This book surveys the theory of defects in solids, concentrating on the electronic structure of point defects in insulat
245 21 40MB
English Pages 977 [980] Year 1985
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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 oneelectron 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. Partlyionic crystals 3.4.5. Raregas crystals SiS) Lattice dynamics 3.5.1. Dispersion curves 3.5.2. Density of states 3.6. The electronphonon interaction 3.6.1. The Hellman—Feynman theorem 3.6.2. Electron—phonon interaction: delocalized case 3.6.3. Local models for electronlattice coupling
IPELECRRONIGISTRUGIUWRE OF ISOLATED, DEFECTS 4. EFFECTIVEMASS THEORY 4.1. Introduction 4.2. Simple effectivemass theory 4.2.1. Fundamental equations for the defect lattice 4.2.2. Approximations of effectivemass theory 4.2.3. The function A(r) 4.2.4. The effectivemass equation 4.2.5. The wavefunction 4.2.6. More complicated band structures 4.2.7. Matrix elements in effectivemass theory 4.3. The quantumdefect method 4.4. Effectivemass theory for a manybody system 4.4.1. Manybody eigenfunctions for the perfect lattice
4.4.2. Fundamental equations for the defect lattice 4.4.3. Approach using exact manybody states as a basis 4.4.4. Approach using HartreeFock states as a basis 4.5. The accuracy of the effectivemass 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 KosterSlater model 5.2.1. The oneband, onesite 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 Tmatrix 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 fewelectron 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. Molecularorbital and valencebond 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. Semiempirical 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. Selfconsistent 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. Longrange distortion from point defects 8.4. The JahnTeller instability 8.4.1. Introduction: the quasimolecular hypothesis 8.4.2. Potential energy surfaces 8.4.3. Static and dynamic JahnTeller 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 slowrotation 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 electronlattice coupling 8.6.3. Intermediate coupling: more advanced models 8.6.4. Strong coupling 8.6.5. The Toyozawa—HakenSchottky 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 lineshape function 10.5. Smakula’s formula and the effectivefield correction 10.6. Einstein coefficients and the principle of detailed balance 10.7. The lineshape function with electronlattice interaction 10.7.1. Introduction 10.7.2. Linear electronlattice 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. Nondegenerate states: linear and quadratic coupling 10.8.3. The configurationcoordinate diagram 10.9. Forbidden transitions: nondegenerate states 10.10. Electronic degeneracy 10.10.1. Introduction 10.10.2. Splitting of optical bands due to degeneracy 10.10.3. The HuangRhys 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. Lowfrequency 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 Tmatrix
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 longwavelength 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 effectivemass 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 electronlattice coupling: moment methods 12.5.1. Introduction 12.5.2. Zerophonon lines 12.5.3. Broadband transitions 12.5.4. Raman effect 12.5.5. The rigidshift hypothesis
3k ELECTRONSPIN RESONANCE 1320 Introduction 132% The spin Hamiltonian 1ESES! The Zeeman effect 13.3.1. Orbital angular momentum 13.3.2. Examples of gfactors 13.4. The zerofield 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. NONRADIATIVE PROCESSES AND INTERACTION OF FREE CARRIERS DEFECTS 14.1. Introduction
466 467 473
WITH
14.2. Nonradiative transitions 14.2.1. General theory 14.2.2. Paramagnetic relaxation 14.2.3. Resonant and nonresonant absorption 14.2.4. Temperature dependence of the zerophonon 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 FCENTRE AND RELATED ONECARRIER SYSTEMS 15.1. Introduction 15.1.1. The hydrogen atom model of the Fcentre 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 onecarrier centres 15.2. The groundstate configuration of the Fcentre 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 lineshapes
555 S25) 5D») 556 558 561 561 562 562 562 571 574 iil 581 583
Xvi
CONTENTS
15.3. Excitedstate configuration of the Fcentre 15.3.1. Introduction 15.3.2. Calculations of the relaxed excited state 15.3.3. Model calculations of the Fcentre excited states
15.4. Perturbed Fcentres 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 Fcentre :trappedhole centres 15.5.1. Trappedhole 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 Mcentre 16.3.2. The F,centre in alkaline earth oxides 16.4. Twohole centres in the alkaline earth oxides
17. THE RCENTRE 17.1. Introduction 17.2. Electronic structure of the Rcentre 17.2.1. The Kern—Bartram model 17.3. The Rcentre and the JahnTeller 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 Rcentre: 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 HCENTRE 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 Hcentre lattice configuration 19.2.3. Interactions with other defects 19.2.4. Electronic structure of the Hcentre 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 (Ucentre) 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 raregas hosts 21.3. Elastic dipole centres: the O; molecular ion 21.4. Offcentre ions 21.4.1. The KCI: Li system 21.4.2. Occurrence of offcentre ions 21.4.3. Oxygen interstitial in silicon
699 703 704 710 710 le 714
2. TRANSITIONMETAL IONS 22.1. Introduction 22.2. Classes of theory and coupling schemes 22.3. The irongroup 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. Rareearth ions 22.4.1. Free rareearth ions
742 742 745
22.4.2. Crystal fields for rareearth ions
23. SHALLOW DONORS SEMICONDUCTORS
AND
ACCEPTORS
IN
23.1. Introduction
751
751
23.2. The application of effectivemass theory :nondegenerate
bands
751
23.2.1. Energy levels
752
23.2.2. Wavefunctions 23.2.3. Response to perturbations 23.3. Applications of effectivemass theory :degenerate bands
755 TSU 758
23.3.1. Energy levels and wavefunctions 23.4. Corrections to effectivemass 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 oneband, onesite 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 lineshapes the oneband, onesite 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 DONORACCEPTOR 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 boundexciton systems 26.3.1. The JahnTeller effect 26.3.2. Optical lineshapes 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. Defectmolecule models 27.4.2. Cluster models 27.4.3. Other methods 27.4.4. The oneelectron 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, semimetals, and heavilydoped 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 IIVI and IIIV 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 closepacked 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 commonlymet 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 closepacked 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
Nearestneighbour distance Lattice parameter
a 2a
Volume per unit cell
2a?
These are cubic crystals, with two atoms per unit cell. The sublattices are both facecentred 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 bodycentred site: this has tetrahedral symmetry, with four cation and four anion neighbours. There are two distinct sites related by inversion; (b) the facecentred site: this has two cation and two anion neighbours. It is readily distorted into the . 16/3,/3 = 30792 a’.
These
are cubic crystals. The
anion
cations lie on a facecentred 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 bodycentred site. This is the bodycentred site of both sublattices. The nearest neighbours are a cube of eight anions. (b) The ‘facecentred’ cubic site (equivalently, the edge centre of
CRYSTALS
O
AND
LATTICE
GEOMETRY
Anion
@® Cation Ml
Edgeand
facecentred
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 Bodyand edgecentred interstitial sites Fic. 1.3. The CaF, structure.
7
8
CRYSTALS
AND
LATTICE
GEOMETRY
1.2.4. Diamond and zincblende structures
Nearestneighbour distance Lattice parameter
a 4a/./3
Volume per unit cell
am, 16/3,/3 ~ 30792 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 facecentred cubic by placing two atoms on each site, one from each sublattice, and separating them along a )°
DY
L=0
M=L
> shells s
Yale, eau i
+
rE
te1— a1}
Fim r
(1.3.1)
Ky
where the coefficients are 4n 8imM = IL+1
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 144. 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 sphericallysymmetric 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
Pointion potential
L r/(nearestneighbour distance)
Fic. 1.6. The pointion potential. The spherically symmetric part of the pointion 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 nearestneighbour 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 174756 for the NaCl structure, 176267 for CsCl, and 163805 for ZnS. In the fluorite structure «
is 176267 at the anion site and ~32761 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 pointion potential of nearneighbours. Expressions for four, six, and eightfold 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 spinlattice 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 nonrelativistic Hamiltonian for a system of electrons and nuclei. Omitting small spindependent terms, the Hamiltonian has the form:
(2.1.1)
KH = HKH.+ Ant An, where the purelyelectronic term is
#,=5. eas
fmt?
2
is
(2.1.2)
Sa Ine}
the purelynuclear 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 oneelectron 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 oneelectron 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 HartreeFock 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 HartreeFock 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 oneelectron functions:
f 94,ree 1ferrioien i — jFi
et oH
 feeaa lr—r Be » Zi TeaRra
meee)
(2.1.8)
The potential which determines the oneelectron function is a sum of three terms: a consistentlychosen Coulomb potential from the electronelectron 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 oneelectron 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 oneelectron 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 oneelectron 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 oneelectron orbitals by an approximate local operator. The form usually adopted is
VX" I= —6a(3p,/4z2)* Ry
—37221ap; 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 partlyfilled band. Both these results are based on estimates of the interaction in a nearlyhomogeneous 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 HartreeFock theory. Thus oneelectron 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. Oneelectron 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 oneelectron 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 oneelectron 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 oneelectron energies. It is not clear yet how good the orbitals obtained are. The nonlocal nature, and hence the momentumdependence 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 HartreeFock 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 independentparticle 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 HartreeFock theory (Lowdin 1959a; Moshinsky 1968). But other properties may be sensitive to correlation, such as the small energy differences between HartreeFock 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 independentparticle models. By an independentparticle model, we mean that the total wavefunction is constructed from oneelectron 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 HartreeFock theory, the manyelectron wavefunction is a single determinant in which the oneelectron 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 twoparticle 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 projectionoperator 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. 945). 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 twoelectron 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 HartreeFock wavefunction ®p, there will always be other HartreeFock wavefunctions ®,, with the same symmetry and spin. The oneelectron 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 electronelectron interaction. The new manyelectron wavefunction is a sum of appropriatelyweighted 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 oneelectron 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 manyelectron problem can be replaced by an exactlyequivalent set of oneelectron 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 (GellMann & 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
—207(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 effectivemass 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, oneelectron, 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 oneelectron potential is not sensitive to the actual occupancies of the oneelectron 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 oneelectron approximation. And thirdly, a ‘muffintin’ potential may be used. In this approximation the volume is divided up into nonoverlapping spheres centred on each atom, and the potential assumed constant between the spheres. The muffintin 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 oneelectron 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 IIVI and IIIV 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 directbandgap 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
nondegenerate 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 pelectrons, for example. The valence band degeneracy can be lifted by spinorbit coupling (e.g. Elliott 1954) or by lowsymmetry 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 spinorbit splitting Aso is largest when the corresponding atomic spinorbit 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 spinorbit

coupling
Cubic, spinorbit
splitting
k
Axial, no spin —orbit
coupling

vain
YOO NaS
Axial, with spin — orbit
splitting
Fic. 2.2. Effects of spinorbit coupling and lowsymmetry fields on band structure.
ELECTRONIC
STRUCTURE
OF
THE
PERFECT
LATTICE
25
The energy surface near the extrema of a nondegenerate 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 effectivemass approximation, with an effectivemass 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 principalaxes representation. The energy surfaces are anisotropic near the extremum for noncubic 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 bandstructure data used in defect calculations is often best obtained by modulation methods which exploit the singularities. Exactly analogous critical points occur in phonon densitiesofstates. 2.2.3. Band structures: wavefunctions A. Band functions and Bloch functions. The oneelectron 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 spinorbit 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 tightbinding 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
OF
THE
PERFECT
LATTICE
29
TABLE 252 Parity of zoneboundary 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 donoracceptor 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 highestenergy 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 tightbinding
30
ELECTRONIC
STRUCTURE
OF
THE
PERFECT
LATTICE
limit, where intraionic terms are dominant, but is large in the nearly
freeelectron 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 spinorbit coupling, which separates E; from E, and E, by A, and by the terms in P. E, and E, form a ‘heavyhole band’ E, and a ‘lighthole 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 tightbinding approximation. For the iodides the spinorbit 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 freeelectron 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 dstates 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 nearlyfreeelectron model, rather than the tightbinding model, is appropriate. One can get reasonable descriptions of the band structures starting from either extreme.
C. Freeion 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 tightbinding approximation. The main differences between freeion and crystalion 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 HartreeFock
calculations of wavefunctions for a free ion in a potential well; the well represents a crystal field. Thus Watson (1958) discussed O?~ in a squarewell 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 Xray structure factors, which measure the charge distribution. These are reviewed by Weiss (1966). Maslen (1967) finds that uncorrected freeion wavefunctions are best for CaF,; Aikala & Mansikka (1970) find symmetricallyorthogonalized freeion 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 Fcentre. 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 Xray 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 bestknown 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 nparticle problem in which the n carriers are supposed to have a positive effective mass and a positive charge. Another example occurs in crystalfield theory (Griffith 1961, p. 245). Suppose we are interested in the states of systems in which the occupancies of one shell are varied. For irongroup transitionmetal 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 electronelectron 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 (Nn) electrons.
34
ELECTRONIC
STRUCTURE
THE
OF
PERFECT
LATTICE
First we consider the expectation value NE
nNn
CSET) = SsCENId+5 y dL Ki
—. Suppose t is the order of the permutation needed to get the identical oneelectron 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
oneelectron 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 electronhole interaction changes sign. If J> and J> differ by two functions k> and /> we find
CSET = (— INCRE — CREEP KD). There is no oneelectron 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 nondegenerate means that y/(r; R) is nondegenerate. 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 JahnTeller 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 offdiagonal elements are found by taking expectation values of the Hamiltonian (3.1.1)
1 IAD > ==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 offdiagonal matrix elements are important. Thus nonradiative 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 groundstate energy (EE,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 shortrange 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 shortrange interactions representing a complicated and interrelated 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 oneelectron 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 twobody central forces of the form ¢,,(r), where J and J refer to the species of ion, and r is their separation. Secondly, there are noncentral twobody forces. These are much less important, and it is often difficult to determine anything but an upper bound on their magnitude. Strictly, the dipoledipole interaction belongs to this category, but it is always treated separately. Thirdly, there are manybody 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, manybody 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 twobody 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 alkalihalide 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 lowestorder 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 secondneighbour interactions (Barr & Lidiard 1970). The third question applies specifically to the shell model, and concerns the division of the shortrange interactions into core—core, coreshell, and shellshell contributions. For ionic crystals the usual assumption is that only the shellshell 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 interatomic 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 stronglydirectional covalent bonding between nearest neighbours with much weaker interactions between more distant neighbours. Nearestneighbour central interactions do not give an adequate description of the forces (Smith 1948; Herman 1959). The interatomic 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 fourbody interactions appear. 3.4.4. Partly ionic crystals
This category includes most IIIV and IIVI semiconductors. The models used are a mixture of models used for ionic crystals and valence crystals. Thus the shell model, valenceforce potentials, and simpler forms like the rigidion 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 rigidion 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 longwavelength optic modes. Thus for GaAs, Z is 004, Z’ is 051, 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 Xray lines, or with descriptions of covalent bonding. Needless to say, these definitions give different values of ionic charges. 3.4.5. Raregas crystals The attractive interaction which dominates in rare gases is the van der Waals interaction. The weakness of the interaction means that threebody 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 raregas crystals. 4. The opticmode frequencies and acousticmode frequencies may overlap, giving a quasicontinuous spectrum. However, there may be a gap between the highest acoustic and lowest opticmode frequencies. This 1s most likely when there are atoms in the unit cell with widelydifferent masses. Thus for a linear chain with nearestneighbour 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 HartreeFock Hamiltonian
for which y> is a singledeterminant closedshell function, and that the Born—Oppenheimer approximation is valid. Some relaxation of these assumptions is possible. Coulson (1971) has extended the theorem to treat openshell cases; the theorem holds for any solution of the full HartreeFock 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 threefolddegenerate valence bands of most cubic crystals the interaction is written in terms of angular momentum matrices L (spinorbit 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 longwavelength lattice vibrations. The essential assumption is that interaction with a longwavelength mode is equivalent to the effect of a locallyhomogeneous 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 intervalley 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 shortranged. 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 IJV 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 ~ (1135) for Group IV valence crystals, ~(1419) for IIIVs, ~(2545) for IIVIs, 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 consistentlychosen 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
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A
II
ELECTRONIC STRUCTURE [ISOLATED DEFECTS
OF
Hf ,
4 EFFECTIVEMASS
THEORY
4.1. Introduction
THE methods known as effectivemass 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 effectivemass 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 perfectlattice potential. An (N+ 1)body problem is reduced to a onebody problem, and the complexity of the perfectlattice potential eliminated, entering only through an effective mass and a dielectric constant. Even manybody effects can be included through these parameters. 4.2. Simple effectivemass theory We begin by deriving the usual effective mass equations in the oneelectron approximation (Kittel & Mitchell 1954). The manyelectron extension will be treated separately later. For clarity we shall first concentrate on nondegenerate 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
EFFECTIVEMASS
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
EFFECTIVEMASS
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 oneelectron approximation but are otherwise quite general. 4.2.2. Approximations of effectivemass 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 kspace; there are corrections if they are not near an
68
EFFECTIVEMASS
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 tightlybound defects, at large distances from the defect. Some problems of matching a discrete region near a deep defect to an asymptotic effectivemass region are discussed by Opik & Wood (1969). Note also that the results can be valid for unbound, positiveenergy 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 effectivemass 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)
ikr 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 effectivemass equation We now make a transformation to reduce (4.2.13) to a realspace 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
EFFECTIVEMASS
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 orderofmagnitude 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 kspace (i.e. if F(r) is diffuse
EFFECTIVEMASS
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 212
= 0(k—k’) ? Ex~
(4.2.25)
m
in which
ee
3
 dr u*(0, r)nu,(0, r) cell
(4.2.26)
(2
EFFECTIVEMASS
THEORY
and mn = p+(1/2mc’)S A VV is an operator which reduces to p when there is no spinorbit coupling. In most cases we can choose the degenerate states u,(0,r) so that there are no offdiagonal 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 freeelectron
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
EFFECTIVEMASS
the order of the degeneracy from (4.2.27)
THEORY
73
and with offdiagonal terms derived
; h2 »:oar [Eo Home} t+ d?k vik—kieak'} te ae Aanarkakytat
= ().
