Physica status solidi / B.: Volume 65, Number 2 October 1 [Reprint 2021 ed.] 9783112501948, 9783112501931

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plrysica status solidi (b)

VOLUME 65 . N U M B E R 2 • 1974

Classification Scheme 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 Alloys. Metallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mössbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraction Investigations See 1 and 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 I m p u r i t y and Defect States 13.5 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Electrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces and Interfaces 14.3.3 Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties (Continued on cover three)

physica status solidi (b) basic research B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, M. C A R D O N A , Stuttgart, W . D E K E Y S E R , Gent, W . F R A N Z , Münster, P . G Ö R L I C H , Jena, E . G R I L L O T , Paris, E. G U T S C H E , Berlin, R. K A I S C H E W , Sofia, P. T. L A N D S B E R G , Southampton, S. L U N D Q V I S T , Göteborg, L. N É E L , Grenoble, A. P I E K A R A , Warszawa, A. S E E G E R , Stuttgart, F. S E I T Z , New York, O. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, J . T A U C , Providence Editor-in-Chief P. G Ö R L I C H Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Bagneux, W. B R A U E R , Berlin, W. C O C H R A N , Edinburgh, R. C O E L H O , Fontenay-aux-Roses, H.-D. D I E T Z E , Aachen, J . D. E S H E L B Y , Sheffield, P. P. F E O F I L O V , Leningrad, G. J A C O B S , Gent, E. K R O E N E R , Stuttgart, R. K U B O , Tokyo, M. M A T Y Ä S , Praha, H. D. M E G A W , Cambridge, T. S. M O S S , Farnborough, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, H . M . R O S E N B E R G , Oxford, S. S H I O N O Y A , Tokyo, W. S T E I N M A N N , München, R. V A U T I E R , Bellevue/Seine

Volume 65 • Number 2 • Pages 429 to 878, K79 to K152, and A9 to A16 October 1, 1974 PSSB 65(2) 429-878, K79—K152, A9—A16 (1974)

AKADEMIE-VERLAG

•

BERLIN

Subscriptions and orders for single copies should be addressed to A K A D E M I E - V E R L A G , 108 Berlin, Leipziger S t r a ß e 3 - 4 or to Buchhandlung K U N S T U N D W I S S E N , Erich Bieber, 7 Stuttgart 1, Wilhelmstr. 4 — 6 or to Deutsche Buch-Export und -Import GmbH, 701 Leipzig, Postschließfach 160

Editorial Note: "physica status solidi (b)" undertakes that an original paper accepted for publication before the 8th of any month will be published within 50 days of this date unless the author requests a postponement. In special cases there may be some delay between receipt and acceptance of a paper due to the review and, if necessary, revision of the paper.

Schriftleiter und verantwortlich für den Inhalt: Professor Dr. Dr. h. c. P . G ö r l i c h , 102 Berlin, Neue Schönhauser Str. 20 bzw. 69 J e n a , Humboldtstr. 26. Chefredakteur: Dr. H . - J . H ä n s c h . Redaktionskollegium: Prof. D r . E . G u t sehe, Dr. H.-J. H ä n s c h , Dr. H. L a n g e , Dr. S. O b e r l ä n d e r . Anschrift der Schriftleitung: 102 Berlin, Neue Schönhauser S t r . 20, Fernruf: 4223380. Verlag: Akademie-Verlag, 108 Berlin, Leipziger Str. 3 - 4 , Fernruf: 220441, TelexNr. 112020, Postscheckkonto: Berlin 35021. — Die Zeitschrift „physica status solidi (b)" erscheint jeweils am 1. des Monats. Bezugspreis eines Bandes 140, — M ; Sonderpreis für die D D R 120, — M. Bestellnummer dieses B a n d e s : 1068/65. Jeder Band enthält zwei Hefte. Gesamtherstellung: V E B Druckerei „Thomas Müntzer" B a d Langensalza. Veröffentlicht unter der Lizenznummer 1310 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Printed in the German Democratic Republic.

Contents Page

Original Papers B . H A Y M A N a n d J . P . CARBOTTE

A Model for the Electron-Phonon Interaction in Polyvalent Metals: AI

439

B . N I K I N , B . S . TOSIÓ, a n d V . M . ZEKOVIÓ

Introduction to an Analysis of Ferromagnets with Dipole-Dipole Interactions

449

E . L . NAGAEV a n d P . A . GRIGIN

E.

L.

H . BRANDT

DAGENS

Screening, Instability of the Uniform State, and Charge Carrier Scattering in Heavily Doped Ferromagnetic Semiconductors . . .

457

The Magnetic Field Distribution of Superconducting Alloys with Small Magnetization

469

Linearized Effective Exchange and Correlation Potentials for OneParticle Property Calculations in Simple Metals

481

G . PAASCH a n d H . WOITTENNEK

J . MONECKE

Critical Investigation of Potentials for Band Structure Calculations

493

Single-Particle Excitations in Mott Insulators

505

J . F . HOULIHAN a n d L . N . MULAY

G.

R . ZELLER

.

