Dynamical Properties of Solids : Phonon Physics The Cutting Edge (Dynamical Properties of Solids)

Dynamical Properties of Solids Volume 7

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Dynamical Properties of Solids Volume 7

Phonon Physics The Cutting Edge

edited by

Amsterdam

G.K. Horton

A.A. Maradudin

Rutgers University Piscataway, U S A

University of California Irvine, U S A

- Lausanne - New York - Oxford - Shannon - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 82262 3 9 1995 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.

Preface

Volumes 1 and 2 of this series, which were published in 1974 and 1975, respectively, contained several chapters devoted to anharmonic properties of solids, to ab initio calculations of phonons in metals and insulators, and to surface phonons. In the twenty years since the appearance of these two volumes each of these important areas of lattice dynamics has undergone significant developments. Consequently, it was felt to be desirable to devote a major part of this volume to a survey of the current status of these areas. A major development in theoretical studies of anharmonic properties of crystals has been the emergence of numerical simulation approaches that can be used in the regime of low temperatures, where quantum effects are large, and where traditional molecular dynamics simulations or classical Monte Carlo methods are inapplicable. One of these approaches, the path-integral quantum Monte Carlo method, originally developed for the study of quantum spin systems, has been applied successfully to the determination of low temperature thermodynamic (static) properties of anharmonic crystals, and to certain dynamical (time-dependent) properties as well. In this method the calculation of low temperature vibrational properties of an n-dimensional crystal is transformed into a classical calculation of these properties in an effective (n + 1)-dimensional crystal. This approach, and results obtained by its use, are described in the chapter by A.R. McGurn. It is a computationally intensive method, which fact has stimulated efforts to find alternative simulation approaches that possess comparable accuracy but which are easier to implement. A significant step in this direction is provided by the effective potential method, in which the atoms in an anharmonic crystal interact via a variationally determined effective potential that incorporates quantum effects in an approximate, yet accurate, fashion. The calculation of static and dynamic properties of anharmonic crystals in the quantum regime become no more difficult than the corresponding classical calculations carried out by Monte Carlo simulations. The chapter by E.R. Cowley and G.K. Horton is devoted to a description of this very promising approach. However, not all the advances in our ability to understand anharmonic properties of crystals have been methodological in nature. New consequences of lattice anharmonicity have been discovered as well. In their chapter A.J. Sievers and J.B. Page discuss the recently intensively studied

intrinsic anharmonic localized modes. These are vibrational modes that are localized about lattice sites of a perfect, i.e. defect-free, crystal by the anharmonicity of the interatomic potential. These modes and their properties have been investigated theoretically by a variety of techniques, all of which are discussed by Sievers and Page. The two topics of ab initio calculations of phonons in metals and surface phonons are combined in the chapter written by A.G. Eguiluz and A.A. Quong. In it are described recent developments in the calculation of bulk phonons and of surface phonons in metallic systems, the application of the results of the latter calculations to the analysis of atom/surface scattering experiments, as well as other properties of such systems in which the screening effects of the conduction electrons play the dominant role. The remaining two chapters are devoted to topics that have not been treated in the preceding volumes of this series. One is phonon transport; the other is phonons in disordered crystals. The chapter by T. Paszkiewicz and M. Wilczyfiski deals with the specific topic of the effects of isotopic and substitutional impurities on the propagation of phonons in harmonic crystals, while the chapter by J.D. Dow, W.E. Packard, H.A. B lackstead, and D.W. Jenkins, is devoted to the vibrational properties of semiconductor alloys, both random and in the form of superlattices, and in their manifestation in experimental data such as are provided by Raman scattering experiments. The work described in the six chapters of this volume testifies to the continuing vitality of the field of the dynamical properties of solids nearly a century after its founding. It bodes well for the discovery of new physics and new methodologies in this field in the years to come. A.A. Maradudin

G.K. Horton

vi

List of Contributors

H.A. Blackstead, Physics Department, University of Notre Dame, Notre Dame, Indiana 46556, USA E.R. Cowley, Department of Physics, Camden College of Arts and Sciences, Rutgers, The State University, Camden, NJ 08102-1205, USA J.D. Dow, Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA A.G. Eguiluz, Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1200, and Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6032, USA G.K. Horton, Serin Physics Laboratory, Rutgers, The State University, Pis-

cataway, NJ 08855-0849, USA D.W. Jenkins, Institute for Postdoctoral Studies, 1128 Almond Drive, Aurora, Illinois 60506, USA

A.R. McGurn, Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008, USA W.E. Packard, Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA J.B. Page, Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504, USA T. Paszkiewicz, Institute of Theoretical Physics, University of Wroctaw, pl. Maksa Borna 9, PL-50-204 Wroctaw, Poland A.A. Quong, Computational Materials Sciences (8341), Sandia National

Laboratory, Livermore, CA 94551-0969, USA

vii

A.J. Sievers, Laboratory of Atomic and Solid State Physics and the Materials Science Center, Cornell University, Ithaca, NY 14853-2501, USA M. Wilczyl~ski, Institute of Theoretical Physics, University of Wroctaw, pl. Maksa Borna 9, PL-50-204 Wroctaw, Poland

~ 1 7 6

Vlll

Contents Volume 7

Preface v List of contributors Contents ix

vii

1 Path-integral quantum Monte Carlo studies of the vibrational properties

of crystals 1 A. R. McGurn 2 Lattice dynamical applications of variational effective potentials in the Feynman path-integral formulation of statistical mechanics 79 E. R. Cowley and G. K. Horton 3 Unusual anharmonic local mode systems A. J. Sievers and J. B. Page

137

4 Influence of isotopic and substitutional atoms on the propagation of phonons in anisotropic media 257 T. Paszkiewicz and M. Wilczyhski 5 Phonons in semiconductor alloys 349 J.D.Dow, W.E. Packard, H.A. Blackstead and D. W. Jenkins 6 Electronic screening in metals: from phonons to plasmons A. G. Eguiluz and A.A. Quong Author index 509 Subject index 523

ix

425

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CHAPTER 1

Path-Integral Quantum Monte Carlo Studies of the Vibrational Properties of Crystals ARTHUR R. McGURN Department of Physics Western Michigan University Kalamazoo, Michigan 49008 USA

9 Elsevier Science B. V, 1995

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

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Contents 1. Introduction

5

2. Classical Monte Carlo methods: inert gas solids 3. Quantum Monte Carlo methods

10

19

3.1. Single particle model 20 3.2. One-dimensional chains 28 3.3. FCC Lennard-Jones crystal 44 4. Time-dependent quantum Monte Carlo

52

4.1. Continued fraction expansion 53 4.2. Moments of the spectral distribution 59 4.3. Gaussian approximation for the spectral density ,

Discussions and conclusions

Acknowledgement Appendix References

72 74

72

64

63

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1. Introduction In this chapter I will look at some of the recent developments in Monte Carlo simulations for the static and dynamic thermodynamic properties of lattice vibrations in quantum mechanical crystaline solids (i.e., the development of quantum Monte Carlo methods as they apply to phonons in crystals). This is a relatively new topic in solid state physics even though the study of the thermodynamics of quantum vibrations in crystaline materials is one of the first branches of modem physics to be developed (Born and Huang 1954; Maradudin 1969; Paskiewicz 1987; Xia et al. 1990). The basis of quantum Monte Carlo simulation techniques as applied to lattice vibrations is the reformulation of the quantum mechanical partition function in terms of a path integral expressed solely in classical (commuting) variables (Suzuki 1976a, b, 1987; Suzuki et al. 1977; de Raedt and Lagendijk 1985; Gubernatis 1986; Negele and Orland 1988; Doll and Gubernatis 1990; Rubinstein 1981; Binder 1984, 1986). In the course of this reformulation, one finds that a d-dimensional quantum partition function is reexpressed as a (d + 1)-dimensional path integral so that the complications of evaluating a partition function formed of non-commuting operators is carried over to the evaluation of a higher dimensional classical problem. Our goal in this chapter will be to present this path-integral reformulation and then to discuss the evaluation of the thermodynamics of the quantum system in terms of the evaluation by classical Monte Carlo techniques of averages formed in the corresponding path-integral formulation. As a final point, contact will also be made with recent efforts to obtain approximate evaluations of the path-integral formulation using variational techniques (Feynman 1988; Samathiyakanit and Glyde 1973; Giachetti and Tognetti 1985-1987; Feynman and Kleinett 1986; Giachetti et al. 1988a, b; Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991, 1993). In our discussions we shall first treat the static thermodynamic properties of vibrational crystaline systems (Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991, 1993; McGurn et al. 1989, 1991; Maradudin et al. 1990) and then consider the more difficult problem of the numerical simulation of the quantum response functions (McGurn et al. 1991; Doll et al. 1990; Freeman et al. 1990; Schtittler et al. 1990; Silver et al. 1990; Cuccoli

6

A.R. McGurn

Ch. 1

et al. 1992a, b, 1993) of these systems. The method of quantum Monte Carlo simulation, as we shall see below, has been applied very successfully to the study of the static thermodynamic properties of a wide variety of quantum mechanical systems, and it should not surprise us to find considerable success in the application of these same techniques to the study of the static thermodynamics of vibrational systems. On the other hand, very little work has been done on the problem of the quantum Monte Carlo simulation of time-dependent response functions, and this area is still very much open as a field in need of more research efforts (Gubematis 1986; Doll et al. 1990; Doll and Gubernatis 1990; Freeman et al. 1990; Schtittler et al. 1990; Silver et al. 1990; Cuccoli et al. 1992a, b, 1993). We shall examine in this chapter just some very rudimental efforts in dealing with the time-dependent properties of quantum systems at finite temperatures. At the present writing, we should also note that a large body of analytic work does in fact exist on both the static and time-dependent thermodynamic properties of vibrational systems. We shall, however, only concentrate in the present chapter on simulation methodology, referring to analytical treatments only for comparisons with simulation data from the quantum Monte Carlo. The interested reader can find good reviews of the analytical aspects of these topics for work done prior to 1969 in the book by Maradudin et al. (1969) and for more recent analytical work in the review of Maradudin et al. (1990) and in the proceedings of a recent topical conference on phonons (Paskiewicz 1987). There are a number of different quantum Monte Carlo methods that have been developed in the last couple of decades, including: 1) The Green's function Monte Carlo which uses the Schr6dinger equation evaluated for imaginary times to determine the ground state properties of many-body systems (Cerperley and Alder 1986; Kalos 1964, 1967, 1970, 1984; Anderson 1975, 1976, 1980; Suhm and Watts 1991). (The Schr6dinger equation for imaginary time is of the form of a diffusion equation and subject to similar Monte Carlo techniques as applied to the study of classical diffusion.), 2) variational techniques based on the well known theorem for determining the ground state of quantum systems (Feynman and Cohen 1956; Boninsegni and Manousaki 1990; Louis 1990), and 3) path-integral techniques (Suzuki 1976a, b, 1987; Suzuki et al. 1977; de Raedt and Lagendyk 1985; Gubematis 1986; Ceperley and Alder I986; Negele and Orland 1988; Doll et al. 1990; Doll and Gubematis 1990; Feynman 1988; Samathiyakanit and Glyde 1973; Giachetti and Tognetti 1985, 1986, 1987; Feynman and Kleinert 1986; Giachetti et al. 1988a, b; Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991, 1993; McGurn et al. 1989, 1991; Maradudin et al. 1990) which arise from the application to the exponential form in the partition function of an identity due to Trotter (Trotter 1959). Of these three methods the path-integral approach is the most readily applicable to the evaluation of the low temperature thermodynamics of vibrational systems and

w1

Path-integral quantum Monte Carlo studies

7

will be the only methodology considered here. The application of pathintegral methods to the study of the properties of vibrational systems has only occurred quite recently in the history of the development of the pathintegral quantum Monte Carlo (Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991, 1993; McGurn et al. 1989, 1991; Maradudin et al. 1990). Path-integral methods have a long history in terms of their applications to quantum spin problems (Suzuki 1976a, b, 1987; Suzuki et al. 1977; Negele and Orland 1988; Giachetti and Tognetti 1985, 1986; Giachetti et al. 1987, 1988a, b; Barma and Shastry 1978; Marcu 1987; Cullen and Landau 1983; Marcu et al. 1985a; Nagai et al. 1986, 1987; Wiesler 1982; Miyake et al. 1986; Suzuki 1985; Suzuki et al. 1987; Gross et al. 1989; Barnes and Swanson 1988; Miyashita 1990; Reger and Young 1988; Okabe and Kikuchi 1988; Manousakis and Salvador 1989; Joanopoulous and Negele 1989; Takahashi 1988; Nomura 1989; Behre et al. 1990; Morgenstern 1990; Deisz et al. 1990; Schtittler et al. 1987), the Hubbard problem (Gubernatis 1986; Suzuki 1987; Negele and Orland 1988; Blankenbecker et al. 1981; Scalapino and Sugar 1981; Hirsch 1983, 1984, 1985, 1987, 1988; Fye 1986; Sugiyama and Koom 1986; White et al. 1988, 1989; Sorrella et al. 1989; Ogata and Shiba 1988; Moreo et al. 1991; Singh and Tesanovic 1990; Zhang et al. 1991; Loh and Gubernatis 1990) and the study of liquid helium (Gubernatis 1986; Negele and Orland 1988; Pollock 1990; Pollock and Ceperley 1984; Berne 1986; Freeman et al. 1986; Schmidt and Ceperley 1992). We shall now briefly outline for the interested reader a little of the background of the development of these other applications of the path-integral quantum Monte Carlo so that he may appreciate the context in which our application of this methodology to study vibrational systems is found. Following this we shall give an overview of the development, presented in Sections 3 through 4, of the quantum Monte Carlo as applied to vibrational problems. The path-integral methodology began with work by Suzuki (1976a, b) on the numerical evaluation of the thermodynamics of quantum spin systems by means of the application of an identity due to Trotter (1959) to the partition function of these models. Since the original work of Suzuki, a large number of Heisenberg spin systems in one- and two-dimensions have been studied by means of the path-integral quantum Monte Carlo (Giachetti and Tognetti 1985, 1986; Barma and Shastry 1978; Marcu 1987; Cullen and Landau 1983; Marcu and Wiesler 1985; Nagai et al. 1986, 1987; Wiesler 1982; Suzuki 1985; Suzuki et al. 1987; Takahashi 1988; Reger and Young 1988; Okabe and Kikuchi 1988; Joanopoulous and Negele 1989; Schtittler et al. 1987; Nomura 1989). In addition to these spin problems, the problem of the evaluation of the thermodynamics of the Hubbard model (Gubernatis 1986; Suzuki 1987; Doll and Gubernatis 1990; Blankenbecker

8

A.R. McGurn

Ch. 1

et al. 1981; Scalapino and Sugar 1981), which in various limiting forms reduces to Heisenberg spin models (Fradkin 1991), and other more general models of fermion systems (Zhang et al. 1991; Imada and Takahashi 1984; Fye and Scalapino 1990; Hoffman and Pratt 1990) have been treated by path integral Monte Carlo methods. The Hubbard model is of particular recent interest as it is thought to be of importance to the study of high-Tc superconductivity (Fradkin 1991; Anderson 1987; Anderson et al. 1988; Schrieffer et al. 1988; Scalapino et al. 1986; Miyake et al. 1986). Another final set of systems which offer great potential for the application of quantum Monte Carlo methods have been Boson systems and in particular the problems associated with helium II (Pollock 1990; Pollock and Ceperley 1984; Freeman et al. 1986; Schmidt and Ceperley 1992). Suzuki (1976a, b; see also Suzuki et al. 1977) introduced the Trotter form of the quantum Monte Carlo in a study of the Heisenberg antiferromagnet in two-dimensions. Improvements on the original Suzuki formulation were made by Barma and Shastry (1978) and more recently by Suzuki (1985), and since the original paper of Suzuki these quantum Monte Carlo methodologies have been applied to a number of magnetic systems in one- and two-dimensions. The two-dimensional antiferromagnet has been of particular interest of late due to its relationship to families of compounds that exhibit high-Tc superconductivity (Reger and Young 1988). One-dimensional spin systems have also been of considerable interest in regards to the Haldane conjecture (Marcu 1987; Haldane 1983a, b) which proposes Certain relationships between the spin quantum number of the magnetic atoms and the magnetic excitation spectra of these systems. In addition, one-dimensional systems are realized experimentally by a number of compounds (Marcu 1987; de Jongh 1974). In all of these formulations for quantum spin systems, the quantum partition function is mapped onto a classical partition function (Ising or vertex models) defined in a higher dimensional space, and the classical partition function mapped onto is evaluated by standard Metropolis sampling techniques (Metropolis et al. 1953). In the evaluation of the classical problems by Metropolis sampling a number of different configurational changes (i.e., local spin flips or changes in the configurations of clusters of spins) have been proposed to facilitate the generation of most probable configurations with which to compute thermodynamic averages (Marcu 1987; Miyashita 1990). Another problem which has received considerable attention for the application of quantum Monte Carlo techniques is that of fermion systems such as the Hubbard model (Blankenbecker et al. 1981; Scalapino and Sugar 1981; Hirsch 1983, 1984, 1985, 1987, 1988; Fye 1986; White et al. 1988, 1989; Sorella et al. 1989; Ogata and Shiba 1988; Loh and Gubematis 1990),

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Path-integral quantum Monte Carlo studies

9

its associated many fermion systems (Zhang et al. 1991; Loh and Gubernatis 1990), and the Wigner solid (Imada and Takahashi 1984). In these systems the Trotter identity is used to obtain mappings of the many fermion partition function onto a classical partition function which can be evaluated with Metropolis sampling. The application of the Trotter identity to many fermion systems is not as straightforward as in spin systems. Complications arising from sign changes associated with the anti-commuting properties of the fermi fields are a major difficulty as well as the necessity in the study of such systems to look at energy scales much smaller than the band width. A number of devices have been tried in order to circumvent these numerical problems. Bosons do not impose as many difficulties in the formulation of efficient quantum Monte Carlo algorithms as do fermion systems (Pollock 1990; Pollock and Ceperley 1984; Beme 1986; Freeman et al. 1986; Takahashi and Imada 1984a, b, c; Schmidt and Ceperley 1992) and they also represent problems which are more closely related to our studies of the vibration properties of Inert Gas Solids. A problem of considerably interest in this light is the path-integral simulation of liquid helium. Quantum Monte Carlo simulations have been applied to liquid helium in two- and three-dimensions, both above and below the lambda transition, for the determination of a number of properties of these systems. Very good agreement between properties computed by these simulation methodologies and experimental results are found. The above list of problems which have been treated by means of the quantum Monte Carlo is meant only to give an idea of some of the applications and techniques that have been developed for this methodology of computer simulation. Our application of the quantum Monte Carlo is different in many respects to the work listed above. The atoms in our vibrational systems are treated as non-identical particles (Samathiyakanit and Glyde 1973) in their quantum statistics and this simplifies the generation of configurations for the Metropolis sampling. However the vibrational properties of crystals have been studied a great deal by other non simulation methodologies (Born and Huang 1954; Maradudin 1969; Paskiewicz 1987; Xia 1990) and the statistical accuracy of our Monte Carlo methods must be very great to surpass the present knowledge of these systems as derived by these other means. We shall in fact make a comparison of our quantum Monte Carlo results with some results of very recent studies of the path-integral problem using variational methods and self-consistent theories (Feynman 1988; Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991, 1993), and will see that the quantum Monte Carlo results agree quite well with some of these altemative approaches. These comparisons will also point out the limitations of our

10

A.R. McGurn

Ch. 1

quantum Monte Carlo methods due to finite size effects of the computer simulation, and we shall discuss means of extracting from the quantum Monte Carlo corrections for the finite size of the system. We shall begin our presentation below by discussing the application of Metropolis sampling to study the thermodynamics of classical non-linear vibrational systems. This will be useful as a review of the ideas upon which Monte Carlo computations of thermodynamic averages are based. Following this discussion of the classical Monte Carlo we shall turn to the quantum Monte Carlo methodology; first considering the elementary applications of the quantum Monte Carlo techniques to study the problem of a single anharmonic oscillator. This study of the single anharmonic oscillator will then be generalized to treat one-dimensional and three-dimensional crystaline systems of atoms with nearest neighbor Lennard-Jones interactions by quantum Monte Carlo methods. The thermodynamic properties of energy, pressure and specific heat of these crystaline systems will be computed as a function of temperature. We shall conclude by discussing the time-dependent (response function) properties of these systems. In the course of these discussions we shall give indications of some of the techniques which have been developed to obtain computer algorithms of high accuracy and efficiency and shall also make comparisons with the same results of analytic treatments of the static and time-dependent properties of these systems.

2.

Classical Monte Carlo methods: inert gas solids

We shall begin our study of quantum Monte Carlo computations of the thermodynamic properties of crystaline solids by reviewing the small body of work which deals with the determination of thermodynamic properties by classical Monte Carlo methods (Squire et al. 1969; Klein and Hoover 1971; Cowley 1983; Day and Hardy 1985). In these works the vibrational properties of crystals are treated by using classical mechanics. This greatly simplifies the problem of computing thermodynamic averages and in addition offers a natural context in which to dicsuss Metropolis sampling methodology (Negele and Orland 1988; Rubinstein 1981; Binder 1984, 1986; Metropolis et al. 1953; Hammersley and Handscomb 1965). As we shall see, Metropolis sampling is also encountered as a component of the quantum Monte Carlo techniques discussed below. The high temperature properties of crystaline systems (temperatures greater than the Debye temperature and for some properties even temperatures a little below the Debye temperature) are well approximated by classical Monte Carlo techniques and hence compliment the quantum Monte Carlo which is most useful in the study of the low temperature quantum vibrational properties of crystaline systems.

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Path-integral quantum Monte Carlo studies

11

The classical mechanical Hamiltonian of a crystaline system formed of N atoms which interact through nearest neighbor pair potentials is given by H = Ho + H1

(1)

for

(2a)

Ho = ~ 2 m i

H1 : ~ r (~,j)

- ~1),

(2b)

where m is the atomic mass, (i, j) indicates a sum over nearest neighbor pairs of atoms interacting through the pair potential r The classical partition function for the system in eqs (1) and (2) is then obtained from (Landau and Lifschitz 1958)

Z=f

d3pi e -~ 2---d~ i=1

(

)e

(~,m)

,

(3)

j=l

where/3 = 1/kBT. A nice feature of the expression in eq. (3) is that the position and momentum variables are given in terms of real numbers and hence the momentum integrals can be evaluated straightaway leaving only the more complicated position space terms to be treated numerically, i.e.,

Z_[27rm]3N/2fN [ /3

H (dr~)e

-~ ~-~ r (,,m)

(4)

j=l

The energy, specific heat, pressure, etc., are then given by the standard thermodynamic identities applied to In Z (Landau and Lifschitz 1958). We find

E-3NkBT+(~ 2

(5) (e,m)

12

Ch. 1

A.R. M c G u r n

C = 3 Nk B +

kBfl2

2

<(

Z

r

- r',~l)



-(

)') (6)

)

etc., where ( ) indicates an average defined for the general function of position A({~}) by N

f

-B ~

I-[ (d3rj) A({r-~I)e

r

(',")

j=l

(A({~/})> = f

N -~ 2 r 1--[ (d3r/) e <"")

(7)

j=l

We shall now turn to a discussion of Monte Carlo techniques which have been developed for the evaluation of the average defined in eq. (7). In the Monte Carlo evaluation of eq. (7) we begin by realizing that N

(A({r~})) = / H ( d 3 r j ) A ( { ~ } ) P ( { F ~ } ) '

(8)

d=l

where p({~})

e

=

f

".')

N

(9)

-/3 ~

[

I-I (d3rj) e

J

j=l

r

-~'m))

<"'>

is a probability distribution for the configurations, {?}}, of the atoms in position space. We then use this fact to try to choose a finite sampling of K random configurations {~}l, {~}2, {f'i}3,..., {Y/}g which best represent P({r~}), for the given finite value of K, so as to approximate 1

K g=l

(10)

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Path-integral quantum Monte Carlo studies

13

It can be shown that for large but finite K the most accurate approximation to (A({ri})) from eq. (10) is obtained by choosing the {g'i}t random configurations through a sequential (Markovian) generation process based on the probability density P({ri}) (Negele and Orland 1988). In this generation process random test configurations are produced through the use of a sequence of random numbers, and a test procedure, developed from the probability distribution P({ r }), is applied to each of these configurations in order to determine whether or not to use them in computing the sum in eq. (10). We shall briefly describe the mechanics of this process for the generation of the random configurations {~'i}1, {r"i}2,..., { r ' i } K , and follow up this discussion with a theoretical justification of the generation process based on the laws of probability and statistics. In generating the random configurations {r'i}l, {r'i}2, 9 9 {r'i}K, we begin by choosing for the first of the sequence of random configurations, {r'i}l, an arbitrary configuration of the N atoms. A random change is then made to the coordinates of one or more of the atoms in {~'i}l and the resulting configuration is accepted or rejected as {~'i}2 on the basis of a set of selection criterion based on P({ri}). If the configuration randomly generated from {r'i}l is rejected in the process, then {?'i}2 is taken to be equal to {r'i}l, i.e., a duplicate of {~'i}l. This sampling process is repeated again and again so that the configuration {~'i}e+l is obtained from the configuration {~'i}e by making a random change to {~'i}e and applying the selection criterion as indicated above. To understand the specific nature of the selection criterion applied to the randomly generated test configurations, let us consider a system in which we have generated the sequence {r'i} 1, {~'i } 2 , 9 9 9, { r " i } e of random atomic configurations and then discuss the generation of a {~'i}e+l configuration of the system to add to this sequence. In this generation we first perform a random change to coordinates in {~'~}e so as to arrive at a new test configuration {g'~}. In the application of the selection criterion to {~}, the configuration {g'~} is taken to be the new {~'i}e+l atomic configuration in the above sequence with a probability defined as PT({~'i}e --+ {gi}). The probability function PT({~}e --+ {~})is derived from P ( { ~ } ) o f eq. (9) and represents the transition probability for the system to go from {r~}e to {~i} during a Monte Carlo sampling. The form of the probability function PT is chosen so as to obtain the most accurate approximation for (A({~'i})) from eqs (9) and (10) that is possible for finite K. With these considerations in mind it is found that, given two configurations of N atoms {g} and {~7}, the probability function PT({ff} --+ {~7}) must obey (Negele and Orland 1988) .....x

(11)

14

Ch. 1

A.R. McGurn

where Hl((u}) and HI({-~}) are the potential energy (eq. (2b)) evaluated for the {g} and {~7} atomic configurations and e -~H~({ u }) = Z P ( { u }). The statement in eq. (11) is just that of microreversibility or detailed balance in the system (Negele and Orland 1988; Landau and Lifschitz 1958). It assures that provided that our generation process is ergodic (i.e., it allows for the possibility of all states of the system to be reached), the states selected from the randomly generated test states by the above criterion will form a sequence of configurations { r ' i } l , {r"i}2, 9 9 9 { r ' } M which in the limit M ~ c~ are distributed by the probability distribution of eq. (9). We shall give a short proof of this limiting behavior below and then discuss the specific solution of eq. (11) known as the Metropolis sampling (Negele and Orland 1988; Metropolis et al. 1953). Let us begin by showing (Negele and Orland 1988) that the probability distribution in eq. (9) remains stationary under transformation by PT, i.e., given the distribution of states P({g}) then ....x

__x

N

P({~7}) = f 1-[ (d3u')P({g}) PT({g}

-~

{~7}).

(12)

i=l

This fact follows by applying eqs (9) and (11) to rewrite eq. (12) as N

P({~7}) = P({~7}) f 1-[ (d3u')Pr( {~7} -~ {g})'

(13)

i=1

where the integral in eq. (13) is equal to one by the definition of Pr as a probability. Hence the distribution of states in equilibrium is not affected by the application of the transformation PT. As a second important property of Pr, it remains only to be shown that the application of ~ to an arbitrary distribution Q({~7}) ~: P({~}) will cause Q({g}) to uniformly converge to P({ff}) of eq. (9). This last fact can be shown by considering the behavior of the norm (Negele and Orland 1988) N

(14)

I, = f I I d3u, IQ({~}) - P((~})[ i=1

under an application of PT to Q and P. Applying PT to Q and P we arrive at the new distribution R given by N

= f l-I i=l

+

(15)

Path-integral quantum Monte Carlo studies

w

15

so that the new norm

N u~ IR({~)) - P({ ~ )) I

i=1 N = f H (d3ui) i=1 N / H (d3vi) (Q({v})- P({v}))PT({V} -+ {if}) i=l N N

(16)

S II (d'u,)f II (d3v,) i=1 i=1

x IQ({~}) - P((~})IPT({~} -+ {~})

= I1,

where we use eq. (11) and the normalization properties of the probability distribution Pr. Since I2 < I1, we see that a successive application of Pr to an initial distribution of states will generate the distribution in eq. (9). Hence the sequence generated from any initial state of our system through the application of Pr, as discussed above, will represent a set of configurations whose distribution is given by P({ri }). One solution of eq. (11) is given by the so-called Metropolis sampling (Metropolis et al. 1953). In Metropolis sampling PT({g} ~ {~7}) = F({g}, {~7})min { 1,

e-~3H((g}) } e_~n({a}) '

(17)

where F({g}, {~7}) = H(IzT- ~71z - 52)

(18)

with H(x)=

1, 0,

x~l, x>l,

(19)

and 6 is a constant which is generally chosen to facilitate the numerical convergence of the averages computed from eq. (7). (In practice ~ is chosen

16

A.R. McGurn

Ch. 1

so that in Monte Carlo simulation programs 40% to 60% of the newly generated Monte Carlo test configurations are accepted as new configurations of the simulated systems, and as a second important point it can be shown that the averages in eq. (7) for N -+ c~ are independent of 6.) One can see that eqs (17) and (18) solve eq. (11) by first considering Pr({~} --+ {~7}) in which e -~H(('7}) < e -~H(('~}), then Pr({z7} -+ {~7}) - e-~H({~}) , e-~H({,~})

(20a)

PT({F} --+ {if}) -- 1

(20b)

for [ ~ - ,/712 ~ ~2 and eq. (11) is solved. On the other hand, if e -3/4(('7}) /> e-~/(('~}) then PT((~} -9 {~7}) = 1,

e-~H({,7})

(21a)

(21b)

for L~ --+ ~12 ~ 52 and eq. (11) is also solved. Hence eq. (17) represents a realization of the classical Monte Carlo methodology and in fact it is the most commonly used realization of Monte Carlo sampling techniques in the study of gases, liquids and solids (Binder 1984, 1986; Wood 1968; Klein and Venables 1976; Klein 1984). The classical Monte Carlo based on Metropolis sampling has been used extensively to study the properties of liquids and even more recently in studies of solid-liquid-gas interfaces (Levesque et al. 1984). In this review, however, we shall only be concerned with the uses of the above Monte Carlo methodologies in the study of the vibrational properties of solids. These studies have concentrated on simulating the thermodynamic properties of inert gas systems modeled as nearest neighbor solids with Lennard-Jones pair potential interactions (Squire et al. 1969; Klein and Hoover 1971; Cowley 1983; Day and Hardy 1985). The first such Lennard-Jones simulation was developed by Squire et al. (1969) in a study of the temperature dependence of the isothermal elastic constants of solid argon (Debye temperature between 75 K to 85 K). The simulation was used to study systems for temperatures between 40 K and 80 K, and a good agreement with the experimental variation of the isothermal elastic constants was obtained over this range of temperatures.

w

Path-integral quantum Monte Carlo studies

17

The motivation for this original study of crystaline argon was the great success of classical Monte Carlo methods in computing the properties of the liquid state of argon and the algorithm used by Squire et al. (1969) was very closely structured on these liquid state algorithms. In a related work Klein and Hoover (1971) later extended the study of Squire et al. to consider the general thermodynamic properties of solid xenon (Debye temperature between 60 K to 65 K), modeled by a nearest neighbor Lennard-Jones system, as functions of temperature. The algorithm used by Klein and Hoover was very similar to that of Squires et al. and full advantage was taken of the improvement in the speed and accuracy of computers to surpass their initial efforts in this field. Specifically, Klein and Hoover calculated the zero-pressure lattice constant, heat capacity, bulk modules and Grtieneisen parameter between 100 K and 140 K, making comparisons with these quantities as computed by a number of different analytical methods and as measured experimentally. The Monte Carlo results allowed Klein and Hoover to judge the validity of the various approximation methods in the high temperature region and a comparison of Monte Carlo results with experimental data also revealed the failure of the Lennard-Jones pair potential as a realistic model of the interatomic force in solid xenon. More recent studies are those of Cowley (1983) and Day and Hardy (1985) for the thermodynamic properties of Lennard-Jones solids with nearest neighbor interactions. These works which take advantage of the further recent increased capacity of electronic computers occur many years after the work of Klein and Hoover and yield results which are believed to be within a fraction of a percent of the value of the correct solutions. An interesting aspect of the works of Cowley and Day and Hardy is that a formulation to adjust for finite size effects of the computer modeled system was used. These adjustments were based on the fact that in the quasiharmonic approximation the thermal energy and specific heat are proportional to N - 1 rather than to the number of atoms, N, and it was assumed that these proportionalities held in the results from the computer simulation (Cowley 1983). In addition, Day and Hardy tried an alternative to the Metropolis sampling method. The sampling method used by Day and Hardy, known as the Gaussian method, was found to be no more efficient than the Metropolis method and we will not discuss Gaussian sampling here. Neither of these studies attempt comparison with the experimental properties of an inert gas system but are rather proposed as accurate numerical simulations of systems which can be used to test analytical approximation methodologies. The most recent classical Monte Carlo work on nearest neighbor Lennard-Jones systems is our work on the spectral moments of response functions and we shall discuss these near the end of w4 (Cuccoli et al. 1993a). In table 1, we present results from Cowley (1983) for the zero pressure lattice constant R; the specific heat at constant volume, Cv; the specific heat

18

Ch. 1

A.R. McGurn

Table 1 Thermodynamic properties calculated by the classical Monte Carlo method for the nearestneighbor Lennard-Jones solid.

kBT/C

R

PV/NkBT

Cv/Nk B

Cp/Nk B

0.125

1,13208

0.225

1 14087

0.0

+ 0.02

2.87 + 0.03

3.09 + 1.1

2.98 + 0.03

-0.01 + 0.02

2.86 + 0.03

3.35 + 0.03

0.3

1 14839

0.02 -i- 0.02

3.01 + 0.03

2.82 + 0.03

3.53 + 0.04

2.97 + 0.03

0.375

1 15692

0.45

1 16680

0.01 + 0.02

2.73 -I- 0.02

3.73 + 0.05

2.90 + 0.03

0.03 + 0.03

2.69 + 0.03

3.97 + 0.06

0.5

1 17454

2.88 + 0.03

0.01 + 0.02

2.66 + 0.02

4.28 + 0.06

2.86 + 0.03

at constant pressure, Cp; and the Gruneisen parameter, 7. These results are computed for a system of 108-atoms with periodic boundary conditions and nearest neighbor interactions described by the Lennard-Jones potential r

= 4e

[ rY __ ( ~ ) 6 ] ( ~ _ ) 12 ff

.

The values in table 1, which have been corrected for finite size effects, are believed to be within 0.5% of the exact results for the N --+ c~ system of atoms. The results in table 1 being those of the classical system, fail to adequately represent the experimental properties of inert gas crystals. In particular the temperatures shown are all much less than the Debye temperature so that the specific heat for example does not approximate the correct functional dependence on temperature of that measured in inert gas solids. However, the results shown are of theoretical interest in themselves as representing the solution of the Lennard-Jones crystal in the context of classical thermodynamics, and until very recently no simulation methods existed which could yield similarly accurate approximations to the quantum mechanic LennardJones model. As we shall see in the next sections, many of the techniques encountered in the Monte Carlo evaluation of the classical mechanical problem are of great importance in the quantum Monte Carlo evaluation of the properties of the quantum mechanical Lennard-Jones system and that is why we have considered the classical Monte Carlo studies here. In addition, it shall be interesting to compare the properties of the quantum mechanical and classical mechanical systems as they are very accurately evaluated by both simulation methodologies. We now turn to a discussion of quantum Monte Carlo techniques.

w3 3.

