to about 13 per cent, that in the halogen hyperfine constant from about 30 per cent to about 26 per cent, and there was no significant change in alkali hyperfine constant. Most of the improvement comes from the 4f admixture. Sammel (1969) has also done a similar calculation going beyond the point-ion model. The crystal ions were represented by free-ion wavefunctions orthogonalized to each other. If exchange, threeand four-centre integrals and the polarization and distortion of the host lattice are ignored then agreement with experiment is limited. Neither the ratio of anion to cation hyperfine constants, nor the difference of the central ion hyperfine constants from those of the free ion are predicted accurately. These corrections can be included approximately, and appear to improve the agreement. Different radial distortions for anions and cations are necessary. The zero-point motion of the hydrogen is also significant. This can be seen from the differences in hyperfine constants for hydrogen and deuterium. The isotropic constant on the nearest cation is 16-5 per cent higher for hydrogen than for deuterium. Spaeth (1969) has estimated the magnitude theoretically. He observes that the hydrogen-host ion separations vary with time through the zero-point motion. The local-mode frequency is much larger than the hyperfine frequencies, so the hyperfine constants should be averaged over the zero-point motion. The local-mode energy is also much higher than thermal at the temperatures of interest, so the excited vibrational states of the hydrogen can be ignored. If the ground-state vibrational wavefunction is

x(r) ~ exp (—4pr’),

(20.2.6)

U-, U,-,

AND

U,-CENTRES

685

where p varies as ./M, M being the mass of the central atom, then Spaeth shows that the mean-square overlaps can be approximated

by S? = Sha

-4 exp (B*/p).

(20.2.7)

Here f is a measure of the decay of the hydrogen-host-ion overlap with distance: S(R) = S(0) exp (— BR). (20.2.8) Two assumptions are implicit here. First, the hydrogen wavefunction is assumed to be unaltered as it moves in its zero-point motion (strictly this is the pseudo-wavefunction; there are changes from orthogonalization). Second, the mean separation R of the host atom from the interstitial is assumed independent of isotope. In fact, one

expects there to be an outward term in the mean distortion which increases with the zero-point motion. The results prove sensitive to the orthogonalization of the host ions to each other. This effect is omitted in model I and included in model II. Results for the isotropic constants are shown in Table 20.2. The effective frequency, derived from the fitted value of p, differs by some 40 per cent from the observed local mode. Given the simplifications, this is acceptable agreement. Finally, we turn to two other spin-resonance parameters, the proton hyperfine interaction and the g-factor. We shall ignore the quadrupole interactions, as the shielding corrections are not known. The observed proton hyperfine constant differs from its value for a free-hydrogen atom by +4-75 per cent (KF), —3-01 per cent (KC\), TABLE

20.2

Isotope effect for KCl: H? (After Spaeth 1969)

Ratios H:D Central ion Anion Cation

Model I

Model II

Experiment

— 1-026 (fit) 1-036

= 1-026 (fit) 1-091

0-9974 1-026 1-165

Ratios H:static

Anion

1-040

1-093

=

Cation

1-13

1-37

a

686

HYDROGEN

IN

IONIC

CRYSTALS:

and —5-88 per cent (KBr). Since the corrections for orthogonality and covalency give increases in all cases (+ 12-7 per cent for KF, 9.7 per cent for KCl, and 7-5 per cent for KBr) there is a discrepancy which is sometimes taken as a major weakness of the model. One obvious possibility is that it is inappropriate to use the free-atom wavefunction in the solid without some correction to the effective Bohr radius and the addition of other terms with zero amplitude at

the proton (e.g. terms transforming like xyz/r? or (x*+y*+z2*—— 3r*)/r*). But Sammel’s work (1965) in the point-ion approximation implies this correction is not enough. A second suggestion, due to Adrian (1960), is that the Van der Waals interaction with the neighbours gives the reduction in spin density at the proton. Essentially, the interaction gives a fluctuating admixture of dipole terms which have zero amplitude at the nucleus. It is straightforward to estimate this change in a form valid for a large separation of the hydrogen atom and its neighbours. The result involves the hydrogen and host-ion polarizabilities and their average excitation energies. Spaeth & Seidel estimate this contribution to be about —10 per cent for KF, KCl, and KBr. Whilst

this leads to very

reasonable agreement with experiment, the problems of the longrange asymptotic form used and of the possible screening of the interaction leave the conclusion uncertain. The orbital contribution to the g-factor comes primarily from three types of excited state: charge-transfer states involving the nearest

anions, similar states

involving

the cations, and excited

states of the hydrogen atom appropriately orthogonalized to the nearest neighbours. This division is only approximate, but it allows use of the ionic spin-orbit coupling constants and mean excitation energies (cf. eqns (13.3.21)—(13.13.3)). The results involve the admixtures of host-lattice core orbitals from orthogonalization and covalency; these may be estimated from the observed anisotropic hyperfine constants. The mean excitation energies were taken as follows. For the anion charge transfer state, E, was a multiple « of the observed optical transition energy Ey. For the cation chargetransfer states, E, was taken as 15 eV, estimated from the Madelung energy and electron affinities. For the hydrogen excited states, Ey was taken as l11eV, ie. the Rydberg minus a polarization correction.

The anion charge-transfer states and the hydrogen excited states dominate in all cases except KF and RbCl, where the cation has

U-,

U,-,

AND

U,-CENTRES

687

larger spin-orbit coupling. The two leading terms have opposite signs, and there is a substantial cancellation sensitive to «. The results for « = 1 and for « = 1-33, fitted to the KCI results, are given in Table 20.3. The agreement for « = 1-33 is particularly good. It suggests that the charge transfer states for the g-factor, involving orbitals, lie higher than those reached optically, which involve o orbitals. TABLE 20.3 g-factors of H®-centres in alkali halides (Spaeth & Seidel 1971) 10* Ag KF

Theory (a = 1) + 0-6

Theory (« = 1-33) —2-2

KCl KBr

+ 16:5 +108

+ 7-0 (fitted) +54

NaCl

+21

+8

RbCl

+ 12:7

+4-6

Experiment — 2-8

+7-0 +39 + 1-3

+ 14-6

20.2.2. Optical properties of H? Optical absorption by interstitial hydrogen atoms has been observed in most of the alkali halides. Fischer (1967) summarizes

most of the experimental results. The main features are the following. There are two bands which are associated with transitions to different excited states. Both of these bands exhibit spin-orbit effects observable by magneto-optics (Cavenett et al. 1968; Ingels & Jacobs 1971). Neither of the bands obeys a Mollwo-Ivey law; the energies seem to be primarily determined by the anions. Thus the main band, which is lowest in energy, lies between 5-0 eV and 5-7 eVin the chlorides, between 4-4eV and 4-8eV in the bromides, and between 3-6 eV and 3-8 eV in the iodides. Its energy decreases with

increasing cation radius. The separation of the two principal transitions increases with the ratio of anion radius to cation radius. The conclusion drawn is that the excited states are not those of the hydrogen atom itself but are charge-transfer states involving the nearest-neighbour anions. Charge-transfer states are commonly found in organic systems, and proposed examples are not uncommon in systems containing transition-metal ions. But these H? systems are among the rare cases in inorganic systems where there is strong

688

HYDROGEN

IN

IONIC

CRYSTALS:

evidence for charge-transfer states. In consequence most ofthe theory has involved molecular-orbital theory, with the use of empirical parameters and symmetry arguments, and not calculations from first principles. The methods are exactly equivalent to ligand field theory (e.g. chapter 22). Schechter (1969) has used a variety of models, all extensions of effective-mass theory and all ones which regard the excited state as a 2p-state of the hydrogen atom. We shall not discuss his work, since the model is not consistent with the interpretation of experiment; further, there are doubts about the application of a continuum theory to such a compact centre. The excited states of the H? are postulated to be a linear combination of the outer p orbitals of the anions. The symmetries which can be obtained in the sodium-chloride structure are: derived from o orbitals—A,,, T,,: derived from z orbitals—

Que Paes dsies

(Cavenett et al. 1968). In the caesium halides if the H? site has four anion and two cation neighbours the symmetries are:

derived from o orbitals—A,,,, B Ba... oe derived from z orbitals—A,,,, By gn? E gn? ASen By Eee

The excited states for optical absorption in the sodium-chloride structure are argued to be linear combinations of the T,,- and T,,-States. The experimental results for optical absorption and magneto-optics can be largely understood by assuming (1) that the transition matrix element to the T,,-state is much larger than that to the T,,-state, and (2) that there is no spin-orbit coupling in the T,,-State (Cavenett et al. 1968 ;Ingels & Jacobs 1971). The nature of the excited state in the caesium halides is less certain, but possibly involves E,,, with a small admixture of E,,, (Ingels & Jacobs 1971). 20.3. The substitutional hydrogen ion H, (U-centre)

20.3.1. Optical properties Optical absorption by the substitutional hydride ion has been observed in most of the alkali halides and in the alkaline earth fluorides. Emission is not observed ; the excited ion tends to decay, giving interstitial hydrogen and an anion vacancy. The free H™ ion presents problems theoretically. There is no Coulomb potential to bind the second electron: electron—electron

U-,

U,-,

AND

U,-CENTRES

689

correlation is vitally important. There appears to be just one bound state, which is approximately described as having two electrons of opposite spins in Is orbitals. Banyard (1968) compares the different calculations,

discussing

the

different

wavefunctions,

correlation

energies, and other properties. For present purposes we note that use of a trial function P(r, 12) = N, (Walt )Wp(t2) + Wilt )Wa(r2)}

(20.3.1)

with yw, and Ww, as hydrogenic 1s functions with different effective Bohr radii, gives a bound state too high in energy by

A ~ 0-39 eV.

(20.3.2)

The radial correlation included in (20.3.1) amounts to about 63-6 per cent of the total correlation energy. The spatial function 'Y of (20.3.1) is appropriate for a spin singlet. It is possible that stable triplet states occur in crystals, with spatial function ALE, 12) =

N3{W(r

y(t.) — Welt Walt 2)}-

(20.3.3)

A. Point-ion calculations. Theoretical energy levels and wavefunctions are available for H; in the alkali halides, caesium halides, and alkaline earth fluorides (Gourary 1958; Spector, Mitra, & Schmeising 1967; Singh, Galipeau, & Mitra 1970). Six different energy levels have been obtained, three being singlet states and three triplets. The trial functions used are as follows: 1. Singlet states.

ground state:

The one-electron functions for (20.3.1) are: Ww, ~ exp (—ar)

W, ~ exp (— fr) first excited state:

W, ~ exp (—6r)

Wy ~ rexp(—yr/2) second excited state: w, ~ exp (— Or)

Wy ~ r(1—yr/6) exp (—yr/3). 2. Triplet states.

ground state:

The one-electron functions for (20.3.3) are:

W, ~ exp(—ar)

Wy ~ (1 — Br/2) exp (— fr/2) excited states:

as for the singlet excited states.

690

HYDROGEN

IN

IONIC

CRYSTALS:

The observed optical transition takes place between the lowest two singlet states. The triplet states could possibly be observed at low temperatures as metastable excited states. The calculations are standard point-ion calculations and ignore lattice distortion and ionic polarization. Since the trial functions are s and p functions only, just the spherically-symmetric part of the potential contributes. The one difference from other calculations comes from the correlation in the ground state, where the energy A of (20.3.2) is subtracted. This is, of course, a little dubious, since it requires the error in the correlation to be the same in the solid as in the free atom. However, the correction is not large. No corresponding correction is considered necessary in the other states, which are unbound in the free ion. The results are as follows. First, the transition energy corresponding to the observed transition follows a Mollwo-Ivey law:

(ay Bei

(20.3.4)

where a is the lattice parameter. For the alkali halides the predicted value of n is 1-14, and the observed value is 1-10. Thus the trend

through the alkali halides is predicted well. The transition between the lowest triplet states is not predicted to follow a rule like (20.3.4); the reason for this is not clear. Secondly, the transitions are typically 10 per cent to 20 per cent less than the observed values. Thirdly, the oscillator strengths are too large by about a factor of 2. Thus the point-ion theory gives general agreement with experiment, but fails in detail. Note that the Hy ion appears to be well described in terms of ionic excited states, whereas the H? excited states appear to involve charge-transfer. There is no adequate theoretical explanation of this difference ;all treatments seem to regard the difference as an empirical fact. This is still true, although not essential, in semiempirical molecular-orbital methods (e.g. Hayns 1972a).

B. Extended ion calculations. Wood & Opik (1967) performed much more detailed calculations on H, in KCl, KBr, and KI. They give results for two degrees of complexity. In the first (Type I) the detailed electronic structure of nearest neighbours was included, but without any polarization. In the second (Type II) the KCI system was treated in more detail: the electronic structure of the

U-, U,-, AND

first three shells of ions were

U,-CENTRES

691

included, and estimates of lattice

distortion and electronic polarization made. Further and more detailed calculations are reported by Wood & Ganguly (1973). One obvious difference from the point-ion models concerns the wavefunctions, especially the orthogonality of the H; function Y(r,,462) and the core orbitals. It is straightforward to ensure the two H, electrons have mutually-orthogonal wavefunctions; this is automatic in the spin-singlet state. Likewise, if the core orbitals are free-ion orbitals it is straightforward to use the L6wdin method to construct new core orbitals which are orthogonal to each other. The orthogonality of the H,; electrons to the cores is more difficult, and Wood & Opik use the ‘strong orthogonality’ condition: far.V(Fo, T2)Perystalll 1 p-++9 Ty = 7 Q,.--, 9) =

0

(20.3.5)

for all n. This includes only that correlation between H™ and core orbitals required by orthogonality; other correlations may be significant, especially for excited states. Correlation of the two H™ electrons, or among the core orbitals, can be treated beyond this. However, Wood & Opik do not add the energy A of (20.3.2), but are content with the radial correlation. The choice of one-electron functions for the two H™ electrons is essentially the same in the point-ion and extended-ion calculations. However, Wood & Opik did include more terms, especially in the p-states for the second calculation (Type II). They observe that f-like states do not seem to be important. The lattice distortion was included by allowing the nearest neighbours to move radially, so as to minimize the total energy. A 2 per cent inward distortion achieves this. Electronic and ionic polarizations were estimated using the interpolation formulae of Toyozawa and Haken & Schottky (§ 8.6.5). These tend to overestimate the screening and the polarization energy for compact functions. The remainder of the calculation uses the standard variational procedures, with some working approximations in the more difficult of the several-centre integrals. The results for KCl are summarized in Table 20.4. The parameters are not greatly changed from the point-ion values, but there is a considerable improvement in agreement with experiment.

692

HYDROGEN

IN

IONIC

CRYSTALS:

TABLE 20.4 Electronic structure of Hy in KCl Transition Energy (eV)

Ground state ed B

Excited state 7 6

Ground state

__ Excited state

Point ion

— 1-094

—0-916

4-84

1-026

0-426

0-823

1-010

Extended ion| Extended ion II

—1:0712 —1-1428

—0-8666 —0-9242

5:57 5:95

1:0213 1:0213

0:5106 0:-5106

0-5602 More

1-0213 1-0213

flexible function

Experiment

5:95

The results for the energies are in atomic units for the absolute energies and electron volts for the difference. The wavefunction coefficients are in ag '. Results are for the two lowest singlet states.

20.3.2. Infrared absorption: the local mode Infrared absorption associated with the local modes of hydrogen and deuterium has been observed in most of the alkali halides and alkaline earth fluorides. Many of the values are listed by Klein (1968, p. 437). The local-mode energies vary systematically throughout the alkali halides and obey a Mollwo-Ivey law quite well:

Ey ~ (0-183 eV)(a A) -1°799,

(20.3.6)

where a is the nearest-neighbour distance. The ratio of the localmode frequencies for hydrogen and deuterium ions is close to the ratio a expected for a mode in which the impurity atom alone moves. The ratio approaches w/e more closely as the mass of the neighbouring cation increases, since heavy near neighbours will participate less in the mode. The ratios for the chlorides and bromides are given in Table 20.5. Anharmonic properties of the local mode have also been studied. These include the intensities and energies of higher harmonics and the temperature dependence of the widths and positions of the infrared transitions. We shall discuss the infrared absorption as a problem in defect-lattice dynamics, as a problem in the electronic structure of a defect, and as a test of some models of

anharmonicity.

A. Dynamics of a lattice with a defect. Page & Strauch (1967) have discussed the local modes of H- in nine alkali halides. Their work gives both the formal theory of lattice dynamics in the breathingshell model and a detailed application of the method. The effects of

U-, U,-,

AND

U,-CENTRES

693

PRABLE: 20:5 Ratios of local-mode frequencies for H, and Dy a

Na

K

Rb

Cl

1-380

1-392

1-400

Br

1-377

1-404

Cs

Anion

=

= 1-414

changes in the long-range Coulomb interactions are included. The results are confined to the harmonic approximation. The H, centre is defined by the changes in terms of five parameters in the Page-Strauch model. These are the change in core mass, in core and shell charges, in the shell-core force constant of the Hy ion (6g), and in the central force constant between the shell of the H, ion and the shells of its nearest neighbours (6k). Since the mode is highly localized, as demonstrated by the isotope effect, it is arguable that changes in force constants involving more distant atoms have negligible effect on the observed energies. Of these parameters, only two are known. The sum of the shell and core charges is unaltered, and the mass change can be determined independently. The others can be estimated from a microscopic theory, but this is not a simple matter. Page & Strauch note that, if there is only a mass change with unaltered charges and force constants, the predicted frequencies are too high by tens of per cent. The results prove insensitive to the choice of defect-shell charge. Thus the problem reduces to a determination of 6g and 6k, and these are chosen to fit the observed frequency. There is now just one undetermined parameter. Some possible choices are these: 1. dg =O (unaltered shell-core force constant); shell charge —|e|. Reductions of 35 per cent (RbCl) to 58 per cent (Nal) in the shell-shell force constant are needed to fit experiment. The Hy

polarizability « varies from 0-19 A® to 0-52 A®, depending on the host. 2. dk = 0 (unaltered shell-shell force constant); shell charge —|e|. Reductions of about 97 per cent are needed in the shell-core force constant. The polarizability « is in the range 2-95 AOtto

5.70 A>, depending on host. 3. Polarizability « = 1-9 A3, fitted to the analysis of LiH by Calder, Cochran, Griffiths, & Lowde (1962); shell charge —|el. Both force constants are significantly weakened, the shell-shell

694

HYDROGEN

IN

IONIC

CRYSTALS:

force constant by 30 per cent to 35 per cent and the shell-core force constant by up to 75 per cent. As it stands, this analysis parameterizes the experimental data. It is not predictive. However, the parameters dg and dk can be found from a microscopic calculation, at least in principle. They also affect some of the other features of infrared absorption and so could be estimated experimentally. Page & Strauch’s results are in qualitative agreement with those of earlier workers (e.g. Sennett 1965; Fieschi, Nardelli,

& Terzi 1965; MacDonald

1966; Timusk &

Klein 1966). B. Microscopic theory of the H,-centre force constants. Wood & Gilbert (1967) have extended calculations of the electronic structure of HT (Wood & Opik 1967) to discuss the local force constants and polarizability of the hydride ion. The basic model is of the ion surrounded by six unpolarizable cations, and with some smaller effects associated with second neighbours. The wavefunction for the hydride ion was taken to have the form (20.3.1), with w, and wy, hydrogenic Is functions having different effective Bohr radii. As the proton is moved away from the centre of the octahedron of neighbours, another contribution of the form (20.3.1) is admixed. In this second contribution, w, is a hydrogenic 2p function and Wy, a ls function. The four wavefunction parameters and the admixture were determined variationally in the extended-ion model outlined in § 20.3.1. The polarization of the hydride ion, described by the admixture, proves important. Likewise, it is essential to go beyond the point-ion approximation. The results can be described in two ways. One is in terms of the local-mode frequency. Wood & Gilbert estimate this using the shellmodel calculation of Fieschi et al. (1965). The calculation differs from the Page—Strauch calculation in its use of a shell model rather than the more successful breathing-shell model and by an approximation which restricts the Fieschi et al. calculation to highly localized modes. The results for KCI show these ratios of predicted frequency to that observed:

no polarization polarization second-neighbour corrections rigid ion, mass change only (Fieschi et al.)

1-28 0-97 1-01 1-59

U-,

U,-,

AND

U,-CENTRES

695

Clearly, the theory leads to rather good agreement with experiment, although this may be fortuitous since no anharmonic terms are included. It is not completely clear how the parameters should be related to those of Page & Strauch. Wood & Gilbert define a polarizability «,,, which is very similar to the polarizability used by Page & Strauch, in the following way. The dipole moment for a given displacement x of the proton is wx) = eC Y(x)[r, +r|P(x)> ~ wyxt...,

(20:3,1)

where ‘P(x) is the variational wavefunction. If the potential energy

U, for the motion of the proton has k = d*U,,/dx?, then we may define a polarizability by Cert

eay hs

(20.3.8)

The values obtained (e.g. 6:95 A? for KCl) tend to be larger than those from the analysis of the lattice dynamics. This theory for H> should be contrasted with the theory for H? in the fluorite structure (Shamu, Hartmann, & Yasitas 1968; Hartmann,

Gilbert, Kaiser, & Wahl

1970). The interaction of the

hydrogen with the nearest-neighbour F ions is treated there as the superposition of interactions estimated from the theory of the free HF

molecule.

Multipole interactions, overlap effects, and cova-

lency all contribute. Agreement with experiment is achieved after allowing for the motion of the neighbours and including the leading anharmonic corrections. The effect of similar changes on the Wood— Gilbert results is not clear.

C. Anharmonicity and the local mode. The anharmonicity in the coupling of the H¢; ion to the host lattice leads to three main effects. The first is the existence of sidebands : infrared absorption is observed with the creation of lattice phonons as well as the local-mode phonon. The second consequence is that transitions to higher vibrational states become allowed, so that higher harmonics of the

local mode are observed. The third effect is a broadening of the local mode which is temperature dependent. Other related questions concern the possibility of dynamic instabilities in the excited state (Matthew & Hart—Davis 1968). The sidebands associated with the local mode are analogous to the phonon sidebands of the zero-phonon line of an electronic

696

HYDROGEN

IN

IONIC

CRYSTALS:

transition. In this picture, the fractional intensity in the main localmode line is exp(—S), where observed values at low temperatures are: sodium halides:

S ~ 0-09

potassium halides:

S$ ~ 0-17

rubidium halides:

S ~ 0:3.

The validity of this analogy has been discussed by Maradudin (1966) and by Hughes (1968c). Hughes discusses the circumstances in which the experimental data should show a reduction in the intensity of the main line with temperature which may be described by an appropriate Debye-Waller factor. The argument observes that there will exist anharmonic terms quadratic in the local-mode co-

ordinate Q, and linear in the host-lattice modes {q;}. Since there is linear coupling to the lattice modes, analogy with the electronic case suggests a Debye-Waller factor for the main local-mode peak. He concludes that the intensity of the main peak falls off as

exp \-byS, coth (ho 247)}, provided (1) that the local-mode frequency w, is much the lattice phonon frequencies and (2) that only the anharmonicity Q” )’; B,q; is important. Of course, other for other properties, such as Qq” terms in the line-width. S; are given by

See es

larger than term in the terms enter The factors

ie

(20.3.9)

where M and m, are the effective masses for the local mode and the lattice modes respectively. The theory shows adequate agreement with experiment. The absolute intensities of the sidebands, as opposed to their temperature dependence, have also been investigated. Page & Dick (1967) found that adequate agreement with experiment could be found on either of two models. The first involved the third-order anharmonicity, as discussed by Hughes. The other mechanism involved

the second-order

dipole moment,

i.e. the terms

in the

dipole moment linear in both Q and {q;}. In both cases a strong decrease was necessary in the shell-shell force constant between the

U-, U,-,

AND

U,-CENTRES

697

hydrogen and its neighbour (6k large and negative in the notation of Page & Strauch). Transitions involving higher harmonics have been studied in some detail by Elliott, Hayes, Jones, MacDonald, & Sennett (1965).

They observed the infrared absorption of hydrogen and deuterium in CaF,, SrF,, and BaF,. The most detailed study was of CaF,, where four transitions were observed for both isotopes: the fundamental, one to the second harmonic, and two to third harmonics,

split by anharmonic terms in the potential. The experimental frequencies are sufficient to fit a model anharmonic potential

V = Ar? + Bxyz+C,(x*+y*4+24)+

+ C,(y?z? +.27x? +x?y’).

(20.3.10)

These are the only independent terms up to fourth order which are allowed by symmetry. When B, C,, and C, are zero, we regain the usual harmonic theory. The local-mode energy is determined by A and by the defect mass. Values for hydrogen are

ho, = 981-1cm~*

|B] = 39600 cm~! A$ C,=—1170cm

VAs

C=, 510. cm

tA*

The deuterium frequencies can be fitted by the same B, C,, and C,. However,

hop = 71021 em" *

(20.3.11) values of

(20.3.12)

is needed, which is a factor of 1-39, rather than 2, lower than ha,,. Thus eight frequencies have been accurately fitted with five para-

meters. A similar analysis of Raman data for KBr and KI has been given by Montgomery, Fenner, Klein, & Timusk (1972). The same model is less successful in accounting for the actual intensities of the higher harmonics. Thus the ratio of second harmonic to first should be about 1:136 for hydrogen and 1:96 for deuterium. Observed values are 1:23 and 1:13-5 respectively. Plausible arguments can be given to suggest that the second-order dipole moment is the source of the extra intensity in the higher harmonic.

698

HYDROGEN

IN

IONIC

CRYSTALS

Elliott et al. have also given a detailed discussion of the width of the local mode as a function of temperature. Inhomogeneous broadening is negligible. The two processes which dominate are the decomposition and scattering mechanisms. The decomposition mechanism involves the decay of the local mode into two or more lattice modes. The interaction is an anharmonic term of the form Qqiq;, quadratic in host-lattice coordinates, as opposed to the Q7q term in the Debye-Waller factor. The rate depends strongly on the ratio of the local-mode frequency to the host-lattice frequencies. It is expected to give deuterium lines broader than those for hydrogen. The process is still efficient at low temperatures, when it should dominate. The scattering mechanism involves the elastic scattering of lattice phonons. It is insensitive to the local-mode frequency, but the hydrogen line should be the wider. The scattering mechanism dominates at high temperatures, where the width should vary as T*. Detailed discussions of these processes are given by Elliott et al. and by Klein (1968, p. 527). Other anharmonic terms have been considered by Olson & Lynch (1971) in their treatment of the CsCl structure. These workers classify the various third, fourth, and fifth

harmonic terms and estimate their effects.

21 THE

REORIENTATION OF IN SOLIDS

DEFECTS

21.1. Introduction

SEVERAL Classes of defect give rise to low-frequency resonances in ionic crystals. These are the electric dipoles such as CN~ and OH", the elastic dipoles such as O;, off-centre ions such as Li* in KCl,

and the oxygen interstitial in silicon. There are many other examples of resonances, but we may demonstrate most features from the groups mentioned. Experimentally, all these defects are conspicuous by their effects on thermal conductivity and by the infrared absorption to which they give rise. Theoretically, their response to perturbations— electric and stress fields especially—is of particular interest. Since the defects are sensitive to these perturbations, internal strains and electric fields in the host lattice can obscure the details of their behaviour. We shall concentrate on two points. The first is the extent to which the theory of § 8.5 explains the observations. The second is the question of when a resonance is expected, for example when a Li* ion is expected to lie off centre, and the related question of the orientation which is favoured. A comprehensive review of experiments on these systems and of their applications is given by Narayanamurti & Pohl (1970).

21.2. Electric dipole centres 21.2.1. CN~

and OH™

in alkali halides

The CN~ molecular ion substitutes for anions in the potassium and rubidium halides. It provides a good example of the hinderedrotor system discussed in § 8.5.2. We shall see that the Devonshire model describes its behaviour well. Measurements of stress-induced dichroism and of the defect contribution to the specific heat indicate that orientation in KI. Weconcentrate on KCI:CN, since its relation to the Devonshire (1936) model is best understood. Most features can be obtained from the infrared spectrum associated with the stretching mode of the CN ~ ion. The transitions observed at high temperatures involve changes An, = 1 in the excitation of this stretching mode, together with excitation

or de-excitation

of rotational

modes

AJ =

+1. Below

about 60K the structure deviates from free-rotor behaviour. At the lowest temperatures one finds transitions An, = 1 with changes in n,, the degree of librational excitation. The lines observed correspond to An, = 0, plus An, = +1 terms allowed because of anharmonicity. Fig. 21.1 shows the change in bandshape as the temperature is lowered. The two-humped structure at high temperatures can be used to give the effective moment of inertia J. In essence, the two peaks correspond to AJ = +1 and AJ = —1 respectively. The rotational energy ofa free rotor varies as J(J + 1)h?/2I. If Jo is the most probable S

3)

2g

=o

jo} [S)

&

Temperature

6. 9

pale

2

5 \

Z :

\

Neve~ssi4 :

Tee

mm

OO mK

Ss) 1K

40 K

ST O50K

ae ----- 75K

Pe ---$ 2K Transition energy relative to 2089 cm7!

Fic. 21.1. Temperature dependence of the CN~ bands in KCl. The change of bandshape with temperature shows the two-humped free-rotor behaviour at high temperatures and librational structure at low temperatures. (After Seward &

Narayanamurti 1964.)