Correlation of Magnetic Susceptibility, Electrical Conductivity, and Structural Parameters of Ti 3 0 5 via E P R Spectroscopy

513

Pseudopotential Calculation of the Piezoelectric Constant of GaAs

521

I . BALSLEV a n d J . M . HVAM

Drift of Electron-Hole Drops in Exciton Density Gradients

. . .

531

H . R . ZELLER a n d P . BRÜESCH

Band Structure of K 2 [Pt(CN) 4 ]Br 0 . 3 o • 3 H 2 0

537

K . SAERMARK a n d J . L E B E C H

Cyclotron Waves in Silver — Interference Effects

543

C. A . POLDY a n d H . R . KIRCHMAYR

The Magnetic Properties and Indirect Exchange in Gd(Ni, Cu) 2 28»

553

Contents

432

Page

R.

PÄSSLER

Quantum-Efficiency of Multiphonon Transitions According t o the Static Coupling Scheme

561

R . H . W I L L I A M S a n d J . I . POLANCO

The Electronic Structure of Chalcogenide Solids: A Photoemission Study of Sulphur

571

J . ALIZON, G . B E R T H E T , J . P . BLANC, J . GALLICE, a n d H . R O B E R T

Spin-Spin and Spin-Lattice Equilibrium in TCNQ T NH 4 h Salt

. .

577

Energy Level Structure of Biexcitons and Related Optical Transitions

591

Trigonal Warping of t h e Valence B a n d Fermi Surfaces of Tellurium

603

F . B A S S A N I , J . J . F O R N E Y , a n d A . QTTATTROPANI

H.

KÖHLER

M . SCHWAB, W . B E R K H A H N , H . G . K A H L E , A . SIMON, a n d W .

WÜCHNER

Magnetic Ordering in H o A s 0 4 Investigated b y Means of Specific H e a t and Spectroscopic Measurements

613

G . SPERLICH a n d W . D . LAZÉ

Estimate of the d - d Overlap of V 2 0 5 from E S R Measurements . . .

625

H . J . W E B E R a n d S. HAUSSÜHL

Electric-Field-Induced Optical Activity and Circular Dichroism of Cr-Doped KA1(S0 4 ) 2 • 12H 2 0 633

W . SZYMANSKA, P . BOGUS&AWSKI, a n d W . ZAWADZKI

Elastic Electron Scattering in Symmetry-Induced Zero-Gap Semiconductors

641

C . CUSUMANO a n d G . J . T R O U P

A Study of the Exchange-Determined Absorption Lineshape outside the Lorentzian Region

655

E . O. MANUCHARYANTS a n d I . P . ZVYAGIN

Green's Functions Approach to the Theory of Hopping Conductivity in Disordered Semiconductors

665

C . - U . TRAUTVETTER

Spin-Wave Spectra below Saturation Magnetization and a t Various Orientations

671

Contents

433 Page

E.

ASCHER

Relativistic Symmetries and Lower Bounds for t h e Magneto-Electric Susceptibility and t h e Ratio of Polarization to Magnetization in a Ferromagneto-Electric Crystal

677

M . D . COUTINHO F I L H O , L . C . M . M I R A N D A , a n d S . M . R E Z E N D E

B. P . PANDEY

Spin Wave Amplication in Ferromagnetic Semiconductors: P l a s m a Effects and s - d Interaction

689

Phonon Dispersion in SiC and ZnS Poly types

699

R . KLUCKER, M . SKIBOWSKI, a n d W . STEINMANN

Anisotropy in the Optical Transitions from the n and a Valence Bands of Graphite

703

S . CIRACI a n d W . A . T I L L E R

Some Applications of t h e Bond Orbital Model

711

C . B E N O I T a n d G . W . C O H E N SOLAL

Induced Scattering of Neutrons b y an Electromagnetic Field

. .

721

Ohmic and Non-Ohmic Transport in Undeformed and Plastically Deformed Tellurium

729

E . THALMAYR, H . KAIILERT, a n d H . KUZMANY

A . MYCIELSKI, A . AZIZA, J . MYCIELSKI, a n d M . BALKANSKI

Frequency-Dependent Relaxation Time of Electrons in PbSe f r o m Infrared Measurements

F . BRIDGES

A

Two-Multiplet Tunneling Model

737

743

D . J . H . COCKAYNE a n d V . Y I T E K

Effect of Core Structure on the Determination of the Stacking-Fault Energy in Close-Packed Metals

751

C . U H E R a n d H . J . GOLDSMID

Separation of t h e Electronic and Lattice Thermal Conductivities in Bismuth Crystals

765

E . G . S H A R O Y A N , 0 . S . TOROSYAN, E . A . M A R K O S Y A N , a n d V . T . G A B R I E L Y A N

E P R and Spin-Lattice Relaxation of Mo 3+ Ions in Corundum . . .

773

S . OLSZEWSKI a n d A . WIERZBICKI

Dynamical Observables in Cubic Crystals from the Standing LCAO Wave Functions

779

Contents

434

Page J . KALINOWSKI a n d J . GODLEWSKI

M.

