Light and Matter Id / Licht und Materie Id

613 Pages·1984·14.194 MB·English

by  H. Bilz, D. Strauch, R. K. Wehner (auth.), Professor Dr. Ludwig Genzel (eds.)

ENCYCLO PEDIA OF PHYSICS
EDITOR IN CHIEF
S. FLUGGE
VOLUME XXV/2d
LIGHT AND MATTER Id
BY
H. BILZ . D. STRAUCH· R.K. WEHNER
EDITOR
L. GENZEL
WITH 139 FIGURES
SPRINGER-VERLAG
BERLIN HEIDELBERG NEW YORK TOKYO
1984
HANDBUCH DER PHYSIK
HERAUSGEGEBEN VON
S. FLUGGE
BAND XXVj2d
LICHT UND MATERIE Id
VON
H. BILZ . D. STRAUCH· R.K. WEHNER
BANDHERAUSGEBER
L. GENZEL
MIT 139 FIGUREN
SPRINGER-VERLAG
BERLIN· HEIDELBERG· NEW YORK· TOKYO
1984
Professor Dr. SIEGFRIED FLUGGE
Physikalisches Institut der Universitiit, D-7800 Freiburg i. Br.
Professor Dr. LUDWIG GENZEL
Max-Planck-Institut fUr Festkorperforschung, D-7000 Stuttgart
ISBN-13: 978-3-642-46435-5 e-ISBN-13: 978-3-642-46433-1
DOl: 10.1007/978-3-642-46433-1
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© by Springer-Verlag Berlin Heidelberg 1984.
Softcover reprint of the hardcover 1st edition 1984
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2153/3130-543210
Preface
The dynamical properties of solids have recently attracted renewed interest
in connection with the increasing understanding of phase transitions and re
lated phenomena. In particular, soft modes or, more generally, phonon 'anom
alies' seem to play an important role in structural and electronic phase tran
sitions, such as ferroelectric or superconducting transitions. The understanding
of the mechanisms responsible for the occurrence of unusually low frequencies in
phonon spectra requires a detailed analysis of the microscopic forces governing
the lattice vibrations. Of particular importance is the influence of the electron
lattice interaction in the adiabatic approximation which in many cases is the
origin of peculiarities in the phonon self-energy.
In this work the vibrational spectra of pure non-metals and of those con
taining point defects are investigated.' In these materials the interrelation be
tween the pseudo-harmonic forces (determining the phonon dispersion re
lations) and the non-linear anharmonic and electron-phonon forces (as they act
in infrared and Raman spectra) is most obvious and can be quantitatively
analysed in terms of appropriate models. The main task is to arrive at a
physically correct treatment of electronic degrees of freedom, as for example in
an electronic 'shell' model, which leads to the description of phonon spectra in
terms of long-range polarizabilities and short-range deformabilities. The pur
pose of our review is to stimulate further investigations which, we hope, will
result in explicit relations between the parameters of the semi-microscopic
models and the matrix elements from the electronic band structure. Our contribu
tion is restricted to vibrational spectra to emphasize the 'phonon' aspects of
infrared absorption and Raman spectra. Therefore, resonant Raman spectra and
related phenomena are bareleey discussed, and the reader is referred to the rich
literature in these fields. The same holds for an explicit analysis of dynamical
aspects of phase transitions and of the interesting Raman spectra of supercon
ductors and valence-mixing crystals. The authors hope that their review provides
a coherent presentation of the basic concepts useful for an understanding of the
dynamical properties of solids as they manifest themselves in their vibrational
spectra.
The authors wish to thank many colleagues and co-workers for stimulating
discussions, helpful criticism and substantial support during the work on the
manuscript.
In particular, they would like to mention some specific contributions to this
article. The essential contents of Sect. 22b on the lattice relaxation are parts of
unpublished results by J.B. Page; the permission to publish them here is grate
fully acknowledged. T.P. Martin kindly agreed to contribute a survey on finite
VI Preface
crystals (Sect. 32) while E. Kiefer-Schroder was kind enough to prepare the
tables of elastic and dielectric constants (Sect. 39).