(4.2.29)
xB
In appropriate cases U will have offdiagonal 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 indirectgap 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 effectivemass 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
EFFECTIVEMASS
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 effectivemass 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.
EFFECTIVEMASS
THEORY
75
4.2.7. Matrix elements in effectivemass theory
Consider
two
states, both well represented
by effectivemass
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/CPP) = {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 slowlyvarying operators. For offdiagonal elements the matrix elements of the Bloch functions give additional selection rules. Thus if kp,, = ko, when n # m then
CP OW,> =
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
EFFECTIVEMASS
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
CPIOP> ~ 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 longrange properties of U(r) are often known rather well, even though the shortrange potential may be both complicated and hard to determine. Thus the potential U(r) for an impurity of charge Ze 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 experimentallyobserved 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
EFFECTIVEMASS
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 sstate 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 shortrange corrections, and a purely shortrange potential for which the Coulomb tail to the potential is not too important. The deltafunction potential would be a typical shortrange 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 shortrange 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. Effectivemass theory for a manybody system The proof of effectivemass results in the oneelectron 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
EFFECTIVEMASS
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 manybody effects are included. 4.4.1. Manybody
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 HartreeFock approximation, and these can clearly be classified by the band indices and wavevectors of the oneelectron 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.
EFFECTIVEMASS
by
THEORY
W®)
It is convenient to introduce a oneparticle 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 oneelectron 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 oneparticle Schrodinger equation, but this does not imply that a oneelectron 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 manybody 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 Nelectron system, without the defect electron, is an insulator. When U; is zero, this requirement can
EFFECTIVEMASS
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 oneelectron case. The second stage involves rewriting of ¢nkU;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(1GoU)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 Tmatrix.
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 effectivemass approximation
the expansion is good when h?q?/2m* is large compared with U. B. The oneband, onesite 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 offdiagonal 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 excitedstate 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 offdiagonal elements of # between different ¢;.
116
6.1.2.
VARIATIONAL
METHODS
The fewelectron 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 Fcentre (essentially an electron bound to an anion vacancy; see chapter 15) or in the simple effectivemass donors (chapter 23) there is just one defect electron. In other cases, such as aggregates of Fcentres like the Rcentre (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 defectelectron wavefunctions are varied (the ‘fewelectron 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 HartreeFock oneelectron Hamiltonian. In these cases the oneelectron eigenvalues of y/(A)> which result are referred to the vacuum level. The energy zero is different from that for the effectivemass 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 Coulombcorrelation corrections and selfenergies (eg. 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 welllocalized 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 atomiclike (§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 nonorthogonal 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 fourcentre integrals occur ae (r,—
aedAA Oe R,)\e7/r; —r,la(r,
—R,,)a,(r,
—R,,)>.
Three and fourcentre integrals are difficult to evaluate accurately and rapidly. They present a serious barrier to accurate calculations for severalelectron 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? = ¢o6>—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 explicitlyorthogonalized 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 crossterms’ 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 freeatom energy levels may be used, and because freeion 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 twocentre 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 «P7H +hP>/ = )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 pointion potential and ionsize 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 longrange 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{UP)YVE)+P(VU)—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 highlyexcited states. Model potentials for isolated ions can be generalized to those for solids by including the displacement of the origin of energy by the pointion 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 pointion 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 pointion 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)
[Jamarenbs 10) 944](I1[
ueissnen
otusso1pAy
suonouny 10) ed4{ ({
ssajdsnp
19}e5
—
—
a
0
uuu)
Ge
aG
[RIUDIOg OBZ
p>41°%—
+4) m/[
d
D
4‘4/0°4 < D
LOY) DS
[JamaIeNbs 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 angularmomentum 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 ‘exactsoluble’ 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 longrange properties of the wavefunction are needed, it may be simpler to use the quantumdefect method (§ 4.3), recognizing the asymptotic validity of effectivemass 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 onecentre form of trial wavefunction (usually possible when we use explicit orthogonalization or a pseudopotential method) or a manycentre form, such as a linear combination of atomic orbitals. Both may involve two or morecentre integrals, but usually the integrals are fewer and easier in the onecentre 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 manycentre 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
14
) Ro,o(r
fr) Ro
SS) &
2:0
) Ro,olt
) ol RoH
aa a
3* oO
Roof)
) oT Ro,
Fic. 6.1. Onecentre versus fewcentre functions. The radial dependence of the sphericallysymmetric part of an L.C.A.O. function is shown. The function is constructed from hydrogenic 1sorbitals with effective Bohr radius a* at a distance R from a common centre. Values of R/a* are given on each plot.
simple onecentre 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 manyelectron, manyatom 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 valencebond (HeitlerLondon) method and molecularorbital 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 molecularorbital approach. These include semiempirical approximations for the integrals which arise in these treatments. Model calculations related to molecularorbital 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 selftrapped holes in ionic crystals (chapter 18) and for vacancies in valence crystals (chapter 27). 7.2. General methods 7.2.1. Molecularorbital and valencebond methods
The two fundamental methods are the valencebond approach and molecularorbital 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 (LonguetHiggins 1948; Coulson 1961, p. 155). Molecularorbital 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 oneelectro functions. determinan of the correspond By contrast, the valencebond 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 pairbond, 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 twoelectron states can be derived from given atomic orbitals in this approach than with the molecularorbital method. The molecularorbital and valencebond 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 valencebond 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 molecularorbital method is best at small separations, the valancebond method at large separations, but that in many cases both methods are comparable. The valencebond method becomes difficult to apply in practice unless all the overlaps of atomic orbitals are very small. The difficulty appears because the manyelectron 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 molecularorbital method uses instead a single determinant whose elements are ortho
normal oneelectron molecular orbitals, and the corresponding difficulty does not arise. It is possible to use orthogonalized atomic orbitals as a basis for the valancebond method (McWeeny 1954, 1955) or to work with a nonorthogonal basis (Newman 1971), but other difficulties may arise. Thus, Slater (1951b) showed
that
AND
MODEL
CALCULATIONS
135
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 spindependent 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
136
MOLECULAR
METHODS
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 twocentre system like H, lying along the xaxis 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, = (3282 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
0®
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 HeitlerLondon functions
oe
Tal
N,)
1 '
Ne Wp (A(F (rz) + br; )a(r)}
(7.2.9)
constructed from wavefunctions proportional to exp (—ar—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 molecularorbital 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
AND
MODEL
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137
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 molecularorbital and bond pictures. The distribution of charge in a bond also appears in the two basic ideas of stereochemistry :the concept of localizedbond 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 fourfoldcoordinated HVI and IIIV compounds (Coulson, Redei, & Stocker 1962). Pauling (1931) suggested the possible formation of hybrids from the 3d), 4s), and 4p) states of transitionmetal ions in ionic crystals. Owen & Thornley (1966) argue that this is not a good picture
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MOLECULAR
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because the energies of the 4s> and 4p) states are usually so much higher than the 3d) energy. However s—pd 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 fourcentre integrals to
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combinations of twocentre 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). Twocentre integrals such as are particularly affected. Secondly, certain finite integrals may vanish because the overlap S,, vanishes. Thirdly, the HartreeFock 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 nonorthogonalized 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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MOLECULAR
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the offdiagonal 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 nonorthogonal 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 onecentre exchange integrals.
(7.3.4b)
Neglect of diatomic differential overlap (NDDO) = 0 for twocentre integrals.
potential
energy (7.3.4c)
Zero differential overlap approximations are usually used in conjunction with semiempirical 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 freeatom 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 plike character of the valence electrons will be strongly affected by orthogonalization. E. Approximation of GoeppertMayer 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 bondorbital singled out in the integral and where all exchange integrals are ignored. The approximation has a history of expedience rather than accuracy (GoeppertMayer & Sklar 1938 ;Daudel, Lebfevre, & Moser 1959; Parr 1964: Coulson &
Kearsley 1957). 7.3.2. Semiempirical approaches
The semiempirical methods use atomic spectroscopic data to simplify some of the matrix elements in the HartreeFock equations. The data are usually ionization potentials, since these can be related to freeatom HartreeFock 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 semiempirical 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 firstprinciples 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 semiempirical 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 HartreeFock 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 oneelectron
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MOLECULAR
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matrix elements %; are approximated. If J; is the oneelectron 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 offdiagonal 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 175. 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 offdiagonal 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 threedimensional shapes were predicted adequately, but that
its spectral predictions were ‘miserable’. Blyholder & Coulson (1968) found that the L.C.A.O. molecularorbital HartreeFock 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 oneelectron 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. Semiempirical 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 longrange Coulomb field in these cases are omitted. The only longrange 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 effectivemass theory would count these as centralcell 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 effectivemass 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 selfconsistent 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 fourcentre 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: (®4Vgl@,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 slike valence orbitals appropriate to the atoms in question. 7.3.3. Model calculations
The semiempirical 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 exactlysoluble models, rather than attempting to approximate the matrix elements of an accurate Hamiltonian.
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Strictly, the use of either effectivemass 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 oneelectron potential of a perfect crystal was represented by an array of attractive or repulsive squarewells, 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. Thefreeelectron 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 onedimensional 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 molecularorbital theory. The method has been used for defect calculations. Fukuda (1965) combined it with the KosterSlater 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 deltafunction 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. Deltafunction 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 longrange binding, and it is reasonable to expect only the one slike bound state as for a 6function 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 oneelectron orbital ¢,. The weak covalency leads to corrections largely described by another oneelectron 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 ‘configurationinteraction’ method, which offers advantage in this case (Rimmer 1965 ; Hubbard,
Rimmer, & Hopgood 1966). The simplest description is the molecularorbital 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 oneelectron Hamiltonian h and the oneelectron 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 spinresonance spectrum, which monitors the spin density. If @, corresponds to a (usually occupied) ligand orbital and ¢y to a (usually empty) transitionmetal 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 molecularorbital 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 HartreeFock approximation. The methods can be extended beyond this, but, as always, at the expense of greater complexity. In the HartreeFock approach the manyelectron wavefunction is a single determinant of one electron functions. Choosing the optimum determinant by using the variational principle, the oneelectron wavefunctions satisfy a set of eigenvalue equations which must be solved selfconsistently. In the method of local orbitals the problems of determining the eigenvalues and of obtaining selfconsistency 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 chemicallyconstant 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
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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 oneelectron wavefunctions is a slight generalization of the standard HartreeFock equation. It can be written
Fo; = 2" ojfu
(7.5.1)
The @; are not necessarily eigenstates of the HartreeFock 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 HartreeFock operator is the oneelectron 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 HartreeFock 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 oneelectron eigenfunctions, we shall often make use of the properties of the Fock—Dirac oneelectron 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 oneelectron operator A
AP = peApe.
(7.5.8)
whose effect on orbitals within the HartreeFock 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 HartreeFock equation. If T is any nonsingular matrix then the transformation
¢Te=¢'
7.5.11)
changes the orbitals and also changes the matrix elements of the HartreeFock operator. But the density matrix is invariant pr > T 'prT = pr,
(7.3.12)
and the eigenvalues of the HartreeFock operator are similarly unchanged. Finally, we note that the HartreeFock 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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MOLECULAR
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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. Selfconsistent solutions We now return to the problem of solving the selfconsistent equations for the orbitals ¢;. There prove to be far more convenient sets of equations for selfconsistent solution than the usual eigenfunction form of the standard HartreeFock equations. The ¢; satisfy not only the equations
Fo, = 0" oifi
(7.5.18)
but also, adding eqn (7.5.9), the modified HartreeFock 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 HartreeFock operator be diagonal, so that
Fy, = &y;.
(7.5.21)
With this choice the &; are just oneelectron energies in the sense of Koopmans’ approximation. The similar condition for the modified HartreeFock 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 selfconsistency problem can then be solved with comparative ease, giving wavefunctions which are linear combinations of the oneelectron eigenfunctions. The eigenfunctions and eigenvalues themselves are found separately by diagonalizing the HartreeFock 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 HartreeFock 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. Welllocalized 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 squarewell 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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MOLECULAR
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eigenstate of F is equal to the number of electrons we wish to localize at site A. The modified HartreeFock 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 freeatom orbitals as closely as possible. The ¢; are then freeatom 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 freeion orbitals. The constraint is already similar to the ‘welllocalized’ constraint with a nonlocal 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. Energylocalized orbitals. Here the Coulomb selfrepulsion 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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and the stationary condition gives E,,, = E,. An equivalent description can be derived without the orthogonality condition, but has the disadvantage that E,,, is nonHermitian in general.
D. Screened orbitals. Here the localized orbitals are chosen to minimize the HartreeFock energy of a part of the whole system, usually a single atom or ion. The HartreeFock energy of the subsystem is extremal, and the modified HartreeFock 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 HartreeFock 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 HartreeFock 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 HartreeFock 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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MOLECULAR
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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 intraatomic overlaps Ciiz
(75,31)
2 Pad) (Saiaj— Ll)pair’),
which are an indirect consequence of the finite overlap integrals, interatomic overlaps
Taz = YY (GaSaibsbajt OBjSpj aiPai> BHA i,j
(7.5.38)
and extraatomic 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 HartreeFock 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 selfconsistent 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 freeion or freeatom 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 HartreeFock 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 firstorder terms, then the approximate matrix elements can be factorized as above. If we choose local orbitals as a basis, rather than HartreeFock 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 freeion HartreeFock orbitals closely, and this further facilitates calculation.
B. Development in powers of overlap. perturbative solutions. The modified HartreeFock equations can be written (Fa t+V,—Ex) Gai =
— PrWocPrPai
(7.5.43)
The righthand 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 onesite 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 defectformation 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)
Higherorder 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. Higherorder terms appear in both VY, and V,. Those from V, are just the anharmonic part of the perfectlattice 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 twobody atomatom 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 perfectlattice 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 secondquantized 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 defectelectron—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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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 forceconstant 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 hostlattice masses is not valid if higherorder 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 zeropoint energy) which result from the change in phonon frequencies and a massdependent admixture of the modes by the higherorder 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 defectelectronlattice 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 defectelectron wavefunction. It includes, for instance, the changes in energy from shortrange repulsive forces when an anion is removed in forming an Fcentre 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+ (higherorder 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
higherorder terms ~3u. B(A,).u+(A—A))A. ut lA’ +5A'(A—Ap)*
2
+...
(8.3.3)
As a rule, both F, and the higherorder 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 longrange 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 defectelectron 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 higherorder 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 forceconstant 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 longrange Coulomb interactions. Kanzaki resolved this difficulty by Fouriertransforming 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)
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F,(/) = ~ DFo@e co
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167
(8.3.12)
and the transform ofthe forceconstant 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 longrange 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 rigidion 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 longrange 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 crossterms 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 higherorder 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 higherorder 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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This can be written more simply in terms of a distortiondependent
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 distortiondependent 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 higherorder 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 secondorder terms can be managed without diffi
culty. Few displacements are large enough to give significant 1igherorder 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 symmetryadapted 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
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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 defectelectron 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 twobody 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 timeconsuming and, in cases where there are longrange forces, especially difficult. A much more satisfactory method has been given by Norgett & Fletcher (1970), using a variablemetric 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 longrange 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 forceconstant 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 moleculardynamics 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 longrange 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 moleculardynamics 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, longrange 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 fullscale 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 shortrange terms will be modeldependent. 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 shortrange 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 JahnTeller effect, and the offcentre 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 JahnTeller 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
interatomic 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 MottLittleton 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 (12). 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
—
+2p.
(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 MottLittleton 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
LATTICE
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177
of an ion in Region  is taken as the electric field multiplied by the ionic polarizability. The point polarizableion 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 nonlinear 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’) falloff 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 Fcentre (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 staticlattice 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 computergenerated response functions. Although the staticlattice 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 forceconstant, i.e. of terms in Vp(u) which are secondorder 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 forceconstant are higherorder 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 secondorder 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)
LATTICE
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179
in which the last term is the perfectlattice 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 JahnTeller coupling the solutions reduce to those of a twodimensional harmonic oscillator whose eigenfunctions are Xnm(P)
=
Rajmi(P) Cue.
(8.4.23)
Here m is an integer less than n. LonguetHiggins 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 lowlying states can be expressed as the sum of three terms. The first is independent of the JahnTeller 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 JahnTeller 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 freerotor 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 spinresonance work. So far our only approximations have been the use of an adiabatic approximation and the neglect of higherorder @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 righthand side of (8.4.22). This should be valid for the lower energy levels with strong JahnTeller coupling. In practice the condition for validity JahnTeller energy > zeropoint energy holds in almost all cases where the JahnTeller 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 lowlying states associated with the upper energy surface (Sloncewski 1963; Miiller 1968a). However the centrifugal term is negligible for the lowlying 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 Estate 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 lowestorder perturbation of the correct
symmetry
V3() = A3(p)p* cos 3
(8.4.28)
in a manner reminiscent of the hinderedrotor 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
DISTORTION
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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(yj3 + 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 slowrotation limit.
I. Tstate equally coupled to E and T modes. This model provides a ‘emarkably good description of the excited state of the Fcentre 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
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Static
Dynamic
oat
limit
—
40 Blaa Fic. 8.5. Effects of anharmonicity on an Estate 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
41
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, JahnTeller 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 Fcentre 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 hreedimensional harmonic oscillator wavefunction and an elecronic T,state; they are independent ofthe electronlattice 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 owlying 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.
208
LATTICE
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IV. Tstates coupled by T modes. Caner & Englman (1966) have analysed numerically the coupling of T modes to Tstates. 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 JahnTeller 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 JahnTeller 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 Tstate 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 strongcoupling limit.
LATTICE
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209
JahnTeller 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 crossterms 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 crossterms with no diagonal elements among the eigenstates of the term involving the principal mode. Thus the effects of the crossterms 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 groundstate 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 Estate 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
210
LATTICE
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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 JahnTeller 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 JahnTeller effect : the slowrotation 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.