Z. ZGIERSKI

A . J . NADOLNY

Singlet Exciton-Charge Carrier Interaction in Anthracene Crystals

789

Herzberg-Teller Interaction and Vibronic Coupling in Molecular Crystals (III)

797

Photo-Induced Electron Spin Resonance (Photo-ESR) in ¡3-Rhombohedral Boron

801

M . SINGH a n d G . S. VERMA

Y.

P.

JOSHI

Scattering of Phonons by Bound Holes and Phonon Conductivity of p-GaSb

813

Resonant Scattering of Phonons b y Two-Level Impurities

. . . .

823

Interserial Phototransitions of Excitons in Semiconductors . . . .

833

P . I . K H A D S H I a n d S . A . MOSKALENKO

Y U . A . BOGOD a n d V . B . K R A S O V I T S K I I

V. A.

SOLOVEV

Experimental Studies of Magnetoresistance Tensor Components in Bismuth a t Low Temperatures

847

The Stress Field near t h e Dislocation Pile-Up Type Defects in Anisotropic Elasticity

857

G . L . KRASKO a n d A . B . MAKHXOVETSKII

The Pseudopotential Method and t h e Problem of Ordering Alloys (I)

in 869

Short Notes V . T . B U B L I K , S. S. GORELIK, A . A . ZAITSEV, a n d A . Y . POLYAKOV

Calculation of t h e Binding Energy of Ge-Si Solid Solution

. . . .

K79

P . MEISSNER a n d 0 . KAXERT

Pulsed Nuclear Magnetic Double Resonance in Ca + + -Doped NaCl K85 Single Crystals

D . HECHTFISCHER, R . KARCHER, a n d K . LUDERS

NMR of V 50 in Single Crystal Vanadium

K89

W . GEHLHOFF a n d W . ULRICI

Trigonal Ni 2 + Centre in CdF 2

K93

Contents

435 Page

T . GOWOREK, W . G U S T A W , C . R Y B K A , a n d J . W A W B Y S Z C Z U K

Influence of Phase Transitions on Positron Annihilation in Phenanthrene K97

H . GOEBEL a n d F . R . K E S S L E R

Plasma Resonance in Heavily Doped Germanium Thin Films . . . K99

V . V . NEMOSHKALENKO,

Y . G . ALESHIN,

A . I. SENKEVICH,

Y U . N . KUCHERENKO,

L . M . SHELUDCHENKO u n d M . T . PANCHENKO

Die Röntgenelektronenspektren und die Zustandsdichte der Valenzelektronen in Cu-Mn-Legierungen K105

V . FILIPAVIÖITJS a n d E . NORMANTAS

Scattering of Holes in Germanium by Long-Range Potential

S. RADELAAR

. . . K109

Zener Relaxation Effect and Its Influence on the Resistivity of AuCu (14 at% Cu) KI13

P . KUMAR a n d G . C. SHUKLA

Effect of Fluctuations on Electrical Conductivity of Zero-Dimensional K115 Superconductors below Tc

F. F.

Sizov,

and K . D . The Faraday Effect in n-Pbo.82Sno.1sTe

V . B . ORLETSKH, G . V . LASHAREV,

TOVSTYUK

K119

R . C. TYAGI a n d V . C. SETHI

Dependence of Impurity Absorption Bands on Lattice Parameters in Alkali Halide Crystals K123

N . N . TRUNOV a n d E . V . B U R S I A N

The Influence of Charge Carriers on the Transversal Mode in Ferroelectrics K129

A.

RAUH

Degeneracy of Landau Levels in Crystals

A . I . M I T S E K , N . P . K O L M A K O V A , D . I . SIROTA, I . N . K A B N A U K H O V , A.

P.

NEDAVNII

K131

and

Orientational Phase Transitions in Uniaxial Ferromagnets

H . DÖHLER, R . NEUBAUER, a n d B . SCHNABEL

. . . .

K137

Theory of NMR Behaviour of a Two-Spin System in Solids under Arbitrary Strong R F Irradiation K141

436

Contents Page

E . B U B Z O , D . P . L A Z Ä R , a n d M . CIORASCTT

Local Environment Effects in Gd(Coj.Nii pounds

Pseudobinary ComK145

H . C. P A N D E Y a n d J . D . P A N D E Y

Compression Study of Higher Alkanes through Pseudo-Grüneisen Parameters K149

Pre-Printed Titles of papers to be published in the next issues of physica status solidi (b) and physica status solidi (a)

A9

Contents

437

Systematic List Subject classification:

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification):