In addition, the authors are grateful for instructive comments and helpful
suggestions by L. Genzel, R. Klein, F.W. de Wette, B. Gliss, W. Kress, D. Smith,
W. Weber and R. Zeyher. The final version of the manuscript benefitted from
a critical reading of parts of the manuscript, concerning physical and linguistic
aspects, by M. Buchanan, R.J. Bell, W. Kleppmann, W. Kress and T.P. Martin.
We also thank Mrs. Eva Genzel very much for the preparation of the subject
index. Finally the untiring help of our secretaries Mrs. R. Ocal, Mrs. E.
Brtigmann, Mrs. A. Wiilti and Mrs. G.!. Keck in preparing the different
versions of the manuscript was essential for the completion of the article.
It is a great pleasure to thank the editors and the publishers, in particular
L. Genzel and H. Mayer-Kaupp, H. Lotsch, K. Koch and K.-H. Winter for
their remarkable patience and co-operation during the time of the production
of the article.
H. BILZ, D. STRAUCH, and R.K. WEHNER
Contents
Vibrational Infrared and Raman Spectra of Non-Metals
By H. BILZ, Max-Planck-Institut flir Festkorperforschung,D-7000 Stuttgart,
Fed. Rep. of Germany, D. STRAUCH, Institut flir Theoretische Physik der Uni
versitat Regensburg, D-8400 Regensburg, Fed. Rep. of Germany, and R.K.
WEHNER, Fachbereich Physik der Universitat Munster, D-4400 Munster, Fed.
Rep. of Germany
A. Introduction . . . . 1
1. Historical survey. 2
2. Outline of the theory of infrared absorption and Raman
scattering . . . . . 4
a) Macroscopic aspects. 4
b) Microscopic aspects . 5
B. Phonons in insulators . . . 9
3. General properties of phonons . 9
a) Dynamic and thermodynamic stability of solids 10
b) The adiabatic approximation . . 15
c) Force constants. . . . . . . . . . 16
d) Symmetry properties of phonons . . 18
e) The pseudo-harmonic approximation 20
4. Ionic crystals . . . . . 21
a) The rigid-ion model. . . 21
b) Dipole models . . . . . 26
c) The breathing shell model 29
d) Ionic deformabilities. . . 31
e) Non-central and many-body forces and the elastic
properties of crystals. . . . . . . . . . . . 38
5. Covalent crystals . . . . . . . . . . . . . . 44
a) Formal force constants and general properties 44
b) Dipole models . . . 44
c) Bond-charge models. . . . . . . . . . .. 46
d) Valence force fields . . . . . . . . . . . . 48
e) Crystals of partially ionic and partially covalent
character. . . . . . . . . 49
f) Sum rule of lattice vibrations. . . . . . . . . 50
VIII Contents
6. Microscopic theory, models, and macroscopic quantities 51
a) Overlap theory . . . . . . . . . . . 52
b) The dielectric function method . . . . . . . 54
c) The direct 'frozen-in' phonon approach . . . 57
d) Charges and polarizabilities of ions and bonds 58
e) Electric fields and effective charges in ionic solids 61
f) Fields and charges in covalent solids . . . . . 70
g) The microscopic description of charges and fields 74
C. Interaction of photons with matter . . . . . . . . 77
7. Theory of interaction of photons with particles 77
a) Non-relativistic theory of inelastic scattering 77
b) Gauge invariance in electromagnetic interaction. 82
c) Dielectric constant of electrons . . . . . . . 86
d) Light scattering by electrons . . . . . . . . 90
e) Interaction of photons with electrons and ions 91
f) Polaritons in the harmonic approximation 92
8. Infrared absorption and dielectric response . . . 95
a) Dielectric susceptibility . . . . . . . . . . 96
b) Absorption of radiation (fluctuation-dissipation theorem) . 99
c) Frequence-dependence and thermodynamic definitions
of the susceptibility, sum rules 101
d) Static susceptibility . . 103
9. Raman scattering of light. . . . 108
a) Introduction . . . . . . . . 108
b) Quantum theory of spontaneous Raman scattering 109
c) Adiabatic representation. . . . . . . . . 112
d) Polarizability theory. . . . . . . . . . . 115
e) Green function theory of Raman scattering. 116