Path-integral quantum Monte Carlo studies

19

Quantum Monte Carlo methods

In this section we will consider the determination of the static thermodynamic properties of vibrational systems by means of quantum Monte Carlo (QMC) methods. We will begin by discussing the application of the QMC to the problem of determining the average energy and specific heat of a simple one-dimensional single-particle anharmonic oscillator (de Raedt and de Raedt 1983; Takahashi and Imada 1984a, b). This system will be useful both as an illustration of the formulation of a QMC algorithm and as a basis for a discussion of some of the numerical techniques which have recently been developed to improve the rate of convergence of such algorithms (de Raedt and de Raedt 1983; Takahashi and Imada 1984b; Suzuki 1976a, b, 1977) to the thermodynamic averages being computed. After these preliminary discussions we will proceed to a treatment of QMC methods as applied to the determination of the average thermodynamic vibrational properties of a one-dimensional chain (Cuccoli et al. 1992b, 1993a; McGurn et al. 1989; Maradudin et al. 1990) and of a fully three-dimensional (Lui et al. 1993; Cuccoli et al. 1992a; Maradudin et al. 1990) solid of coupled atoms. In our latter treatment of one- and three-dimensional vibrational systems we shall choose model systems for study whose properties have been subject to extensive investigations by a variety of other theoretical techniques than the QMC (Klein and Venables 1976; Klein 1984). Specifically, we shall assume that the atoms in our one- and three-dimensional systems interact with one another by short range interactions based on the standard (12-6) Lennard-Jones potential (Klein and Venables 1976; Klein 1984). The choice of the Lennard-Jones potential is based on the fact that it is commonly used to model and/or fit the vibrational properties of inert gas solids. These are some of the most extensively experimentally studied vibrational systems available (Born and Huang 1954; Maradudin 1969; Paskiewicz 1987; Klein and Venables 1976). In addition, all available theoretical results on one- and three-dimensional vibrational systems can be easily evaluated for the Lennard-Jones potential (Cuccoli et al. 1992a, b, 1993a; Maradudin et al. 1990; McGurn et al. 1989; Gursey 1950). For the one-dimensional chain, systems with both nearest neighbor and next nearest neighbor atomic interactions will be studied, while the threedimensional system will be assumed to be simple face centered cubic (fcc) with only nearest neighbor atomic interactions. We will use the QMC to determine the average energy, pressure, volume and specific heat of these fully anharmonic many-body systems as functions of the temperature and comparisons with other available theoretical results will be made.

20

A.R. McGurn

Ch. 1

3.1. Single particle model In this subsection we will use single particle models to illustrate the fundamental ideas upon which the QMC is based and then give a discussion of various techniques which have been used to improve the numerical convergence of computer algorithms designed to generate the thermodynamic averages of quantum mechanical systems. Once understood, the generalization of these ideas and techniques to higher dimensional many-body systems is straightforward. 3. I. 1. Fundamental ideas

Let us begin our study of QMC methods by considering the determination by these techniques of the average thermodynamic properties of a single quantum mechanical particle moving in a one-dimensional anharmonic potential (de Raedt and de Raedt 1983). We will take the Hamiltonian of our system to be given by H = Ho + H1,

(22)

where h2 Ho =

d2

2m dx 2

(23a)

and H1 = v(x)

(23b)

with v(x) being some arbitrary anharmonic potential. We will assume that v(x) is well behaved enough that the partition function for the Hamiltonian of eq. (22) exists and has well defined, non-pathological, properties. The thermodynamic problem posed by eq. (22) has little significance to the theory of solids and suffers from other difficulties associated with it not being a true many-body problem (Landau and Lifschitz 1958), but it is very well suited to illustrate the application of the QMC. The partition function of the system, defined by eqs (22) through (23) above, is given by Z = Tr e -~(H~

(24)

w3

Path-integral quantum Monte Carlo studies

21

where Tr indicates a trace involving all the eigenstates of H. In order to cast the partition function in eq. (24) into a form which we will find to be accessible to evaluation by Monte Carlo methods, it is useful to apply to eq. (24) an identity developed by Suzuki (1976a, b) from the work of Trotter (1959). This identity states that for (A1, A2,..., An} a set of noncommuting operators

If we apply eq. (25) to eq. (24), taking H0 and H1 as a set of two noncommuting operators, we find M

Z-

lim

M--+cx3

(26) = Tr M-,~lim [(e-~H~

-~H'/M) (e-~H~

]

M

where we have interchanged the order of the trace and limit in eq. (26). The form of the partition function written on the far right hand side of eq. (26) is very useful as it allows us to rewrite the partition function for H as a path integral. This is done by introducing, between each successive product of exponentials in the term on the far right hand side of eq. (26), a position space representation of the identity operator. Hence between the pair formed from the ith and (i + 1)th exponentials in eq. (26) we insert

/

I=,>d=,(=,l

(27)

= 1,

for all pairs i - 1 , 2 , . . . , 2M to obtain from eq. (26)

Z

lira f

M---~cx:~

dxldx2..,

dX2M(X2Mle-~H~

[gEl) (28)

• (X2M_l le-~',mlz2M).

A.R. McGurn

22

Ch. 1

The matrix elements whose products form the integrand in eq. (28) can be easily evaluated so that the resulting expression for Z takes the form of a path integral written completely in terms of commuting position space variables. To evaluate the matrix elements involving H1 = v(x), we use the fact that the potential energy, v(x), is diagonal in the position space representation to write

(x~le -aH'/M Ix,+l ) - 6(xi

-

Xi+l )e -ov(z')/M.

(29)

The evaluation of matrix elements of the form (xi [e-~H~ IXi+l ) is a little more difficult as these elements are off diagonal in the position space representation. To evaluate (xiLe-OH~ we insert momentum space representations of the identity operator to write

(xile-flH~

f dkl f

dk2

(30)

x (xilkl)(kl[e-BH~ Using the fact that (klx) = 1 / v ~ e mentum space representation with (kl

le-- H~

ikx and that Ho is diagonal in the mo-

= ~(kl - k2)e -~Ek/M,

(31)

where Ek = h2k2/(2m), eq. (30) reduces to

_rhzk2 (xi le -~n~

Ixi+l) = f dk 27r e -

-

2m

e~k(Zi+l 9 _~,) (32) \

_

m

exp ( -

2rh 2

(Xi+I -- Xi)2)

/

with r = ~/M. From the matrix elements in (29) and (32) we can now write the partition function of our system explicitely in the form of a path integral. Substituting eqs (29) and (32) into the expression for Z, given in eq. (28), we have Z =

lim

M-~oo

] M/2 f dxl dx3 d x 5 . . , 27rrh2 m ... m

dx2M-1 (33)

M

x exp ( - 2rh2 Z

i=l

(z2i+l - z2i-1):2

-

i=I

w

Path-integral quantum Monte Carlo studies

23

with X2M+I -- X l , or relabeling the dummy variables in eq. (33) gives

] M/2 Z =

lim

m

M--+oo

27rTh2

x exp

(

f dxl d x 2 (Xi+l

2rh2 i=1

dx3

9 9 9dXM

"

xi) 2 - ,

v x,,,

)

(34)

i=1

where XM+I = Xl. The expression for the partition function in eq. (34) is now composed solely of commuting position space variables. The average energy, E = -~-~ In Z obtained from our path-integral form for the partition function, is r"

E =

lim /

M

M--,~ [

2/3

(35)

mM M

1 M

i=l

i=1

2 f l 2 h2

i1

where ( ) indicates a weighted average, defined for a general function A({xi}) as

(A) =

lim

f (i=~l d-i)A({xi})Q({xi}) ,

(36)

where

Q({x,})

exp

m 2Th 2~ i=1

(Xi+I

(37)

Xi) 2 i=1

Upon taking the temperature derivative of E in eq. (35) we then find the following expression for the specific heat C:

24

Ch. 1

A.R. M c G u r n

,am{ +/3--~

2

i=1 (38)

2/32h2 y~ (Zi+l- Zi)2+ "~ Z'O(Zi)

i=1

i=1

2/3h2 y~ (Xi+l- xi)2-+- ~ Z v(xi) i=1 i=1

.

The averaged expressions appearing in eqs (35) and (38) for the quantum mechanical energy and specific heat are of the general from given by eq. (36) with the weight function of eq. (37). Equations (36) and (37) are very similar to those encountered in the study of the thermodynamic averages computed in classical mechanical theories of polymer conformations (Flory 1989). Specifically, our averages are those of M --+ c~ classical particles with positions x~ along a one-dimensional axis and a weight function, eq. (37), of the form e - ~ n were /~H=

M

m 2rh2 ~

i=1

M (zi+l - z i ) 2 + r y ~ v(a:i).

i=1

(39)

These averages are directly accessible, for large but finite M, to computation by means of the standard classical Monte Carlo methods discussed above in w2. In practice, one finds that good approximations to the M --+ c~ averages can be arrived at through the Monte Carlo study of chains with M<50. At this point it is important for the reader to realize that the formulation of the QMC that we have been discussing above is not necessarily the only possible one for our quantum system. Other equally good mappings of the quantum mechanical partition function, composed of non-commuting operators, onto corresponding partition functions of classical mechanical systems are possible. These alternative formulations can be developed by making different choices of the H0 and H1, than those given in eq. (23), with which to represent H and by using different representations of the identity operator than that in eq. (27) to insert between the exponentials in eq. (26). Each of these alternative formulations, though, is found to yield the same results for the thermodynamic averages of the original quantum mechanical

w

Path-integral quantum Monte Carlo studies

25

system in the (M -+ ~ ) thermodynamic limit. The possibility of developing a number of alternative QMC formulations, through the use of the two above mentioned devices, becomes more evident in higher dimensional many-body systems where it is also found in the mapping of partition functions for quantum spin and fermion systems onto the partition functions of various classical Ising and vertex models. For finite N (number of particles) and M (Trotter slices) computer computations, a good choice of the QMC mapping to a corresponding classical problem, is very important as it will often save computer time by increasing the rate of convergence of the Monte Carlo averages to those of the exact values of the quantum system being studied. Once the choice of the QMC mapping to be used is made there are still other techniques that can be applied to facilitate numerical convergence. We will now turn to a discussion of these techniques.

3.1.2. Improve numerical convergence In this subsection we shall outline some additional techniques which improve the convergence of computer algorithms to yield accurate estimates of the thermodynamic properties of quantum mechanical systems. We shall not, however, discuss techniques used to improve the sampling methods for the classical Monte Carlo performed on the mapped-onto-classical partition function but shall concentrate, specifically, on methods used to quicken the convergence with M to the M --+ c~ limit of eqs (34) through (38). In particular, all of the methods which we shall discuss arise from the study of higher order corrections in filM to the Trotter approximation M

e x p [ - / 3 ( H o + H1)] ~ [exp ( - ~

Ho)exp ( - ~

H1)I

(40)

For finite M, Suzuki (1976a, b, 1977) has shown that to terms of order (l/M) n e x p [ - / 3 ( H 0 + HI)] -~ PMn -- [exp (-- 13HolM)exp(- flHIIM) x exp (~2C2/M2)...exp

((-~)nCn/Mn)]M,

(41)

where C2--~

1[H1 , H0] ,

(42a)

c3-

1102 , H1 + 2H2]

(42b)

26

A.R.

Ch. 1

McGurn

and for general n =

--~.

-~-~

( e - "V" - l c n - , . . . e - "X2 C2 e - "XB e - "XA e "x( A + B ) )

] ,~=0

(42c)

For each M the exact partition function, Z, can then be approximated by ZM~

= Tr P M , ~ ,

(43)

lim ZM,~ M---~oo

(44)

such that Z =

for all n > 1. Suzuki has demonstrated that, for fixed M, Z M n + I is a better approximation for Z than is Z M n so that in the composition of QMC algorithms a certain optimization of computational efforts is achieved by choosing n larger for a given value of M. Continuing with these efforts, de Raedt and de Raedt (1983) have looked for improvements which can be made upon the Trotter approximation of eq. (40) and accomplished by the addition of higher order terms in 1/M to the fight hand side of eq. (40). They found it advantageous in these approaches to introduce higher order corrections in such a way as to maintain the hermitian form of the operator taken to approximate exp [-13(H0 + H1)]. (We note that PMn of Suzuki's expansion is not necessarily Hermitian in form.) The main advantage of retaining hermiticity is that it is often easier to evaluate the matrix elements, resulting from the insertion of representations of the identity operator between the product exponentials in the Trotter identity, during the transformation of the quantum partition function to the form of a path-integral. Specifically, de Raedt and de Raedt have studied the approximation exp[-/~(Ho + H,)] [e -~

xp

(-/3 H~ M 2

x exp

M

H1)exp([~-~Mfl] 2

(--fl H1)exp( -~ H~ M M

2

M

2

'

(45)

w

Path-integral quantum Monte Carlo studies I

-0.8

27

-!

I

-0.9 9 E (m) m o

-I.01 oo

Elm41

I

I

40

20

iI

m

Fig. 1. The average energy of the system H = p2/2- 2z 2 + z 4/2 at/3 = 5 versus the Trotter slice number m = M for: a) E~ ) given by the approximation of eq. (40). b) E(m 4) given by the approximation in eq. (45). The figure is taken from de Raedt and de Raedt (1983). where C3 is defined in eq. (42b). The approximation in eq. (45) is correct to order ( 1 / M ) 5 and has been successfully tested by de Raedt and de Raedt on single spin, single harmonic and single anharmonic oscillator problems. Takahashi and Imada (1984a, b) have also used eq. (45) as the basis for a formulation of the QMC for quantum many-body systems and we shall discuss their work in a later section dealing with many-body systems. For the single particle Hamiltonian given by H = p 2 / 2 - 2 x 2 + x4/2, de Raedt and de Raedt (1983) have compared the QMC result for the energy at /3 = 5, as a function of M, computed using the simple Trotter approximation of eq. (40) and the approximation in eq. (45). A plot of the comparison of the energy computed by using these two approximations is shown in fig. 1. We emphasize in presenting these results that, though both eqs (40) and (45) agree in the M --+ ~ limit, the convergence to the M --+ oc result is much quicker for the Monte Carlo based on eq. (45) than for that based on eq. (40). The rapidity of the convergence with M of the thermodynamic averages, evaluated by the above path-integral techniques, to their M --+ c~ limits is important due to aspects of Monte Carlo sampling other than the limitation

A.R. McGurn

28

Ch. 1

on computer time. If we look at the expression in eq. (35) for the average energy, we see that as M --+ 0o the average energy should be independent of M. This means that the term

I

mM M 2/32h2 Z

( z , + , - z,) 2

i=1

i=1

becomes large with increasing M. Hence it must be determined with a high degree of statistical accuracy in order to yield accurate values of the thermodynamic energy upon its addition to -M/2~ in eq. (35). A similar problem is seen in eq. (38) for the specific heat. Here, again, the specific heat is independent of M but is represented as a sum of average quantities which must each be of order M or M 2. A rapid convergence with increasing M to the M --+ c~ limit is necessary to facilitate the development of effective QMC computer algorithms. In addition to this last point, we note that there is an inherent complication with numerical accuracy in the computation of the specific heat from eq. (38). The specific heat computed from this expression is essentially the statistical variance (second moment) of the energy and hence all of the numerical inaccuracies encountered in the computation of the average energy as a function of M are amplified in the use of eq. (38) to compute the specific heat. Recently Maradudin et al. (1990), in studies of three dimensional systems, have gotten around this last mentioned source of error by using a method which involves fitting data for the average energy versus temperature by a polynomial form and obtaining the specific heat from this form by differentiation with respect to the temperature. This procedure is only reliable when one can choose the polynomial used to fit the average energy based on broad theoretical considerations such as a knowledge of the general form of the high temperature or low temperature expansions of the average energy in terms of the temperature. We shall consider this problem in detail when we discuss below the vibrational properties of the face centered cubic Lennard-Jones system. 3.2. One-dimensional chains We next turn to a consideration of the thermodynamic properties of a one-dimensional chain of quantum mechanical atoms (McGum et al. 1989). This system is one to which the QMC can be easily applied and which begins to display many of the interesting properties, related to atomic correlations, which are distinguishing features of the physics of many-body systems. One-dimensional quantum systems still, though, represent rather

w

Path-integral quantum Monte Carlo studies

29

primitive problems in many-body physics as, in the absence of infinite ranged interactions, such models can never exhibit long range order (Landau and Lifschitz 1985). This last point is important to us as the atomic chains we shall consider in this section are never really ordered in the sense that at T - 0 a correlation in the positions of two atoms on the chain exists for an infinite atomic separations (i.e., due to the presence of disordering ,quantum fluctuations, the atoms are not positioned on a one-dimensional lattice). This is not a great difficulty in studying phonons in one-dimensional chains as at low temperatures a large amount of short range order exists in these systems and this short range order is capable of supporting phonon-like excitations. A quite similar situation occurs in the study of magnons in one-dimension and in the short range order of the paramagnetic phase in two- and threedimensional magnetic systems which occurs immediately above the critical point (Freeman et al. 1990; Mori 1965a, b). Again in our presentation below, we shall first discuss the formulation of the problem of the one-dimensional chain in the QMC and then discuss techniques which aid in the computer generation of thermodynamic averages.

3.2.1. Formulation The specific system that we shall now consider is that of a quantum chain of atoms in which the atoms are constrained to move only along the axis of the chain and to interact with one another through nearest neighbor only Lennard-Jones pair potentials of the form qS(r) = 4~ [(ra-) 1 2 - ( ~ ) 6 ] .

(46)

In eq. (46) r is the separation along the chain of the pair of interacting particles, and e and cr are parameters with dimensions of energy and length, respectively. We shall consider the atoms in our model to be distinguishable particles and to retain their sequential ordering along the chain. Modification of our QMC treatment of the one-dimensional chain of atoms to systems of identical bosons or fermions will be discussed at the end of this section (Takahashi and Imada 1984a, b, c). The quantum mechanical Hamiltonian of the above described system is H =

h2 ~ i~2 cx~ Z..,, + ~ r 2m i=l O2ri i=l

- ri-1 [)

(47)

where m is the mass of an atom in the system and {ri} are the positions of the sequentially ordered atoms on the chain. (In this notation r~_l and

A.R. McGurn

30

Ch. 1

ri+l are the nearest neighbor positions to ri.) To make the system described in eq. (47) treatable by computer methods it is convenient to approximate the infinite atomic system of eq. (47) by treating it as a system of N < c~ panicles confined to a one-dimensional chain segment of length L which is subject, at its ends, to periodic boundary conditions. This last restriction arises due to the finite storage space available, for particle coordinates, in the computer memory. The Hamiltonian of the N particle system is then H=

h2~0

2

N

+ y ~ q~([ri2m i=l Or/2 i=l

ri+ 1 I),

(48)

where 0 < r l < r2 < ... < r N < L and r N + 1 = r l + L. The partition function for the system in eq. (48) is Z = Tr e -t~H =

lim Tr [e-~H~ M---+e~

-~H'/M] M,

(49)

where on the far right hand side of eq. (49) we have used the Trotter identity, eq. (25), with

no=

(50a)

,=,

N H, = ~ r i=1

r,+, 1)-

(50b)

As in the evaluation of eq. (26), it is now convenient to introduce, between pairs of successive exponentials in the product on the far right hand side of eq. (49), position space representations of the identity operator. For the one-dimensional chain, these position space representations are of the form

N

(51) i=1 where f' = ( r l , r 2 , . . . , r N) for 0 < 1"1,1"2... r N < L is an N-dimensional vector composed from the N atomic positions on the segment of length L. Once this has been accomplished the partition function will be seen, as in the case of eq. (28), to be formed of a product of position space matrix

w

Path-integral quantum Monte Carlo studies

31

elements which can easily be rewritten into the form of a path integral for Monte Carlo (Metropolis) evaluation. Using eq. (51) in eq. (49) we then find a representation of Z in terms of the integral (Takahashi and Imada 1984a, b, c) 2M

Z

lim

__.

M-+cx~

f II i=1

(52)

x (#'(1)ie-eZCo/U 1#'(2)) (~(2)le-em/u 1#'(3)) 999 x (~'(2M

-

1)ie-eH~

1~'(2M))

(e'(2M)le-nmlU I~(1))

where ~'(i) = (rl (i), r2(i),..., rN(i)). The matrix elements of the form

(#'(i)le -~Ho/M le'(j)) can be evaluated to find:

(#'(i)le -~Ho/M = ~

l~'(j))

(g(i)l~; } exp (

M/3 2m hE k2)(~;lr'(J)}

(53) = LN ~

.

k

exp i fr (~'(i) - ~'(j)) - ~ k2 2Mm '

-' 1--J--ei~"#'(i) with ~e= T 2~r( n l , n2,.. -, nN) and the {ni} each where (~'(i)lk)= LN/2 range over the entire set of integers. The sum over k on the far right hand side of eq. (53) can be obtained by using the identity 1 cr 27r ~ n--

einOexp

(en2) --~

1 m 2~r Z

--OO

n----

( exp

1 ) ~ee (0 + 27rn)2

(54)

OO

(see Appendix I for a proof of this identity) to find

(#'(i)le -e/co/M Mm =

l~'(j)) ~ .=_oo

exp

Mm

- ~ 2/~h2 (~'(i) - ~'(j) - ~L)2

),

(55)

A.R. McGurn

32

Ch. 1

where ~ = ( h i , n 2 . . . . . nN) and each of the ni range over the entire set of positive and negative integers and zero. The remaining matrix elements which are of the form

(~(i)le-~H'lMI~'0")) (56)

= 5(r-'(i)- ~'(j))exp

- ~ E r

(i) - re-l(i)l)

~=1

are diagonal in the position space representation. From eqs (52), (55) and (56) we find that M

Z = /II

dNr(j) W (~'(j), ~'(j + 1)),

(57)

j=l

where ~'(M + 1) = ~'(1) W (~'(j), ~'(j + 1)) = f (~'(j), ~'(j + 1)) exp - ~ v(~'tj))

(58)

with

f(e, e')= (ele- o/Mle'),

(59a)

N

v(f') = ~

r

r,+l l)

(59b)

i=1

and where the N components of ~' satisfy 0 ~< r l < r 2 < . . - < r N ~ L. Using the expression in eq. (55) for f(~', ~") in eqs (58) and (57), we establish a mapping of the partition function for the one-dimensional quantum mechanical system of atoms onto the partition function of a two-dimensional system of classical (commuting operators) particles. As we shall now see, thermodynamic averages of the quantum system can be obtained in terms of averages computed in the corresponding two-dimensional system described by eqs (57) through (59). An approximation to f(~', ~") in eq. (59) can now be made which greatly facilitates the computer generation of the thermodynamic averages from

w3

Path-integral quantum Monte Carlo studies

33

eq. (57). From eq. (57) and the matrix elements of eq. (59) involving the potential energy, the integral defining the partition function is seen to be dominated by configurations of 0 ~ rl(j) < r2(j) < ... < rN(j) ~ L in which the ri+l(j)- riO'), differences are roughly equal to one another. As a result of this dependence of the integrand on the {~'(g)}, configurations of {~'(g)} in which Iri(j + 1 ) - ri(j)l -~ L are, also, found to give small contributions to the partition function integral. This can be seen from the fact that a large value of ri(j + 1 ) - ri(j), due to the 0 ~ rl(g) < r2(g). 99< rN(g ) ~ L inequality, would lead to a significant rearrangement (clumping or unclumping) of the particles on the chain in either or both of the ~'(j) or ~'(j + 1) and the resulting contribution to the integral in eq. (57) from these configurations would be significantly decreased due to the potential energy matrix elements in eq. (59). In the light of these facts we see that, as used in eq. (57), we can approximate eq. (55) by

f (~'(i), g(j)) "~

[

Mm

exp

2rrfl h2

(

Mm 2/3h2 (~'(i)- ~'(j))

)

2

.

(60)

In eq. (59) the approximation of eq. (60) is found to, in fact, lead to errors in the thermodynamic averages which are small of order exp [-(Mm/2~h2)L2]. We shall now see how thermodynamic averages of the quantum system can be obtained in terms of averages computed in the two-dimensional system described by eq. (57) with f(~', ~") approximated as in eq. (60). The energy and specific heat of the system of atoms are obtained by taking the appropriate derivatives of the free energy from the partition function in eq. (57), using the approximation in eq. (60) for f(~', ~"). The quantum mechanical energy, E = - ~ In Z, is given by

E/Ne

--

lim (2-~

-+-

U---+ cx~

1

MN

(Ul)

)

(61)

'

where r = ~e/M and

Ul --

1 (~l[y(j)_y(j+l)]2

+ [ y ( M ) - y(1)] 2)

2T2 \ j = l

(62) 1 M

+-Zv(y(o), i=1

A.R. McGurn

34

Ch. 1

where ri(j) = c~oyi(j) and c~2 = h2/me. The quantum-mechanical specific heat C/N, where C = aE/OT, is then obtained from eq. (61) as

C = lim MT2 ( 1 1

Nk B

2T 2

M--+co

M N ((u2)

-- (u2) -'l-(Ul) 2)

)

,

(63)

where

1 (~l[yfj)_y(j+l)]2

+ [y(M) - y(1)]:) .

(64)

u =7 \j=l In both eqs (61) and (63), we have used a notation in which ( ) denotes the average defined, for a general variable A({r(j))), by

fi

l~dr~(j)

j=l

i=1

(A) =

AW({rdj))) (65)

j=l ~=1 where 1~ ((~'(j)}) is defined in eq. (58) with f(~(i), ~(j)) approximated as in eq. (60). The integration variables in eq. (65) are limited by the inequality 0 < rl(j) < r2(j) < . . ' < rN(j) <~L. Another quantity of thermodynamic interest in many-body systems which can be computed in the QMC is the pressure-volume-temperature, equation of state, relation. The pressure of the one-dimensional quantum mechanical chain of length L is obtained from standard thermodynamic relations as P =

In Z,

(66)

3 0L where Z is the quantum partition function expressed either in terms of H or as a path integral. An expression for the pressure defined in eq. (66) can be straightforwardly calculated from eq. (57) by making the change of integrai ) where the new integration variables, tion variables defined by ri(j) = L(p (2

w3

Path-integral quantum Monte Carlo studies

35

{p~)}, are dimensionless and satisfy 0 ~< p?) < p2~) < . . . < p~) ~< 1. Differentiating the explicit L dependence of In Z in eq. (66) we then find

P=

2N

M

L

2~

N 2/3T E E [ Y i ( J + I ) - - Y ' ( J ) ] 2 ~=1 j=l

11

N( j=l

i•k=l

0vl,

/

(67)

~[Yi(j)--Yk(j)]

where ~'(M + 1) = ~'(1). The numerical evaluation of the energy, specific heat and pressure obtained above in eqs (61), (63) and (67) can now proceed using standard techniques of classical Monte Carlo sampling based on the partition function in eq. (57). For finite values of N and M the system considered in eq. (57) represents a well defined classical problem presented on an N • M lattice. By studying the behavior of the Monte Carlo results for varying values of N and M, the behavior of the N --+ c~, M --+ c~ system of interest can be inferred. The Monte Carlo computer routine used in the evaluation of the thermodynamic averages of the above discussed N • M systems proceeds by passing site by site sequentially through the N • M lattice upon which the systems are defined. At each particular site, (i, j), sampled on the lattice, a random number is added to the site variable ri(j) of the system and the resulting new configuration of the N • M system is tested, using the standard classical Monte Carlo (Metropolis) criterion, to see whether it or the configuration without the random change in ri(j) is to be kept as the resultant configuration of the system. In general the random increments to the ri(j) are selected from an interval centered about zero whose size is such as to assure a 40-60% rate of configuration change in the system. Before we present results for the energy, specific heat and pressure of our Lennard-Jones system of atoms and make comparisons with other techniques for studying this system, we turn to a short discussion, similar to that presented in w3.1.2, regarding techniques for speeding up the convergence of machine generated averages, calculated in the QMC, to their correct values.

3.2.2. Techniques to aid computer evaluation As of this writing only one method has been presented which facilitates the generation of QMC computer codes for the thermodynamic averages in many-body vibrational systems which are more rapidly convergent in the

A.R. McGurn

36

Ch. 1

Trotter slice number M. The method in question is that of Takahashi and Imada (1984b) and is based, like that of Suzuki (1976a, b, 1977) and of de Raedt and de Raedt (1983), on the inclusion of higher terms in 1/M to the argument of the exponential integrand forming the functional integral of the QMC partition function (see eq. (57) above). These higher order terms do not contribute to the thermodynamic averages in the M --+ c~ limit but they do make the convergence with M of the finite M averages to their M --+ c~ limits much quicker. Specifically, Takahashi and Imada (1984b) show that

exp ( -

13 H1) exp ( M 2

/3 ( H 0 + H 1 ) ) = e x p (

x exp

-

~

~

M

2 (68)

2

+ 0.M-4/35.,()

x exp M

exp

M

2

where

R=24mi=l

~

"

(69)

Using eqs (68) and (69), the partition function in eq. (49) can be rewritteo as

Z

=

lim

M--+cx~

• exp

9r{ex.

/ -0/ M

2

exp

-

(70) M

• exp

M

2

exp

M

2

where the M -~ c~ limit in eq. (70) is the same as that in eq. (49).

w

Path-integral quantum Monte Carlo studies

37

For finite M the expression on the right hand side of eq. (70) can be evaluated using similar techniques to those used in eqs (49) through (59) for the evaluation of the partition function defined in eq. (49). Upon doing this we find that, for finite M, Z is given by eqs (57) through (59) with the expression in eq. (59) for v(r) replaced by N

~(,) = Z ~(I,, - r,+l I) i=l

(71)

2

24mM

Z r i=l

j=l

- ri+ll)

The average energy, specific heat and pressure are then calculated from this new expression for the partition function using the standard relations" E=

i~/3

lnZ,

C=

aT

and

P=

/3 a i

lnZ.

Using the correction given in r (71), McGurn et al. (1989) and Maradudin ct al. (1990), have found that the quantum mechanical energy of the nearest neighbor Lennard-Jones chain is given by

['

E / N e = lim M-~

1 1 ~T + M N (ul) '

1

(72)

where for ri(j) = cmyi(j), with a2o = h2/me, and r = 13elM,

Ul =

M-1

1

2r 2

] [ ~ ( j ) - ~(j + 1)] 2 + [ ~ ( U ) - if(l)] 2

j=l

(73)

1 M

+ ~ Z ['1 (~(o) + 3~(~(0)], i=l

where ~ ' - c~0ff, N

~,(~) = Z * ( I , , -

(74a)

r,+l t/,

i=1

, h2 v 2 ( r ) - - 24 m

2k(~ i=1

2

(74b) ~r/

38

Ch. 1

A.R. McGurn

and the average ( ) is defined for a general variable A by

]

1-I driQ/') AW({ri(j)})

=1 i=l

/I N ]

(A) =

(75)

=1 i=1

with

rO,)]:, • exp

{_ZM

-~- Z (Vl ({r'(j)}) q- '02 ({r'(j)}))

}

(76) 9

j=l

The quantum mechanical specific heat C/N, obtained from the temperature derivative of eq. (72), is given by

C =

Nk a

lim M T : [ 1 27-2

M---too

1 ((u2)-(u 2)+(ul)2)]

MN

(77)

where

u2 = ~-~ ~

]

[if(j)- ff(j + 1)] 2 + [if(M)- ~7(1)]2

j=l

(78)

6 M

+ ~-v ~ v~.((~(o~), i=1

and ( ) is again defined by eqs (75) and (76) above. The pressure of the one-dimensional chain, which is related to the partition function by P =

/3 i)L

In Z,

(79)

w

Path-integral quantum Monte Carlo studies

39

is found by McGurn et al. (1989) to be

p=2N{ L

M N 213T

2~ 1

1

M

N

/

~ ~ [Yi(j + 1 ) - yi(j)]2 ~=l j=l

(80)

C)('O1-k-'/)2) /

2M j=l E i#k=l E \ [Y~(j)- Yk(j)] i}lye(j)- Yk(j)]

'

where ( ) is defined in eq. (75). These three modified expressions for the energy, specific heat and pressure in eqs (72), (77) and (80), respectively, are found to converge with M to their M --+ oc limits much faster than do the expressions without the modification of eq. (71) for these same respective quantities. We shall now turn to a discussion of the numerical results obtained by McGurn et al. (1989) for the Monte Carlo evaluation of eqs (72), (77) and (80).

3.2.3. Results and discussion McGum et al. (1989) (MRMW) have evaluated eqs (72), (77) and (80), defined on an N • M lattice, using standard Metropolis sampling techniques applied to the position space variables of the N atoms on the chain. The chain is subject to periodic boundary conditions in its length, L, and the convergence of eqs (72), (77) and (80) in both M and N to their M --+ ~ , N --+ c~ limits have been investigated numerically. The ability of the QMC to yield accurate (to within a couple per cent) estimates of the M -+ cc thermodynamic properties of the one-dimensional chain for moderate values of M (M = 10 to 15) and for N = 5, 10, 15 is observed in the nearest neighbor Lennard-Jones system for Metropolis samplings involving up to 250 000 000 configurations on the N • M lattice. MRMW have computed the energy, specific heat and pressure of the chain over a range of temperatures which interpolate the T --+ 0 and T --+ c~ limits and for a fixed nearest neighbor separation of 21/6o ". This separation is the T = 0 nearest neighbor separation of the classical mechanical chain with nearest neighbor only interactions. (More general average nearest neighbor separations can be easily handled by the computer algorithm and the value 21/6o. was only chosen to exemplify the temperature behavior of the system for a fixed nearest neighbor separation.) In addition to depending on the standard thermodynamic variables of temperature and average nearest

40

Ch. 1

A.R. McGurn

neighbor separation of atoms in the chain, QMC results for the energy, specific heat and pressure being quantum mechanical must depend on Planck's constant. For the Lennard-Jones systems considered, this addition quantum dependence is characterized by the parameter c~ = h / ~ which indicates the importance of quantum fluctuations. The value of a is seen to increase as quantum effects increase in the system. Results for the QMC are then presented as functions of T, a and the nearest neighbor atomic separation. In fig. 2 we present QMC results from MRMW for the energy and specific heat of the one-dimensional chain for c~ = 0.03 (appropriate to solid argon) (Klein and Venables 1976) and c~ = 0.10. Comparison is made with results of the quantum mechanical harmonic (phonon) approximation (dashed lines) and the exact classical mechanical (a = 0 limit) solution. Chains of N = 10 atoms and M = 10 and 15 were used for the QMC points presented in the figure, and the size of the simulation points represent the spread in the data in going from M = 10 to 15. (A detailed study of the dependence on M and N of the simulation data can be found in McGurn et al. (1989) where the data results of the individual runs from which the averages and standard deviations drawn in fig. 2 were obtained are presents.) As for fixed N the average values of the energy and specific heat change by less than a couple of per cent in going from M = 10 to M = 15 runs, we feel it reasonable to expect that the M --+ oo limit is represented accurately to within a few per cent by the results shown in fig. 2. The increasing importance at low i

0.5

Z

0.0

tJJ

-0.5

-I.0

,. 0.0

I.

I

AI

(Q)

As~s1 i

0.5

i

1.0

i

1.5

1

2.0

kBT/E

Fig. 2a. The internal energy, E / N s , versus absolute temperature, T, for Lennard-Jones chains. Quantum Monte Carlo results for a = 0.03 (open circles) and a = 0.1 (triangles) are shown. The low temperature harmonic approximation results for a = 0.03 and 0.1 are indicated by dashed curves. The classical result is given by the solid curve.

w

Path-integral quantum Monte Carlo studies 1 . 5 1-

(b)

/ Z {J

0.5

i

i

~ ~

~

o

a

!