LATTICE
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211
The ‘displacedoscillator hypothesis’. For a strong JahnTeller 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 ‘displacedoscillator 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 electronlattice coupling is diagonal, that is for nondegenerate states and the exceptional case of Tstates 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 Estate 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 JahnTeller terms and appropriate higherorder 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 higherorder terms, the 0mode 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 approximatelyorthogonal wavefunctions obtained in the same way as one constructs Wannier functions from atomic orbitals. In special cases, such as Tstates 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 linearlyindependent electronic functions. If the electronic degeneracy Np is less than Ny, this is not possible. Thus for Estates coupled to E modes Ny = 3 and Np = 2; for Tstates 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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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 JahnTeller 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 Estates 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 Tstate 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 offdiagonal matrix element B, of the form (8.4.53), determines the splitting. As the JahnTeller 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 JahnTeller effect. The sign of B is important, since it determines whether the lowest state is degenerate or nondegenerate. For a Tstate coupled to E modes f is identically zero, and there is no tunnel splitting. For Estates coupled to E modes and for Tstates 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 Estates 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 ¢6Q\@> term contributes, and this is negative because of the —4 coefficient in ¢,,>. In the same way for the Tstates 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 nonpathological system should be nondegenerate, 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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appears because of the novel boundary conditions. Thus for the Estate 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 JahnTeller case, however, the doublydegenerate vibronic state has fewer nodes than the nondegenerate state. That the degenerate state lies lower than the nondegenerate one also fits in with the fastrotation 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 Estate coupled to E modes the results depend on the detailed form of the higherorder 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 zeropoint energy. Caner & Englman (1966) evaluated the tunnel splitting for a Tstate coupled to T modes, and found it to be
4Bry X 132Ey, exp(—124E,q/ho,),
(8.4.61)
where Ej; is given in (8.4.10). We now discuss the timedependent 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 JahnTeller 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 displacedoscillator approximation. At the lowest temperatures, these occupation numbers will be zero,
and there will be zeropoint ‘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. 1434 and 1767), 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 twophonon 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 displacedoscillator model, when the lattice potentialenergy 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 JahnTeller 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 (slowrotation limit), the excited states are best described by a dynamic effect (the fastrotation limit). Again, we expect thermallyactivated 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 JahnTeller 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 onephonon 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 nonradiative 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 spinorbit 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 offdiagonal elements (~ # f) the reduction factor may be much less than unity. Offdiagonal elements among vibronic states may be much smaller than the elements between the corresponding electronic states. This reduction of offdiagonal elements is known as the Ham effect. It is easily seen that the reduction will be greatest when there is a strong JahnTeller 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 JahnTeller 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 JahnTeller interaction reduces operators of different symmetry. For even if we diagonalize the JahnTeller coupling, offdiagonal 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 offdiagonal electronic matrix elements among these states, one must consider the admixture of other vibronic states. In essence, the secondorder 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 offdiagonal 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 offdiagonal
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. Symmetryadapted 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 JahnTeller effect. It is convenient to introduce a number of standard matrices M: four for the (2x2) matrices of Estates and nine for the (3 x 3) matrices of Tstates. 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 Estates %, = —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 Estates i
U, =
1
0
0)
1
—1
oa
i
1
0
0
0
Cae
+1
0
(ie

1
0
Matrices for Tstates 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 Estate 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 JahnTeller coupling
at ee
(8.4.73)
q ~ 12S
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 (—1974 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 0484 < q < 05. He conjectures that 0484 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 spinorbit coupling or Zeeman interactions give a term in .%,. Some of these cases will be discussed in more detail later. The case ofa Tstate 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 JahnTeller 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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both E and T modes when they both produce equal JahnTeller
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 gfactor and the spinorbit 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 JahnTeller systems which could take up different equivalent orientations. In this section we contrast the two situations. Electronic degeneracy plays no role in nonJahn—Teller systems. This has two main consequences. First, the nuclear potential energy surface is singlevalued, 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 JahnTeller systems are most apparent when we discuss reorientation. In both cases it is easiest to discuss two extreme limits—the fastrotation limit, when the reorientation
is rapid, and the siowrotation 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 twodimensional 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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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 JahnTeller 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 JahnTeller systems and others in the slowrotation limit occurs because the number of potential minima for nonJahn—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 JahnTeller systems, as for cases of accidental degeneracy. The large number of minima means that more independent transition probabilities must be treated. In principle the lowlying 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 slowrotation limits A quantitative comparison of the fast and slowrotation limits can be made which shows the way the two extremes approach each other. Although we derive the results for systems with singlevalued potential energy surfaces, the general features are probably valid for JahnTeller systems too. The fastrotation limit starts with freerotor energy levels and the effects of lowsymmetry potentials are considered. This gives the ‘hinderedrotor’ picture (Devonshire 1936; Sauer 1966: Jain & Tewary
1972; Beyeler 1972). The slowrotation 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
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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 hinderedrotor 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 hinderedrotor model constrains the defect to a sphere on which the potential energy varies as Vir)=
Si
lxtayt+203
oa
a
3)
(8.5.1)
Beyeler has given results for a hindered rotor with sixthorder 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
—3K.
(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 displacedoscillator 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?
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(a)
(b)
Fic. 8.8. Contrast of hinderedrotor and tunnellingmodel 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 welldepth and J, the moment of inertia of the rotor. A. Relative energy splittings for (100) minima (k > 0). We denote by A, the tunnellingmodel 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 JahnTeller 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)
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227
®
0
20
40
60
80
,
@&
Fic. 8.9. Energies in the fast and slowrotation 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 largek limit of the hinderedrotor model. Agreement is not significantly
improved by the small modeldependent 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 largek limit of the hinderedrotor 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 hiaderedrotor
picture becomes
inefficient
228
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E(Ax,)—E(Aig)} { (Ay,)} {E(P)—£
Fic. 8.10. Energies in the fast and slowrotation 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 slowrotation 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 hinderedrotor 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 hinderedrotor 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 slowrotation 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 JainTewary
2 —
11273 14214
0558 0547
0270 0240
0136 0115
0072 0059
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 JahnTeller 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 freerotor 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 twodimensional (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, higherorder 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 centreofmass 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 slowrotation 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 latticestrain 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 effectivemass theory of chapter 4 to include electronlattice 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 IIIV 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 selfenergy (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 boundpolaron 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 timedependent 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 zeroorder Lagrangian %. Then (Loxact Lo) is treated as a perturbation on the motion. Unfortunately, very few convenient % are known, and the lowestorder 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 lowestorder corrections to the groundstate energy may be found by secondorder 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 ptype. 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 highspin states are favoured because of the signs of the important electronelectron interaction integrals. Loosely one might argue that electrons with parallel spins must have orthogonal spatial oneelectron 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 twoelectron 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 transitionmetal ions in crystals where degenerate levels are only partly filled. Other cases are the twohole centres found in oxides. Whereas twoelectron centres have singlet ground states, twohole 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 intraatomic exchange interaction which gives the triplet ground state. These systems also demonstrate that configuration mixing may be important in determining the groundstate 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(v3»¥4)?
(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 modeldependent. 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 everincreasing 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 angularmomentum 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 wellknown form ¢ JQ*(OD.
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
>—Ip+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 nonlinear effects. Then, for atom I, the operator Q is
Q=—
JY
0
cos (o(t—215/¢)} =,
(10.1.10)
Vs
electrons s
and the crosssection 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 electricdipole 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 crosssection 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 crosssection 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*? = 3R,,?. 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 highfrequency 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 crosssection 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 lineshape 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 linewidth 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 lineshape 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 Fcentre in
KCl. Most other theoretical checks have been made on the related
OPTICAL
PROPERTIES
279
‘rcentroid’ 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 Fcentre 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 Fcentre. His results are given in Table 10.1. Other estimates of the accuracy of the Condon approximation can also be estimated from the stressdependence of parameters, e.g. the photoelastic constants (Aggarwal & Szigeti 1970), which measure straininduced 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 14+0:3 12+06 12+0:3
2:4+02 5140:3 34+04 33+0:3 71+07
Results
taken
from
Schnatterly
(1965).
A
is the
integrated absorption and E the mean energy of the Fcentre transitions.
280
OPTICAL
PROPERTIES
10.4. The lineshape function
The crosssection is usually written in terms of a lineshape function G(q). It is this function which is actually calculated when there is electronphonon 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{nnrilmt}> = €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}>? = 93dQ 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)
OPTICAL
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 crosssection (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 Qdependence 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 electronphonon 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, nonradiative 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 lowerlying 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 socalled zerophonon transition. Secondly, many authors have attempted to derive lineshapes by weighting FranckCondon 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 Estates coupled to E modes (O’Brien 1965, 1971, Longuet—Higgins et al. 1958) where predicted lineshapes are too symmetric and have a spurious
zero. Other
slightlydifferent
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 lineshape 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 lineshape function as G(@)
=
Nd, [TotNy» Ny 3 Gj, ae (w =
WF nn’)
(O17)
298
OPTICAL
PROPERTIES
The importance of the HuangRhys 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)? = 1S+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 electronphonon 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 welldefined 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 electronphonon 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 electronlattice 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 linewidths 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 configurationcoordinate diagram One of the few strongcoupling models which can be solved analytically for the lineshape function is the Huang—Rhys model. The qualitative features of this model—e.g. the temperature dependence of the linewidth—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 configurationcoordinate 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 electronphonon 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 configurationcoordinate 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 lineshapes 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 spinallowed magnetic
330
OPTICAL
PROPERTIES
dipole transition *A, +*T,. The HuangRhys 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 welldescribed by the Pekarian form (10.7.41), apart from interference terms. The sharp band is the spinforbidden 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 spinallowed states associated with *T, by the spin—orbit coupling. When the spinorbit 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 lineshape can be found by our earlier methods, or equivalently by writing down the timedependent Schrédinger equation for the state amplitudes. The Schrédinger equation proves easier to generalize to include the Estate when spinorbit 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 zerophonon 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 spinorbit 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 spinorbit 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 raregas 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 ~ 13, and the intensity of the structure is very close to that predicted from the known spinorbit 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 freetobound 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 lineshape 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 oneelectron 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 freecarrier 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, nondegenerate, and parabolic,
E(k) = h?k?/2m*.
(10.11.1)
These approximations will be relaxed in later sections. The absorption is proportional to a lineshape 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?km . 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/dfzx,,
(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 constantenergy surface, rather than a sphere
334
OPTICAL
PROPERTIES
in kspace. 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 nonparabolic 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 phononbroadening, 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 lineshape 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 effectivemass theory and associated with band i. Band i is distinct from the conduction band c since it is not consistent to assume that a groundstate i) is principally derived from the conduction band and yet that the conductionband 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 quantumdefect method (§ 4.3). This, and its analogues (e.g. Adelman 1972), also exploit the feature that the longrange 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 deltafunction 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 lineshape, it is useful to introduce X, defined by
X = (vka*)? = 2m*(hw— E,)/mF Eg.
(10.11.13)
The lineshape 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 longrange 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 deltafunction 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.
Deltafunction limit
o nN
Hydrogenic limit
S(v,X)/S absorption Normalized max aeie 00
O72
ere LL Teeeal 2)04 80:6 F08OnTl
Ee
18 Omi
14
Fic. 10.11. Dependence of the photoionization crosssection on initial state. Results from the quantumdefect 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 effectivemass 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 planewave 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 shortrange potential. The second change includes the effects of longrange 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 squarewell 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 squarewell perturbation is reasonable, for example, for the F’centre in alkali
halides, transitionmetal ions in ionic crystals, and trapped electrons in metalammonia solutions. The potential has resonances, and these affect the lineshape. 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 pstates, which dominate in the optical absorption from an slike 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)
Crosssection
02 04 06 08 Transition energy (eV)
10
(b)
Crosssection
00
IS Transition
oa
T re
oH
energy (eV)
Fic. 10.12. Dependence of the photoionization crosssection on final state. Calculations of Kajiwara et al. contrast results for planewave final states (I) with those (II)
including the eflect of a squarewell potential of radius 5 A and depths (a) 0564 eV, (b) 0939 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 crosssections using planewave final states with those using correct eigenfunctions of the effectivemass 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 moresymmetric 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 crosssection (hw,,—E,) and E, depends on the details of the model. In the quantumdefect 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 shortrange 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 Fcentre. 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 deltafunction 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 lineshape 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 twostep process: first the absorption of a photon and then a separate transition involving phonons. Both transitions are met in practice. Singlestep transitions occur when the nophonon 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 ‘momentumconserving’ 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 twostep 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 twostep 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 crosssection a; is of order 7,5: But for the twostep process, the defect makes a direct transition with crosssection o,,,. The subsequent nonradiative 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 twostep 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 inband 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. Inband 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 lowlying 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 lowlying 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, Xray or neutron scattering, the Mossbauer
effect,
or
spinlattice
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 doubletime thermal Green’s functions, as pioneered by Zubarev (1960). In this approach thermallyaveraged 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 JahnTeller 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 threedefect 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 runningwave 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 defectelectron 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 onedimensional 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 nearestneighbour 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 nonvanishing 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)7o(@,)? , 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 wellknown 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 Fcentre 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 firstorder 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 spreadout states corresponding to the K band. We shall consider moment analyses of Raman spectra in § 12.5.3, and discuss in chapter 14 the nonradiative 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 spinorbit 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?+5Etr)(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 fielddependent 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 higherorder 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 spinorbit coupling is important. Moreover
€(r), which describes this, is dominated
by the regions
close to nuclei, and it is common to have a manycentre problem. This manycentre feature can make it hard to retain the desired gaugeinvariance. We discuss these three difficulties later in connection with spinresonance. 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 magnetooptic phenomena such as Faraday rotation. 12.3.2. Zeeman effect in atoms and effectivemass 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 effectivemass 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. Lowand highfield limits. Twoextreme cases may be distinguished, depending on the relative sizes of the spinorbit 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 weakfield 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)
=) (857)
—
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 2pstate are
Quartet (L = 1,5 =4,J = 3)
g = 4742,
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 strongfield limit we must decide which J states will be admixed. For the hydrogenic centre described above the two multiplets derived from the 2pstate (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 spinorbit 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 smallfield limit is observed for a number of rareearth ions, and the highfield 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 lowfield behaviour of a pstate. The figure is drawn for a spin— orbit coupling A > 0 and for y < g,/2. In the highfield limit the states L,,S,> are given; the corresponding energies are BH(g.S,+yL,).
lattice. For an sstate (L = 0) there is no direct orbital contribution to the magnetic moment, but the spinorbit coupling causes deviations from the freespin gvalue. Calculations of gfactors 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 gfactor for an electron in the conduction band is given
in terms of the band gap E, = (E,o—£E,), and spinorbit splitting of the valence band A by ex
2A El
e
os i). m*
(12.3.9)
The correction to the freeelectron 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 conductionband 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 gfactor 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 spacevariation 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 gfactor 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 spinorbit splittings and possibly splittings from noncubic 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
LinChing & Wallis (1969). C. Excited pstates 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 spinorbit 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 spinorbit 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 2pstate 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 effectivemass Hamiltonian has the form Zz
=
sat V(r) +4ho(xp, — yp,) + 3moz(x? + y’) m
(12.3.19)
for a field along the zaxis 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 twodimensional 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 spinorbit coupling, so the highfield theory discussed earlier applies. The splitting is then described by a gfactor 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 zaxis, 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 lowfield 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 largefield 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 onedimensional 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 lowfield wavefunctions go over into Gaussian highfield functions depends on the system. Horii & Nisida (1970, 1971) cite F ~ 015 for donors in Ge. The intermediatefield 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 (Is2p,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 electronphonon coupling is compared with Stillman, Wolfe, & Dimmock’s results for GaAs. In (b) Larsen’s calculations for « = 04 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)
00 03 05 07 10 15 2.0 25 30 50 25.0 1000
1.000 126 139 151 1661 187 2:04 2.19 2:32 275 4.733 7457
0.2500 EN; = = 09118 = = ae = L717 3183 5261
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 nonparabolic 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 parabolicband value at large fields. Further, Larsen has discussed the effects of the electronphonon 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 Zeemansplit 2p level and the 1s level of the donor may be very close to the longitudinaloptic 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 highfield 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 lowfield limit. However, it is useful to have results for a donor with small axial
Continuum
G
iSat eT
ee
ae 4
"Continuum
ABBB
c
Seed  SA
Lefthand
Righthand
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 unboundelectron gfactor. 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 electricdipole 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 Twentyone 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 (Opila>
2
rea (12.5.21)
where the incident polarization and final polarization are i and j respectively. The total crosssection is
8 d Cro = ul (sal
(12.5.22)
ij
and is independent of i. Henry (1966) has discussed the total crosssection 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 Fcentre 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 firstorder Raman effect and is the one we want. Other terms give higherorder Raman contributions,
e.g. twophonon spectra and corrections of order {E,,,,/(E—,)} to the onephonon 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?)? = 0665 x 107 ®A’, the Thomson crosssection for freeelectron scatter, and f is the oscillator strength, including localfield 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 twolevel 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 welldeveloped algebra of angular momenta can be exploited. The spin Hamiltonian formulation consists of constructing a spin operator Q,, whose matrix elements within the effectivespin 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 lowlying states. This concept of an effective spin is exactly analogous to the isotopic spin of elementaryparticle 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 nonKramers’ doublet in an electric field, where one must choose between the simplest form without the desirable symmetry properties (Bleaney & Scovil 1952) and other, lesssimple 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 freeatom 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 principalaxes 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 effectivespin operators and externalfield 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 loworder 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 effectivespin and I a nuclear spin. The terms are, respectively, the Zeeman energy, the hyperfine interaction, the zerofield 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 magnetomechanical 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 defectelectron 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 spinorbit coupling
Heo = Y ElrL; . S,
(13.3.5)
which is important here. The spinorbit Hamiltonian is shortranged, and is only significant close to each nucleus. We shall use
ELECTRON
SPIN
RESONANCE
443
RussellSaunders coupling, so that the electronelectron 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 crystalfield energies or electronelectron 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 gaugeinvariant. 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 gaugeinvariance, 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 gaugeinvariant, i.e. independent of x in (13.3.6). Slichter (1963) has shown that if the wavefunction can be written as a sum of nonoverlapping 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 gaugeinvariant combinations. He derives an expression for the gfactor of an orbitally nondegenerate 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 spinorbit coupling, unlike Griffith, and do not need to divide the wavefunction into nonoverlapping 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 gaugeinvariant 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> pstates) 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 spinorbit coupling can remove the quenching to give L a finite expectation value. The point is that offdiagonal elements of L may be finite. The spinorbit coupling AL. S, can mix in excited states e> into the original state 0>
0> + 6> = O > le>
(13.3.9)
Eo
The crossterms 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
(135201)
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/?,
(133913)
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 dstates by
keel
Pyes TRIS ae
(13.3.14)
For any reasonable admixture, k < 1. Most experimental results lie in the range 085 to 100 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 spinorbit 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 gfactors
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 spinorbit 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 irongroup ions in MgO. The gfactor 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 nontrivial. Two points have emerged. One freespin 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 gfactor.