f) The law. . . . . . . . . . . 120
0)4
g) Polariton picture of light scattering . . . . 123
h) Resonant Raman scattering (RRS). . . . . 124
i) Rayleigh, Brillouin, and Hyper-Raman scattering 125
D. Expansion theory of susceptibilities and polarizabilities 125
10. General lattice potential . . . . . . . . . . . 126
a) The undeformed lattice . . . . . . . . . . 126
b) The lattice in a static electric field and under deformation 127
11. Lattice dipole moment . . . . . . . . . . . . . . . . . 132
a) The undeformed lattice . . . . . . . . . . . . . . . 132
b) The lattice in a static electric field and under deformation 132
12. Lattice and electronic susceptibility 133
a) Formal expansion of the susceptibility 133
b) The harmonic approximation. 134
c) Anharmonic susceptibility . . . . . 137
Contents IX
. d) The anharmonic dispersion oscillator 138
e) The damping function. . . . . . . 140
f) The renormalized dipole moment . . 141
g) The general form of the lattice susceptibility 141
h) Coupling of dispersion oscillators . . . . . 142
i) Anharmonic coupling parameters . . . . . 142
j) The susceptibility under external pressure and
in a static field . . . . . . . . . . . . . . 142
13. Lattice polarizability and Raman scattering . . . 143
a) Formal expansion of the electronic susceptibility 143
b) Harmonic approximation . . . . 144
c) Anharmonic treatment. . . . . . 145
d) Raman scattering in cubic crystals. 145
e) Raman coupling parameters . . . 146
f) Effects of static fields and external pressure. 147
E. Interpretation of experimental spectra . . . . . . . . . . . 148
14. Model theory of infrared absorption and Raman scattering 148
a) General features of infrared and Raman processes. . 148
b) Microscopic and model treatment of electron-phonon
interaction . . . . . . . . . . . . . . . . . . . 151
c) Shell model treatment of Raman scattering. . . . . 153
d) Bond charge and bond polarizability in infrared and
Raman processes . . . . . . . . . . . . 156
15. Infrared spectra of ionic crystals. . . . . . . . . . . 157
a) Qualitative classification of infrared spectra. . . . . 157
b) The infrared spectra of alkali halides: anharmonic effects. 163
c) Critical point analysis . . . . . . . . . . . 167
d) Density of states approximation. . . . . . . 169
e) The effect of short-range cubic anharmonicity . 171
f) The effect of quartic and higher anharmonicity 172
g) Coulomb anharmonicity . . . . . 180
h) Absorption at very low frequencies 183
i) Non-linear dipole moments. . . . 186
j) The effect of ionic polarizability. . 190
k) Final states interactions of phonons: anharmonic
broadening and bound states . . . . . . . . . 190
1) Line widths of dispersion oscillators and temperature-
dependence. . . . . . . . . . . . . . . . . 191
m) Discussion of other diatomic ionic crystals . . . 198
n) Cubic crystals with three and more ions in a cell 203
16. Infrared spectra of covalent crystals . . . . 208
a) General features of the spectra . . . . . . 208
b) Spectra of crystals with diamond structure . 209
c) Covalent crystals with linear dipole moments 214
x
Contents
17. Infrared spectra of crystals with mixed ionic and covalent
character. . . . . . . . . . . . . . . . . . . . . 214
a) The concurrence of anharmonicity and non-linear dipole
moments. . . . . . . . . . . . . . . . 214
b) Spectra of crystals with zincblende structure 216
c) Spectra of perovskites . . . . . . . 219
d) Spectra of low-symmetry crystals . . 224
e) Spectra of amorphous semiconductors 227
18. Raman scattering from ionic crystals. . 229
a) Raman spectra of cubic ionic crystals 230
b) Other diatomic ionic crystals 244
c) Perovskites. . . . . . . . . . . . 245
d) Other ionic crystals . . . . . . . . 246
e) Photoelasticity and Raman scattering 246
f) First-order Raman scattering. . . . 251