0.0

!

41

0.0

~,t

I

t I

0.5

I 1.0

kaTIE

.

I 1.5

2.0

Fig. 2b. The specific heat at constant length, CL/Nk B, versus absolute temperature, T, for Lennard-Jones chains. Quantum Monte Carlo results for c~ = 0.3 (open circles) and a = 0.1 (triangles) are shown. The low temperature harmonic approximation results for c~ = 0.03 and 0.1 are indicated by dashed curves. The classical result is given by the solid curve.

temperature of quantum fluctuations in the system as a goes from a = 0.03 to 0.10 is seen in the departure of the QMC and harmonic approximation results from the classical solution in these regions. The deviation from the classical (a = 0) limit increases as a increases and for a - 0.10 and kBT/r <~ 0.5 we see that the QMC and harmonic approximation results begin to differ significantly at low temperatures due to the non-linearity of the system which is ignored in the harmonic approximation but is correctly treated in the QMC. At high temperatures (kBT/e >> 1) the QMC points for both the energy and specific heat are seen to approach the limits of the classical solution, i.e., 1 1 E/Ne = ~kBT/e and CL/Nk B = ~. This is a standard result in statistical mechanics, i.e., that the T --+ ~ limit of a quantum system reduces to the high temperature limit of an appropriate classical mechanical model. For kBT/e << 1 on the other hand, the energy and specific heat seem to be well represented (aside from effects from the non-linearity of the interatomic interaction) by the harmonic (phonon) approximation. This later point is less to be expected because the assumption of long range order, upon which the harmonic approximation is based, is known to be incorrect for onedimensional systems with finite ranged interactions. At low temperatures, however, the one-dimensional chains possess a significant amount of short range order in which the atomic positions are highly correlated over large segments of the chain. Under these conditions phonon-like modes, for modes with wavelengths less than the atomic pair correlation length along the chain,

A.R. McGurn

42

Ch. 1

15.0

I0.0

-

o"

P'T"

n

A

~ t.5

~ 2.0

_

= 5.0

o.0 o.o

i o.5

.............. t ~.o koT/e

Fig. 3. The pressure Po'/e, versus absolute temperature, T, for Lennard-Jones chains. Quantum Monte Carlo results for c~ = 0.03 (open circles) and c~ = 0.1 (triangles) are shown.

The classical result is given by the solid curve.

are found to exist in the correlated segments and have a similar dispersion relation to that obtained in the harmonic (phonon) approximation. This accounts for the success of the harmonic approximation in approximating the results of our QMC for kBT/e << 1. A similar type of effect, where modes of excitation propagate in regions of short range order, occurs in the low temperature excitations of one-dimensional magnets and in higher dimensional magnetic systems at temperatures above the critical temperature where spin wave like modes are found to exist on the short range order of these systems (Freeman et al. 1990; Mori 1965a, b). In fig. 3 we present QMC results for the one-dimensional Lennard-Jones chain with N = 10 from MRMW for the pressure versus temperature at a fixed average nearest neighbor atomic separation of 21/6cr. Again, agreement with the classical system is obtained at high temperatures but at low temperatures the QMC differs markedly from the classical result. At T = 0 the non-zero pressure of the system is an artifgct of the zero point motion of the atoms in the chain. As with the above discussed results for the energy and specific heat the QMC points presented in fig. 3 are expected to be within a couple of per cent of the M ~ c~ limits.

3.2.4. Systems of identical particles In the above discussion of the one-dimensional chain we have treated the atoms of the chain as being distinguishable particles. We have done this because our interest in this chapter is in the study of models which represent

w

Path-integral quantum Monte Carlo studies

43

the vibrational properties of pure crystaline systems with monatomic bases. In such models of crystaline solids, the atoms are treated as distinguishable particles even though they are in reality indistinguishable. The reason for this is that in these systems the atoms forming the crystal, for temperatures below the melting point, have vibrational motions about their equilibrium lattice sites in which they seldom move farther than some small percentage (<15%) of the lattice constant. For such motions the possibility of an exchange of atoms between different sites of the crystal is extremely small (exceptions to this are solid H and He where atomic vibrations can be large enough for exchange effects to become important) so that only states of the identity atomic permutation operator are important in the partition function, and the system of identical atoms behaves as if it were formed from distinguishable particles (atoms). Even though the one-dimensional system considered in this section has no long range order, a significant amount of short range order exists so that the distinguishable particle model is of interest in our development of theoretical techniques which will eventually be applied to higher dimensional crystaline systems. The generalization of the partition function in eq. (49) to treat a system of identical bosons or fermions is of some theoretical interest, however, due to problems cited above associated with the Hubbard model, Wigner solid, liquid He 3 and He 4, solid H and He. Such a generalization is quite straightforward and expressions for the partition functions of these systems expressed in terms of path integrals have been given by Takahashi and Imada. For completeness we will turn to a brief consideration of these generalizations for the one-dimensional system before taking up a discussion of the vibrational properties of the three-dimensional fcc solid. Given the density matrix (Feynman 1988) for the N distinguishable particles of our one-dimensional Lennard-Jones system in eq. (49)

if(e,

=

(81) i

where (~i(r-')} are a complete set of normalized wavefunctions of the N atom system and (w~} are their respective statistical weights, then the partition function of eq. (57) can be written in terms of r ~') as

Z = / d U r ~(~', ~").

(82)

Using the standard results of density matrix theory, the partition function for the system in eqs (48) and (82) in which the N atoms are treated as

44

A.R. McGurn

Ch. 1

identical bosons is then

1 /

Z B = ~ . v ~p

dUr((~';P(r-3),

(83)

where ~-'~p sums over the permutations of components of ~' = ( r l , r 2 , . . . , and in which the N atoms are treated as identical fermions is then ZF__ rl. ~ ( _ P

1)E / dN r ~ (~.; P(r-O)

rN), (84)

where

(_l)p_ { 1 -1

for even permutation, for odd permutation.

(85)

In terms of the notation in eq. (57) we then have ZB = ~

~p _

d N rj j=l

W (P_.~M, " ~1) Z W(~/, r~+l) i=1

and

(fi

ZF : ~1

J=~ dNrj

)

(86)

W(Pr' M, fl) i=IH W(~/, fi+l). (87)

The arguments presented in eqs (81) through (87) are really quite general and can be easily extended to treat the three-dimensional systems discussed below. We shall leave to the reader such considerations for the three-dimensional systems which are now addressed.

3.3. FCC Lennard-Jones crystal For a Lennard-Jones crystal of N atoms defined on the three-dimensional face centered cubic lattice, the hamiltonian has the form (Maradudin et al. 1990) H = Ho + H1, Ho =

~2 2 2m ~ V i ' i

H1 = ~ ~(l~'i - ~J ), (i,j)

(88) (89) (90)

w

45

Path-integral quantum Monte Carlo studies

where is the Lennard-Jones pair potential defined in eq. (46), the sum in eq. (90) is restricted to be only over nearest neighbor pairs of atoms which interact via the ~(1~'1) potential and the atoms of our system are allowed a three-dimensional motion in space. Again, due to limitations on computer memory, considerations are restricted to a finite array of atoms (32 atoms) subject to periodic boundary conditions and the investigations of the thermodynamic properties of eqs (88) through (90) are based, as are those in w167 3.1 and 3.2 above, upon the formulation of the QMC given by Imada and Takahashi (1984). We shall, in the application of this formulation below, assume that the atoms in our system can be treated as distinguishable particles. This last assumption is valid for virtually all solids. The QMC for the fcc crystal is obtained, as with the one-dimensional chain, by applying the Trotter identity of eq. (26) to the trace of the partition function and performing the trace and matrix multiplications in the position space representation. In the position space representation, the two relevant matrix elements for the evaluation of the trace of the partition function are

exp ( /3h2 2Mm

~2)

= f ~d3k exp(ifr (2rr) 3

_

--

_ ~-.,)

[ ]3,2 ( Mm

27r/3h2

E exp n=-oo

/3h2 k 2) 2Mm Mm

2/3h2

_.,

(~'

r

(91)

) ~,L) 2

where f' and ~" are the position vectors of single atoms and

/

t

# 'l,

exp

- ~ r

- ~'2[)

)i ) rl, r 2

= 2rrS(< - f"l)2rr5(~'2 - F ; ) e x p

- ~ r

(92) - f'2[)

,

where ~ - nx~ + n j + n, la and L in the length of an edge of the cubic array of N fcc atoms and f'l, ~'2 are again atomic position vectors of individual atoms.

A.R. McGurn

46

Ch. 1

Using the fact that the atoms in our system are distinguishable particles with motions localized, respectively, about their average lattice sites, we write (f']exp ( 2 M ~h2 m V2) ~'')

(

Mm 2r/3h2

= f(~', ~")=

exp _

~

(~,

2flh2

~.t)2

(93)

)

so that in terms of the matrix elements eqs (91) and (92) above we have the partition function expressed as a functional integral

H d3ri(j) W({r*i(j)}),

f

Z - M~colim

j=l

(94)

i=1

where W ({~'i(j)}) =

H f (~(J + 1), ~/(j)) i=l j = l

)

x exp j=l (~,t)

)

(95)

and ~(M + 1) = ~(1). As with the one-dimensional chain model the rate of convergence of the fight hand side of eq. (94) for increasing M m the M -+ cr limit can be increased by modifying the exponentials involving the pair potential terms in eq. (95) to include in their arguments higher order corrections in 1/M and the pair potentials. To this end, following a similar line of reasoning m that outlined in eqs (68) through (71) above, Takahashi and Imada make the replacement

(i,~)

(i,~) (96) 1 h2 ( / 3 ) 2 k

+~~ ~

j=l

vj E ~(!~,- ~l)l: (i,e)

w

Path-integral quantum Monte Carlo studies

47

in the exponential argument on the fight hand side of eq. (95). We have found in our studies of fcc Lennard-Jones systems discussed below that Monte Carlo simulations of finite N and M arrays, based on the new form of W({ri(j)}) obtained by this replacement, appear to be convergent to within a few percent of the M --+ c~ limit for M as small as M -- 10 (Maradudin et al. 1990). Given the path-integral form of the partition function in eq. (94) with W({ri(j)}) of eq. (95) modified to include the higher order corrections described in eq. (96), the thermodynamic properties of energy, specific heat and pressure can be obtained from the standard relations involving appropriate derivatives of the partition function. The original three-dimensional quantum mechanical problem now appears rewritten in the form of a four-dimensional functional integral involving only classical position space variables. As with our discussion in w167 3.1 and 3.2 above the classical functional integral and its averages for finite M and N can be approximated using standard Metropolis sampling techniques based on W({ri(j)}) as the weighing function of the statistical sampling. Maradudin et al. (1990) have performed a Metropolis sampling for N = 32 and M = 10, based on 106 configurations, computing the average energy for an fcc system at zero average pressure. In these computations of pressure and energy, the fcc lattice constant of the system is varied until a zero average pressure is computed. A simple linear interpolation based on runs for the average pressure computed, respectively, at two arbitrary chosen lattice constants usually sets the pressure to zero to within 0.1%. Results from Maradudin et al. (1990) for the energy versus temperature computed in this manner for h/v/mea = 0.0295, appropriate to solid argon, are shown in fig. 4 with the size of the plotted points indicating the uncertainty of the averaged values. The solid line in the figure represents a fit of the low temperature energy form E/Ne

= eo + e l T 4 + e2 T6 -k- e3 T8

(97)

to the experimental data (Maradudin et al. 1990). As a by product of the above computations, the variation of the zero pressure lattice constant with temperature is obtained using the particular values of the lattice constant for the temperatures of the Monte Carlo points shown in fig. 4. A crude expression for the variation of the lattice constant for 0.05 < kBT/e < 0.20, obtained (Maradudin et al. 1990) by fitting the simulation data with a simple linear form yields

a(T) o"

= 1.134701 +

O.046097(kBT/e).

(98)

A.R. McGurn

48

(a)

-5.1

..........

-5.2

-

-5.5

-

-5.4

-

Ch. 1

I

!

I

i

o.zo

Z

I,I

0.00

(b)

2.5

I

I

0.05

o.~o

o.t5

I

i

kaT/~

I

2.0-

/

m

/ / m

1.5-

z

(j

/ 1.0-

/

/

/

/

/

Z

0 . 5 -

0.0 ~ . - ~ " o.oo

!

0.05

I

o.~o

I

0.15

kaT/~ Fig. 4. QMC results (solid curves) for the a) internal energy, E/Ne, and b) specific heat at constant pressure, (]/Nk B, v e r s u s temperature, T, of a 32 atom fcc Lennard-Jones crystal, plotted for the system at zero pressure. The dashed curve in b) is the result of the harmonic approximation evaluated for the zero pressure lattice constants obtained from QMC results.

A more accurate representation of a(T) may be obtainable by using other functional forms for fitting a(T) based on low temperature perturbation theory, but these details have yet to be considered by us (Maradudin et al. 1990). The specific heat at constant pressure along the zero pressure curve of the fcc system can be easily calculated from the energy results by taking the temperature derivative of the fitted form in eq. (97) (Maradudin et al. 1990).

w

Path-integral quantum Monte Carlo studies

49

The resulting dependence of the specific heat on temperature, computed in this way, is shown in fig. 4(b) where we also display results from the quasiharmonic approximation for the specific heat at constant volume in which the atomic spring constants are computed using the temperature dependent lattice constant, a(T), given by eq. (98). Computing the specific heat in the usual way, based on a simulation of 0 C =

0

0T 0/3

In Z,

is much less efficient than using the method discussed in the above paragraph because the computation involves evaluating an expression which is essentially the statistical variance of the energy. The statistical variance requires many more configurations to obtain an accurate estimate of than does the simple average of the energy. In addition, as we have seen above in previous sections, the specific heat in the path-integral formulation for C =

aT ~#

In Z

is computed as the difference of two large numbers which must be very accurately determined in order to obtain an accurate value of their difference. A disadvantage of the present approach though is that only the specific heat at constant pressure is computed. In addition to the original work of Maradudin et al. cited above, two other more recent quantum Monte Carlo studies of the fcc nearest neighbor Lennard-Jones system have appeared. These are studies of Cuccoli et al. (1992a) and Liu et al. (1993). Both of these studies extend the size of the system considered in the Monte Carlo simulation to 108 atoms subject to periodic boundary conditions and Cuccoli et al. (1992a), in addition, consider modifications of the Metropolis sampling to treat sampling configurations generated from multiple Trotter slice movements for a given atom. (Metropolis sampling involving multiple Trotter slice movements for a given atom were also considered by Maradudin et al. (1990) but were not found to yield significant improvement in the Metropolis sampling over the single slice treatment and were, hence, not reported in their work.) The work of Cuccoli et al. (1992a) and Liu et al. (1993) also used the quantum Monte Carlo results generated by them to compare with results obtained from a variational approach for approximating the path-integral formulation of the partition function. These variational methods offer hope for extending pathintegral methods to treat complex atomic systems and for the evaluation of

A.R. McGurn

50

Ch. 1

Table 2 QMC results of Cuccoli et al. for Ar. Temperature

Pressure

(~/k B)

(6/~ 2)

Energy (~)

0.08347 0.1669 0.339 0.5008

0.077 0.089 0.111 0.111

-7.775 -7.699 -7.343 -6.853

Specific heat (constant V) (kB) 0.54 1.54 2.36 2.61

specific heats and dynamical properties using programs which may be more computationally efficient than those based on standard QMC methods. (We shall have more to say about variational methods in w5.) We now briefly consider a discussion of the work of Cuccoli et al. and Liu et al. In the work of Cuccoli et al. (1992a) a system of 108 atoms subject to periodic boundary conditions was considered for a = 0.0294 appropriate to solid Ar and QMC results for this system were generated as function of 1/M in order to extract the M --+ c~ limit. In their work corrections were added to the QMC results for the nearest neighbor system to account in an approximate way for the further neighbor interactions that would be present in the unrestricted Lennard-Jones solid. These corrections were made by adding to the QMC nearest neighbor results the potential energy for further neighbor interactions computed using the equilibrium atomic separations in the system. These corrections in large part account for the differences between the results of Cuccoli et al. presented below and those of Maradudin et al. presented above. The energy and the equation of state were investigated by Cuccoli et al., and found to converge rapidly with increasing M to the M -+ oo limit and to be in surprisingly good agreement with experimental values for solid Ar. The specific heat was found to be most difficult to compute and Cuccoli et al. suggested that the specific heat could be more efficiently computed as the numerical temperature derivative of the QMC energy data rather than using Metropolis sampling techniques on the operator form for the specific heat. In table 2 we present some of the results of Cuccoli et al. for the properties of solid Ar computed in the QMC. In generating the results shown in table 2, Cuccoli et al. consider Metropolis samplings which entailed a simultaneous motion of all Trotter slices associated with any given atom in the system. This differed from the work of Maradudin et al. which only allowed for a change associated with one Trotter slice and one atom at any given sampling. The improvement in the sampling procedure in the studies of Cuccoli et al. was made to increased the rate of convergence with number of sampled configurations to the average

w

Path-integral quantum Monte Carlo studies

51

thermodynamic properties of the system. Similarly, types of multiple slice sampling procedures have been employed by Pollock and Ceperley (1984) in studies of solid and liquid He systems, and an interesting discussion is given in Pollock and Ceperley (1984) illuminating of difficulties associated with Monte Carlo samplings for path-integral partition functions. The path-integral partition function has features in common with partition functions of systems of polymers and these features suggest increasing program efficiency through multiple Trotter slice sampling. Few details, however, regarding the specifics of the CPU times involved in various QMC procedures have been presented in the literature so that a detailed comparison of different sampling techniques is not currently possible. It would be very helpful in the future for work presented on the QMC to include such details. One important additional object of the work of Cuccoli et al., which shall be discussed later in the context of time-dependent properties, was to test a new and very promising approach, the effective potential-Monte Carlo (EPMC), at approximating the thermodynamics of the path-integral Lennard-Jones partition function. This was done by comparing results for the thermodynamics of the Lennard-Jones model for solid Ar generated using the EPMC with corresponding data obtained using the QMC. The EPMC was found, in particular, to be valuable in computing the specific heat and the frequency moments of the spectral densities of time-dependent response functions (Cuccoli et al. 1992b) in a quick and accurate manner. In another series of studies, Liu et al. (1993) treated the nearest-neighbor Lennard-Jones model of Maradudin et al. (1990) considering parameters appropriate to Ne 22 at zero pressure (c~ = 0.0759). Both the QMC for a system of 108 atoms subject to periodic boundary conditions and EPMC were computed by Liu et al. as well as results for the improved self-consistent (ISC) theory which can be described in terms of a diagrammatic perturbation expansion in the anharmonic interaction. The QMC results were found to converge rapidly with increasing M and are believed for M = 20 to be within less than 1% of the M --+ c~ limit. The results for the QMC computed by Table 3 Summary of QMC results (Liu et al.). Temperature Spacing Number Trotter Millions Pressure (e/kB) cr of atoms number of configs. (e/or 3) 0.134 0.249 0.421

1.16816 108 1.17075 108 1.1834 108

30 20 10

13 26 14

Energy (e)

-0.08 4- 0.01 -4.426+0.006 -0.02 4- 0.01 -4.329 -t- 0.005 0.01 4- 0.02 -3.9844-0.006

52

A.R. McGurn

Ch. 1

Liu et al. (1993) are shown in table 3. These results are for the nearestneighbor only Lennard-Jones system and do not contain corrections, as do those of Cuccoli et al. (1992a), for further neighbor interactions. The results shown in table 3 were used by Liu et al. (1993) as standards to test the validity of the EPMC and ISC approximations. In general the ISC energy and lattice constants as functions of temperature were found to agree well with those of the QMC while the EPMC energy and lattice constants agreed less well with the QMC results. Approximations for the specific heat are possible in both the ISC and EPMC but the QMC does not yield reliable specific heat data. A further discussion of the comparison of the EMPC and ISC with QMC data made by Lui et al. will be treated below in w5. The importance of the QMC, here as in Maradudin et al. and Cuccoli et al., is that it yields reference values for well defined theoretical models and these reference values can be used to test other perhaps less computationally demanding methods based on approximation schemes. This becomes more important as attempts are made to compute more computationally demanding properties such as specific heat and time-dependent properties as compare to the less computationally demanding treatments of the average energy and lattice constant.

4.

Time-dependent quantum Monte Carlo

In this section we will consider time-dependent thermodynamic averages computed in vibrational systems by means of QMC methods (McGurn et al. 1991; Cuccoli et al. 1992a, b, 1993a). Specifically, we shall be interested in the evaluation of the time-dependent atomic displacement pair correlation functions related to the study of inelastic neutron scattering events with the creation of single (phonon) quasi-excitations. The spectral density of these correlation functions, for vibrational systems, are expected to be quite simple, consisting of pole singularities at frequencies appropriate to the creation or destruction of single phonon excitations. For this simple type of time dependence we shall see that a successful representation of the shifting and broadening of the phonon poles can be obtained by studying a limited number of the frequency moments of the spectral representation (McGurn et al. 1991; Cuccoli et al. 1992a, b, 1993a). We have chosen to study the atomic displacement pair correlation functions because of their simple (pole singularity) spectral densities. In particular, these types of simple pole spectral densities yield to a continued fraction representation, involving a limited number of frequency moments of the spectral density, as an accurate approximation of their overall behavior. On the other hand, methods which have been previously proposed

w

Path-integral quantum Monte Carlo studies

53

to treat, in the context of the QMC, general time dependent functions with more complicated spectral densities have as a whole only received a very limited success. This is due primarily to the fact that the general form for time-dependent averages involves the evaluation of functional integrals, (similar in form to those considered in w3 above as representations of the partition function), which include extra imaginary terms in the exponential arguments of their integrands (Berne 1986; Doll et al. 1990; Freeman et al. 1990). These functional integrals are very difficult to evaluate numerically because of the rapid phase variation in the path integral exponential and often require a great number of frequency moments of their spectral densities for an accurate representation in terms of a continued fraction expansion. Special techniques, however, have recently been developed by Doll et al. (1990) for the direct evaluation of path integrals which are functions of both time and temperature. These involve various filtering transformations, applied to the path integral during its Monte Carlo evaluation, to reduce the effects of rapid phase variations. To date the only vibrational systems that these techniques have been applied to have been single particle anharmonic oscilators and we shall not consider them further in this review. The reader is referred to Doll et al. (1990) for a more complete review of direct evaluation techniques and to their application to the single particle problem as well as to other type of quantum many-body problems. The procedure we shall use to study the atomic displacement correlation functions is based on a continued fraction representation of the spectral density in terms of its frequency moments (McGurn et al. 1991; Cuccoli et al. 1992b, 1993a; Mori 1965a, b; Lovesey 1971; Lovesey and Meserve 1972; Tomita and Mashiyama 1972; Lindenberg and West 1990). The frequency moments of the spectral density are easily expressed as static averages of atomic position and momentum variables and their time derivatives at zero time. These averages are directly accessible to the static QMC techniques outlined in w3. We shall outline below techniques for computing the pair correlation function, first describing the expansion in moments and then the evaluation of the moments using the static QMC. The results of these evaluations are then applied to the linear chain problem discussed in w3.2.

4.1. Continued fraction expansion In this section we shall discuss the representation of spectral densities in terms of continued fractions (McGurn et al. 1991; Cuccoli et al. 1992b, 1993a; Mori 1965a, b; Lovesey 1971; Lovesey and Meserve 1972; Tomita and Mashiyama 1972; Lindenberg and West 1990). The theoretical foundation of these types of representations for the thermodynamic response

54

A.R. McGurn

Ch. 1

functions of finite temperature systems was originally established in a series of papers by Mori (1965a, b). Since Mori's original papers, his ideas about continued fraction representations have been used to study a variety of electronic and magnetic systems (Lovesey 1971; Lovesey and Meserve 1972; Tomita and Mashiyama 1972; Lindenberg and West 1990). We shall give below a brief overview in words of the development of Mori's ideas regarding the continued fraction representation of spectral densities; referring the reader to the original references for a complete mathematical exposition (Mori 1965a, b). We shall then use these ideas to develop a continued fraction representation of the density of states for the one-dimensional chain system described in w3. From quite general thermodynamic considerations Mori showed that the spectral density of time-dependent correlation functions for general equilibrium many-body systems can be represented by an infinite continued fraction developed in the frequency. The analysis given by Moil is based on a description for the time development of the dynamical variables in terms of an optimally defined set of orthogonal basis states. Specifically, the basis used in this representation is obtained from a Gram-Schmidt process based on employing thermodynamic averaging to define the inner product in the space of the dynamical variables of the system. The resulting inner product is then used to define a projection operator which projects onto a dynamical variable of the system the components of the equal time time-derivative of that dynamical variable. Applying this projection technique, Mori starts with a given dynamical variable of the system and projects its time derivative into the dynamical variable itself and into the complementary space. Mori then repeats this process on the projection in the complementary space, resolving the time derivative of the complement variable into a projection into the variable itself and into its own complementary space. By repeating this procedure an infinite number of times a sequence of relations is arrived at which can be used to write the spectral density for the time-dependent pair correlation function of the original (first projection) variable as a continued fraction. Mori goes on to further show the important fact that the coefficients in this continued fraction representation can be written solely in terms of the frequency moments of the spectral density. This representation is then applied to study the dynamics at finite temperatures of a number of classical and quantum mechanical systems. As we shall see below, these techniques are quite accessible to the vibrational system we have discussed in w3. We shall now turn to a brief discussion of the mathematical structure of this continued fraction representation and how it can be used, along with the QMC, to obtain a representation for the spectral density associated with inelastic neutron scattering from the one-dimensional chain of w3.

w

Path-integral quantum Monte Carlo studies

55

For the one-dimensional system of N atoms described by eqs (46) and (48), the inelastic neutron scattering is related to the quantum mechanical spectral density (McGurn et al. 1991)

Ck(w) =

F

(rk(t)r_k(O))e-i~tdt,

(99)

oo

where for lattice constant a

1

N

(100)

rk = v/--N =j~lrje-ik(ja)

and k is specified by k = 27rn/Na for n = -t-1, + 2 , . . . . The spectral density in eq. (99), while directly related to the scattering from the system, does not however yield directly to a useful continued fraction representation, and it is in fact better to study the continued fraction representation of a form which is closely related to the spectral density in eq. (99) but slightly different from it. In particular, if we define an even function in frequency, Ck(w), 0 by (McGurn et al. 1991)

c%o) =

1 [Ck(w) + Ck(--w)] (101)

= 1 [1 + e ~h~] Ck(w), 2 we find that C~ which is simply related to Ck(w), is much easier to study in a continued fraction format than is Ck(w). The reason for this is that, it is easier to develop continued fraction representations for spectral densities which are even functions of frequency because, in general, a smaller number of moments are required to accurately approximate a function over either positive or negative frequency axes than over both positive and negative frequency axes. In the following we shall specifically concentrate on considerations of C~ 0 For an even function of frequency such as Ck(w), the continued fraction development of Mori, allows us to re-express C~ as

C0(w)= 2Re[r

dw' Ck(w'), oo 27r

(102)

A.R. McGurn

56

Ch. 1

where ek(w) is an infinite continued fraction of the form

1 a2(k) a2(k) ek(w)=iw+

iw+

iw+

(103)

and the {a2(k)} are related to the frequency moments (wEt) k --

F

dw Ck(w)w 2l

(2O

by =

a~(k) =

(104a)

(~4)k

(032)k

( 2)k

(w~

,

(104b)

and so on. (For a general expression for a2(k) in terms of the (602/)k see Mori (1965a, b).) The expansion in eq. (102) which yields an exact representation of C~ can also be used as a basis for approximations of C~ Approximations to C~ can be made by terminating the continued fraction in eq. (103) after a finite number of terms. We shall concern ourselves with such termination schemes in our discussions below. In addition, for systems which posses only a finite number of non-zero an2 (k) an exact representation in terms of eq. (102) can easily be given. We shall discuss such an exactly representable system below (i.e., the continued fraction representation of the harmonic approximation of the one-dimensional chain of atoms) and then go on to discuss continued fraction termination schemes which yield approximations for the spectral density of an anharmonic one-dimensional chain of atoms. As a specific example of the continued fraction representation of 0 it is interesting to note that in the harmonic approximation to our LennardJones chain of atoms we find that a2(k) = O. Hence from eq. (103) we see that ek(W) is terminated at the a~(k) term. For this case it follows from r (102) through (104) that

co(w ) : #o(k) [5(o) - w k ) + 5(w + wk) ] 2

(105)

w

Path-integral quantum Monte Carlo studies

57

where wk -

12.2-1/6~/-~

a1 [1

-

cos(ka)] 1/2

(106)

and ,o(k) =

=

7rh ~2t.O k

(107) coth

i '1

so that eq. (105) is the exact result for C~ in the harmonic approximation. In the limit of small anharmonicities we expect that C~ will be little changed from its representation in eq. (105) and that a reasonably simple termination approximation for Ck(W) in eq. (103) can be given for the anharmonic case. When anharmonicity is introduced into the linear chain system we expect the peaks in C~ given by eq. (105) to develop widths and the frequencies at which the maximum values of these peaks occur to be renormalized. These two effects will show up in the frequency moments of the spectral densities and it is one of the hopes of those who apply continued fraction methods to the study of response functions that, if the effects of the anharmonicity is small, a reasonable approximation to C~ can be obtained from the knowledge of a few of the lower order moments, along with some intelligently chosen termination of the continued fraction sequence. A number of termination schemes have been proposed as means of obtaining, from a knowledge of a finite number of moments, a good continued fraction representations of the spectral densities. Among these methods are included: the Gaussian approximation of Tomita and Mashiyama (1972) the three pole approximation of Lovesey (1971) and various terminators associated with recursive methods (Pettifor and Weaire 1985). In the preliminary results of our moment studies which we present below, we have only considered the Gaussian approximation of Tomita and Mashiyama (1972). This method has found considerable success in the study of the time dependent correlation functions of one-dimensional magnetic systems and is also easily applicable to our vibrational systems. In future work, we hope to pursue a detailed comparison of the different termination schemes with a view towards determining the most appropriate one for the study of vibration systems. For now we shall just outline our studies using the Gaussian termination.

A.R. McGurn

58

Ch. 1

Specifically, Tomita and Mashiyama (1972) in their method of Gaussian termination propose to terminate the infinite continued fraction Ck(w) in eq. (103), approximating 2 (k) an(k) an+l

iw+

iw+

9..

(108)

in eq. (103) by the Fourier transform of a Gaussian function of time chosen to yield the correct (w2n+2)k moment of the spectral density. In the cases that we consider below for the spectral density of our chain of atoms, we shall consider the n = 2 approximation to Ck(w). We find for n = 2 that Wk(w) ~

1

iw + a2(k)f (w, a2(k))

,

(109)

where

f (w, a2(k)) =

fOx)

dt e-aZz(k)t2/2e-i~~

1 a2(k)

_y2/2

ie-

]

(110)

ds d s2/2

u=-a2(k)

The termination given by f(w, a~(k)), then, replaces the infinite continued fraction multiplying a2(k) in eq. (103) by the Fourier transform of a Gaussian function whose spectral width gives the correct fourth frequency moment of the spectral density. It is interesting to note that in the limit that a~(k) --+ 0+ eqs (102), (108) and (109) reduce to the harmonic approximation in eq. (105). Since eq. (109) reduces to the correct form in the harmonic limit we expect that eqs (102), (108) and (109) will form a basis from which to represent spectral densities which may be approximated by two poles with reflection symmetry through w = 0 in the complex frequency plane or reflection symmetry through the line defined by Re (w) = 0 in the complex frequency plane. Before we can use the above discussed approximate equations of the continued fraction representation to study the time-dependent correlations in our system, we need to have a method for obtaining the static frequency moments of the spectral densities. As we shall see, these moments are simply related to the static QMC averages of well defined expressions involving the atomic coordinates on the chain. We shall now turn to a discussion of the determination of the moments using the techniques of static QMC averaging.

w

Path-integral quantum Monte Carlo studies

59

4.2. Moments of the spectral distribution The frequency moments required for the evaluation of the continued fraction representation of the spectral density can be written entirely in terms of static thermodynamic averages. From the expression for C~ given in eq. (101) we find that

(6o2n)k --

F

dw 602ncO(6d)

oo

= 27r(- 1)" (r(k2")(O)r_k(O)), (w2n+l)k

(111)

-- O,

where

d2nrk(t) r(2n) (0) =

dt zn

(112) t=O

The time derivatives of r k, which are obtained in the usual way from the Schr6dinger equation of the one-dimensional chain of atoms, when substituted into eq. (111), yield expressions for the frequency moments of the spectral densities in terms of the dynamic variables of the atoms on the chain. A useful identity in evaluating the expressions in eq. (111) is obtained from the stationarity property (Cuccoli et al. 1992b)

(r~m)(t)rj(O)) = (r~m)(O)rj(-t)).

(113)

Differentiating both sides of eq. (113) n times with respect to t, we find for t = 0 that

(r~m+n)(O)rj(O)) -- (--1) n (r~m)(O)r~)(O)),

(114)

so that from eq. (111) we have (w2~)k = 27r(r(k~)(O)r(_~(O)).

(115)

A.R. McGurn

60

Ch. 1

In particular, the first four frequency moments are given by (Cuccoli et al.

1992b) (w~

= 27r(rk(O)r_k(O)),

(w2)k =

m

E eikea

e

(116a)

dre

E eik'a e dre

(W4)k -- ~

v

r_ k(0) ,

V)(Ee-ikna drnV)>'

(116b)

(116c)

where V = ~--~qS(lri- ri+,]), i

(117)

and we have used r~l)_ mp~, 1 r(21 _

1 ~

(118/ 0 V.