is that the deviation of g from the where J is the spinorbit 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 spinorbit coupling has one direct application. The spinorbit 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 gshift 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 holecentres. 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 spinorbit 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 gfactor 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 electroncentres is no longer valid. B. Symmetry and the gtensor. We now relate the anisotropy of the gtensor 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 gfactor we are calculating more than the other states, which are mixed in by the spinorbit coupling. For the Fcentre, an electron trapped at an
450
ELECTRON
SPIN
RESONANCE
anion vacancy in cubic symmetry, the ground state is nondegenerate. Under uniaxial stress the gfactor becomes anisotropic because the degeneracy of the excited pstates 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 gfactor. 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 singledout 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 nondegenerate 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. Secondorder 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 zerofield 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 spinorbit 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) {82335(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 dipoledipole 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 twoelectron 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 gfactor in eqns (13.4.4}13.4.6) is strictly 2.0023, the freespin 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 gfactor appropriately and to let S be the effective spin. The argument is that the gfactor 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 spacefunction 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 lowestorder configuration interaction, two types of configuration can be admixed: those (J>, say) differing from the dominant configuration 0> by a single oneelectron state and those (\II>, say) differing in two such states. Other states are admixed only by higherorder perturbation in the Coulomb interaction between electrons. Since #4, of (13.4.4) is a twoelectron 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 intraatomic 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 oneelectron 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 pointion part of the interatomic 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). Offcentre ions, like Lit and Cu* in KCl, can be treated by natural extensions of this case (Wilson & Blume 1968). The Fcentre 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 nearestneighbour 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 powerlaw relationship and the neglect of changes in force constant. The powerlaw fails if there are both longrange and shortrange 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 rareearth ions in alkalineearth 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 Fcentre hyperfine structure. Longrange 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 NONRADIATIVE 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 nonradiative transitions between bound states. Freecarrier recombination is particularly important in two contexts. The freecarrier lifetime is an important parameter in the possible use of systems in electronic devices. Further, the relative importance of nonradiative and radiative processes affects both the efficiency of luminescent devices and the possible creation of defects by optical irradiation (e.g. Pooley 1966). Nonradiative 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. Nonradiative 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 nonadiabatic 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)
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NONRADIATIVE
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. Configurationcoordinate diagram for nonradiative processes.
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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 timedependent 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 zerothorder 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 timedependent 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 lowestorder 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 nonadiabatic terms in the Hamiltonian or from the configuration dependence
480
NONRADIATIVE
PROCESSES
AND
INTERACTION
THE
of the wavefunctions (cf. § 10.9). We shall give expressions for the nonadiabatic terms, since the other (Herzberg—Teller) case can be obtained readily by comparison. The nonadiabatic 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 spinorbit coupling, rather than the nonadiabatic 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,
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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
NONRADIATIVE
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 weakcoupling 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
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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 ‘energygap law’. Riseberg & Moos (1968) have verified the dependence over a range of 10° in W in extensive work on rareearth ions in ionic crystals. They have also given a simple derivation of the general form. Suppose the probabilities of nth and (n—1)thorder 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
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
lowest energeticallyallowed 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 onephonon 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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485
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
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
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 Fcentre. 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 isotopesubstitution, 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 HuangRhys theory has been applied to radiative and nonradiative recombination in alkali halides (Pooley 1966). The matrix elements for the nonradiative 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
OF
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487
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 nonradiative 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 Mcentre) configuration of the excited state. The large value of So, affects the nonradiative 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 strongcoupling 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)
488
NONRADIATIVE
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AND
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INTERACTION
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 thermallyactivated 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 nonadiabaticity 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 thermallyactivated transition probability, it must not be assumed that thermal activation always appears. In the weakcoupling 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 thermallyactivated 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
OF
FREE
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WITH
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489
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 fewphonon 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 6function
{
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 onephonon terms
yal{os +o)+5(@,—@)},
— (14.2.48)
490
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the third term gives twophonon 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 lowtemperature limit, the sums )’; reduce to a powerlaw 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 nonradiative transitions which have been studied in most detail quantitatively are those of paramagnetic ions in insulators. The transitions studied are basically spinreversals, in which the ion goes from one Zeeman level to another. Typical energy separations are in the range 0110 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 externallystimulated 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 03cm~', 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 spinorbit or crystalfield 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 spinlattice relaxation.
range 201000 cm’. These excited states may be assumed unoccupied at the temperatures at which experiments are carried out. A. Nature of the nonradiative 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 onephonon processes, a single phonon of energy 6 is absorbed or emitted. In twophonon 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 staticlattice 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 spinlattice 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 freeion Hamiltonian, Vj represents the crystalline environment, “%pQ is the spinorbit coupling, and #7, and #7; are the orbital and spin Zeeman energies. Occasionally other divisions
492
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are more convenient. Thus, if an ion has an orbitallydegenerate ground state in the absence of spinorbit 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 resonantRaman or Orbach process. We discuss these in turn. Nonresonant twophonon 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 nonKramers’ 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 timereversal 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
Aha,
adh pen Ost Onda 71)
498
NONRADIATIVE
The Raman
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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 latticeinduced transitions between eigenstates of the sum of “%,, and the singleion 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
energyconserving delta function, and the densityofstates 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 nonKramers’ 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 nonresonant contributions to the twophonon 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 nonresonant 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 crossterms in M,M,. These are predicted to show a characteristic dispersionlike 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 ratedetermining 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 nonresonant 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 nonresonant 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 nonresonant absorption acoustic field: internal friction. Thus for dielectric loss one might have resonant absorption from purely electronic transitions and nonresonant 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
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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 nonresonant behaviour can be seen.
A. Nonresonant 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 (spinspin 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 weakdamping 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) (212) (ae)
The timedependence 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 zdirection and a smaller field of constant magnitude rotating in the xy plane. We now seek the steadystate 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 zerophonon line
A. Linewidths and lifetimes. The zerophonon line contains contributions from all transitions in which the total number of phonons is unaltered. The mean energy and linewidth of the zerophonon 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 temperatureindependent contributions may be important. One comes from random internal fields, which broaden the zerophonon 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 lineshape 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 selfenergy operator of the oneparticle Green’s function. The linewidth I is obtained from a twoparticle Green’s function. There proves to be a certain class of terms in I’ which violate
506
NONRADIATIVE
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the WeisskopfWigner 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 linewidth. A second general result concerns the shifts of zerophonon 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 energylevel 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 Rlines 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 coworkers (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 zerophonon 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 secondorder 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 longwavelength 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. Linewidths and shifts. Two main classes of methods are used here; the first includes moment methods, and the second uses
508
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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 electronlattice coupling so that only the shape of the zerophonon line is considered. Only the nonresonant 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 zerophonon 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 linewidth 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 zerophonon line should be Lorentzian, but with a faster falloff in the wings of the line. This shape can be represented by a cutoff Lorentzian (Kittel & Abrahams 1953) which is zero beyond a particular separation from the centre of the line. The cutoff 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 cutoff 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 lineshifts 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 configurationcoordinate 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 BoseEinstein value. Thus
A(T) = nh(w, —@,),
(14.2.100)
which corresponds to(14.2.98) apart from a temperatureindependent 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 nondegenerate states, the centroid of the broad band and the zerophonon line shift by the same amount with temperature. The linewidth 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 crosssection 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
)
510
NONRADIATIVE
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THE
This result is derived using approximations leading to (12.5.24) for the Raman crosssection. Both resonant and nonresonant cases (Fig. 14.4) are readily treated in the Green’s function approach, where one examines the effect of the electronlattice coupling on the oneelectron 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 oneelectron 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 nontrivial 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 oneelectron 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 zerophonon line. The simplest contributions to Aand TI are the ‘direct process’ terms, which appear because there
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S11
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 higherorder 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 resonantRaman processes of spinlattice 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 zerophonon lineshape 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 oneelectron 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 halfwidth 6; wp lies at energy A below the
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energy of the electronic transition in the absence of coupling. The resulting lineshapes 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 lineshape depends on 6, the width of the band of phonons to which coupling occurs, relative toits peak value I). As 6is reduced, the threepeak 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 threepeaked 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 crosssection o, by
ty! = Now,
(14.3.2)
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513
where there are N defects per unit volume and v is the electron velocity. The effective crosssection for use in transport problems o, is related to the differential crosssection 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 crosssection depends only on the angle through which scatter occurs, not on the absolute orientation in space. The corresponding expression for the total crosssection 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 crosssections are identical for swave scattering, in which No is the only finite phase shift, 4n ay
=
OTono
=
.
psi’
No
(14.3.8)
We shall now calculate crosssections 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
514
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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 distortedwave 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 longrange parts of the perturbation when the shortrange terms can be solved exactly. In the opposite extreme of slow electrons, effectiverange 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 swave 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 crosssection 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 swave 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 lowenergy electrons, typically those whose energy satisfies S16) ee Sik, (14.3.15)
the crosssection 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.
516
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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 crosssection is different for carriers with spins parallel (o;) or antiparallel (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 crosssection 1s
o = (1—P.P,)o,+4(3+ P.P,)o, + spinindependent terms. (14.3.19) The Erginsoy result A ~ 20 is the spinaveraged result
6 = 46,+36,. However, Schwartz (1961) has obtained crosssections separately
(14.3.20) the singlet and
k?a5 = 488 (ka*) k?o_ = 104 (ka*) }. k?é = 200 (ka*) A number been made.
of measurements For semiconductors
of spindependent (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 Fcentres 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 antiparallel 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 crosssection 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 smallangle 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 crosssection. 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 crosssection 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,
(
)
518
NONRADIATIVE
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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 crosssection 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 phaseshift analysis by Blatt (1957) examines this lowenergy case. Typically, the Born expression is good above 100K, but then falls off too fast with temperature. The conductivity crosssection 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 (singlevalley) 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 effectivemass 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 Swave 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 crosssection 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 threevalleyed 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 effectivemass 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)
36 x 10?
Neutral defect (14.3.16)
E~? (low energy)
13x 10°
E~? In {1+(2E/E)?} E7In {(1+E/E,)—EM(E+E,)}
15 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 = 0136 eV with these figures.
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NONRADIATIVE
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is probably good enough for trends, the Born approximation gives a differential crosssection (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 crosssections are shown in Table 14.2.
14.4. Capture of electrons in solids 14.4.1. Introduction
We now discuss the nonradiative 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 crosssections 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 electroncapture crosssection. 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 crosssection may exceed the geometric area of the
orbit 2(a*)?, typically 10? or 10° A?. TABLE 14.3 Typical capture crosssections
Type of trap
Example
Typical crosssections
(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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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 onephonon processes are possible, in principle, and other cases where multiphonon processes are essential (although not always ratedetermining). 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
BonchBruevich & 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 crosssection. But neutral centres and centres with a repulsive Coulomb interaction may have similar crosssections; 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 powerlaw 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 nonradiative 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 energydistribution function from the thermalequilibrium 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 crosssection 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 singlecarrier—singletrap problem; the screening radius and the mean separation between traps are both very large. We also assume
the electronphonon
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 selftrapped carriers like the V,centre (chapter 18). In the present cases the length for comparison is an orbital radius, roughly fixed by
OF
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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 singletrap 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 crosssections.
A. Cascade processes. The cascade hypothesis asserts that the electron is captured into some highexcited state and subsequently loses energy, mainly by a cascade of onephonon 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 lowlying sstates. We discuss both, since it is likely that the correct picture contains elements from each of the two concepts. The cascade mechanism predicts the crosssection 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
524
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E into an excited state x), with crosssection 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 crosssection 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 crosssection is measured and that capture into certain lowlying excited states is also effective.
The AscarelliRodriguez 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 crosssection ¢,(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 sstate 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
deformationpotential constant, and v, is the velocity of sound. The use of Coulomb wavefunctions is particularly important. If planewave
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 temperaturedependent. Using the parameters for Ge at 4K, Ascarelli & Rodriguez find R, = 00061, R, = 0011, R, = 0020,
and R, = 0053 for the first few sstates. Planewave states given an underestimate of the capture rate. Specifically, R,w, is given by the righthand 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)
526
NONRADIATIVE
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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 socalled 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 sorbitals 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 nonsstates 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 4sstates. 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 onestep 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 nonsstates 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 highlyexcited 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 quantummechanically. 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 crosssection 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 singletrap 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
528
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
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 firstorder timedependent perturbation theory, and treats the initial and final electronic states as if they were freeelectron 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 crosssection. 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 freeelectron 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
OF
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WITH
DEFECTS
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 orderofmagnitude 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 crosssection 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 hydrogenatom value
y~ 2 m
Ry)2.
(14.4.19)
530
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
It is usual to fix A from this equation and R from a very simple estimate of the boundstate 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 acousticphonon 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 steadystate 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 (boundstate 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
OF
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WITH
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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 longrange 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 sstates, one pstate, and one dstate. 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 groundstate energy as a cutoff when calculating the initial capture crosssection. 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
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
in groundstate 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 discretelevel 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
CARRIERS
WITH
DEFECTS
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 crosssections: 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 crosssections than are observed. One check of 105
:
5 x 104F
S
a er.
(A?) Crosssection
3 ee
105
4
oe
el
(Se Ne NS
Same Se
Temperature
(K)
Fic. 14.6. Capture crosssections 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.
534
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
the mechanisms discussed is to compare the capture crosssections of a variety of donors or acceptors. If the longrange 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 crosssections
System
Electron  Ge:As traps  Ge:Sb
Crosssection
Temperature
Temperature
(Ome cms)
(K)
dependence
10*10° 10*10°
39 39
10*
Hole traps
4
Si:Zn7
IO
Ge:(?)
125x 103
GaP :(Zn, O)
Si:B
@p. the Debye temperature. It also requires a more complicated strongcoupling condition so that the saddle point can be found
538
NONRADIATIVE
PROCESSES
readily. With a HuangRhys 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 crosssection 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 = 003 eV (@p ~ 350K),
S = 10eV,
Ey = 38 hap ~
0114 eV, E, = 035 eV or 024 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 038 my to
TABLE 14.7 Crosssections (A?) for multiphonon processes
E, = 0:35 eV
T=10,
T = 10,
T=0)
a* = 34A a* = 415A
51 x 1074 34x 1075
2:8 x 103 2:8 x 107°
19x 107? 48x 1075
(b) Dependence on temperature and ionization energy
a* =3.4A
E, = 024eV E, = 035 eV
= 10,
T = 10,
T=,
345.107 4 Suilseil@r
42x 107! 2:8 x 1073
6:8x 107! 19x 102
OF
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WITH
DEFECTS
539
Mo produces a change of 10° in the crosssection. 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 meanfield (or HartreeFock) interaction which is important. The transitions occur between eigenstates of the HartreeFock
Hamiltonian,
which includes the meanfield
interaction as part of the oneelectron 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
NONRADIATIVE
PROCESSES
AND
THE
INTERACTION
Assumptions for wavefunctions. It can be seen from the list of processes that three types of state are involved : freecarrier states, boundcarrier states, and excitons. Further differences between the
approaches are found in the choice of wavefunctions. The freecarrier states are usually taken to be perfectcrystal 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 boundcarrier states are usually those from the simplest form of effectivemass 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 lsstates considered, and all treatments seem to assume an isotropic crystal with the band extrema at the zone centre. Strictly, the boundstate functions should be orthogonal to the freecarrier 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 effectivemass theory, nor the pseudowavefunction 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
FREE
CARRIERS
WITH
DEFECTS
54]
1963, pp. 11719). 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 electronlattice coupling. This follows from arguments equivalent to those used in § 10.4 for the lineshape function. There is no experimental evidence for a Condontype 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 freeelectron concentration n and the freehole 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 freecarrier 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 lowtemperature approximation,
542
NONRADIATIVE
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;;
BonchBruevich &
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 freecarrier 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 temperaturedependent 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 144 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 Isstate
¢o= m*/m*. For capture
y,=1
y2 = {20/(1+0)}* yy = TH QUL4
T)}4
y4 = T? {20/(1+0)}*
(14.4.32)
OF
FREE
CARRIERS
WITH
DEFECTS
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 ntype Ge. Koenig,
Brown,
& Schillinger (1962) find T, ~ 15x10~73cm®s~!
for
shallow donors Ge:Sb and Ge:P. The predicted value is 5 x 10°7? F,? cm®s~', using m* = 025m, and E; = 012eV. Karpova & Kalashnikov (1962) estimate T, ~ 10°?°cm°®s~! (Eg—E;) ~ 04eV.