DEREBCRSIINSSOLEIDS

701

initial value of J then the separation of the peaks should be (25+

+ 1)h7/I. But Jo tends to \/{kT/(h?/I)} at high temperatures, so that the splitting is

az ~2

kTh? || i .

(212.1)

The oT dependence is observed, and the value of J which results corresponds to a C-N separation of 1-4 A. The distance is plausible, but should not be taken too seriously because some of the moment of inertia will be associated with host-lattice atoms affected by the CN™

motion. Indeed, values of J tend to be lower when the host

ions are heavier, suggesting a host contribution. Results at low temperatures indicate a librational transition at 12 cm~ *. This appears to correspond to a transition from the lowest T,, level to the T,, level above it (see Fig. 21.1). The Devonshire model can be used to predict K, the parameter characterizing the cubic potential. The value of 20cm? is consistent with the temperature dependence of the infrared absorption below 60K, but must be modified now it is known that ¢111>, not ¢100), directions are favoured. Given

the values

of K and

J, the Devonshire

model

predicts

the splittings of the lowest A,,, T,,, and E, levels. These splittings are not resolved optically, probably because of random strains. However, they can be obtained from specific-heat anomalies. The results are shown in Table 21.1. They are rather good for KCI:CN , but for the analogous system in RbCl the agreement is worse both qualitatively and quantitatively. A possible reason has been suggested by Pompi & Narayanamurti (1968), who observe that a small tetragonal term in the potential explains anomalies in both the TABLE 21.1 Tunnel splittings for CN” in KCl and RbCl

E(T,)—E(Ay,)

E(E,)— E(T,) g

(a) KC1:CN~ Devonshire model Observation (b) RbCI:CN™ Devonshire model Observation

1-4 1-15

1-0 0-85

0:49 0-1

0-25 0-5

Values are given incm '.

702

THE

REORIENTATION

OF

energy levels and in the electric-field induced dichroism. The tetragonal term is sufficiently small that it could come from random strains:

Say

Vy -/(j

peas

[553-1

(21.2.2)

where K’ is k. 0-04°K for KCl and k. 0-03°K for RbCl. The systems containing OH™ have also been studied extensively by infrared absorption, thermal conductivity, and para-electric relaxation. The para-electric relaxation results are affected very strongly by the internal strains. General agreement with the Devonshire model is obtained, especially for the lowest levels. However, some bands are observed which are not predicted. The origin of these lines is not clear, since their infrared spectra and response to electric fields are complicated. Wedding & Klein (1969) discuss the possibility of a resonance associated with centre-of-mass motion of the OH” in KBr. The most promising interpretations involve both vibrational tunnelling of the OH” ion and vibrational motion of the centre of mass, with possible coupling between the two (Kirby, Hughes, & Sievers 1970; Keller & Kneubiihl 1970). This phenomenon is clearly shown by other systems discussed in § 21.2.2. The simplest model of the tunnelling levels is that of Gomez et al. (1967), which was compared with Devonshire’s hindered-rotor model in § 8.5.2. The levels are then given in terms of tunnelling matrix elements I'y, giving jump probabilities through an angle ¢. Knop & Kanzig (1972) have shown that 180° jumps are completely negligible in KCl:OH _, in line with the general rule that the shortest jump dominates. Among the rare exceptions to the rule are V,centres in CaF, and SrF, (180° jumps instead of 90°) and off-centre

Ag* in RbCl and RbBr (90° jumps instead of 60°—Kapphan & Liity (1972)). When an external stress is applied, the values change

Te

i

bese:

(21.2.3)

where e is the strain tensor. Dick (1970) has made the interesting observation that the 6, can probably be neglected except for the g@ = 0 terms. For practical purposes long-wavelength lattice modes do not mix states with different orientations. Dick’s result is demonstrated by the qualitative temperature dependence of the reorientation rate of KCl:OH~. It is also verified by Knop & Kanzig’s observations of reorientation under stress in KC1:0H~. One

DEFECTS

IN

SOLIDS

703

possible reason for the &, being neglected when ¢ ¥ 0 is that the matrix element involves a lattice overlap, like the Ham reduction factor in Jahn-Teller systems. The simplest models, either tunnelling or hindered-rotor, ignore the random strains in the lattice and assume a static host potential. The random strains cannot be ignored experimentally, although they present no excessive theoretical problems. One unusual and more subtle result of the random internal fields of these dipole centres is that, at very low temperatures, the reorientation rate can increase with decreasing temperature (Knop & Kanzig 1972; Janssen 1972). The static-host potential is a more subtle assumption, since some of the effects of host-ion motion can be included phenomenologically. There is very little clear evidence for important dynamic effects for CN, although there are hints from the variation

of I with host and small values of 6, that motion of host ions is important. Such ‘polaron’ effects are more clearly shown by the elastic dipole centres discussed in § 21.3. They are also discussed by Shore & Sander (1972) who give a detailed small-polaron treatment of RbBr:OH~.

21.2.2. Hydrogen halides in rare-gas hosts In § 21.2.1, the dipole centres in alkali halides were affected primarily by the crystal field, and less by the coupling between their translational

and rotational

motion.

For HCl, HBr, and DCI in

rare gases the translation—rotation coupling dominates (see Turrell 1972, p. 313 et seq.; Friedmann & Kimel 1965a, b; Pandey 1965; Bowers & Flygare 1966). Thus the natural formalism is a perturbation expansion in A, a parameter which measures the separation of the centre of mass from the centre of interaction about which the

molecular impurity rotates. Friedmann & Kimel show that the coupling of translation and rotation is quadratic in A in this perturbation limit. If the potential for translational motion of the centre of interaction is very hard, giving a large frequency v; for this motion satisfying

h

vr > ——J,

*

4ni

(21.2.4)

then expressions for the transition energy simplify further. Here / is the moment of inertia of the molecule relative to its centre of mass and J is the rotational quantum number in the initial state. The

704

THE

REORIENTATION

OF

transition energy becomes

AE(J > J’) = 2nh{B(1— MA?/I)} {J'(J'+1)-—J(J +}.

(21.2.5)

It is as if the rotational constant B = h/4zI were simply reduced by a factor (1—MA7?/I), where M is the molecular mass. Thus the effective rotational constant depends on isotope in a completely different way from the hindered rotor, since both J and M are isotope-dependent. The crystal field may also give some extra splitting of the coupled energy levels. Results for HCl and DCI in rare-gas hosts suggest that the centre of interaction lies on the hydrogen side of the centre of mass, the separation A being just under 3 per cent of the separation of the two ions in the free molecule. 21.3. Elastic dipole centres: the O, molecular ion

The O; molecular ion enters alkali halides substitutionally on anion sites, where it is oriented along one of the directions. The defect is of interest as an impurity system, but our concern will be with O, as the rotational analogue of the polaron. Its electronic structure is discussed by Zeller & Kanzig (1967) and Shuey & Zeller (1967). Its optical absorption and associated phonon structure are analysed by Rebane (1970) and Rolfe & Ikezawa (1973). The interactions between O,; ions are discussed by Beyeler (1970), Kanzig et al. (1971), and Giovannini & Muggli (1972). The reorientation of the O; ion is most easily studied in the presence of an external uniaxial stress. The advantages are partly that the behaviour under different stress magnitudes and directions allows one to sort out the transitions and partly that the external stress can overwhelm the random internal stresses which otherwise complicate the results. The mechanism of reorientation involves non-radiative transitions from one state to another and is sometimes described as phonon-assisted tunnelling (e.g. Sussmann 1964). The theory (Sussmann 1964; Silsbee 1967 ; Pirc, Zeks, & Gosar 1966;

Gosar & Pirc 1966) is very similar to the discussions of hopping processes in § 8.4, of spin-lattice relaxation in § 14.2.2, and of the V,-centre motion in§ 18.6. The two processes which seem to dominate for O, in the potassium halides are (Pfister & Kanzig 1969): 1. the one-phonon process, in which the transition occurs with the absorption or emission of a single phonon; 2. reorientation through excited librational levels.

DEFECTS

Other

possible

processes

are

INZSOLIDS

two-phonon

705

and

many-phonon

processes ;we shall not consider them in detail because they are not

important in the best-documented cases. The O, centre is usefully contrasted with the V,-centre, since both are defects observed to reorient among the various ;.

(21.3.4)

The crystal density is p, and v; is the appropriate phonon velocity. The corresponding result for absorption of a phonon of energy A is obtained by changing Ato — A, giving a denominator {exp (A/kT) — — 1}. Gosar& Pirc have derived the analogous transition probability to all orders, and find

yes 5 rc T Ws = Wg exp }-2(9+7)} +s(oa.z}t

(21.3.5)

The details of f(x, y) need not concern us, although we note that f tends to zero when A and T are small. The important feature is the

factor exp {—2(J + T?/T3)\. The factor reduces the effective value of the T, to 2 I, = 1, exp }-[46+]}

(21.3.6)

Cw)

and represents the feature that the lattice distortion alters when reOrientation occurs. The factor decreases the rate of rotation in just the way that polaron effects enhance the effective mass and decrease the mobility of carriers. It is also directly analogous to the Ham reduction factors in Jahn-Teller systems. The two parameters J, and Tj can be defined in terms of the displacements Q;, in mode A when the Oj ion has orientation i. The self-trapping distortion

is then 32,Ma3Q},, where M is an ionic mass. The two parameters are A

Se

5

Moy

2

AQ, /hw, 2

(21.3.7)

DEFECTS

IN

SOLIDS

707

and

[z} =a2 Me T

lias

\2

1

ee,

23

A ae

2

2n(w,)/ha@,,

(21.3.8)

where AQ, is the change (Q;,—Q,) in the transition. In the Debye model, for T« Op

T2 = @33n7J,.

(21.3.9)

We have not yet defined with any precision the local states between which transitions occur. Sussmann (1971) has argued that they should be regarded as tunnelling states localized by random internal strains and by the applied stress. Certainly these features are important, the applied stress especially so, but the inertia associated with the local lattice distortion can be important enough to give sensibly localized states by itself. This is the case for the V,-centre. The position is not clear for O;, but probably the coupling is too weak and Sussmann’s argument holds. Pfister & Kanzig (1969) and Silsbee (1967) have made a detailed comparison of experiment and theory for O; in the potassium halides. The experimental results of main interest are the reorienta-

tion rates of '°O; and of '8O, as functions ofstress, stress direction, and temperature. Sample results are shown for KI hosts in Figs. 21.2 and 21.3. In Fig. 21.3 the reorientation of }°O; is shown asa function of [100] stress. Clearly the one-phonon theory gives reasonably good quantitative agreement. The contribution of higher-order processes becomes more important at higher temperatures and stresses, but is never dominant. By contrast, the rate as a function

of[111]stress isnot described nearly so well, although it is qualitatively of the form expected. The temperature dependence at constant [100] stress, shown in Fig. 21.4, verifies the linear temperature dependence expected when A > kT, and also shows a rapid rise at higher temperatures from other mechanisms. Pfister & Kanzig have analysed their data in terms of the Gosar— Pirc model. We now summarize their results for the KI:O; system. The parameters

T,, ,, and J, can be estimated, and their values

are given in Table 21.2. It is obvious that the I, factors favour 60° jumps strongly and that the reduction factors I'3/, = exp (—J,)

THE

708

0

2

REORIENTATION

4

6

Fic. 21.2. Reorientation of '8O; in Kl asa

8 LOR AU (K)

OF

2

eal

function of (100) stress. Results of Pfister

& Kanzig are shown for three temperatures, as a function of the stress splitting AU.

(CS) egy

T (K) Fic. 21.3. Reorientation of '°O; and '8O; as a function of temperature. Results of Pfister & Kanzig are shown.

DEFECTS

IN

SOLIDS

709

TABLE 21.2 Parameters for K1:O; LO

ies Is aries

NEOs

23x 1072 21x 1074 10-2

Jeo

1-45 x 1072 1-0 x1074 8-1 10-5

ws

5-7

The J, are dimensionless, and the other parameters are in degrees Kelvin.

TABLE 21.3 Effective potential parameters and librational frequencies 60° jump

90° jump

V, (K)

165

206

wf (103 s~*)

13

1-0

TABLE ER 2 Lead Effective moment of inertia for K1:O;

bsg 18Q> Results are in units moment of inertia.

60° jump

90° jump

6-0 5-7

2.2 2:1

of the

free

molecule’s

are very small indeed. Another way of describing these data is to construct an effective potential

V, 180° 6 V9 = i+ cos | ; .

(21.3.10)

where the parameters are chosen so that I$ are correctly predicted by the W.K.B. method. Results are given in Table 21.3 for V, and for w?, the associated librational frequency. Further, one can estimate the effective moment of inertia of the O,. The results (Table 21.4) show the inertia associated with lattice distortion: clearly the isotope effect in [*/I is rather small. Finally, at higher

temperatures the '$O; system appears to reorient by transition to

710

THE

REORIENTATION

OF

a higher librational state and subsequent reorientation. The contribution from this mechanism varies as

12 (45% 10" exp —95°- K .

(21.3.11)

6

The activation energy k. 95°K can be written as a frequency 1-25 x

Se1032 so Clearly, most of the theory of O; reorientation has been phenomenological in that qualitative predictions have been parameterized. The fact that such sensible values emerge is an indication that the processes are largely understood, although it would be nice to verify this by a proper microscopic calculation. 21.4. Off-centre ions

It is remarkable that small substitutional ions in alkali halides may occupy sites different from the ion for which they substitute. This behaviour is found for cations (e.g. KCl:Li) and for anions (e.g. NaBr:F). The reason for this novel behaviour was first given by Matthew (1965). It is the result of two effects. The first is the attractive interaction between the charged defect ion and the electric dipoles induced in its neighbours. The second is that the small defect ion can move rather close to a lattice ion before their repulsive interaction is appreciable. The reduced repulsion and enhanced attraction leads to an asymmetric equilibrium configuration in some cases, and it is these we discuss now. We shall discuss the existence of off-centre configurations later. First we discuss the observed properties of KCI:Li, since this has been studied in greatest detail. 21.4.1. The KC1:Li system

The lithium ion tunnels between sites displaced along the eight equivalent ¢111) directions from a cation site in KCl. This is demonstrated by its effect on the various components of the elastic compliance, by the quadrupole splitting of Li’ under stress, and by specific-heat data. Since minima occur, the Devonshire parameter K of (8.5.1) will be opposite in sign to that for 33> 2, Ys Dae Ds

38 Ts 0,

O>

Fic. 21.5. Energy levels of the oxygen interstitial in silicon. Results of Bosomworth

et al. not drawn to scale; the separations shown are of order hy,, typically 40 cm™ ' and much less than hy, (517 cm_') or hv, (1136 cm ').

716

THE

REORIENTATION

OF

DEFECTS

IN

SOLIDS

The anharmonicity is such that the oxygen lies off the silicon— silicon axis, but there are no detectable deviations from axial symmetry. A potential for the oxygen can be constructed as a function only of r = dcos a, the distance of the oxygen from the axis,

V= 5Mow? +A exp | 7/2) The energy levels shown in Fig. 21.5 can be written |N, />, where N is the quantum number appropriate to the harmonic part of the potential and / is the angular momentum about the defect axis. Parameters can be chosen which give a good description of the observed energy levels. The same parameters give a good description of the stress response of the lowest transition if it is assumed that the bond angles, not the bond lengths, change.

22 TRANSITION-METAL

IONS

22.1. Introduction

THE transition metals to be discussed belong to two main groups. These are the iron group, with partly-filled 3d shells and the rare earths, with partly-filled 4f shells. Other transition-metal ions have electronic configurations with different incompletely-filled d or f shells. The incompletely-filled shells are responsible for the magnetism and colour of gemstones and other doped crystals, which are major reasons for interest in these systems. Experimental work on transition-metal ions has treated an enormous number of different combinations of ion and of host. This, in turn, has lead to theoretical work

at all levels, from the

most empirical to formidably complicated calculations. This is not the place to discuss the theory in detail, since there are many reviews of the field; instead, the theory will be described in broad

outline, and an attempt will be made to relate it to the general scheme of work on defects which is described in this book. Theories oftransition-metal ions in crystals make two fundamental assumptions common to almost all methods. First, the ion in the crystal is taken to have a strong resemblance to the free ion. In ionic crystals the resemblance requires that the inner, filled shells are not significantly affected by the host and that the unfilled shell of electrons contains the same number of electrons in the free ion and in the ion in the crystal. Further, the one-electron wavefunctions for the unfilled shell have the same symmetry properties under the point symmetry of the site as the free-ion wavefunctions would have

had.

For

instance,

such

one-electron

wavefunctions

of an

iron-group ion in an ionic crystal will transform like d functions under the point symmetry of the site. This assumption isjustified by experimental evidence (Griffith 1961, p. 187); it is plausible, but there is no obvious theoretical argument to justify it in general. Essentially the same results hold in highly covalent crystals. The major difference is that a small number of the electrons from the

718

TRANSITION-METAL

IONS

unfilled shell may participate in bonds. The remainder of these electrons have similar symmetry properties to those in the unfilled shell of the free ion. A consequence of the similar symmetry properties is that there is a correspondence between many of the terms in the Hamiltonian of the free ion and the terms in the Hamiltonian of the ion in a crystal. Much of the formalism for free ions can be carried over, with modifications such as changes in magnitude of spin-orbit coupling or of electron-electron interaction integrals. This assumption points to one major difference from other defect systems. For transition-metal ions reasonable zero-order wavefunctions are available without a complicated variational calculation. The second fundamental assumption is that the properties of the ion are determined by a small part of the crystal only. This complex consists of the transition-metal ion and its nearest neighbours (the ‘ligands’). In a few special cases more distant neighbours are included, but the complex is still small. There is some theoretical justification for the assumption. Most calculations agree that the wavefunction of the transition-metal ion falls off very rapidly in space and that overlaps with any but the closest neighbours are very small. It is less clear that one can neglect the Coulomb potential from distant neighbours in an ionic crystal, though this is often done. The main reason why this assumption works is that it is covalency, not pointion interactions, which dominate in the important properties. 22.2. Classes of theory and coupling schemes

The many calculations of the properties of transition-metal ions can be classified in two ways. The first is the level of complexity, the second is the relative magnitudes of the terms in the Hamiltonian. The four main levels of calculation are the following: 1. Crystal-field theory describes the approximation in which the effects of the host lattice are represented by a unique one-electron potential V(r). This potential is usually taken to be the potential of the nearest neighbours, regarded as point charges. 2. Ligand field theory recognizes the electronic structure of the neighbours and uses a qualitative molecular-orbital treatment of the transition-metal ion and its ligands. It is a phenomenological theory : one uses symmetry, intuition, and experiment, not specific calculation, to order energy levels.

TRANSITION-METAL

IONS

719

3. The semi-empirical molecular-orbital methods use the models discussed in chapter 7 to make rough quantitative estimates of the energy levels and wavefunctions. These are one-electron calculations so, at best, they could be equivalent to Hartree-Fock theory. 4. The a priori theories go beyond the semi-empirical approaches in making an attempt to achieve self-consistent wavefunctions and in including the important effects of correlation. Since the main terms in the Hamiltonian can be found reasonably accurately, these methods should always be preferred to the approximate, but easier, semi-empirical methods. The coupling schemes involve the relative magnitudes of three terms. These are the electron—electron interaction “, measured by the various electrostatic matrix elements, the spin-orbit coupling Ao, and the crystal field, measured by the splittings of degenerate free-ion levels by the host. It is convenient to divide the crystal field into three terms:

V, = VigtVi, + Vip.

(22.2.1)

The first term is spherically symmetric, and its main effect is to shift energy levels without affecting their relating separations. The second term is the dominant part of the crystal field which lowers the symmetry from spherical; the third term contains any small terms of different symmetry. The schemes are these: (a) Rare-earth scheme. The terms diagonalized first are #,+ Ho. The crystal field V, is then treated as a perturbation of these freeion orbitals. The approach is suitable for the rare-earth series, and possibly for the actinide series. (b) Weak field coupling. The electron-electron interaction H is diagonalized first, followed by a part V., of the crystal field. The spin-orbit coupling and the remainder of the crystal field (%o0+ V.2) are treated last. Many iron-group ions suit this scheme. (c) Strong field coupling. The major part of the crystal field V,, is diagonalized first, followed by the electron-electron interaction H,. Again the spin-orbit coupling and the remainder of the crystal field (#o+ V2) are treated last. Many of the iron-group ions suit this scheme. These schemes apply to all the four levels of calculation outlined before.

720

TRANSITION-METAL

IONS

22.3. The iron group in cubic symmetry

22.3.1. The cubic field splitting We now consider ions with unfilled 3d shells in sites which have cubic symmetry. Qualitatively similar results hold in tetrahedral symmetry, where the lower-symmetry odd-parity terms in the Hamiltonian are usually too small to produce major changes. The one-electron states derived from the 3d levels can be classified by the way they transform under the operations of the cubic group. The various possibilities, together with polynomials which transform in the same way under the cubic group, are

E states: |e>

(x?—y?)/2

\0> (22? x? — y*/2,/3 T, states: |€>

yz

In>

2x

IC>

xy.

In cubic symmetry, the two E-states have the same energy, and the three T-states are degenerate. The cubic field splitting A is the difference

nd ae ae

(22.3.1)

It is a major parameter in all the theories of iron-group ions. If positive, the one-electron energy levels are split, as shown in Fig. 22.1.

le>,lo>

2A/5

lE>,|ln>|C> Fic. 22.1. Cubic field splitting of d-electron orbitals. The sign of the splitting is appropriate for octahedral coordination.

TRANSITION-METAL

22.3.2.

IONS

721

Theories of the cubic field splitting

A. Crystal-field theory. The simplest theory regards the splitting of the d levels as arising from the electrostatic potential of the nearestneighbour ions. The sign of A can be seen by inspection. The E orbitals are concentrated along the cube axes, whereas the T orbitals are mainly along the three-fold axes. In octahedral symmetry the nearest neighbours are in the cube directions and will affect the E-states most. In eight-fold or four-fold coordination the T-states are most affected by the neighbours. Usually the ligands are negatively charged, and repel the electrons. Thus we expect A to be positive in octahedral symmetry, and negative in the other cases. Explicit calculation gives

A 28Rie ty Ree) 2

4

RR

(octahedral)

_~(tetrahedral)

(22:3:2)

(2233:3)

Here

(22.4.2)

electrons i

Since

the parameters

are empirical,

they contain

automatically

some, at least, of the many-electron effects. Secondly, one concen-

trates on the 4f shell alone and assumes free-ion states with the same radial dependence for all components. In consequence, the V,; may be written as a sum of angular terms only, the coefficients being functions of the environment and of the radial dependence of the 4f electrons. Thirdly, Russell-Saunders coupling may be used. As mentioned earlier, intermediate coupling is appropriate although more complicated to apply. The fourth assumption is that the crystal field does not mix states with different total angular momentum J. This is valid if the crystal field is small compared with the spin—orbit coupling, a result which becomes increasingly good for the heavier rare earths. Perturbation corrections are often adequate. Finally, lattice distortion is ignored, and, in particular, one neglects the changes of lattice configuration with the state of the rare-earth ion or from ion to ion in a given host. The components V, can be broken into angular components in quite a number of ways. The results are confusing, since the notation is not unique. We shall follow Wybourne (1965) and Newman (1971) in writing the V; as a sum

v= >

BtZ™,

(22.4.3)

where the average over radial functions is included. The Z‘ are

746

TRANSITION-METAL

IONS

related to the spherical harmonics by

(Y,,-m—¥,ce) a

2n+1

=

Va

ree O

an

(Yh n,—m ae 1)OY

nr

0a

2)

and the B", are the real parameters which define the crystal field. Relations between the sets of parameters are given by Wybourne (1965, Table 6.1), Dieke (1968, Table 8) with errors listed by Newman (1971, p. 202). Note in particular that Dieke’s B” parameter is not the same as here; his are defined by

V, = ¥ BrP* (cos 6;) exp (im@).

(22.4.5)

The electrostatic model, together with the assumption that the rareearth electrons do not overlap the charge distribution producing the field, suggests an expansion

ELBA)

(22.4.6)

If all the 4f electrons have the same radial dependence then this scheme suggests as parameters £,,,,(r">. The expansion is, of course, more general. However, the electrostatic model suggests methods of calculating the parameters analogous to estimates of the cubic field splitting for iron-group ions. We emphasize once more that the electrostatic model often gives good order-of-magnitude agreement with experiment; the agreement is deceptive, and does not reflect the mechanisms involved. Whilst the schemes described are more general than the electrostatic model, their structure is clearly rooted in it. Another model,

which includes the electrostatic one as a special case, makes it easier to incorporate more subtle contributions. This is the ‘superposition model’, based on the assumption that the total! crystal field is a sum of separate contributions from each of the ions in the crystal (e.g. Bradbury & Newman

1968).

The assumption immediately increases the number of free parameters, since small distortions can change the contributions of ions significantly ;indeed, in some cases there may be more parameters

TRANSITION-METAL

IONS

747

than observables. The tests of a fit are thus, first, that there be small residual errors and, secondly, that the parameters chosen are

physically sensible. Criteria include the signs of some of the terms involved, the dependence on distance of the interaction, and local distortions which are small (if necessary) and consistent with any

other data from spin-resonance or X-ray experiments. Once a fit is achieved,

some

extra information

is available. The deviations

give a measure of the importance of three-body interactions. The results may resolve ambiguities of signs of parameters or of orientations of axes, and, in very special cases, they may give a guide to the local distortion. It is possible to include a number of sophistications of the crystalfield model. These include the generalization of the spin—orbit coupling and Coulomb integrals to reflect the anisotropy of the environment (cf. the iron-group ions in § 22.3.3). Further, the Coulomb correlation of the electrons will be modified by the crystal field. We shall not discuss these generalizations here, but merely note that they require the introduction of extra parameters.

B. Ab initio calculations. There are three obvious differences between the rare-earth crystal fields and those for iron-group ions. First, the crystal field is weaker for the rare earths. This means that the mixing of free-ion configurations (as for Co?*, § 22.3.5) can be ignored. Secondly, the 4f shell is not the outermost electron shell: the 5s and Sp electrons can shield some of the interactions. Thirdly, there are more crystal-field parameters available to provide a check on the theory. Apart from these features, there is a striking resemblance in principle (if not in detail) between the theories for the two transition series. We shall concentrate on the system PrCl;, where there are four independent crystal-field parameters. Experimental values and their sources are given by Newman (1971, Table 12):

ARCr a4

7c

a

Ao 0-1. One other important feature is that the levels are not monotonic functions of y. Non-monotonic behaviour occurs when levels attempt to cross each other, but avoid doing so because of finite matrix elements coupling the states

concerned. Very complete comparisons of the theory with experiment have been given for donors in silicon and germanium by Faulkner and for 1

0-0

SBN

~0-1 EEA i

5-05

ee

=

—0:3h4¢

my —1-0

S04

x

oa

SSE

ae

310s é

—0:5+ —1:5

0-0 0-2 0406 0810 (a)

0-

ees

sere

000204060810 Hs

(b)

== Mean

ns ie

ee

0002 04 06081-0 7

(c)

Fic. 23.1. Donor levels and mass anisotropy. Results for 1s, 2p), and 2p, levels in effective-mass theory. p is mt/m#. (After Faulkner.)

754

SHALLOW

DONORS

op

AND

iP

eeAse Sb

——

ce pe

pe

— 10/4P3Po i

ACCEPTORS

Bigs

ee

Sees

=

ee

SS

OS

I0K S

E Nell oii ate 23

E+T—

sige

—E—T\=T,

oS

a

1

—40-

—A, —— AY



50+

—A,

a (zie

me

70

Fic. 23.2. Donor levels in silicon. Results from effective-mass theory are compared with experiment. (After Faulkner.)

0,

rheory P+—

Li

seme 5Po

(meV) Energy

—15

\

(b) Fic. 23.3. Donor levels in germanium. Results from effective-mass theory are compared with experiment. (After Faulkner.)

IN

donors in GaAs

SEMICONDUCTORS

by Fetterman,

ISS

Larsen, Stillman, Tannenwald,

&

Waldman (1971). The results for silicon and germanium are given in Figs. 23.2 and 23.3. The theory is confirmed with remarkable accuracy for the excited p-states; indeed, the analysis of these results yields more accurate dielectric constants ¢ than those obtained by direct measurement. The 1s-states show substantial deviations from theory, which we shall discuss in § 23.4.

23.2.2. Wavefunctions The effective-mass equations predict the energy levels and the envelope wavefunction of the shallow centre. In spin-resonance experiments the full wavefunction is measured—the product of the envelope function and a Bloch function appropriate to a band minimum. For silicon and germanium, where there are several equivalent minima, the wavefunction is a sum of the form

Wr) =), Fi(r)u(ko;,¥).