(119)

i Where pi is the momentum of the ith atom and eqs (118) and (119) are obtained directly from the Schr6dinger equation. For the one-dimensional chain system, eqs (116) can be evaluated using eqs (58), (65) and the Monte Carlo methods discussed in w3. Recently, Cuccoli et al. (1992b) have discussed the evaluation by quantum Monte Carlo methods of eqs (116) for the one-dimensional chain system of w3.2 at wave vectors of ka = 0.2, 0.5 and 1.0 (see table 4). Results for kBT/e = 0.1, 0.2, 0.3 were obtained for M = 4, 8, 16 by using several million Monte Carlo sampling configurations. The values for (w~ (wz)k, (w4)k obtained in these studies and shown in table 4 are quite interesting as they indicate, for fixed kBT/e and ka, a rapid convergence with increasing M to the M - ~ limit of these respective quantities. We expect that, in fact, the M dependence of the moments may be fitted by some polynomial form (e.g., A + B / M 2 + C ' / M 4 + .. .) and that an appropriate fitting of such a form to the data in table 4 will allows us to obtain a very accurate estimate of the M --+ c~ exact moments. The investigation of such methods for extracting the M -+ c~ moments from the data in table 4 is currently underway and will be published elsewhere.

w

Path-integral quantum Monte Carlo studies

61

Table 4 Quantum Monte Carlo results for a chain of 20 atoms for Trotter numbers M = 4, 8, 16 and kBT/e = 0.1, 0.2, 0.3. Results for (w~ (w2)k and (wn)k are presented in units of 0"2, elm and (e/(ma)) 2, respectively. A) ka = "rr/5 M=4

kBT/e

(W0)k

(W2)k

(W4)k

0.1

0.0299 4- 0.0015

0.705 4- 0.038

24.6 + 1.0

0.2

0.0547 4- 0.0050

1.297 4- 0.100

50.0 + 3.0

0.3

0.0733 4- 0.0020

1.810 -l- 0.060

78.2 4- 2.0

kBT/e

(W0)k

(W2)k

(tO4)k

0.1

0.0296 4- 0.0010

0.725 4- 0.010

29.7 4- 1.0

0.2

0.0570 + 0.0020

1.330 + 0.040

53.4 4- 1.0

0.3

0.0781 4- 0.0040

1.665 4- 0.065

75.2 4- 2.0

kBT/e

(w~

(W2)k

(W4)k

0.1

0.0288 4- 0.0010

0.732 4- 0.010

32.0 4- 1.0

0.2

0.0631 4- 0.0085

1.567 + 0.150

63.1 -4- 4.0

0.3

0.0813 + 0.0040

1.894 4- 0.020

90.2 4- 7.0

kBT/e

(w~

(W2)k

(W4)k

M=8

M=16

B) ka = 7r/2 M=4

0.1

0.00780 4- 0.00005

0.989 4- 0.006

165.6 4- 1.0

0.2

0.01170 + 0.00010

1.512 + 0.030

291.4 4- 5.0

0.3

0.01623 4- 0.00100

2.031 4- 0.060

453.9 4- 10.0

kBT/e

(~~

(~2) k

(w4)k

0.1

0.00859 4- 0.00010

1.170 4- 0.010

217.1 + 5.0

0.2

0.01129 + 0.0040

1.555 4- 0.074

318.7 + 15.0

0.3

0.01850 + 0.00900

2.143 -1- 0.115

497.0 4- 17.0

M=8

A.R. McGurn

62

Ch. 1

Table 4 (continued) M=16

kBT/e

(600)k

(W2)k

(W4)k

0.1

0.00871 4- 0.00008

1.200 4- 0.010

233.9 + 3.0

0.2

0.0132 -I- 0.0040

1.623 -1- 0.025

337.6 4- 5.0

0.3

0.01715 -1- 0.00120

2.190 4- 0.040

503.5 4- 20.0

C) ka = 7r M=4 kBT/e

(wO)k

(W2)k

(W4)k

0.1

0.00464 + 0.00005

1.191 4- 0.004

382.3 4- 4.0

0.2

0.00680 4- 0.0010

1.763 4- 0.060

664.3 4- 7.1

0.3

0.00892 4- 0.0020

2.336 + 0.075

1000.6 -1- 15.2

M=8 ]eBT / e

(wO)k

(W2)k

(034)k

0.1

0.00543 4- 0.00024

1.504 -1- 0.052

523.8 + 11.0

0.2

0.00707 4- 0.00010

1.946 4- 0.028

762.1 4- 10.0

0.3

0.00890 + 0.0021

2.422 4- 0.110

1024.2 4- 15.2

M=16 k BT/e

(wo)k

(w2)k

(w4)k

0.1

0.00555 4- 0.00010

1.572 + 0.012

573.4 + 60

0.2

0.00805 + 0.00016

2.290 4- 0.060

887.4 + 31.0

0.3

0.00977 4- 0.0008

2.452 4- 0.150

994.1 + 40.0

The moments (w~ (w2)k and (604)k can also be evaluated in the h --+ 0 classical limit using the classical Monte Carlo techniques discussed in w2. Cuccoli et al. (1992b) have performed this Monte Carlo evaluation for the classical one-dimensional Lennard-Jones chain of atoms, and we present the results of this evaluation in table 5. It is seen that for kBT/e = 0.2 and 0.3, there is a similarity in the values of the classical and quantum mechanical moments of the spectral density and their dependences on wave vector and temperatures. The numerical values of these moments (classical

Path-integral quantum Monte Carlo studies

w

63

Table 5 Classical Monte Carlo results for a chain of 40 atoms. Results for (w~ (W2)k and (W4)k in the classical limit are presented in units of a 2, elm and (e/(mor))2, respectively.

A) k a = kBT/e

7r/5

(wO)k

(W2)k

(094)k

0.2

0.0614 4- 0.0020

1.291 4- 0.049

44.3 4- 5.8

0.3

0.0836 4- 0.0140

1.824 4- 0.299

81.4 4- 24.1

B) ka='tr kBT/e

(w~

(602)k

(tO4)k

0.2

0.00584 -4- 0.00014

1.276 4- 0.024

433.5 4- 11.7

0.3

0.00857 4- 0.00005

1.888 4- 0.044

749.1 4- 20.0

as compared to quantum) for a given wave vector and temperature, however, can differ by as much as 25%. We expect that these differences will show up as significantly different forms in the spectral densities as functions of frequency for the classical and quantum mechanical systems. We shall now turn to a brief discussion of some preliminary results for C~ obtained by McGurn et al. (1991) from values of (w~ (w2)k and (w4)k determined from the quantum Monte Carlo and the Gaussian approximation in eqs (102) through (104).

4.3. Gaussian approximation for the spectral density McGum et al. (1991) have used quantum Monte Carlo results for (w~ (wz)k and (w4)k and the Gaussian approximation in eqs (102) through (104) to compute C~ Specifically, C~ for ka = 7r and for kBT/e = 0.175, 0.2 and 0.3 were computed and the results of these computations are shown in fig. 5. Since the computations shown in fig. 5 were presented, a more thorough computation of the moments (presented in table 4) at more general wave vectors and temperatures has been done. The results in fig. 5 are highly preliminary in nature but indicate a shifting of phonon (peak) frequencies to higher frequencies and a decreasing of phonon lifetimes (indicated by increasing peak widths) with increasing temperature. A more thorough analysis of the determination of C~ from its frequency moments is currently underway. Results for C~ from these studies will be published at a future date.

A.R. McGurn

64 3.400

1

kBT/e

0.003

0.001

I

-

0.17'5

3.300

0.002

Ch. 1

0.2

-

-

I

i..--

I0

15

20

25

Fig. 5. Plot of C~ for the one-dimensional chain versus w. The wavevector k is at the Brillouin zone boundary, the average nearest neighbor atomic separation is a = 21/6o ", and results for temperatures kBT/e = 0.175, 0.2, and 0.3 are shown. The plots presented at each kBT/e are made using the average values of po(k), a2(k), and a2(k) from the simulation. The errors in peak frequency and in the full width at half maximum of these curves are given in McGurn et al. (1991).

5.

Discussions and conclusions

In this final section we present a brief summary of the main conclusions arrived at in the course of this review and give an indication as to what we consider to be the most important directions for future research efforts in the quantum Monte Carlo determination of the thermodynamic properties of vibrational systems. We shall also discuss some recent analytical results for the evaluation of the quantum Monte Carlo path-integral partition function and its averages based on variational methods. A comparison of these results from variational methods, some results from diagrammatic perturbation theory, and the results for our quantum Monte Carlo simulation will be made. We have discussed some of the applications of quantum Monte Carlo methods to the study of the thermodynamics properties of crystalline vibrational systems. Both the static and time-dependent (response function)

w

Path-integral quantum Monte Carlo studies

65

thermodynamics have been treated. We have, also, cautioned the reader that the work presented on time-dependent quantum Monte Carlo properties is preliminary in nature and is in fact only in the process of being perfected at the present writing. The static thermodynamic properties presented for the single particle system, the one-dimensional chain and the fcc solid are found to be well converged in N and M and we believe give highly accurate (to within a few percent) representations of the thermodynamic average energy and pressure in the N, M --+ c~ limit. Our confidence in the accuracy of these resuits comes from studies of the rapidity in the convergence with increasing M to the M --+ ~ limit for a number of finite M formulations based on 1/M expansions of the Trotter partition function. The formulation specifically studied by us was chosen so as to optimize the rate of convergence of the quantum Monte Carlo routine to the limiting values of the thermodynamic averages by using expansions in 1/M developed by Takahashi and Imada. Comparisons of these studies were also made with respect to classical thermodynamic solutions and to the results for the harmonic (phonon) approximation, and the simulation results appear to exhibit a reasonable behavior in regards to these two limiting forms. Another possibility of speeding up quantum Monte Carlo routines (aside from those discussed above in the text) with an attendent improvement in the accurate determination of thermodynamic averages is to improve the Metropolis sampling. This sampling method, however, has been used for the last forty years with few really successful improvements, adapted to particularized problems, being found to be of value. It appears most likely that future developments in improving the efficiency of quantum Monte Carlo algorithms will come from advances in the methodology for application of the Trotter formula such as those that we have discussed above rather than from developments relate to classical sampling methods. We have also discussed the time-dependent (response function) properties of our systems and their computation using quantum Monte Carlo methods. The moments of the spectral densities are used to obtain continued fraction representations for the spectral densities of the quantum response functions. While the moments that we presented can be computed with reasonable accuracies by static quantum Monte Carlo methods, the problems involved with obtaining an accurate spectral density from a small number of moments is still in the process of being resolved by us. We expect that this area of time-dependent properties will be one of the primary focuses of future research in quantum Monte Carlo techniques. As a final point in this review, we would like to mention some very recently presented analytical approaches to the evaluation of the path-integral

A.R. McGurn

66

Ch. 1

partition functions of eqs (34), (57) and (94) for the vibrational thermodynamic properties of crystalline solids and give a comparison of the results from these approaches with the results of the Monte Carlo evaluations discussed in w167 3 and 4 above. These recent analytical approaches to the path-integral partition function are all based on obtaining variational approximations to the free energy as derived from the path-integral partition function and represent improvements on some ideas originally proposed by Feynman (1988). We shall give a brief sketch of some of the theoretical points upon which the variational analysis rests (for a detailed discussion of these points see the chapter by Cowley and Horton in this volume) and then discuss the comparison of the simulation and analytical treatments. The path-integral partition functions of the single particle (eq. (34)), onedimensional chain (eqs (57) through (60)) and fcc solid (eqs (93) through (95)) are all of the form (Feynman 1988) Z = Tr e s = e - 0 F ,

(120)

where the trace is over a set of classical position variables and F is the free energy of the quantum mechanical system. Feynman (1988) showed that an approximation to F in eq. (120) could be obtained by using a variational technique. To do this he chose a functional form So defined over the same position variables as S and such that Zo = Tre s~ = e -0F~

(121)

could be evaluated analytically. (So is also taken to involve certain undefined variational parameters.) From eqs (120) and (121) it then follows that

e_~(F_Fo) =

Tre s TreSo

(122)

and Tre s =

Tr[eS-S~ s~

Tr e so

= (eS-S~

(123)

Tr e so

where ( ) represents a statistical average with respect to the e s0 probability distribution. From eqs (120) through (123), we find that e-0(F-F~ =

(eS-S~

(124)

Path-integral quantum Monte Carlo studies

w

67

and for e s0/> 0 over the range of the position space configurations of So e <s-s~ ~< <es - s ~

(125)

so that

e(S-S~ <<e-f~(F-Fo).

(126)

Taking the logarithm of both sides of eq. (126), then gives

F< Fo- 51 ( S - So)

(127)

and by minimizing F o - 51 <S - So)

(128)

with respect to the variational parameters in So an upper limit to the free energy F is obtained. An approximation for F can then be made by taking F to be equal to the upper limit given by eq. (128). A common form taken for So in evaluating eq. (128) is that which gives the so-called first-order self-consistent phonon approximation (Liu et al. 1993). In this approximation So is taken to be of the form of a harmonic oscillator for the single particle model of w or a system of harmonic oscillators for the chain of particles (w3.2) or the fcc solid (w3.3). The masses of the particles in So are taken to be the same as those in the S, but the elastic constants in So are taken as the variational parameters used in determining eqs (127) and (128). The first-order self-consistent phonon approximation turns out to give a poor description of systems, such as Ne, which have large quantum fluctuations (Liu et al. 1993). However, a diagrammatic representation for the first-order self-consistent phonon approximation can be made in terms of the phonon fields and by the inclusions of some additional higher order diagrams an improved approximation known as the improved self-consistent method (ISC) is obtained (Liu et al. 1993). (See the paper by Goldman et al. (1969), for an explanation of the improved self-consistent method in terms of a diagrammatic perturbation theory.) A better approach to obtaining a variational approximation, based on eq. (128), to the free energy of the path-integral partition function is that known as the method of the effective potential formulation (EPMC) (Giachetti and Tognetti 1985, 1986, 1987; Feynman and Kleinert 1986; Giachetti

68

Ch. 1

A.R. McGurn

et al. 1988a, b; Cuccoli et al. 1990, 1992a, b, 1993a; Liu et al. 1991). In this method So is taken to be of the form of an harmonic oscillator with an additional constant potential shift. The path-integral averages in eq. (128) are then performed by summing over all paths (in the Trotter slice numbers) for the atomic displacements as a function of the Trotter slice numbers with fixed given average atomic position values. This is them followed by an integration over the average atomic position values of the various average paths. After this the partition function can be written in the form of a classical partition function for a system of classical particles which interact with one another through effective potentials. A detailed explanation of this effective potential method can be found in the chapter by Cowley and Horton. In fig. 6 we present a comparison of results obtained using the effective potential method (EPMC) for the one-dimensional chain (Cuccoli et al. 1990) with the quantum Monte Carlo results discussed in w3.2. A fairly good agreement is obtained between the two methods and the computer time involved in the evaluation of the effective potential expressions is a very small fraction of the CPU required for the Monte Carlo simulation. This indicates that the effective potential method may be quite useful in the quick determination of the thermodynamic properties of models such as that of the

I.Or.,,.

I

,

C

,

'l'

~'~.

0.8

/

0.4[

.T,

17 0.0

I

..2---L---I

s ~ '~

/

OOL

I

#z

.

I

0.5

.

ee=

I

.

.......................... =

.

I

.

1.0

.

I

I

1.5

I

(b)

I

t

Z.O

Fig. 6. Specific heat C per atom for the infinite Lennard-Jones chain for c= = 0.1. The solid line is the results of the effective potential calculation; the dotted line is results from some other approximation techniques (see Cuccoli et al. (1990) for details); the dashed line is harmonic approximation results; the dashed-dotted line is classical results; the triangles are the quantum Monte Carlo results of MRMW.

w

Path-integral quantum Monte Carlo studies

69

one-dimensional chain. It also strengthens our faith in the accuracy of the data obtained from the Monte Carlo simulations. In fig. 7 we again present some recent results of both quantum Monte Carlo and variation approaches to the study of the average nearest neighbor atomic spacing and energy per atom plotted versus temperature for an FCC nearest neighbor Lennard-Jones solid at zero pressure (Liu et al. 1993). The parameters of the Lennard-Jones potential were given values which are commonly used to model solid Ne 22. Specifically, we have taken m = 21.9914, e = 72.09 • 10 - 1 6 ergs and tr = 2.7012 A. The results in fig. 7 are obtained from the quantum Monte Carlo simulation, analytical results from the improved self-consistent theory and the effective potential method. The quantum Monte Carlo program used was the same one as was used to obtain the results in fig. 4 for the energy and specific heat of solid argon at zero pressure but now modified in the present case to treat a 108 atom cell with periodic boundary conditions rather than the 32 atom cell used in fig. 4. A variety of Trotter numbers from M - 10 to 30 were used in the quantum Monte Carlo and typically 13 to 40 million configurations were sampled. Again we find good agreement between the quantum Monte Carlo, improved self-consistent theory and the effective potential method. This is quite significant as the quantum fluctuations in solid Ne are very important in determining the low temperature thermodynamic properties. In general the improved self-consistent theory agrees best with the quantum Monte Carlo at the low and intermediate temperatures of the figure. The effective potential method (EPMC), on the other hand, seems to work best at the intermediate and higher temperatures in these plots. (The break down in the EPMC near kBT/e = 0 is also evidenced by a comparison of the QMC results in Cuccoli et al. (1992a) for the kinetic energy of solid Ar with the EPMC results in Cuccoli et al. (1993b) for the same properties.) In any case the agreement found between these approaches is quite satisfying and indicates that the current computational accuracies that are achieved by quantum Monte Carlo methods can make valuable contributions to our theoretical understands of the low temperature vibrational properties of solids. The most recent success of the variational approach (EPMC) to the pathintegral formulation has dealt with computations of the time-dependent properties of the one-dimensional chain of atoms (Cuccoli et al. 1992b, 1993a). In these studies the EPMC was used to compute the frequency moments of the spectral densities for the time-dependent response functions. These frequency moments were found to be in good agreement with the moments computed using QMC methods. The EPMC moments, which require less computer time to determine than the QMC moments, were then used to generate a continued fraction representation of the spectral density of the response functions related to inelastic neutron scattering from the chain of

70

A.R. McGurn

(O)

I. 185

1

,~ 1.180

............ i

Ch. 1

~"

-

I

/,,,'1 /," I

O. !.175 (.3

~

I'~

1.170

----.-"

I. 165

oi,

0.0

,

,

o.z

0.3

Temperature

(b)

-3.9

.....

-4.0

"

i

I

, 1 0.4

(~/ke) i ..............

i

/

I

-4.1 0 oz.

-4.2

i~ -4.3

/v

-4.4 "......- - - - ,

0.0

....

O. I

,

0:2

...., ......

0.3

Temperature (,~/ke)

0.4

Fig. 7. a) Nearest neighbor distance in Ne22. Circles are QMC results, dashed line is improved self-consistent approximation and solid line is effective potential method. The square at 0 K is the third-order effective potential result, b) Internal energy per atom in Ne22. Circles are QMC results, dashed line is improved self-consistent approximation and solid line is effective potential method.

w

Path-integral quantum Monte Carlo studies

71

atoms. In the latest study by Cuccoli et al. (1993a, b) for the response functions of the quantum mechanical chain of atoms the EPMC moments were used in conjunction with an improved termination scheme for the continued fraction representation to yield what are thought to be very accurate response functions for the atomic chain system. In this improved termination scheme a continued fraction representation is developed for the classical mechanical chain of atoms using the frequency moments for the classical system. The termination of the continued fraction for the classical mechanical chain is then adjusted so that the continued fraction representation fits molecular dynamics data for the response function of the classical chain of atoms. The response functions of the quantum mechanical chain of atoms are then generated as a continued fraction representation involving the EPMC determined quantum mechanical frequency moments, but the termination of the continued fraction is taken to be the termination obtained from fitting the molecular dynamics data of the classical mechanical chain using the classical mechanical moments. As the quantum fluctuations are expected to have a small effect on the termination function of the quantum chain, the quantum response functions generated by this procedure are expected to be quite accurate. (The last described method of Cuccoli et al. (1993a, b) has also been applied by Cuccoli et al. (1994) to study the quantum response functions of the quantum mechanical chain for the case in which the Lennard-Jones pair potentials are replaced by their expansion to fourth order in the atomic displacements from equilibrium. These results are more easily compared to low temperature perturbation theory results than are the results obtained directly from the use of the Lennard-Jones pair potentials.) These efforts are, however, secondary to our considerations here as the EPMC (not the QMC) is used to generate the solutions. In conclusion: we see that the quantum Monte Carlo can be successfully applied to study the vibrational properties of solids and is accurate enough to distinguish between the various analytical approximation techniques which currently exist. At this writing I feel that the two major problems which most urgently face future work in this field are the determination of methods to improve the convergence of thermal averages with increasing Trotter numbers and the development of a formalism which will yield time-dependent response functions for these systems. With the development of more powerful computers and more efficient quantum Monte Carlo codes we expect that a wide variety of problems in phonon physics (e.g., more complex crystal structures with longer range interactions, amorphous materials, surface waves, systems with impurities and polymer systems) will become accessible to a deeper understanding and, consequently, the determination of and a more accurate theoretical modeling of their properties.

A.R. McGurn

72

Ch. 1

The quantum Monte Carlo method as a general tool to study all manner of solid state phenomena surely offers spectacular possibilities for the quick and accurate computation of the thermodynamic properties of quantum manybody systems. In the years to come great advances in determining accurate models for the description of all types of solid state effects will no doubt be made. Access to these more accurately defined models should in turn provide for a deeper understanding of many-body effects which are now only crudely explained in terms of approximation methodologies and simulations based on classical mechanics approaches.

Acknowledgement Preparation of this manuscript was supported in part by NSF Grant No. DMR 92-13793.

Appendix

The identity in eq. (54) 1--~-einO er

E

n~-

~(:K~

27r

/2 =

E

_

~ e

1 (O+2.n.n)2

~

(A1)

7'1,-"~(:K:}

can be obtained from the Poisson summation formula

1 f eF(z) z - 1 dz,

E n--

(A2)

F(i27rn)= ~ i ~

OK:}

where c is a closed contour encircling the imaginary axis but not enclosing any of the poles of the general function F(z). Let us take zO

F ( z ) = ~1 e ~ e - 2 27r

e

()2 z

(A3)

~i

in eq. (A2) to find e

()2 z

zO

1 einOe_~n2/2 = 1 f e-2 ~ e2~ -n = - ~ 27r (2702 ez - 1

dz.

(A4)

Path-integral quantum Monte Carlo studies

73

Making the change of variables iv = z we have

~-~

1 einO e-en2 /2 =

n = - ~ 27r

1 (2,n-) 2

eV2 i0___v feoe 8~2e2~ dv,

e iv - 1

(A5)

where now co encircles the real axis. To evaluate eq. (A5) we must determine two integrals

I1

1

f~-i6

lim ! 6~o (2702 J - ~ - i 6

C1)2 iOv e 8~2e2~

dv

(A6)

e8,~2e. 2,~ dv. e' ~ - 1

(A7)

e ~v- 1

and

ev_.2.~ i O_..v I2 = lira 1 [-~+i6 6--+o (2702 a~+i6

In I1 we have [eiv[ /> 1 so that we can expand

ei v - 1

=e-iV(l+e -iv+e-i2v+...),

(A8)

and in I2 we find leiv[ ~< 1 so that

ei v - 1

= - ( 1 + e iv -+-ei2v + . . . ) .

(A9)

Using eqs (A8) and (A9) in eqs (A6) and (A7) and the integral identity

1 f~

~V2

(O+27rn)2

i 19

e 8~rZe2~rVeinvdv =

e

(270 2 J _ we find that eq. (A1) follows from eq. (A5).

2e

(A10)

74

A.R. M c G u r n

Ch. 1

References Anderson, J.B. (1975), J. Chem. Phys. 63, 1499. Anderson, J.B. (1976), J. Chem. Phys. 65, 4121. Anderson, J.B. (1980), J. Chem. Phys. 73, 3897. Anderson, EW. (1987), Science 234, 1196. Anderson, EW., G. Baskaran, Z. Zow, J. Wheatley, T. Hsu, B. Shastry, B. Doucot and S.D. Liang (1988), Physica C 153-155, 527. Barma, M. and B. Shastry (1978), Phys. Rev. B 18, 3351. Barnes, T. and A.S. Swanson (1988), Phys. Rev. B 37, 9405. Behre, J., S. Mujashita and H.J. Mikeska (1990), J. Phys. A 23, Ll175. Berne, B.J. (1986), J. Stat. Phys. 43, 911. Binder, K. (1984), Applications of the Monte Carlo Method in Statistical Physics (Springer, Berlin). Binder, K. (1986), Monte Carlo Methods in Statistical Physics (Springer, Berlin). Blankenbeckler, R., D.J. Scalapino and R.L. Sugar (1981), Phys. Rev. D 24, 2278. Boninsegni, M. and E. Manousakis (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena, Ed. by J.D. Doll and J.E. Gubernatis (World Scientific, Singapore). Born, M. and K. Huang (1954), Dynamical Theory of Crystal Lattices (Oxford Univ. Press, New York). Ceperley, D. and B. Alder (1986), Science 231, 555. Cowley, E.R. (1983), Phys. Rev. B 28, 3160. Cuccoli, A., V. Tognetti and R. Vaia (1990), Phys. Rev. B 41, 9588. Cuccoli, A., A. Macchi, M. Neumann, V. Tognetti and R. Vaia (1992a), Phys. Rev. B 45, 2088. Cuccoli, A., V. Tognetti, A.A. Maradudin, A.R. McGurn and R. Vaia (1992b), Phys. Rev. B 46, 8839. Cuccoli, A., V. Tognetti, A.A. Maradudin, A.R. McGurn and R. Vaia (1993a), accepted for publication in Phys. Rev. B. Cuccoli, A., A. Macchi, V. Tognetti and R. Vaia (1993b), Phys. Rev. B 47, 14923. Cuccoli, A., V. Tognetti, A.A. Maradudin, A.R. McGurn and R. Vaia (1994), Phys. Lett. A 196, 285. Cullen, J.J. and D.P. Landau (1983), Phys. Rev. B 27, 297. Day, M.A. and R.J. Hardy (1985), J. Phys. Chem. Solids 46, 487. Deisz, J., M. Jarrel and D.L. Cox (1990), Phys. Rev. B 42, 4869. de Jongh, L.J. and A.R. Miedema (1974), Adv. Phys. 23, 1. de Raedt, H. ~ind A. Lagendijk (1985), Phys. Rep. 127, 234. de Raedt, H. and B. de Raedt (1983), Phys. Rev. A 28, 3575. Doll, J.D., D.L. Freeman and T.L. Beck (1990), Adv. Chem. Phys. 78, 61. Doll, J.D. and J.E. Gubernatis (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena, Los Alamos National Laboratory, Aug. 8-11, 1989 (World Scientific, Singapore). Feynman, R.P. (1988), Statistical Mechanics (Addison-Wesley, Redwood City). Feynman, R.P. and H. Kleinert (1986), Phys. Rev. A 34, 5080. Feynman, R.P. and M. Cohen (1956), Phys. Rev. 102, 1189. Flory, P.J. (1989), Statistical Mechanics of Chain Molecules (Hansen Pub., Oxford Univ. Press). Fradkin, E. (1991), Field Theories of Condensed Matter Systems (Addison-Wesley, Redwood City). Freeman, D.L., R.D. Coalson and J.D. Doll (1986), J. Stat. Phys. 43, 931.

Path-integral q u a n t u m M o n t e Carlo studies

75

Freeman, D.L., T.L. Beck and J.D. Doll (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena, Ed. by J.D. Doll and J.E. Gubernatis (World Scientific, Singapore), p. 58. Fye, R.M. (1986), Phys. Rev. B 33, 6271. Fye, R.M. and D.J. Scalapino (1990), Phys. Rev. Lett. 65, 3177. Giachetti, R. and V. Tognetti (1985), Phys. Rev. Lett. 55, 912. Giachetti, R. and V. Tognetti (1986), Phys. Rev. B 33, 7647. Giachetti, R. and V. Tognetti (1987), Phys. Rev. B 36, 5512. Giachetti, R., V. Tognetti and R. Vaia (1988a), Phys. Rev. A 37, 2165. Giachetti, R., V. Tognetti and R. Vaia (1988b), Phys. Rev. A 38, 1521. Goldman, V.V., G.K. Horton and M.L. Klein (1969), J. Low Temp. Phys. 1, 391. Gross, M., E. Sanchez-Velasco and E. Siggia (1989), Phys. Rev. B 39, 2484. Gubernatis, J.E. (1986), Proc. Conf. on Frontiers of Quantum Monte Carlo, Los Alamos National Laboratory, Sept. 3-6, 1985, J. Stat. Phys. 43, 5-6. Gursey, F. (1950), Proc. Cambridge Philos. Soc. 46, 182. Haldane, ED. (1983a), Phys. Lett. A 93, 464. Haldane, F.D. (1983b), Phys. Rev. Lett. 50, 1153. Hammersley, J.M. and D.C. Handscomb (1965), Monte Carlo Methods (Methven, London). Hirsch, J.E. (1983), Phys. Rev. B 28, 4059. Hirsch, J.E. (1984), Phys. Rev. Lett. 53, 2327. Hirsch, J.E. (1985), Phys. Rev. B 31, 4403. Hirsch, J.E. (1987), Phys. Rev. B 35, 1851. Hirsch, J.E. (1988), Phys. Rev. B 38, 12023. Hoffman, G.G. and L.R. Pratt (1990), in: International Workshop on Quantum Simulation of Condensed Matter Phenomena, Ed. by J.D. Doll and J.E. Gubernatis (World Scientific, Singapore), p. 105. Imada, M. and M. Takahashi (1984), J. Phys. Soc. Jpn 53, 3770. Joannopoulos, J.D. and J.W. Negele (1989), Phys. Rev. B 39, 4435. Kalos, M.H. (1970), Phys. Rev. A 2, 250. Kalos, M.H. (1964), Phys. Rev. 128, 1791. Kalos, M.H. (1967), J. Comput. Phys. 2, 1967. Kalos, M.H. (1984), Monte Carlo Methods in Quantum Problems (Reidel, Dordrecht). Klein, M.L. and W.G. Hoover (1971), Phys. Rev. B 4, 539. Klein, M.L. and J.A. Venables (1976), Rare Gas Solids (Academic Press, New York). Klein, M.L. (1984), Inert Gases: Potentials, Dynamics and Energy Transfer in Doped Crystals (Springer, Berlin). Landau, L.D. and E.M. Lifschitz (1958), Statistical Physics, Transl. by E. Peierls and R.E Peierls (Pergamon Press, London). Levesque, D., J.J. Weis and J.P. Hansen (1984), in: Applications of the Monte Carlo Method in Statistical Physics, Ed. by K. Binder (Springer, Berlin), p. 37. Lindenberg, K. and B.J. West (1990), The Nonequilibrium Statistical Mechanics of Open and Closed Systems (VCH Publishers, New York). Louie, S.G. (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena, Ed. by J.D. Doll and J.E. Gubernatis (World Scientific, Singapore). Loh, E.Y. and J.E. Gubernatis (1990), in: Electronic Phase Transitions, Ed. by W. Hanke and Y.V. Kopaev (Elsevier, New York). Lovesey, S.W. (1971), J. Phys. C 4, 3057. Lovesey, S.W. and R.A. Meserve (1972), Phys. Rev. Lett. 28, 614. Lovesey, S.W. (1986), Condensed Matter Physics: Dynamic Correlations, 2nd edn (Benjamin/Cummings, New York).

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Liu, S., G.K. Horton and E.R. Cowley (1991), Phys. Lett. A 152, 79. Liu, S., G.K. Horton, E.R. Cowley, A.R. McGurn, A.A. Maradudin and R.E Wallis (1993), Phys. Rev. B 45, 9716. Maradudin, A.A. (1969), Lattice Dynamics (Benjamin, New York). Maradudin, A.A., A.R. McGurn, R.E Wallis, M.S. Daw and A.J.C. Ladd (1990), in: Lattice Dynamics and Semiconductor Physics, Ed. by J. Xia et al. (World Scientific, Singapore). Manousakis, E. and R. Salvador (1989), Phys. Rev. B 39, 575. Marcu, M. (1987), in: Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems, Ed. by M. Suzuki (Springer, Berlin), p. 64. Marcu, M. and A. Wiesler (1985), J. Phys. A 18, 2479. Marcu, M., J. MOiler and F.K. Schmatzer (1985a), J. Phys. A 18, 3189. Metropolis, N., A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller (1953), J. Chem. Phys. 21, 1087. McGurn, A.R., P. Ryan, A.A. Maradudin and R.E Wallis (1989), Phys. Rev. B 40, 2407. McGurn, A.R., A.A. Maradudin and R.E Wallis (1991), in: Microscopic Aspects of Non-linearity in Condensed Matter, Ed. by A. Bishop and V. Tognetti (Pergamon, New York). Miyatake, Y., Y. Yamada, K. Nishino, M. Toyonaga and O. Nagai (1986), J. Magn. Magn. Mater. 54--57, 687. Miyashita, S. (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena (World Scientific, Singapore), p. 238. Miyake, J., S. Schmidt-Rink and C. Varma (1986), Phys. Rev. B 34, 6554. Moreo, A., D.J. Scalapino and E. Dagotto (1991), Phys. Rev. B 43, 11442. Morgenstern, I. (1990), Z. Phys. B 80, 7. Mori, H. (1965a), Progr. Theor. Phys. 33, 423. Mori, H. (1965b), Progr. Theor. Phys. 34, 399. Nagai, O., Y. Yamada, K. Nishino and Y. Miyatake (1987), Phys. Rev. B 35, 3425. Nagai, O., Y. Miyatake, Y. Yamada and H.T. Diep (1986), J. Magn. Magn. Mater. 54--57, 681. Negele, J.W. and H. Orland (1988), Quantum Many-Particle Systems (Addison-Wesley, Redwood City), Chapters 7 and 8. Nomura, K. (1989), Phys. Rev. B 40, 2421. Ogata, M. and H. Shiba (1988), J. Phys. Soc. Jpn 57, 3074. Okabe, Y. and M. Kikuchi (1988), J. Phys. Soc. Jpn 57, 4381. Paszkiewicz, T. (1987), Physics of Phonons (Springer, Berlin). Pettifor, D.G. and D.L. Weaire (1985), The Recursion Method and Its Applications (Springer, Berlin). Pollock, R. (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena, Ed. by J.D. Doll and J.E. Gubernatis (World Scientific, Singapore), p. 279. Pollock, E.L. and D.M. Ceperley (1984), Phys. Rev. B 30, 2555. Qiang, H.Q. and M.S. Mikivic (1990), Phys. Rev. Lett. 64, 1449. Rubinstein, R.Y. (1981), Simulation and the Monte Carlo Method (Wiley, New York). Reger, J.D. and A.P. Young (1988), Phys. Rev. B 37, 5978. Samathiyakanit, V. and H.R. Glyde (1973), J. Phys. C 6, 1166. Scalapino, D.J. and R.L. Sugar (1981), Phys. Rev. B 24, 4295. Scalapino, D.J., E. Loh and J.E. Hirsch (1986), Phys. Rev. B 34, 8190. Schmidt, K.E. and D.M. Ceperley (1992), in: The Monte Carlo Method in Condensed Matter Physics, Vol. 71, Ed. by K. Binder, Topics in Applied Physics, (Springer, Berlin). Schrieffer, J.R., X.G. Wen and S.C. Zhang (1988), Phys. Rev. Lett. 60, 944. Schtittler, H.B., D.J. Scalapino and P.M. Grant (1987), Phys. Rev. B 35, 3461.

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Schtittler, H.B., C.X. Chen and A.J. Fedro (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena (World Scientific, Singapore), p. 329. Silver, R.N., D.S. Sivia and J.E. Gubernatis (1990), in: International Workshop on Quantum Simulations of Condensed Matter Phenomena (World Scientific, Singapore), p. 340. Singh, A. and Z. Tesanovic (1990), Phys. Rev. B 41, 614. Sorella, S., S. Baroni, R. Car and M. Parrinello (1989), Euro. Phys. Lett. 8, 663. Squire, D.R., A.C. Holt and W.G. Hoover (1969), Physica 42, 388. Sugiyama, G. and S.E. Kooin (1986), Ann. Phys. 168, 1. Suhm, M.A. and R.O. Watts (1991), Phys. Rep. 204, 294. Suzuki, M. (1976a), Progr. Theor. Phys. 56, 1454. Suzuki, M. (1976b), Commun. Math. Phys. 51, 183. Suzuki, M. (1977), Commun. Math. Phys. 57, 193. Suzuki, M., S. Miyashita and A. Kuroda (1977), Progr. Theor. Phys. 58, 1377. Suzuki, M. (1987), Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems (Springer, Tokyo). Suzuki, M. (1985), Phys. Lett. A 111, 440. Suzuki, M., S. Mijashita and M. Takasu (1987), Phys. Rev. B 35, 3569. Takahashi, M. (1988), Phys. Rev. B 38, 5188. Takahashi, M. and M. Imada (1984a), J. Phys. Soc. Jpn 53, 3871. Takahashi, M. and M. Imada (1984b), J. Phys. Soc. Jpn 53, 963. Takahashi, M. and M. Imada (1984c), J. Phys. Soc. Jpn 53, 3765. Tomita, H. and H. Mashiyama (1972), Progr. Theor. Phys. 48, 1133. Trotter, H.E (1959), Proc. Amer. Math. Soc. 10, 545. White, S.R., R.L. Sugar and R.T. Scalettar (1988), Phys. Rev. B 38, 11695. White, S.R., D.J. Scalpino, R.L. Sugar, E.Y. Lob, J.E. Gubernatis and R.T. Scalettar (1989), Phys. Rev. B 40, 506. Wiesler, A. (1982), Phys. Rev. Lett. A 89. Wood, W.W. (1968), in: Physics of Simple Liquids, Ed. by H.N.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (Wiley, New York). Xia, J. (1990), Lattice Dynamics and Semiconductor Physics (World Scientific, Singapore). Zhang, X.Y., E. Abrahams and G. Kotliar (1991), Phys. Rev. Lett. 66, 1236.