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 electronhole annihilation occurs) and deep centres. The results are, of course, sensitive to both the details of the traps and to their
544
NONRADIATIVE
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 Fcentre in alkali halides. The results are needed to interpret two observations: the removal of Fcentres 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 nonradiative 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 electronhole 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 Fcentre 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 nextnearest neighbour to the Fcentre 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 510 per cent of the width ofthe # band. Dexter & Heller conclude that the Auger processes do indeed provide an adequate mechanism for the destruction of Fcentres by excitons. Nelson et al. use a result derived for the internal conversion problem in nuclear physics (Blatt & Weisskopf 1952, p. 617) with
OF
FREE
CARRIERS
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 Kshell 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* ~ 025mo, ¢ = 102, and w ~ 35x 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 internalconversion
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 slowlyvarying 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
NONRADIATIVE
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 nonradiative 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? ~ 0013) and Cu,O (p? ~ 03) 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 0027 eV, m* = 02 mo, m* = 116 mo, and ¢ = 93, he finds the nonradiative process should exceed the radiative for 4x 10'° centres per cubic centimetre. For Cu,O, with m* = m* = 05 mp, € = 625, and a trap depth of 06 eV, the radiative processes are overwhelmed at 3 x 10° defects per cubic centimetre. The associated crosssections are large: for CdS, taking a mean exciton
velocity of 10’ cms~!, the crosssection is 45 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 nonradiative transitions to dominate, which is equivalent toa crosssection 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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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 timedependence 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 timedependent 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 crossterm 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 nonradiative processes compete, and illustrate some of the processes which may be important. For simplicity we use the configurationcoordinate picture. The example is a twolevel 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 nonradiative decay between the different vibrational levels associated with one electron state (intralevel transitions). Finally, there are interlevel nonradiative transitions, in which the electronic state changes. For many common defects the ordersofmagnitude for the lifetimes of states for the first two processes are
Allowed optical transitions:t ~ 1077107* s Intralevel transitions:t ~ 107 11107}? gs,
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549
Whilst these estimates are very rough, it is clear that intralevel transitions are usually much more rapid unless the electron—lattice coupling is weak (see § 12.2.6). The interlevel 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 interlevel transition are particularly rapid. There are ‘vertical’ transitions like AB, which are favoured because of the large lattice overlap
Fic. 14.7. Interlevel transitions in a configurationcoordinate 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 neardegeneracy of levels associated with the two electronic states. This resonance effect has been demonstrated clearly by Sussmann (1967a). The behaviour of the interlevel 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 Fcentre 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
NONRADIATIVE
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. 1215; Heims 1965; Kutz, Davidson, & Konzak 1971), but is less common. The conditions for simple decay are discussed by Strandberg & Share (1973), with
emphasis on spinlattice relaxation. The solution of nonlinear rate equations is more difficult, and numerical solutions are usually essential. Such equations occur in crossrelaxation (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 donoracceptor 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 donoracceptor separation. Thus both the decay rate and the spectral distribution of the emitted light change during the decay, since the closepair 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 threelevel 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
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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 spinlattice relaxation of Fe?* in MgO (Lewis & Stoneham 1967). This is a threelevel system, whose return to equilibrium is characterized by two relaxation times. The resonance lines are strainbroadened, 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
50
75
(E,/ahwy/M,hw
Fic. 15.7. HuangRhys factors for optical absorption by Fcentres. Results from Dawson & Pooley are compared with the combination of Fband energy, effective frequency and cation mass discussed in the text.
ONECARRIER
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585
Analysis of the lineshape 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 lowenergy 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 lineshape. The earliest detailed calculations of the Fcentre 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 longitudinaloptic 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 Fcentre wavefunction is used, rather than a continuum
function. This should allow the
JahnTeller terms to be included, although at present a proper calculation of the lineshape (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 lineshape 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 HuangRhys model to treat coupling to a full spectrum of perfectlattice modes. Lemos &
586
THE
FCENTRE
AND
RELATED
ABER 1541.0 Theories of the Fcentre lineshape 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 Pointion Pointion
Cluster modes Cluster modes
Experimentt Experimentt
Deformation dipole Shell model
Pointion 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 configurationcoordinate 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 pointion model is not used for the zeroorder 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
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587
(6.2.9}(6.2.10). The approximations will be more important in this lineshape problem than for transition energies, since highorder derivatives of the energy with respect to displacements (up to third order in the Benedek & Mulazzi paper) are involved. Benedek & Mulazzi discuss changes in centralforce constants for the first (a, 0,0), second
(a,a,0), and fourth (2a,0,0) neighbours
and non
central firstneighbour force constants. By contrast, Kern & Bartram concentrate on the centralforce 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 pointion potential for secondneighbours. 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 Fband energy on lattice parameter found from the MollwoIvey 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 Fcentre is compact. The ionsize terms, however calculated, fall off rapidly for the same reason. The shortrange repulsive terms between the host ions are also very weak except for neighbouring ions (§ 3.4). Thirdly, the degree of ‘softness’ of the Fcentre 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 perfectlattice configuration. Kern & Bartram find a 17 per cent reduction for KCl. These anharmonic corrections are most apparent in the onephonon lineshape, although Warren (1967) finds evidence for them in the full lineshape. The onephonon lineshape cannot be determined directly from Fcentre 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
THE
FCENTRE
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RELATED
in Fig. 15.8 show how the onephonon sideband moves to lower frequencies as the approximations improve. The conclusions agree with those of McCombie et al. The general features of the lineshape are predicted rather well, and reasonably accurate predictions made of the HuangRhys factor and the temperature dependence of the linewidth. The lineshape is very nearly Gaussian, with a slight asymmetry. Both theory and experiment are conveniently compared with a ‘doubleGaussian’, in which different parameters are used on the high and lowenergy sides of the peak. The logarithm of Kern & Bartram’s normalized lineshape I(E)/I,,,, proves very nearly linear in (E—E,,x)*, showing that the approximation works well. Calculations of the lineshape 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 JahnTeller energies can be made from the contributions of
06
ft
0:54
r ! 1
S£:
Zerophonon line
=> Ww
xS,(@) 104 (eV) S tr
—y ¢.
01
0
5
10 1
15 1
20 25 wx 10!2( §1)
30
35
40
Fic. 15.8. Onephonon sidebands for the Fcentre in KCl. Kern’s calculations are for three cases :(a) an unperturbed Fcentre with neither force constant changes nor lattice distortion, (b) the case of altered force constants, and (c) altered force constants with groundstate distortion.
ONECARRIER
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589
asymmetric modes to the second moment of the lineshape. Thus, if a single effective frequency determined from the temperature dependence of the linewidth 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 Amode coupling. The nonGaussian lineshape 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 nondegenerate mode. When S is large and S, is zero, R is 2:1. The observed value of 248 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 Fcentre, gives rather poor lineshapes. The apparent conflict of a broad band and a welldefined zero
phonon line can also be resolved. There are two features. First,
ONECARRIER
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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 Fcentre. (a) Optical absorption data (Escribe & Hughes) for the (oneelectron) Fcentre ;(b) O’Brien’s predictions fora system with equal Eand Tmode coupling in the excited state. These theoretical curves use S = 545 for the JahnTeller coupling and Amode coupling corresponding to a width 083 phonon energies.
594
THE
FCENTRE
AND
RELATED
O’Brien has shown that the fractional intensity in the zerophonon
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
$S(hw)?, rather than the S(hw)* for nondegenerate modes. In the present case, for a given lattice relaxation energy, the linewidth will be substantially larger. Of course, neither feature explains the smallness of S,. The real test of the ‘equa!lcoupling’ model of the CaO Fcentre is quantitative. One can examine the various data to see if consistent values of S,S,, and hw can be deduced from them. Values of S and S, can be determined in five ways:
1. From fractional intensity in zerophonon line: S,+S—In(S) = 37 (consistent with: Sx = 16.)
2. From R = M,/M3 and M,:
S, = 164,
3. From M, and the centroid—zerophonon line separation: Sc 4. From the observed stress splittings of the zerophonon line:
5. From magnetic circular dichroism (Merle d’Aubigné & Roussel 1971):
S.=933):
S$ = 328,
Léon
ks = B23:
Sites Leys
Se—24:0!
Ser 2055.
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 onephonon 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; ~ 31050 cm *), The temperature dependence of the first and second moments in absorption suggests an effective phonon energy of about 09 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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(d) Temperature dependence of the zerophonon
line: There is no convincing explanation (Escribe & Hughes 1971).
hw = !140cm7/. of the last value at present
15.3. Excited state configuration of the Fcentre 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—HakenSchottky—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 slike state lying just below the 2p*state, as expected from boundpolaron 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 2pstate 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 ‘extended2p’ 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
slike
contribution.
(Mollenaur & Baldacchini
Measurements
of hyperfine
constants
1972, Baldacchini & Mollenauer
1973)
suggest an extended 2pstate for KI and MBr. Conclusions from gfactors (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 opticallyexcited Fcentre 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 JahnTeller 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 001002 eV. If these are to be estimated accurately, it cannot be reasonable to drop substantial JahnTeller energies, which Schnatterly’s results (Table 15.11) suggest are about 01 eV. A. Calculations without the JahnTeller 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 JahnTeller terms and the use of the Toyozawa— HakerSchottky—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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597
Fowler’s (1964) thesis is that the lattice distortion in the relaxed
excited Fcentre is so great that the wavefunction changes from being a compact function to being a very diffuse function. Thus his work concerns the ‘extended 2pstate’ 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 twofold: 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 extendedion effects and their use of very flexible variational functions. The weaknesses are consequences of the Toyozawa—HakenSchottky—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 totallysymmetric
598
THE
FCENTRE
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 Fcentre. 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 discretelattice to an effectivemass 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 069 A in absorption but merely 024 A in emission, since the 1s—2p overlap is reduced. However, the ¢2sz2p> element is increased from 180 A in absorption to 544 A in emission, because of the increase in extent of the 2s and 2p functions. The conclusions of the WoodOpik work are the following. First, the relaxed 2s and 2pstates may be very close in energy. They predict separations of about 006 eV to 001 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 extendedion effects and the importance of very flexible variational functions. Finally, we observe that Sak (1971b; see § 8.6) has shown that the 2sstate of a bound polaron with weak coupling lies just below the 2pstate. The coupling for the Fcentre is too strong for this result to be used directly.
B. Calculations including the JahnTeller effect. Very few calculations have attempted a quantitative calculation of the JahnTeller 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 ionsize corrections to the defect electron— lattice coupling and the discussion of centres like the Fcentre 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.
600
THE
FCENTRE
AND
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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 extended2p 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 shortrange forces between host ions from the removal of the anion, a term from the pointion interaction of the electron and host ions, and a contribution from ionsize effects. The first two
contributions present no difficulties. The ionsize 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 pointion 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 ionsize 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 rigidion model) is not good enough. Detailed results were given for the 2p excited state of the Fcentre 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 05 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
ONECARRIER
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shift of 11 eV. However, there are qualitative conclusions of some
importance. First, the ionsize 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 rigidion model. One expects similar results, differing only in detail. But different symmetry distortions are found in the different cases. For example, with the neutrondata shell model a tetragonal distortion is favoured ; with the shell model fitted to longwavelength 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 divalentanion vacancy. The relaxation energy then contains a large term associated solely with the forces F, from the net charge, a term solely from the shortrange forces discussed above, and a crossterm involving both sets of forces. The term in F, alone is independent of the electronic state and is not important, but the crossterms are difficult because they are so very modelsensitive. 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 hostlattice 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 BuimistrovPekar 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 shellshell part of the forceconstant 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
602
THE
FCENTRE
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terms associated with the net charge of the defect. This expression assumes that the shell motion follows the defectelectron 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 totallysymmetric distortions were included. Only the nearest neighbours to the Fcentre 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 extendedion 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 105 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 Fcentre 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 nonvibronic treatment of coupled s and pstates of Bogan & Fitchen (1970) and Stiles, Fontana, & Fitchen (1970). A. Ham’s model in the strongcoupling 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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603
operator. The eigenvalues involve three energies: the A,—T, separation A in zero distortion, the JahnTeller 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 slike (J = 0, A’ = 1) and the first excited state plike (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 hydrogenatom model of the Fcentre. The advantages of identifying the s and pstates with those of Ham’s model are as follows. First, it removes the need for an acci
dental near degeneracy of 2s and 2pstates, with the 2sstate lower by a mere 001002 eV. One can imagine neardegeneracies 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 fieldinduced change in lifetime are no longer so directly related. Hetrick & Compton’s (1967) observation of stressinduced 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 straininduced splitting of the electronic pstates. Further, the verysmall temperatureindependent 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 electricfield 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 Fcentre 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 spinlattice relaxation, give splittings which are of order hw (0018 eV for KCI and 0017 eV for NaF and KF). Further, in the strongcoupling limit, the radiative lifetime of the
lowest levels becomes independent of J. Thus there should not be
604
THE
FCENTRE
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RELATED
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 01 eV, in the weak to intermediatecoupling 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 harmonicoscillator 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 temperatureindependent part of the magnetic circular dichroism is —2g,BH/(A+hw,). If g,, the orbital gfactor, has the same value as in absorption (g, ~ 1) then (A+/ha ) is about 019 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 longitudinaloptic phonons dominate, hw, is known and A is found to be 015 eV for KF and 009 eV for KCl. As in the strongcoupling theory, there is no need to assume a particularly close 2s—2p degeneracy. 3. The JahnTeller 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 0040 eV for KF and 0014 eV for KCl, consistent with weak coupling. The ratio of the radiative lifetime to that of a pure pstate 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 opticfield vectors proves to be opposite in direction and weaker than the predicted blue shift. However, this may be a result
ONECARRIER
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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 JahnTeller and spinorbit couplings. Available data seem to suggest that the JahnTeller coupling is greatly reduced, and that coupling to trigonal modes is essentially nonexistent. Further, the spinorbit 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 2pstate with a nearby sstate and coupling to lattice vibrations of appropriate symmetry. Whilst the sstate is naturally taken to be the electronic 2sstate (as, for instance, in the WoodOpik
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 JahnTeller 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
FCENTRE
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 nonradiative 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 meansquare amplitudes of T,,, modes. Once repopulation is important, thermal ionization dominates. Honda & Tomura
also note that, in four cases (NaF, NaCl, KF, RbF),
direct nonradiative transitions from §> to the ground state appear to be important, since the quantum yield is less than at low temperatures.
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TABLE 13.12 Admixture «? and s—p splittings for Fcentre 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 : +02 COR
026473 046402
: 02
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 strongcoupling adiabatic model of Bogan & Fitchen and the weakcoupling 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 strongcoupling 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 TypeI 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 Fcentre, describing the observed lineshapes, including the intensity of the zerophonon 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 BoganFitchen models. Only one Fcentre is known where the model clearly applies, but in other cases there is evidence that the E and Tmode couplings are similar in strength. This is shown by the estimates in Table 15.11. Whilst the strong coupling to A modes
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THE
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RELATED
hides the most conspicuous consequences of the model, it is possible that the theory has wider application. 15.4. Perturbed Fcentres 15.4.1.
Introduction
Several defects occur which can be regarded as an Fcentre in an alkali halide with an adjacent impurity or other defect. Examples are the following. 1. The F,centre. Here one of the nearestneighbour cations is replaced by a different alkali ion. In almost all observed cases the impurity is smaller than the replaced host ion. Thus the Fcentre 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 Fcentre—Rareearth complexes. Most of the theory treats centres consisting of an Fcentre 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. Fcentres with vacancies at the nearest anion or cation sites. The Fcentre 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 Fcentre 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 Fcentre with a nearestneighbour 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 twoelectron 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 plike state along the axis of the centre which passes through the defect ion.
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The F,, band includes transitions to the two plike 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 JahnTeller modes. The transition energies provide an important test of theories which go beyond the pointion model. The F,, band lies close to the F band. As a rule the separation is less than 01 eV. The F,, band is shifted by a larger amount, but usually only about 10 per cent of the Fcentre 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 Fcentre,
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 boundexciton « 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 linewidths and Stokes shifts are comparable to those of the Fcentre. 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, lowenergy 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 TypeI 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 01 eV. The electronic state profoundly affects the reorientation rate. More remarkable still, the reorientation
of the TypeII 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 TypeII centres. We shall concentrate on the absorption results.
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THE
FCENTRE
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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 pointion 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 hostion 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 tightbinding 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 pseudowavefunction is assumed to vary slowly over the cores. Thus one expects, as for the Fcentre, that a reduction factor corresponding to the « = 053 of § 15.2.2 should be necessary in the pseudopotential coefficients. Both papers agree that the energy changes in going from the Fcentre 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 pseudowavefunction varies too rapidly in space. It is less surprising that « is closer to unity than for the Fcentre, 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 vacancycentred functions, which are linear combinations ofSlater functions, with radial dependencer” exp (— (7).
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611
These functions are usually worse trial functions for the Fcentre (invariably so in the pointion 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 Fcentre. He found that the F—F, shifts were given well by lowestorder perturbation theory and that the results did not vary much if one used different trial functions. For example, perturbation results using pointion wavefunctions and those using the Bartram,
Stoneham,
& Gash
wavefunctions
gave very nearly the same results. In his work on spinresonance 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
vacancycentred
function Experiment
7?
Theory: floating function
0036
0119
0119
0045
0103
0094
0051
0112
0085
The experimental values are compared with those for vacancycentred functions and those for ‘floating’ functions centred on a point 0185 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
:3Iy [TVD
AIG :
19q2M % AIC :
ISGaAM 7
eee
sonjeA pojonb o1e JOJ JY}
PAM
se
A3I9U9
sooualayip 9 =
aes
09 ea = me
(SIUSZOIPAY)
xq
99
58 £9 68 =
001 99 68 LL
110d
jGeur 8°61 891 661 SLI
B8dM =
G
ST cel 91 cvl
{—uUoNIsueN (UoMIsuey pue“9 =
‘suoHOUNJaA
I

Ov 96 70
€9 Gk cS aa:
a
‘SUOOUNJIARM
(P9}19X9) eM
AB19U9) Jo @Vy
L
‘suoTjOUNJOAeM
a
0
0
07 07 07
Of Of Of Oe
Vv
"SUOTIOUNJIARM
(P9}9x9) SJ
(TUDSOIPAY)
(Punos3) % [TWO
(punoys) ¥ [TVD
TVD
[VD
ee a
$0:0—
Lae ; 60 10
06
S
*e
A8iou9) Jo °Vy
a
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
Z7LZ00 (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?