(23.2.8)

We now consider how well this wavefunction describes spinresonance measurements. As shown in §§ 4.2.5 and 4.5, the wavefunction W(r) is less accurately described by (23.2.8) than the eigenvalues are given by the effective-mass equation. Another difficulty occurs for cases of multiple minima, since there are interference effects. Clearly expectation values contain cross-terms in several minima

(HOW> = ¥ |dr FOU NOF(hulk) +

+ ¥ [drFeruttky OFfrie),

(23.29)

and these make the analysis of experiments difficult, since it becomes hard to determine to which shell of atoms a particular spin-resonance signal corresponds. The interference effects can be used fruitfully, however, to determine the position of the band minima Kg; in the Brillouin zone. Similar interference between contributions from different valleys is observed in the exchange between donors pairs (Cullis & Marks 1970). Detailed spin-resonance and ENDOR measurements are available for donors in silicon, together with less accurate data for germanium (see, in particular, Feher 1959; Hale & Mieher 1969;

756

SHALLOW

DONORS

AND

ACCEPTORS

Castner, Hale, & Craven 1970). They confirm that the simplest theory, based on wavefunctions of the form y(r), is inadequate. The inadequacies are the following. First, the amplitude of the wavefunction on the impurity itself differs appreciably in magnitude from that expected. Of course, this is closely connected with the corrections to effective-mass theory discussed later in this chapter. Castner (1970) notes that the effects of a central-cell perturbation can be discerned at quite large distances from the defect. Secondly—a related phenomenon—the spin-resonance parameters for a given shell of atoms near the defect do not change monatonically as the donor atomic number increases. Thirdly, use of the simple continuum function implicitly assumes inversion symmetry; this is not valid for the real system with tetrahedral symmetry. Fourthly, it is not possible to fit the data consistently for silicon with any single value of Ko;. There have been no a priori calculations of shallow defect wavefunctions which are adequate to treat all these points. Some calculations treat the hyperfine interaction at the defect itself, and these are described later. Castner, Hale & Craven have attempted to analyse the data in another way. They argue that the leading corrections to the simple wavefunction should correspond to the admixture of functions like w(r), but coming from other minima in the Brillouin zone, since the density of states is greatest near extrema. Thus W(r) = » » a, F)(r)uj(k;;, ¥),

(23.2.10)

where I labels the set of minima, and i labels the member of the set.

The major contribution for silicon comes from the {100} minima at point A, in the zone; another important contribution comes from the {111} minima at L,. Castner, Hale, & Craven also include the contributions from the K, and U, valleys. They treated the admixture coefficients «, as empirical parameters determined by the best fit to the isotropic hyperfine constants for the identified shells of neighbours. The results improved agreement of theory and experiment for these shells considerably, and also lead to a consistent value of ky for the lowest minima. These improvements are, of course, only to be expected from the extra parameters introduced. However, the improvement appears to be significant, and many of

IN

SEMICONDUCTORS

7157

the anomalies mentioned earlier can be resolved. Thus we need only explain one set of the anomalies ;the others, for different shells, are automatically explained. Hale & Mieher (1971) have adopted a different approach, using a molecular description of the silicon host rather than a conventional band picture. An equivalent-orbitals description is used to parameterize the orbitals ; it is not necessary to use an L.C.A.O. approach, and indeed the special case of L.C.A.O. sp? hybrids gives rather poor agreement with experiment. However, with admixture of d-like orbitals (essential for certain sites), a reasonably general envelope function, and choice of plausible values for three parameters,

good agreement with experiment is achieved. 23.2.3. Response to perturbations We shall only mention a few points of particular interest here, since most of the theory was given in chapter 12. The response to perturbation appears to be well described by effective-mass theory. This is especially true for the Zeeman effect for donors in GaAs (Stillman, Larsen, Wolfe, & Brandt

1971), where a very accurate

variational calculation was found to give extremely good agreement with experiment. The response of donors to stress can be used to estimate deformation potentials for the host. Results for silicon, for which data are especially complete, show some discrepancies (e.g. Tekippe, Chandrasekhar,

Fisher, & Ramdas

1972). One

set of values are

found from the direct observation of the effects of stress on the band extrema and from the piezo spectroscopy of donor excited states. Another set, some 30 per cent higher, is obtained from the stress-

response of spin resonance in donor ground state. Since the spinresonance data appear to depend on the species of donor, they are presumably less reliable, but the differences are not satisfactorily understood. Alloying one host with another can also be considered a perturbation. Onton & Chicotha (1971) have studied the interesting case of In, _,Ga,P with Te donors on the P site. The host has a direct gap for x ~ 0 and an indirect gap, with conduction-band extrema at the edge of the Brillouin zone, for x ~ 1. The donor level follows the band extremum near the crossover ; there is no significant change in binding energy, despite the large intervalley splitting for GaP.

758

SHALLOW

DONORS

AND

ACCEPTORS

23.3. Application of effective-mass theory: degenerate bands The degeneracy of the valence band is important for acceptors in diamond, silicon, and germanium, where the valence-band extrema are at the centre of the Brillouin zone. The valence-band degeneracy is complicated by spin-orbit coupling, since the band at the extremum is of predominantly p-like character. In other cases, as in wurtzite structures, there may also be a crystal-field splitting, as in Rige2.2: The effect of the spin-orbit coupling is to split the p-like band into an upper four-fold-degenerate band (J = 3) and a lower two-folddegenerate band (J = 3). For germanium the spin-orbit coupling is sufficiently large that the J = + band may be ignored (Mendelson & James 1964; Schechter 1962). However, for diamond (Bagguley, Vella-Colleiro, Smith, & Summers 1966) and silicon (Mendelson & Schultz 1969; Schechter 1962) this simplification is not possible, and we shall not use it here. To

make

our

description

clearer, we

write

the effective-mass

wavefunction in matrix form:

V(r) = F(r). O(n),

(23.3.1)

in which @ gives the six Bloch functions (including spin) at the band extremum, and F(r) gives the associated envelope function. The effective-mass equation for F and for the eigenvalues is then

HF = EF.

(23.3.2)

The effective-Hamiltonian matrix (§ 4.2.6) contains four classes of term. There are ‘kinetic energy’ terms, which appear in the corresponding equation for cyclotron resonance, and the Coulomb attraction gives a term proportional to the unit matrix. The spin— orbit coupling appears, and attempts to split the J = 4 and J = 3 components, and finally there are any crystal-field terms needed for non-cubic crystals. The ‘kinetic energy’ terms are particularly complicated, and we shall not reproduce the matrix here. It is given in full by Schechter (1962) and Mendelson & Schultz (1969). 23.3.1. Energy levels and wavefunctions Exact solutions of (23.3.2) do not seem possible. As for donors, the most accurate calculations are variational. The trial functions

IN

SEMICONDUCTORS

759

are taken in the form

Fir) = VY AGA)

(333)

Lk

where the A‘ are linear combinations of spherical harmonics of order |; k labels the different and distinct combinations of spherical

harmonics of given order which transform in the same way under the symmetry operations of the crystal. In the effective-mass approximation (or indeed any continuum approximation) the Fr) have definite parity; they form a basis for a Ig, 7, or T's representation ofthe full double tetrahedral group T,. Consequently, we may label an even-parity state of symmetry I, as |n+ 5, and its odd parity form as |n—)». The general methods for constructing

these A”) in various cases have been outlined by Schechter (1962): Mendelson & Schultz give the results for all these states, including all terms up to third-order spherical harmonics. The functions f,,(r) depend only on the magnitude of r. They can be found numerically or taken as sums of orthogonal polynomials multiplied by an exponential factor. The variational calculation involves varying the exponent and the weights of the various polynomials, and is done numerically. Lipari & Baldereschi (1970) and Baldereschi & Lipari (1973, 1974) have given an alternative approach, based on an analogy with atomic systems. The advantage of their scheme is that it is much simpler to apply than the other, very complex methods. Thus one can discuss other systems or modifications of parameters with relative ease and without great loss in accuracy. The acceptor Hamiltonian in the limit of strong spin-orbit coupling can be written in terms of the hole momentum p and the angular momentum operator J for a particle with spin 3 (Luttinger 1956). Both p and J appear in second order only. Lipari & Baldereschi suggest writing the Hamiltonian as a term with full spherical symmetry and terms with lower (cubic) symmetry. The cubic terms prove to be small, so the solutions for purely spherical symmetry can be used to classify the eigenstates and to give reasonable estimates of the energy. In most cases the cubic terms are adequately given by first-order perturbation theory or by degenerate perturbation theory within degenerate zero-order levels. Unfortunately, the parameters used by Lipari & Baldereschi differ sufficiently from those of Mendelson et al. to make a detailed comparison impossible. But the agreement seems quite good; this is

760

SHALLOW

DONORS

AND

ACCEPTORS

especially so for Ge, where the parameters are particularly favourable to the simpler method. We shall describe more recent results only, since the data used,

like effective masses, are more reliable. In all cases there are problems about the assignment of the excited states. A. Acceptor states in diamond (Bagguley et al. 1966). This acceptor was believed to be aluminium for a long time; it is now considered most probably boron. Bagguley et al. found the energy levels by degenerate perturbation theory rather than a variational calculation. They treated the spin-orbit coupling exactly, but treated the offdiagonal ‘kinetic energy’ terms (the ‘band warping’ terms) by perturbation theory. Thus the zeroth-order problem gives simply the separable equations for s- and p-states with spin—orbit coupling, and the wavefunctions are hydrogenic with an appropriate averaged mass tensor. To the next order, the off-diagonal terms were taken equal to their expectation values in these states. The resulting secular equations were then solved analytically, and the band parameters adjusted to give the best agreement with experiment. Quite good agreement was achieved with parameters similar to those from cyclotron resonance. Further verification of the model has been given by Anastassakis (1969). B. Acceptor states in silicon (Mendelson & Schultz 1969). The fit of the theoretical levels is by no means as good as for the donors. Some of the levels (e.g. the two lowest |8 — ) states) are sensitive to the mass parameters. Indeed, it is not easy to place the experimental and theoretical levels into a one-to-one correspondence without ambiguity. The most probable form (Table 23.1) requires corrections to effective-mass theory which change the order of excited states, like the lowest |6—) state and the fourth and fifth |8—) states. Moreover, this re-ordering is the same for all the acceptors. Further, the intensities of some lines are strong when they are predicted to be weak, like the transition between the two lowest |8 + > states. These discrepancies do not seem to bea result of the approximations in solving the effective-mass equation; they probably come from inadequate input data (¢ = 12 is used instead of the presently accepted value 11-4) or from corrections to simple effective-mass theory. C. Acceptor states in germanium (Mendelson & James 1964). The situation is rather more satisfactory than for silicon. In all cases, of

IN

SEMICONDUCTORS

761

TABLE 23.1 Binding energies of shallow acceptor states in silicon

State

Theory

8+ ye

37-1 18-7

Experiment (some assignments uncertain) B In Al Ga

44.4 14-0 9-8

68-9 14-0 10-4

1-8

Line label or or description

72:7 14-5

156-2

5-6

6-4

3

4-8 4-0

4.8 4.5

5.4 5-1

4 4A

3:8 2-6

4:3 29

3-4

4B 5

2:2

2-2

29

6

14.2 10-4

Ground state

1 2

Theory values are from Mendelson & Schultz; some predictions of still higher excited states are omitted. Energies are in meV, and states |n+

> transform like

I, with the appropriate parity.

4

Tl

| oeN 1

Al

=

|

|

£

meoO ° 5
states. The B line does not seem to have a final |7 — » state; |8 — > or some more complex combinations is more probable.

D. Acceptor states in III-V compounds (White et al. 1973; Kirkman and Stradling 1973). The theory of Baldereschi and Lipari has been particularly successful in predicting the lowest optical transitions in GaAs. Observations on C, Be, Mg, Zn, Si, and Cd acceptors show

systematic chemical shifts, the value for C being close to predictions without any central-cell corrections. 23.4. Corrections to effective-mass theory

Many of the predictions of effective-mass theory have been confirmed with great accuracy. Evidence for corrections is most clearly seen in the ground states of donors or acceptors, since they can have appreciable amplitude at the defect site itself. The relevant observable features are as follows. First, the ground-state ionization energies differ from values extrapolated from the differences in energy of excited states. Secondly, the ionization energy and the amplitude of the wavefunction at the defect both depend on the species of donor or acceptor. Thirdly, the ground state is split when the host band structure is multi-valleyed. The splitting from the multivalley structure is shown in Fig. 23.5 for a number of common

ee

Rises

ee

Fic. 23.5. hee mee in on hosts. The three cases are (a) six (100) minima, as in diamond or silicon, (b) three (100) minima, as in GaP, and(c c) four (111) minima, as for germanium.

IN

SEMICONDUCTORS

763

TAB DEI 3 32 Experimental data for donors and acceptors in silicon and germanium Silicon Donors

Lowest state E,-E, E,-E, 2s-state

Germanium

ie

As

Sb

Bi

Ip

As

Sb

Bi

14.04 11:73 1:37

22.24 21:09 1-41 0-28

11:24 9-86 2-42

39.20 38-08

3:09 2:83

4:38 423

0:52 0-32

2-98 2-87

1-73

1-18

140

0:17

0-69

B

Al

Ga

In

ish

1:19

1-52

169

2-63

3-82

—0-05 —0-03

3s-state

\¥(0)|? from EXPER:

0-43

Acceptors

B

Al

Ga

In

73

31:8

35-6

119-1

Lowest state

2-15

We quote increases in binding energies relative to the best available theoretical values, or alternatively valley—orbit splittings as in Fig. 23.5. The theory values are: Donors in silicon: Donors in germanium: Acceptors in silicon: Acceptors in germanium:

31-27(1s), 8-83(2s), 4-75(3s) 9-81(1s) 37-1 (ground state) 9-28 (ground state)

All units are meV. Spin—orbit coupling causes complications for Si: Bi.

systems. So far, such splittings are best known for ground states; hardly any excited-state splittings have been detected. Data for Si and Ge hosts are given in Table 23.2. Theoretically, it is convenient to distinguish between four classes

of correction. There are corrections to compensate for the approximations made in deriving the general effective-mass equation. These were described in our earlier chapter ; there is still considerable uncertainty in the magnitude of these terms. Secondly, there are the corrections which are properties of the host. Thirdly, there are terms which depend in detail on the electronic structure of the defect. Finally, there are corrections which depend on the electron-lattice interaction. We shall not discuss dynamic effects which give temperature-dependent shifts (e.g. Nishikawa & Barrie 1963) since these shifts seem to be accounted for completely by the temperature dependence of the dielectric constant in the systems we shall discuss (Faulkner 1969). The various predictions are summarized in Table

O33, There is only one general rule in this field: almost every worker has felt obliged to make the contribution he calculated comparable

764

eg

SHALLOW

ep

DONORS

AND

ACCEPTORS

TABLE 23.3 Predictions of corrections to simple theory ay ne ea ae Donor

Contribution and author

As

Sb

Bi

8 8 11 11 1 1 (Negligible)

8 11 1

8 11 1

P

TERMS DEPENDING ON THE HOST ALONE Wavevector-dependent dielectric constant Simple band (Muller) Intervalley coupling A d (Baldereschi)

Host spin-orbit coupling (Appel)

TERMS DEPENDENT ON THE DONOR ELECTRONIC STRUCTURE Detailed theory with matching (Nara & Morita) A-state T-state E-state Delta function perturbation based on Nara— Morita form (Schechter) Thomas—Fermi theory (Czavinsky) Model pseudopotential (Jaros) Empirical covalency (Phillips)

9-0 1-4 0.2

19-0 0-4 —0-4

21-0 —0-2 —1-2

9:8 6-4 4 16

25-9 10 —6 10

3-2 1-5 0 (14)

16

14

TERMS INVOLVING LATTICE DISTORTION Short-range forces (this book) Early approximate work (Weinreich ;

Less than

Shinohara) Polaron correction (Myskowski & Gomulka) Dynamic correction (Barrie & Nishikawa) Experiment (A-state)

1

1

13-5

42

About 10 About 0:8 About 1 16-3

24:5

Values in meV are given for donors in silicon. The contributions are not strictly additive, and the underlying theory is disputable in several cases. The list is not exhaustive.

to the whole observed effect, irrespective of terms omitted. We shall see that differences between the detailed electronic structure of donor and host dominate, but that there are a number

of other

contributions which cannot be neglected. The outstanding puzzles are convincing reasons why simple trends with defect atomic number are not found. The few attempts at understanding the Si:Sb data are far from satisfactory. 23.4.1. Terms which depend only on the host

Certain corrections are independent of the chemical species of the donor or acceptor, and depend only on the properties of the host. These include intervalley coupling, alterations in the Coulomb

IN

screening

close

SEMICONDUCTORS

to the defect,

and

765

a number

of smaller

effects.

One could include the effects of the anisotropy of the effective mass in this category, but such effects are straightforward and were discussed in § 23.2. The work we discuss now generalizes the effective-mass equations to include a wavevector-dependent dielectric constant. The effective screened potential is (§ 2.2.6):

1

U(r) (ny

{ d?q{Uo(a/e(q)} expliq.).

(23.4.1)

We emphasize that our proof of the effective-mass equation (and all proofs with which the author is familiar) assumed that U(q) was negligible for any but the smallest values of q. The wavevector dependence of « is only significant for quite large wavevectors, and only affects U(r) very close to the defect (e.g. Srinivasan 1969). Thus the use of (23.4.1) in the effective-mass equation, whilst plausible, cannot be justified in detail. The actual magnitudes of both the Miiller and Baldereschi corrections, discussed here, must be treated with caution. Miiller (1964, 1965) used the modified U(r) to estimate corrections for a donor in a host with a conduction-band minimum at the zone centre, and U(r) correspondent to the «(q) of silicon. Several analytic approximations were used for the space-dependent dielectric constant defined by U 1/e(r) = ae :

iL

1

L/e,(r) = 1- fa) =

forr < ro

1/00);

(23.4.2)

r >To

Here ro is rather arbitrary, but plausibly chosen as the nearestneighbour distance.

2

1/e,(r) = e “+ A(1—e 8")+

B(I1—e~”),

(23.4.3)

with parameters fitted by Azuma & Shindo (1964) or by Miiller; Sf,

Le,(7) = 1/e.(f) = Ie(oo)

1 < a

(23.44)

7 > 74

The radius r, was chosen so that the two regions of ¢; matched. The important feature of ¢, is that the energy levels can be found analytically. This choice can be exploited in case (3) by choosing r, so that

766

J, rr

SHALLOW

DONORS

AND

ACCEPTORS

‘(ey '—e3*) vanishes. Accurate energies for case (3) can

then be obtained by perturbation theory from the results obtained using ¢,. In practice, the first-order perturbation should be small, and it is neglected by Miiller. His final answer gives a correction of about 8-107 1eV. He also estimates the effect on the wavefunction amplitude at the nucleus, which is enhanced. Baldereschi (1970) concentrates on the intervalley coupling. It is this coupling which leads to the ‘valley—orbit’ splittings d and D for siliconand A for germanium. He treats the coupling as a perturbation, using single-valley wavefunctions in the variational form (W(r)~

exp —{(x?+ y*)/a? +2z7/b?}) of Kohn and Luttinger. The terms in the potential which couples the different valleys uses Nara’s (1965) wavevector-dependent dielectric constant. The results are quite sensitive to the separation of the valleys in the Brillouin zone, and very sensitive to the wavevector dependence of the dielectric constant.

For

silicon,

Baldereschi

estimates

D = 10-6x1072eV

and d = 1-1 x 1073 eV; for germanium A is 0-6 x 107? eV. Other corrections have been proposed. Sham (1966) argued that there should be an r~? term from various higher-order corrections. These include dipole effects, and terms which occur because of the effect of the defect on the chemical potential. However, Sak

(1968) showed that this r~? term cancelled with other terms and

that an r-* term was the one of next order to the dominant r7! potential. less than splittings these are

Experiment suggests that both terms are negligible and the experimental errors. They would lead to anomalous of the excited states, independent of the defect species; not observed.

23.4.2. Corrections which depend in detail on the electronic structure of the defect

The most important of the corrections to effective-mass theory in practice come from the fact that the potential associated with a donor or acceptor is not purely Coulombic. The corrections, which are dominated by the changes in the detailed electronic structure close to the defect, are very short-ranged. Consequently, the potential varies too rapidly in space for effective-mass theory to be applied directly. There are two methods for attacking this problem. The most obvious is a detailed first-principle calculation for some small region near the defect, combined with effective-mass theory for the remaining

IN

SEMICONDUCTORS

767

region. These calculations will be described as a priori. The alternative is to avoid direct solution of the Schrédinger equation in the regions where it is most difficult. Instead, one attempts to relate observables, such as the hyperfine constants or the changes in binding energy from simple theory, to some characteristic of the correcting potential. One may then try to connect the deduced perturbation with a direct calculation, or at least attempt to show its magnitude is plausible. Such calculations are called a posteriori. Almost all the theory has been for substitutional donors. Some interstitial donors have been observed, e.g. Si: Li (Watkins & Ham 1970) and Si: Mg° (Ho & Ramdas 1972). These are simpler because no host atom need be removed

in their formation. Instead, there

are other complications : the Jahn-Teller effect for Liand the doubledonor nature of Mg®, which makes it an analogue of the helium atom rather than the hydrogen atom. The only calculation of central-cell corrections for interstitials is for Si:Li(Nara & Morita 1968) and is for the incorrect interstitial site. A. A posteriori methods. We begin by discussing the a posteriori approaches, since they make more apparent the systematic behaviour from one impurity species to another. An observer ignorant of the chemical species of the donors or acceptors would notice the monotonic variation of a number of properties with the ionization energy. These include the hyperfine constant on the central atom, the dissociation energy of an exciton bound to the neutral centre (Haynes 1960) and the ratio of the intensities of the zero-phonon and one-phonon lines in the recombination of an exciton bound to the neutral donor or acceptor (Dean, Haynes, & Flood 1967). Regrettably, the observer would not find such simple trends with atomic number for donors in silicon, germanium, or donors on the Ga site in GaP (Lorentz, Pettit, &

Blum 1972). The earliest a posteriori approach was Kohn & Luttinger’s (1955a) discussion of Si:P. They examined the matching of the effectivemass solution with the observed binding energy to a solution in the cell based on simple assumptions about the potential. From this it was possible to deduce consistently, if not exactly, hyperfine constants which were in better accord with experiment with those from simpler effective-mass theory. A subsequent paper (Kohn & Luttinger 1955b) used similar arguments, together with the quantum-defect

768

SHALLOW

DONORS

AND

ACCEPTORS

method (§ 4.3), to relate the central-cell corrections of the various s-states. If simple hydrogenic theory gives a 1s binding energy R*, then they argue that the binding energies are given by: R*

E,,ns ==; (n—6)?

23.4.5

(

where 6 may be fixed from the n = | state. When correction to the binding energy varies as

6 is small the

{e.—45] a 1) +o(°)}. n

n

)

(23.4.6)

n

Thus the correction for the 2s-state should be about % of the Is correction. To this order, the correction is proportional to the square modulus of the wavefunction on the central site, and hence to the

hyperfine constant for the central atom. However, Nassau, Henry, & Shiever (1970) find a ratio z4¢ for donors in CdS rather than the § predicted. The position is not clear for silicon (Dean et al. 1967; Kleiner & Krag 1970), but the agreement is probably worse. A more general approach (Stoneham 1970, unpublished work) relates the observed hyperfine constant at the impurity and the deviation of the ionization energy from its value for the simple theory to a parameter which measures the deviation of the defect potential from Coulombic. We start from a general result that the change in ground-state energy A, caused by a perturbation V can be written (Kittel 1964)

Se

_ V0>

(23.4.7)

Gy

where |0> is the unperturbed solution and |» the perturbed ground state. This result is exact. For present purposes |0> is the solution for a pure Coulombic defect potential, V is the short-range correction to this potential, and |» is the exact wavefunction for the correct potential. The exact solution can now be simplified enormously by two weak approximations. The first is that |), like |0>, represents a very diffuse

wavefunction,

and

that

the

dominant

fraction

of both

charge distributions lies outside the range in which V(r) is appreciable. It is then easy to show by the quantum-defect method (or otherwise) that

= 1=g(A).

(23.4.8)

IN

SEMICONDUCTORS

769

Provided A is not too large, g(A) can be neglected. This should be adequate for most cases of interest to us; in other cases, it is trivial

to extend our discussion. The second approximation is to assume that V(r) varies much more rapidly in space than either |) or |). Then we write, approximately,

V(r) = v Ar)

(23.4.9)

= v WE(O)W(0).

(23.4.10)

Whilst there are special technical problems associated with genuine delta-function potentials (Velenik, Zirkovic, de Jeu, & Murrell 1970:

Kumer & Sridhar 1972; Edwards 1972) they can be avoided here without difficulty. If, instead, we had assumed that V varies much more rapidly than the envelope function only, we would have found

ti

us (r)V (r)uo(r)

lieder em vae=o [eno

23.4.11 fae

The effect of these two approximations is to give:

Ao = vw5(0)W(0).

(23.4.12)

But |y/(0)|* is proportional to the hyperfine constant A, so

Ao/./A = v x (terms independent of the defect).

(23.4.13)

In principle, (23.4.13) could be checked by comparison of values of A and A for different s-states of the same donor. Such data are not available at present. However, it is interesting to observe that, if v varies systematically with the impurity through the periodic table, we should expect A,/,/A to vary similarly, even though neither Ay nor ,/A separately vary in a simple way. This allows us to understand (although not to give a detailed explanation of) the unusual results found for Sb in the group V donors in Si and Ge. The Sb donors has very small values of Ay and A compared with values obtained by interpolating on Ay or on A as a function of atomic number.

But, as shown

in Table 23.4, Ao/A

varies much

more

smoothly than Ag or A. The unexpected feature is that v, which measures the short-range perturbation, decreases towards higher atomic numbers; the opposite trend is expected from simple

770

SHALLOW

DONORS

AND

TABLE

23.4

ACCEPTORS

Donor dependence of A/,/ A Donor

Silicon

Germanium

Pp As Sb

2:01 1-60 0-96

3-72 3-14 (No hyperfine constant available)

Bi

1

1

Here A is the excess donor binding energy over Faulkner’s predictions from effective-mass theory, and A is the isotropic hyperfine constant for the donor. The values A/,/A should vary simply with atomic number, apart from corrections for longer-range effects like elastic strain. Results are normalized to unity for the Bi donor.

pseudopotential models. This suggests that the problem is even more complicated than this fairly general treatment has achieved. One of the most accurate measurements of central-cell terms was for donors in GaAs under strong magnetic fields (Fetterman et al. 1971). Whereas we have been discussing only the effects of a shortrange perturbation, the magnetic field affects the whole of the donor wavefunction; the short-range perturbation enters only because the local amplitude of the donor wavefunction has been changed. Detailed comparison with theory in this case shows that A and |¥(0)|? are linearly related under changes in magnetic field. This does not conflict with (23.4.13), since both w,(0) and y(0) are modified by the field.

B. A priori methods. The most detailed a priori approaches have been by Morita and Nara (Morita & Nara 1966; Nara & Morita 1966, 1968). They match the effective-mass solutions outside a sphere of radius rg with those from direct solution of the Schrédinger equation inside the sphere. To see how this is done we consider donors in Si or Ge, where the conduction band has several minima. The conduction-band wavefunctions W(k, r) are solutions of the Schrédinger equation: h2

ia

+ ran}W(k, r) = E,(k)w(k, r),

(23.4.14)

10)

where Vo is the perfect-crystal potential and mg is the free-electron mass. The band has minima at k = k;. Effective-mass theory for the defect is used outside the sphere |r| = ro, where the defect-electron

IN

SEMICONDUCTORS

77

wavefunction is approximately

out) = 2) CIF (nW(k;, 1) = F(ryya(n).

(23.4.15)

Here « labels the symmetry ofthe state (a = A,E, or T for Si, ora = A, T for Ge) and the C? are numerical coefficients fixed by symmetry. The envelope function F(r), assumed spherically symmetric, is a solution of the effective-mass equation with a potential —e?/e,r and an appropriate effective mass in the kinetic-energy term. Morita & Nara simplify their treatment by using an average isotropic effective mass. To describe the wavefunction inside the sphere it is convenient to introduce functions #(k;, r) which are orthogonal to the donor core functions |d>. These functions are related to the band functions of the crystal by

Wk;,1) = ¥ki.n)+

DY —

DY ld>.

Si cores c

— (23.4.16)

donor coresd

An approximate solution of the Schrodinger equation h2

ep

as Vo(r)+ U(r) |W(r) = EW (r)

(23.4.17)

is needed. Morita & Nara assume a function of the form:

inl) = Ds C2 (nW(k;,r) = f(NWa(r).

(23.4.18)

The C? are the same numerical coefficients as in Wy, and f(r) is assumed to be spherically symmetric. The potential in the perfect crystal Vo(r) can be eliminated, together with the perfect-crystal eigenvalue E,(k,). This is most easily done by comparing the equations for Wi_(r) and for W$(r). Subtraction gives

Wi 2

Vs 3V7Y%)

= (U)—eWeaWs,

(23.4.19)

where «, is E*, referred to the band minimum. We now turn to the matching problem. We hope to satisfy

d

d

qp nik (NWolr)} = Fn

a

SOV)

(23.4.20)

772

SHALLOW

DONORS

AND

ACCEPTORS

at all points of the sphere |r| = ro. Rewriting this equation, the condition becomes:

d ©in {FO} = “fin yay) —In Yate} = Q(r).

(23.4.21)

It can only be satisfied if Q(r) is constant on the sphere. With this assumption, Q(r) can be found from our equation in wf, and 6:

[_

@rvavevs—vavevn) = |

in sphere

do vav50l).

(234.22)

surface

This gives, with (23.4.19),

Oro) = aobee nore dF (UC) =8Win feceres do Win

(23.4.23)

0

The matching equation is still intractable unless we further assume U(r), the difference of the potential from the perfect lattice value, is spherically symmetric. We then expand w%, and wW% in Kubic harmonics: |—

in = DRI). Ki, 9)

We = DRI). K,0, 9)

.