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CHAPTER 2

Lattice Dynamical Applications of Variational Effective Potentials in the Feynman Path-Integral Formulation of Statistical Mechanics E. ROGER COWLEY Department of Physics Camden College of Arts and Sciences Rutgers, the State University Camden, NJ 08102-1205 USA

GEORGE K. HORTON Serin Physics Laboratory Rutgers, the State University Piscataway, NJ 08855-0849 USA

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

9 Elsevier Science B.V., 1995

79

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Contents 1. Introduction 2. Formalism 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. ,

83 85

Path-integral formulation of the partition function 85 Variational procedure and the effective potential 87 The quadratic variational potential 90 The Ginzburg parameter 96 The low coupling approximation 98 Many-particle systems 99 Thermodynamic averages 106

Applications to linear chains

109

4. Three-dimensional Lennard-Jones model

111

5. Improvements in the three-dimensional results 6. Dynamic effects, moments and phonons 7. A model ferroelectric- static effects

120

124

8. The model ferroelectric- soft mode behaviour 9. Conclusions

130

Acknowledgements Note added in proof References

132 133

134

81

119

128

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1. Introduction While there exist several formalisms to calculate the lattice dynamical properties of crystals, no one approach is universally superior. The original harmonic, or quasi-harmonic, formalism is valid only if the amplitudes of the atomic vibrations are small. For most materials this restricts the method to low temperatures. For quantum crystals, the harmonic approximation is never satisfactory because even the zero-point motion of the atoms is too large. Anharmonicity can be included through a perturbation expansion and this extends the temperature range over which the theory can be applied. However, the successive terms in the perturbation expansion become more and more complicated, and in some cases the expansion appears to be only asymptotically convergent. Self-consistent phonon theories were originally devised to deal with the quantum crystals. The harmonic force constants are replaced by their averages calculated self-consistently over a trial set of harmonic oscillator functions. The theory can be shown to be equivalent to a summation of an infinite subset of the perturbation theory terms. The great triumph of the first order self-consistent phonon theory (SC1) is that it provides a physically reasonable starting point for the description of quantum crystals such as helium, but when careful, quantitative, comparisons are made, the first order theory is usually found to be inadequate. Probably the best lattice dynamical formalism is the improved self-consistent theory (ISC), which adds on to the first order theory a non-self-consistent calculation of the first missing perturbation theory term. It has been found to give good agreement with experiment in many applications but is clearly not a complete theory. At some temperature, which is not easily predicted, the omitted terms must become significant. This leaves us with the simulation techniques which have been widely used in recent years. Molecular dynamics and the classical Monte Carlo method each yield values for a wide range of average quantities which are in principal exact if quantum mechanical effects are neglected. As more powerful computers have become widely available, the statistical uncertainties in the averages have been greatly reduced. These theories seem to us to provide the most satisfactory description of the thermal properties of solids at high enough temperatures. When quantum effects must be included, at 83

84

E.R. Cowley and G.K. Horton

Ch. 2

low temperatures in any material and at all temperatures for quantum crystals, the most powerful method is the quantum Monte Carlo formalism (see the article by A.R. McGurn in this volume). Quantum Monte Carlo (QMC) calculations are extremely time consuming with the present state of computer power, and relatively few accurate results for crystals have actually been obtained. We describe in this article a procedure for including quantum mechanical effects by a modification of the potential energy in a classical formalism of statistical mechanics. This seems to us to combine the best of all worlds at the present time, since classical Monte Carlo techniques, which give very accurate results, can now be applied at all temperatures. The effective potential method was first devised by Richard Feynman (Feynman 1972; Feynman and Hibbs 1965) as an illustration of the use of the path-integral form of the partition function in statistical mechanics. In the last eight years the method has grown from this small (though brilliant) beginning into a computational technique which rivals any of the established approximations, such as self-consistent phonon theory, and which can often achieve results comparable with quantum Monte Carlo calculations at a small fraction of the cost. A key step in its development was the improvement of the variational function used, by the inclusion of quadratic terms (Giachetti and Tognetti 1985, 1986; Feynman and Kleinert 1986). The method has been applied to one-dimensional chains interacting with sine-Gordon (Giachetti et al. 1985, 1986, 1988a), ~b4 (Giachetti et al. 1988b, c), and Lennard-Jones (Cuccoli et al. 1990; Zhu et al. 1992) potentials, as well as to a Toda lattice (Cuccoli et al. 1992c) and a ferromagnetic chain (Cuccoli et al. 1992e). It has been used to calculate static correlation functions as well as the thermodynamic quantities (Cuccoli et al. 1991). With some approximation it has been applied to three-dimensional crystals interacting with Lennard-Jones potentials (Liu et al. 1991a, b, 1992; Cuccoli et al. 1992b) and it has been used to describe the phase transition in a model ferroelectric crystal (Cowley and Horton 1992). A generalization of the theory which is applicable to other Hamiltonians, including magnetic systems, has been given by Cuccoli, Tognetti, Verrucchi, and Vaia (1992a). While the method was originally devised to calculate equilibrium thermodynamic averages, it has recently been shown how the technique can be combined with the continued fraction formalism of Mori (1965a, b) to give frequency dependent correlation functions (Cuccoli et al. 1992d). In our own recent work, the approximations which had previously customarily been made in applying the method to three-dimensional systems have been reduced or eliminated (Acocella et al. 1995). We should also point out that there have been a number of papers in which the effective potential method has been applied to single particles moving in

w

Lattice dynamical applications

85

a variety of potentials (Janke and Kleinert 1986; Janke and Cheng 1988; Srivastava and Vishwamittar 1991), as well as a number of attempts to improve the variational principle on which the method is based (Kleinert 1986a, b, c, 1992, 1993). We do not discuss these here because our theme is the application of the method to lattice systems and the new formalism has not yet been applied to such cases. It remains to be seen whether it will represent an improvement over the original Feynman and Kleinert formalism. We would also like to draw attention to the related article on quantum Monte Carlo techniques, by A.R. McGurn, which appears in this volume. In this article, for the benefit of readers unfamiliar with the formalism, we first give an introduction to the path integral method and the development of the effective potential approximation. We have tried to introduce the key ideas one at a time. Further details have been given by Liu (1992). A number of the results given in w2.6 have not, as far as we are aware, been written out before. We then discuss the calculations which have been made of the thermodynamic properties of solids, first in one dimension and then in three dimensions, the use of the method to calculate spectral functions, and its application to a model ferroelectric.

2. Formalism 2.1. Path-integral formulation of the partition function For simplicity we write the equations for a single particle moving in one dimension. We shall leave the generalization to many particles and three dimensions to w2.6, because we find that the physical ideas emerge more easily in the one-dimensional presentation. The partition function is written (Feynman 1972)

Z = e -zF = / p(x, x; ~) dx,

(1)

where p(x, x;/3) is the diagonal element of the density matrix. This is evaluated as a path integral

p(x, x;/3)- /e-S[*(")179x(r). The path integral is a summation of the contributions from all possible paths with the same starting and ending point, x, and S[x(r)] is the action s - - l fm o /~~2h([r ) h

2

+ V(x(v))] dr.

(2)

E.R. Cowley and G.K. Horton

86

Ch. 2

The integration variable -i- has the dimensions of time. The integral over r and the summation over all possible paths incorporate into the partition function the quantum mechanical fluctuations of the particle about its classical position. As the temperature becomes large, the upper limit of the integral in eq. (2) becomes small so that x(v) is almost constant and the potential V(X(T)) can be replaced by its value at the beginning and end point of the path. The partition function then takes the classical form

Zc1--i ~2 ~r h2 f e-oV(z)dx"

(3)

It is not difficult to see that the free energy calculated in this way is an upper bound to the complete value (Feynman 1972). For some of the paths V(x(r)) will surely have lower values somewhere along the path than it has at the end point, giving larger contributions to the density matrix and therefore resulting in a smaller free energy. The path integral can be evaluated at finite temperatures for a number of important special cases. For a single free particle the value of the path integral is

p(x, x; 3 ) =

27r/3h2 9

(4)

When this is used in eq. (1), with limits of integration corresponding to the size of the container, the usual result for the partition function is, of course, obtained. For a harmonic oscillator with angular frequency w, the path integral gives

p(x, x;/3) =

i

7//zo

27rhs-inh(2f)

exp

mw x2 tanh(f)], h

where

f--

13hw 2

The well-known result for the partition function Z =

1

2 sinh(f)

(5)

is easily obtained if this expression for p(x, x; B) is used in eq. (1). The limits of integration in this case are from - o o to +oo.

w

Lattice dynamical applications

87

2.2. Variational procedure and the effective potential One approach now for problems where the path integral cannot be evaluatied analytically is to perform the integration over r numerically, replacing the integration by a summation over a discrete set of values. This is of course the basis of the quantum Monte Carlo method (McGum, this volume). If the numerical procedure can be made sufficiently accurate, this method is in principle exact. In practice it is expensive. To get a more analytic approximation useful at low temperatures, Feynman developed a variational approximation to Z based on the inequality (Peierls 1938)

F <~Fo+-~1 ( S - So)o, where the subscript 0 indicates that the free energy and the average are to be evaluated using a trial action So. In our applications the trial action will take the form So=~

1 fOOh [ rn:i:2(r) + ~(X(T))] dT, 2

and in this case the inequality reduces to the more familiar form F ~< Fo + ( V - V0)0. As we mentioned earlier, there have been some attempts recently to sharpen the Peierls inequality (Kleinert 1992, 1993). It also turns out to be useful to focus attention on the average value of the displacement of the particle over a particular path

1 f0

- -~

X(T) dT

rather than the value of x at the beginning and end point. The calculation of the partition function then involves integrating over all paths with a given average value, and afterwards integrating over the average values. In fact, corresponds closely to the classical meaning of x. In his first application, Feynman (Feynman 1972; also see Feynman and Hibbs 1965) took as the trial action

So =

lfoZh[m2(r) 2

+

dr.

E.R. Cowley and G.K. Horton

88

Ch. 2

That is, the potential along any path is approximated by a constant value which depends only on the average value of the position for that path. The path integral is now that of a free particle, eq. (4). The trial free energy is then related to a trial partition function Zo=e -~F~

m f e -~W(~)d2, 2rr/3h2 J

(6)

V

which is exactly of the classical form, except that the potential is replaced by the variational function W(y:). It is also necessary to evaluate the correction term

! <s- So>o, which in this case reduces to ( V - W(2))o.

(7)

The averages involve a path integral over all paths with a given 2, followed by an integration over 2. However, the variational condition to find the optimal value of the parameter W(2) is applied before the final integration over 2 is performed. After the path integration the correction term is found to be

f (K(2) - W(2)) e -~W(e) d2 (v-

=

f e -~W(e) d2 where K(y:) is the potential V(x) averaged over a Gaussian smearing function

K(2)=

[

~6[3:2JV(x)ex p -

6m(2 - x)2 ]

~ h2

dx.

It is now clear that the choice of the variational function W(2) which minimizes the trial free energy is W(~) = K(2)

{}2

89

Lattice d y n a m i c a l a p p l i c a t i o n s

1.o 0.8 .c

tTI k-

0.6

0.4

c

ok_ la_

0.2

-

/

/ f

0.0-0.2

' 0.2

0.0

'

0.4

' 0.6

' 0.8

Temperature (heo/k)

1.0

Fig. 1. Free energy of a harmonic oscillator. Solid line is the exact result, dashed line is the classical approximation, and dots are values from the original Feynman effective potential. and that with this choice the correction term ( V - W(~))0 vanishes. The best approximation to the partition function is thus given by eq. (6) and the variational function W(y:) plays the role of an effective potential. Both the smeared function K(:~) and the trial partition function can be calculated analytically for the case of a harmonic oscillator. In that case we have V ( z ) _- _1 mW2X 2,

2 K ( i ) _ _1 row2 ( ~2 q_ 12m

2

1 Zo = /3hw exp

[

'

/~2h2602

-

24

1 [ /~2h2w2 ] Fo = ~ ln(flhw)+ 24 "

(8)

This last equation is to be compared with the exact expression for a harmonic

Ch. 2

E.R. Cowley and G.K. Horton

90 oscillator

F = _1 ln{2 sinh(f)},

(9)

/3hw

f--

2

which is obtained from eq. (5). In fig. 1 we plot the free energy values given by eqs (8) and (9), together with the high temperature value, which is just the first term on the fight side of eq. (8). The effective potential procedure succeeds in including most of the effects of the quantum fluctuations down to a temperature of about one quarter of hw/k. We emphasize that this result comes from the free particle variational action. The improved variational function discussed next would give the exact result for the harmonic oscillator.

2.3. The quadratic variational potential An improved version of the effective potential was introduced by Giachetti and Tognetti (1985). They extended the variational function to include terms quadratic in the fluctuations of the particles around the average values on the path. Even in their original work, Giachetti and Tognetti applied the method to a linear chain of particles, moving in a sine-Gordon potential. For simplicity we develop the formalism first for a single particle. The trial action is therefore written

s0=

1

1 fo~n [ mz(r) 2 + W(~) + ~ r

]

- ~)2 dr.

The additional variational function is a force constant, r which is a function only of the average position along the path. We have not included a term linear in the fluctuation. Such a term can readily be included but it drops out from the calculation. With this more complicated function the path integrals can still be evaluated, using results given by Feynman (1972). The trial partition function is now found to be

Z0 =

~/

m f f -~w(~) 27rflh2 sinh(f) e dy:,

w

Lattice dynamical applications

where f hence f into the the trial

91

--/3hw/2 as before, but now with W 2 - " r Notice that r and are functions of Y:. It is convenient to move all of the integrand exponent to become parts of an effective potential, so we rewrite partition function as

fexp{

m

Zo-

[ W ( 2 ) + ~1 ln(sinh(f))- 1 ln(f)] } d2.

-/~

The limits of integration in all the integrals over ~ are from - o e to ~ . The correction term, eq. (7), in the trial free energy now takes the form

(V-

l/b)o = ~oo

27rfTh2

x exp {

-

K ( 2 ) - W(2)

maw22]

fl [W(:~)+ ~1 ln(sinh(f))- 1 ln(f)]}d~.

Here K(2) is a new type of smeared average potential

K(5:)=

('i ~

V ( 2 + y ) exp

{ "'} -~

dy

(10)

with c~ given as cz --

h{ 2mw

coth(f) -

i}

.

(11 )

The quantity c~ is the difference between the quantum mechanical and classical expectations of the squared displacement for a harmonic oscillator, or for any particle evaluated in a Gaussian approximation. Giachetti and Tognetti refer to it as the "quantum renormalization parameter". Note that in the early papers the definition of c~ was larger by a factor of two. We have adopted what seems to be the most recent notation. The value of W(2) which minimizes the trial free energy is again the value which causes the correction to be zero. It is

W(~c) = K(2)

mO~60 2

E.R. Cowley and G.K. Horton

92

Ch. 2

and with this value the trial partition function once more takes the classical form

Zo=e-~F~

m/e-~288

(12)

but with an effective potential 1 1 : ln(sinh(f))- -= l n ( f ) -

Veff(~) = K ( s

mc~w 2

(13)

It still remains to determine the value of the second variational function, r It is easily shown that the value of r which minimizes the trial free energy is

r

-- < d2----~Y> dx 2

1 -

fd2V ~+y exp{ y2}dy dx---S

so that r is a smeared force constant. Since a is itself a function of w and hence of r eqs (11) and (14) must be solved self-consistently before the free energy can be evaluated. The equations (13) and (14) for the effective potential Veff and the variational force constant r bear a strong resemblance to the results of the first-order self-consistent phonon theory (SC1), (Choquard 1967; Gillis et al. 1968; Klein and Horton 1972), which is also sometimes called the meanfield approximation when applied to a single particle. We shall use the name SC1 for continuity with later sections. The SC1 results for the free energy and force constant are

FSCl = (V)scl =

1

2

( i

dx,

1

27rc~'

,

a =

moJa) t2

(V)scl + ~ ln(2 sinh(f')) -

h

2mw'

V(x)

exp

{

-

2a'

coth(f'),

f' = ~hw'/2, = Vr

1 f 27ra'

d2V dx 2

exp{

x2 }dx, 2a'

w

Lattice dynamical applications

93

where the primes emphasize that these quantities will generally differ from the corresponding terms in the effective potential theory. In SC1, the width c~' of the smearing function is the complete expectation value of z 2, whereas, in the effective potential smearing, eq. (11), only the difference between the quantum mechanical and classical values is used. Similarly, the effective potential, eq. (13), contains the difference between the logarithmic term in Fscl and its classical value. At zero degrees the classical contributions disappear. Also, at zero degrees, the value of the free energy given by the effective potential theory is just the value of Veff at the configuration where it is a minimum since classically the fluctuations of the particle about this configuration disappear. The two theories thus give identical results in this limit. However, the zero degree limit is usually the most favorable case for the SC1 theory. At any finite temperature, the mean square displacement is greater and the omitted terms in the cumulant expansion of the free energy become important. In particular, there are no contributions included, either in the free energy or in the smeared potential, which arise from odd derivatives of the potential. At high temperatures, SC1 often gives disappointing results. The effective potential theory, on the other hand, has the correct classical limit built into it, and, as Giachetti and Tognetti (1985) have shown, it also contains correctly the first term in the Wigner expansion for the quantum mechanical corrections. It can only improve as the temperature increases. The concepts of smeared energy and force constants have been used in self-consistent phonon theory for many years. Most often, the smeared quantities have been calculated by numerical integration but series expansion has also been used. The question of the existence of the integrals when a hardcored interatomic potential such as the Lennard-Jones form is used, and of the possibility that the expansion of the smeared averages is only asymptotically convergent, have been discussed (Choquard 1967; Homer 1974). Homer's ansatz modifies the Gaussian pair distribution function in such a way as to remove the singularities in the integrands while preserving certain key sum rules. We have shown that this gives excellent results in the case of Ar (Cowley and Horton 1987). As far as we are aware, these difficulties have not yet been addressed in the very similar application to the effective potential, except that the series expansion has been restricted to only a rather low order. As an example of the use of the quadratic variational function, Giachetti and Tognetti (1986) applied the method to the case of a single oscillator moving in a potential which contains both quadratic and quartic terms. Feynman and Kleinert (1986), who arrived independently at the same theory, used a similar example. For flexibility in later applications we shall write this in a

94

E.R. Cowley a n d G.K. Horton

Ch. 2

general form, as

1

1

2

24

v ( z ) -- + - a x 2 +

(15)

b x 4.

If the negative sign is used in the quadratic term, the potential is a double well, while, if the positive sign is used, it is a single, anharmonic well. For each value of 9 the trial force constant is found from the set of equations

r

= =k=a+

(y~ + y)2 exp

-

dy

22v2-

1

-- -k-a -k- ~ b(:2 2 -k- ce),

Ot =

2 }.

h {coth ~hw 2row 2

/3hw

These can be solved with a simple iterative procedure. Feynman and Kleinert show that while r may sometimes be negative in the case where the quadratic term is negative, there is always a solution such that a is positive. This is another reminder that w is not the complete vibration frequency and that the phonon-like excitations we shall meet later do not have the same frequencies as phonons measured in neutron scattering experiments. The partition function and free energy are then calculated as a numerical quadrature over the Boltzmann factor of the effective potential

Zoe-' o mf' " =

=

e-

d~,

2~'/3h2

Veff = + --1 2

1

a('~, 2 + Or) -4r

b (~74 + 6y:2a + 3a 2)

1

+ ~ ln(sinh(f))- ~ ln(f)

Tr~t~6o2

w

Lattice dynamical applications

2.0

....... , .... 0.2-"'TIS; "'''. f

1.01.5

.5 t~

, ..... ~...... ,,,,

0.0

.

0.4

0.0

-0.5

0"6Ill

- 1.0

0.81.0I 0

1

2

3

4

Reduced

5

0.0

/

-

tD)

0.2

0.5

-1.5

95

9 ",,

f--~

-

v

:

" ' ' ~

, 0.5

\\\i t

.....

1.0

I1.

,

1.5

2.0

Temperature

Fig. 2. Free energy of quartic oscillators with (a) positive quadratic term, and (b) negative quadratic term. Solid lines are the exact results, dashed lines are the classical approximations, dots are the quadratic effective potential results, and triangles are the SC1 values.

We have evaluated these formulas and compared them with the SC1 results for two cases, one with a positive quadratic term in the potential and one with a negative term. It is possible to scale lengths and energies so that only one independent parameter remains in the model. We have taken this to be A-

hb

.

(16)

W/18m{a3{ Either a low mass, corresponding to large quantum fluctuations, or a large value of b, corresponding to large anharmonicity, results in a large value of A. We have taken the value of A to be 1. This is roughly the value corresponding to a one-dimensional version of Ne. It is also possible to calculate exact results for this potential, by expanding the eigenfunctions of the Hamiltonian in terms of a large number of harmonic oscillator wave functions, diagonalizing the resulting matrix, and summing directly the contributions to the partition function. Figure 2(a) shows the results for the free energy for the case where the quadratic term in the potential is positive. The solid line is the exact result, the solid circles are the effective potential

E.R. Cowley and G.K. Horton

96

Ch. 2

results, and the triangles are the SC1 results. In this case, the effective potential results are indistinguisable from the exact values on the scale on which the graph is plotted, and the SC1 results are also very similar. The classical result, obtained by numerical integration of the classical partition function, is shown by the dashed line. The quantum mechanical effects cause substantial corrections which have still not disappeared at the highest temperature plotted. Figure 2(b) shows the results for the case where the quadratic term in the potential is negative, for the same value of A. This model can be viewed, qualitatively, as a one-dimensional model of solid He. At zero degrees the effective potential and SC1 results, which are identical, are too high. The true ground state wave function is double peaked and is poorly represented by a single Gaussian. The SC1 free energy remains well above the exact value over the entire temperature range. However, the effective potential values quickly move into close agreement with the exact results as the temperature increases. This simple model is the starting point for the ~b4 model discussed in w3 and the ferroelectric model discussed in w6.

2.4. The Ginzburg parameter While a numerical solution to the equations can probably always be developed in the case of a single oscillator, it is helpful to look at other approaches which will become more necessary when we consider systems of coupled particles. In the calculation of the smeared potential energy K(~), we can expand the potential about the average value (3O

V(Yc + z)= E ---1 v(n)zn n=0 n !

where V ('~) is the nth derivative of V, evaluated at 2. When the Gaussian averaging is performed, the odd terms disappear and we find

oo,

(o).

(17)

n--O

A similar expansion can be made for the smeared force constant, r giving 1

r

=

n-1

V(2,~)(a ) ( n - 1)' 2

n-1

(18)

w

Lattice dynamical applications

97

These two expansions can now be inserted in the expression, eq. (13), for the effective potential, with the result ~

(n-l)

Veff(~) = V ( ~ ) + ~ l n ( s i n h ( f ) ) - ~ I n ( f ) - Z

1

1

nv

n

n=2

It is noteworthy that the term corresponding to n = 1 has cancelled from the two expansions. A similar cancellation occurs in SC1 theory. It is also important that 4~, which is equivalent to mw 2, appears in the effective potential multiplied by an additional factor of a (see eq. (13)). In most of the early applications of the method, this expansion has been used, truncated to a low order. In a first-order theory, Veff is calculated to first-order in a, which requires that ~b should be calculated to zero-order. In a second-order theory, Veff is needed to second-order, so 4~ must be calculated to first-order, and so on. The first-order theory is particularly appealing, since the variational force constant 4~ is set equal to the actual second derivative V ~2~ so that the iterative solution is not necessary. If we write out the beginning of the expansion for 4~ ~(ffT)- V (2) -t- V (4) o~ -+- 9 9 9

2 -

V (2)

1+

V(4)~

+---

/

.

2 V (2)

The magnitude of the second term in the bracket, which is called the Ginzburg parameter, clearly determines the suitability of the expansion process. We shall see later that, in realistic applications, the equivalent quantity is indeed small for all of the heavier inert-gas crystals, having the value 0.24 for Ar at zero degrees, much smaller values at all other temperatures and for all the heavier elements. However, for Ne, it has the value 0.4 at the melting temperature and 0.88 at zero degrees. The convergence of the expansion is very questionable in this case, particularly at low temperatures. For He the second derivative of the bare potential is negative and even calculated with the smeared values the Ginzburg parameter is greater than unity. In such a case, the expansion of the effective potential is useless, but there is no reason that the full smeared expressions would not be valid, provided that the integrals exist for the particular potential used. We believe that a theory for He is fully feasible and we have made some preparations to address this very interesting problem.

E.R. Cowley and G.K. Horton

98

Ch. 2

2.5. The low coupling approximation In the following section we shall consider the application of the formalism to systems of N particles. In that case, even the first-order theory involves the calculation of the normal mode frequencies of the system, which requires the diagonalization of an N x N matrix at an arbitrary configuration. On the other hand, for a crystal the static equilibrium configuration is a periodic one, so that the normal modes can be labelled by wave vectors, and the dynamical matrix largely block-diagonalized. It is therefore tempting to look at the possibility of expanding the solution to the variational equation about this configuration. The low coupling approximation (LCA) assumes that one can neglect the change in the quantum renormalization parameter, a, as the atoms move from their equilibrium positions. For a material which shows only weak quantum effects this should be a very good approximation. At low temperatures, the atoms perform small amplitude oscillations around the equilibrium positions so that the assumption of a constant a is close to correct. At high temperatures, the atoms make larger excursions but the quantum correction in the effective potential is small anyway. Remember that the true potential appears exactly in Veff. It is only the quantum corrections which are being approximated and these go to zero at high temperature. It is possible to use the LCA without making any other approximations. The self-consistency eqs (11) and (14), for the smeared force constants, can be solved iteratively at the equilibrium configuration and the value of so obtained used in the calculation of the smeared potential, eq. (10), for other configurations. In practice, the LCA has been combined with the low-order series expansion of the smeared integrals. Giachetti, Cuccoli and co-workers (Giachetti et al. 1988b; Cuccoli et al. 1990) have shown that the two approximations can be combined to give a very convenient expression for the effective potential. We recognize first that the change in the force constant q~ as the atoms vibrate is of the same order as the change due to renormalization, so that even in a second-order theory for Veff we need consider only first order changes in 4~ and hence first order changes in the squared frequency. If the two logarithmic terms in eq. (13) for the effective potential are expanded, the term first order in the change in the force constant is found to be

1

+ ~ a0~6, where a0 is the value of a at the equilibrium position (z = 0 for the single particle model we are using). This combines with the last term in the

Lattice dynamical applications

w

99

expression for Veff in eq. (13) to give

1 ao5r 2

-

-

1 2

-

am~

2 ~

1 2

-

ao~r

1

ao(r

+ ~r

CeO~O. Also, in the two expansions for K(5:) and r eqs (17) and (18), we replace c~ by c~0. The result is 1

1

Veff(5:) = V ( ~ ) + -= ln(sinh(fo))- -= ln(fo) /J /J oo

1

c~0 n

(19)

+ ~ ~.I {y(2n)(37)- nV(2n'(o)}(-2 ) n=l

Here f0 is the value of f at the equilibrium position. There is a small error in the corresponding equation, eq. (12), of Cuccoli et al. (1990). 2.6. M a n y - p a r t i c l e s y s t e m s

We now consider the extensions to the formalism required to apply it to systems of N particles. In fact the form of the equations remains remarkably similar, though for the most general case, where the form of the normal modes is not fixed by symmetry and where the LCA is not made, the numerical work involved in their implementation is formidable. For a system of N atoms moving in d dimensions, the trial action becomes

So - - -

miuio ZO~

~

1

] dT,

io,,j~

where ~' represents the average positions of the whole set of atoms, and u~,~(r) is the instantaneous displacement of atom i in the c~ direction, measured from its average position ui~(r)

=

ricO-)

-

~.

E.R. Cowley and G.K. Horton

100

Ch. 2

We have included the possibility that the masses of the atoms are not all equal. This would allow the application of the formalism to a crystal with several atoms in the unit cell. The masses are easily removed from the notation by a transformation Via Uia --"

which leaves the action in the form

=

-;

vi~

ia

Via(T) riO(T)] dr. ia,j~

The trial potential energy is now diagonalized by an orthogonal transformation dN

via = ~

Uia,s~Ts,

s---1

where I

ia,j~ V/mimj

Uia , r U j ~ , s

"-- ~ O s2 ~ r s

9

The kinetic energy and quadratic terms in the trial energy are then

'%(~) + ~ which is a sum over independent terms of the earlier type. The trial partition function can now be evaluated by the same methods used previously. It is

) 9

d/2

2~-f~h 2

x/exp{-~[W(#)+~~{ln(sinh(L)-ln(L)}l}d~, fs =

2

(20)

w

Lattice dynamical applications

101

This exponential also contains a sum of contributions from the normal modes of the trial potential, labelled by the index s. The prefactor has to be written as a product to allow for the possibility of the masses not all being equal. The value for W(~') which minimizes the free energy is 1 -

-

"

W(~') = K(~') - -~ E ~

2

8

where c~s =

coth(fs)

2w~

(21)

]7

and

K(#) = H 8

(')/ v/27rc~;

V (# + g(rT)) exp

{ r -

~

}

dr~

(22)

Here 77 represents the set of dN normal coordinates and ~707-3is the set of atomic displacements related by the transformations which have been made 1

Yi~

The smeared average of any other function of the atomic positions can be expressed in a similar way, including the smeared force constants

r

= II ( 8

v/2rroe,

aui,~uj~ #+y

exp { - ~

~

} drT,

(23)

which minimize the free energy. Suppose that the potential energy contains a term v(r~) which depends on a single coordinate. If this is expanded as a power series, its smeared value is

n

n!

where ui~ is the deviation of the coordinate from its average value ei~ and the derivatives v ('~) are evaluated at r--/~. The odd powers of u ~ average to

102

E.R. Cowley and G.K. Horton

Ch. 2

zero, and the even powers follow the usual results for Gaussian averaging (Choquard 1967). The result is __1 v(2n)(Di~)'~ n! -'~

(v(ri~)) = ~ n

where we have defined

Dia = m----~ZU2a,sOls. 1 8

Di~ is the purely quantum mechanical contribution (that is, the quantum mechanical value minus the classical value) of the mean square displacement of the atom i in the c~ direction. Exactly the same expansion is obtained if the smeared value is written as (v(ri~)) = v/2rDi a , v(~ia + y) exp

2Di~

dy.

(24)

We can thus picture the smearing of the potential energy term equally well in rl-space and in u-space, but the u-space picture involves only a singledimensioned integral. A similar result is obtained if a term in the potential is a function of the difference of two atomic coordinates. This would include the case of a pair potential in one-dimension. The contribution to the potential energy Can be expanded as a power series in the deviation of its argument from the average value and integrated term by term. The result is

1 v(2n)(Dija)n n

where Dija

~

v/_~_7

n

This is again equivalent to an integral in real space (v(ri.

- rj.))

1

f

X/'27rnij,~

J

=

y2 •

2D~j,~ } dy.

(25)

w

Lattice dynamical applications

103

The important case of a pair potential in three-dimensions can be handled with only slight extra complication. If the potential energy term is a function of the difference in positions of two atoms and g is the vector change in their separation rij

=

ri

-

rj

-=r'i-ry+ff

=~'ij +g, then the function can be expanded and averaged term by term ~nv

1 ~

8ua ...Su~.

Uot I " " " Uotn ) 9

The derivatives are evaluated at f'ij. The averages of the products of odd numbers of u's vanish. Possible approximations are to assume that the different components of g are uncorrelated or that one can project out longitudinal and transverse terms. These are not generally exact however. It is really necessary to define a tensor correlation function

I Uiot,$

(26)

In terms of D~3 the first few terms in the expansion of v are -

1

3

"+"~.I Z

(2) D . .

.(4) D ij,o~f~ Dij-y~ +'-" ,

w o~f~5

The general term in the expansion is cumbersome to write out but the terms written here give v correct to second order in D~j. The corresponding expansion for the force constants need be carried out only to first order. To write this as a smearing integral in real space, it is helpful to introduce the orthogonal matrix R which diagonalizes D~j

xy

E.R. Cowley and G.K. Horton

104

Ch. 2

where the ,~x are the eigenvalues of D~3. The smeared potential can now be written

(v(~'ij))

=

1 / v(/ij

v/87r31DI

+ Y(O) exp

{-

~

~2}d3{,

~

(27)

where (x are the three integration variables along the principal axes of the D~j tensor and

R~(~. X

Except for the (important) change in meaning of the a, to remove the classical contributions, eq. (21), this is identical in form with the smearing usually used in self-consistent phonon theory. In the LCA the various correlation functions defined in the previous paragraph are evaluated at the equilibrium positions of the crystal. It is helpful in this case to make contact with the formalism of lattice dynamics (Born and Huang 1954), since many of the transformations are close to standard forms. We consider a crystal with N unit cells (notice the change in definition) each containing s atoms. The unit cells are labelled by an index I and the atoms in each cell are labelled by an index k. The equilibrium position of atom lk is R(Ik). Because the equilibrium configuration has translational periodicity, the normal modes are waves which can be labelled by a wave vector q', distributed through the first Brillouin zone, and a polarization index j. It is common notation to diagonalize the potential energy by means of a unitary transformation

u,~(lk) = v/Nmk ~.. e,~(k, ~j)e'r qJ

where Q((j) is a complex normal coordinate and e~(k, qj) is a component of a normalized sd-component eigenvector for the q'J mode. The reality condition on the atomic displacements imposes the condition

Q ( - ( j ) = Q* (qj).

(28)

To recover a set of real normal coordinates we can write

1 Q(gj) - - ~

[T]l(q'j)+

ir/2(~'j)]

w

Lattice dynamical applications

105

for vectors in one half of the Brillouin zone, and then the coordinates for the other half of the zone are given by eq. (28). The elements of the orthogonal matrix U are then given by u.qk, r

ea(k, ~j)(ei~'h(lk) + e-iq-"R(lk))

=

v/2Nmk for the/71 coordinates and

i

Uo,(lk, ~j) = v/2Nmk

e,~(k, ~j)(e i~'~(lk) -- e -iq-'h(lk))

for the ~72 coordinates. The sum over wave vectors is taken over one half of the Brillouin zone and there are two normal coordinates with the same value of w~ for each branch index. These expressions can be substituted into the formulas for the various correlation functions and the sums over the two types of coordinate performed. For the three-dimensional case, eq. (26), the result is

Do,~(lk, l'k')

2 {1

e~(k, Cj) e~(k, 0"J) + ~

1

e~(k', 0"j) e~(k', Cj)

q3 1

V/ink ink,

{ ~(k, r

x cos [q'" (ff~(lk)-/~(l'k'))]

r + ~(k, r

r

}aCj.

The prime on the sum indicates that it is restricted to one half of the Brillouin zone. This equation is easily iterated with eq. (21) to give a self-consistent solution at the equilibrium configuration. If we assume that the potential consists entirely of a sum of pair terms

v - ! ~ ~(r 2 lk,l' k'

r

then the series-expansion LCA combination, eq. (19), can be generalized and the effective potential written as a constant term plus a sum of effective

106

E.R. Cowley and G.K. Horton

Ch. 2

pair interactions Veff = ~... q3

12]

1 ~ { ln(sinh(fo"~)) - ln(fo'~)} - -~ a Cj w 4j

+

v ff(elk, , k'), lk,l'k'

-

-

1

=

vo,~Do,~(lk, l'k' )

3

-t- ~.. E - ( 4"O)ott37 61-1ot13 ra elk ~, , l'k')D,y~(lk, l'k') + . . . The potential derivatives are evaluated at the instantaneous positions but all of the terms depending on the normal mode frequencies, w~j, a~.j, fcj, and D a # ( l k , l'k'), are evaluated at the equilibrium configuration. The frequencies are calculated self-consistently from the smeared force constants at the equilibrium configuration, which can be calculated as integrals, similar in form to eq. (27), or can be calculated as series expansions

1 Ca13(lk, ltk t) = V(2ot)~(lk, l 'kt) --Jr-~ E vo~fl"/6(4) (Ik, l ' k ' ) D ~ ( l k , l'k') + . . . .,/6 to first order in D. In the LCA, these need to be evaluated only at the equilibrium configuration. This should be an adequate approximation for a majority of applications. While the effective potential has been written as a sum of contribution from pairs of atoms, the individual contributions cannot in the general case be written in terms of an effective potential which is a function only of the distance between the two atoms. As far as we are aware, no LCA calculation of this generality has yet been carried out but we do not see any reason why it should not be done. While we have considered only pair potentials between neighboring atoms, the theory can be certainly applied in principle to other potentials, including three-body forces and long range (e.g. Coulomb) potentials.