FCENTRE
v0
a v0 a sal
ION PN JE PN: 109dPN 1d9aPN:
97
R8dM
sanyea ,_QT)‘Ne=
$a4juao¥
THE
ada
(E> (EG HE URE=
Do YT: Ja VT: YT1O9a JaqTT:
139dxq
eo
1sh¢wa
*e
aTavy T VIS uo1pISUDL]SaiZ4aua 4of ay)J
612
ONECARRIER
CENTRES
613
If the impurity is on the z axis, then the F,, transitions are to x and y plike states which have nodes at the impurity. Thus 6, represents the change in the energy of the ground slike state. The splitting of the F, bands 6, is dominated by the effect of the impurity on the z plike 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 Fcentre with a trivalent rareearth ion on a nearestneighbour 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 rareearth 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 defectelectron wavefunction is a linear combination of orbitals on the Fcentre site and on the rareearth site. The most important admixtures for the excited pstate of the Fcentre are with the 5d orbitals rather than the partlyfilled 4f shell. Rather than use unperturbed Fcentre wavefunctions, one must correct for the effects of the rareearth ion. These effects include two terms obtained for the F,centre, those from lattice distortion and from ionsize 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 pstates of the Fcentre, since the repulsive ionsize and attractive pointion 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
FCENTRE
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 ¢ = 233 (cf. ¢,, = 2:19) and R = 396A (cf. the anionanion separation of 444 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 2pstates. This is discussed in S56. 3:1. Berezin (1966) has made estimates of the energy of the lowest transition using a pointion 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 Fcentre 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 alkalineearth 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 pointion model only, and found very poor agreement with experiment: the pstates of the Fcentre were split by a much larger energy than observed. Weber & Dick included ionsize 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 Fcentre in absorption. There is a MollwoIvey 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 Fcentre 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 230 eV in absorption and
ONECARRIER
CENTRES
615
1215 eV in emission. The Z, bands in absorption are at 2.106eV (Sa, Sr) and 2107 eV (Ba); the emission 0855 (Sr), and 0701 eV (Ba).
values are 0833 eV (Ca),
15.5. Antimorphs of the Fcentre: trappedhole 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 Fcentre, 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 Fcentre 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.
Trappedhole centres have been seen in the alkali halides (Kanzig 1960a), various oxides (see the review of Hughes & Henderson (1972)) and in other IIVI 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 IIVI compounds differs from the alkalihalide 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
Trappedhole 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
FCENTRE
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 crystalfield 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 Estates. Their results give variational parameters ¢ for each basis function (or pair of functions for Estates). Linear combinations of these optimized basis functions are then used to give oneelectron wavefunctions and energy levels ¢,,, where T = A,,E and k = 1, 2,3. As expected, the lowest oneelectron level has A, symmetry.
B. Threeelectron states. The nine oneelectron wavefunctions lead to 53 distinct threeelectron 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 oneelectron 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 Rcentre in KCl One electron Configuration
S, WCt 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
— 00119 0:1662 02471 0:2590 02625 02709 04133
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
04252 04942
+ Configurations included by Silsbee and Wang & Chu are marked S, WC.
634
THE
RCENTRE
calculations singled out just three configurations by treating only the lowest A, and Estates; their selection is not adequate and omits a number of important states. New orbital parameters ¢ are needed for the neutral Rcentre. 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 threeelectron states of the same symmetry. Parameters for oneelectron 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 oneelectron
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 Rcentre Wavefunction
R24
Lowest 7A,
Lowest *A,
Lowest *E
Woo
061
0405
0499
0.444
oneelectron < W,
065
0456
0626
0673
046
Not present
0363
0342
A,
orbitals
Wo
E
Uae
oneelectron
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 stressinduced 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 Rcentre Relative oscillator strength
Transition
Band
Theory
Experiment
7E + 7E
Rn R, Re R, Rx Ry R*
025 391 044 {100} 001 009 0:26
037 + 007 120+012 oo [100] 018 + 004 0:29 +003 a
7E > 7A, 7E > 7A, *A, > *E
THE
RCENTRE
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
001
002
1
003
1
0:04
4
Energy (eV)
Fic. 17.3. R, band in KCl. The observed lineshape (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 onephonon peak (002 eV above the zerophonon line) is associated with A, modes; the other peaks are probably associated with modes of different symmetry.
638
THE
RCENTRE
17.3. The Rcentre and the JahnTeller effect 17.3.1.
Introduction
The electronic states of the R*,R~, and R’centres have A,, A>,
or E symmetry. The Estates couple to lattice modes of the same symmetry, giving a JahnTeller effect. The JahnTeller 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 zerophonon 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 Estate and an Astate. 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 JahnTeller 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 nonJahn—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 Rcentre use a particular
THE
RCENTRE
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 potentialenergy 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 twodimensional harmonicoscillator 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
RCENTRE
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 harmonicoscillator functions, using (8.4.24). Thus lattice overlaps can be found in terms of standard harmonicoscillator overlaps. When we treat the effect of external fields on the lowest vibronic states associated with an electronic Estate, 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 Rcentre z axis), q Ill R(E,¢) (electric fields or stress components in the plane of the Rcentre),
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 phononassisted 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 JahnTeller coupling. Fig. 17.4 shows the two cases compared with R’centre data for LiF. There is general agreement, especially for the twohumped structure (§ 10.10.2) in the A> E transition. However, there are anomalies. In emission (E > A), there is a sharp peak at 0049 eV, corresponding to an E mode. It should be present, slightly displaced from vibronic effects, at about 0054 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 = 0049 eV.
THE
RCENTRE
643
Fetterman estimated the ratio of the intensity in the zerophonon line to that in the onephonon sideband. This decreases with increasing JahnTeller 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 Zerophonon strength line
Fic. 17.5. Relative intensities of the zerophonon and onephonon 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 phononassisted 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 HuangRhys 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 JahnTeller effect. Comparison of the results from different experiments then indicates just how well the LHOPS model describes the defects. We
644
THE
RCENTLRE
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 bestdocumented 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 Estate, 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 Rcentre in an otherwiseperfect crystal proves inadequate. Random strains split the Estates by amounts compar
able with the fieldinduced terms, and must be included in the analysis. The LHOPS model has cylindrical symmetry about the z axis normal to the Rcentre; all terms in the potential which have just the threefold 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 Estates are partly quenched by two effects. The first is the Ham effect (XLz1Y> vibronic
a
DOXA
tenets ,
Gide st l )
THE
RCENTRE
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 Estates 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 Rcentre by stresssplitting of zerophonon lines. This work confirmed the F, model and also established the symmetries of some of the states involved. Silsbee also examined the stressinduced dichroism in the broad phononassisted 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 xstate 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 Emode phonons by noting that o transitions are only allowed when accompanied by emission of an odd number of E phonons. Thus Fetterman found opolarized peaks for the R’centre in LiF at 0049 eV, 0076 eV, and 0083 eV. These correspond to an E phonon + These are vibronic states referred to axes in which the normal to the Rcentre 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
RCENTRE
Fic. 17.7. Splitting of the A, where L, is the sum of electronic angular momenta. The electronic spinorbit 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 spinorbit 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 spinorbit 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
spinorbit
coupling
Strain splitting
(ytiex)a
Spinorbit and
Zeeman
energies
Orbital Zeeman
energy Zero strain
Large strain
Fic. 17.8. Zeeman effect for the Rcentre: 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 spinorbit coupling is weak and can be regarded as providing an effective magnetic field A¢S_>/f. The gtensor 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 Estates by the spinorbit 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 Rcentre. These indicate fsa
2
tenia
gy = 048.
dena
The term (g, — g,) is positive: the spinorbit 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 Fcentre. Estimates of the JahnTeller coupling can be made from the stressdependence of the gtensor. The internal strains must be included here. The results suggest k? ~ 3+05 and lead to
SoA = —— O26 CMe
(17.3.16)
If the ratio of go and 4 is correctly predicted then this implies Zo
——
A=
11
—48 cm.
(L7Sithz)
Duval et al., Burke, and Shepherd all observed the magnetic circular dichroism of the Rcentre in KCI. The important measurements are of changes in the total intensity and the first moment of the left and righthand polarizations for both the zerophonon line and the broad phononassisted 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
RCENTRE
649
There are three contributions to the changes in area of the components of the zerophonon line: population effects, the admixture of higher vibronic levels of the electronic Estate by the spinorbit coupling and magnetic field, and the admixture of higher vibronic states associated with the electronic Astate 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 Astate. If A,, A_ give the area of the zerophonon line for left and righthand 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 Aderived 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= 0055 and RA = 0:24cm poe The ratio A/gy is about —44, 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? ~ 36. Similar calculations can be made for the broad band. The calculations are more complicated because of the JahnTeller effect, but can be carried through without difficulty. The area changes and firstmoment changes are given by
bea A,
= (*As] 2® band
(17.3.21)
zpl
and CAE
hand
=
CAE s Dzpi®,
(173722)
650
THE
RCENTRE
where only the populationchange contribution to the zerophonon 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 Rcentre contains the Ham reduction factor q. For the strength of JahnTeller 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
RCENTRE
651
The second effect of an applied field is to admix the / = 3A, and A,States into the lowest vibronic Estate. 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 Estate. 17.3.5. Comparison of LHOPS theory with the Rcentre: 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 lineshape Stressinduced polarization of broadband 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 Temperatureindependent magnetic circular dichroism (assume A = 05)
k? Bix [> 7,). The transition energies are interpolated from Gilbert & Wahl’s (1971) HartreeFock 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
404 370
348 3:30
338 330
315 269
328 2:65
340 2:20
172 187
165 160
BAND
Experiment Jette et al. theory INFRARED BAND Experiment Jette et al. theory
~
339 2:37 165 123
EPR BAND (MEAN OF AE, AND AE,)
Experiment (f, = 1) Jette et al. theory
2:57 2:55
263 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 freeion separation than the calculations of § 18.2 predict. Preliminary results allowing more extensivelatticerelaxation(Norgett, Lumb, & Stoneham, unpublished
660
SELFTRAPPED
HOLES:
© CaF,
o LiF
°oKC
Energy levels (eV)
5 4
3
Rese
:
E. E,
16, Fluorides
Chlorides
B32 34 36 3:8 4:0 4:2 44 46 48 50 5:2 54 56 5:8 60 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 freeion value for KF and 95 per cent for KCI. The potentialenergy 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 dipoledipole interaction. Ikenberry, Jette, & Das (1970) have shown that exchange polarization is important in the isotropic interaction. It is not sufficient to concentrate on the singlyoccupied 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 — 669
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 crystalion 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
SELFTRAPPED
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 spinresonance properties. It proves possible to fit optical data and the gfactor (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 crystalfield distortion or an admixture of valenceband functions. But it is also possible that better wavefunctions than the present HartreeFock 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 selftrapping 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 selftrapping energy has the form
Ey = $¥M,w?0?
(18.6.1)
and is the sum of an increase in latticestrain 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 selftrapping distortion during the jump from one site to the next is AQ, = Q,—Q,,, then
Ex = pyeM oAO7215
(18.6.2)
Ex is the lattice activation energy; this is the energy required to distort the lattice to a type of saddlepoint 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 thermallyactivated : 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 selftrapping distortion do not change significantly as the relaxation proceeds, and in consequence the normal modes and their frequencies w, are not affected by the selftrapping 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 longitudinaloptic modes of a single frequency 10 (Flynn 1971). The result for the alkali halides is that
0838 eV A=
a
2
: (e, +29(4 *II,, with x polarization. These features are observed. Thus, in KCl the transition energies are Hcentre o transition m transition
369 eV 2:38
Theory
V,centre
453 2:16
339 165
The theory results come from interpolation of Gilbert & Wahl’s (1971) HartreeFock calculation to the 2.54 A separation predicted by Dienes et al. In CsBr effects of spinorbit coupling are observed, and the z band is split (Chowdari et al. 1971). The gfactor 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 gtensor by
&\ = 80> where ¢ is the spinorbit coupling. For KCl, Kanzig & Woodruff (1958) give g, ~ 2:0023, as expected, and g, = 20227, g, = 2.0219. Using these values with € = 440 cm! (Slichter 1963) one obtains AE ~ 545 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 GilbertWahl calculation shows no such crossing for any separation of Cl; that they consider. Of course, it may be that a smaller effective spinorbit 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 Hcentres. 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 gfactors.
central ions by factors 68 (LiF), 83 (KCl), and 84 (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 spinresonance 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 spinresonance 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 43 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
HCENTRE
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, electronhole 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. Longrange motion of interstitials in Ge has been postulated (see the review of Mackay & Klontz (1971)), and there is some evidence for lowtemperature 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 bodycentred 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 multiplescattering theory reviewed in §5.4, and predicted an activation energy: Enex— Etetra = 0:85 eV (diamond) 051 eV (silicon)
044 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 shortrange repulsive interaction. Indeed, his estimates suggest that the hexagonal site is stable for the positive interstitial: 0 = Enex—Etetra tetra
~
= — 025 (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 oneelectron
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 bondcentred interstitials are the most favoured in diamond. For the neutral interstitial the energies relative to that for the tetrahedral interstitial are: hexagonal
—91eV
split
—217eV
bond centre
—217eV.
680
THE
HCENTRE
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 —194
Bondcentred
— 209
—74 — 242 — 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 molecularorbital 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 nonmagnetic 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 bondcentre 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 Ucentre, 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 NaClstructure alkali halides has tetrahedral symmetry, in the CsClstructure 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 bodycentred 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 Hcentre 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 spinresonance 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 hostion 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
168 463
0732 0:796
(b) Effect of covalency (After Spaeth & Seidel 1971) Theory/experiment
Nearest cation
Nearest anion
Nextnearest cation
Covalency No covalency
055 144
0595 0825
126 051
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 0961 ay and 354 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 nearestcation 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 threeelectron 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 oneelectron 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 spreadout 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 freeatom 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 pointion 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 pointion model. The crystal ions were represented by freeion wavefunctions orthogonalized to each other. If exchange, threeand fourcentre 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 zeropoint 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 165 per cent higher for hydrogen than for deuterium. Spaeth (1969) has estimated the magnitude theoretically. He observes that the hydrogenhost ion separations vary with time through the zeropoint motion. The localmode frequency is much larger than the hyperfine frequencies, so the hyperfine constants should be averaged over the zeropoint motion. The localmode 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 groundstate 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 meansquare overlaps can be approximated
by S? = Sha
4 exp (B*/p).
(20.2.7)
Here f is a measure of the decay of the hydrogenhostion 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 zeropoint motion (strictly this is the pseudowavefunction; 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 zeropoint 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 spinresonance parameters, the proton hyperfine interaction and the gfactor. We shall ignore the quadrupole interactions, as the shielding corrections are not known. The observed proton hyperfine constant differs from its value for a freehydrogen atom by +475 per cent (KF), —301 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
— 1026 (fit) 1036
= 1026 (fit) 1091
09974 1026 1165
Ratios H:static
Anion
1040
1093
=
Cation
113
137
a
686
HYDROGEN
IN
IONIC
CRYSTALS:
and —588 per cent (KBr). Since the corrections for orthogonality and covalency give increases in all cases (+ 127 per cent for KF, 9.7 per cent for KCl, and 75 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 freeatom 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 pointion 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 hostion 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 gfactor comes primarily from three types of excited state: chargetransfer 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 spinorbit coupling constants and mean excitation energies (cf. eqns (13.3.21)—(13.13.3)). The results involve the admixtures of hostlattice 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 chargetransfer 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 spinorbit coupling. The two leading terms have opposite signs, and there is a substantial cancellation sensitive to «. The results for « = 1 and for « = 133, fitted to the KCI results, are given in Table 20.3. The agreement for « = 133 is particularly good. It suggests that the charge transfer states for the gfactor, involving orbitals, lie higher than those reached optically, which involve o orbitals. TABLE 20.3 gfactors of H®centres in alkali halides (Spaeth & Seidel 1971) 10* Ag KF
Theory (a = 1) + 06
Theory (« = 133) —22
KCl KBr
+ 16:5 +108
+ 70 (fitted) +54
NaCl
+21
+8
RbCl
+ 12:7
+46
Experiment — 28
+70 +39 + 13
+ 146
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 spinorbit effects observable by magnetooptics (Cavenett et al. 1968; Ingels & Jacobs 1971). Neither of the bands obeys a MollwoIvey law; the energies seem to be primarily determined by the anions. Thus the main band, which is lowest in energy, lies between 50 eV and 57 eVin the chlorides, between 44eV and 48eV in the bromides, and between 36 eV and 38 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 chargetransfer states involving the nearestneighbour anions. Chargetransfer states are commonly found in organic systems, and proposed examples are not uncommon in systems containing transitionmetal ions. But these H? systems are among the rare cases in inorganic systems where there is strong
688
HYDROGEN
IN
IONIC
CRYSTALS:
evidence for chargetransfer states. In consequence most ofthe theory has involved molecularorbital 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 effectivemass theory and all ones which regard the excited state as a 2pstate 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 sodiumchloride 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 sodiumchloride structure are argued to be linear combinations of the T,, and T,,States. The experimental results for optical absorption and magnetooptics 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 spinorbit 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, (Ucentre)
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 ~ 039 eV.