(23.4.24)

and obtain the matching condition:

“in {F(r)} = Q(r) dr

_ 2m Yio dr(ee—URERE HS RA r)RH(ro)

(23.4.25)

It is this equation which is used by Nara and Morita, who varied ¢, to obtain consistency. The main assumptions are the spherical symmetry of Q(r), of U, and of the envelope functions f(r) and F(r). The results depend on the precise value of ro. We have not yet discussed the perturbing potential U; this can include any of the corrections mentioned earlier, including lattice deformation within the sphere. Nara & Morita use a screened potential like (23.4.1), based on an unscreened potential U,(r), which

IN

SEMICONDUCTORS

773

is the difference between the impurity core potential and the host core potential. For singly charged donors in silicon, for instance,

Uolr) = Ur, Z$)— U(r, Z3);

(23.4.26)

here, U,(r, Zy) is the potential of the N-fold ionized atom. In practice this is taken to be the simple form

Ur, Zy)

= —{Z—(Z—N)exp(—or)}e?/r,

— (23.4.27)

with o is chosen to give a best fit to the Hartree-Fock potential. Z is the atomic number. Of course, the valence electrons are not con-

sidered here; their presence is felt through the screening by e(k). One unusual feature, resulting from the rather complicated forms of Uo and ¢k), is that U is not constant in sign, having repulsive regions for Si: Sb.

C. Model potentials. We now discuss the work of Jaros (see also Jaros & Kostecky 1969) who uses a model potential approach. The unscreened potential V,,(r) is the difference between the defect and the host model potentials (§ 6.2.4), and is defined in the form: Var =



> A,P,

ip =< Ru

l

Ze Vay i

a

Ryu;

(23.4.28)

r

where Z, is the valence of the defect. The coefficients A, for angular momentum state | were obtained, by the method of Animalu and Heine (1965), from free-ion energy levels linearly extrapolated as a function of energy. The matching radius Ry was chosen so that the resulting screened potential had the simple form —Z,e?/reo for r > Ry. Further, the screened potential for r < Ry

ah EAs eee om)exp (ay) 1 Ae AAG i

eee Eo

r

Eo

a

was approximated by the exponential form — A? exp (2r/a). The resulting potential gives a Schrédinger equation, which can be solved analytically in both regions. The matching then gives the corrections to the energy. Jaros’ approach contains three main approximations. The first is the assumption of isotropy, which is used by most workers. The second is the assumption that the whole of the model potential

774

SHALLOW

DONORS

AND

ACCEPTORS

is screened. It is clearly correct to screen the potential which appears because of the difference in charge distribution between the defect and the host. However, the model potential is a pseudopotential, and includes a term representing the effects of orthogonality; this part should not be screened. The error is least when the defect core is most similar to the host core, as for Si: P or Ge: As where the

orthogonality term in the potential is least. The third approximation is that a pseudo-wavefunction may be matched to an effective-mass envelope wavefunction. This approximation becomes better for effective masses m* close to the free-electron value, but care is still

necessary with regard to energy zeros. The general arguments for using pseudopotential methods in effective-mass calculations have been given by Hermanson & Phillips (1966). The conclusions are reasonably clear: the pseudopotential cancellation does occur, so that the orthogonality constraint can indeed be represented by a repulsive term which cancels a part of the impurity potential. The restrictions in actual calculations are also clear, as mentioned above, and care must be taken over the form of screening used and the boundary condition. Ning & Sah (1970, 1971) have used related ideas to examine the model potential needed to fit observed transition energies. They verify that these potentials exhibit similar trends to atomic Hartree— Fock potentials.

D. Empirical covalency schemes. Phillips (1970a) has argued that the important quantity in determining the energy shift is the change in the valence-bond energy

E, = J(E2+C’),

(23.4.29)

where E,, is the bandgap of the host group IV semiconductor and C is a measure of the ionicity of the system considered. He suggests that the shift can be written

AE(D) = K|E,(AyD)—E,(H)|

(23.4.30)

where H refers to the host crystal and A,,D to the compound of Ay, the acceptor next to H in the Periodic Table, and the donor D. The

constant K is found by fitting the experimental value for D,,, the donor next to H in the periodic table. Thus

AE(D) = AE wD)

E,(AyD) — E,(H)

AD) BAR|

(23.4.31)

IN

SEMICONDUCTORS

Was

Phillips also attempts to derive K and to justify the modulus signs from the first principles. However, his argument contains so many empirical elements that it is more reasonable to regard these two equations for AE as postulates. Another approach of Phillips (1970b) has been to suggest that the varied and complicated corrections can be replaced by a simple modification of the effective-mass tensor. The effective-mass tensor involves matrix elements of the type

A eno

ipl? head

tj

(23.4.32)

ig

Phillips postulates that the modified tensor simply corrects these coefficients :F;; is changed by f,; AE;;. The coefficient f,; is obtained from the difference between the effective-mass tensor of the host and of the binary compound of the host and the defect species, and the AE;; are obtained from covalency theory. The ionization energies are then fitted in the form

E,— Eo = 6E\ + power series in AE

ij?

(23.4.33)

where dE;9 comes from the changes in the F;;. Again, the justification is very incomplete. Both the schemes (23.4.31) and (23.4.33) are in quite good accord with experiment. E. Spin-orbit splitting. Two donors in silicon (Sb and Bi) exhibit spin-orbit splitting of the T,-states derived from the 1s envelope states. Krag, Kleiner, & Zeiger (1970), who observed this splitting,

comment that it agrees in magnitude and sign with an estimate by Roth (1962, unpublished), using an extension of Liu’s orthogonalized plane-wave calculations. The interaction splits the states into a doublet (I,) lying below a quartet (I,). The observed separa-

tions are 0:29 x 1073 eV (Si:Sb) and 1-00 x 10> 3eV (Si: Bi); theor-

etically, 0-9 x 10° eV is predicted for Si: Bi. Spin-orbit coupling is more than a mere perturbation for acceptors, and was included explicitly in the discussion of § 23.3. The effective value of the coupling constants is close to the value for the conduction band of the perfect crystal, slightly modified by terms associated with the envelope function. 23.4.2. Lattice deformation

Host-lattice atoms are displaced from their positions in the perfect crystal by the presence of a donor. This distortion, in turn, affects

776

SHALLOW

DONORS

AND

ACCEPTORS

the transition energies. It is most unlikely that the deformation gives a large change in the ground-state energy of the donor, but the deformation contribution is of interest for other reasons. It gives a perturbation which is long-ranged, and not merely confined to a central cell: thus excited states with zero amplitude at the donor are affected. The small deformation contribution to the ground-state energy must be known and subtracted before experimental data can be used to give information about the short-range corrections; the long-range strain field may alter scattering cross-sections and affect electron mobilities in semiconductors.

A. The strain field. There are three main contributions to the distortion, which we describe by generalized forces Fsp, Fpc, and Fpg on the lattice. We concentrate on substitutional donors, although equivalent treatments hold for interstitials. The first source of strain comes from the differences of the wavefunctions of the donor’s core and valence electrons from those of the replaced host atom. The force Fp is short-ranged, and is often described in terms of host and donor ionic radii. Secondly, there is the perturbation, given by Fpc, due to the net charge of the donor, excluding the defect electron.

Foe comes from the electron-lattice interaction of the donor electron; only Fp, depends on the electronic state of the donor. The Coulombic parts of Fpc and Fp, cancel at large distances from a neutral donor. We simplify initially by representing all these contributions by a short-range force. The strain field at large distances can then be found by the methods of § 8.3 and related to AV, the crystal-volume change per donor in the appropriate state. Thus, in isotropic elasticity the strain tensor is given by (8.3.65):

I,

(23.4.34)

where A is AV{(1+v)/1272(1—v)}, v being Poisson’s ratio. This expression is not adequate, mainly because it gives a zero dilatation. In isotropic elasticity with forces applied at a point, there are two contributions only to the dilatation. One is singular at r = 0, and cannot be used with effective-mass theory. The other contribution appears because the tractions at the free crystal surface must vanish ; it is constant in space, and does not affect splittings of energy levels.

IN

SEMICONDUCTORS

777

In real crystals, the dilatation contains two important contributions at large r:

RO

BB \[x*+y*+2* ye = | = | | ee Px Pur

3 5

(23.435)

The first term is a result of elastic anisotropy. For weak anisotropy it is proportional to:

= 375 (Cyy + 2€12)(C11 — C12 — 2€ 44) AV. 8x Bent2e54-4c,,)7

—(23.4.36)

The second term appears because the defect forces are not applied at a point but at a radius R roughly equal to the nearest-neighbour distance; here

Be Re

105

/[3\c,,+2c

s(3I

ft

ea

2

AV

23.4.

oa)

is the constant of proportionality. It appears that elastic anisotropy is dominant in silicon and germanium, so we set B’ equal to zero. B. The perturbation. We shall use the expressions for the elastic strain with the deformation potentials discussed in § 3.6.2. The precise form depends on the band structure. For a non-degenerate band with its minimum at the zone centre, the perturbation

V(r) = Sy(e,,

(23.4.38)

+eyy + 2)

involves only one of the deformation potentials. In this case the perturbation vanishes using (23.4.34); only the higher-order terms (23.4.35) lead to finite contributions. For donors in silicon and germanium there are two important contributions. One is the term just discussed, including the elastic anisotropy; the other is associated with the anisotropy of the conduction-band extrema. In all,

Gy

corresponds to (3.6.3) for an extremum along unit vector K from the centre of the Brillouin zone. In this second term it is probably adequate to use isotropic elasticity, so our perturbation becomes:

| eee V(r) =

=,(5

r*

|45

+2,(4)a ~3 cos? 0), ir

(23.4.40)

778

SHALLOW

DONORS

AND

ACCEPTORS

with cos @ = (r. K)/r. This expression is valid for donors. Morgan (1970b) has discussed strain effects for acceptors, but unfortunately omits the important term from elastic anisotropy. This term enters

in a qualitatively different way in his model, and it is not easy to see how his conclusions are affected. It is probable that deformation causes important modifications for acceptors, and further calculations are needed. C. Effects on energy levels. We begin by calculating the effect of V(r) on the splitting (E, —E))) of the 2p, ,and 2pp excited states of donors in Si or Ge. This splitting should be very insensitive to central-cell

corrections,

and

yet can

be measured

tolerably

ac-

curately. This allows some check of our expressions as well as spectroscopic estimates of parameters like AV. We then estimate the contribution of lattice distortion to the ground-state energy. This requires a modification of (23.4.39), because the expectation value of 1/r? over a function finite at the origin diverges. A suitable change is to replace r? by (c+r)~ 3. Whilst there are several possible choices of c, they are all similar to the nearest-neighbour distance; we shall

use this value. The corrections to the energies prove small, so first-order perturbation theory should be adequate. We follow Kohn (1957) in choosing effective Bohr radii different in the directions parallel (a7) and perpendicular (a*) to the symmetry axis of the band extremum. The expectation values can then be evaluated as functions of (af/a*), c, and two proportionality constant 2,B/(a*a*t*) and &,A/ (a, a**). It appears that the elastic anisotropy contribution dominates, but both terms will be included whenever we compare results with experiment. Significant deviations from effective-mass theory values of the 2p splitting (E, —E)) are found for Si:Bi and Ge:Bi. In other cases, experimental errors make interpretation even less certain. The results are further complicated in silicon by a near-resonance of the Is—2p, transition

with a phonon

(§ 14.2 and Onton,

Fisher, & Ramdas

1967a). In consequence, |(E, — E,)| is increased from the effectivemass value for Ge: Bi, as predicted from the strain terms, but decreases for Si: Bi. The resonance correction can be estimated only crudely from the observed stress dependence of the transition. The results are consistent with the predicted value and sign of the strain term. With Kohn’s (1957) values for a* and aj, the strain terms give

IN SEMICONDUCTORS

7719

these corrections:

Increase in 2p,—2p, splitting: silicon: 0-35(AV/Vo) meV germanium: 0-11(AV/V,) meV Lowering of Is energy: silicon : 1-62(AV/Vo) meV germanium: 1-54(AV/V,) meV,

where Vo is the volume per atom of the host crystal. The observed 2p)—2p, correction in Ge:Bi is (0-06 +0-02) meV, suggesting that AV/Vo is about (0-54 +0-2) and that there is a 1s correction of about 1 meV. The resonance in Si:Bi makes quantitative results less certain, but they are consistent with a resonance correction of order 0-5 meV, a volume change of about V, and a lowering of the Is energy by 1-2 meV. The 1s corrections are very small and cannot be measured directly. The predictions are much smaller than those from the approximate treatments of Weinreich (1959) and Shinohara (1961). Here, the predictions can be checked because they amount to a spectroscopic measurement of the volume change. Observations of AV are incomplete and often contradictory, as shown for example, by the extensive measurements of Bublik, Gorelik, & Dubrovina (1969). A simple rule that seems to give consistent results is to say that AV is the difference between V, of the host and the volume per atom V, of the tetrahedral group IV element nearest in atomic number to the donor. Thus for donors P, As, and Sb one uses, for V,, volumes per atom of Si, Ge, and Sn.

The values of AV/Vo for silicon are zero (Si:P), 0-13 (Si: As), and 0-86 (Si: Sb), and are in line with the deduced value of order unity for Si: Bi. Observed values are —0-11 (Si:P, Cohen 1967) and 0-15 (Si:As, Moyer 1968), and are similar to ours. Values for Si:C agree very well with the prescription. The other available values are for Si:B, and conflict with each other in magnitude and sign (+ 0-07, Horne 1955; —0-34, Cohen 1967; —0-85, Bublik et al.). Corrections to the energy levels will be very small, except possibly for Sb donors, where accurate spectroscopy could prove useful. The prescription for AV also seems to work well for N in diamond, where line-width results imply rather small distortions (Davies 1970).

D. Wavefunction-dependent distortion. Calculations of the 1s-energy changes due to the interaction of the donor electron with the host

780

SHALLOW

DONORS

AND

ACCEPTORS

have been given by Mycielski (1962) for an isotropic band structure and by Myskowski & Gomulka (1964) for silicon and germanium. These contributions (essentially from the Fpc and Fp, mentioned earlier) ignore the terms previously discussed which depend on the short-range forces Fez. Strictly, the two contributions are not additive, but for rough estimates we may consider them separately. The results with deformation-potential coupling show that there is a small reduction in the 1s energy for donors in silicon (~0-8 meV) and a reduction for germanium which is completely negligible (~0-025 meV). If isotropic bands are assumed, energies 30-40 per cent higher are found. These values are related to the ‘dynamic corrections’ of Barrie & Nishikawa (1963), and roughly correspond in magnitude. We have assumed here that the observed transitions in silicon and germanium are zero-phonon lines, so that no Stokes shift is expected. There has been controversy over the exact nature of the transitions, largely because of small differences between optical and thermal ionization energies (Brooks 1955; Lopez & Koenig 1966); no phonon structure is observed. It is unlikely that the differences in ionization energies correspond toa Stokes shift. Instead, mechanisms observed for other shallow centres are more probable. Thus for donors and acceptors on the Ga site of GaP, discrepancies seem to be a consequence of the different concentrations at which the two energies are measured (Casey, Ermanis, Luther, Dawson, & Verleur

1971). Stillman, Wolfe, & Dimmock (1971), discussing analogous differences for donors in GaAs, argue that the higher excited states on different donors overlap to give an impurity band which merges with the conduction band. The thermal ionization energy will then be less than the value of the optical energy extrapolated from lower levels.

24 ISOELECTRONIC

IMPURITIES

24.1. Introduction

ISOELECTRONIC impurities are those in which a host ion is replaced by an impurity of the same charge and similar electronic structure. Free carriers can be trapped by their short-range interaction with the defect; there is no long-range Coulomb interaction. Most of the theory of isoelectronic defects has been for III-V systems (e.g. Faulkner 1968), and so it is worth stressing the range of systems in which isoelectronic defects have been studied. In the valence crystals like diamond, Si, and Ge, work has been purely theoretical. Callaway & Hughes (1967) have treated the neutral vacancy in silicon as an isoelectronic defect; we shall discuss this system later in § 27. Larkins (1971e) has discussed energy levels of C in Si and of Si in diamond, but there is no experimental evidence for localized states. In III—V systems the most important cases have been N or Bi substituting for the anions in GaP (Thomas & Hopfield 1966; Faulkner 1968), in InP (Dean et al. 1971) and in alloys In, _,Ga,P (Scifres, et al. 1971b) or GaAs, _, P,. (Scifres et al. 1971a). Important examples in II-VI compounds are ZnTe:O (Hopfield, Thomas,

& Lynch

1966) and

CdS:Te

(Aten & Hoenstra

1964).

Excitons bound to isoelectronic defects have been observed in silver halides (e.g. Kanzaki & Sakuragi 1970) and in rare-gas solids (Baldini

1965;

Hermanson

&

Phillips

1966;

Webber,

Rice

&

Jortner 1965). In addition to these systems there are others which are arguably isoelectronic traps but which, for various reasons, we treat separately. The neutral vacancy in valence crystals is one example, where both the neutral defect and the defect with a trapped carrier are rather too complicated to treat by the methods of this chapter. Dipolar centres too are best treated as a different kind of defect, despite similarities. The F’-centre in alkali halides is another case: a second electron is trapped at an F-centre. The F-centre, an electron trapped at an anion vacancy, is electrically neutral; the strong

782

ISOELECTRONIC

IMPURITIES

binding of the extra electron arises partly from correlation in the motion of the two electrons. It is remarkable that isoelectric traps are almost universally impurities on the anion sublattice. None of the systems listed in the survey of Czaja (1971) involve cation impurities, irrespective of host ionicity, the direct or indirect nature of the band structure, or the

trapped-carrier charge. The only cation traps seem to be in the alkali halides where the hole is self-trapped (chapter 18) and can be bound by ions like Tl or even by Li in NaF (Bass & Mieher 1968) or Li or Na in KCl (Schoemaker 1970). Three charge states of isoelectronic defects are important. These are the empty neutral centre, which we denote by ©, the centre with one trapped carrier ([© e] or [© h], as appropriate), and the bound exciton state [© eh]. We shall be interested in the effects of the neutral defect © on optical absorption and carrier mobility, and also in the bound states and resonances of the [© e] and [© h] systems. Thus we consider the bound exciton states of GaP:N; the bound state [© e] may exist in this case, but evidence is lacking. However, the state [© h] does exist for GaP : Bi.

24.2. Binding mechanisms In this section we discuss the form of the perturbation by which the carriers are trapped. The potential need only be attractive to one of the two carriers. Thus, if an electron can be bound, a hole can be added to form a bound exciton, attracted by the Coulomb interaction between electron and hole. If a hole is bound, the same Coulomb interaction can trap an electron. In some cases a defect may be attractive to both carriers. Our discussion distinguishes between two main classes of mechanism: those which depend on distortion of the lattice by the impurity and those which do not.

24.2.1. Mechanisms independent of lattice distortion The mechanisms operative here are essentially those which gave central-cell corrections for donors and acceptors: the difference in detailed electronic structure between the impurity and the host. No method currently available seems completely satisfactory, mainly because the energy levels are so very sensitive to the perturbation: for GaP:N the binding energy can be varied by +100 per cent simply by changes of about 1 per cent from the value

ISOELECTRONIC

IMPURITIES

783

perturbation giving the correct binding energy (Faulkner 1968, § IV D). We describe some of the models used. The very simplest approaches treat the binding as a direct consequence of differences in electro-negativity. Thus Larkins (1971) has used extended Hiickel theory to discuss Si:C and Si in diamond. The defect differs from the host atom only in its ionization potential and in its overlaps with host atoms. Whilst naive, extended Hiickel theory can give qualitative insights, and even quantitative ones if it is used as an interpolation relation by incorporating experimental results. More sophisticated approaches use either atomic wavefunctions or pseudopotential coefficients. We discuss Faulkner’s calculation for GaP:N based on atomic functions. This calculation derives unscreened pseudopotentials for nitrogen and phosphorus, and regards the difference as a perturbation. It makes two of the simplifications discussed in chapter 6: the pseudo wavefunction |W) is assumed to vary slowly over the core orbitals, and the core orbitals on different sites are assumed orthogonal. The pseudopotential is then given by the Phillips-Kleinman form:

Viw(x) = V(x)W(x)— 3 (E.— E(x) {d°y x b*(y)W(y).

;

(24.2.1)

The slow variation of the wavefunction allows one to write

Viw(x) = V(x)W(x)— >) (E.— E)b.x)W0) |d°y $2(y) Cc

~ {Ys ¥ (E:—B)bx) Icdr) woo (24.2.2) which gives an approximate local pseudopotential

V, = V(x)— ¥(E.—E)b.(0) |dydt(y).

(24.2.3)

Using the same approximations systematically in the second term gives a result of the type discussed in § 6.2:

Vi, = V(x)— Y(E.-B) {d*y $.(y) : 5(x).

(24.2.4)

Further, if the pseudo-wavefunctions W(y) vary much more slowly

784

ISOELECTRONIC

IMPURITIES

than V(x), then we obtain an effective potential: 2

V em fos Viy)- > (EB)

fey .(y)

6(x). (24.2.5)

Faulkner used V, of (24.2.3) and further approximated by taking E to be an average of the outer atomic s and p levels and by smoothing

(VI(x)—VE(x)) for regions near x = 0, using an approximation of the form B(x/b)* exp (—4x/b). The pseudopotential is not sensitive to E because the core levels are so deep. We shall discuss the accuracy of V, later. Here, we merely observe that it appears to be too large by about a factor of 2. There are at least three possible reasons for this. The first is that the wavefunction does not vary sufficiently slowly over the cores. This is the most likely explanation; errors of almost exactly this type and magnitude were found in analogous pseudopotential coefficients by Bartram et al. (1968) (discussed in chapters 6 and 15). The second possibility is discussed shortly; it is Phillip’s suggestion that lattice polarization and distortion conspire to reduce the effective binding in this manner. In his review, Baldereschi (1973) also argues that screening is important, but for different reasons. The polarization and distortion may well be important, but there are other reasons to doubt this mechanism as the sole effect. In the work of Bartram et al. the polarization and distortion were included separately and did not contribute to the observed reduction of the pseudopotential. A third possibility is that, since we are concerned with systems which only just give bound states, we may have to go beyond the Born—Oppenheimer approximation to consider subtle dynamic effects. Expressions for the perturbation similar to Faulkner’s result have been given for rare-gas systems by Hermanson & Phillips (1966) and Hermanson (1966). Their analysis differs in three ways. First, they use the Cohen—Heine pseudopotential rather than the Kleinman-—Phillips form. Secondly, in constructing matrix elements among Wannier functions, free-electron Wannier functions were used. Thirdly, the potential V(x), which forms part of V,(x), was taken as the screened potential rather than the bare potential. Whilst it is clear that the potential in a solid is not exactly the sum of free-atom potentials, it is not clear whether screening using a model dielectric function appropriate to the host is justified. Hermanson & Phillips observe that core-polarization is important.

ISOELECTRONIC

IMPURITIES

785

Allen (1971) noted that both empirical and model pseudopotentials are available for ions which are consistent, in the sense that

the same V,(q) are appropriate for a range of substances (Cohen & Heine 1970). He then asked how the properties of these V(q) can be used to decide whether bound states are likely for different defects in a given host. In the one-band, one-site model a bound state is expected when the strength of the perturbation

AAV (g)V 5G)

(24.2.6)

reaches a critical value determined by the band structure. Allen argues that a similar criterion can be given in more complex band structures. The critical value in such cases must be fixed by a more difficult calculation. However, it is clearly true that one can get some idea of which systems have bound states in this way. The potentials V,(q) have a simple dependence on wavevector q. At small q they tend to a negative value —|V,(0)|, and they rise with increasing q. At a value |q| = qo the potential passes through zero before reaching a maximum. A convenient parameterization is

V,Pp (q) = — 91V,(0y 2224/49)

(Bq/4o)° ’

(24.2.7)

where J, is a Bessel function and B = 5-1356. The Fourier transform

of V, is a potential which is proportional to ,/(1 —(rgo/f)) and zero forr > B/qo. Within a given group of the Periodic Table, and excluding the first row, it is found that the empirical pseudopotentials vary little from element to element, that the model pseudopotentials are also similar from element to element but different from the empirical pseudopotentials, and that both qo and |V,(0)| increase as one goes from group II to group VI. The first-row elements tend to have V,(r) which are deeper and narrower than corresponding elements. Allen suggests that the perturbation can be characterized by the expectation value of the change in V,(r) between hydrogenic 1s functions with effective Bohr radii a,,, which approximate Wannier functions. To a good approximation the expectation value is the change in A re

=

— 0017072,

V(0)\do4n ’;

(24.2.8)

where Q,, is the atomic volume. Thus A,r, depends on the host through « and on |V,(0)|qo. The result differs from simple electronegativity arguments which, in essence, would also involve changes

786

ISOELECTRONIC

in a. Values of Q|V,(0)|qo in atomic Allen also observes that heavy-atom coupling terms and that an impurity atom it replaces should be attractive

IMPURITIES

units are listed in Table 24.1. impurities give large spin-orbit with a larger coupling than the to holes.

TABLE 24.1 Effective strengths for isoelectronic defects Group IV

Species

(e

Si

Group V

Ge

Sn

N

70

78

80

61-8

—_



Group VI

P

O

Syseuic

91

99-6

50

126

71-3

80-4





Empirical

potential Model potential

— 60-3

Results are those of Allen (1971), and are in atomic units. They should be compared only within each row. In a silicon host, Callaway & Hughes (1967a) have estimated that the critical values for a bound state is about 20 per cent larger than the appro-

priate values for Si cited here.

Baldereschi & Hopfield (1972) also assume that for each host there is a critical value of the effective perturbation to give a bound state. Their results differ mainly on the questions of screening and of lattice distortion, and they give values of the parameter A of (5.2.1), calculated both from pseudopotential and covalency theory arguments. The corrections for distortion are common to both calculations of A. Suppose an impurity X substitutes for a Y ion in host YZ. Then the perturbation can be calculated for spacings appropriate to pure YZ (x = 0) or for pure XZ (x = 1). Linear interpolation gives effective forces of the defect on the lattice. Elastic restoring terms are proportional to x, so that the equilibrium value of x can be found as in (8.2.4). The detailed treatment was primitive, but the predicted values in the range x ~ 0-3-0-5 are plausible. The effect of relaxation is to make the potentials more attractive to holes. The pseudopotential calculations proceed differently for the two values of x. For x = 0, the difference between

the bare atomic

pseudopotentials for X and Y was taken, and subsequently screened. For x = 1, the screening problem was avoided by using differences between the empirical pseudopotentials for XZ and YZ. The screening for x = 0 appears to screen the orthogonalization part of the potentials (cf. §23.4). Baldereschi & Hopfield propose a modification

ISOELECTRONIC

IMPURITIES

787

of the screening by the host dielectric constant usually chosen, arguing that a local electron-density correction is needed. This somewhat ad hoc procedure is mainly designed to correct the trend in which values of A are systematically too large for heavy impurities, especially in more ionic crystals. The improvement which results is very slight. In the dielectric method, favoured by Baldereschi & Hopfield, the value ofA was obtained from the differences of the ionization potentials for XZ and YZ, averaged over the T, X, and L points of the Brillouin zone. Corrections for the different lattice parameters were made using deformation potentials for the valence band, and certain kinetic corrections were subtracted in a free-electron approximation. The results for this model prove more successful. The approaches are compared in Table 24.2, from which the important role of distortion is clear. 24.2.2.

Mechanisms which involve lattice distortion

The binding of an exciton to an isoelectronic defect is such a finely balanced phenomenon that lattice distortion can have a profound effect. This is confirmed by contrasting the recombination of excitons bound to isoelectronic defects with those bound to neutral donors, as in Dean’s (1970) comparison of the GaP: N and GaP:S systems. One obvious difference is that the oscillator strength for the isoelectronic defect is 0-1, two orders of magnitude larger than for the donor. This occurs because an indirect exciton is involved, the

amplitude of the Fourier component w,(k ~ 0) of the electronic envelope function determines the rate, and it is much larger for the isoelectronic case than for the neutral donor (§ 5.2). The sensitivity of the isoelectronic system to lattice vibrations is shown by the fraction of the intensity in the phonon-assisted bands: 90 per cent for the isoelectronic system, but only 6 per cent for the neutral donor. However, the observed response to pressure (Merz, Baldereschi, & Sergeant 1972) is not extreme. The observed derivations of the binding energy with respect to

pressure are: dE,/dP = —1-12(GaP:N) = —0-29 (GaP : Bi),

in units of the bandgap change at the same pressure (~6 x 10°4 eV atm” !). If the binding potential is measured by A (cf. § 5.2), Merz

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ISOELECTRONIC

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788 IMPURITIES

ISOELECTRONIC

IMPURITIES

789

et al. note that dA/dP is very similar in both systems. The predicted values of dE,/dA are in the ratio 0-7(N) to 0-2(Bi), giving values of dA/dP differing by only 10 per cent. This similarity is surprising in view of the other arguments given that local lattice distortion is responsible for the differences in dE,/dP. These arguments require N to be a small, elastically-hard defect, Bi to be a large, elastically-

soft defect, and local lattice expansion to favour hole binding. Whilst qualitatively plausible, no simple relation between the values of dA/dP is implied. Phillips (1969) has argued that the standard effects of polarization and distortion of the host by the defect lead to a binding energy which is close to zero. Suppose the perturbation of the defect is gVi(r), where g is a coupling constant. Then Q, the localized change in the cell containing the defect, is a function of g. This is shown in Fig. 24.1: there is no binding (Q = 0) when g is below a critical

Q ett

Q

Fic. 24.1. Relation between strength of perturbation g and charge localized in the central cell Q. I shows the expected trends as a function of g; II shows the effects of lattice polarization on the perturbation. Self-consistent solution gives g.,, and Qer,.

value g,; above this value Q(g) increases monotonically in some way. Now the localization of the charge causes polarization and distortion of the host and this, in turn, affects g: g is now a function g(Q). Phillips argues that, as a rule, the signs of terms are such that g decreases from g(0) = go. The values of g and Q appropriate in any case are found by solving the two relations consistently. To this point, the argument is similar to the standard ones used in chapter 8. However, Phillips suggests that the consistent solution corresponds to very small values of g and Q. There seems to be no direct evidence for this, and the only indirect evidence offered is that predicted perturbations suggest larger binding energies than observed.