1.2

K97, K137

1.3

553

4

711, 869, K79

6

439, 561, 641, 665, 699, 721, 797, 813, 823, K109, K129

8

613, 765, 813, K149

10

857

10.1

751, K113

10.2

729, K123

12

857

12.2

K149

13

439, 641, 665, 711, 729, 779, K79, K105,

13.1

493, 505, 521, 537, 571, 591, 641, 703, 833, K119

13.2

481, 543, 603

13.4

743, 823, K93

13.5.1

531, 591, 789, 797, 833

13.5.2

689, 737

13.5.4

457, 689

14

505, 779

14.1

439, 481, 543, 847, K113

14.2

469, K115

14.3

457, 513, 603, 665, 689, 729, K109

14.3.1

K99

14.3.4

729, 789

14.4

521

14.4.1

677, K129

15

765

16

801

17

571

18

737, K119

18.2

449, 457, 553, 655, 677, K137

18.2.1

671

18.3

K145

K131

18.4

505, 513, 553, 613

19

513, 577, 625, 655, 671, 773, 801, K85, K89, K93,

20

721, K105, K119

20.1

537, 591, 613, 633, 703, 737, 797, 833, K99

20.3

561, 591, 789

K141

438

Contents

21

439, 481, 493, 751

21.1

493, 553, 671, K89, K105, K113, K145

21.1.1

493, 671

21.2

493

21.3

493, 869

21.4

553, K145

21.6

543, K113

21.7

765, 847

22

457, 561, 641, 665, 711, 737, 833, K119

22.1

703, 801

22.1. 1

K79, K99, K109

22.1. 2

K79

22.1.3

571, 603, 729

22.2.1

521

22.2.3

813

22.3

699

22.4.1

561, 591, 699

22.5

591

22.5.2

743, K85, K123

22.5.3

721, K93

22.6

513, 625, 773

22.8

537, 633, 689

22.8. 1

613, K129

22.8.2

677

22.9

577, 655, 789, 797, K97, K149

The Author Index of Volume 65 Begins on Page 879 (It will be delivered together with Volume 66, Number 1.)

Original Papers phys. stat. sol. (b) 65, 439 (1974) Subject classification: 6 and 13; 14.1; 21 Physics Department, McMaster University, Hamilton, Ontario

A Model for the Electron-Phonon Interaction in Polyvalent Metals : Al By B . H a y m a n and J. P . C a b b o t t e Bross and Bohn have suggested a form for the electron-phonon matrix element which avoids the serious difficulties that are associated with the one OPW approximation for the case of a Fermi surface that intersects zone boundaries. Using this suggested form, we have calculated the electron-phonon mass enhancement, the transport scattering times, and the electrical resistivity. A discussion of the merits and limitations of the Bross-Bohn matrix element is included. Bross et Bohn ont suggérés une forme pour l'élement de matrix de l'interaction electronphonon qui circonvient les sérieuses difficultés associées avec l'approximation d'une O P W dans le cas où la surface de Fermi intersecte les plans de Bragg. Nous utilisons cette forme dans des calculs de changement de masse, de temps de diffusion et de resistivité due a l'interaction entre electron et phonon. En conclusion nous donnons une discussion des limitations et mérites de la forme donnée par Bross et Bohn. 1. Introduction

I t is a well known fact [1] that when one is considering the electron-phonon interaction in systems where the Fermi surface intersects zone boundaries (either in polyvalent metals or monovalent metals with severely distorted Fermi surfaces), the one OPW (Orthogonalized Plane Wave) electron-phonon coupling constant gkk-x, given by (1)

becomes infinite whenever the momentum transfer q{ = k' — k) becomes exactly equal to a reciprocal lattice vector G. This divergence occurs because of the phonon frequency factor wqx which appears in the denominator of (1). When q = G, ooqx (which is determined through qlei = q — G) becomes equal to zero for acoustic phonons. In (1) h is Planck's constant over 2n, M is the ion mass, N the ion number density, W(q) the electron-ion pseudopotential form factor and sqx the phonon polarization vector for the q)Sth mode with A the branch index. The standard procedure to eliminate the difficulty with the one OPW approximation is to expand the electron states |fe> and |fc'> in terms of multiple OPW's, the number of terms in the expansion being determined by the symmetry requirements of the particular state. When this is done giek'x, in the limit of small qled, becomes proportional to qK&l^a>qx instead of ql^(oqx and the divergence is removed. The reader is referred to Sham and Ziman [1] for a detailed treatment of the problem. It is quite obvious, however, that the numerical difficulties and computational time involved become rather prohibitive once one is dealing with a multiple, as opposed to a single, OPW system.

440

B . H A Y M A N a n d J . P . CARBOTTE

Bross and Bohn [2] (BB) have proposed a simple device which enables one to eliminate the divergence while at the same time retaining the economics of a one OPW system and a spherical F.S. Their method consists of merely replacing the Wigner-Seitz sphere which occurs in the familiar Bardeen [3] matrix element with the true Wigner-Seitz atomic polyhedron. When this is done, the spherically symmetric "shape factor" G(\q\) which appears in the standard Bardeen matrix element is replaced by a new function G'(q) which has the interesting property of being equal to zero whenever q = G. This, of course, eliminates the divergence in g^xIn Section 2 we describe the model of Bross and Bohn for the electron-ion matrix element and fix the one parameter that enters into it. In Section 3 we make a comparison with the familiar Heine-Abarenkov pseudopotential form factor and point out the important differences that arise. We then show explicitly how the model of Bross and Bohn eliminates the divergence that comes into the electron-phonon coupling when the one OPW approximation is employed. Results for specific quantities of interest are presented in Section 4 which deals with the electron-phonon mass enhancement, the transport scattering times and the electrical resistivity. A final Section 5 contains a discussion. 2. The Model The bare potential entering the Bardeen matrix element is [4]