(20.3.2)
The radial correlation included in (20.3.1) amounts to about 636 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. Pointion 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 oneelectron 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 oneelectron 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 pointion calculations and ignore lattice distortion and ionic polarization. Since the trial functions are s and p functions only, just the sphericallysymmetric 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 MollwoIvey law:
(ay Bei
(20.3.4)
where a is the lattice parameter. For the alkali halides the predicted value of n is 114, and the observed value is 110. 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 pointion 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 chargetransfer. 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 molecularorbital 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 pointion 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 mutuallyorthogonal wavefunctions; this is automatic in the spinsinglet state. Likewise, if the core orbitals are freeion 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 oneelectron functions for the two H™ electrons is essentially the same in the pointion and extendedion calculations. However, Wood & Opik did include more terms, especially in the pstates for the second calculation (Type II). They observe that flike 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 severalcentre integrals. The results for KCl are summarized in Table 20.4. The parameters are not greatly changed from the pointion 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
— 1094
—0916
484
1026
0426
0823
1010
Extended ion Extended ion II
—1:0712 —11428
—08666 —09242
5:57 5:95
1:0213 1:0213
0:5106 0:5106
05602 More
10213 10213
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 localmode energies vary systematically throughout the alkali halides and obey a MollwoIvey law quite well:
Ey ~ (0183 eV)(a A) 1°799,
(20.3.6)
where a is the nearestneighbour 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 defectlattice 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 localmode frequencies for H, and Dy a
Na
K
Rb
Cl
1380
1392
1400
Br
1377
1404
Cs
Anion
=
= 1414
changes in the longrange 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 PageStrauch model. These are the change in core mass, in core and shell charges, in the shellcore 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 defectshell 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 shellcore force constant); shell charge —e. Reductions of 35 per cent (RbCl) to 58 per cent (Nal) in the shellshell force constant are needed to fit experiment. The Hy
polarizability « varies from 019 A® to 052 A®, depending on the host. 2. dk = 0 (unaltered shellshell force constant); shell charge —e. Reductions of about 97 per cent are needed in the shellcore force constant. The polarizability « is in the range 295 AOtto
5.70 A>, depending on host. 3. Polarizability « = 19 A3, fitted to the analysis of LiH by Calder, Cochran, Griffiths, & Lowde (1962); shell charge —el. Both force constants are significantly weakened, the shellshell
694
HYDROGEN
IN
IONIC
CRYSTALS:
force constant by 30 per cent to 35 per cent and the shellcore 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 extendedion 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 pointion approximation. The results can be described in two ways. One is in terms of the localmode 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 breathingshell 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 secondneighbour corrections rigid ion, mass change only (Fieschi et al.)
128 097 101 159
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, +rP(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 nearestneighbour 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 localmode 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 zerophonon 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 ~ 009
potassium halides:
S$ ~ 017
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 DebyeWaller factor. The argument observes that there will exist anharmonic terms quadratic in the localmode co
ordinate Q, and linear in the hostlattice modes {q;}. Since there is linear coupling to the lattice modes, analogy with the electronic case suggests a DebyeWaller factor for the main localmode peak. He concludes that the intensity of the main peak falls off as
exp \byS, coth (ho 247)}, provided (1) that the localmode 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 linewidth. 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 thirdorder anharmonicity, as discussed by Hughes. The other mechanism involved
the secondorder
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 shellshell 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 localmode energy is determined by A and by the defect mass. Values for hydrogen are
ho, = 9811cm~*
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 139, 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 1:135 respectively. Plausible arguments can be given to suggest that the secondorder 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 hostlattice coordinates, as opposed to the Q7q term in the DebyeWaller factor. The rate depends strongly on the ratio of the localmode frequency to the hostlattice 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 localmode 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 lowfrequency resonances in ionic crystals. These are the electric dipoles such as CN~ and OH", the elastic dipoles such as O;, offcentre 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 stressinduced 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 deexcitation
of rotational
modes
AJ =
+1. Below
about 60K the structure deviates from freerotor 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 twohumped 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 twohumped freerotor 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 CN separation of 14 A. The distance is plausible, but should not be taken too seriously because some of the moment of inertia will be associated with hostlattice 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 specificheat 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
14 115
10 085
0:49 01
025 05
Values are given incm '.
702
THE
REORIENTATION
OF
energy levels and in the electricfield induced dichroism. The tetragonal term is sufficiently small that it could come from random strains:
Say
Vy /(j
peas
[5531
(21.2.2)
where K’ is k. 004°K for KCl and k. 003°K for RbCl. The systems containing OH™ have also been studied extensively by infrared absorption, thermal conductivity, and paraelectric relaxation. The paraelectric 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 centreofmass 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 hinderedrotor 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 offcentre
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 longwavelength 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 JahnTeller systems. The simplest models, either tunnelling or hinderedrotor, 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 statichost potential is a more subtle assumption, since some of the effects of hostion 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 smallpolaron treatment of RbBr:OH~.
21.2.2. Hydrogen halides in raregas 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 isotopedependent. The crystal field may also give some extra splitting of the coupled energy levels. Results for HCl and DCI in raregas 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 nonradiative transitions from one state to another and is sometimes described as phononassisted 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 spinlattice 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 onephonon 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
twophonon
705
and
manyphonon
processes ;we shall not consider them in detail because they are not
important in the bestdocumented 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 JahnTeller 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 selftrapping 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 onephonon theory gives reasonably good quantitative agreement. The contribution of higherorder 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 102
Jeo
145 x 1072 10 x1074 81 105
ws
57
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
10
TABLE ER 2 Lead Effective moment of inertia for K1:O;
bsg 18Q> Results are in units moment of inertia.
60° jump
90° jump
60 57
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 125 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. Offcentre 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 offcentre 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 specificheat 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 TRANSITIONMETAL
IONS
22.1. Introduction
THE transition metals to be discussed belong to two main groups. These are the iron group, with partlyfilled 3d shells and the rare earths, with partlyfilled 4f shells. Other transitionmetal ions have electronic configurations with different incompletelyfilled d or f shells. The incompletelyfilled 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 transitionmetal 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 oftransitionmetal 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 oneelectron wavefunctions for the unfilled shell have the same symmetry properties under the point symmetry of the site as the freeion wavefunctions would have
had.
For
instance,
such
oneelectron
wavefunctions
of an
irongroup 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
TRANSITIONMETAL
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 spinorbit coupling or of electronelectron interaction integrals. This assumption points to one major difference from other defect systems. For transitionmetal ions reasonable zeroorder 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 transitionmetal 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 transitionmetal 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 transitionmetal 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. Crystalfield theory describes the approximation in which the effects of the host lattice are represented by a unique oneelectron 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 molecularorbital treatment of the transitionmetal ion and its ligands. It is a phenomenological theory : one uses symmetry, intuition, and experiment, not specific calculation, to order energy levels.
TRANSITIONMETAL
IONS
719
3. The semiempirical molecularorbital methods use the models discussed in chapter 7 to make rough quantitative estimates of the energy levels and wavefunctions. These are oneelectron calculations so, at best, they could be equivalent to HartreeFock theory. 4. The a priori theories go beyond the semiempirical approaches in making an attempt to achieve selfconsistent 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, semiempirical 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 spinorbit coupling Ao, and the crystal field, measured by the splittings of degenerate freeion 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) Rareearth 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 rareearth series, and possibly for the actinide series. (b) Weak field coupling. The electronelectron interaction H is diagonalized first, followed by a part V., of the crystal field. The spinorbit coupling and the remainder of the crystal field (%o0+ V.2) are treated last. Many irongroup ions suit this scheme. (c) Strong field coupling. The major part of the crystal field V,, is diagonalized first, followed by the electronelectron interaction H,. Again the spinorbit coupling and the remainder of the crystal field (#o+ V2) are treated last. Many of the irongroup ions suit this scheme. These schemes apply to all the four levels of calculation outlined before.
720
TRANSITIONMETAL
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 lowersymmetry oddparity terms in the Hamiltonian are usually too small to produce major changes. The oneelectron 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 Estates have the same energy, and the three Tstates 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 irongroup ions. If positive, the oneelectron energy levels are split, as shown in Fig. 22.1.
le>,lo>
2A/5
lE>,ln>C> Fic. 22.1. Cubic field splitting of delectron orbitals. The sign of the splitting is appropriate for octahedral coordination.
TRANSITIONMETAL
22.3.2.
IONS
721
Theories of the cubic field splitting
A. Crystalfield 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 threefold axes. In octahedral symmetry the nearest neighbours are in the cube directions and will affect the Estates most. In eightfold or fourfold coordination the Tstates 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 manyelectron effects. Secondly, one concen
trates on the 4f shell alone and assumes freeion 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, RussellSaunders 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 rareearth 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
TRANSITIONMETAL
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 irongroup ions. We emphasize once more that the electrostatic model often gives good orderofmagnitude 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
TRANSITIONMETAL
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 spinresonance or Xray experiments. Once a fit is achieved,
some
extra information
is available. The deviations
give a measure of the importance of threebody 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 irongroup 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 rareearth crystal fields and those for irongroup ions. First, the crystal field is weaker for the rare earths. This means that the mixing of freeion 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 crystalfield 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 crystalfield parameters. Experimental values and their sources are given by Newman (1971, Table 12):
ARCr a4
7c
a
Ao 01. One other important feature is that the levels are not monotonic functions of y. Nonmonotonic 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
00
SBN
~01 EEA i
505
ee
=
—0:3h4¢
my —10
S04
x
oa
SSE
ae
310s é
—0:5+ —1:5
00 02 0406 0810 (a)
0
ees
sere
000204060810 Hs
(b)
== Mean
ns ie
ee
0002 04 060810 7
(c)
Fic. 23.1. Donor levels and mass anisotropy. Results for 1s, 2p), and 2p, levels in effectivemass 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 effectivemass 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 effectivemass 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 pstates; indeed, the analysis of these results yields more accurate dielectric constants ¢ than those obtained by direct measurement. The 1sstates show substantial deviations from theory, which we shall discuss in § 23.4.
23.2.2. Wavefunctions The effectivemass equations predict the energy levels and the envelope wavefunction of the shallow centre. In spinresonance 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 effectivemass equation. Another difficulty occurs for cases of multiple minima, since there are interference effects. Clearly expectation values contain crossterms 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 spinresonance 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 spinresonance 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 effectivemass theory discussed later in this chapter. Castner (1970) notes that the effects of a centralcell perturbation can be discerned at quite large distances from the defect. Secondly—a related phenomenon—the spinresonance 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 equivalentorbitals 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 dlike 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 effectivemass 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 conductionband 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 effectivemass theory: degenerate bands The degeneracy of the valence band is important for acceptors in diamond, silicon, and germanium, where the valenceband extrema are at the centre of the Brillouin zone. The valenceband degeneracy is complicated by spinorbit coupling, since the band at the extremum is of predominantly plike character. In other cases, as in wurtzite structures, there may also be a crystalfield splitting, as in Rige2.2: The effect of the spinorbit coupling is to split the plike band into an upper fourfolddegenerate band (J = 3) and a lower twofolddegenerate band (J = 3). For germanium the spinorbit coupling is sufficiently large that the J = + band may be ignored (Mendelson & James 1964; Schechter 1962). However, for diamond (Bagguley, VellaColleiro, 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 effectivemass
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 effectivemass equation for F and for the eigenvalues is then
HF = EF.
(23.3.2)
The effectiveHamiltonian 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 crystalfield terms needed for noncubic 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 effectivemass 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 evenparity 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 thirdorder 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 spinorbit 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 firstorder perturbation theory or by degenerate perturbation theory within degenerate zeroorder 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 spinorbit coupling exactly, but treated the offdiagonal ‘kinetic energy’ terms (the ‘band warping’ terms) by perturbation theory. Thus the zerothorder problem gives simply the separable equations for s and pstates with spin—orbit coupling, and the wavefunctions are hydrogenic with an appropriate averaged mass tensor. To the next order, the offdiagonal 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 onetoone correspondence without ambiguity. The most probable form (Table 23.1) requires corrections to effectivemass theory which change the order of excited states, like the lowest 6—) state and the fourth and fifth 8—) states. Moreover, this reordering 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 effectivemass equation; they probably come from inadequate input data (¢ = 12 is used instead of the presently accepted value 114) or from corrections to simple effectivemass 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
371 187
Experiment (some assignments uncertain) B In Al Ga
44.4 140 98
689 140 104
18
Line label or or description
72:7 145
1562
56
64
3
48 40
4.8 4.5
5.4 51
4 4A
3:8 26
4:3 29
34
4B 5
2:2
22
29
6
14.2 104
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 IIIV 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 centralcell corrections. 23.4. Corrections to effectivemass theory
Many of the predictions of effectivemass 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 groundstate 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 multivalleyed. 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, 2sstate
Germanium
ie
As
Sb
Bi
Ip
As
Sb
Bi
14.04 11:73 1:37
22.24 21:09 141 028
11:24 986 242
39.20 3808
3:09 2:83
4:38 423
0:52 032
298 287
173
118
140
0:17
069
B
Al
Ga
In
ish
1:19
152
169
263
382
—005 —003
3sstate
\¥(0)? from EXPER:
043
Acceptors
B
Al
Ga
In
73
31:8
356
1191
Lowest state
215
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:
3127(1s), 883(2s), 475(3s) 981(1s) 371 (ground state) 928 (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 excitedstate 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 effectivemass 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 electronlattice interaction. We shall not discuss dynamic effects which give temperaturedependent 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 Wavevectordependent dielectric constant Simple band (Muller) Intervalley coupling A d (Baldereschi)
Host spinorbit coupling (Appel)
TERMS DEPENDENT ON THE DONOR ELECTRONIC STRUCTURE Detailed theory with matching (Nara & Morita) Astate Tstate Estate Delta function perturbation based on Nara— Morita form (Schechter) Thomas—Fermi theory (Czavinsky) Model pseudopotential (Jaros) Empirical covalency (Phillips)
90 14 0.2
190 04 —04
210 —02 —12
9:8 64 4 16
259 10 —6 10
32 15 0 (14)
16
14
TERMS INVOLVING LATTICE DISTORTION Shortrange forces (this book) Early approximate work (Weinreich ;
Less than
Shinohara) Polaron correction (Myskowski & Gomulka) Dynamic correction (Barrie & Nishikawa) Experiment (Astate)
1
1
135
42
About 10 About 0:8 About 1 163
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 effectivemass equations to include a wavevectordependent 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 effectivemass 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 effectivemass 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 conductionband minimum at the zone centre, and U(r) correspondent to the «(q) of silicon. Several analytic approximations were used for the spacedependent 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 firstorder perturbation should be small, and it is neglected by Miiller. His final answer gives a correction of about 8107 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 singlevalley 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) wavevectordependent 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 = 106x1072eV
and d = 11 x 1073 eV; for germanium A is 06 x 107? eV. Other corrections have been proposed. Sham (1966) argued that there should be an r~? term from various higherorder 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 effectivemass 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 shortranged. Consequently, the potential varies too rapidly in space for effectivemass theory to be applied directly. There are two methods for attacking this problem. The most obvious is a detailed firstprinciple calculation for some small region near the defect, combined with effectivemass 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 JahnTeller 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 centralcell 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 zerophonon and onephonon 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 effectivemass theory. A subsequent paper (Kohn & Luttinger 1955b) used similar arguments, together with the quantumdefect
768
SHALLOW
DONORS
AND
ACCEPTORS
method (§ 4.3), to relate the centralcell corrections of the various sstates. 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 2sstate 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 groundstate 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 shortrange 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 quantumdefect 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 deltafunction 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 sstates 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 shortrange 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
2:01 160 096
372 314 (No hyperfine constant available)
Bi
1
1
Here A is the excess donor binding energy over Faulkner’s predictions from effectivemass 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 longerrange 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 centralcell 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 shortrange 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 effectivemass 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 conductionband 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 perfectcrystal potential and mg is the freeelectron mass. The band has minima at k = k;. Effectivemass theory for the defect is used outside the sphere r = ro, where the defectelectron
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 effectivemass equation with a potential —e?/e,r and an appropriate effective mass in the kineticenergy 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 perfectcrystal 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 Nfold 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 HartreeFock 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 freeion 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 pseudowavefunction may be matched to an effectivemass envelope wavefunction. This approximation becomes better for effective masses m* close to the freeelectron value, but care is still
necessary with regard to energy zeros. The general arguments for using pseudopotential methods in effectivemass 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 valencebond 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) = KE,(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 effectivemass tensor. The effectivemass 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 effectivemass 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. Spinorbit splitting. Two donors in silicon (Sb and Bi) exhibit spinorbit 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 planewave 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 100 x 10> 3eV (Si: Bi); theor
etically, 09 x 10° eV is predicted for Si: Bi. Spinorbit 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
Hostlattice 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 groundstate energy of the donor, but the deformation contribution is of interest for other reasons. It gives a perturbation which is longranged, 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 groundstate energy must be known and subtracted before experimental data can be used to give information about the shortrange corrections; the longrange strain field may alter scattering crosssections 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 shortranged, 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 electronlattice 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 shortrange force. The strain field at large distances can then be found by the methods of § 8.3 and related to AV, the crystalvolume 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 effectivemass 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 Bent2e544c,,)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 nearestneighbour 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 nondegenerate 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 higherorder 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 conductionband 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 centralcell
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 groundstate 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 nearestneighbour distance; we shall
use this value. The corrections to the energies prove small, so firstorder 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 effectivemass 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 nearresonance 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: 035(AV/Vo) meV germanium: 011(AV/V,) meV Lowering of Is energy: silicon : 162(AV/Vo) meV germanium: 154(AV/V,) meV,
where Vo is the volume per atom of the host crystal. The observed 2p)—2p, correction in Ge:Bi is (006 +002) meV, suggesting that AV/Vo is about (054 +02) 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 05 meV, a volume change of about V, and a lowering of the Is energy by 12 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), 013 (Si: As), and 086 (Si: Sb), and are in line with the deduced value of order unity for Si: Bi. Observed values are —011 (Si:P, Cohen 1967) and 015 (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 (+ 007, Horne 1955; —034, Cohen 1967; —085, 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 linewidth results imply rather small distortions (Davies 1970).