790

ISOELECTRONIC

But, as mentioned

IMPURITIES

in § 24.2.1, other explanations

are available.

Moreover, the result appears to require a larger Stokes shift than found experimentally. Allen (1971) has also discussed lattice strain contributions to the binding. In most systems the strain due to the defect provides a weak, slowly varying perturbation given in terms of the deformation potentials and strain. Intervalley terms are negligible, and the change in binding energy given by the expectation value of the perturbation with the envelope function of the defect electron. The change may be attractive or repulsive, depending on the size of the defect ion and the sign of the deformation potential (as in chapter 23). Allen notes, however, that if intervalley coupling is strong the correction is always attractive. No system is known to show this, but it is more likely in systems for which self-trapping of carriers can occur. 24.3. The one-band, one-site model

One-band, one-site models were discussed in §5.2 and §5.3. The major simplification is that matrix elements of the perturbation U among Wannier functions is assumed to have the form Ke n|Ula, m»

=

Ado, Sor 1m Oin>

(24.3.1)

which is finite only for site 0 and band /. The model is used phenomenologically. A is fixed from one measurement, such as a binding energy, and is then used to predict other properties. We shall discuss optical absorption, scattering of carriers, and the binding energies of pairs of defects. 24.3.1.

Band structure

Whilst the perturbation is treated phenomenologically, it is usual to attempt to treat the band structure accurately. For shallow bound states, Faulkner’s approximation (5.2.17) is adequate. The Green’s function is real, and it can be split into one part independent of energy and another part adequately given by effective-mass theory. For convenience we define the part independent of energy as

(HED = = |@Pk{EMO)-' 1

(24.3.2)

where the bottom of the conduction band has zero energy, Baldereschi & Hopfield (1972) cite values of (1/E,>~! as — 1-8 eV (Si),

ISOELECTRONIC

IMPURITIES

791

—1-4eV (GaP), and —0-9 eV (ZnS). For a bound state to exist, Q,

given by

Q = (1/A)+ 1)A is zero and

A = DX{1—,/(1-X~?)}.

(24.3.6)

A is proportional to the density of states. These approximations have been used in an extended form by Scifres et al. (1971a, b) for the mixed crystals GaAs,_,P, and In,_,Ga,P. For GaAs (x = 0) and InP (y = 0) the bandgaps are direct whereas GaP(x = y = 1)

has an indirect gap; the changeovers occur at x = 0-4 and y ~ 0-71. The extension by Scifres et al. consisted of fits analogous to (24.3.5) (24.3.6) for each of the band individually and for three regions of the Brillouin zone separately: the centre of the zone (0 < kj/(2m/a) < 0-125), the intermediate part (0-125 < k,/(2m/a) < 0-5), and the

remainder of the first Brillouin zone. In principle, any theoretical model for the band structure may be used. In practice, it is convenient to use an empirical pseudopotential method which incorporates as much experimental information as possible (Brust, Phillips, & Bassani 1962, Brust, Cohen, & Phillips

1962, Cohen & Bergstrasser 1966). This method can be extended to mixed crystals. For example Scifres et al. (1971a) scale the lattice parameter and the pseudopotential form factors V,(GaAs, —,P,) = xi(GaP)+(1—x)K(GaAs).

(24.3.7)

792

ISOELECTRONIC

IMPURITIES

24.3.2. Energy levels: bound states and resonances Values of A predicted from observations of the isolated isoelectronic trap GaP:N have been used to discuss states of pairs of nitrogen ions in GaP (Faulkner 1968) and of isolated nitrogen ions in mixed crystals GaAs, _,P, and In, _ ,Ga,P (Scifres et al. 1971a, b). The bound exciton transitions observed experimentally for GaP :N are the allowed J = 1 exciton (A line) and forbidden J = 2 (B line) exciton transitions. The A line is at 0-021 eV below the gap energy, i.e. 0-011 eV below the free-exciton line. If the electron-hole binding is not affected by the defect then 0-011 eV would be the binding energy of the exciton to the defect. Thermal-decay measurements (Cuthbert & Thomas 1967) suggest a binding energy of 0-008 eV. Probably the electron is the bound carrier, but there is no independent evidence of a [© e] state. Faulkner estimated Q from the binding energy of an exciton to the isoelectronic defect and obtained

(24.3.8)

Q = 0-022 (eV) “'.

The electron and hole wavefunctions w,(k) and w,(k) have the form

W. ~ (E(k)+£,)~' and yw, ~ (E,(k)+ E,) 7.Here E, and E, refer to the conduction and valence bands, and the parameters E, and E,, are found variationally :

E, = 0-0104 -

(24.3.9)

E,, = 0.0096 eV)’

corresponding to orbital radii of the order of tens of angstroms. The theory of an electron bound to two nitrogen ions in GaP can be obtained as a natural extension of the isolated impurity problem. If one defect is at the origin and the other at R then the perturbation matrix has the form A

= qu! + expi(q—k). R}, where / denotes the rane

(24.3.10)

band. If we introduce a function

exp (iq. R)

S(E, R)= akeOe Q,

Zz

Eq)



E

24:3:11 (

)

then the eigenvalue equation reduces to

1+A{f(E, 0)+ f(E, R)} = 0.

(24.3.12)

There are two solutions, analogous to the bonding and antibonding

states of the hydrogen molecule. The function f(E, R) can be quite

ISOELECTRONIC

IMPURITIES

793

sensitive to the orientation of R when the conduction-band extrema are not spherical. For GaP the effective masses m, and m, differ by a factor of about 5-7. Faulkner’s calculations verify the weakness of one-band, one-site calculations: the energy levels of nitrogen pairs are not given correctly, nor are the excited states of the single defect described satisfactorily. But these results are more a criticism of the simplifying approximations used rather than the basic method. Scifres et al. (1970a,b) found that sharp emission lines were observed with energies larger than the direct bandgap in nitrogendoped alloys of GaP. These correspond to the A line in pure GaP. The emission is associated with a resonant state of the isoelectronic trap. Their analysis of the theory involves two main features. One is the parameterization of the band structures as a function of composition. The alloy is considered homogeneous with a composition-dependent potential and lattice parameter. The direct and indirect gap energies are shown in Fig. 24.2. The other feature is the 2-9

(a) Oe

Direct gap

(LT) DSS)

= 2

L

Indirect gap (X)

Ba o

i 2:2

a

0-5

:

InP

1-0 GaP

% 0:03 0-02

(b)

g

= 0-01 0

i

1

i

Newel

0-5

It

Fic. 24.2. Bound states and resonances in In,_,Ga,P:N.

1-0

Results of Scifres et al.

show the indirect bandgap at X, the direct bandgap at I’, and bound states/resonances

associated with isoelectronic nitrogen. (a) Give the position of the resonance relative

to the bandgaps and (b) the full width of the resonance.

794

ISOELECTRONIC

assumption

of the

one-band,

IMPURITIES

one-site

model

with

a strength

parameter chosen to give a binding energy for [© e] of 0-008 eV in GaP. Thus they ignore the changes in electron-hole interaction with band structure. The methods outlined in chapter 5 then give the position and width of the resonance. Results are shown in Fig. 24.2. The model confirms that the sharp level persists as the direct gap becomes lower in energy, and shows that the width of the resonance grows rapidly from zero at the crossover. The shift in binding energy with composition is less in the model systems than observed for both GaP,,As,_,:N and In,_,Ga,P:N and indicate that the strength parameter A increases with alloying. But the qualitative behaviour seems to be correct and unlikely to be affected by sophistications of this sort. 24.3.3. Optical line-shapes

We now consider the optical absorption of an isoelectronic defect. Three processes are considered, following Faulkner’s analysis of GaP:N. These are: A. creation of a bound electron and a free hole; B. creation of a free electron and a free hole; C. creation of a free exciton.

In all cases the initial state consists of a crystal, perfect apart from the presence of the isoelectronic impurity ;there are no free carriers or carriers trapped by the impurity. Weare interested in the optical absorption near the threshold, and we shall ignore all latticedynamic effects until later. The first stage is the calculation of the final-state wavefunctions. A. Bound electron and free hole. The physical assumption made is that the bound-electron wavefunction does not depend on the presence of the hole and that the free-hole wavefunction is not affected by the defect. This is a natural extension of Faulkner’s treatment of the bound exciton, where it is assumed that the iso-

electronic defect traps an electron by the short-range interaction and the hole is trapped by Coulomb interaction with the electron. In the bound-electron, free-hole case this Coulomb interaction is ignored.

This simplification should be poor at threshold, but become increasingly good as the hole kinetic energy becomes large compared with the Coulomb terms.

ISOELECTRONIC

IMPURITIES

The electronic function is derived by standard

795

Koster—Slater

methods, and has the form:

Wi(k) ~ {E(K)+E,}7!,

(24.3.13)

Be sles] ae 32 ae

(24.3.14)

where

The final-state wavefunction for process (1) has the form:

YPa(k,,k2) ~ wa(k,) 6(k, —k,),

(24.3.15)

where hk,, is the final hole momentum. Effective-mass theory is used to relate the hole energy and momentum. B. Free electron and free hole. This system is marginally more complicated, since we cannot ignore the short-range interaction of the electron with the isoelectronic defect. The effect of this interaction on a state with energy E = E,(Ko) is given by the Schrédinger equation:

ay =e As in bound-state introducing

pee problems,

f(B)=

I

TV)

A

{d°qwe(q).

the equation

— (24.3.16)

can be simplified by

{ dq meee

(24.3.17)

and by noting that the y,,(k) are orthonormal in the sense

[ea vittaws.ta) = otk, -k,)

(24.3.18)

The electron wavefunction becomes

1

A

|

Wilk) = koko) E—E,(k)+io Q, {1+ Af(B)}

24.3.19

(

This should be compared with the result using the Born approximation:

A VEk) = dk—ko)+ EE Sis G,"

(24.3.20)

796

ISOELECTRONIC

IMPURITIES

obtained by putting y,, ~ d(k—ko) in the right-hand side of the Schrédinger equation (24.3.16). The extra factor (1+ Af(E))~* has a profound effect if there is a bound state close to the threshold. The Born approximation could underestimate the optical absorption by several orders of magnitude. Optical absorption singles out the states with k ~ 0. We are interested mainly in absorption to energies E above the indirect minima in GaP, but below the direct extremum at energy E,,. The wavefunction simplifies here, since the term 6(k—Kk,) does not appear and the term io can be ignored:

Wilk ~ 0)

A 1 1 ~ Q,1+Af(E) E- Eca

The final state wavefunction for process (2) has the form:

Walk, ,k2) = we(k,) d(k, —k,).

(24.3.21)

Again the Coulomb interaction of the electron and hole is ignored, an approximation which becomes increasingly good at higher energies.

C. Free exciton. This system is more complicated still, since the Coulomb interaction V,,, of the electron and hole cannot be ignored. The problem proves tractable if this interaction is retained only to the lowest order which is meaningful. Thus, if the Schrédinger equation for the electron and hole wavefunction is:

{E— E.(h,) —E(k) —Van}Wells Ke) = k,)+

E—E,(k,)—E,(k,) +io

«=A |aa Yea, ka)+ +(terms in V,,),

(24.3.23)

then the terms in V,, are ignored. The Coulomb interaction is involved in the usual free-exciton function ¥@, but all cross-terms

ISOELECTRONIC

IMPURITIES

797

in the short-range potential and in V,, are dropped. Again the approximation improves at energies well above threshold. The approximation yields an expression for Y< in terms of the freeexciton function: ¥e(k, > k,) =

Pek, ’ k,)+

"BoE(k,)—E£(k,) 410 14Af(E Bk} A Soi id3q ¥2(q, k,). The factor [1+ Af{E—E,(k,)']~' approximation.

(24.3.24)

would

be unity in the Born

It now remains to estimate the free-exciton wavefunction. This is a product of two factors. One describes the motion of the exciton as a whole, with effective mass M = m*+m* and momentum hx. The other factor describes the relative motion of the electron and hole, with a reduced mass p = m*m*/(m* + m*) and momentum in the centre-of-mass frame hk, }:

kp = W(k’,/m* +k,/ms). Here

k’, is measured

from

the

(24.3.25)

appropriate

conduction-band

minimum, and differs from k which is measured from the centre of the zone. In all, the free-exciton function is:

Pe(k, Kk.) = d(k; —k —«) (ky 2).

(23.4.26)

The factor ¢ describing relative motion was assumed to have a simple hydrogenic envelope function, described by an effective

Bohr radius a,, expressed in terms of the exciton binding energy

ieee

te(Quay)

The component of (k,,k,) with optical absorption. This is given by

ES

k, =k,

is important

in

1

PERE) 1

A

$a(k+H *

“T+ Af{E— Eb} Q,



(24.3.27)

798

ISOELECTRONIC

For small values approximation is

of total exciton

1

¥c(k, k) — 1F

IMPURITIES

momentum

hx an adequate

1

: Eg 1+ Af(E—k2/2m#) Q,

afk

En

We k

n(24308)

when h?x?/2M —|E,,| is small compared with E,,, the difference between the direct and indirect gaps. D. Optical cross-section. The optical cross-section is calculated by the standard methods given in chapter 10. Here it is essential to introduce the band functions: the wavefunctions we have been considering are essentially envelope functions. Since the hole wavefunction are all localized about k, ~ 0 the assumption is made that the matrix element may be replaced by 0, and here Faulkner used a model band structure as in (24.3.6}(24.3.7). He assumed

CX (E) = (F =u VE),

(24.3.34)

with the value y{1/E,> equal to 0-82 (eV) 7. The final form of S, varies as (w—W)? at threshold and approaches (w—.«)* asymptotically. Since t(x) is positive definite, $,(w) is bounded above by its value with t = 0:

a sy

(24.3.35)

0 and / > 0). Still more types of spectrum are possible in the wurtzite lattice. The line intensities depend on both the number of donor— acceptor pairs with a given separation and on the capture mechanisms (§ 25.3.3) if the transitions are not saturated. Williams (1968) has surveyed calculations of the distribution of pairs. These vary from random distributions to ones which allow for the discreteness of the lattice and for the association which results from interactions between the defects. The distribution G(R;) calculated is the product of two factors: the probability that a donor lies at position i from the acceptor under consideration and the probability that no donor lies closer than site i to the acceptor. We define p, the number of donors per unit volume, and p(R,), their pair distribution function with respect to the acceptors. The probability that there is a donor in one of the N(r;) equivalent sites i is thus p p(R;). This is usually rewritten as CoN(R;) exp {— E(R;)/kT }, where E(R;) is the interaction energy between the donor and acceptor, which is dominated by the Coulomb term e?/e)R;, and T is the lowest temperature at which the ions are effectively mobile. C is a normalization. Low concentrations are implicit here, since there is no constraint that keeps two defects off the same lattice site. In this limit p « 1 the distribution function takes the form

G(Rj) = C exp { — E(Rj)/kT} exp {-s ») Z|{exp (fZ;)— 1},

_ where f is just p divided Association of defects separations; since the separations appropriate in ZnS but not in GaP.

(25.1.4)

by the number of unit cells per unit volume. will always be important for very short interaction energy is large. At the larger to pair emission, association is important

25.2. Transition energies

25.2.1. One carrier bound to afinite dipole

This problem is one of the few which are exactly soluble in quantum mechanics. The solution has been given by Wallis, Herman, & Milnes (1960), whose work we now summarize. These authors discuss a specific system with Z = 1, m* = mo, and ¢ = 1, so that

D* = |\e|R for charges separated by R. We shall discuss later the scaling necessary for using their tables for other systems.

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS _ 809

The Schrodinger equation is solved after transformation into elliptical coordinates. Thus, if the carrier is at distances r, and r_ from the two fixed changes we use 1 = (r, +r_)/R,w = (r,—r_)/R, the azimuthal angle @ as coordinates. The wavefunction has the general form

P(A, Hs) = Ln (AM (HWOn(P).

(25.2.1)

The Schrédinger equation separates, and the equations for L and M solved numerically. The eigenfunctions are classified by (n,, ny, m). In the limit of large separations the system resembles a weaklyperturbed donor state with principal quantum number n: n=ny,+nytm+i.

(252.2)

A typical correspondence between the states is shown by (n_,ny,m): (000)

— n = 1; |s-state

(1 00) (0 1 0)

—>n

2; 2s-, 2p-states

M2523)

(00 +1) When using the results of Wallis et al. in practice, the first stage is to find E, the energy eigenvalue. The scaling necessary because Z,m*/mo, and e differ from unity can be done in two simple ways. One is to use systematically lengths measured in the effective Bohr radius and energies measured in the effective Rydberg. The other is to find the eigenvalue for the correct value of D* and then to note that, at constant dipole moment, the product ER? is invariant (strictly, m*ER? is a unique function of D*). Given the eigenvalue, one can then use recursion relations quoted by Wallis et al. to

derive the wavefunctions. It is not difficult to show, for example, that the ground state (000) does indeed reduce to a hydrogenic ls-state at large separations. The convergence of the energy levels on the hydrogenic levels at large separations is shown in Fig. 25.1. The convergence of the ionization energy is much slower than usually assumed, mainly because of the direct interaction of the electron (regarded as a point charge) with the negative ion. In excited states, the spread of the wavefunction can be quite considerable. This can be shown

810

DIPOLAR

SYSTEMS

AND

Pe (200) (100)

33

PAIRS

DONOR-ACCEPTOR ae

- 3s,p,d ’ + 2s,p

20-51-

Oo

Nts

(000)

SlkOle

re

(a)

3

hese

Fak

ed SA Chala Pio acai eS (000)

ras Is

Be

(001)

2 10°

(010) 73s,p.d (200) (b) (10) 10)

=

=}

3 e jos)

=

(002)

10

(011)

ae, dad Uy

yf Wy

A

AY Wee

10s Oks eam 10 Son

De

|

SaeOS

ee ee

+ 2s,p

ed

S

|

Fic. 25.1. Levels of an electron bound to a finite dipole. Energies are in Rydbergs and the separation of the charges +e is in atomic units. The total energy (a) is needed in calculations of the polarizability, and the ionization energy is given in (b). The results of Wallis, Herman, & Milnes are labelled (n,, ny, m), and the limits at infinite separation are indicated.

from the separations, in effective Bohr radii, at which the ionization

energy is 50 per cent of its value at infinite separation: (000)

— (Is)

(1 00) (010) $—+(2s,2p) (00 +1)

R ~ 4a4

R ~ ILat R~ 14a%* R ~ 17 a%

(25.2.4)

By contrast, the wavefunctions converge more rapidly. We may see this from the polarizabilities discussed later. At the largest separations, when the effect of the second fixed charge is merely a weak perturbation, the energies can be found by assuming that the system consists of an extra charge which produces a uniform electric field and a polarizable hydrogenic defect. In this limit one can go beyond the simplest effective-mass theory. O’Dwyer & Nickle (1970) have included the Frohlich coupling in a

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

811

Hartree model. Their calculation is appropriate for defects in ionic crystals, where ¢,, and é are different. If ey and ¢,, were the same, one would estimate the polarization energy in terms of a4, the polarizability of a free hydrogen atom, as follows. The polarization energy isd. E, where |E| = Ze/e)R? is the electric field at separation R. The dipole moment d of the hydrogenic part of the system is aE, where a(é9, €,,)1sits polarizability. Thus the polarization energy varies

as wE*, or «Ze7/e3R*. The R~* dependence gives the leading correction to the energy at large separations. The polarizability can be scaled from the hydrogenic value o,, = 4-5 a.u. by noting

| CN

ie

(25-25)

O’Dwyer & Nickle’s work shows that, when é and «¢,, differ, A(Eq ’ Eo)

Maroy) WE0,8) = Se \(1 + 50/8)

5

Da, eee)

where v = (é9—€,,)/é,,. Their result is not exactly the hydrogenic value when €) = &€,, = 1, but the difference is not important. For GaP the difference between &€,) and «,, leads to a reduction ofa(éo, éo) by a factor of 1-67. In essence, the electric field must change not only the electronic wavefunction but also the corresponding ionic polarization. One example of a single carrier bound to a short dipole is the Fz -centre in MgO (To et al. 1969). This centre consists of an electron bound to adjacent cation and anion vacancies. It is a centre, and its observed properties are the hyperfine constants on the

neighbouring 7>Mg** ions and the energy at which the centre may be optically bleached. The centre seems to be described reasonably well by effective-mass theory, using ¢,, as the dielectric constant and my, as the carrier mass. The system is predicted to have one deep bound state, 2-4eV below the conduction band; other states

may occur within 0-01 eV of the band extremum. This binding energy should be compared with the bleaching efficiency, which rises from zero at about 3 eV to a maximum at 3-6eV. The peak should be at a higher energy than the binding energy because of the increasing density of final states for photoionization away from the band extremum. Comparable results are found for the hyperfine constants. Here, however, one needs the Bloch functions, as mentioned in § 4.2.7 and § 13.5. As a crude approximation, the band

functions were taken to be plane waves orthogonalized to free-ion

812

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

cores. Isotropic hyperfine constants for three sites were given. For the site next to the anion vacancy and opposite the cation vacancy, theory (18-1 gauss) and experiment (17-5 gauss) agree well. For the other sites next to the anion vacancy, theory (9-1 gauss) and experiment ( [@ OJ+ER)

(25.2.8)

in the limit in which the components of the dipole have a large separation R. Such transitions occur in semiconductors with resolvable zero-phonon lines and give a diversity of detailed experimental data. To a first approximation, the initial state may be regarded as a neutral donor some distance from a neutral acceptor. This is valid at separations large compared with the effective Bohr radii. The transition energy is then obtained by considering the cycle of processes | to 4:

ie

[Be

Oh] —[@e]+[C hb],

(25.2.9)

where it is assumed that the interaction of the two neutral systems is negligible;

2. [(@e]+[O h]+FE,+Ep

oy

- ©+0O+e+h;

et+th- Eg,

(25.2.10)

@5241)

814

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

and the final step of returning the ionized donor and acceptor to their initial sites:

4.

O+O > [@ O]te2/eoR.

(25.2.12)

Here, éq is the true static dielectric constant. The total recombination energy is found by summing the equations and gives E(R)

==

Egte?/eoR—E,—Ep.

(25.213)

This equation shows that the transition energy for distant pairs increases as their separation decreases. The observed spectra are so sharp in some cases, e.g. in GaP, that (25.2.13) is not sufficiently accurate. The numerous small corrections must be estimated. Again, it proves easier to keep track of the terms by using the sequence of steps | to 4. The corresponding corrections can be written A,,A,,A3, and A,, and contain the following contributions. 1. Separation of neutral donor and acceptor. (a) Deviation from zero of the interaction of the electron and hole with each other and with the donor and acceptor impurities. The term A,,, includes the effects of the overlap and distortion of the donor and acceptor wavefunctions. (b) Van der Waals correction Avy. (c) Electric multipole contributions Ay,. These are usually derived empirically, and may include parts of A), . (d) Corrections A,, from the wavevector dependence of the dielectric constant. (e) The change in polaron energy A, and the effects of static lattice distortion Ap,. Our sign convention is that A, is positive for attractive donor— acceptor interactions. 2 and 3: Electron—hole recombination and ionization of the separated defects. Here the electron—lattice interaction leads to absorption or emission of phonons. This term is straightforward and will not be discussed further. 4. Returning ionized donor and acceptor to their initial sites. (a) Electric multipole interactions Ayq. (b) Corrections A,, from the wavevector dependence of the dielectric constant. (c) Elastic interactions between the defects Ap,.

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

815

Our sign convention is again that A, is positive for attractive donor— acceptor interactions. Thus, if no phonons are absorbed or omitted

in stages 2 and 3, the recombination energy is given by

E(R) = Eg+e7/egR—E, —Ep—A,+Ag.

(25.2.14)

In some cases it is not possible to be certain which of the contributions is observed. However, many of the terms can be separated from three types of observation. These are the direct spectroscopy of the transitions we have been discussing, the temperature dependence of these transitions, and anomalies in intensities of the lines which reflect the energetics of the capture of carriers by ionized donors and acceptors. Excitons bound to very close neighbour donor-acceptor pairs have been observed in the systems GaP :(Cd, O) and GaP :(Zn, O) (Morgan, Welber, & Bhargava 1968; Henry, Dean, & Cuthbert

1968; Dean, Henry, & Frosch 1968). These have strong central-cell corrections associated with the oxygen, invalidating application of the simpler theories. Remarkably, Dean et al. observe that for nearest neighbours the Coulomb term e?/é9R of 0-55 eV is very close to the observed values of 0-553 eV (Cd,0) and 0-6 eV (Zn,0). However, the energies on this simple model do not agree with observation for other close-neighbour positions. A. Deviation of the electron-hole interaction from the point-charge value. The simplest contributions to A,,; come from the overlap of the donor and acceptor wavefunctions and the associated distortion of their form. These terms have been discussed by Williams (1960) and Mehrkam & Williams (1972) using a simple wavefunction of the form: ¢,(r.)@;(",). This form is only appropriate at rather large separations; at closer donor-acceptor separations the electron and hole become

correlated, as in an exciton, and the assumption

of

independent-particle functions breaks down. Hopfield (1964) has commented on this pronounced change of behaviour from donor— acceptor pair to bound exciton. As the donor—acceptor separation decreases, E(R) increases until, at some critical separation R,, it reaches the free-exciton binding energy. E(R) does not increase at smaller separations, and there may be an abrupt change in the variation of E(R) with R. Hopfield estimates R, ~ 1-41 a*, when the electron and hole masses are equal. Whilst this is almost certainly an overestimate, the productrial function $,¢, is unlikely to be accurate at significantly smaller distances.

816

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

To lowest order, A,, can be found by taking the expectation value , Where # includes the various Coulomb interactions and kinetic energies. The kinetic energy terms can be related to the isolated donor and acceptor energies if @, is taken as an unperturbed donor function and @, is taken as an unperturbed acceptor function. The remaining Coulomb terms can be obtained using the standard methods for molecular integrals. Williams (1960) shows that, for hydrogenic Is functions and equal electron and hole masses, E(R) is reduced by terms which fall to zero exponentially at separations large compared with the Bohr radius. The general features are not altered by modest differences between the electron and hole Bohr radii. Mehrkam & Williams (1972) extended these calculations to allow some modification of d, and ¢,, from wavefunctions appropriate to isolated defects. The product form $,¢, was retained, but the components were generalized to be admixturest of hydrogenic s and p functions: g(r) ~ exp (—r/a*)(1+

60 Z/a*).

(25.215)

The normalization is omitted in this expression. For simplicity, the same effective radii a* are used in both the s and p parts. Functions @, for electrons and @, for holes were taken to be identical, with the same values of a* and o. Further, Mehrkam & Williams argue that a* does not vary significantly with R and that the value appropriate for large separations may be used. Qualitatively, the magnitude of Aj; is increased at small R relative to the value for overlap alone. The sign of the p admixture in the electron and hole wavefunctions is such as to decrease f(r) between the donor and acceptor, relative to the values for pure s functions. Bindemann & Unger (1973) used a different type of trial function, adopting spherically-symmetric functions, centred at variationally determined points along the donor-acceptor axis. These ‘floating’ functions have the advantage that the energy behaves better at very

small separations, where the transition energy should be close to the free-exciton recombination energy. Further, experience for H; shows that the analogue of the Bindemann—Unger calculation (Hurley 1954) gives a significantly better energy than that of the t+ Mehrkam & Williams refer to the variation of one-electron functions as ‘configuration interaction’. This confusing description is unique to their paper. No correlation is implied, contrary to the long-established and standard usage of § 2.1.5.