2.7. Thermodynamic averages So far we have developed an approximation for calculating the partition function and hence for the free energy - k T In Z. Because the expression for

w

Lattice dynamicalapplications

107

the partition function takes the classical form, we can use any of the methods which have been applied classically. In particular, we can evaluate averages weighted by the Boltzmann factor of the effective potential by means of a Monte Carlo procedure. This is referred to as the effective-potential Monte Carlo method (EPMC). However, in setting up the averages, we must allow for the fact that the effective potential depends on the temperature and on the volume of the sample. The internal energy, for example, must be calculated as a derivative of the partition function 0 In(Z) E

__

__

O#

dNkT< 2

+

Veff+ #

aVeff )

a#

where d is the number of dimensions, 1 or 3 in the applications discussed here. Similarly for the heat capacity

Cv = k fl 2

Z

i~fl 2

k/32 ~2Z Z lct~2 ~2 Z Z

0/~2

~~

k ~ 2 E 2,

O~ 2

dNk 2 -'F -2

Z

OVeff a#

~-/3

Veff + fl Off ~2Veff ~ 2 >"

In a first-order theory, the frequencies w~ are independent of temperature and the only temperature dependence in Veff is the explicit dependence in Veff itself and in the a~. In a second order theory, the temperature dependence of the frequencies must also be included. If the LCA is used this is not difficult. The effective potential is calculated as a preliminary step before a simulation is performed and the temperature dependent terms can be calculated at that time. Notice that many of the contributions to Veff are in fact independent of the configuration in LCA. In a calculation which does not make some type of

E.R. Cowley and G.K. Horton

108

Ch. 2

LCA or expansion approximation, it will be much more difficult to calculate the heat capacity directly. Of course, the heat capacity can also be obtained by numerical differentiation of the internal energy (Cuccoli et al. 1992b). Two features of the formulas given above are worth comment. In a classical calculation the heat capacity approaches the value dNk at low temperatures whereas in a quantum mechanical model it goes to zero as T 3. The expression for the heat capacity includes a contribution of dNk/2 from the prefactor in the partition function, arising from the integration over the momentum coordinates. The integration over the position coordinates, which might be performed as a simulation, gives the remaining factor of dNk/2. These two contributions need to be cancelled in the effective potential result and this is achieved by the extra contributions to Cv above. In particular the -(1//3)In(f) term in the effective potential, eq. (19), gives an exact contribution of - d N k in the heat capacity. In an EPMC calculation the cancellations will not be exact. The configuration integral has a finite value and a correspondingly large statistical uncertainty, which propagates through the calculation. In consequence, the overall value for the heat capacity at low temperatures has a much larger relative uncertainty than might have been expected. An accurate calculation therefore requires much longer simulation runs than if only the energy and pressure are calculated. A redeeming feature is that the remaining logarithmic term in the effective potential, (1//3) ln(sinh(f)), when summed over all the modes, gives rise to the proper quantum mechanical temperature dependence of the heat capacity. Even if the EPMC calculation is being carried out with quite a small number of atoms (typically 108), the sum of the normal mode contributions over the Brillouin zone can be carried out for a much larger sample, and the T 3 behaviour properly obtained. There are also extra contributions associated with the volume derivatives. The pressure is usually calculated in Monte Carlo simulations from the virial theorem formula p

where ~b is the potential energy and V is the volume. The volume dependence of the effective potential again causes extra terms to be present in the expression. When the series-expansion LCA is used, these can be calculated in a straightforward fashion. Some of the additional terms are evaluated only at the equilibrium configuration and the rest form a series expansion whose coefficients are found as part of the self-consistent solution. However, to date no calculation has included the bulk modulus, or other elastic properties, which would involve second volume derivatives. We see no reason why such a study in the effective potential formalism is not feasible.

w3

Lattice dynamical applications

109

When averages of other quantities are calculated in a simulation, the pathintegral smearing can also generate additional contributions. Consider, as a rather obvious example, the mean square displacement of an atom. For simplicity we consider the case of a single particle. Then the value of X(T) averaged over the quantum mechanical fluctuations around 57 is

X(T) e-S[~(~)lDX(T) = ~/ ~ l f

(57 +

y)2 e_y2/2c~dy

= 572 + Ct(57). In a simulation, the additional term c~(57)must also be included and averaged over 57. Cuccoli et al. (1992a) have derived a very general formalism for calculating such averages.

3. Applications to linear chains As is often the case, there are substantial mathematical simplifications which can be made when one-dimensional models are used. If the interaction is restricted to nearest neighbors, the partition function can be evaluated by a transfer matrix technique (Takahashi 1942; Gursey 1950). While the final integration must often be carried out numerically, this is a much less demanding calculation than, for example, a Monte Carlo evaluation of averages. In addition, the smearing integrals can be evaluated analytically in many cases. Of course, these latter methods are dead ends, in the sense that they cannot be extended to three dimensions. However, the results obtained point the way to what is physically important and to which approximations work. Giachetti and co-workers (1985, 1986, 1988a, b, c) have studied a number of Hamiltonians in which the interaction between adjacent atoms is harmonic, all the anharmonicity being in the single site terms, so that the potential energy is V__

7m Z x S jxj + 9 Z z3

i

These include the sine-Gordon model and the q~4model. The classical statistical mechanics of these models has been discussed by Schneider and

110

E.R. Cowley and G.K. Horton

Ch. 2

Stoll (1980). At low temperatures the thermodynamic properties are determined by the phonons associated with the minima in the potential, with anharmonic renormalization, but at intermediate temperatures there are additional contributions ascribed to solitary wave or soliton solutions to the equations of motion. Since the inter-site terms are harmonic, they are unchanged by the smearing process and the single-site smeared contributions to the potential are all of the form of eq. (24), which can be evaluated exactly for either of the forms used. In the LCA the U matrix is just the transformation matrix to running waves which diagonalizes the B matrix, and is independent of temperature. At low temperatures the renormalized phonon picture is obtained, atthe level of approximation of SC1. Numerical solutions at higher temperatures show results qualitatively similar to those found by other methods (see Schneider and Stoll 1980), but which contain both quantum mechanical corrections and soliton-soliton interaction effects. As an example of a one-dimensional system interacting with a pair potential, Cuccoli, Tognetti and Vaia (1990) studied a model with a Lennard-Jones potential between nearest neighbor atoms. The nearest neighbor model has the substantial advantage that the partition function can be calculated easily, to high numerical accuracy, using a transfer-matrix method (Takahashi 1942; Gursey 1950). In order to avoid questions concerning the stability of the one-dimensional lattice, Cuccoli et al. constrained the lattice spacing to the value which minimized the potential energy. They gave results for the internal energy and specific heat for two choices of parameters, corresponding to one-dimensional versions of Ar and Ne. They used the LCA approximation carried out to first and second orders. For the Ar parameters there is very little difference between the two sets of results, but for the Ne parameters the difference is easily visible. They also make comparison of their results with QMC calculations for the same models (McGurn et al. 1989). The agreement is satisfactory considering the very large uncertainties in the QMC results, arising both from statistical fluctuations and the short length (15 atoms) of the chain used. This work was extended to an all-neighbor model by Zhu et al. (1992). The transfer matrix method is not applicable in this case, so an EPMC procedure was used instead. The classical Monte Carlo method is very fast for a one-dimensional system so that Zhu et al. were able to obtain very well converged results for the energy and better than 1% accuracy for the heat capacity at most temperatures. They also used a system of 100 atoms, much larger than the one used for the QMC work. Another instructive one-dimensional model is the Toda lattice (Toda 1967a, b). In this model the atoms interact with an exponential repulsive potential between nearest neighbors and a linear attraction. The useful

w

Lattice dynamical applications

111

feature of this model is that many of the calculations can be carried out analytically. The smeared potential, eq. (25), can be evaluated analytically and in the LCA the partition function can be expressed in terms of a digamma function. An effective potential treatment of the model was given by Cuccoli, Spicci, Tognetti and Vaia (1992c). The results of an LCA calculation of the specific heat were in excellent agreement with a calculation using a Bethe ansatz (Hader and Mertens 1986). One general conclusion which can be drawn from the one-dimensional models is that the combination of the low-coupling approximation with the Ginzburg expansions usually gives good results. Only in the case of the model with parameters corresponding to neon was there a large difference between the first and second order theories. This is supported by the threedimensional results discussed in the next section. The Ginzburg expansion should be reliable in all applications except for calculations on the materials usually described as quantum crystals. Even in these cases it is quite possible that the LCA used without the Ginzburg expansion will be adequate.

4.

Three-dimensional Lennard-Jones model

The first application of the EPMC method to a three-dimensional solid was made by Liu, Horton and Cowley (1991a, b). They used a model of an inert-gas crystal in which the atoms interact with their nearest neighbors only, through a Lennard-Jones potential r

= 4e ( ( r(7 )12

__(~)61 (7 9

(29)

This model has been used in many other calculations, with the parameters of the potential determined by fitting to the lattice spacing and binding energy at zero degrees (Horton 1962, 1976). It thus provides a good comparison of the method with anharmonic perturbation theory (Shukla and Cowley 1985), self-consistent and improved self-consistent theories (Goldman et al. 1968), and with classical Monte Carlo theory (CMC) (Cowley 1984). In order to preserve as strong a similarity as possible with the one-dimensional analysis and for simplicity, Liu et al. used a rather over simplified form of the variational nearest-neighbor force-constant matrix -

(30)

An advantage of this approximation was that it allowed the solution of the low-coupling-approximation equations to high order without the tensor

E.R. Cowley and G.K. Horton

112 1.16

...i

!

Ch. 2 '

i

.......... i

b U tO

-,-, ._~

1 17

Xe

1.16

C]I 0 L

= J--

.o

1.]b

t--

J

S

-

~

j " "*"

.

~

._~ z

.,., 1.14 o'1 L 0 z

1.13

___1.12

0.0

=

=

O. 1

0.2

I ....

0.3

,

0.4

i

0.5

Temperoture ( k T / r

Fig. 3. Nearest neighbor distance in Xe, in units of or. Dots are the second-order effective potential results, solid line is CMC, dashed lines are perturbation theory to order )~2 (lower line) and )~4 (upper line). formalism of w2.6. Liu et al. were able to solve the first-order and secondorder equations fully consistently, and, at zero degrees, were able to get a solution to the third-order equations, which, though not iterated to selfconsistency, was probably adequate. For the heavier inert-gas crystals, their results are a striking illustration of the power of the effective potential method. The results for the nearest neighbor distance, internal energy, and heat capacity, for Xe are shown in figs 3-5, and similar results for Ar are shown in figs 6-8, all at zero pressure. The values are all given in reduced units: cr for the nearest neighbor distance, e for the internal energy per atom, and Nk for the heat capacity. The temperature is plotted in units of e/k. For the nearest-neighbor model, the melting points of all the inert-gas solids except He are then close to 0.5. The values of the parameters used are given in table 1. We should remember that these parameter values are fitted to experimental data, usually corresponding to zero degrees, and that a particular method of calculation had to be used for the fitting process. If a different method of calculation is used with the same parameters, even if it is a more accurate method, the agreement with experiment will not be so good. Partly for this reason, we do not compare the results with experimental values. In the figures the EPMC results are shown as points, and are compared with anharmonic perturbation theory results, shown by dashed lines, and with CMC results shown as solid

w

Lattice dynamical applications i

i

i

. v

//~ i

113

/

1

Xe

-4.5

E

0 El

f//~J

-

t_

a. >~ OI

-5.0

...7"

I.

t-

El

E

"

-5.5

tP

-6.0

.

I

0.0

0.1

I

I

I

0.2

0.3

0.4

Temperature

(kT/~)

0.5

Fig. 4. Internal energy per atom in Xe, in units of e. Symbols are as in fig. 3.

3.0 ~

i

~

2.5 "

//

2.0 .

.~ g.

1.5

~

1.0

~

0.5 0.0 - " 0.0

,e~.

i

~_-__ ~

•

i

/

,

-

II

!

I

i J

i 0.1

i 0.2

i 0.3

i 0.4

I 0.5

Temperature ( k T / r

Fig. 5. Heat capacity per atom in Xe, in units of k. Symbols are as in fig. 3.

lines. Anharmonic perturbation theory results are shown for the lowest order ()~2) and next lowest order ()~4) (Shukla and Cowley 1985). As the temperature decreases the two sets of perturbation theory results converge,

E.R. Cowley and G.K. Horton

114

Ch. 2

1.18

v

b Q) U c o

,4J m

1.17

1.16

a

t..

o .13 to1 q) Z 4J

M

1.15 1.14

P o

Z

1.13 1.12

0.1

0.0

0.2 0.3 Temperature (kT/e)

0.4

0.5

Fig. 6. Nearest neighbor distance in Ar. Symbols are as in fig. 3. ..........

W. . . . . . . . . . . . . . . . .

!

.....

! .

.

.

.

.

!

....

]

Ar v

E

-4.5 /

9

o

o k. q) o.

-5.0

131

s

I.

~

Q)

er

I) 0

E

-5.5

qtl l.

-6.0

................... ,

0.0

,1

0.1

.....

!

i

0.2 03 Temperature (kT/r

i

,,

0.4

0.5

Fig. 7. Internal energy per atom in Ar. Symbols are as in fig. 3.

which shows that the truncated perturbation expansions are accurate, and the temperature at which they begin to diverge gives some idea of the limit of their usefulness. This typically happens at one-quarter to one-third of

Lattice dynamical applications

w

Ii

li

-~

E

i

Ar

3.0

115 i

J

2.5

0

L Q)

2.0

13.

~u

1.5

0 0 I o0

-

0 ID

0.5-

/

/

I

I

I

I b

/

/

/

/e 0.0

*

"-J

0.0

0.1

I

I

0.2 0.3 Temperoture (kT/r

I

0.4

0.5

Fig. 8. Heat capacity per atom in Ar. Symbols are as in fig. 3. Table 1 Lennard-Jones potential parameters for the inert gas solids. e(10 -16

Xe Kr Ar 22Ne

452.55 325.2 235.95 72.09

erg)

tr(A) 3.8469 3.5600 3.3043 2.7012

the melting temperature. The CMC results, on the other hand, are valid only in the high temperature limit, roughly down to the Debye temperature. In the case of Xe there is a small overlap between the temperature ranges in which perturbation theory and Monte Carlo results are reliable, while in Ar there is a substantial gap between the two. In both cases, the EPMC results pass smoothly from agreement with anharmonic perturbation theory at low temperatures to agreement with CMC results at high temperatures. The theory thus encompasses the whole temperature range, and it may be the first which does so this successfully. We now consider Ne. For Xe, Kr, and Ar, the first and second order theories gave very similar results, even at zero degrees. Also the two orders of anharmonic perturbation theory agreed at low temperatures. We could therefore believe that all of the results obtained were converged. In the case of Ne, however, there were differences between the first and second order

E.R. Cowley and G.K. Horton

116 1.0

L

q)

E

a

L.

a ~L.

u

i

Ch. 2

i

"f

I 0.3

"'I 0.4

0.8

Ne

0.6

crl

I.

n N e-

0.4

Ar

0.2

0.0

o.o

I o.1

I 0.2

0.5

Temperature (kT/e) Fig. 9. Ginzburg parameters for Ar and 22Ne.

results even at the highest temperature and at zero degrees the differences were substantial. Also the anharmonic perturbation theory does not give sensible results for Ne (Shukla and Cowley, 1985). The Ginzburg parameters for Ar and for 22Ne are shown as functions of temperature in fig. 9. The values for 22Ne at all temperatures are greater than the zero degree value for Ar. It is not clear that the Ginzburg expansion can even be used in this case. Liu et al. (1992) made a careful comparison of the effective potential results for 22Ne with improved self-consistent phonon theory (ISC) and QMC values. Results for the nearest neighbor distance and the internal energy are shown in figs 10 and 11 respectively. The long-dashed line in the figures shows the first-order effective potential result and the solid line shows the second-order result. The increase in the value of the spacing at very low temperatures is hinted at in earlier effective potential results but is much larger for Ne than for the heavier elements. We assume that it indicates the breakdown of the Ginzburg expansion to the given order. The increase is much less in the second-order theory than in the first-order calculation. Liu et al. were also able to calculate a third-order point at zero degrees. In fact, at zero degrees, the effective potential result for the free energy is the same as the SC1 value. We can easily modify a standard SC1 program to make the same isotropic approximation for the force constants as was made in the EPMC calculation, eq. (30), and calculate a zero-degree lattice spacing

w

Lattice dynamical applications 1.185

. . . . . . . . . . . . . . . . . . . . . .

1.180

-

o o EL

1.175

-

0

1.170

~.

,

.

.

.

.

.

.

1.165

.

.

.

.

.

.

.

.

.

.

/,'/

/ . , /

-

,4,,'/

.

~.-"

.

.

.

.

.

..... 0.0

.

/,9'

.__.

.

117

."

-

.

' 0.1

, ............... I 0.2 0.3

Temperature (~:/kB)

, 0.4

Fig. 10. Nearest neighbor distance in 22Ne, expressed in units of the parameter cr of the Lennard-Jones potential. Long-dashed line is first-order EPMC, solid line is second-order EPMC, short-dashed line is ISC and points are QMC results. The square at 0 K is the fully converged effective potential result from the modified SC1 calculation.

and this is also shown in the figure, as a square. The third-order values for both the nearest-neighbor distance and the energy were very close to the modified SC1 values. This seems to indicate that the isotropic approximation is not the cause of the rise in lattice spacing. The fully self-consistent point at zero degrees is also very similar to the minimum value on the secondorder line, so it is plausible that the fully converged curve would join the minimum of the second-order curve horizontally to the vertical axis. If this is the case, the zero degree spacing would be about 1.168cr. We can also run the SC1 program, without the isotropic approximation. It turns out that the energy is very little affected, but that part of the disagreement for the nearest-neighbor distance does arise from this extra simplification. The figure also shows results for three QMC points as dots and for the ISC theory as the short-dashed curve. The ISC result agrees very well with the lowest temperature QMC point and we can therefore use it as a reliable guide at even lower temperatures. We then have to conclude that the effective potential results are not quite as reliable as ISC at the low temperature end. The results for the internal energy lead to a similar conclusion. However, at the highest temperature QMC point, a careful comparison indicates that the EPMC result is in fact in better agreement than the ISC result. Notice that Ne is still far from classical at its melting temperature. Ultimately, at

E.R. Cowley and G.K. Horton

118 -3.9

""

i

l

i

0.3

0.4

Ne

-4.0 v E 0 -~

i

Ch. 2

-4.1 -4.2

k_

-4.3 t~ - 4 4~ k-

W t-

ta

~J

-4.5

0.0

0.1

0.2

Temperoture (~/ks) Fig. 11. Internal energy per atom in 22Ne, in units of the well-depth e. Long-dashed line is first-order EPMC, solid line is second-order EPMC, short-dashed line is ISC and points are QMC results. The square at 0 K is the fully converged effective potential result.

sufficiently high temperatures, the EPMC results must coincide with QMC results. We emphasize that the results have been plotted on greatly expanded scales in order to show the differences at all. It is gratifying to find that we have two theories which give predictions in such excellent agreement with those of QMC for a material with such large quantum effects. A second comparison of EPMC and QMC results was made by Cuccoli, Macchi, Neumann, Tognetti and Vaia (1992b). They used a model of Ar, with Lennard-Jones parameters determined from gas phase data. Interactions with neighbors beyond the first shell are not negligible with this potential and were represented in a static approximation. That is, the simulations were carried out including only nearest neighbor interactions and then the static lattice contributions to the energy and pressure from more distant neighbors were added. The advantage of using this potential is that comparison with experiment becomes more reasonable. It would be even better to use one of the modem realistic potentials. The disadvantage is that the results cannot be directly compared with the large number of earlier calculations which use the nearest neighbor potential. Cuccoli et al. used the LCA approximation and the first-order expansion of the smearing integrals. The frequencies used are then the usual quasi-harmonic frequencies of the model. They did not use the isotropic trial function of Liu et al. but separate

Lattice dynamical applications

w 1.100

. . . . . . . . . ti"

,u,,,,,,,,!

" I

........

|

J

119 !

1.075

1.050 IO lq) C) I(P

E

1.025

1.000

Z

0.975

0.950

0.0

~ I. . . . . . . . . . . . . . 0.1 0.2

= 0.3

.. . . . . . . 0.4" 0.5

, ...... 0.6 0.7

Reduced Temperature

Fig. 12. Density as a function of temperature in Ar, after Cuccoli et al. (1992b). Points are experimental values, solid line is EPMC, and dashed line is CMC. parameters for longitudinal and transverse branches. This procedure, which had been suggested by Feynman and Kleinert (1986), is undoubtedly a step in the fight direction but is still not completely general. In the same study, Cuccoli et al. also report extensive QMC calculations of Ar, and compare both types of calculation with each other and with experiment. An example of their results is shown in fig. 12. This shows the experimental curve of density as a function of temperature, together with the EPMC results as a solid line and CMC results as a dashed line. Results from QMC are not shown since they were carried out for only a limited number of volumes which did not correspond to zero pressure. However, EPMC and QMC results at the same volume and temperature were in excellent agreement.

5. Improvements in the three-dimensional results We have implied that the full set of equations describing the effective potential theory, eqs (20), (21), (22) and (23), cannot be implemented fully. In an EPMC calculation, the effective potential needs to be calculated selfconsistently at each instantaneous position of the atoms, where there are no simplifications from symmetry. Numerical simplifications, principally the LCA and the Ginzburg expansion, have been made in all three-dimensional

E.R. Cowley and G.K. Horton

120

Ch. 2

applications carried out so far. It is clear from the results described in the previous section that a calculation at this level of approximation is barely adequate to describe neon and it would therefore surely fail for helium. The question is whether a more complete calculation can be performed. The elimination of the Ginzburg expansion is not difficult. We have indicated this in the description of the formalism earlier. We are presently carrying out calculations for neon, retaining the LCA but making no other approximations in the formalism. The present results are very encouraging. We expect to present the final results shortly (Acocella et al., to be published). It also seems worthwhile to attempt a full implementation of the theory. If it achieves nothing else, this should allow us to test the approximate versions. In fact, we believe that a full calculation is possible on a modem massively parallel computer, such as a Connection Machine. The form of the equations lends itself to such an architecture. We have a program presently under development, and it seems at the moment that it will require substantially less time than a full QMC calculation. We hope to report results soon.

6.

Dynamic effects, moments and phonons

The effective potential formalism is based on an variational approximation for the partition function. It can therefore be expected to give optimal values only for the free energy and its derivatives. It is certainly possible to use the effective potential to calculate other quantum mechanical equilibrium averages such as static correlation functions (Cuccoli et al. 1991) and a mean square displacement (Cowley and Horton 1992). It is highly desirable to have a procedure for calculating time-dependent correlation functions and the associated frequency-dependent Fourier transforms. Cuccoli, Tognetti, Maradudin, McGum and Vaia (1992d) have applied such a procedure based on Mori's continued fraction representation of spectral functions (Mori 1965a, b; Lovesey and Meserve 1973). Any spectral function C(w) can be written

C ( o.)) - "['~{ ~1 (~0 / ( Z -t- (~1/ ( Z -F (~2 / ( Z

-F'" ")))}z=iw

where the 8i are related to the even moments #2n of the spectral function.

#2n =

F oo

w2nC(w) dw.

Lattice dynamical applications

w

121

In particular (Jhon and Dahler 1977)

#0 -- (~0, /1;2 -- (~0(~1' /1'4 -'- (~06162 -l-"6052. The moments #:n can be obtained directly from the equations of motion as equal time averages and can therefore be calculated any of the techniques of equilibrium statistical mechanics, including QMC or EPMC. (See also the accompanying article by A.R. McGum.) Scattering properties of crystals are usually expressed in terms of a function S(Q,w) which, for a monatomic material, can be defined as (van Hove 1954)

S ((~, w) = / F ((~, t) e iwt dr, where

F(Q, t) is the intermediate scattering function

F (0, t) - (p(0, t). p* (0, 0)), p((~, t ) = Z

eiQ'r'i(t)" i

Here Q is the scattering vector and p(Q, t) is a sum involving the instantaneous positions fi(t) of the atoms. Often it is sufficient to consider a one-phonon approximation to the scattering function, which includes the displacements of the atoms from their equilibrium positions only to first order. Then El ((~, t) = Z eio" [h,-/i~] <((~. ij

~i(t))((~" ~Tj(0))),

: Z eiQ'[h'-h~]Z QaQ~(uia(t)ui~(O)). Actually the usual definition of the one-phonon term sums extra contributions to include a Debye-Waller factor but we shall omit this. The even moments

E.R. Cowley and G.K. Horton

122

Ch. 2

of S(t~,w) are equal to the successive even time derivatives of F(t~,t), evaluated at t = 0. The moment #0 thus requires the evaluation of equal time correlation functions of the type

#0

e i~" [/~'-/~]

(uiauj#),

ij

the moment #2 involves

#2"~ Y~ d 2eiQ'[/~'-/~] u i a i( j dt 2

ujo)

: ~--'~ eiQ. [/~,-/~] 1 ij m2 (Pic,Pjf~)

and so on. Detailed expressions for #4 and #6 have been given by Cuccoli et al. (1992d) for a very similar one-dimensional formalism. In order to produce a result with only a finite number of moments, it is necessary to terminate the continued fraction in some way. The question then arises as to whether the final results are sensitive to the termination used. This in turn depends on how deeply the expression is terminated, which is fixed by the number of moments which can be calculated. An examination of the equations shows that 81 is the mean square frequency of the spectral function and that 82 measures the spread of the distribution around its center. Clearly these two numbers represent the minimum information about the frequency and its width that we must include, and to calculate 81 and 82 requires a knowledge of the moments up to #4" Aubry (1975) has studied the convergence of the continued fraction representation for a one-dimensional system, and reached discouraging conclusions. In extreme anharmonic situations the method was quite unreliable. However, the model Aubry was testing was itself very anharmonic, displaying an instability at zero temperature. It may well be the case that, in realistic situations where the anharmonicity is less extreme, useful predictions can be obtained with only a few moments. Cuccoli et al. (1992d) applied the formalism to a linear chain with nearest neighbor Lennard-Jones interaction and show a number of peaks calculated both classically and with quantum mechanical effects included. More recently (Cuccoli etal. 1993) they have performed molecular dynamics simulations for a Lennard-Jones chain in order to obtain comparison peaks,

w

Lattice dynamical applications

123

Fig. 13. Classical and quantum relaxation functions for the Lennard-Jones chain, at ka = 0.2zr and (a) kBT/e = 0.1, (b) kBT/e = 0.3, and (c) kBT/e = 0.8. The continuous and dashed lines are the classical and quantum results, respectively. The long dashed and dotted lines are the classical and quantum 4-pole results obtained by using the value of "r4 fitted to classical molecular dynamics data. After Cuccoli et al. (1993).

though only in the classical limit. They find that, by fitting a parameter, they can obtain good agreement between the results from the simulation and a moment expansion terminated at quite a low order. They then use the same parameter values to terminate the quantum mechanical moment expansions. Examples of their results are shown in fig. 13. The relaxation function F(k,w) is similar to what we have called S(Q, to). Results are shown for a wave number ka = 0.27r at three temperatures. The solid and short dashed lines show the classical and quantum results obtained from the moment expansion with no fitting. The long dashed and dotted lines are the results obtained with the fitting parameter. A limitation of this approach is that an independently calculated result is necessary before reliable results can be obtained from the moment expansion method. It is to be hoped that, perhaps if the moment expansion is extended to a higher order, this limitation can be eliminated.

E.R. Cowley and G.K. Horton

124

Z

Ch. 2

A m o d e l f e r r o e l e c t r i c - static effects

The effective potential technique should be useful in any situation where there are large anharmonic effects so that lattice dynamical approaches are inadequate, at low enough temperatures so that classical simulation techniques cannot be used. Displacive and order--disorder phase transitions certainly involve large anharmonic displacements of the atoms and in many ferroelectric crystals there are transitions which occur at temperatures such that kT is small compared with many of the phonon energies. Indeed, these materials have been described as quantum crystals (Werthamer 1969). As a first trial of the effective potential method in this field, we have calculated the properties of a simple model which has been extensively studied using other techniques (Cowley and Horton 1992). The model consists of a set of one-dimensional oscillators moving in potential wells containing both quadratic and quartic terms. In order to produce a phase transition in a one-dimensional system, the coupling between them must be of infinite range. Gillis and Koehler (1974) studied this model using SC1 theory and also by calculating the exact eigenstates of the molecular field Hamiltonian and summing the contributions to the partition function. They wrote the Hamiltonian in a scaled form

~2 d2 Z

9

]

+ 4u~ + 4u 4 -

2 dx 2

1

X,juiuj,

~ .

.

where, to achieve the phase transition

Xij = x / N ,

N - + oo.

While this might seem a primitive model, it is currently used in the analysis of experimental data (Salje et al. 1991). If the minus sign is used in the potential, the atoms move in a double well and the model describes an orderdisorder phase transition, while if the plus sign is used the model describes a displacive transition. The terms describing the one-particle potential energy can be obtained from the form used in w2.3, eq. (15), by a transformation to new units. Lengths are measured in units of v/12a/b and energy is measured in units of 3e2/2b. The scaled model has two parameters: )~, defined in eq. (16), measures how quantum mechanical the system is, and X measures the strength of the coupling. The infinite range of the interaction allows us to use a molecular field ansatz. The terms in the potential involving atom i are

17/- +4u~ + 4u4 -Xs

w

Lattice dynamical applications

125

where 1

N

J

The value of ~ is calculated self-consistently from the potential V~. In the classical case

f u~ e -zV` dui =

.

(31)

f e -~v~ dui The model displays a second-order phase transition from a low temperature structure with a finite average value of the order parameter ~ to a high temperature structure in which the order parameter is zero. Gillis and Koehler (1974) made calculations for both the + and - signs in the potential, for two values of A of 0.2 and 1.0, and for a range of values of X. Using the in-principle exact procedure of expanding the wave functions in terms of a large basis set of harmonic oscillator wave functions, they found a second-order transition in all cases. They also made a careful analysis of the SC1 theory applied to the model. They showed that, while for some ranges of the parameter values it gave quite good results, in other cases it was completely unreliable, predicting a first-order transition. We have applied the effective potential formalism to this model (Cowley and Horton 1992). The quadratic and quartic terms in the potential are exactly the same as discussed in w2.3. The model is in fact a close relative of the t~4 model discussed in w3. The coupling term is linear and is not affected by the smearing. The order parameter can therefore be calculated exactly as in the classical case, eq. (31), but with the effective potential. The results were found to be in very good agreement with the numerical quantum mechanical results of Gillis and Koehler. The largest disagreement occurred at high temperatures and for small values of the quantum parameter A. Since this is just the region where the effective potential method should be most reliable, we repeated Gillis and Koehler's calculation including more terms in the sums, and now found excellent agreement in that regime also. Gillis and Koehler had calculated the eigenvalues and eigenstates by expanding them in terms of the lowest 200 of a set of optimized harmonic oscillator wave functions. However, they included only the lowest 50 of the functions in the sum over states. When we included all of the upper energy levels in the sum over states the agreement with the effective potential results was near perfect. We find that, while the higher energy levels are not given as

E.R. Cowley and G.K. Horton

126 0.8

Co)

Ch. 2

1.0

0

0.6

=

0.2

-

10

0.8

I.. t) ,,i.a m

E o

I1.

0.4

0.6

),--0.2 X = 1.0

L Q) "0 I,.

0.4

o

0.2

X"5

0.2

0.0

0.0

.

I

0.2

.

0.4

.

I

0.6

.

i

0.8

.

1.0

0.0

0

2

' ;

.

I

i

4

.

t! --

6

.

8

Temperature

Fig. 14. Order parameter in the order-disorder model ferroelectric with ,k = 0.2. Solid line is the effective potential result, solid dots are our recalculation of Gillis and Koehler (1974), and hollow dots are SC1.

accurately as the lower ones, it is better to include them in the sum than to discard them completely. Some results are shown in figs 14 and 15 for the double-well orderdisorder model and in fig. 16 for the single-well displacive model. The solid lines are the results of the effective potential calculations and the solid dots are our recalculation of Gillis and Koehler's values. Also shown in some cases are classical values, calculated directly from eq. (31), shown as dashed lines, and SC1 values shown as hollow circles. Where the line of SC1 results stops at a finite value, a first-order transition occurs. For )~ = 1.0 there are large quantum mechanical effects and the effective potential method includes most of them accurately. The only remaining discrepancy is for the double well case at zero degrees and for large )~. In this case the ground state wave function is double peaked and is poorly represented by the single displaced Gaussian of the variational function. However, the discrepancy quickly disappears as the temperature is increased. It seems that enough thermal tunnelling is then included to remedy the deficiency. It is also possible to calculate a susceptibility for the model. To do this we include in the Hamiltonian and extra term - F n i and define the susceptibility

Lattice dynamical applications

w i

i

-

-- . . . .

i

'

\

0.6

|

(o~ % a,

0.8

L

i

,,

:h = 1.0 X=4

\

127 !

!

|

-

(b)

1.(3 " ~-'~,,

~=1.0 X = 8

\\\

0.8

\

\

9

\

E

\

0 t... t2

0.6

I1.

L.

0.4

"lO

o

It !

0.2

0.0

0.4

0.2

0.0

0.4

0.8

1.2

9

0.0

1.6

i

0

.

i

1

.

i

2

.

xl

3

.

5

4

Temperature Fig. 15. Order parameter in the order-disorder model ferroelectric with )~ = 1.0. Solid line is the effective potential results, dots are our recalculation of Gillis and Koehler (1974), and dashed line is classical calculation. 0.8

,

,

,

,

,

,

(0)

0.6

-

1.0 ) , = 1.0 X = 16

~

(b)

9 %

~

~

~

X '= 1.0 X " 24

~

0.8

%

\

L I)

\

ID

E

o

0.4

0.6

\

\

0

n

\

t,.

\

\\

0.4

0

0.2

~ 0.2

It ! 0.0

'

0.0

0.5

'

1.0

1.5 2.0

-

2.5

'

3.0

I

3.5

0.0

I

0

2

4

6

8

10

Temperature

Fig. 16. Order parameter in the displacive model ferroelectric. Solid lines are the effective potential results, solid dots are our recalculation of Gillis and Koehler (1974), dashed lines are classical calculations, and hollow dots are SC1.

E.R. Cowley and G.K. Horton

128

Ch. 2

as the derivative of ~ with respect to F, at F equal to zero. The result is Z{(u2>c1 -

OF

(u>2}

1 -/3X{(u2}c1- (u} 2}

where by (U2}C1 we mean an average calculated using a classical expression but with the effective potential f u 2 e -~ 88 du

(U2)C1 = f e -z 88 du Two results are shown in fig. 17, both for the susceptibility and its inverse. The inverse clearly shows the Curie-Weiss behaviour near the transition. In general, it is a fair comment that figs 14-17 show that the effective potential method gives a reliable approximation to an exact calculation for this model ferroelectric.

8.