W0(q) =

4 nZe2N

(2) + U0 G(\q\) 2 1 with N the number of atoms per unit volume and U0 a parameter to be fixed later. The factor G(\q\) is the "shape factor" mentioned above, and is essentially [4] the overlap of the electron states \k) and |fe + qr> over the Wigner-Seitz sphere. Introducing the screening due to the electron gas, one obtains for the screened form factors (3)

where s(q) is a suitable dielectric constant. BB retain all the essentials of the original Bardeen matrix element except that wherever the Wigner-Seitz sphere enters into the picture (in the definition of the potential, in the integrations, etc.) it is replaced by the Wigner-Seitz polyhedron. They obtain identical expressions for W(q) except that G(\q\) is replaced by G'(q) given by G'(q) = i

J

e-^-'d3r.

(4)

Wigner-Seitz a t o m i c polyhedron

G'(q) is now obviously the overlap of |fc> and |k + 9 ) over the atomic polyhedron. For a f.c.c. structure (we will be considering the case of A1 in some detail) (4) can be rewritten as W2 + V2 + w2 [ (u — V)2

'U+ V + w' sin u — sin I

A Model for the Electron-Phonon Interaction in Polyvalent Metals

u + v— w

— sin-

u — v— w

sin -

u — v

(u

2 u — sin v — sin + v)2 — wsm

441

w

four additional cyclicly permuted terms

(5)

where u = ^ a / 2 , v = qval2, and w = q^a/2 (a is the lattice constant). G'(q) as given by the above expression or by (4) has the following behaviour which appears to be merely a geometrical feature of the Wigner-Seitz cell,

q = G#= 0 , q= 0 .

(6)

The only parameter which occurs in the pseudopotential is the quantity U0 which B B show is essentially the difference in energy between the bottom of the conduction band and the potential on the surface of the atomic polyhedron, and can be estimated from band structure calculations. B B fit the resistivity at 300 °K to obtain a value for U0; however, we will take a value directly from the band calculations of Segall [5]. For the dielectric function s(q) we will use the familiar Lindhard expression because of its great simplicity. 3. Illustration of the Rectification of the One OPW Approximation In this section we will illustrate the failure of the one O P W approximation and its rectification in A1 by looking at the behaviour of the electron-phonon effective phonon frequency distributions - ojx(k - k'))

(7)

and

) = N(0) J ^ Z

S(co - ax(k - fe')) .

(8)

These two functions differ only by the factor 1 — cos (kk) which is needed to weigh more heavily larger angle scattering for the case of transport phenomena. In (7) and (8), N(0) is the density of electronic states at the Fermi surface. The integration is over a solid angle at the Fermi surface. These functions contain in them all the information that is needed about the electron-phonon interaction to be able to discuss several interesting properties of Al. They refer to a specific state on the Fermi surface and hence take into account possible anisotropy with position on the Fermi surface. To start with, in Fig. 1 we show a plot of the Heine-Abarenkov (HA) [6] pseudopotential in Al together with the B B pseudopotential obtained if one uses a value of 0.335 R y d for the parameter U0. We have chosen this value on the basis of the band calculations of Segall [5]. One notes first of all that the B B pseudopotential is not spherically symmetric and the form factors depend on q not merely |g[. We have shown plots of W(q) for q in the three major symmetry directions. The zeros of the B B potential occur at values of q for which q — G.

442

B . H A Y M A N a n d J . P . CARBOTTE

Fig. 1. A plot of the H A and BB potentials in Al. For the BB potential the three different plots are for q in the three high symmetry directions. On the g-axis the three regions are: N the exclusively normal processes, U the exclusively umklapp processes and N + U a mixture of the two

Also on the (¿-axis we have indicated the regions of exclusively normal (N), exclusively umklapp (U), and a mixture of the two ( N + U), processes. To do this one merely looks at the largest and smallest values of |«jr| for which the top of the q lies on a zone boundary. An interesting feature to note here is that due to the polyvalent nature of Al, the umklapp processes are going to be even more dominant here than in the case of the alkalis [7]. One also notes that the BB potential is not too severely different from the H A potential. Using the BB and H A pseudopotentials (with a spherical F.S. and in the one O P W approximation) mentioned above and the Born-von Karman force constants as measured by Gilat and Nicklow [8] we have computed the directional ) Fk(m) and (Xtik(o)) Fk(m) functions at 33 points on the irreducible 1/48'th of the F.S. of Al. These are the same points as used by Leavens and Carbotte [9] in a discussion of the anisotropy in the electron-phonon contribution to the electron effective mass. The technique used to evaluate these functions is exactly the same as that used by Leavens and Carbotte and will not be repeated here. For those values of k which lie close to the points of intersection of the free electron Fermi sphere and the FBZ, one can expect the divergence of the one O P W gitk'x to be most obvious; as for these fc's these will be many values of q for which q G. Accordingly, then, we illustrate the divergence by showing plots of ocfc(co) (obtaining by dividing (x'i(oj) Fk(u>) by the directional frequency distribution Fk(a>) versus frequency co, for initial points k = ( k y , 8 = 27°, (p = 0°), and k = ( k v , 6 = 30°,