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIR'S

817

Mehrkam—Williams method (Rosen 1931). As the donor-acceptor separation R increases, the effective Bohr radii a*(R) of the floating functions increase and the centres of the s-like functions move further apart. The separation of the centres exceeds R by an amount which passes through a maximum at R ~ 4a*(co). Both variational calculations agree that corrections to the simplest formulae start to be important when R ~ 2a*(oo) and that the initial corrections are such as to reduce the electron and hole densities in the overlap

regions. Both calculations are very restricted variational functions, and are likely to be accurate over restricted regions of R. They improve only slightly the calculations first given by Hopfield (1964). One obvious omission is the electron-hole correlation, which is clearly important at small R where the pairs resemble an exciton bound to an isoelectronic defect. Mehrkam & Williams offer a primitive interpolation estimate of this term, but clearly better variational functions are desirable. Much more flexible calculations have become available recently through the SEMELE programme of Stoneham and Harker (1975). This programme exploits analogies between donor-acceptor pairs and molecular systems and developments in molecular Hartree— Fock programmes developed over the last few years. The main differences between molecular and semiconductor systems lie in the masses of carriers (m* #4 m* # my in general), in dielectric screening, in the difference of the electron and hole charges, and in the selection rules on matrix elements which appear because of the conduction- and valence-band Bloch functions in the full electron and hole wavefunctions. These changes are incorporated systematically, giving a programme which can use very general wavefunctions and which can treat complicated systems with many more than the two centres involved here. There is another advantage too: the central cell corrections can be included by putting in appropriate core electrons on each site.

B. Van der Waals interaction. Another simple contribution to E(R) is the Van der Waals interaction energy Ayw between the polarizable neutral donor and acceptor. The crucial question for this term 1s dielectric screening, and this is quoted incorrectly in almost all sources. The correct form in a nonpolar crystal, excluding any

818

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

possible local-field corrections, is

Avw = Go jegR”

(25.2.16)

at very large separations (McLachlan 1965; Israelachvili 1972). a, which has dimensions of length, will be discussed shortly. The quadratic dependence on €, is easily understood from the nature of the interaction. An instantaneous dipole on the donor induces a dipole on the acceptor with moment lowered by é9, because of screening. The interaction of the induced and instantaneous dipoles is proportional to the product of the moments (hence ~ éQ ') and is

itself screened, giving é 7 in all. The original discussion of the Van der Waals correction for GaP (Trumbore & Thomas 1965) found that Ayy improved the agreement of theory and experiment. Values of« were of the order ofthe effective Bohr radius, and decreased as expected with increases in E, and Ey. However, Dean & Patrick (1970) cast doubt on the interpretation, arguing that (25.2.16) is a poor representation at small separations and negligible at large separations. A large part of the improvement attributed to Ayw arose from a poor value of ¢). We can verify Dean & Patrick’s conclusions directly using London’s formula (see

Tosi 1964 or Born & Huang 1954). Instead of (e7a°/e3) one has 3a,0, E,E,/(E,+£E,), where a; and E; are the polarizability and ionization energy of defect i. Using the measured polarizabilities of Dean & Patrick (which presumably already include any necessary local-field correction), we predict values of Ayw which are 1-2 per cent (Zn, S), 1-3 per cent (Zn, Se), 2:9 per cent (Cd, S), and 3-3 per cent (Cd, Se) of the values required by Trumbore & Thomas. The

small Van der Waals term is completely negligible for all separations for which (25.2.16) is appropriate. C. Multipole fields. Spectra for donor-acceptor pairs in GaP show more lines than there are equidistant shells (Patrick 1969; Dean & Patrick 1970). The shell substructure has been attributed to there being sets of sites within each shell which are crystallographically inequivalent, e.g. (731) and 553) sets. Patrick considered the extra term in E(R) resulting from the lowest-order multipole corrections: “

Il (BOCE

7

eX cisheny Gt 7

4p:

ha

ot

ellie

(25.2.17)

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS.

819

TABLE 25.2 Multipole terms for donor—acceptor pairs in GaP Type of pair

Type I Type Il

Values

The

Species

k,

GaP:(C,O) GaP: (Zn, O) GaP: (Cd. O)

are taken

from

Dean

k,

(+)24 —24 — 24

~0 19 28

& Patrick

units of k; are meV(nm)*

and

(1970).

of k, are

meV(nm)°.

Parameters are given in Table 25.2 for defects in GaP. The two terms in (25.2.18) suffice to give a very good fit, although some lines seem to be missing; but this discrepancy may be experimental in origin. These are multipole interactions between the ionized donor and acceptor. Patrick argues that k, may be associated with the oxygen ; if so, the associated octopole moment corresponds to charge distributed along the bond directions. The term in k, may be due to distortion of the cation sublattice by the cation Zn or Cd. D. Wavevector dependence of the dielectric constant. Rashba (1971) observed that the static dielectric constant was wavevector-dependent and that this would affect E(R). As shown by Srinivasan (1969), for example, the wavevector dependence is only likely to be important at separations of the order of the lattice parameter, and so it is unlikely that the correction is important in the asymptotic limit which we are considering. The leading term is of the form

A.4(R) =

1S ee. (e eas 2 eee eR R?

X44 Y*+Z*

ree Rt

3 SAVE Hel wk | (

where (e’/e,) is a measure of the anisotropy of the dielectric constant at small wavevectors. Note that the angular dependence is the same as that of the second multipole correction in (25.2.17). E. Lattice polarization: the polaron shift. The ionic polarization changes in the recombination. The energy of the zero-phonon line for the transition is altered by the polaron shift discussed in § 8.6. Duke & Mahan (1965) have given an exact expression for the shift as a function of R. Their result is valid for a hydrogenic donor and acceptor, and includes both deformation-potential and piezoelectric electron—lattice coupling.

820

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

The resulting shift is the sum of a term A,(R) and of the polaron shifts for an isolated donor and acceptor. The contributions of the isolated defects are effectively included in the (Ep+£,) term of (25.2.13), and so we ignore them here. The extra term A,(R) is the sum of three contributions. The first comes from the piezoelectric contribution

to the static dielectric constant;

in effect, e?/eyR of

(25.2.13) is modified. The second term has components which fall off as exp (—2R/a*) or exp (— 2R/ax). They are associated with the spread of the electron and hole charge-distributions, which are measured by the effective Bohr radii a* and a*. Finally, there is a term proportional to 1/R which occurs only when the piezoelectric interaction is unscreened; when there is screening from mobile carriers the term is exponentially damped. F. Lattice distortion. The polaron shift, just discussed, arises from the coupling of the electron and hole to the lattice. Related phenomena have been considered by Dishman & DiDomenico (1971) who have constructed configuration-coordinate diagrams for GaP :(Zn, O) and GaP :(Cd, O). We now turn to terms which do not depend on this coupling but which involve a lattice distortion because of differences in impurity and host radii or because of the charge of the ionized donor or acceptor. Probably the dominant contribution is that given by Hutson (1973 and private communication). The important experimental results are those of Morgan & Maier (1971), who were able to isolate contributions to Ap, from the splitting of Zn and C acceptor levels by the strain fields of O, S, Se, and Te donors in GaP. Hutson argued that the electric field associated with elastic strain in these piezoelectric crystals was most important. The field E is related to the displacement D and strain e by D, =>

EE; + €;je jx

(25.2.19)

For GaP only one component of the couplings is finite (¢,, = é ~ 0-16e A~?). In a region where there are no free charges, V.D = 0,

and we define the potential ¢ by E = — V@. This gives V*o =- : Sere —e,. e Ql axe

292.20 (2222

ijk#

The potential @ can then be found if e is known. The simplest assumption is to assume displacements u = AR/R® appropriate to

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

821

a simple point defect in an elastically-isotropic crystal. This leads to an expression

yy

Ve XYZ 5 “4 Re

252.21)

very similar to the multipole terms discussed earlier in (25.2.17). A quantitative analysis can be done in two ways. One is simply to fit this expression for ¢ (multiplied by the electronic charge) to the observed splittings. The strengths A are then related to the change in crystal volume due to the defects. Hutson shows that the derived volumes are reasonably in agreement with those expected from known ionic radii. The C acceptor and O donor cause lattice contraction, whereas the S, Se, and Te donors appear to cause rather similar expansions. The second treatment is to recognize that corrections to the form of u lead to modifications of ¢ similar to the multipole corrections, and so to fit data for many shells to see if this model can explain the multipole corrections. Since the nextorder multipole has even parity, the modification of importance to u comes not from anisotropic elasticity (which leaves @ odd) but from non-local forces; the displacements due to four radial forces

at the nearest neighbours (instead of point dipoles at the origin) give displacements of both parities, and these lead to even terms in o. If the nearest-neighbour distance is a, the leading corrections are of order

(a/R*); {a(X* + Y*+Z* —2)/R"\ ; (aX? Y*Z?/R?); (even)

(a2X YZ/R"); {a2X YZ(X4+ Y44+Z*—3)/R"}.

(odd).

A provisional fit over available data for Type I spectra gives acceptable agreement for (fitted) values of a between | A and 2 A. Morgan & Maier’s original discussion ignored the piezoelectric terms, using instead deformation-potential coupling only. Whilst this is adequate qualitatively, the inferred parameters (either strengths A or deformation potentials) were much larger than seems plausible. Rashba (1971) included piezoeffects, but concentrated instead on the alteration in the separation of the ionized donor and acceptor from the value in an undistorted crystal. If there is a change in spacing u then there is a contribution u. V(e?/eR) in the energy. Rashba considered the displacements induced by the electric field of the ionized centre through the piezocoupling. The leading

822

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

term has the correct multipole form but is about an order of magnitude less than that observed. All these approaches make simplifications: the use of isotropic elasticity, lack of allowance for the spread of the wavefunctions, and long-range forms of the displacements; but it seems likely that Hutson’s approach has identified the major contributions, and that other sophistications are not of special importance. 25.3. Transition probabilities

We consider now the probability of the transitions

([@ e][O h]) > [@ OJ) +h

Oso)

for donor-acceptor pairs. Three features are important: the absolute rates, the rate as a function of separation, and the rate as a function of the transition energy. 25.3.1. Basic theory Two calculations of the radiative transition probability have been made (Shaffer & Williams 1964; Novotny 1969). There are some technical differences, but the basic assumptions are very similar. Shaffer & Williams use a Heitler-London description. The transition corresponds to an electron leaving the donor state fp(r) to occupy the empty state at the acceptor described by the hole wavefunction wa(r’). Standard time-dependent perturbation theory then gives the transition probability in terms of the dipole matrix element

d = |e {d?r Hp(r)rda(r—R),

(25.3.2)

where R is the separation of the components of the pair. The actual form of the matrix element depends on whether R is a lattice vector and on whether we assume r is a slowly varying operator within effective-mass theory (cf. § 4.2.7). The two wavefunctions @p and o, can be written as

v(t) == u(t) iFFp(t) \ alt) = u(r)F 4(r)

25.3.3)

where F, and Fp are envelope functions and u,, u, are Bloch functions for the band extrema at the zone centre. Since the envelope

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

823

functions are nearly constant over the unit cell,

d ~ |e| ) Fo(r,)Fa(r;—R) |d*ru(r—R)u,(r)r.

(25.3.4)

Let R; be the lattice translation connecting r to the equivalent point in the cell containing the donor. Then, ifr) = r—R, J?

d ~ |e| > Fo(t))Fa(t;—R) *"s J

x

{dor. u(to — R)u(Fo)ro +

eR: idr u(to — R(t).

(25.3.5)

The two integrals over u, and u, are independent of j, so that we may rewrite d in terms of the two overlaps

Se = { d3r Fp(r)F 4(r—R) ,

(25316)

.

(25.3.7)

Sy = |dr u(r —R)u,(r) and two matrix elements of the form

d, = |e| | d3r Fp(r)rF4(r — R)

* dy = lel |d3r u(r — R)ru,(r) giving d =

Sy-dy + Syde.

(25 3.8)

If R is a lattice translation the Sy is zero from the orthogonality and periodic properties of Bloch functions (eqn (2.2.11)). If the variation ofr over the unit cell is neglected then dy vanishes for the same reason. Thus the matrix element can be sensitive to the details of the problem. Novotny uses a different formalism—a second quantization method which is nearly equivalent to the Shaffer—Williams approach. The wavefunction for the spin-zero excited state is written

824

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

in the form

kw P=PYkz Vicky k3 ar VAS) ’ ry

(ki, Ye (k2, + (253.9)

(ky ’ r) >

where the spin functions are suppressed and the w,(k, r) and (k, r) are band functions. There is also a separate factor corresponding to the ‘vacuum’ state, where the valence band is full and the con-

duction band empty. The antisymmetry included explicitly in y leads to an exchange term in the transition energy, not included by Shaffer & Williams. The coefficients c(k,,k,) are given by an equation equivalent to the effective-mass equation. However, Novotny uses an unscreened electron-hole interaction. All other interactions are screened by the same constant ¢ used by Shaffer & Williams. Novotny’s final expressions give transition probabilities for both dipole and quadrupole transitions. The quadrupole transitions are relevant if the Bloch functions u,(r) and u,(r) at the band extrema have the same parity. It is possible to eliminate the band functions using the known oscillator strengths f, and transition energies E, of free excitons. The dipole transition probability for the spin-zero case is

(peat ALDDod &

d

ey. paste)

;

,

(25:3210)

The factor |¢,(0)|* reduces to 1/na? for a hydrogenic exciton, where a, is the appropriate Bohr radius. The factor D(R) gives all the dependence on R, apart from the slight dependence of the transition energy hw. Even for hydrogenic functions, D(R) is complicated. If the envelope function vary as exp (—r/a*) for the electron and exp (—r/af) for the hole then D(R) has the expected variation as exp (—2R/a%), when the electronic radius is much larger than that for the hole, and varies as exp(—2R/a*) in the opposite extreme. Novotny compares his predictions with the results of Colbow (1966) for CdS. Here a* » a*, so the recombination rate can be written approximately as

W, = Wo exp(—2R/a*),

(25.3.11)

where R is the mean separation. With the values m* = 0:2 mo, my = 1-1 mo (an average of the anisotropic mass tensor), ¢ = 9-3,

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS _ 825

andf, = 0-512 x 1073, the values are

Wo (theory)

= 4-7

10° se:

Wo (experiment) = (44+2)x 10% s~!)_

(25.3.12)

The agreement is very satisfactory. The results are altered slightly when the conduction-band minima lie away from the centre of the Brillouin zone, as in silicon. Enck & Honig (1969) have described results in this limit due to Miller & Friedman. Weshall consider silicon explicitly, where there are six conductionband minima at (k,,0,0) and equivalent positions. The minima have anisotropic masses, which give distinct effective Bohr radii a, and a). The results to be given are valid in the limits R > ay,a, > I/ko. The first inequality ensures small overlaps of the donor and acceptor wavefunctions, and the second inequality ensures that the donor wavefunction in reciprocal space has no significant contribution from near the centre of the zone. The transition probability takes the form

W(R) = Wa exp(—2R/a,)}

ee ssins (Ri).

(25:3.13)

i=x,y,z

where W, varies with the effective Bohr radii as (apaja))~ '. The I allow for the anisotropy of the effective mass

lr, = exp [—2(R/a,){(1 + 12a)?— 1}],

(25.3.14)

in which I, is R,/R and « is (a,/a,)? — 1. Note that W(R) depends on the donor Bohr radius, even when the acceptor radius is larger. This is because of the oscillatory terms introduced by the minima being away from the centre of the zone. Enck & Honig (1969) have used this model successfully for analysing data for Si(B,P). They observe that, since the oscillating factors vary very rapidly, it is the average over a cycle of W(R) which is of most interest; the individual oscillations are not resolved experimentally. The detailed expressions for W(R) are usually unnecessary. It is merely necessary to know that W falls off exponentially with the exponent related to the effective Bohr radius of the more extended centre.

826

DIPOLAR

25.3.2.

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

Time dependence of recombination

When a system of donor-acceptor pairs is excited, the luminescence decays after the excitation ceases. The time dependence of the intensity depends on the relative distributions of donors and acceptors and on the dependence of the transition probability on separation. Further, the spectral distribution of luminescence will be time-dependent, since the transition probabilities and energies both vary with donor-acceptor spacing. These effects were observed and analysed by Thomas, Hopfield, & Colbow (1964), and Thomas, Hopfield, & Augustyniak (1965). We begin by considering Q(t), the probability that a given acceptor still has a hole on it after a time t since the excitation ceased. For simplicity the acceptor is assumed to be shallow and the donor to be compact. The appropriate kinetic equations were discussed in § 14.5; they correspond to the case where intersystem transitions dominate (Grant 1971). The probability Q is given by a configuration average of the form

O10) =e fry... fary Ples,..sevexp {Weert (25.3.15) where v is the total volume, N/v = p= is the density of defects, P is the probability of a given configuration, and W is the corresponding transition probability. The result can be simplified by two assumptions. First, the probability P(r,,...,1ry) is assumed to factorize into a number of independent factors p(r,)... p(ry). This assumption should be reasonable at low concentrations, but ignores the restriction that only one defect can occupy a given site. The factor p(r) is essentially the pair-distribution function of the donors relative to the acceptors. The second assumption is that W(r,,..., ry) is just a sum of independent terms W(r,)+ ... +W(ry), one for each donor for a particular acceptor. The function Q(t) is then given by some simple manipulation, since it factorizes in the form

Q(t) malts = -{e r p(r)

W(r)t} wort)

exp { —

,

= [i—— [ar pent — exp (— wi ne

N

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

1 |dr pint exp {— Weel

PAIRS

827

N

exp | —p |d*r p(r)[1 —exp {— weer). (253:16) in the limit of large N. The total intensity J of emitted light is then proportional to dQ/dt

dQ) _ E|d?r ar

p(r)W(r) exp {— woo Q(t).

(25.3.17)

The intensity at a given energy (Eg—E,—Ep+A) can also be calculated if we assume all the emission associated with separation R occurs at the one energy with A = A(R) = e?/eR. Thus phonon sidebands are ignored, and the effects of anisotropy will be dropped. We transform the integral in Q(t) to an integral over A, and examine the contribution to —dQ/dt from the term between A and A+daA. The result is complicated unless A(r), p(r), and W(r) are functions of r only, in which case

(A) = 52Weadplrsvexp {—Wey}O,

253.18)

defining r, through A ~ e?/er,. Thomas, Hopfield, & Colbow and Thomas, Hopfield, & Augustyniak have calculated the total intensity and J(A) numerically for the special case in which p(r) = 1,a random distribution, and W = W,,,, exp(—r/Ro). Very satisfactory fits were obtained for the total intensity and for the changes in the spectrum in time. The parameters W,,,, and Ro also had plausible values. Since a strong longitudinal-optical phonon-assisted band was also present, the spectrum was fitted at one time and its changes with time compared with observation. As the time since excitation increases shift to lower energies the transition energy (larger donor— acceptor separations) and the spectra become sharper. 25.3.3.

Other transitions

The capture mechanisms which precede radiative decay are of interest, both as a preliminary to luminescence and because of effects on mobility (e.g. Reiss et al. 1956). No detailed calculations are available, so a qualitative analysis alone will be given (Dean & Patrick 1970).

828

DIPOLAR

SYSTEMS

AND

DONOR-ACCEPTOR

PAIRS

The dominant capture mode seems to be the capture of one free carrier by one ionized defect, followed by capture of the other free carrier by the other defect, e.g.:

[@ O]+e > [Pe S]+E,

}

[Be O]t+h>[(Ge Oh]+E,

(25.3.19)

These are non-radiative transitions. The factors of importance are the following. First, what one-carrier states are bound? At very small

separations there are no bound states, and capture is presumably by the direct capture of an exciton; at intermediate spacings the details of the bound states matter. Thus, secondly, are any of the energy spacings close to some phonon energy? If so, rapid nonradiative transitions are expected. Also, the appearance of an extra excited state permits new capture modes, so that discontinuous changes in capture cross-section may occur. Presumably, at large spacings, cascade capture dominates (§ 14.4.2). As a ‘rule of thumb’ the capture radius will be about half the donor—acceptor separation (and certainly E, must be positive). Thus the capture cross-section should increase as the square of the donor—acceptor separation for large spacings. DiDomenico, Dishman, & Sinha (1971) have compared the rate of the two-step process (25.3.19) with direct exciton capture:

e Oh] +Es. [ o]+[eh]>[G

(25.3.20)

Their calculations show that the two-step process (25.3.19) dominates at high temperatures. The ratio of the probabilities was found using effective-mass theory, with hydrogenic or Coulombic envelope functions for the hole and an exciton envelope function of the form Ske) fy(k,). The ratio of the two rates was found to be: rate of first stage of (25.3.19)

rate of (25.3.20)

= (rte) sie eer E x

p is the free-hole density, E,, is the acceptor binding energy, a* is the associated effective Bohr radius, and E, is the exciton binding energy. The first of the two factors on the right represents the enhancement associated with the Coulomb wavefunctions.

26 BOUND

EXCITONS

26.1. Introduction

A BOUND exciton is an excited state ofa crystal containing a defect. The concept includes those states which can be described rather accurately as a free exciton localized by weak interaction with the defect. It is a useful concept when the defect merely localizes the exciton without any significant change in exciton character. Like many simplifications, the notion is often used beyond its validity. However, cases have been found in many classes of crystal. Most examples to be discussed are for II-VI, IIJ-V, or group IV semiconductors. Others occur for ionic crystals: the « and B bands of alkali halides (e.g. Schulman & Compton 1962) and the exciton binding to isoelectronic impurities in silver halides (Tsukashi & Kanzaki 1971). Bound excitons can also occur in molecular crystals (Merrifield 1963). These excited crystal states are usually observed in luminescence. One symptom is an emission line narrower than the value of about kTexpected in the recombination of free carriers. However, detailed identifications depend on Zeeman and piezospectroscopy, which exploit the relation between the free and bound excitons. 26.2. Quantum chemistry of the bound exciton 26.2.1. Introduction The details of bound exciton systems vary as much as the many free exciton systems discussed by Knox (1963). To provide a basis for discussing data, we treat the system which proves particularly amenable to analysis (Hopfield 1964). Here the host crystal is a nonpolar isotropic semiconductor for which the simplest effectivemass theory is appropriate. The defect to which the exciton is bound has only minimal effect on the exciton; central-cell corrections are ignored or treated by modifying effective masses. The various possible bound excitons cause problems of notation, so we begin by discussing simpler systems. Our notation includes

830

BOUND

EXCITONS

TABLE 26.1 Bound exciton and related systems DONOR STATES AND THEIR ANTIMORPHS Ionized donor

(@] (@ e] [Bee]

Neutral donor Electron bound to neutral donor

(Antimorph [©]}) ([© h])) ({© hh})

[@eeh]

Exciton bound to neutral donor

([© hhe})

{@ eh]

Exciton bound to ionized donor

([© he})

OTHER

CASES

{eh] [eheh] [© eh]

Free exciton Excitonic molecule Exciton bound to isoelectronic defect

{© eheh] {© hhhe]

Excitonic molecule bound to isoelectronic defect Exciton bound to neutral double-acceptor

explicitly all electrons in the conduction band (e) and holes in the valence bands (h), bound or free. The defect involved is then described by its charge: an ionized donor by @, an isoelectric trap by ©, and an ionized acceptor by ©. In each of these cases all covalent

bonds are considered complete and any free or bound carriers are shown separately. Groups of particles bound together will be shown bracketed thus: [X YZ]. The cases we shall discuss are given in Table 26.1. Transitions occur among the various configurations, and the energies involved are defined by an equation analogous to thermochemical equations: [X YZ] + Eyyz)> [X]+L[YZ].

(26.2.1)

26.2.2. Basic transition energies

The host crystal we consider has effective masses m* and mf, a static dielectric constant €), and negligible electron—lattice coupling. We define o = (m*/mx). This dimensionless parameter determines many of the bound-exciton properties. The donor binding energy is

Ep

fe (m¢/mo)

£6

Ry.

(26.2.2)

We shall express all energies in terms of Ep. Thus the acceptor binding energy is

Ee, = (tlio) py £6 yr

(26.2.3)

BOUND

EXCITONS

831

Similarly the free-exciton binding energy G, defined by

[eh] + G > [e]+[h], has the value

G = Ep/(1 +0).

(26.2.4)

The exciton energy E, is given by

En = Ee G,

(26.2.5)

where Eg is the bandgap. We now consider energies of more complicated systems, derive expressions of the form E; ~ Epf(o) to compare observation. Of course, the use of Ep as a reference energy introduce some asymmetry in the results for electron and systems.

and with does hole

26.2.3. Electron bound to neutral donor and its antimorph The configuration [@ ee] and its antimorph [© hh] are qualitatively similar to the negative hydrogen ion H . The binding energy of the extra electron in H™ is 0-75 eV (Bethe & Salpeter 1957). Thus,

if the screening of the carrier—carrier interaction is the same as that of the electron—donor or hole—acceptor interaction, we may scale the results for H™:

E(@ eye - Evo nh

Ep

Ex

These ratios do not depend on o = mé/m*. However, since E,/Ep depends on the mass ratio, the energies can be written

Eveye= 0-055 Ep

(26.2.6)

Evonn = 0-055 Ep/o.

(26.2.7)

These defects have not been observed directly in semiconductors,

although their effects can be inferred. Thus, Dean, Haynes, & Flood

832

BOUND

EXCITONS

(1967) discuss the radiative capture of holes at [@ ee] systems in silicon. These systems should influence transport properties by providing charge-trapping states, and may be intermediaries in the formation of bound exciton states. The systems [@ eee] and [© hhh], with yet another carrier bound, are not stable with just the Coulomb interaction. The result (Levy— Leblond 1971) occurs because the carrier furthest from the defect always has a positive potential energy ; the result is independent of the Pauli exclusion principle and of the mass ratios.

26.2.4. Exciton bound to ionized donor, and its antimorph The exciton bound to an ionized donor [@ eh] is one of the simpler three-particle systems. As such, it has received more detailed attention than other systems; data are available giving binding energies, oscillator strengths, and measures of correlation. A. Mode of decay. We begin by showing that the defect is unstable against loss of the hole, rather than loss of the exciton. To see this, consider the transitions leading to three separate components

[@®], le], and [h]. This decay can be accomplished in two ways. The first is loss of the exciton and its subsequent decay: (® eh] + Egcen) > [©] + [eh]

[eh] + G — [e]+[h] or, in total, [® eh] + Egeny

+ G > [@]+[e]+ [h].

The second mode involves loss of the hole, followed by loss of the electron,

[® eh] + Evo en > [@ €] + [h] [@ e]+ Ep > [@]+[e],

so that, in total,

[® eh]+ E.o@entEp > [(®]+ [e] + [h]. Comparing the two sequences, we see Ev@en = E@ceny t+ (G— Ep) =

Eocen)——

| Eo:

(26.2.8)

BOUND

EXCITONS

833

using (26.2.4). The second term in the equation is always negative, so that E(e.), 18 always less than Eg m* when o > 1. The electron is then strongly localized near the donor, and the hole spread over a more distant region in which the field (varying as r-* at large distances) is too weak to trap the light hole. For this system there is a critical value of o such that the exciton bound to an ionized donor is stable if o < o,,;, and unstable if ¢ > o,,;, (Levy-Leblond (1969) has shown that there are no other disjoint ranges of o which lead to a stable system). Values of o,,;, can be obtained in several ways. The earliest (Lampert 1958) exploits an analogy with the Hj molecular ion. Since it neglects certain dynamical effects—the electron is assumed to follow the hole adiabatically—an approximate upper bound results, ert

X exp {—(Are+ Biren + Ciry)} + alle, tn),

(26.2.12)

i=1

where the extra function

exp (—Cr,)— exp (— a eta.)

=

Xne

em

(—rl00)

; h

(26.2.13) represents a hole loosely bound to a donor with effective Bohr radius dp. This second term is particularly important when the binding is weak. The variational parameters are the X;, 4;, B;,C;, C’, and C”; usually, p= 2 and N = 4 were chosen. The envelope function is normalized so that

(26.2.14)

VU Cemle= 2

The binding energy in the ground state E,o,,, is shown in Fig. 26.1 as a function of a. The binding energy is a monotonic function

° als

as i In (E(@e)n/Ep)

—)

0

4

0-2

= et

0-4

6

Fic. 26.1. Binding energy of [@ eh] as a function of c. Results are from Skettrup et al. Energies are in units Ep. The limit ¢ = 0 corresponds closely to the Hz molecule.

BOUND

EXCITONS

835

of o (Levy-Leblond 1969), falling to zero at o equal to o,,,,. At the other extreme, when a is zero, the results can be related to the inl

eigenvalues. The two energies Egy) and Eoem) are then equal to 0-20 Ep. The oscillator strength of the exciton can be extremely large. It can be written in terms of the free-exciton properties: the freeexciton Bohr radius a,,, transition energy E,.,, and oscillator strength f.,.

=f. =

(26.2.15)

Ie (R, R)

Bex fieen}

ae a

Q. is the volume of the unit cell and E’ is the recombination energy of the bound exciton. Matrix elements involving the Bloch functions are contained in f,,. The factor |)’ f(R, R)|’, involving the envelope function, is known as the enhancement factor. It is unity when the complex is unstable and equivalent to a donor plus a free hole. However, when mf > m, it becomes very large, as shown in Fig. 26.2. The oscillator strength can be 10* times larger than the value Sex for a free exciton. Andreev & Sugakov (1972) have also discussed asymptotic expressions for the oscillator strength.

Relative oscillator strength 0

0-2

04

oa

Fic. 26.2. Enhancement of the [@ eh] oscillator strength. Results of Skettrup et al. give oscillator strengths in units of the integrated strength of the recombination of a free hole with a bound electron.