The model ferroelectric- soft mode behaviour

While the Curie-Weiss law behaviour of the susceptibility is a good indication of soft mode behaviour, the actual calculation of a phonon spectral function is much more difficult. The moment expansion technique described in w6 offers a possible method of calculation. This has not yet been applied to the effective potential model but we have calculated the moments #0 through #1o of the one-phonon spectral function, both for finite and zero wavevectors, in the Classical limit (Cowley et al. 1994). This provides additional information on the convergence of the moment expansion. We find that, at this level of termination, the peak shapes are quite insensitive to the actual termination procedure used. For this model, all finite wave vectors give the same result and the averages needed can all be calculated from the molecular field Hamiltonian. For the zero wavevector case, it is necessary to include the terms describing the correlated motion of the atoms, which become infinite at the transition point. Some results for the spectral functions are shown in fig. 18, and the temperature dependence of the peak centers is shown in fig. 19. The soft mode behaviour is evident. We hope to include quantum mechanical effects by means of the effective potential formalism shortly.

Lattice dynamical applications

w 20

i

!

I

5

i

(o)

I

Order-Disorder X=0.2

I

16

I I

X

=

1.0

5

i

i

i

129 ,

i

,

,

,

4

3

I

g 15~ (/1

3

t I

r-

~--~

|

8

| t

2

2

1

1

0

0

m "D

-

~og~ ~~

I

4 -

.~~,

~

0.2

0.4

0.0

:3 m

t.--~

0

20 <

I

ip u M :3 tn

25

Displacive X=l.0 X = 24.0

4

I

~ 12 .IQ

,

(b)

0.6

0.8

Temperoture

Fig. 17. Susceptibilities and inverse susceptibilities for (a) order-disorder, and (b) displacive model ferroelectrics.

1400

.

.

.

.

700

I

l

(o) 1200

600 : ~

X=16

i

X "

16

500 -

1000

:a

l

(b)

T = 3.30

3.25

800

400

II (3m 600

300 - 200 /

400 200

~T \\

i

0 0.0

I

0.2

= 3.33

= 3.24 - 100 j

I

a

0.4

0.6

Frequency, =

3.20 i 0.8

1.0

0.0

T = 3.36

0.2

0.4

0.6

0.8

3.50 i 1.0 1.2

Frequency,

Fig. 18. S(q = 0, w) for the ferroelectric model at several temperatures (a) below and (b) above the Curie point, for a X of 16. For this value of X the transition occurs at Te = 3.2632 K.

E.R. Cowley and G.K. Horton

130 3.0

i

~

,,I

!

'

Ch. 2

I

2.5

X==36

2.0 1.5 1.0 0.5 0.0

1.4 1.2

o>,

t,.-

1.0

X=16

o"

0.8 0.6 0.4

12.

0.2 0.0

0.6 Z=9

0.4 0.2 0.0 L . 0.8

0.9

1.0

1.1

1.2

T/Tc Fig. 19. Temperature dependence of the value of w at which the spectral function is a maximum, for three values of X.

9. Conclusions The results already obtained show that the effective potential method accurately accounts for quantum mechanical effects, while still retaining the simplicity of a classical formalism, in a wide variety of applications. The theory goes over to the classical form at high temperatures and contains correctly the first term in the Wigner expansion. At zero degrees it is equivalent to the first-order self-consistent phonon theory. It is thus pinned down at both ends of the temperature range to exact results at one end and to a good approximation at the other. The classical form of the partition function

w

Lattice dynamical applications

131

allows one to use any of the standard methods of evaluating thermodynamic averages in a non-quantized system, including, most importantly, a classical Monte Carlo procedure. In a number of one-dimensional systems which have been studied, even this step was not necessary. The effective potential and the partition function could be calculated with essentially no further numerical uncertainty. The excellent results obtained leave no doubt of the validity and usefulness of the general method. The aesthetic aspect of the theory also seems to us remarkable, in that the equations have an elegant and inevitable look to them. The SC1 theory had a similar quality when it was first derived. While the basic results of the path integral theory were given in Feynman's books (Feynman 1972; Feynman and Hibbs 1965), their application to the quadratic variational function involves a considerable amount of algebra. Anyone who has worked through the pages of manipulations to see the physically appealing form of the effective potential emerge can only applaud those who pioneered the theory (Giachetti and Tognetti 1985, 1986; Feynman and Kleinert 1986). While the effective potential theory shares this satisfying appearance with SC1, it is a much more powerful formalism. We expect that, of the calculations reviewed here, the application to ferroelectric and other phase transitions will be extended. This is clearly a very anharmonic situation which often arises at temperatures sufficiently low that quantum mechanical effects cannot be ignored. Many real ferroelectrics have complicated crystal structures. While the interatomic potentials seem to be quite well known, the usual methods of calculation are either quasiharmonic lattice dynamics or classical simulation techniques. The effective potential technique provides a way of including quantum mechanical effects at the accuracy of the harmonic approximation into a simulation which can accurately handle the anharmonicity. The recent attempts to calculate spectral functions also open up a very important extension of the method which we expect will be active in the near future. The moment expansion method may turn out to be the most powerful of the techniques proposed to extract time dependent quantities from a quantum simulation and the use of the effective potential method offers a considerable speeding up of the calculations. A much more speculative comment is that it might be possible to use an effective potential approach to include quantum effects in some kind of molecular dynamics formalism for the spectral functions. We have only mentioned briefly attempts to improve on the basic Peierls inequality which is the starting point of the variational calculation (Klein-' ert 1992, 1993). There is a parallel between these improvements and the development of self-consistent phonon theory. At present the best practical

132

E.R. Cowley and G.K. Horton

Ch. 2

self-consistent scheme for calculating thermodynamic properties is the improved self-consistent phonon theory (Goldman et al. 1968). In this scheme, the cubic anharmonicity is included in a non-self-consistent fashion as the lowest order perturbation correction to SC1, evaluated using the SC1 frequencies and smeared potentials. More elaborate formalisms have been developed, including Choquard's full second-order theory (Choquard 1967), but only a partial implementation of this has been carried out (Kanney and Horton 1974). Both of these theories are based on the cumulant expansion of the free energy. A less systematic attempt to include short-range correlations, especially in the calculation of phonon frequencies, was made by Homer (1974). It seems likely that improvements to the present effective potential formalism along these lines will be both necessary and possible. It is also appropriate to mention here a related theory. Doll, Coalson, and Freeman (1985) have developed an acceleration procedure to improve convergence of QMC calculations. The usual QMC procedure of dividing the integration over ~- into a sum over a relatively small number n of discrete values is equivalent to including the first n Fourier components of the particle trajectories through the path integral. Doll et al. included an approximate evaluation of the higher Fourier components based on a free-particle density matrix, and showed that the convergence of the QMC values as n was increased was greatly improved. They also mentioned that they had carried out similar calculations using a harmonic oscillator density matrix, which would be very similar to the formalism described here. This work predates the papers on the improved effective potential theory which we have cited, so that it might be classed with what astronomers call pre-discovery observations of new phenomena. It certainly seems to be a powerful technique. As computers continue to gain in power, and as that power becomes more generally available, we can expect that quantum Monte Carlo calculations, which are in principle exact, will become more common. However, there will always be problems which are just beyond the attainable horizon of QMC. For these, the cutting-edge problems, the effective potential formalism will represent a valuable if approximate technique.

Acknowledgements We have benefitted from many discussions with our colleagues Dominic Acocella, Eugene Freidkin, Shudun Liu, and Zizhong Zhu. We would like to thank Drs R. Giachetti, V. Tognetti, A. Cuccoli, and R. Vaia, and their collaborators, as well as A.A. Maradudin, A.R. McGurn, and R.E Wallis, for keeping us informed of their work through preprints. This work was partially supported by the U.S. National Science Foundation under Grant No. DMR 92-02907.

Lattice dynamical applications

133

Note added in proof There has been substantial progress in the application of the effective potential method since this article was completed, early in 1994. A number of possibilities mentioned in the article have been realized. We give here a brief description of the work both of our own group and of others. The elimination of the Ginzburg expansion and the evaluation of the smeared potential as an integral (see eq. (27)), were described by Acocelia, Horton and Cowley (Phys. Rev. B 51, 11406 (1995-1)). The isotropic approximation which had been made earlier was also eliminated. The only remaining approximation was the LCA. The calculations were made for neon since the previous methods were adequate for the heavier rare-gas solids. The largest discrepancy remaining was in the internal energy at low temperatures. This was anticipated, since the EPMC procedure becomes equivalent to the SC1 approximation at zero degrees, and that approximation is known to suffer from its neglect of odd terms, particularly cubic terms, in the expansion of the anharmonic potential. A successful correction to SC1 is the improved-self-consistent theory (ISC), which adds on to the SC1 free energy a correction term involving the smeared cubic derivatives. The smearing is carried out at the SC1 level, so that the calculation is not completely selfconsistent. It has proved to be very successful at low temperatures. We have applied a similar philosophy to the effective potential calculations. We add to the free energy at zero degrees a term identical with the ISC correction. As the temperature increases, this term must be phased out, since in the high temperature limit the potential is already fully included to all orders in the simulation part of the calculation. We therefore subtract from the ISC correction its high temperature limiting value. This gives a difference of terms, similar to the expressions, eqs (11) and (13), of effective potential theory. The procedure is undoubtedly arbitrary, though plausible, and its justification lies in the excellent results it gives. Results have been given by Acocella et al. (Phys. Rev. Letters 74, 4887 (1995)). The most striking agreement is with the earlier QMC results. We would like to stress that the calculation of the cubic correction is very fast compared with the EPMC simulation, so that the increase in computer time is negligible. We have called this procedure the improved effective potential (IEP) method. While this calculation goes beyond the effective potential method in one sense, it still retains the low-coupling approximation. We have long felt that it was necessary to test the validity of the LCA. There is no difficulty in principle of performing a calculation without the LCA, but the computational demands are vastly increased. The frequencies and the smearing matrix are calculated from the eigenvalues and eigenvectors of a 3N • 3N matrix, and

134

E.R. Cowley and G.K. Horton

Ch. 2

this needs to be done for every step of the Monte Carlo simulation. We have now completed such a calculation, for the method of neon (Acocella, Horton and Cowley, submitted to Phys. Rev. Letters). The results are fascinating. At zero degrees, they agree, as they must, with SC1, and at high temperatures they agree with the QMC calculation. In between, there is a cross-over region where the results are, inevitably, quite unphysical. The heat capacity, for example, is negative. The computer time required for the calculation was also a disappointment. It is at least as expensive as a QMC calculation. What the calculation does firmly demonstrate is that the low-coupling approximation is inadequate. Except at zero degrees, there are substantial differences between the values calculated with and without the LCA. All other aspects of the two calculations are the same, so that the difference is entirely due to the changes in smearing as the atoms move. The full EPMC theory is not a practical approach, since it is so expensive of computer time. However, a comparison of the results of the various approximations does show clearly the amazing accuracy of the IEP method. It seems that the form of the cubic correction which we adopted corrects for the deficiencies in the LCA method at the same time that it is including the effects of the cubic terms. The method is both accurate and fast. We believe that it is the most powerful formulation of the effective potential method yet devised. There has been substantial progress also made recently in the calculation of time and frequency dependent quantities by the effective potential method. Macchi, Maradudin and Tognetti (Phys. Rev. B 52, 241 (1995)) have applied the moment method to a three-dimensional Lennard-Jones crystal. The even moments through #6 were calculated as functions of temperature, but no spectral functions were given. A different approach has been taken by Cao and Voth (J. Chem. Phys. 100, 5093, 5106, 101, 6157 (1994)). They have attempted to apply the effective potential directly to the calculation of the dynamics of the particles. This could lead the way to an extension of the effective potential method to the field of molecular dynamics. It is clear that the limits of the effective potential method have not yet been reached. We expect to see much more work in this field. References Acocella, D., G.K. Horton and E.R. Cowley (1995), Phys. Rev. B 51, 11406. Aubry, S. (1975), J. Chem. Phys. 62, 3217. Born, M. and K. Huang (1954), Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford). Choquard, P.E (1967), The Anharmonic Crystal (Benjamin, New York). Cowley, E.R. (1984), Phys. Rev. B 28, 3160. Cowley, E.R., E. Freidkin and G.K. Horton (1994), Ferroelectrics 153, 43.

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Cowley, E.R. and G.K. Horton (1987), Phys. Rev. Lett. 58, 789. Cowley, E.R. and G.K. Horton (1992), Ferroelectrics 136, 157. Cuccoli, A., V. Tognetti and R. Vaia (1990), Phys. Rev. B 41, 9588. Cuccoli, A., V. Tognetti and R. Vaia (1991), Phys. Rev. A 44, 2734. Cuccoli, A., V. Tognetti, P. Verrucchi and R. Vaia (1992a), Phys. Rev. A 45, 8418. Cuccoli, A., A. Macchi, M. Neumann, V. Tognetti and R. Vaia (1992b), Phys. Rev. B 45, 2088. Cuccoli, A., M. Spicci, V. Tognetti and R. Vaia (1992c), Phys. Rev. B 45, 10127. Cuccoli, A., V. Tognetti, A.A. Maradudin, A.R. McGurn and R. Vaia (1992d), Phys. Rev. B 46, 8839. Cuccoli, A., V. Tognetti, A.A. Maradudin, A.R. McGurn and R. Vaia (1993), Phys. Rev. B 48, 7015. Cuccoli, A., V. Tognetti and P. Verucchi (1992e), Phys. Rev. B 46, 11601. Doll, J.D., R.D. Coalson and D.L. Freeman (1985), Phys. Rev. Lett. 55, 1. Feynman, R.P. and A.R. Hibbs (1965), Quantum Mechanics and Path Integrals (McGraw-Hill, New York). Feynman, R.P. (1972), Statistical Mechanics (Benjamin, New York, 1972; Addison-Wesley, Reading, MA, 1988). Feynman, R.P. and H. Kleinert (1986), Phys. Rev. A 34, 5080. Giachetti, R. and V. Tognetti (1985), Phys. Rev. Lett. 55, 912. Giachetti, R. and V. Tognetti (1986), Phys. Rev. B 33, 7647. Giachetti, R., V. Tognetti and R. Vaia (1988a), Phys. Rev. A 37, 2165. Giachetti, R., V. Tognetti and R. Vaia (1988b), Phys. Rev. A 38, 1521. Giachetti, R., V. Tognetti and R. Vaia (1988c), Phys. Rev. A 38, 1638. Gillis, N.S., N.R. Werthamer and T.R. Koehler (1968), Phys. Rev. 165, 951. Gillis, N.S. and T.R. Koehler (1974), Phys. Rev. B 9, 3806. Goldman, V.V., G.K. Horton and M.L. Klein (1968), Phys. Rev. Lett. 21, 1527. Gursey, E (1950), Proc. Cambridge Philos. Soc. 46, 182. Hader, M. and EG. Mertens (1986), J. Phys. A 19, 1913. Horner, H. (1974), in: Dynamical Properties of Solids, Vol. 1, Ed. by G.K. Horton and A.A. Maradudin (Elsevier, New York). Horton, G.K. (1962), Amer. J. Phys. 36, 93. Horton, G.K. (1976), in: Rare Gas Solids, Ed. by M.L. Klein and J.A. Venables (Academic Press, New York). Janke, W. and H. Kleinert (1986), Phys. Lett. A 118, 371. Janke, W. and B.K. Cheng (1988), Phys. Lett. A 129, 140. Jhon, M.S. and J.S. Dahler (1977), J. Chem. Phys. 68, 812. Kanney, L.B. and G.K. Horton (1974), in: Proc. Conf. on Quantum Crystals, Ed. by E.L. Andronikashvilli (Tbilisi, USSR). Klein, M.L. and G.K. Horton (1972), J. Low Temp. Phys. 9, 151. Kleinert, H. (1986a), Phys. Lett. A 118, 195. Kleinert, H. (1986b), Phys. Lett. A 118, 267. Kleinert, H. (1986c), Phys. Lett. B 181, 324. Kleinert, H. (1992), Phys. Lett. B 280, 251. Liu, S. (1992), Thesis, Rutgers, the State University. Liu, S., G.K. Horton and E.R. Cowley (1991a), Phys. Lett. A 152, 79. Liu, S., G.K. Horton and E.R. Cowley (1991b), Phys. Rev. B 44, 11714. Liu, S., G.K. Horton, E.R. Cowley, A.R. McGurn, A.A. Maradudin and R.E Wallis (1992), Phys. Rev. B 45, 9716. Kleinert, H. (1993), Phys. Lett. A 173, 332.

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Lovesey, S.W. and R.A. Meserve (1973), J. Phys. C 6, 79. McGurn, A.R., P. Ryan, A.A. Maradudin and R.E Wallis (1989), Phys. Rev. B 40, 2407. McGurn, A.R. in: Dynamical Properties of Solids, Vol. 7, Ed. by G.K. Horton and A.E. Maradudin (Elsevier, Amsterdam), p. 1. Mori, H. (1965a), Progr. Theor. Phys. 33, 423. Mori, H. (1965b), Progr. Theor. Phys. 34, 399. Peierls, R. (1938), Phys. Rev. 54, 918. Salje, E.K.H., B. Wruck and H. Thomas (1991), Z. Phys. B 82, 399. Schneider, T. and E. Stoll (1980), Phys. Rev. B 22, 5317. Shukla, R.C. and E.R. Cowley (1985), Phys. Rev. B 31, 372. Srivastava, S. and Vishwamittar (1991), Phys. Rev. A 44, 8006. Takahashi, H. (1942), Proc. Phys. Math. Soc. Jpn 24, 60. Toda, M. (1967a), J. Phys. Soc. Jpn 22, 431. Toda, M. (1967b), J. Phys. Soc. Jpn 23, 501. van Hove, L. (1954), Phys. Rev. 95, 249. Werthamer, N.R. (1969), Am. J. Phys. 37, 763. Zhu, Z., S. Liu, G.K. Horton and E.R. Cowley (1992), Phys. Rev. B 45, 7122.

CHAPTER 3

Unusual Anharmonic Local Mode Systems

A.J. SIEVERS

J.B. PAGE

Laboratory of Atomic and Solid State Physics and the Materials Science Center Cornell University Ithaca, NY 14853-2501 USA

Department of Physics and Astronomy Arizona State University Tempe, Arizona 85287-1504 USA

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

9 Elsevier Science B.V., 1995

137

This Page Intentionally Left Blank

Contents 1. Introduction

141

1.1. Impurity modes in crystals 141 1.2. Localized modes in perfect anharmonic lattices ,

143

Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag + 147 2.1. Initial experiments 147 2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics 182 2.3. Pocket gap mode experiments 193 2.4. The (3, 3', 3") and the quadrupolar deformability models 199 2.5. Discussion and conclusions 203

,

Intrinsic localized modes in perfect anharmonic lattices 3.1. One-dimensional monatomic lattices 207 3.2. One-dimensional diatomic lattices 239

4. Speculation

243

4.1. ILMs and anomalous defect properties 4.2. ILMs and other properties 247 5. Conclusion

251

6. Acknowledgements Note added in proof References

251 251

252

139

244

206

This Page Intentionally Left Blank

We dedicate this chapter to LUDWIG GENZEL and the late HEINZ BILZ for their pioneering experimental and theoretical studies of numerous key aspects of lattice dynamics in crystals

1.

Introduction

1.1. Impurity modes in crystals The early far infrared measurements with grating instruments on the low temperature properties of defects in alkali halides produced a variety of sharp features associated with local modes (Sch~ifer 1960), gap modes (Sievers et al. 1965) and resonant modes (Sievers 1964; Weber 1964). The sharp resonant modes were somewhat of a surprise, even though the possibility of strongly coupled impurity-induced lattice resonances had been proposed in earlier theoretical work (Brout and Visscher 1962; Visscher 1963; Dawber and Elliott 1963). The observation of these low-lying sharp spectral features was quickly followed by more accurate measurements using the fourier transform IR technique (Sievers 1969) and somewhat later by Raman scattering studies. (Klein 1990 has recently reviewed this topic.) A variety of defect systems has been studied in detail, and much of that work has been reviewed from both the experimental (Barker and Sievers 1975; Bridges 1975) and theoretical (Stoneham 1975; Bilz et al. 1984) points of view. The findings are that most localized vibrational modes in defect-lattice systems behave in the expected fashion; that is, the modes identified with impurity motion can be interpreted from the experimental studies as harmonic or slightly anharmonic oscillators. However, for some systems the modes have anomalous properties that place them outside of the bounds of that interpretation. The vibrational properties of the light Li + impurity in KC1 or KBr are two defect-lattice systems that have received a great deal of attention over the years. KCI:Li + has been studied mainly because the impurity ion is off-center in the (111) directions so that low frequency tunneling states play a prominent role (Narayanamurti and Pohl 1970; Kirby et al. 1970; Devaty and Sievers 1979; Wong and Bridges 1992). On the other 141

142

A.J. Sievers and J.B. Page

Ch. 3

hand, an equal effort has gone into examining the resonant mode properties of KBr:Li + because now the impurity ion is barely stable at a normal lattice site (Sievers and Takeno 1965; Page 1974; Kahan et al. 1976). The near instability demonstrated by the single ion spectrum is fullfilled for Li + pairs in KBr which are found to be off-center and to tunnel coherently from site to site (Greene and Sievers 1982, 1985). Although these point defect systems have received ample attention in the literature and are thought to be understood there are other anomalous systems which have not played a central role since the paucity of available data did not allow their unusual features to be recognized as crucial. There are two "simple" heavy defect ion-lattice systems in particular, which display unusual vibrational properties: KI:Ag + and RbCI:Ag +. At low temperatures both of these systems show defectinduced lattice modes in the far IR. Early measurements showed that the Ag + impurity in KI was on-center at a K + lattice site, while the Ag + impurity in RbC1 was off-center in a (110) direction and exhibited tunneling between equivalent sites (Barker and Sievers 1975). The dynamics of this heavy impurity had been assumed to be similar to those found for the Li + impurity in KBr and KC1, respectively. In the last decade, paraelectric resonance studies of the tunneling modes in RbCI:Ag + as a function of hydrostatic pressure indicate unusual intensity dependences for some transitions. These have been interpreted as a signature for the existence of an on-center configuration nearby in energy (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). For KI:Ag +, the early observation that the IR active resonant mode and the later observation that the entire T = 0 K impurity-induced spectrum disappear as the temperature is increased to 25 K has been shown to be associated with the Ag + ion moving from the on-center to an off-center position with increasing temperature (Sievers and Greene 1984). These results for KI:Ag + may indicate that at T = 0 K there is both an on-center configuration and an off-center one with nearly the same energy. The very unusual behavior of these two lattice-defect systems has called into question the underlying fundamentals of standard defect phonon theory. Where is the flexibility in the Lifshitz theory (Lifshitz 1956) for the existence of multiple elastic configurations? To address this fundamental lattice dynamics question, much theoretical and experimental effort has gone into reexamining in greater detail the T = 0 K properties of the simplest of the two systems, namely on-center KI:Ag +, which may have at least two low-lying configurations. The approach that has been used is first to apply the harmonic shell model in an attempt to explain the observed spectroscopic properties of this lattice defect system, excluding the temperature dependence. This has proved to be quite successful, and with the addition of some anharmonicity as described

w1

Unusual anharmonic local mode systems

143

here, it gives a quantitative description of the local dynamics of the KI:Ag + system at T = 0 K. The resulting comparison between theory and experiment in unprecedented detail has led to the prediction and verification of an unexpected new class of defect modes, called "pocket" gap modes (Sandusky et al. 1991). These defect modes have the unusual property that the maximum vibrational amplitude is not at the impurity but is localized at lattice sites well removed from the impurity. The experimental investigation of the isotope effect (Sandusky et al. 1991; Sandusky et al. 1993a), stress shifts (Rosenberg et al. 1992) and electric field Stark effect (Sandusky et al. 1994) provide sufficient experimental information to demonstrate that a dynamically-induced electronic deformability of the Ag + impurity plays an essential role in the dynamics. Upon analysis of much of the data, one finds that a convincing explanation of the T = 0 K experimental far IR and Raman defect-induced spectra can be made within the Lifshitz perturbed phonon framework, in terms of a slightly anharmonic shell model in which the coupled defect/host system is nearly unstable. In contrast, while a variety of phenomenological anharmonic models have been introduced to account for the observed temperature effects, it is still not possible to present a consistent dynamical picture of the temperaturedependent behavior of this system. One purpose of this review is illustrate how the temperature dependence models fail. The end result of these tightly woven arguments will be that the observed anomalous temperature dependence of the spectra appears to fall truly outside the bounds of current defect mode theory. A primary function of this review is to gather together all of the experimental and theoretical data on this particular on-center system, so that the reader can inspect, understand and appreciate what lies beyond the framework of current impurity dynamics theory. 1.2. Localized modes in perfect anharmonic lattices While it is not surprising that the loss of periodicity in defect crystals leads to localized vibrational phenomena, the standard description of the dynamics of defect-free periodic lattices in terms of plane wave phonons is so deeply ingrained that it was surprising to many researchers when it was argued theoretically in 1988 (Sievers and Takeno 1988; Takeno and Sievers 1988) that the presence of strong quartic anharmonicity in perfect lattices can also lead to localized vibrational modes, henceforth called "intrinsic localized modes" (ILMs). These papers studied the classical vibrational dynamics of a simple one-dimensional monatomic chain of particles interacting via nearestneighbor harmonic and quartic anharmonic springs, and they used a "rotating wave approximation" (RWA), in which just one frequency component was

144

A.J. Sievers and J.B. Page

Ch. 3

kept in the time dependence. For the case of sufficiently strong positive quartic anharmonicity, it was found that the lattice could sustain stationary localized vibrations having the approximate mode pattem A(..., 0 , - 1 / 2 , 1 , - 1 / 2 , 0 . . . . ), where A is the amplitude of the "central" atom. This pattem is odd under reflection in the central site and is "optic mode"-like, in that adjacent particles move 7r out of phase. As with all nonlinear vibrations, the ILM frequencies are amplitude dependent. The ILMs can be centered on any lattice site, giving rise to a configurational entropy analogous to that for vacancies, and some thermodynamic ramifications were explored in the preceding two references and by Sievers and Takeno (1989), along with speculations about the possible presence of ILMs in strongly anharmonic solids such as solid He and ferroelectrics. A provocative, but still unproven speculation was that these modes might be involved in the anomalous thermally-driven low temperature on-center --+ off-center transition of the nearly unstable KI:Ag + system, which is the other main subject of this chapter. It is straightforward to check the theoretical predictions using molecular dynamics simulations to solve the equations of motion numerically. This of course avoids the RWA and other approximations used by Sievers and Takeno (1988), and fig. 1 shows the results for a 21-particle monatomic chain with periodic boundary conditions and harmonic plus quartic anharmonic nearest-neighbor interactions. Each panel shows the time-evolution of the particle displacements for the same initial displacement pattem, namely that of the predicted ILM pattem given above, centered on the middle particle. The initial velocities were zero. The two panels give the subsequent displacements for the case of (a) purely harmonic interactions and (b) harmonic plus strong quartic anharmonic interactions. As expected, in (a) the initial displacements of the purely harmonic system spread into the lattice, since they are a linear combination of independently evolving homogeneous plane-wave normal modes. In sharp contrast, for the anharmonic system (b), the initial displacement pattem is seen to persist, and the theoretical predictions are strikingly verified. In the few years since the appearance of the 1988 Sievers and Takeno papers, there has been a rapidly increasing number of theoretical studies published related to ILMs. Interestingly, during the preparation of the present chapter, we discovered an earlier, brief paper (Dolgov 1986), which obtains the Sievers-Takeno solution and the analogous even-parity ILM solution, derived independently by Page (1990). The Dolgov paper has gone unnoticed by subsequent workers.

w

Unusual anharmonic local mode systems

145

(a) harmonic 0

o it} o

EL

9

.... J T

T - - _ -

....

-

5 0

r

tO EL

----------x

10 t 0

~

5

-

-

~ 10

15

(b) anharmonic 10 rr

._o

5

O

0

to

-5

EL

"s

EL

-10

0

~i

1'0

15

time (units of 2~'(Om)

Fig. 1. Molecular dynamics simulations for a monatomic chain with (a) purely harmonic and (b) harmonic plus quartic anharmonic nearest-neighbor interactions. For both panels, the initial configuration is the odd-parity ILM displacement pattern A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ), with the amplitude on the central particle being 0.1 of the equilibrium nearest-neighbor distance a. The initial velocities are zero. The masses are 39.995 amu, the lattice constant is 1 ,~ and k2 = 10 eV/,~2. The dashed lines in (a) are guides which delineate the spreading of the initial localized displacement pattern into the harmonic lattice. For (b) the value of the anharmonicity parameter A4 = k4A2/k2 is 1.63, well within the range when this ILM should be a valid solution. An implementation of the fifth-order Gear predictor-corrector method was used for the MD runs (Allen and Tildesley 1987), with the time step taken to be 1/180 of the period of the maximum harmonic frequency ~Om. In both panels, the particle displacements are magnified, for clarity.

T h e I L M p a p e r s p u b l i s h e d since 1988 r o u g h l y divide into t w o b r o a d and o v e r l a p p i n g c a t e g o r i e s . T h e first o f these i n v o l v e s the relation o f the n e w excitations to the g e n e r a l b e h a v i o r o f discrete n o n l i n e a r systems, with e m p h a s i s on c o n n e c t i o n s to soliton-like b e h a v i o r in lattices. T h e s e c o n d c a t e g o r y is m o r e tightly f o c u s e d on g e n e r a l i z a t i o n s and e x t e n s i o n s o f the 1988 papers, in o r d e r to d i s c o v e r the p r o p e r t i e s o f h i g h l y - l o c a l i z e d l a r g e - a m p l i t u d e I L M s and also to assess their likely i m p o r t a n c e for real solids. T h e general diffi-

146

A.J. Sievers and J.B. Page

Ch. 3

culty of dealing with complex nonlinear dynamical systems is reflected in the diversity of theoretical approaches used, which range from purely analytic studies to completely numerical molecular dynamics simulations. In our opinion the former should, whenever possible, be accompanied by the latter, since the unfamiliarity of the phenomena can easily lead to unwarranted approximations being made in analytical work. Computer studies of the dynamics of nonlinear lattices extend back to the pioneering work on one-dimensional chains by Fermi, Pasta and Ulam (1955), and they have been a vital adjunct to much of the subsequent work on solitons in anharmonic lattices. Discussions of both analytic and numerical aspects of soliton behavior in one-dimensional lattices with intersite cubic and quartic anharmonic interactions are found in Flytzanis et al. (1985) for the monatomic case and in Pnevmatikos et al. (1986) for the diatomic case. Of particular interest for ILMs is their relationship to lattice envelope solitons. Stationary lattice solitons were studied theoretically more than 20 years ago by Kosevich and Kovalev (1974) for anharmonic chains with cubic and quartic onsite and intersite anharmonicity. The emphasis was on onsite anharmonicity, and solutions were obtained for low-amplitude stationary envelope solitons whose spatial extent is broad compared with the lattice constant. Later MD simulations by Yoshimura and Watanabe (1991) for linear chains with quadratic plus quartic interactions showed that the stationary envelope soliton solutions account well for ILMs whose spatial widths are sufficiently broad, e.g., 10 or more lattice constants. However, as was subsequently emphasized by Kosevich (1993a), the stationary envelope soliton solutions do not describe the large-amplitude highly localized ILMs, such as that of fig. 1. Some recent papers dealing with general aspects of localized dynamics in discrete nonlinear systems and ILM-related soliton studies are: Flach and Willis (1993), Flach et al. (1993), Kivshar and Campbell (1993), Chubykalo and Kivshar (1993), Claude et al. (1993), Kosevich (1993b, c), and Cai et al. (1994). Some representative papers on ILM solutions in lattices with nearest-neighbor intersite anharmonicity and not already cited are: Takeno et al. (1988), Burlakov et al. (1990a, b, d), Kiselev (1990), Takeno (1990), Bickham and Sievers (1991), Takeno and Hori (1991), Takeno (1992), Fischer (1993), Aoki et al. (1993), Chubykalo et al. (1993), and Kiselev et al. (1993). Some representative studies discussing traveling ILMs are given in Takeno and Hori (1990), Bickham et al. (1992), Hori and Takeno (1992), Sandusky et al. (1993b), Bickham et al. (1993) and Kiselev et al. (1994b). A detailed study of the stability of ILMs was given by Sandusky et al. (1992), and the relationship between ILMs and the stability of homogeneous lattice phonon modes has been investigated by Burlakov et al. (1990c), Burlakov and Kiselev (1991), Dauxois and Peyrard (1993), Kivshar (1993),

w

Unusual anharmonic local mode systems

147

Sandusky et al. (1993b), and Sandusky and Page (1994). Energy transport and/or the effects of defects or disorder are discussed by Bourbonnais and Maynard (1990a, b), Takeno and Homma (1991), Kivshar (1991), Kiselev et al. (1994a, b) and Zavt et al. (1993). An important recent development has been the generation of stable ILMs in the presence of cubic anharmonicity (Bickham et al. 1993), and the subsequent inclusion of realistic potentials such as Lennard-Jones, Morse, or Born-Mayer plus Coulomb (Kiselev et al. 1993; Sandusky and Page 1994; Kiselev et al. 1994a, b). A primary effect of including the odd-order potential energy terms is that the ILMs are accompanied by localized DC lattice distortions. It is not our aim here to give a critical assessment of the rapidly growing literature in this area. Rather, we wish to describe some of the important phenomena and results as simply as possible, referring to the literature for details. We will emphasize the work of our own groups, and we will restrict our attention to one-dimensional monatomic or diatomic chains of particles interacting via quadratic, cubic and quartic springs, or via realistic potentials. Throughout, we will stress the phenomena themselves rather than the theoretical methods, which are well-described in the original papers.

0

Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag +

2.1. Initial experiments

2.1.1. Intrinsic temperature dependent absorption in KI A number of investigators have shown that the far infrared transmission of pure alkali halide crystals at frequencies below the reststrahlen region is controlled by two-phonon difference band energy-conserving absorption processes (Stolen and Dransfeld 1965; Eldridge and Kembry 1973; Eldridge and Staal 1977; Hardy and Karo 1982). The probability of absorption of a phonon is proportional to the phonon occupation number n~ and that of emission of a phonon is proportional to (1 + n~). For a two-phonon difference process in the absorption spectrum, the absorption of a photon is accompanied by the emission of one phonon (1 + n l) and the absorption of another phonon n2, giving a temperature-dependent factor (1 + n l)n2. The net absorption is obtained by correcting for spontaneous photon emission by subtracting the reverse process, namely (1 + n2)nl, so that the resulting temperature dependence is proportional to In2- nl I. Since thermal phonons are required to generate such an absorption process, pure crystals are extremely transparent at low temperatures and low frequencies, far from the

A.J. Sievers and J.B. Page

148

Ch. 3

reststrahlen band. However, since we are interested in examining the temperature dependence of defect induced absorption, these temperature-dependent properties of the pure crystal need to be quantified as well. As long as the sample does not have parallel sides, the absorption coefficient c~(w) is determined from the transmission expression It --

(1 =

-

R) 2

exp[-c~(w)/]

=

t(w)

Io

,

1 --

R 2

( 2 . 1 a)

exp[-2o~(w)/]

where the transmission coefficient t(w) is defined as the ratio of the transmitted to the incident intensity, R is the reflectivity coefficient and l is the sample length. When R is small and/or ~(w)l is large, the second term in the denominator can be ignored and eq. (2.1a) reduces to

t(w) ~ (1

-

R) 2

exp[-a(w)/].

(2.1b)

Figure 2 shows the temperature-dependent absorption coefficient a(w) for KI over a wide range by using three different ordinate scales versus frequency (Love et al. 1989). For the samples used here, the frequency and temperature-dependent absorption coefficient a(w, T) is obtained by dividing the spectra taken at the higher temperature by one at 4.2 K. From eq. (2.1b) this gives 1

c~(w, T) - c~(w,4.2 K) = -7 In[It(w, T)/It(w, 4.2 K)].