) functions in Al at 33 points on the irreducible 1/48'th of the F.S. using the one O P W approximation with the BB pseudopotential. We will use these functions to discuss the anisotropy in the scattering times, the phonon-limited resistivity (both in terms of the scattering time and first order variational solutions to theBoltzmann equation), and finally the anisotropy in the electron-phonon contribution to the electron effective mass. In Fig. 4 and 5, we show plots of the ) Fk(oj) and «trit(w) Fk(co) functions in A1 for k in the three high symmetry directions. In each plot we have included the functions obtained if umklapp processes are arbitrarily excluded from the integrations. This is to demonstrate the crucial role played by umklapp processes in determining the difference between the distributions and hence the anisotropy. I t is interesting to note that the umklapps are more dominant in ottrfc(co) Fk(oj) as opposed to tx2F. This is, of course, due to the extra q term which appears in «trfc(w) Fk(o)) as opposed to regions of the ) Fk(oj) will be determined exclusively by normal processes, in analogy with the alkali metals. F o r those points which lie on or near points of intersection of the F . S . and Bragg planes, although umklapps can now contribute down to the very lowest frequencies, a proper treatment of the scattering ensures that these contributions will vanish as q m \, and consequently m, approach zero. Thus one is assured that at the lowest temperatures, which implies the lowest frequency regions of the«trfe(), the scattering is dominated by normal processes or very weakly weighted umklapps. Thus the anisotropy will disappear at low T. Since the above arguments are equally valid even on a distorted F . S . , one should expect to see essentially the same behaviour in all metals. Finally, in Fig. 9, we show a plot of the variation of Xu, the electron—phonon mass enhancement parameter, over the F . S . Xu is given by (Leavens and Carbotte [9]; Carbotte et al. [12])

-/

a>

For comparison, we also show the results of a one OPW calculation, using the HA potential, performed b y Leavens [13]. The dotted line indicates the regions of the F . S . for which the divergence in guwx becomes crucial. This is where the free electron F . S . intersects zone boundaries. In these regions Leavens [13] employs a renormalization technique which consists of arbitrarily changing the functional dependence of the «&(&>) Fk(a>) function for low &> so that tx|(oj) no longer diverges. Of course, for the B B potential we have employed, this problem does not occur.

50

Fig. 8. The effective number of carriers n*/n versus T in Al. All calculations are done with the BB potential

100 T(°K)

•

447

A Model for the Electron-Phonon Interaction in Polyvalent Metals Fig. 9. The variation of Xk, the electronphonon mass enhancement parameter, over the 1/48'th of the F.S. in Al. the unrenormalized calculations of Leavens [13]. •

= 2 2 i ° , and O cp = 45°

0°

20'

e

60'

One notes that the B B potential gives somewhat less anisotropy and higher values of Xk than the HA potential. The larger values for Xk occur because for most q the B B potential form factors are larger than the HA form factors. Having presented some of the results obtained with the B B potential in Al, we would like to conclude this section by noting that this potential offers an interesting and very economic way of dealing with the divergence in the one OPW matrix elements for a F.S. that intersects zone boundaries and, in addition, illustrates that when a reasonably correct approach to anisotropy is employed in polyvalent metals, essentially the same behaviour as monovalent metals is obtained. However, it suffers from some fairly serious defects. Since it is essentially a Bardeen matrix element, it has the same shortcomings as the original. The major weakness [1] is the assumption that the field of the ion can be represented by a sharp-edged square well effective potential, and a sharplydefined volume of charge (the atomic polyhedron) for the valence electrons. Also, the behaviour of the form factors around 2kv is rigidly determined by the function G'(q) and the parameter U0 is relatively ineffective in altering the shape of the "pseudopotential". It is somewhat of a happy coincidence that in Al, the B B potential turns out to be quite similar in shape to the HA potential, which is based on somewhat firmer physical grounds. However, in other metals, it is not clear that the B B potential would not be radically different from other empirical potentials. 5. Discussion The objective in this paper has been to investigate the B B potential as an interesting device designed to overcome the difficulties encountered in the one OPW approximation in a very economical way. It has also hopefully demonstrated that the consequences of anisotropy in a polyvalent metal are essentially the same as those encountered in the alkali metals — one expects to see the anisotropy vanish at high and low temperatures and reach a maximum in some intermediate temperature region. In regard to the electrical resistivity, we have also shown that a conventional one OPW calculation leads to serious discrepancies at low temperature which can be resolved by treating the divergence of the one OPW matrix elements. There is no doubt, however, that a completely reliable and detailed multiple OPW calculation is necessary to discuss all the details of the above effects. We do feel, however, that in a qualitative sense, the above calculation demonstrates the essential features of the problem. 29«

448

B. HA YMAN a n d J . P . CARBOTTE: E l e c t r o n - P h o n o n I n t e r a c t i o n in AL

References [1] L. J . SHAM a n d J . M. ZIMAN, Solid S t a t e P h y s . 15, 263 (1963). [2] H . BROSS a n d G. BOHN, p h y s . s t a t . sol. 2 0 , 2 7 7 (1967).