The electron and hole spins can be parallel (triplet exciton) or anti-parallel (the singlet exciton, observed optically). These two states are separated by the exchange splitting A. If J is the exchange integral for conduction and valence band Wannier functions in the same cell, then

Nea

If(R, R)|?. R

(26.2.16)

BOUND

836

EXCITONS

A correction for exciton—exciton interactions has been omitted. The correction gives an extra splitting into longitudinal and transverse states ;it is usually negligible. Finally, we mention the mean interparticle distances as a function of c. These give a measure of the correlations in the motion. When oc is small the electron is at roughly equal distances from the donor and the hole, and the donor and hole are further apart ea

PAO RIS Ss

(26.2.17)

When a is large, the hole lies far from the electron and donor, which are close to each other: CE

OUSICT

ee Sa ES

(26.2.18)

Clearly there is little point in calling the complex [@ eh] a bound exciton in the limit o > 1. The system does not resemble an exciton unaltered apart from weak binding. Skettrup et al. obtain the results for CdS and ZnO, shown in Table 26.2. There are no adjustable parameters, although there is

TABLE 26.2 Exciton bound to ionized donor (after Skettrup et al. 1971) CdS (¢ = 0-20) Observable

E’-E,, (change inenergy from free exciton

ZnO (o = 0-12)

Unit Theory

Experiment

Theory

Experiment

10°*eV

64

4-6 to 7-10

13-4

11-3 17:0

10°2eV

0-49

0:31

0-83

0-9



10x 105

large

0-69 x 10°

large

transition)

Exchange splitting A

20:9

(any concentrationdependent correction would reduce this by about 10 per cent at

107 '° per cm?) S(+envfex (exciton A band (?))

some ambiguity in the experimental values used. This comes partly from uncertainties in the data, and partly from the choice of observables. If there were no simplifications in the theory the choice would be immaterial, but the final results may be affected because

BOUND

EXCITONS

837

different terms in the theory are affected differently by the approximations. Nevertheless, the results are in rather good accord with experiment. D. Variational calculations with electron-iattice coupling. Several authors (Schroder & Birman 1970; Elkomoss 1971; Mahler & Schroder 1971) have considered modifications of the simple ¢, screening of the electron—donor, hole-donor, and electron-hole interactions. All use approaches related to the Toyozawa—Haken-— Schottky model of § 8.6.5. The most important point arising is that the stability condition is no longer just a function of a, as in (26.2.10); a more complicated condition involving the electron—lattice coupling is needed. Elkomoss observes that large values of é/e,, favour binding. The corrections to the screening are written by replacing «9! by an expression of the form: ot

= &o ‘+(e! —&

')o{ exp (—x/a)+ exp (—x/b)}.

Here x is the separation of two components (e-h, e-@, or h-@) and a, b are polaron radii appropriate to these components: (h?/2m* x

x hw )* for the electron, (h?/2m*hw,)* for the hole, and oo for the donor. In addition, Mahler & Schroder corrected the carrier masses

in the kinetic-energy terms for polaron effects in the region in which their separation was greater than the polaron radii. At smaller separations there is some cancellation of the lattice polarization effects of the electron and hole. The most detailed discussion is that of Elkomoss (1971a), who performed a formidable variational calculation. The trial function was the product of two terms. One depended exponentially on the three interparticle separations; the other was an expansion in Laguerre polynomials whose arguments were perimetric linear combinations of the separations. As a simplification, average effective dielectric constants were used, being expectation values of €or, Over the exponential part only of the trial function. Elkomoss and Mahler & Schroder all emphasize that their predictions are very sensitive to the electron—lattice coupling, and both papers claim good agreement with experiment for a wide range of crystals. Elkomoss gives a particularly careful discussion of the host-lattice data used.

838

BOUND

EXCITONS

26.2.5. Exciton bound to a neutral donor, and its antimorph We now consider the two systems [@ eeh] and [© hhe] with three bound carriers. In the limit of a large hole mass (o = m*/m* — 0), the exciton bound toa neutral donor resembles the H, molecule. The comparison can only be made quantitative if all the interactions are screened by the same amount, usually an acceptable approximation. The dissociation energy of H, is 45eV, so we estimate that when a is zero

AS

EM etl ey

(26.2.19)

The variations of this ratio with o near o = 0 can also be found from the known isotope dependence of the H, dissociation energy. In the other extreme (o = m*/m* — oo) the hole will have a large orbit. We can relate the exciton binding energy to other energies by the sequence [®

eeh] + E\ @eeyn

[® c€]+

Fe L®

ee] +h;

Ev@ecre aa [® elare;

e+h—G - [eh].

In total, the exciton binding energy is given by: Ev@ey(eny = (E(@eeyn — G) +E(@ee-

The first term is negligible when m* > m*: in both parts, the binding of a free exciton and in the binding of a hole to [@ ee], we have a hole in a large orbit attached to a massive negative charge. Thus in this limit we may use the analogue between [@ ee] and the H~ ion, giving E(@eyeh)

ae

Ev@ec)e

ae

0-055.

(26.2.20)

However, Munschy (1972) has argued that [@ eeh] is unstable against loss of an exciton whenever o exceeds a critical value less than 2.

A. Variational calculations. Variational results and other bounds without electron-lattice coupling have been given by Sharma & Rodriguez (1967), Bednarek & Adamowski (1972), and Munschy

BOUND

EXCITONS

839

(1967, 1972). These calculations for [@ eeh] are less detailed than those for [@ eh], and no one treatment is best for all ranges of a.

Sharma & Rodriguez note that the dissociation energy need not vary monotonically with o. The energy expectation value (in units of Ep) varies monotonically with o, but the dissociation energy is the difference of two monotonic functions and need not be monotonic. Bednarek & Adamowski discuss the binding of two excitons with different hole masses: the [® eeh] system corresponds to o, = a, o, = 0. They note that, if W is the dissociation energy of a biexciton with equal hole masses and o = (o, +4,)/2, then the binding energy for unequal masses is W(c,,0,), where W(o,,02)

< W+(1+o,)>'+(1+o,)°'—2/{1+4(6,+6,)}.

Results from the different calculations are summarized in Fig. 26.3.

The behaviour of E,.)en) is still not monotonic in a, but the deviations are in the range of larger o where the calculations are least accurate.

Binding energy

Fic. 26.3. Binding of an exciton to a neutral donor. The results shown are in units Ep and are those of Bednarek Rodriguez (SR).

& Adamowski

(BA),

Munschy

(M), and

Sharma

&

840

BOUND

EXCITONS

The binding in the special case o = 1 can be deduced from Houston & Drachman’s (1973) calculations on the positronium hydride system. Their very detailed calculations give a value 0-0497 for the binding in units of the donor energy, a result inconsistent with the stronger binding obtained by Sharma & Rodriguez. B. Haynes rule. Exciton binding to neutral donors and acceptors has been observed in many systems and shows systematic trends known as Haynes rule. In its simplest form, if E,@eyenm/Ep = f(a) and E,onyen/Ea = f(1/c) are measured for a variety of defects in a given host or for a variety of hosts with similar ratios mf/mx, then the values of f(a) deduced are essentially the same in all cases. The remarkable feature is that the exciton binding energy and donor or acceptor binding energy remain proportional even when there are substantial central-cell corrections; the results seem valid beyond the model. Examples are shown in Fig. 26.4 for II-VI compounds and for silicon. For the II-VI compounds o ~ 0-25 is typical, and the results suggest (0-25) ~ 0-2, f(4) ~ 0-1, consistent with the theory.

Cug”

He A

=z 0-050

Abepou4

83

oo

2 0:020F 20

in

as

:

= 0-010

ee

oO

Ag

Te. Ag 4,7 Re\

Pew

0-005-

seit

A ZnTe acceptors

Al

@ ZnSe acceptors

aid

v CdTe acceptors

7

@ CdS acceptors @ CdSe acceptors

yy 00g”

02

x Si donors

+ Si acceptors

4%, Se

|

005

J

0-1

ae

0:2

1

0:5

a=

4

1:0

Ep (eV)

Fic. 26.4. Haynes rule. The dotted line shows binding energies of excitons to neutral donors or acceptors of 0-1 of Ep or E,, as appropriate. Only precisely identified defects are labelled.

However, the simple direct proportionality fails for several IIJ]-V hosts like GaP (Dean et al. 1971; Dean 1973), GaAs (White et al. 1973) and InP (White, Dean, Taylor, & Clarke 1972). Linear relations

between Ep and Evgeyen) OF E,onyen are still found, although in

BOUND

EXCITONS

841

exceptional cases like InP: Zn and InP: Cd the energies do not even increase together (White et al. 1972). The proposed explanations all invoke central-cell corrections, and their content is equivalent to the assumption that the donor energy can be written as

Ep = Ep+Vopen and the exciton binding energy as E(@e\eh) = E(eeyen) + VPcx:

Here energies with superscript 0 are from effective-mass theory alone, V, is a measure of the central-cell perturbation, and pep, Pex are the charge densities in the central cells. The observed energies can now be related in a variety of equivalent ways to their effectivemass equivalents. Thus, eliminating V,, we find

0 Ev@eyien) Pcx E(+eyeh) = Ep(Pcx/Pcp) + Ep Pia Reed D

Pcp

For a given host and relatively weak central-cell corrections (so that pcx and Pcp are not altered greatly from their effective-mass values) this expression has the form desired

E(@eyen) = AdEpt+ Bp,

(26.2.21)

where A and B are independent of the impurity. The values of the parameters for acceptors in the same host (A,, B,) will be different, of course, since they are functions of |/o rather than o. For GaP it appears that Bp > 0 and By, < 0. Despite the general plausibility of (26.2.21), it is not clear that the details are correct, nor is it under-

stood why the central-cell terms give large values of Bp in some cases but not in others. Programmes like SEMELE (§ 25.2.2A) should be able to resolve the central-cell questions. But, in any case, the rule

still provides a very useful aid in identification of centres. C. Oscillator strengths. The oscillator strength of excitons bound to neutral donors can be large, as in the [@ eh] systems considered earlier. In both cases the oscillator strength is essentially the freeexciton oscillator strength per unit cell multiplied by the number of unit cells over which the bound exciton extends. One major difference is that the non-radiative Auger processes ($14.4) may dominate by 10? to 10* for [@ eeh] and [© hhe] systems. Henry & Nassau (1970) have given a detailed discussion of the oscillator strength of an exciton bound to a neutral donor in CdS, and Hwang

842

BOUND

EXCITONS

& Dawson (1972) cite a similar analysis for GaAs. Their treatment leads to an expression analogous to (26.2.15) but with an altered amplification factor. The complex is treated exactly like an exciton bound to a neutral

donor. Thus the envelope function contains a product of three factors: one for the donor electron ¢,(r,), one for the relative motion of the electron and hole of the exciton @,,(r. —r,), and one describing the motion ofthe exciton as a whole F,,(R = (or, +1)/(o + 1)). There is a slight increase in complexity since there are now two electrons, and the wavefunction must be antisymmetrized. If the two terms in the antisymmetrized wavefunction have only negligible overlap then some simple approximations lead to the expression

f= fazQ. ||PR FalR)

2

(26.2.22)

for the oscillator strength of the singlet transition. The amplification factor can be estimated for simple model systems, including an exciton bound to a donor-acceptor pair (when the oscillator strength depends on the donor-acceptor separation) and an exciton bound by a delta-function potential with binding energy

E,. If A* = h*/{2(m* +m*)E,} then, in this case,

ipa pleaa KOap

(26.2.23)

which can be large for weak binding. 26.2.6. Other systems

We have now considered the more important systems. Two others are also relevant: double-donor or acceptor states and the bound excitonic molecule; both are four-carrier systems. The double-donor or acceptor states are especially interesting because of the way in which the valence-band degeneracy enters (Hopfield 1964). Thus a neutral double-donor[ + + ee]is analogous to the He atom; an extra electron can only be weakly bound because the Is-like shell is full. However, a hole can be strongly bound to a neutral acceptor [ — — hhh] because of the extra states associated with the valence band degeneracy. Further, exciton binding to a neutral double-acceptor should be stronger than to a neutral acceptor, since the correlation ofthe extra hole with the other carriers

allows it to be only partly shielded from the attractive nuclear charge.

BOUND

EXCITONS

843

The binding of two free excitons to form an excitonic molecule has been discussed

by Sharma

(1968), Adamowski,

Bednarek,

&

Suffczynski (1971, 1972), Akimoto & Hanamura (1972), Huang & Schroder (1972), and Bobrysheva, Miglei, & Shmiglyuk (1972). The binding energy, in units of the exciton Rydberg, increases monotonically with o and has a second derivative with respect to o which is never positive. Whilst there are no calculations of very high accuracy, the different approaches agree that Ejeyyen) 18 Of order 0-1 Ep for o = 1. This result is relevant for GaP, where Merz, Faulkner, & Dean

(1969) have observed

the binding of a second

exciton to the isoelectronic defect GaP:N. The first exciton is bound by 11 meV, and the second by 9 meV. The binding of the second exciton will come partly from the binding between the two excitons (about 5 meV) and partly from the direct attractive interaction of the extra carriers with the isoelectronic trap. There seems no reason to invoke, like Merz et al., special effects of the electron— phonon coupling based on Phillip’s suggestions (§ 24.2.2).

26.3. Other properties of bound-exciton systems We now consider properties for which the quantum chemistry approach needs extending. The most important factors are the effects of electron—lattice coupling and effects which depend on the detailed band structure of the host lattice. 26.3.1.

The Jahn-Teller effect

In most host lattices of interest to us the hole states are derived from the p-like states at the top of the valence band. If there is significant spin-orbit splitting, we may concentrate on the four I’, levels which lie highest in energy. Morgan (1970) (see also Onton & Morgan 1970) has observed that the Jahn-Teller coupling within these states affects both free- and bound-exciton properties. He has examined

the effect for the system GaP:Bi,

observe both free- and bound-exciton

where

one may

properties. There are two

bound-exciton lines: the A line, a dipole-allowed transition, and the dipole-forbidden B line, which becomes important at helium

temperatures because the initial state of the transition lies lower in energy. The exciton levels are obtained by considering the states of an electron in the conduction band (S, = 4, L = 0) and the hole in the IT’, valence band. The eight levels are split into two groups by the

844

BOUND

EXCITONS

exchange splitting: J = 1 (state |A>) and J = 2 (state |B>). The Jahn-Teller coupling has two effects. It may mix the J = | and J = 2 levels, but, more important, it can give terms equivalent to

J4+J}+J%, which split the J = 2 level into a doubly-degenerate

level I’; and a triply-degenerate level I',. Morgan argues first that the Jahn-Teller coupling is strong. One reason is that there are fairly strong and broad phonon-assisted bands, a symptom of strong electron—lattice coupling. Another is that he shows the ratio of the A-line and B-line oscillator strengths, including sidebands, should be given by

(eae

(26.3.1)

in the limit of strong coupling to tetragonal or trigonal modes. This is very close to the observed ratio. In the absence of electron— lattice coupling,f, is zero. Further, a consistent set of Ham reduction factors can be obtained from comparing bound- and free-exciton properties. The properties include the deformation potentials for tetragonal and trigonal strains, the anisotropy ratio of the stressinduced valence-band splittings, the K and L parameters in the gfactor of the hole (where the sign of the anisotropy of the Zeeman interaction depends on the relative coupling strengths), and the ratio of the crystal-field splitting of the B line to the separation of the A and B lines. The ratio of tetragonal to trigonal deformation potentials is reduced by a factor of 2:5, and the spin-orbit coupling by a factor of about 3 from free-exciton values. The deformation potentials may also be affected by factors other than the Jahn-Teller effects, such as local changes in elastic constants and the compact nature of the wavefunctions. The Jahn-Teller coupling seems consistent with a relaxation energy of about 0:1 eV. It may be much smaller in other boundexciton systems; for GaP:Nf, is about 100f,, and for ZnTe:O fy is about 30f,. 26.3.2. Optical line-shapes

We now consider how electron—lattice coupling affects the shapes of optical transitions. In general terms the results were given in chapter 10: there is a zero-phonon line and there are phononassisted transitions. If the coupling is not too strong, there will be a zero-phonon line and phonon-assisted transitions with structure corresponding to critical points in the phonon spectrum.

BOUND

EXCITONS

845

The intensity of the zero-phonon line involves different factors when the bound exciton is indirect rather than direct. If the host has an indirect gap, then the recombination requires conservation of quasi-momentum. For free excitons this is achieved by absorption or emission of an appropriate momentum-conserving phonon, and a zero-phonon line is not observed. For bound excitons, the coupling to the defect provides another mechanism, so transitions involving no momentum-conserving phonons are also seen. One expects this zero-phonon line to become more important as the binding to the defect increases. This is observed for excitons bound to donors and acceptors in silicon (Dean 1969), and is shown in Fig. 26.5. Transitions involving other phonons will also vary in importance with the binding, through the effect of the binding on the wavefunction. 4

In

No

radiation honon

radiation One-phonon

1

0-005

0-010

——

0-015

os

0-020

E @ eyen(meV) Fic. 26.5. Zero-phonon line intensity and dissociation energies. The ratio of the zero-phonon and one-phonon intensities varies systematically with E.g (en). Results are for Si. (After Dean.)

In special circumstances the line-shape for phonon-assisted transitions of a direct exciton can be predicted exactly (Hopfield 1962: Duke & Mahan 1965, § VII), as for an exciton bound to a neutral defect with linear electron—lattice coupling. Both papers include the unscreened

piezoelectric interaction; Duke

& Mahan

also include deformation-potential coupling. The host lattice was assumed to have acoustic modes, with an isotropic Debye spectrum, and non-degenerate isotropic valence and conduction bands. Two

846

BOUND

EXCITONS

special assumptions which are probably more restrictive to the application of the results also prove necessary. First, the Coulomb correlation of the electron and hole in the exciton is ignored; the system is considered as an electron and a hole independently bound to the neutral defect. Secondly, the deformation-potential constant for the hole D, is assumed equal in magnitude but opposite in sign to that for the electron

Doe)

(26.3.2)

The main features of the line-shape are a zero-phonon line, and wings whose form is sensitive both to m¥/m* and to the relative strengths of deformation-potential and piezoelectric coupling. The coupling also causes polaron shifts of the energy levels; the shift is not just the sum of electron and hole contributions but contains

an extra cross-term. Hopfield’s calculation shows good agreement for CdS, correctly predicting a zero-phonon line and asymmetric wings on either side. The shapes of these phonon-assisted wings agree well in general shape and in their trends with temperature, but this may be partly fortuitous, since Duke & Mahan have shown that the deformation-

potential coupling causes significant changes. Duke & Mahan also quote results for CdTe. One notable feature is that the fractional intensity of the zero-phonon line varies exponentially with temperature overa much wider rangethanexpected,namelyfor T => 15 K. Henry & Hopfield (1972) have observed interference effects in the radiative recombination

1® eeh] > [@e]+hv of excitons to neutral donors in CdS and CdSe. The phonon replica associated with longitudinal-optic modes of the host is roughly Lorentzian, but contains a sharp notch near the centre of the line, corresponding to the local modes of § 8.6.6. The interference results from the couplings of three classes of state with the same energy: the donor in its ground state in the presence ofa host-optic phonon or ofa local-mode phonon and the donor in an ionized state without phonons present. 26.3.3.

Undulation spectra

Undulation spectra are observed in GaP containing neutral acceptors and isoelectronic nitrogen. The structure is observed in

BOUND

EXCITONS

847

luminescence at energies below that of the GaP:N zero-phonon line. It appears to be a consequence of the conduction-band minima being at points Kg away from the centre of the Brillouin zone (Hopfield, Kukimoto, & Dean 1971). The spectra are associated with an exciton bound to both the isoelectronic nitrogen (binding energy Ey ~ 11 x 10~* eV in isolation) and to a neutral acceptor (binding energy roughly Ey/2 in isolation). If R is the separation of the two defects, and if b = ./(h?/m*.En) is a measure of the scale of the wavefunction, then the binding energy of the complex is of the form

E.o onyen(R) ~ Ex+C exp (—2R/b).

(26.3.3)

We want to know also how the emission intensity varies with R, so that we may relate transition energy and intensity. The emission is determined by the k ~ 0 part of the wavefunction. The amplitude of this component has the form

A’ + B' exp (iKy . R),

(26.3.4)

where A’ and B’ vary in space about as fast as the envelope function. For a free exciton the amplitude (26.3.4) is zero. The intensity of the emission line will vary as A” + B” cos (K,.R+¢) or, averaged over angles,

sin (Ky. R+¢)

I(R) ~ A+B

(26.3.5)

[Kol RI

Ifthe spectrum from defects with various separations R is unresolved, then there will be peaks in intensity separated by changes in KoR of 2n. But Kg is 27/a, where ais the lattice parameter, so the peaks occur at intervals of ain R. The transition energies should form a sequence separated from the energy appropriate to isolated nitrogen alone by

AE ~ C exp { —(2a/b)n} = C exp { —an}.

(26.3.6)

For GaP a is 5-45 A and b is typically 40 A, so « should be of order 0-27. Good fits are obtained for various species of acceptor with the values: GaP:N,C

a = 0-204

(

)

GaP oNeMe

@=

1

)

GaP:N,Zn

a=0-24

(0-645);

GaPcNP Cadi

ria

x

022,

.0:262

)

848

BOUND

EXCITONS

The bracketed results are the binding energies of an exciton to the corresponding neutral acceptor in units of Ey. As one would expect, a and Een) increase together. Morgan, Lorentz, & Onton (1972) have disputed the theory of Hopfield et al. mainly on the grounds that undulations are expected but not observed in absorption, that the amplitude of the undulations predicted is far too small, and that the relative intensities of the J =1 and J = 2 lines are affected in the wrong way. They propose instead that the undulations are a manifestation of a dynamic Jahn-Teller effect, and correspond to Jahn-Teller energy levels

(h?/21I{(j+4)?-1}

with j = 4,3,....

The

rotational

constant

(h?/2I) can be related to deformation potentials and to the effective

Bohr radius of the acceptor. The origin of the undulations is still controversial at the time of writing, and other explanations have been given, involving the discrete lattice structure.

26.4. Response to external fields 26.4.1. Zeeman effect

The bound-exciton description of defects is only useful when the state concerned resembles a free exciton localized but only weakly perturbed by a defect. The response to external fields, as reflected in both optical selection rules and g-factors, is primarily a measure of host-crystal properties. The localizing potential may render some transitions weakly allowed instead of forbidden, and may lead to some extra splittings with small g-factors, but can be negligible in its effect. Since the host’s band structure is so important, there are important differences between cases like CdS and like GaP. The wurtzitestructure CdS has band extrema at the centre of the zone, and the

degeneracy of its valence is lifted by the combination of spin-orbit coupling and the axial crystal field. By contrast, GaP is more complicated: the valence-band degeneracy is only partly lifted by the spin-orbit coupling, and there are three conduction-band minima in (100) directions which can give rise to valley-orbit splittings of electron states. The increase in complexity is shown by Table 26.3, which gives the degeneracies of the various configurations ofinterest in transitions such as [® eh] o @+hv

[® eeh] o[@e]+hv

BOUND

EXCITONS

TABLE

849

26.3

Degeneracies of states in bound-exciton transitions System

Cds GaP GaP! GaP"

[@ e]

[9 h]

[® eh]

[© eh]

[®eeh]

[© hhe]

2 6 12 ;

2 4 4 4

4 24 48 8

4 24 48 t

2 60 264 4

2 36 72 t

+ Depends on the details of the central-cell correction. The degeneracies quoted include spin degeneracy. The three models for GaP are (a) the usual model of three conduction-band minima at the (100) zone boundaries, (b) GaP', which assumes six minima in (100) directions close to the zone boundary, and (c) GaP" where all valley-orbit states but one are ignored. The CdS valence band is assumed to have only Kramers degeneracy. Table from Thomas, Gershenzon, & Hopfield (1963).

and their antimorphs. Clearly results for CdS present only simple problems, but a detailed theory is impractical for GaP. The theory of the Zeeman effect for bound excitons in CdS has been discussed by Thomas & Hopfield (1962), using the principles outlined in § 12.3. Bound excitons in GaP lead to more subtle problems (Thomas, Gershenzon, & Hopfield 1963 ; Yafet 1963), which we shall consider here.

& Thomas

The problems in GaP stem from the conduction-band structure and the associated valley-orbit splitting. Simplifications can occur for several reasons, and allow one to concentrate on the totally symmetric valley-orbit state. First, in the optical transitions ®+hv-[@

eh]

we expect only the totally-symmetric valley-orbit state to have a reasonable transition probability, since it is the only valley-orbit state with an appreciable admixture of k = 0 terms in the wavefunction. Second, when we consider the Zeeman effect in oneelectron systems like [@® e] or [@ eh], the orbital contribution to the electron g-factor is zero. This is true when the valley-orbit splitting is large, leaving an orbitally non-degenerate state lowest. It is also true when there is no valley-orbit splitting, for the matrix elements of BH. L between the different degenerate states are reduced by the overlaps of the Bloch functions for the different minima (Dean, Faulkner, & Kimura 1970); this is an electronic analogue of the Ham

effect. The arguments are less straightforward for other systems like

850

BOUND

EXCITONS

[® eeh] and excitons bound to ionized or neutral acceptors. For [@ eeh] it is important to know if the two electrons prefer to be in different valleys (favoured marginally by their mutual Coulomb interaction for anisotropic valleys) or the same valley (favoured by an attractive central-cell correction). It is likely that the central-cell correction dominates, and that only the totally-symmetric valleyorbit state is important in the transitions of interest for both absorption and emission. Results for acceptor centres are not clear because the sign of the central-cell correction is not obvious, and simplification is not possible.

26.4.2. Response to stress Again the principles involved are essentially those discussed in chapter 12. However, two special aspects have been considered. P. T. Bailey (1970) has discussed the stress response of boundexciton systems in direct-gap zincblende-structure hosts like GaAs. He observed that, whilst the antimorphs [@ eh] and [© he] show similar response to stress, the antimorphs [@ eeh] and [© hhe] show different splitting patterns. This occurs because there are two identical carriers subject to the exclusion principle and because of the differences in conduction and valence band structures. In the second study, Morgan & Morgan(1970) have considered the response to stress of an exciton bound to a donor-acceptor pair, where the defect lowers the symmetry. White, Dean, Taylor, & Clarke (1971) have observed the effects of random internal strains on excitons bound to neutral acceptors in InP.

P| VACANCIES

IN VALENCE

CRYSTALS

27.1. Introduction

THERE is no adequate theory of the defects discussed in this chapter. This is not just a question of detail. It is essential to realize that many of the assumptions which one would like to make break down. The centres we consider are the various charge states of vacancies in diamond, silicon, and germanium and in related crystals like SiC. Far and away the best experimental results are those for silicon, although

most

theoretical

work

has

been

on

diamond.

The divacancy has been observed in silicon, and probably in diamond, germanium, and silicon carbide. Whilst many of the points raised here also apply to the divacancy, we shall not discuss it in detail. The main problems can be understood from the simplest model of the neutral vacancy V°. In this picture, the atoms of the perfect crystal form sp? hybrids, which give directed bonds. When an atom is removed, four bonds are broken. The electrons in the four sp?

hybrids on the neighbours to the vacancy, and which previously participated in bonds to the removed atom, rearrange themselves. In this simple model, the ‘defect-molecule model’, the behaviour of these four defect electrons in the ‘dangling bonds’ determines the properties of the centre. For this reason, we have a four-electron problem. The first difficulty, which we discuss later, is the controversial question of the use of the one-electron approximation. The

second difficulty concerns the Jahn-Teller effect. Both theory and experiment indicate that it is a large effect. Indeed, Watkins has argued that lattice distortion is so large that results for the undistorted lattice have little meaning. Certainly, there is evidence for nonlinear terms in the electron—lattice coupling and very clear evidence of interlevel Jahn-Teller coupling. The interlevel coupling leads to mixed trigonal and tetragonal distortions for the negative vacancy V~ in silicon, and may be responsible for the unusual stress-splitting of the GR1 band in diamond.

852

VACANCIES

IN

VALENCE

CRYSYALS

After reviewing experimental evidence for vacancy centres, we shall summarize the present knowledge of the theory and its problems. Recent reviews of aspects of these areas have been given by Watkins (1964), Fan (1967), Stoneham (1971), and Coulson (1973). 27.2. Vacancy centres in silicon

Our understanding of vacancy centres is based almost entirely on the spin-resonance work of Watkins (1963, 1964, 1967, 1968).

This, together with the simple model which Watkins developed to describe the data, constitutes the major contribution to the field. A. Charge states of the vacancy. Four charge states of the vacancy are observed or, when they have zero spins, inferred from vacancy diffusion through the lattice. All states can be produced by electron irradiation of silicon at low temperatures. Clearly, the dominant charge state depends on the position of the Fermi level. This is shown in Fig. 27.1. The charge states are V*~(S = 0), V-(S = 3), V°(S = 0), and V*(S = 4). Conduction

band

Wes

(motion)

v1:18 eV

(EPR: S= 4)

Ve

(motion)

(EPR: S=!)2 Valence

i

— < 0-05 eV

band

Fic. 27.1. Charge states of the vacancy in silicon. The ranges of Fermi energy over which given charge states are stable are shown. Also indicated are the experiments used to establish these ranges. (After Watkins.)

Other defects are observed which consist of an impurity next to the vacancy. There are three main classes. First, there are defects in which there is a Ge atom next to the vacancy. These systems have an impurity of the same valence as the atom replaced. Three charge states, GeV~, GeV°, and GeV*, are known. Note that here

the impurity is larger than the host atom; in ionic crystals, the analogous F,-centre only appears to form with smaller impurities.