19. Raman spectra of covalent and partially ionic crystals 252
a) Spectra of diamond and its homologues 252
b) Spectra of III-V and II-VI compounds. 258
F. Lattices with point defects . . . . . 262
20. Types of defects and their effects . 262
a) Introductory remarks . . . . 262
b) Point defects, vacancies . . . 264
c) Defect-induced infrared and Raman spectra. 266
d) Localized modes, gap modes . . . . . . . 267
e) Resonant modes . . . . . . . . . . . . 271
f) Off-center and molecular defects: Tunnelling motion. 280
g) Internal vibrations of molecular defects . . . . 283
h) Interstitials. . . . . . . . . . . . . . . . . 286
i) Effects of defect clusters and defect concentration 288
j) Dislocations, surfaces . . . . . . . . . . 289
21. Information contained in defect-induced spectra . 289
22. Lattice dynamics of impure lattices. . . . . . . 295
a) Introduction: Molecular model - the nature of
perturbations due to a defect . . . . . . . 295
b) Lattice distortions - method of lattice statics 297
c) Equation of motion of the perturbed lattice. 300
d) Symmetry considerations. . . . . . . . . 303
e) Lifshitz method for the solution of the equation of motion
- localization of perturbations . . . . . . . . . 309
23. The Green function of the harmonic perturbed lattice. 315
a) Real Green function and T matrix. . . . . 315
b) The complex Green function . . . . . . . . . . 317
c) Resonances: Localized and resonant modes. . . . 320
d) Eigenvalue treatment of the Green function and T matrix
in the impurity space . . . . . . . . . . . . . . . . 323
Contents XI
24. Properties of the perturbed harmonic lattice Green function . 324
a) Kramers-Kronig transform. . . . . . . . . . . . . . 324
b) Normalization of the perturbed resonance-mode eigenvectors:
An effective mass of the resonance vibration . . . . . .. 325
c) Approximate form of the Green function and of the T matrix
near a resonance frequency: Width and intensity . . . .. 326
25. Applications of Green functions: Phonon spectra in
perturbed crystals . . . . 328
a) Phonon density of states 328
b) Dielectric susceptibility 331
c) Raman scattering . . . 341
d) Resonance Raman scattering 347
26. Dynamics of lattices with interstitial or molecular defects 351
a) Formulation of the problem . . . . . . . . . 351
b) Standard procedure - application to interstitials. . . 352
c) Formalism modified for molecular defects . . . . . 355
27. Shell-model treatment of the dynamics of perturbed lattices
and the model theory of infrared-absorption and Raman-
scattering spectra . . . . 358
a) The use of shell models . . . . . . . . . . 358
b) Effective force constants . . . . . . . . . . 361
c) Shell-model extension of the Lifshitz formalism 363
d) Shell-model interpretation of the effective charge 373
e) The higher-order dipole moments . . . . . . . 376
f) Shell-model interpretation of the Raman scattering intensity 377
28. Concentration effects. . . . . . . . . . . . . . . . . 382
a) Introduction: Diagrammatic expansion . . . . . . . 382
b) Low-concentration single-site scattering approximation. 386
c) Self-consistent approximation. . . . . . 388
d) Coherent-potential approximation (CPA) . 388
e) Applications . . . . . . . . 389
29. Mixed crystals. . . . . . . . . 394
a) One- and two-mode behaviour 394
b) Theoretical models . . . . . 399
c) Changes in the lattice constant and Ivey relation 405
30. Anharmonic effects in perturbed crystals . . . . . 406
a) Introduction: Resonance modes in analogy to the Rest-
strahlen or Raman oscillator . . . . . . . . . 406
b) Qualitative aspects of the anharmonic self-energy
in perturbed crystals. . . . . . . . . . . . . 408
c) Diagonal and off-diagonal elements of the perturbed
self-energy . . . . . . . . . . . . . . . . . . . 409
d) Low-order contributions to the self-energy . . . . . 413
e) Approximate form of the anharmonic Green function 416