[3] J. BARDEEN, Phys. Rev. 52, 688 (1937). [4] J. ZIMAN, Electrons and Phonons, Clarendon Press, Oxford 1962. [5] B . SEGALL, P h y s . R e v . 1 2 4 , 1 7 9 7 ( 1 9 6 1 ) ; 1 8 1 , 121 (1963).

[6] V. HEINE a n d I . ABARENKOV, Phil. Mag. 9, 451 (1966). [7] B. HAYMANH a n d J . P. CARBOTTE, J . P h y s . E 2, 915 (1972). [8] [9] [10] [11] [12]

G. GILAT a n d R . M . NICKLOW, P h y s . R e v . 1 4 3 , 4 8 7 (1966). C. R . LEAVENS a n d J . P . CARBOTTE, A n n . P h y s . 70, 3 3 8 (1972). R . S. SETH a n d S. B . WOODS, P h y s . R e v . B 2, 2 9 6 1 (1970). B . HA YMAN a n d J . P . CARBOTTE, P h y s . R e v . B 6 , 1 1 5 4 (1972). J . P . CARBOTTE, R . C. DYNES, a n d P . N . TROFIMENKOFF, C a n a d . J . P h y s . 4 7 ,

(1969). [13] C. R. LEAVENS, Ph. D. Thesis, McMaster University, 1970. (Received

May

31,

1974)

1107

B.

NIKIN

et al.: Analysis of Ferromagnets with Dipole-Dipole Interactions

449

phys. stat. sol. (b) 65, 449 (1974) Subject classification: 18.2 Boris Kidri6 Institute

of Nuclear Sciences,

Belgrade

Introduction to an Analysis of Ferromagnets with Dipole-Dipole Interactions By B . N I K I N 1 ) , B . S . TOSIC, a n d V . M . ZEKOVIC 2 )

The harmonic spectrum of a ferromagnet with dipole-dipole interactions is investigated. It is shown that the dipole-dipole interaction can be treated as a perturbation only after preparing of the Hamiltonian of the system. This preparation consists in elimination of linear terms of the Hamiltonian and in evaluation of the correct harmonic spectrum w i t h respect to the fact that the system does not conserve the total number of quasi-particles. The approach suggested is valid only in the presence of strong external electromagnetic fields. B paSoTe nccjiejiyeTCH ciiera-p ajieiweHTapHbix B036y»c;ieHHH B (JieppoMarHeTMitax C nHnOJlb-HHIIOJlhHblMH B3aHMOJieiiCTBHHMH. Il0Ka3aH0, HTO HHnOJIb-HHnOJIbHOe B3aHMoneHCTBHe Hejib3H yqHTbiBaTb MeTonaMH TeopHH B03MymeHHH 6e3 n p e a BapHTejibHott nepepaCoTKH RAMHJIBTOHHAHA CHCTeMbi. IlepepaSoTKa COCTOHT B OCB060H«HeHHH OT JlHHeHHblX IJieHOB B raMHJIbTOHHaHe C nOMOmbK) KaHOHHqecKoro npeo6pa30BaHHH onepaTopoB H B nocjienyioiueM BbmHCJiemiH npaBHJibHoro cneKTpa 3jieMeHTapHbix B036y>KHeHHH c yqeTOM Toro $aKTa, UTO IHCJIO nacTHU B cHCTeMe He coxpaHHeTcn. IIpeaaaraeMbiii n o j i x o n npHMeHHM jinuib B cjiytiae Kor«a (JteppoiwarHeTHK HaxonHTCH BO BHEINHEM MarHHTHOM n o n e .

1. Introduction Dipole-dipole interactions in ferromagnets are by two or three orders of magnitude weaker than the exchange interactions. Consequently, the Hamiltonian of the dipole-dipole spin interaction can be treated as perturbation with respect to the exchange part of the Hamiltonian. I t must be noticed that in this case the perturbation method cannot be applied in a straightforward way. The complete Hamiltonian of the system must be previously "prepared" due to the following reasons: a) The Hamiltonian of the system contains terms linear in creation and annihilation spin operators, i.e. the vacuum state of the system is not well-defined. b) The Hamiltonian of the system does not commute with the total occupation number operator. I t means that the number of quasi-particles in the system is not conserved. In this case we can make calculations using the perturbation methods, but, as it was shown in [1], the zero-order energy spectrum of the approximate second quantization (ASQ) method, leads to incorrect results. Consequently, the perturbation methods can be used in the analysis of systems only after elimination of linear terms from the Hamiltonian and after finding the correct zero-order energy spectrum. !) Present address: School of Enginering, Mechanical Department Zrenjanin, Yugoslavia. 2 ) Present address: Department of Physics, University of Novi Sad.