VACANCIES

IN

VALENCE

CRYSTALS

853

Secondly, there are vacancies with substitutional group V donors

(P, As,Sb) or group III acceptors (Al, B,Ga) adjacent. Thirdly, there is one case (V+0;)” of an interstitial impurity next to the vacancy.

B. Watkin’s model of the vacancy. Watkins (1964) has shown that the observed properties of the vacancies may be understood in terms of three basic principles. First, one may apply the defectmolecule model, and concentrate on the dangling bonds. Secondly, electrons in these orbitals pair off to have spin zero whenever possible, and thirdly, the Jahn—Teller effect is strong. In particular, for silicon it is assumed the Jahn-Teller energies are larger than any corrections to the one-electron model. In the defect-molecule model there are four dangling-bond orbitals (|a>,|b>,|c>,|d>) one on each of the neighbours to the vacancy. One may construct four molecule orbitals from them. The lowest in energy has A, symmetry:

oe la>+|b> +Ic> +14)

2,/(1 +3S?)

(27.2.1)

where S = —|b> +1e>—Id) Be

2,/(1 — S?)

|Ja> +|b> —le> —|4> ly ~ 21-8)

27.22 (27.2.2)

ise |]a>—|b> —|e> + 14> 2,/(1 —S?) The various ground states of the differently charged vacancies can be found by populating these one-electron orbitals in turn and allowing for a Jahn-Teller effect because ofthe orbital degeneracy of the T,-state. This is shown in Fig. 27.2. Whilst we have assumed the states to be strongly localized, Watkins (1973) has observed that most features continue to hold when one uses delocalized states with only the spin localized. There are three defect electrons in V*. Two occupy the A, orbital ;the third is unpaired and goes into a T,-state. A tetragonal distortion results, lowering the symmetry to Dj,. The unpaired

854

VACANCIES

IN

VALENCE

CRYSTALS

T, We

Fic. 27.2. Electronic structure of vacancy centres. Watkin’s model is shown. The symmetries of the one-electron orbitals are indicated ;energy levels are not to scale.

defect electron for a distortion along a z axis has wavefunction |z>, and the electron is distributed equally on all four of the neighbours of the vacancy. Spin resonance confirms these features, and shows that about 65 per cent of the unpaired electron’s wavefunction can be associated with these nearest neighbours. The fourth defect electron in V° also goes into the |z> orbital, giving spin zero. The centre is not observed in spin resonance, although it can be shown from experiments in which the charge state is altered that V° also shows a tetragonal Jahn-Teller distortion. The negative vacancy V™ can be observed in spin resonance. The extra electron would naturally go into one of the degenerate |x> or |y> orbitals separated from |z> by the tetragonal distortion. Watkins argues that a second Jahn-Teller distortion removes the degeneracy. The unpaired electron observed is in an orbital concentrated on only two neighbours (la) —|d>),/{2(1 —S)}. About

60 per cent of the wavefunction is associated with these neighbours.

VACANCIES

IN

VALENCE

CRYSTALS

855

The symmetry of the centre is now C,,, and the Jahn-Teller effect has led toa mixed trigonal and tetragonal distortion. The description of this distortion is different from our earlier discussion (§ 8.4) of mixed distortion from interlevel coupling, but the physical origin of the phenomenon is the same. The interlevel-coupling description is necessary when handling more complicated models of the electronic structure. The charge state V*~ is not seen in spin resonance. The sixth defect electron presumably pairs up with the electron in the {|a> —|d>] orbital. It can be seen that Watkins’s model gives a consistent picture of the spins, symmetries, and unpaired-electron wavefunctions for all of the vacancy charge states. We shall compare his analysis with a priori theories in later sections. C. Jahn-Teller energies.+ The Jahn-Teller energies and coupling coefficients can be obtained from the stress response of the observed spin-resonance signals and the associated repopulation of the different defect orientations relative to the stress. Results can also be obtained for V° by changes of charge state under appropriate conditions. The data are analysed in terms of asimple model in which motion of the nearest neighbours alone is important. For modes of each symmetry I (totally symmetric, tetragonal, or trigonal) the energy may be written as a function of distortion amplitude Q E(Qr) =

—F,Q-+4K,Qr.

(27723)

The F; are obtained from the stress data, ignoring any changes in local elastic constants. The effective force constants Ky can be found from the model of Lidiard & Stoneham (1967), in which only the nearest neighbours to the vacancy are allowed to move. The model includes the effect of the vacancy on the force constants, and so predicts different values, K; for tetragonal modes and Ky for trigonal modes:

Ke Ky

II

11-9eV A- i

Il 595eVA~?

(27.2.4)

+ Some of the data and their analysis given here had not been published at the time

this chapter was written. I am greatly indebted to Dr. Watkins for providing this important information.

856

VACANCIES

IN

VALENCE

CRYSTALS

In this model K,; = 2Ky. If one writes Ky

= Ma;, then we is

85 per cent of the Raman frequency ofthe host, and wy is 60 per cent. The observed F, describe the total response of all the defect electrons to the applied stress. Theoretically, one is more interested here in the changes in the one-electron energies. If these one-electron coupling coefficients are independent of charge state, and if the Ky are the same in all cases too, then one can predict the variation of F, from one charge state to another. Under a given tetragonal distortion, one T, orbital is lowered in energy by 26 and the other two are raised by 6. Putting one, two, and three electrons into these orbitals, subject to the exclusion principle, gives total energy contributions —26, —46, and —36 respectively. These are in ratios 3:6:4-5, very close to those observed for both the vacancy and for Ge-vacancy complexes. Results are shown in Table 27.1. By contrast, TABLES2 7.1 Jahn-Teller energies and coupling coefficients Tetragonal Distortions Fy(eV A')

E¥;(eV)

Wr

29

0-35

Ve wee

6:0 4-5

1-51 0-85

Trigonal Distortions Fy(eV Ast)

Ej,(eV)

(1-58)t 3-8

1-21

+ This is not an experimental value, but deduced assuming K, and the one-

electron coupling coefficients are the same as for V. A tetragonal distortion is observed for V*.

either the trigonal coupling or K; must show a pronounced dependence on charge state. F; can only be measured for V~, and use of the same data for V* implies a trigonal relaxation energy about four or five times larger than the tetragonal relaxation energy. However, a tetragonal distortion is observed. The detailed explanation is not yet clear. For Ge: V* the trigonal coupling appears to be much weaker than for Ge: V~, and probably the same feature

occurs for V* and V~. D. Motion and reorientation. Vacancies in silicon are striking in that they reorient and diffuse at low temperatures. We summarize the data here (Watkins 1967; Elkin & Watkins 1968). Two classes of reorientation are observed. In both the symmetry axis of the defect changes without any motion of the vacancy itself.

VACANCIES

IN

VALENCE

CRYSTALS

857

The first class involves a rotation so that a vacancy with a given initial tetragonal axis changes to a different tetragonal axis. This behaviour can be studied for three charge states, giving effective activation energies of 0-013 eV (V*), 0-23 eV (V°), and 0-072 eV (V-). Similar values are found for the Ge-vacancy complexes. The second class of behaviour only occurs for V~. The unpaired-electron wavefunction can switch from one pair of atoms (e.g. (la) —|d))) to another (e.g. (b> —|c))). The observed activation energy is very low, a mere 0-008 eV.

Vacancy motion has been analysed for V° and V2~. The results show the two striking features also found in reorientation, namely a charge-state dependence of the activation energy Ey, which also has a very low magnitude. Observed values are 0:33 eV (V°) and 0-18 eV (V_). Gregory (1965) has commented that vacancy motion can be directly affected by electron injection in p-type materials. These results allow one to estimate the formation energy E, of the vacancy. If donors diffuse by a vacancy mechanism, their motion allows one to measure the sum (Ey,+ E, — E,), where E, is the magnitude of the vacancy-donor binding. If E, is small (Watkins estimates about 0-3eV from a simple model) then these results suggest (Ep +E,) to be about 4eV. About SeV is obtained from self-diffusion data at comparable temperatures. 27.3. Other observations of vacancy centres A. Vacancies in diamond. The primary defects in diamond have still not been identified with any certainty. We consider the several defects which are absent in unirradiated diamond, which are observed with intensity linear in dose of electrons or neutrons, and

which show no correlation with known chemical impurities. These defects are likely to be simple and intrinsic. The most prominent of them are responsible for the GR1 optical band, reviewed by Clark & Mitchell (1971) and Clark & Walker (1972), and the spin-resonance systems surveyed by Owen (1965) and Whippey (1972). The GR1 band has a zero-phonon line at 1-673 eV, with a broad band of phonon-assisted transitions peaked at about 2eV. The oscillator strength is of order 0-1. The broad band has vibronic structure, and there is also a weak, sharp satellite to the zerophonon line, which can be interpreted in terms of a 70 cm ! groundstate splitting. The zero-phonon line shows a linear Stark effect (Kaplyanskii, Kolyshkin, Medvedev, & Skvortsov 1971), and has a

858

VACANCIES

IN

VALENCE

CRYSTALS

complex response to stress (Runciman 1965; Clark and Walker 19734 Davies and Penchina

1974). The most recent data show that the

lowest GR1 state has E symmetry; an A state lies slightly higher, giving a zero-phonon line displaced by 8 meV. The excited state to which transitions occur has T symmetry. Since the GR1 band anneals out at higher temperatures than most of the spin-resonance systems, it is usually assumed that the GR1 band comes from a vacancy centre: interstitials are usually more mobile than vacancies. If this assumption is correct, then the GRI1 centre is probably V° or V-. The GR1 line is not seen in semiconducting diamond

until the dose is sufficient to compensate all the impurity acceptors, so the GR1 defect must be a donor and is unlikely to be V*. It is not easy to settle the question of V° or V~ by spin resonance. Watkins has commented that the high Debye temperature and low spin-orbit coupling tend to lead to defects whose spin-resonance signal is easily saturated, so that the absence of spin-resonance does not rule out V-. Owen lists four centres which appear to be simple intrinsic defects. These give the a(A’), b(C), c(B), and d(A) systems. All annealout at lower temperatures than the GR1 band. In only one case is there resolved hyperfine structure, namely the a system which Baldwin (1963) has identified with V*. This system is very similar to the V* in silicon observed by Watkins, having S = 4.anda[100] axis. The resonance is excited by light above 2-83 eV in energy, suggesting that this is the V° ionization energy. If so, this is additional evidence that the GR1 line is V°, for Walker, Vermeulen, & Clark (1974) have shown that a number of bands (notably GR2 at 2:881 eV and GR3 at 2:88 eV) are associated with the GR1 centre and that these lines give rise to photoconductivity. Owen has argued that V* is unlikely to be the a system because of its high predicted energy relative to V° and V~ in Coulson & Kearsley’s work. But the more recent and detailed work by Coulson & Larkins and Lidiard (1973) shows that

V* cannot be eliminated in this way. Both the b and c systems have spin 1, a g-value of 2:00, and a zero-field splitting of 0-140 cm~'!. They may well be related defects. The b system has a [994] axis, suggesting complexity, and we shall not consider it further. The c system has a spin-zero ground state; the observed [100] spin-1 state lies higher by about 300cm™'. The d system shows a similar phenomenon, and Whippey has identified an S = 3 state lying 550 cm~! above an S = 4 state. Identifica-

VACANCIES

IN

VALENCE

CRYSTALS

859

tions are difficult, although tentative associations have been made with theory. Thus the c system resembles the 'E- and >T,-states of V° or the neutral interstitial, and d the *T,- and *A,- or *T,-states OF Nis: Probably the most striking feature is the appearance of three defects with splittings of order 10? cm~ '. This may be compared with Watkins’s observations of defects in silicon with low reorientation energies. Indeed, the splittings may be motional in origin, as Lannoo & Stoneham (1968) have suggested for the GR1 satellite. Analysis of the recent stress work suggests strongly that the lowest E and A states derive from the same E electronic state, with a vibronic splitting. The evidence (Stoneham 1974, unpublished) comes partly from the fact that a consistent set of parameters can be found which describe a range of observations, and partly from the relations between the stress coupling coefficients within the E state (Ggg say) and those between the A and E states (G, say). If the E and A states were derived from distinct electronic states then, because only the E state alone would have a large Jahn-Teller distortion, Ham reduction factors would ensure |Gp,/G,| > 1. This is not found. What is found is that |Gg,/G,| lies within the fairly narrow limits appropriate for a vibronic splitting. Thus the GR1 centre exhibits a dynamic Jahn-Teller effect in its ground state.

B. Silicon carbide. Silicon carbide crystallizes in a large number of polytypes. In each the Si and C atoms are tetrahedrally coordinated, and in all cases the bond length is close to 1-9 A. Balona & Loubser (1970) have observed a variety of spin-resouance spectra in electronand neutron-irradiated cubic 3C and hexagonal 6H polytypes. Certain of the spectra appear to be associated with vacancies. The vacancies have only been seen in the 6H polytype, which is hexagonal (Verma & Krisha 1966, p. 81). In the defect-molecule model the tetrahedral coordination is, of course, more important than the crystal symmetry. The spectra and models proposed all involve

carbon

vacancies,

although

silicon

vacancies

appear

to

occur in more complicated centres. In each case hyperfine structure is resolved. The S =} E spectrum is identified with V,. It is only observed when the temperature is high enough to ionize the S= 4 Bspectrum,

identified with V,. The A spectrum, identified with V2, has S = 1 and a zero-field-splitting tensor with an unusual symmetry axis at

860

VACANCIES

IN

VALENCE

CRYSTALS

45-4° to the hexagonal axis. Balona & Loubser argue that this is a result of an asymmetric lattice distortion in which one of the silicon neighbours (not the one on the hexagonal axis) relaxes radially outwards. Their model for V2 is rather closer to the models for twohole centres in II-VI compounds (§ 16.4) than to Watkins’s model, although there are features in common. C. Germanium.

There

has been

no

successful

optical

or spin-

resonance identification of the vacancy in Ge. However, vacancies

are often postulated in analysing radiation damage (see the review of MacKay & Klontz 1971) and have been detected through their effects on group V donors. Gershenzon, Goltsman, Emtsev, Mashovets,

Ptitsyna, & Ryvkin

(1971) have argued that the state V* does not occur and that only V° and negatively-charged vacancies should be stable. This idea is based on the fact that complexes of group V donors and vacancies are observed, whereas no complexes of group III acceptors and vacancies are found. If V* existed, the Coulomb attraction to accep-

tors should lead to complexes, as for V~ and donors. 27.4. Electronic structure in the undistorted lattice

From

our

earlier comments

it is clear that lattice distortion,

especially that from the Jahn-Teller effect, plays an important part in the vacancy centres. Calculations for the undistorted lattice are of interest because they are an essential preliminary to calculations of the distortion and because they provide a check of certain special features. Three questions are of special importance: what are the likely ground states, how well is the defect wavefunction localized, and is the one-electron approximation valid?

27.4.1. Defect-molecule models Most of the work in this category was stimulated by the pioneering paper of Coulson & Kearsley (1957). The basic idea is that the vacancy properties are primarily determined by the electrons in the ‘dangling bonds’ on the neighbours to the vacancy. The energy levels of this system can be determined by standard molecular methods. In particular, this model allows one to treat the electron—

electron interaction energy rather well. This is necessary when one has a few-electron problem instead of a one-electron problem: for example in the case of V° where there are four defect electrons.

VACANCIES

IN

VALENCE

CRYSTALS

861

However, it is not possible to place the energy levels accurately relative to the energy bands of the perfect crystal. A. One-electron states. Two classes of one-electron wavefunctions have been used. In the first, the L-C.A.O. wavefunctions of (27.2.1) and (27.2.2) were used. Coulson & Kearsley and Yamaguchi (1962)

used

simple

Slater

functions

for the atomic

orbitals;

Larkins

(197 1a, b), Larkins & Stoneham (1970), and Coulson & Larkins (1971) have used free-atom Hartree-Fock atomic functions, and Kunz, Larkins, & Stoneham (1972, unpublished) have used local orbitals

(cf. §7.5) which take into account the effect of the vacancy on the electronic structure of its neighbours. In the second class, the oneelectron wavefunctions were taken to be vacancy-centred functions of the appropriate symmetry. Such functions have been used by Gourary & Fein (1962), Stoneham (1966), and Schroeder (1972, unpublished). In principle, very flexible trial functions can be constructed in this way, and they are simple to handle. So far, very simple functions have been used, and the results are almost certainly less accurate than the L.C.A.O. treatments. However, the radial extent of the defect electron wavefunction can be found variationally from the one-centre treatments, and shows that the centre should indeed be

compact, as observed. In most respects, the two choices of one-electron wavefunctions lead to very similar results. They agree in the general order of the energy levels, the order of magnitude of their separation, and, as we discuss later, they agree that the one-electron approximation is not adequate. The details of the energy levels do depend, of course, on

the details of the one-electron

wavefunctions,

but the lowest

energy states can be predicted by general arguments; we illustrate

this for V°. B. Many-electron states. The one-electron orbitals consist of a lowest A,-state, with a triply-degenerate T,-state higher in energy by A. For V° we consider the way in which four electrons can be distributed over these orbitals. There are three factors which affect the relative energy of the states. One is the one-electron energy. There is an energy gain A if an electron is placed in the A, orbital rather than a higher T, orbital. The second factor is the exchange energy, which favours parallel electron spins. It is this interaction which leads to Hund’s rules for free atoms. The third factor is the reduction ofthe electron—electron interaction if the wavefunction

862

VACANCIES

IN

VALENCE

CRYSTALS

is one in which the four electrons tend to reside on different neighbours of the vacancy. In such a case the intra-atomic exchange energy, which is primarily responsible for the ‘Hund’s rule’ configurations, becomes less important. There appears to be a delicate balance between these effects in diamond. If A is small and the exchange interaction significant (as in weak-field ligand field theory, §22.2) then the °A-state is favoured. This is shown in Fig. 27.3. There are three electrons in the T,-state and just one in the A,-state, so the configuration is [vt*] in terms of the states (27.2.1) and (27.2.2). If A is large, the [vt?] configuration is favoured, with two electrons in the A,-state. The °T,-state is favoured if Hund’s-rule coupling favours parallel spins for the electrons in the T,-state, and a ‘'E-state is lower if the electron— electron interaction term is dominant. A singlet ground state may seem

unlikely

at first, although

other

cases

are

known.

Ander-

son’s model of kinetic superexchange for antiferromagnetism argues that the loss of exchange energy for parallel spins is more than outweighed by the decrease in kinetic energy, because the spatial wavefunctions need not be orthogonal for anti-parallel spins.

Sap Gb eap pS * = + (a)

(b)

(c)

+ (d)

Fic. 27.3. Potential ground states of the neutral vacancy.

All the calculations for the neutral vacancy agree that the 'E,

°T,-, and °A,-states are among those lowest in energy. C. Calculation of matrix elements. Most of the elements can be evaluated without difficulty. Approximations are necessary in some cases, notably in the three- and four-centre electron—electron interaction integrals. These are estimated using the Mulliken approximation (7.3.1). However, Coulson & Kearsley have argued from experience with other molecular calculations that two one-centre integrals are likely

VACANCIES

IN VALENCE

CRYSTALS

863

to be inaccurate. These are the ‘penetration integral’ P,

P = 914 Pick, R. M., 48, 183, 914 PIKuS, G. E., 407, 812 PINES, D., 234, 247, 908, 914 Pirc, R., 704, 705, 706, 707, 900, 914 PISERI, L., 921 PITAEVSKI, L. P., 135, 900 PLATZMAN, P. M., 233, 236, 248, 914 PODINI, P., 561, 583, 914 POHL, R. O., 224, 230, 389, 397, 699, 912, 915 POKROVKSY, Y. E., 534, 915 POLINGER, V. Z., 324, 921 POLLAK, M., 103, 915 Pompl, R. L., 760, 915 POOLEY, D., 477, 486, 487, 583, 584, 668, 892, 896, 915 POPLES J) AS oy 141 4450120805 Popov, Y. M., 915

Popova, A. P., 892 PORTER, R. N., 638, 651, 915 PORTO, Sa bys74554923 POWELL, M. J. D., 170, 898, 915 PRATHER, J. L., 11, 915 PRATT, G. W., 84, 96, 915, 916 PRATT, R., 234, 903 PREIER, H., 915 PRENER, J. S., 915 PREuss, H., 134, 892

PREZIOSI, 891

QUEISSER, H. J., 295, 915 QUIGLEY, R. J., 175, 915

B., 104, 108, 109, 334, 571,

RABIN, H., 399, 400, 436, 625, 894, 915 RACCAH, P. M., 915 RADOSEVICH, L. G., 389, 915 RAJNAK, K., 745, 915

PALPH, H. I., 391, 915 RAMDAS, All Ke O11, Sil2. 7/5 75eOue 778, 902, 913, 920 RAO, B. K., 461, 903 RASHBA, E. I., 819, 821, 915 Ray, D. K., 493, 915 RAY, T., 441, 493, 915 RAYLEIGH, LORD, 184, 915 READ, W. T., 550, 918 REBANE, K. K., 130, 297, 308, 400, 401, 704, 915 REBANE, T. K., 892 REppy, T. R., 474, 890 REDEI, L. B., 137, 897 REDFIELD, D., 57, 392, 897 REICHERT, J. F., 396, 915, 921 REIF, F., 471, 894 RENK, K. F., 347, 899 REYNOLDS, D. C., 253, 916 Ruys, A., 294, 479, 481, 485, 486, 585, 586, 903 RHYS-ROBERTS, C., 540, 541, 542, 907 RIBBING, C. G., 87, 893 RICE, B., 161, 181, 894 Rice, S. A., 121, 337, 781, 904, 922 RICHARDS, P. M., 505, 895 RICHARDSON, J. W., 724, 916 RICKAYZEN, G., 537, 916 RIMMER, D., 146, 148, 723, 724, 738, 903, 916 RISEBERG, L. A., 483, 484, 916 RITTER; J. T., 57,585, 586, Sol.o005 916

AUTHOR ROBERTS, D. A., 409, 901 ROBINSON, D. A., 621, 622, 623, 624

>

909 INoauccie4,

Sey Sil,

SOW,

Sill,

Ss)

52455255926, 52755325535. 7.62, 838, 839, 840, 890, 893, 916, 917 ROHRLICH, F., 291, 904 Roltsin, A. B., 396, 916

ROLFE, J., 704, 903, 916 ROLLEFSON, J., 713, 916 ROMESTAIN, R., 223, 433,

436,

624

897, 910, 916 ROOSBROOK, W. VAN, 290, 916 ROOTHAAN, C., 119, 916

Ros, P., 724, 897 ROSEI, R., 393, 432, 569, 581, 894 ROSEN, N., 817, 916 ROSENBERGER, F., 393, 397, 916 ROSENSTOCK, H. B., 916 ROSENTHAL, J. E., 461, 916 ROSENTVEIG, L. N., 180, 908 Ross, J. A., 103, 894, 895 ROSIER, L. L., 341, 916 ROSTOCKER, N., 109, 906 RoTH, L. M., 84, 405, 775, 916

>

INDEX

SYNEE (CC, 3s, AOL Sil, We, COW. Caley 916, 920 SAFFREN, M. M., 262, 916 SAGAR, A., 103, 915 SAK, J., 235, 599, 766, 916 SAKURAGI, S., 781, 905 SAKAMOTO, Y., 916 SALZMAN, W. R., 260, 891 SALPETER, E. E., 339, 621, 623, 831, 891 SAMMEL, B., 684, 686, 916 SANDER, L. M., 703, 918 SANGSTER, M. J. L., 358, 493, 915 SARFATT, J., 198, 916 SATO, 52, 916 SAUER, P., 224, 227, 228, 904, 916 SAXON, S. D., 145, 916 SCHAAD, L. J., 895 SCHAWLOW, A. L., 291, 903, 921 SCHECHTER, D., 70, 688, 758, 759, 764, 917 SCHERR, C. W., 145, 916 SCHERZ, U., 110, 917 SCHIFF, L. I., 337, 520, 917 SCHILLINGER, W., 533, 543, 906

ROUHANI, D., 870, 916

SCHIRMER, O. F., 461, 917

ROUSSELL, A., 574, 594, 911 Royce, E., 396, 916 ROZENFEL’D, Y. B., 310, 324,

SCHMEISING, H. N., 689, 919 SCHMID, D., 454, 464, 917 SCHMID, D., 877, 917 SCHMIDKE, H. H., 134, 892

905,

921 Ruch, E., 267, 916 RUDGE, M. R. H., 120, 910 RUEDENBERG,

K., 139, 145,

155, 897,

916 RUEDIN, Y., 560, 916 RUNCIMAN,

39

W. A., 413, 415, 422, 423,

645, 668, 743, 858, 897, 903, 904, 915, 916 RUSANOY, I. B., 413, 415, 422 RUSSELL, G. A., 559, 896 RUSSELL, J. D., 143, 896 RUTHERFORD, D. E., 19, 916

RYSTEPHANICK, R. G., 506, 890 RyvkKIN, S. M., 860, 899

SAARI, P. M., 401, 915 SABISKY, E. S., 494, 495, 916 SACHS, R., 45, 262, 916 SACK, R. A., 198, 199, 200, 223, 297, 324, 631, 638, 641, 642, 643, 644, 645, 646, 647, 909 SACKS, R. G., 909

SCHNATTERLY, S. E., 279, 281, 323, 429, 433, 437, 569, 577, 580, 902, 917 SCHNEGG, P.-A., 560, 596, 916 SCHNEIDER, P. M., 22, 109, 907 SCHOEMAKER, D., 782, 917 SCHOLZ, A., 877, 917 SCHON, M., 917 SCHONHOFER, A., 267, 916 SCHONLAND, D. S.; 470, 898 SCHOTTKY, W., 250, 251, 691, 837, 901 SCHRODER, U., 48, 837, 843, 903, 910, 913, 917 SCHROEDER, K., 861, 873 SCHULMAN, J. H., 436, 829, 915, 917 SCHULTZ, D.R., 758, 759, 760, 761,911 SCHULTZAT. D.23355.0755 LAS 9165917 SCHWARTZ, C., 469, 917 SCHWARZ, K., 17, 917 SCHWINGER, J., 96, 257, 908, 917 SCHWOERER, M., 454, 917 SCIFRES, D. R., 781, 791, 792, 793, 917 SCLAR, N., 545, 917

940

AUTHOR

Scom PAL. 4995 907 ScoviL, H. E. D., 480, 892 SEARLE, T. M., 618, 917 SEATON, M. J., 76, 917 SEDUNOV, B. I., 340, 906 SEEGER, A., 180, 877, 910, 917 SEGAL, G. A., 109, 144, 901, 915 SEIDEL, H., 454, 682, 683, 686, 687, 917,919 SERDZs hey oy Ol SENGUPTA, D., 748, 917 SENGUPTA, S., 145, 919 SENNETT, C. T., 360, 371, 694, 697, 897, 901, 917 SERBAN, T., 142, 917 SERGENT, A. M., 787, 911 SEWARD, W. D., 700, 917 SHAFFER, J., 822, 823, 824, 917

SHAM, L. J., 17, 122, 766, 890, 906, 917 SHARE, J. H., 551, 920 SHARMA, R. R., 337, 838, 839, 840, 843, 917 SHAMU, R. E., 695, 917 SHEINKMAN, 543, 544, 917, 921 SHEKA, D. I., 74, 917 SHEKA, V. I., 74, 917 SHEKTMAN, V. L., 921

SHEPHERD, I. W., 646, 648, 918 SHINAGAWA, K., 441, 992: SHINDO, K., 765, 890 SHINOHARA, S., 764, 779, 918 SHIEVER, J. W., 768, 911, 912 SHIREN, N. S., 918 SHLITA, L. M., 522, 891 SHMIGLYUK, M. I., 843, 892 SHOCKLEY, W., 56, 290, 550, 916, 918, 920 SHOLL, C. A., 471, 918 SHORE, H. B., 703, 918 SHORTLEY, G. H., 443, 727, 743, 894 SHUEY, R. T., 675, 704, 918 SHYU, W. M., 461, 899 SIDONOV, V. I., 533, 908 SIEMS, R., 173, 918 SIEVERS, A. J., 229, 373, 374, 703, 711, 712, 889, 894, 904, 905, 913, 918 SIGGIA, E. D., 918 SILIN, A. P., 918 SILSBEE, R. H., 218, 265, 283, 506, 632, 633, 634, 635, 636, 645, 646, 648, 649, 651, 704, 705, 707, 712, 907, 918 SIMANEK, E., 738, 739, 740, 741, 918

INDEX SIMON, B., 257, 258, 259, 265, 918 SIMPSON, J. H., 177, 239, 918 SINANOGLU, O., 20, 22, 918, 920 SINGH, B. D., 57, 918 SINGH, R. K., 918 SINGH, R. S., 689, 918 SINGHAL, S. P., 678, 680, 918 SINHA, K. P., 828, 896 SITZMAN, U., 364, 918 SKETTRUP, T., 833, 834, 835, 836, 918

SKLAR, A. L., 140, 141, 863, 900 SKLENSKY, A. F., 522, 534, 918 SKvorTsov, A. P., 393, 857, 905 SUATER J@. 342s 1S) ali ONO OROSs

134, 147, 148, 349, 906, 918 SLICHTER,

Cy P53235'°325.8429

438°

437, 443, 455, 464, 467, 661, 676, 902, 918 SLONCEWSKI,

J. C., 201, 209, 222, 652;

918 SMAKULA, A., 281, 981 SMITH, C. S., 47