D. Wavefunctiondependent distortion. Calculations of the 1senergy 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 shortrange forces Fez. Strictly, the two contributions are not additive, but for rough estimates we may consider them separately. The results with deformationpotential coupling show that there is a small reduction in the 1s energy for donors in silicon (~08 meV) and a reduction for germanium which is completely negligible (~0025 meV). If isotropic bands are assumed, energies 3040 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 zerophonon 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 shortrange interaction with the defect; there is no longrange Coulomb interaction. Most of the theory of isoelectronic defects has been for IIIV 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 IIVI 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 raregas 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 Fcentre. The Fcentre, 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
trappedcarrier charge. The only cation traps seem to be in the alkali halides where the hole is selftrapped (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 centralcell 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 electronegativity. 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 PhillipsKleinman 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 pseudowavefunctions 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 raregas 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, freeelectron 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 freeatom potentials, it is not clear whether screening using a model dielectric function appropriate to the host is justified. Hermanson & Phillips observe that corepolarization 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 oneband, onesite 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 = 51356. 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 firstrow 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 QV,(0)qo in atomic Allen also observes that heavyatom coupling terms and that an impurity atom it replaces should be attractive
IMPURITIES
units are listed in Table 24.1. impurities give large spinorbit 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
618
—_
—
Group VI
P
O
Syseuic
91
996
50
126
713
804
—
—
Empirical
potential Model potential
— 603
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 ~ 0305 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 electrondensity 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 freeelectron 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 01, 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 phononassisted 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 = —112(GaP:N) = —029 (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 07(N) to 02(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, elasticallyhard 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. Selfconsistent 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 selftrapping of carriers can occur. 24.3. The oneband, onesite model
Oneband, onesite 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 nUla, 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 effectivemass 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 — 18 eV (Si),
ISOELECTRONIC
IMPURITIES
791
—14eV (GaP), and —09 eV (ZnS). For a bound state to exist, Q,
given by
Q = (1/A)+ 1)A is zero and
A = DX{1—,/(1X~?)}.
(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 = 04 and y ~ 071. 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) < 0125), the intermediate part (0125 < k,/(2m/a) < 05), 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 0021 eV below the gap energy, i.e. 0011 eV below the freeexciton line. If the electronhole binding is not affected by the defect then 0011 eV would be the binding energy of the exciton to the defect. Thermaldecay measurements (Cuthbert & Thomas 1967) suggest a binding energy of 0008 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 = 0022 (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, = 00104 
(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 conductionband extrema are not spherical. For GaP the effective masses m, and m, differ by a factor of about 57. Faulkner’s calculations verify the weakness of oneband, onesite 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 compositiondependent potential and lattice parameter. The direct and indirect gap energies are shown in Fig. 24.2. The other feature is the 29
(a) Oe
Direct gap
(LT) DSS)
= 2
L
Indirect gap (X)
Ba o
i 2:2
a
05
:
InP
10 GaP
% 0:03 002
(b)
g
= 001 0
i
1
i
Newel
05
It
Fic. 24.2. Bound states and resonances in In,_,Ga,P:N.
10
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
oneband,
IMPURITIES
onesite
model
with
a strength
parameter chosen to give a binding energy for [© e] of 0008 eV in GaP. Thus they ignore the changes in electronhole 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 lineshapes
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 finalstate wavefunctions. A. Bound electron and free hole. The physical assumption made is that the boundelectron wavefunction does not depend on the presence of the hole and that the freehole 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 shortrange interaction and the hole is trapped by Coulomb interaction with the electron. In the boundelectron, freehole 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 finalstate 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. Effectivemass 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 shortrange 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 boundstate 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 righthand 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 freeexciton function ¥@, but all crossterms
ISOELECTRONIC
IMPURITIES
797
in the shortrange 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 freeexciton 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 centreofmass frame hk, }:
kp = W(k’,/m* +k,/ms). Here
k’, is measured
from
the
(24.3.25)
appropriate
conductionband
minimum, and differs from k which is measured from the centre of the zone. In all, the freeexciton 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 crosssection. The optical crosssection 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 082 (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* = \eR for charges separated by R. We shall discuss later the scaling necessary for using their tables for other systems.
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
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; sstate
(1 00) (0 1 0)
—>n
2; 2s, 2pstates
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 lsstate 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
DONORACCEPTOR ae
 3s,p,d ’ + 2s,p
2051
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 effectivemass theory. O’Dwyer & Nickle (1970) have included the Frohlich coupling in a
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
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,, = 45 a.u. by noting
 CN
ie
(2525)
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 167. 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 effectivemass theory, using ¢,, as the dielectric constant and my, as the carrier mass. The system is predicted to have one deep bound state, 24eV below the conduction band; other states
may occur within 001 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 36eV. 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 freeion
812
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
PAIRS
cores. Isotropic hyperfine constants for three sites were given. For the site next to the anion vacancy and opposite the cation vacancy, theory (181 gauss) and experiment (175 gauss) agree well. For the other sites next to the anion vacancy, theory (91 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 zerophonon 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
DONORACCEPTOR
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
DONORACCEPTOR
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 donoracceptor 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 centralcell 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 055 eV is very close to the observed values of 0553 eV (Cd,0) and 06 eV (Zn,0). However, the energies on this simple model do not agree with observation for other closeneighbour positions. A. Deviation of the electronhole interaction from the pointcharge 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 donoracceptor separations the electron and hole become
correlated, as in an exciton, and the assumption
of
independentparticle 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 freeexciton 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, ~ 141 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
DONORACCEPTOR
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 sphericallysymmetric functions, centred at variationally determined points along the donoracceptor 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 freeexciton 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 oneelectron functions as ‘configuration interaction’. This confusing description is unique to their paper. No correlation is implied, contrary to the longestablished and standard usage of § 2.1.5.
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
PAIR'S
817
Mehrkam—Williams method (Rosen 1931). As the donoracceptor separation R increases, the effective Bohr radii a*(R) of the floating functions increase and the centres of the slike 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 electronhole 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 donoracceptor 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 valenceband 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
DONORACCEPTOR
PAIRS
possible localfield 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 localfield correction), we predict values of Ayw which are 12 per cent (Zn, S), 13 per cent (Zn, Se), 2:9 per cent (Cd, S), and 33 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 donoracceptor 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 lowestorder multipole corrections: “
Il (BOCE
7
eX cisheny Gt 7
4p:
ha
ot
ellie
(25.2.17)
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
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 wavevectordependent 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 zerophonon 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 deformationpotential and piezoelectric electron—lattice coupling.
820
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
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 chargedistributions, 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 configurationcoordinate 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 ~ 016e 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
DONORACCEPTOR
PAIRS
821
a simple point defect in an elasticallyisotropic 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 nonlocal 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 nearestneighbour 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 deformationpotential 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
DONORACCEPTOR
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 longrange 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 donoracceptor 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 HeitlerLondon 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 timedependent 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 effectivemass 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
DONORACCEPTOR
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 =
Sydy + 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 spinzero excited state is written
824
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
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 effectivemass equation. However, Novotny uses an unscreened electronhole 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 spinzero 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 = 11 mo (an average of the anisotropic mass tensor), ¢ = 93,
DIPOLAR
SYSTEMS
AND
DONORACCEPTOR
PAIRS _ 825
andf, = 0512 x 1073, the values are
Wo (theory)
= 47
10° se:
Wo (experiment) = (44+2)x 10% s~!)_
(25.3.12)
The agreement is very satisfactory. The results are altered slightly when the conductionband 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
DONORACCEPTOR
PAIRS
Time dependence of recombination
When a system of donoracceptor 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 timedependent, since the transition probabilities and energies both vary with donoracceptor 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 pairdistribution 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
DONORACCEPTOR
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) _ Ed?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 longitudinaloptical phononassisted 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
DONORACCEPTOR
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 nonradiative transitions. The factors of importance are the following. First, what onecarrier 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 crosssection 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 crosssection should increase as the square of the donor—acceptor separation for large spacings. DiDomenico, Dishman, & Sinha (1971) have compared the rate of the twostep process (25.3.19) with direct exciton capture:
e Oh] +Es. [ o]+[eh]>[G
(25.3.20)
Their calculations show that the twostep process (25.3.19) dominates at high temperatures. The ratio of the probabilities was found using effectivemass 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 freehole 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 IIVI, IIJV, 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; centralcell 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 doubleacceptor
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 boundexciton 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 freeexciton 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 075 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= 0055 Ep
(26.2.6)
Evonn = 0055 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 chargetrapping 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 threeparticle 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,,;, (LevyLeblond (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
02
= et
04
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 (LevyLeblond 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 020 Ep. The oscillator strength of the exciton can be extremely large. It can be written in terms of the freeexciton 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
02
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 antiparallel (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 (¢ = 020) Observable
E’E,, (change inenergy from free exciton
ZnO (o = 012)
Unit Theory
Experiment
Theory
Experiment
10°*eV
64
46 to 710
134
113 17:0
10°2eV
049
0:31
083
09
—
10x 105
large
069 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 electroniattice coupling. Several authors (Schroder & Birman 1970; Elkomoss 1971; Mahler & Schroder 1971) have considered modifications of the simple ¢, screening of the electron—donor, holedonor, and electronhole 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 (eh, 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 kineticenergy 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 hostlattice 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
0055.
(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 electronlattice 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 00497 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 centralcell corrections; the results seem valid beyond the model. Examples are shown in Fig. 26.4 for IIVI compounds and for silicon. For the IIVI compounds o ~ 025 is typical, and the results suggest (025) ~ 02, f(4) ~ 01, consistent with the theory.
Cug”
He A
=z 0050
Abepou4
83
oo
2 0:020F 20
in
as
:
= 0010
ee
oO
Ag
Te. Ag 4,7 Re\
Pew
0005
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
01
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 01 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 centralcell 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 effectivemass theory alone, V, is a measure of the centralcell 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 centralcell corrections (so that pcx and Pcp are not altered greatly from their effectivemass 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 centralcell 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 centralcell 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 nonradiative 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 donoracceptor pair (when the oscillator strength depends on the donoracceptor separation) and an exciton bound by a deltafunction 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: doubledonor or acceptor states and the bound excitonic molecule; both are fourcarrier systems. The doubledonor or acceptor states are especially interesting because of the way in which the valenceband degeneracy enters (Hopfield 1964). Thus a neutral doubledonor[ + + ee]is analogous to the He atom; an extra electron can only be weakly bound because the Islike 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 doubleacceptor 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 01 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 boundexciton 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 JahnTeller effect
In most host lattices of interest to us the hole states are derived from the plike states at the top of the valence band. If there is significant spinorbit 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 JahnTeller coupling within these states affects both free and boundexciton properties. He has examined
the effect for the system GaP:Bi,
observe both free and boundexciton
where
one may
properties. There are two
boundexciton lines: the A line, a dipoleallowed transition, and the dipoleforbidden 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 JahnTeller 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 doublydegenerate
level I’; and a triplydegenerate level I',. Morgan argues first that the JahnTeller coupling is strong. One reason is that there are fairly strong and broad phononassisted bands, a symptom of strong electron—lattice coupling. Another is that he shows the ratio of the Aline and Bline 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 freeexciton properties. The properties include the deformation potentials for tetragonal and trigonal strains, the anisotropy ratio of the stressinduced valenceband 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 crystalfield 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 spinorbit coupling by a factor of about 3 from freeexciton values. The deformation potentials may also be affected by factors other than the JahnTeller effects, such as local changes in elastic constants and the compact nature of the wavefunctions. The JahnTeller 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 lineshapes
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 zerophonon line and there are phononassisted transitions. If the coupling is not too strong, there will be a zerophonon line and phononassisted transitions with structure corresponding to critical points in the phonon spectrum.
BOUND
EXCITONS
845
The intensity of the zerophonon 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 quasimomentum. For free excitons this is achieved by absorption or emission of an appropriate momentumconserving phonon, and a zerophonon line is not observed. For bound excitons, the coupling to the defect provides another mechanism, so transitions involving no momentumconserving phonons are also seen. One expects this zerophonon 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 Onephonon
1
0005
0010
——
0015
os
0020
E @ eyen(meV) Fic. 26.5. Zerophonon line intensity and dissociation energies. The ratio of the zerophonon and onephonon intensities varies systematically with E.g (en). Results are for Si. (After Dean.)
In special circumstances the lineshape for phononassisted 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 deformationpotential coupling. The host lattice was assumed to have acoustic modes, with an isotropic Debye spectrum, and nondegenerate 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 deformationpotential 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 lineshape are a zerophonon line, and wings whose form is sensitive both to m¥/m* and to the relative strengths of deformationpotential 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 crossterm. Hopfield’s calculation shows good agreement for CdS, correctly predicting a zerophonon line and asymmetric wings on either side. The shapes of these phononassisted 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 zerophonon 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 longitudinaloptic 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 hostoptic phonon or ofa localmode 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 zerophonon line. It appears to be a consequence of the conductionband 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 545 A and b is typically 40 A, so « should be of order 027. Good fits are obtained for various species of acceptor with the values: GaP:N,C
a = 0204
(
)
GaP oNeMe
@=
1
)
GaP:N,Zn
a=024
(0645);
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 JahnTeller effect, and correspond to JahnTeller 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 boundexciton 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 gfactors, is primarily a measure of hostcrystal properties. The localizing potential may render some transitions weakly allowed instead of forbidden, and may lead to some extra splittings with small gfactors, 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 spinorbit coupling and the axial crystal field. By contrast, GaP is more complicated: the valenceband degeneracy is only partly lifted by the spinorbit coupling, and there are three conductionband minima in (100) directions which can give rise to valleyorbit 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 boundexciton 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 centralcell correction. The degeneracies quoted include spin degeneracy. The three models for GaP are (a) the usual model of three conductionband 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 valleyorbit 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 conductionband structure and the associated valleyorbit splitting. Simplifications can occur for several reasons, and allow one to concentrate on the totally symmetric valleyorbit state. First, in the optical transitions ®+hv[@
eh]
we expect only the totallysymmetric valleyorbit state to have a reasonable transition probability, since it is the only valleyorbit 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 gfactor is zero. This is true when the valleyorbit splitting is large, leaving an orbitally nondegenerate state lowest. It is also true when there is no valleyorbit 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 centralcell correction). It is likely that the centralcell correction dominates, and that only the totallysymmetric 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 centralcell 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 directgap zincblendestructure 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 donoracceptor 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 ‘defectmolecule model’, the behaviour of these four defect electrons in the ‘dangling bonds’ determines the properties of the centre. For this reason, we have a fourelectron problem. The first difficulty, which we discuss later, is the controversial question of the use of the oneelectron approximation. The
second difficulty concerns the JahnTeller 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 JahnTeller 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 stresssplitting 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 spinresonance 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
— < 005 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 JahnTeller energies are larger than any corrections to the oneelectron model. In the defectmolecule model there are four danglingbond 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 ~ 218)
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 oneelectron orbitals in turn and allowing for a JahnTeller 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 oneelectron 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 JahnTeller 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 JahnTeller 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 JahnTeller 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 interlevelcoupling 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 unpairedelectron wavefunctions for all of the vacancy charge states. We shall compare his analysis with a priori theories in later sections. C. JahnTeller energies.+ The JahnTeller energies and coupling coefficients can be obtained from the stress response of the observed spinresonance 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
119eV 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 oneelectron energies. If these oneelectron 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 3:6:45, very close to those observed for both the vacancy and for Gevacancy complexes. Results are shown in Table 27.1. By contrast, TABLES2 7.1 JahnTeller energies and coupling coefficients Tetragonal Distortions Fy(eV A')
E¥;(eV)
Wr
29
035
Ve wee
6:0 45
151 085
Trigonal Distortions Fy(eV Ast)
Ej,(eV)
(158)t 38
121
+ 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 0013 eV (V*), 023 eV (V°), and 0072 eV (V). Similar values are found for the Gevacancy complexes. The second class of behaviour only occurs for V~. The unpairedelectron 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 0008 eV.
Vacancy motion has been analysed for V° and V2~. The results show the two striking features also found in reorientation, namely a chargestate dependence of the activation energy Ey, which also has a very low magnitude. Observed values are 0:33 eV (V°) and 018 eV (V_). Gregory (1965) has commented that vacancy motion can be directly affected by electron injection in ptype 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 vacancydonor binding. If E, is small (Watkins estimates about 03eV from a simple model) then these results suggest (Ep +E,) to be about 4eV. About SeV is obtained from selfdiffusion 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 spinresonance systems surveyed by Owen (1965) and Whippey (1972). The GR1 band has a zerophonon line at 1673 eV, with a broad band of phononassisted transitions peaked at about 2eV. The oscillator strength is of order 01. 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 zerophonon 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 zerophonon 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 spinresonance 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 spinorbit coupling tend to lead to defects whose spinresonance signal is easily saturated, so that the absence of spinresonance 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 283 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 gvalue of 2:00, and a zerofield splitting of 0140 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 spinzero ground state; the observed [100] spin1 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 JahnTeller 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 JahnTeller 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 19 A. Balona & Loubser (1970) have observed a variety of spinresouance spectra in electronand neutronirradiated 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 defectmolecule 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 zerofieldsplitting tensor with an unusual symmetry axis at
860
VACANCIES
IN
VALENCE
CRYSTALS
454° 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 IIVI 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 negativelycharged 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 JahnTeller 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 oneelectron approximation valid?
27.4.1. Defectmolecule 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 fewelectron problem instead of a oneelectron 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. Oneelectron states. Two classes of oneelectron wavefunctions have been used. In the first, the LC.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 freeatom HartreeFock 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 vacancycentred 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 onecentre treatments, and shows that the centre should indeed be
compact, as observed. In most respects, the two choices of oneelectron 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 oneelectron approximation is not adequate. The details of the energy levels do depend, of course, on
the details of the oneelectron
wavefunctions,
but the lowest
energy states can be predicted by general arguments; we illustrate
this for V°. B. Manyelectron states. The oneelectron orbitals consist of a lowest A,state, with a triplydegenerate 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 oneelectron 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 intraatomic 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 weakfield ligand field theory, §22.2) then the °Astate 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’srule coupling favours parallel spins for the electrons in the T,state, and a ‘'Estate 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 antiparallel 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 fourcentre 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 onecentre 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 RHYSROBERTS, 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