450

B . N I K I N , B . S. TOSIC, a n d V . M . ZBKOVIC

As it will be seen later, the solution of the mentioned problems is not a trivial procedure. W e shall investigate the special case when the ferromagnet is in a strong external magnetic field 3€, such that ¡x^X (/uB magnetic moment of a isolated atom) is of the same order of magnitude as the exchange integrals. W e shall also limit our investigations to the range of small wave vectors. I t means that the results obtained could find practical application for weak ferromagnets in the low-temperature regions.

2. Stabilization of the Hamiltonian The Hamiltonian of the ferromagnet with dipole-dipole spin interactions (see [2]) has the following form: H =

-

E 1

- i s IigStSg lg

- i z Dfg[S,Sg lg

3 (eigS,)

-

(efgSg)],

(2.1)

A

where / and g are the lattice point vectors, S the spin operators, Itg = Igf exchange integrals, and

e1a=i\r=i\'

a

=

(

w

)

•

(2-2)

The coefficients Df g characterize dipole-dipole spin interactions. The Hamiltonian (2.1) can be written in the form which is more convenient for further calculations, i.e. H% + Hz + H, ,

H = H0 + Hx +

(2.3)

where H0 =

-fiBXSN

z

-

i NS* (J0

+y0),

t

H2 =

+ S(J0

+ y0)] Z (S 1

3})

- i

£ (I/g 1,9

+ ccfg) SfS+g

+

1,9

s3

lg

#4 =

-

[dfgsj

(S -

Si)

+

i Z {Itg + Ylg) (S lg

Jo

= H hi, i

« *

=

Yo = i

- D * { 1 - T

[(g) ;

Q0 =

i

)],

Z

0o/ >

.

+ i 4 A )

6fg = Wig

,

(efge}g +

ie%e}g) .

v

(2

i B

3

fg =

/9

=

L 1

We note t h a t a and /? are exact u p to terms quadratic in small parameters £0. The stable Hamiltonian has the following form H = H0 + H2 + H3 +

Hi,

where 1

H0 = N H2=Y

1 (•A> + y0) - 1 7

N |0OI2 (9?oBo - doBo 16 /¿ B X ie0l IA.I

Z QjQ( + £ XfgQIQg + - Z (TfgQ}Qt + TfgQgQt) ; / fg fg

y = r + r w + y, YW

=J

2

3|0§| J 0 16(^^)2

y< 2) A

/g — A/g

+

1

+

V(0) — _ —— ~ 1,T Afg fg > Ay ( 2 )

/9

=

2 2/nBJe M * + OS ) ,

fg » y(

i)

=

1 Ifg\e0\2 16 (/lib36)2 ' 4/UB3€ (Qofyg + 00%) ,

27, = 2/9 fg + i" j/bi f>

(2-12)

Introduction to an Analysis of Ferromagnets with Dipole-Dipole Interactions

,

(3.1)

Q +Q = B +B - B+B+BB , the relevant contributions to ASQ spectrum give the fourth- and sixth-order parts of the boson Hamiltonian with equal number of creation and annihilation operators in first-order perturbation and the third-order part as well as fourthorder parts with non-equal number of creation and annihilation operators in the second order perturbation. The quadratic part // 2 b of the boson Hamiltonian has the following form:

#2B = Y Z BfBf + £ Xfg BfBg 1

+1Z

1,9

1,9

(Tfa B}B+ -+ n0 Bf Bg )

(3.2)

and can be diagonalized by the use of the transformation

B, = - L ^ (Uk Ck + VtC +k ) e ikf , IIN k

(3-3)

where Ck and C£ are the new Bose operators, and

Uk = l + ^\Qk \ 2 + 0(\Q \3) Qk Tfe

rH = - e t - QM + 0(i(?|3

Tu > b '

oik — y 0

Jk — Jo

(3.4)

Jk = Z Jot G ikf, 1 Tk = L Tof e ikf. 1 The formulae obtained are exact up to quadratic terms in the small parameter Qk. After substituting (3.3) in the fourth- and sixth-order terms of the boson Hamiltonian and reduction of these terms to normal products we obtain the correction to the ASQ spectrum. After the corrections caused b y the third-order terms of the Hamiltonian and by fourth-order terms of the type C +G +C +C | c w e obtain as poles of the Green's function n ) is connected with the polarization operator IJ(q, ia>„) by the Dyson equation /

47re2 \

K(q, iu>n) = U{q, ia>„) + II{q, ia>n) ( —

K{q, iw„

where /? = 1 ¡T; wn = 2nnT; T is the imaginary time-ordering operator, is the operator of density fluctuations in the Heisenberg representation for imaginary time. The dielectric function e(q, w) is given by the expression

o,(t)

4TIB^1

e(q, w) = 1 + — n(q, a>) , (8) M' where II(q, w) is the analytical continuation of IJ(q, icon ) from the upper half-

plane to the real half-axis a> 0. Some graphs for the polarization operator are presented in Fig. 1, where solid broken lines correspond to the Green's functions of the free electron Gp(i