(2.2)

l

In fig. 2(a), (b) the absorption coefficient at 4.2 K is small enough on both of these scales that it can be ignored, hence these data give the temperature dependence of the difference band absorption coefficient directly. The observed spectral structure agrees with that predicted by early workers, including the weak difference band contribution at 11.3 cm -1 which stems from vertical transitions between the s163 symmetry branches in the frequency region where the dispersion curve slopes are nearly parallel. The sharp edge at 31 cm -1 in fig. 2(c) is due to relatively strong ZI(LA)-Z4(TO) two-phonon difference band transitions across the acousticoptic phonon gap.

2.1.2. Resonant mode IR absorption Because of the freezing out of the intrinsic difference band absorption processes for frequencies below the reststrahlen region in alkali halides at

w

Unusual anharmonic local mode systems . (0)

i

i

i

i

I

I

I

/82K

I

40-

-

.

/64K

20

~/47'

K.-

,

"T

(b)

E

2

149

55K

5K

8

C

._~

-

llJ O O

/

4-

J]

e--

O .IQ

<

S

O

(c) 1.6-

82K

0.8-

0

20

40 Frequency

60

80

( c m -a)

Fig. 2. Temperature dependence of the absorption coefficient of pure KI at different temperatures. The sample temperature is varied from 4.2 to 82 K. The spectral resolution is 0.44 cm-1 for frequencies between 3 and 26 cm -1 , and is 0.35 cm -1 at higher frequencies. The spectral features are produced by two-phonon energy-conserving difference band processes. (After Love et al. 1989).

low temperatures, transmission measurements on impurity-doped crystals at p u m p e d liquid helium temperatures give directly the impurity induced absorption coefficient c~i(w). The Ag + ion was the first impurity to be studied at far infrared frequencies in these hosts below the reststrahen region (Sievers 1964; Weber 1964). A simple method in which the lattice-constant

A.J. Sievers and J.B. Page

150

Ch. 3

dependence of an impurity resonant mode can be seen is to plot the log of the observed mode frequency for a fixed impurity ion versus the log of the host lattice constant. This is presented for Ag + in fig. 2. The straight line through all but one of these frequencies indicates that this defect resonant mode obeys a Mollwo-Ivey-like rule (Hayes and Stoneham 1985) up to at least the large lattice constant region (Sievers 1968). This straight line implies that the dependence of the frequency upon the lattice constant can be written as

0 lnw 01na

3A = w '

(2.3)

where a is the lattice constant, w is the resonant frequency and A is the hydrostatic coupling coefficient. For five of the alkali halide hosts in fig. 3, the quantity 3A/w = 3; however, KI:Ag + is seen to give a much lower frequency than that projected from this straight line. The A coefficient of KI:Ag + can be estimated from uniaxial stress measurements, with the result that the quantity 3A/w in eq. 2.3, is equal to 68. This corresponds to a very different slope, as shown by the line at the KI:Ag + point in fig. 3. It is perhaps fortuitous that the resonant mode for KBr falls on the 3A/w = 3

! 601

I- oc,

~

I

I

,o F

I >'351

"-"

~,,~No I

"~Br

~ 3o

25

20-

16 z'~

3.2s LATTICE CONSTANT(A) I

3.00

I

,

,

.

t KI 3.50

Fig. 3. Center frequency of the Ag+ resonant mode for different alkali halide crystals as a function of the host lattice constant. For five different alkali halide lattices the resonant mode frequency can be fit to a straight line. The line centered at the KI:Ag+ point is from uniaxial stress measurements. (After Sievers 1968).

w

Unusual anharmonic local mode systems

151

line, since RbC1, which has nearly the same lattice constant, is an off-center system (Kirby et al. 1970). [The multiple absorption bands associated with excited tunneling transitions in RbCI:Ag + (Barker and Sievers 1975) are not included in fig. 3]. The resonant mode absorption bands associated with the Ag + ion, excluding KI, are found to be relatively temperature independent, at least for thermal energies up to the resonant mode energy. One possible reason that the KI:Ag + resonant mode frequency does not follow the 3A/w = 3 line in fig. 3 is that KI:Ag + might not be an on-center defect system. This possibility has been eliminated by studying the uniaxial stress dependence of this absorption line. A typical splitting for stress applied in the [100] direction for two different polarizations of the radiation is shown in fig. 4. The area under each of the three curves appears to be roughly the same. By making such measurements for uniaxial stress along the [100], [110] and [111] crystal directions, it is possible to distinguish between a frozen off-center system, a tunneling system or an on-center system (Nolt and Sievers 1968). From the number of components observed and the absence of any stress-induced dichroism, it can be concluded that the Ag + defect has either Oh or Td symmetry. In addition, these measurements provide experimental values for the three even-symmetry stress coupling coefficients between the resonant mode and the lattice: A(hydrostatic), B(tetragonal), and C(trigonal). 2

v

I

i

!

I

q .--,.

.

E

I

~' [o~o]

I

'

I

K I : Ag

4-

(b)

[I 00]

Zero Stress

Stress

(J r

._e .u_ ' r. - I

---I I'--

(o)

/

0

(.)

~' Doo]

\ \ 0

I

I 16

,

n

IB

Frequency

(cm"l)

20

J22

Fig. 4. KI:Ag + absorption coefficient for a [100] stress of 3.3 kg/mm 2. The sample temperature is 1.5 K. The instrumental resolution is 0.3 cm -1. The polarizations are identified in the figure. The integrated line strength is 1.03, 1.23 and 1.1 4-0.1 cm -2 for curves (a), (b) and (c), respectively. (After Nolt and Sievers 1968).

A.J. Sievers and J.B. Page

152

Ch. 3

Electric field measurements on the IR active resonant mode confirmed that the defect had Oh symmetry but in addition produced surprises (Kirby 1971). The electric field induced absorption coefficient for KI:Ag + is shown in fig. 5. The negative Aa region identifies components associated with the zero-field spectrum that have changed. The complex difference spectrum in the resonant mode region arises from the Tlu mode at 17.3 cm -1 mixing with an even mode of Eg symmetry at 16.35 cm -1. These measurements also show another E-field induced line at 25 cm -1, which has been interpreted as an A lg symmetry mode. In addition, absorption features at larger frequencies, namely 30 and 44 cm -1, decrease in strength with applied Efield. The presence of the Eg symmetry mode was confirmed by Raman scattering studies, but no trace of the Alg mode was found. An example of the temperature-independent behavior found for the Ag + doped hosts that obey the Mollwo-Ivey-like rule is seen in fig. 6, which presents the spectrum of NaI:Ag + at 1.2 and 24.2 K (Greene 1984). At the higher temperature the resonant mode centered at 36.7 cm -1 appears on a two-phonon difference-band background spectrum. The resonant mode strength itself is expected to be temperature independent if the mode behaves as a harmonic oscillator. From the Kramers-Kronig dispersion relations, a general f sum rule for the absorption coefficient can be found (Smith 1985):

cx3

f0

71"~ dw c~i(w) = 2c = constant,

(2.4a)

where the exact expression for the plasma frequency squared or oscillator strength depends on the specific model employed. As long as the magnitude of the effective charge which produces the absorption coefficient is temperature independent and the integral is taken over the complete IR absorption spectrum, then the above sum rule holds. In particular, it applies to anharmonic as well as to harmonic oscillators. When anharmonicity is included the transition of interest may shift, broaden and change in strength, but the anharmonicity will also produce other lines in the spectrum (e.g. sidebands, overtones) which will ensure the constant sum rule (Bilz et al. 1984). A specific form of eq. (2.4a), appropriate to impurity-induced absorption over a frequency interval encompassing a line for the case of a host with dielectric constant e0 is (Sievers 1964)

dw Oq(W)

--

2c

=

CMQx/C~

3

'

(2.4b)

A.J. Sievers and J.B. Page

154 32 F

KBr:Li//'~

30~

/''o jw " ~ ' ~ l :cu "//~'~~

-'" 281 o ' E 26

~

P

~' 241-

," 18e~

I I

A

/o

_.I

i6~- ,,e~,~ 14~"~ 0

Ch. 3

~

~h--e-.-14

I , 2 Pressure,

I 3

,

I I 4 in kilobars

CsI : TI I 5

i

I 6

I

I 7

I

Fig. 7. Hydrostatic pressure-induced shift in the resonant mode frequency of four different resonant mode systems. The experimental uncertainties are +0.1 cm-] in the resonant mode frequency and -t-10% in the pressure. The sample temperature is 4.2 K. (After Patterson 1973). where e* is the effective charge and MQ is an effective mass associated with the mode. The fact that the resonant mode shown in fig. 6 is nearly temperature independent indicates that this defect system is not anharmonically coupled to other degrees of freedom, for if it were the area under the resonant mode transition would decrease with increasing temperature and the area difference would transfer into its sidebands such that the total area remained constant (Klein 1968; Barker and Sievers 1975). Hydrostatic pressure measurements up to 7 kbars on the resonant mode feature provide additional evidence that the simple absorption spectrum is not associated with a tunneling defect, since none of the complex spectral behavior previously found for the tunneling system KCI:Li + was observed here (Kahan et al. 1976). Figure 7 shows the hydrostatic frequency shifts observed for different on-center defect systems (Patterson 1973). For all of these systems only one IR active resonant mode is found at each pressure. One curious feature is that the effective resonant mode force constant for KI:Ag + varies nonlinearly with strain (for Aa//a up to 1.6%), while all other low lying resonant modes show a linear dependence over the same range. Figure 8 shows the resonant mode absorption spectrum c~(w) for a KI:Ag + sample at atmospheric pressure and at 1.9 kbars, respectively, as measured with respect to an empty pressure cell (Patterson 1973). Because of the different transmission charactistics of the empty cell, the ordinate gives the absorption coefficient in arbitrary units; however, since the absorption spectra at the two pressures were measured sequentially, the absorption strengths can be compared. Within experimental accuracy the area under the line

w

Unusual anharmonic local mode systems I

!

0.3 . - .

tJ

0.2

........

'

I

'

I

'

KI:Ag+ 175 kV/cm 4.2OK -'11"-

1

I

J

o.~

153

-

j

o

/'-"~_/~

'"

-o.i -0.2

!

IO

i

20

I

t

I

=

30 40 FREQUENCY ( cm-I)

~

-

50

Fig. 5. Electric-field-induced absorption coefficient for KI:Ag +. A negative Aa indicates a decrease in absorption with applied field, while a positive Aa indicates a field-induced increase. The complex line shape in the 17 cm -1 region indicates that two resonant modes of opposite parity are being mixed by the odd-parity electric field. Here Ei~llEdcll[100], and the instrumental resolution is 0.55 cm -1. (After Kirby 1971).

3.0

NaI +

A

3E v

!

0.2 %

..... 1 . 2 K

AgI

I

........ 24.2 K

O

..,./-

,.e...

,-" 2 . 0

.~. r

-

NI,O (..) C:

.9 .a,...

1.0

O. i,_ 0 U') ,JO

<:[ 0 ,, 20

I

30 Frequency

1

40

50

(cm -I)

Fig. 6. Temperature dependent Ag+-induced absorption in the resonant mode frequency region of NaI. The strength of the resonant mode centered at 36.7 cm -1 is relatively temperature independent, while the two phonon difference band absorption clearly increases between the two temperatures shown. The instrumental resolution is 0.52 cm -1. (After Greene 1984).

w

Unusual anharmonic local mode systems

r 0 "On O. t,. 0 r JO

155

9 ~

9 00 4~ OIbO0000 O0 Ollo

O0

O0

Oo~

| OOO O

13

I, 15

1 7 3 c m -I

216cm -t

I 17

I 21

I 19

Frequency,

I 25

co

I 25

I 27

I 29

(cm -I)

Fig. 8. Strength of the KI:Ag + resonant mode at two different hydrostatic pressures. The pressures are 0 and 1.9 kbars, respectively. (After Patterson 1973).

appears to scale with frequency over this pressure range. This pressuredependent area is a surprising result since according to eq. (2.4a, b) for a single absorption line the area should remain fixed, independent of frequency (ignoring the small decrease in e0 with increased pressure). The uniaxial stress measurements, admittedly at lower pressures, did not appear to give such an effect. Although the eigenvector of the resonant mode would be expected to change with hydrostatic pressure, the magnitude of the effective charges in eq. (2.4b) are expected to remain fixed over this pressure range. Clearly, the possibility of this pressure-dependent effect should be examined with more experimental precision. 2.1.3. Anomalous temperature dependence of the resonant and gap modes

Figure 9 clearly indicates that the temperature dependence of the KI:Ag + resonant mode strength is qualitatively different from that observed for the other Ag + resonant mode systems. The strength of the mode decreases rapidly with increasing temperature until by 21.6 K it seems to have disappeared into a slightly larger background spectrum (Sievers and Greene 1984). This temperature dependence of the line strength is independent of the Ag + concentration. Notice that between 1.2 and 9.3 K where about half of the strength has been lost, the center frequency of the mode remains nearly unchanged.

A.J. Sievers and J.B. Page

156

I

1.0

I

I

Ch. 3 I

_ KI" AgI

E O.8

"5 0.6

1.2 K

--it-

U

~,

0.4

9.3 K

o

<~ 0.2

f2,.6 K

,,~..~.,.....~...~.~....,r I0

15 Frequency

20 (cm "l)

25

Fig. 9. Temperature dependence of the absorption spectrum of KI + 0.2 mole% AgI in the frequency region of the T]u resonant mode. The three temperatures are identified in the figure. At high temperatures the broad nonresonant absorption coefficient is about 3% of the low temperature resonant mode peak value. (After Sievers and Greene 1984). For emphasis it is helpful to contrast this unusual temperature dependence of the resonant mode strength with the temperature-independent properties of a related defect system. Each of the five temperature-independent Ag + induced resonant mode frequencies in fig. 3, when normalized to the appropriate host lattice Debye frequency, is much larger than observed for KI:Ag + (Wr/WD = 17.3/91 = 0.19). Perhaps below a certain value of this ratio a temperature-dependent strength occurs? The NaCI:Cu + resonant mode (Weber and Nette 1966) centered at 23.5 cm -1 provides a good test of this idea since now the ratio Wr/WD is 23.5/223 = 0.11. The temperature dependence of this mode's absorption is shown in fig. 10, where the temperature ranges from 4.2 K for curve (a) up to 77 K for curve (d). The resonant mode peak can easily be seen at each temperature. The center frequency and the linewidth are temperature dependent, as expected for a slightly anharmonic oscillator, but for at least the two lowest temperatures the strength appears to increase slightly with increasing temperature 8(21.5 K) /S(4.2 K) = 1.15. Because of the large width at still higher temperatures it is difficult to separate unambiguously the resonant mode and the two-phonon difference band absorption. At 42.6 K (curve c) the intrinsic two-phonon difference band absorption at the resonant mode center frequency is only 0.25 cm -1 (Love et al. 1989), compared to the resonant mode peak value of

w

Unusual anharmonic local mode systems 8

'i .........

157

I

._

89

caL , , "

:

:"lt

j

'G

~ u 0

9

.o

(b)

0

**

(a) .

!o

.

.

.

.

.

.

.

.

.

.

.

20

, i , ,

5o Frequency (crn-a)

|

40

Fig. 10. Temperature dependence of the absorption coefficient of NaCI+0.5 mole% CuC1 in the frequency region of the resonant mode. The four temperatures are (a) 4.2 K; (b) 21.5 K; (c) 42.6 K and (d) 77 K. At high temperatures the resonant mode resides on top of the intrinsic two phonon difference band absorption. The resolution is 0.43 cm-I . (After Sievers, unpublished). about 4 cm-1 but it is nevertheless clearly observable on the high frequency of the line in fig. 10. The unusual temperature-induced absorption strength change of the KI:Ag + resonant mode can be examined in a number of different ways to separate out the different contributions to the spectrum. One of these is to examine the strength change over AT jumps. Such changes can be seen with precision in fig. 11, where the absorption coefficient difference between neighboring temperatures as a function of frequency is presented (Greene 1984). The largest strength changes are seen to occur in the 6 K to 13 K region. Compared to the rapid resonant mode strength change with temperature, the line broadening and center frequency shift in this representation are again seen to be quite small. The broadening and shift also appear comparable to those measured in fig. 10 for the resonant mode of NaCI:Cu + at the two lowest temperatures. This strength change was first observed in 1965, with less precision (Takeno and Sievers 1965). In that work it was proposed that the resonant mode, through anharmonicity, is linearly coupled to other low-lying phonon

A.J. Sievers and J.B. Page

158

:(a)

Ch. 3

; all.2)-a(2.91

I I I

/,,%.~A

-~'-

A

~

-]

__1_

; ~

:

@

9

:~

-

(c)

IE

%;rJ

x .

eJ

l

I

I

.

I

I

a(8.2)-a(9.3)

,

-

.;

,

0 - ~

"1" II

a (10.9)'-a 3.7)ll (I .,,..,~

I /~

i!

IIII1~ ~ l l

Ii

. . . . . . . . . . . . . .

V

t I

~

(2(18.1)-Cl(21.6) -'

.

g

A

tl

V V .,_,-

-

',

I

.(d) .[ .,L

I

] i I

, I

I

I

r

I

I

I

L~-'"-,,~J_ !

I

I

I

,

1

~5

20

, I

~o

.....

Frequency

I ....

(cm -I)

--

:

v-

.

l

Fig. 11. Absorption coefficient differences between adjacent temperatures, vs. frequency for KI:Ag +. The instrumental resolution is 0.47 cm -]. (After Greene 1984).

modes. Such a coupling could remove strength from the "zero phononlike" resonant mode transition without contributing to the temperature dependence of the center frequency and linewidth. The temperature-dependent strength data fit a Debye-Waller-like factor with a T 2 in the exponent. If the sidebands extended over a large enough frequency interval they would be negligible in the experimental measurement. The temperature dependence produced by such a Debye-Waller-like factor has the form

I(T) = exp

-

SO~ 2

1+ 4

~

JO

e :~ -- 1

'

(2.5)

where S'0 = 9W2c/4Mv2hw 3 contains an acoustic spectrum cutoff frequency We which is less than the Debye frequency coD. Here 0c = hwc/kB, v is

w

Unusual anharmonic local mode systems

159

the Debye sound velocity, and M is the average ion mass appropriate to the Debye modes. For Oh symmetry the effective strain coupling parameter A is determined by the three coupling coefficients A(hydrostatic), B(tetragonal), C(trigonal) for the even-parity symmetry types. Note that it is only at very low temperatures that the T 2 behavior contributes to the exponent. When the measured strain coupling coefficients from the uniaxial stress measurements are used to determine A, the value is an order of magnitude too small to agree with the temperature dependent strength data (Alexander et al. 1970). Besides linear coupling between a resonant mode and a Debye spectrum, Alexander et al. (1970) also attempted to fit the temperature dependence of the absorption strength with a linear coupling to a single even-parity resonant mode of frequency DE. In this case the intensity of the zero phonon line is given by I(T) = e-'YEIo(CE),

(2.6)

where 7E = SE(2~E + 1)

(2.7)

CE = SEcsch(hf2E/2kBT).

(2.8)

and

Because Sz is now of order unity instead of N -1, where N is the number of atoms as for extended lattice modes, one must include the modified Bessel function Io(Co) in the modulation analysis. One of the conclusions of the work of Alexander et al. (1970) was that eq. (2.6) provides a much better fit to the observed temperature dependence than does eq. (2.5), given the known strain coupling coefficients. This work, together with the experimental work of Kirby (1971) on the presence of both odd and even-parity resonant modes in the KI:Ag + spectrum, seemed to indicate that the vibrational properties of point defects were indeed well understood. But it was then discovered by Greene (1984) in the early 1980's that the reported good agreement between the linearly coupled odd and even mode analysis of the temperature-dependent spectrum was in fact due to an erroneous numerical calculation. No set of model parameters in eqs (2.6), (2.7) and (2.8) can produce agreement with the observed temperature dependence of the KI:Ag + resonant mode. In retrospect, it is now easy to see why the linear coupling model cannot explain the observed temperature dependence. For linear coupling to either extended band modes or to localized modes, the temperature

160

A.J. Sievers and J.B. Page

Ch. 3

dependence in the exponent comes from a (2n + 1) occupation factor. At temperatures large compared to the characteristic modulation frequency of the phonons or the low lying even mode this term must vary as kBT which, even in the exponent, is too slow to account for the rapid disappearance of the line strength by 25 K. When the E-field dependence of the resonant mode was investigated by Kirby (1971), other silver-induced features in the absorption spectrum were found, most notably a gap mode centered at 86.2 cm -l but also other much weaker but identifiable peaks located at 29.8, 44.4, 51.0, 55.9 and 63.6 cm -1. The interconnection between these different features remained unexplored until temperature-dependence measurements on the complete impurity-induced spectrum were made (Greene 1984; Sievers and Greene 1984). Some key features can be identified in the Ag+-induced absorption spectra shown in fig. 12. The solid curve in fig. 12(a) shows the impurity-induced absorption coefficient c~(w) versus frequency at 1.2 K. The assignment of the different features is the same as that given earlier by Kirby (1971). One surprising result is that the entire impurity-induced spectrum is very temperature dependent, as shown by the dotted absorption curve for 11 K in fig. 12(a). With increasing temperature the strength of the resonant mode, combination bands, and gap mode all decrease, while a broad absorption band associated with intrinsic two-phonon difference band processes increases at high temperatures. Precise measurements of the temperature dependence of the absorption coefficient in the low-frequency region around the resonant mode show that the decrease in the resonant mode strength is accompanied by a corresponding increase in a broad absorption which appears to be nearly frequency independent from the smallest wave number measured (3 cm -1) up to at least 25 cm -1. The magnitude of this non-resonant absorption is proportional to the low temperature strength of the IR active resonant mode over a factor 40 in Ag + concentration (Sievers and Greene 1984). There are other spectral features which appear at elevated temperatures, as can be seen in fig. 12(b), where the difference in absorption coefficient Ac~(w) between a given temperature and the 1.2 K reference temperature is plotted versus frequency. The solid curve is for T = 3.4 K, and the dashed one is for 10 K. A positive Aa(w) in this figure indicates that the sample absorbs more at high temperatures than at 1.2 K at that frequency. In addition to the nonresonant absorption (n) at low frequencies there are two additional new impurity-induced features which appear at elevated temperatures, namely a density of states peak (d) at 69 cm -1, and a gap mode (g) at 78.6 cm -1. Note that the KI phonon gap extends from 70 to 96 cm -1. The evolution of the excited state transitions with temperature can be seen more clearly in fig. 13, which gives absorption difference traces at a number of temperatures

A.J. Sievers and J.B. Page

162

0.6

----r

~

!

!

Ch. 3

r'""'T'-

q

/ 0.4 IE r

<1

0.2

-

s TK

30

J / !

50 70 Frequency (era-I)

Fig. 13. Temperature dependence study of the excited-state transitions for KI:Ag+. The ordinate shows the absorption coefficient difference between the temperature given in the figure and 1.2 K. The two features at 69 and 78.6 cm-1 are seen to grow with increasing temperature, as does the intrinsic two phonon difference band absorption. (After Greene 1984). (Greene 1984). All three of the high-temperature absorption processes (n, d and g) have similar temperature dependences: they are not observable at 1.2 K and grow in magnitude with increasing temperature. The common low-temperature behavior indicates that all of these absorptions may stem from the same elevated energy state. An important experimental finding is that the temperature dependence of the two strong modes, labeled r and g in fig. 12(a) are essentially the same. The strength I(T) of the IR active gap or resonant mode at a particular temperature T is given by eq. (2.4a, b), but with the frequency integration only across the line itself. The temperature dependence of the measured strength I(T) of each line is plotted in fig. 14(a). The growth of the excited state gap mode at 78.6 cm-1 is shown in fig. 14(b). The strength of this "hot" band is normalized to the low temperature strength of the gap mode at 86.2 cm -1. At first sight it might appear that these temperature dependences could follow from simple Boltzmann population effects for anharmonic oscillators, but this is not the case. One of the simplest explanations would be to ascribe the temperature dependence to the anharmonicity of the resonant mode itself, treated as an independent oscillator. If the 0 --+ 1 transition does not coincide with the 1 --+ 2 and higher transitions then only the 0 --+ 1 transition contributes to the unshifted resonant mode transition, and its intensity would decrease as the ground state is thermally depopulated (Alexander et al. 1970). Assuming that the anharmonicity is not too large

w

Unusual anharmonic local mode systems

161

Fig. 12. Temperature dependence of the Ag+-induced absorption spectrum in KI. The vertical dot-dashed lines divide the figure into three parts: The concentration in the center region (2 • 1018 Ag+/cm 3) is twice that of the two end regions. Assignment of the different features: r = resonant mode, c = combination band, d = density of states peak, g = gap mode, and n = nonresonant absorption. (a) The absorption coefficient vs. frequency for two temperatures: solid curve, 1.2 K; dashed curve, 10 K. (b) absorption coefficient difference between a given temperature and 1.2 K vs. frequency for two temperatures: solid curve, 3.4 K; dashed curve, 10 K. Note that the ordinate in the center region of (a) is expanded 5 times. (After Sievers and Greene 1984).

w

Unusual anharmonic local mode systems

163

Fig. 14. Normalized absorption strength vs. temperature for the Tlu resonant and gap modes in KI:Ag +. (a) Gap and resonant mode data for a number of Ag + concentrations. Solid and dashed curves are the predictions of the three dimensional double anharmonic oscillator model. The dotted curve, which is 1 minus the nonresonant growth curve data, is taken at a frequency of 4 cm -1. (b) Excited-state gap mode data. The solid curve shows the predicted temperature dependence of the excited-state strength for the same anharmonic model as used in (a). (After Sievers and Greene 1984). so that the h a r m o n i c oscillator partition functions can still be used, then for a t h r e e - d i m e n s i o n a l oscillator in w h i c h only o n e e x c i t e d state is i m p o r t a n t for e a c h polarization, o n e has

I(T)/I(O) - (1 - e - h l 2 r / k T ) 4

_ Zr4

(2.9)

w h e r e Zr is the partition function of a o n e - d i m e n s i o n a l oscillator. We n o w s u p p o s e that there is a t h r e e - d i m e n s i o n a l d o u b l e a n h a r m o n i c oscillator with e x c i t a t i o n n u m b e r s labeled (g,r). T h e t e m p e r a t u r e - d e p e n d e n t intensity o f two transitions are o f interest:

A.J. Sievers and J.B. Page

164

Ch. 3

(a) resonant mode (0, 0) ~ (0, 1) with intensity -,~ (Zg)-3(Zr) -4, (b) gap mode (0, 0) ~ (1,0) with intensity ~ (Zg)-n(Zr) -3. The dashed curve in fig. 14(a) gives the predicted temperature dependence for the gap mode and the solid curve gives the predicted dependence for the resonant mode. The solid line in fig. 14(b) shows the corresponding prediction for the excited state gap mode (0, 1) ~ (1, 1) transition. In each case the temperature dependence is not fast enough to fit the experimental data. Another missing feature required by this model at elevated temperatures is the (0, 1) ~ (0, 2) transition. Figure 15 shows that such a feature does not appear in the spectrum; instead the high temperature component consists of nonresonant absorption over the entire frequency region (Greene 1984). Here the difference between the absorption coefficient at the temperatures shown and that at 20.6 K is plotted versus temperature. Since the nonresonant absorption is not present at 1.2 K and the resonant mode absorption is essentially gone by 20.6 K, the lowest spectrum in fig. 15, cff l.2 K ) - c~(20.6 K), shows both the resonant mode and the nonresonant absorption components fully developed. 1

I

T

13.7 K

o

\

U

'

0

0 5

I0

15 20 Frequency (cm-I)

25

Fig. 15. Temperature-induced change in the absorption coefficient for KI:Ag+. Ac~ = c~(T) - c~(20.6 K) and T for each trace are given. Both the resonant mode and nonresonant absorption are evident in the lowest trace. The spectral resolution is 0.43 cm-1. (After Sievers et al. 1984).

Unusual anharmonic local mode systems

w

9

0.6

9

9

0..0~0

165 1.0

O0

A

o, 0.4

A

w, to to

/

A


e-

0.5 ~ , I--v

/"

_

i-

i:I

/"

0.2

,L" 0

-.,~-.~'~

0

I

I0

I

I

20

Temperoture (K)

I

i

30

Fig. 16. Normalized nonresonant far IR absorption coefficient for KI:Ag + at 4 cm -1 vs. temperature (solid curve). The normalization factor is the high temperature value. Also shown (dots) is the measured temperature dependence of T A 1 defined later by eq. (2.15) in the text. This quantity is proportional to the population in the off-center configuration. The initial increase of the data with temperature follows an energy gap law with a gap of 24 K. (After Hearon and Sievers 1984).

In order to understand better the role of the nonresonant absorption, its contribution was examined at a frequency far removed from the resonant mode itself. A broad band millimeter wave spectrum with intensity maximum centered near 4 cm-] and a full width at half maximum of 4 cm-1 was used to measure the nonresonant absorption growth curves as a function of temperature. The experimental results normalized to the high temperature data are represented by the solid line shown in fig. 16 (the ordinate for this trace is on the right). By subtraction of the normalized growth curve from unity, the data can be compared directly with the temperature dependence of the resonant and gap mode strengths. These data are represented by the dotted curve in fig. 14 (a). It appears that the strength lost by the gap and resonant modes with increasing temperature is transferred to the nonresonant absorption (Sievers et al. 1984).

2.1.4. Two elastic configuration model When the unusual strong temperature dependent properties of KI:Ag + are contrasted with the temperature-independent properties of NaI:Ag +, it seems clear that the original Lifshitz approach (1956) would have qualitative difficulties in serving as a basis to explain these temperature dependent features,

A.J. Sievers and J.B. Page

166

I cz

Cl

I

[ i|

Ch. 3

ii

I

I

I

I I

I d g

I I

I i

i

I I I

I 0

I

Fig. 17. Schematic representation of the two configuration model. C1 identifies the on-center elastic configuration, with only the gap mode transition shown. C2 shows the higher energy (~) off-center elastic configuration with the excited state density of states and the gap mode transitions. One possible source for the relaxation time r is shown.

since the same impurity is surrounded by the same nearest neighbors in two alkali halides having the same symmetry. To describe in a phenomenological manner the temperature-dependent data, it was proposed that under certain circumstances a second elastic configuration could occur in addition to the ground state configuration of the defect-lattice system (Sievers and Greene 1984). Figure 17 illustrates such a possible two-configuration arrangement. Here C] identifies the ground state on-center configuration and C2 the offcenter one at a higher energy ~. A few of the assignments of the observed transitions are also shown. The rapid temperature dependence observed for KI:Ag + requires that the second configuration contain a low-lying multiplet with a large degeneracy in comparison to the on-center configuration, perhaps related to the displacive tunneling of the Ag + defect among many equivalent minima centered about a normal lattice site. This possibility is not so unusual, since a twelve-component spectrum with a tunneling splitting of ,,~ 0.1 cm -1 has been identified for the normal ground state configuration of Ag + in RbC1 (Kirby et al. 1970; Kapphan and Luty 1972; Holland and Luty 1979). As long as the thermal and electromagnetic energies are much larger than the tunnel splitting, the frequency dependence of the absorption coefficient for such an arrangement could have the simple Debye form considered below.

w

Unusual anharmonic local mode systems

167

The Clausius-Mossotti equation for a medium with intrinsic dielectric constant eo that contains Ni polarizable defects per cm 3 is (Hearon and Sievers 1984) A

~'- 1 ~"+ 2

=

eo + , 4 - 1 eo + ,~ + 2

=

eo- 1

47rNi~i

)

eo + 2

,

(2.10)

3

A

where ,4 is the defect-induced contribution to the complex dielectric constant and ~i is the dipolar polarizability of the defect. In the limit of a small offcenter concentration Ni, the impurity-induced dielectric constant becomes (Sievers et al. 1984)

Z~ = 47rNi~i ( e~ + 2 ) 2.

(2.11)

3

In the Debye approximation, the dielectric response at frequency w is

3 = A 1 -I--iA2 -- A0(1 - iw'r)- 1,

(2.12)

where

A0 = 4"n'Ni ~

3

'

(2.13)

p is the dipole moment, k is Boltzmann's constant, T is the temperature and ~- is the relaxation time. According to eq. (2.12) for the far IR limit (wT >> 1), the nonresonant absorption coefficient is

O~n

CV~

~

,

(2.14)

which is independent of frequency and is proportional to the number density of off-center ions. If the Debye model describes the nonresonant absorption data, then in addition to the far IR signature in the temperature-dependent absorption coefficient there should also be a radio frequency signature in the real part of the impurity-induced dielectric function.

A.J. Sievers and J.B. Page

168

Ch. 3

2.1.5. Radio frequency dielectric constant measurements Measurements of the real and imaginary parts of the dielectric function at 10 kHz as a function of temperature are shown in fig. 18 (Hearon and Sievers 1984). An important observed feature of these measurements is that only the real part of the dielectric function is temperature dependent; no temperature dependence is found for the imaginary part. Although e l ( T ) - ~1(1.4 K) shows a maximum at about 10 K, no corresponding loss peak is observed in e2(T). For ground state tunneling systems such as RbCI:Ag +, a peak in el (T) as a function of temperature is always accompanied by a concomitant peak in e2(T) as r(T) is swept through the wr = 1 condition with increasing temperature (Holland and Luty 1979); moreover, the radio frequency spectra shown in fig. 18 are independent of excitation frequency w. Both of these results indicate that wr << 1 over the entire temperature region of these KI:Ag + measurements. Equations (2.12) and (2.13) show that in this limit

A I = A oo(

Ni(T) T

and

A2=0,

0.04

(2.15)

I

z~

I

0.02 x~ x

F--

X

X

x

0.00 0

20

40

Temperature (K) Fig. 18. Temperature dependence of the impurity-induced dielectric constant for KI:Ag + as obtained from radio frequency measurements, el ( T ) - e1(1.4 K) is plotted vs. temperature for the single-crystal samples of 0.5 and 0.2 mole% AgI in KI in the melt (triangles and crosses respectively.) Also shown is the temperature dependence of e o ( T ) - e0(1.4 K) for pure KI (the z's). e 2 ( T ) is temperature independent and has the same value for all three samples (,-~ 1.5 x 10-3). The measuring frequency is 10 kHz. (After Hearon and Sievers 1984).

w

Unusual anharmonic local mode systems

169

so that the temperature dependence measurement of the real part of the dielectric function provides a direct determination of the population in the off-center configuration. The temperature dependence of the population in this second configuration is found as follows. The data in fig. 18 for the 0.5 mole% AgI are subtracted from those for pure KI, the result is multiplied by T and plotted vs. temperature, resulting in fig. 16. At high temperatures the data points in this figure indicate that the off-center population approaches a near constant value. Information about the reorientational relaxation time for Ag + in this offcenter state can be obtained by combining the experimental results for the dipole moment from the radio-frequency dielectric constant measurement with the measured temperature dependence of the nonresonant absorption coefficient c~n in the far IR spectral region. Inspection of fig. 16 shows that the far IR nonresonant absorption data follow Ni(T), while according to eq. (2.14) the data should vary as Ni(T)/[TT(T)]. The consequence is that TT(T) -= constant. These radio-frequency measurements appear to support the picture that at least two elastic configurations exist for this lattice defect system. At low temperature the Ag + ion is on-center with its associated on-center spectral features. When the temperature is increased, a second elastic configuration becomes populated. This configuration has an energy with respect to the on-center ground state of at least 24 K. The large increase in the defect contribution to the dielectric constant indicates that the Ag + ion is off-center in this second, higher-energy elastic state. Because the off-center population appears to approach a constant value at elevated temperatures, the available data seem to indicate that only these two elastic configurations are close by in energy. The large difference in the temperature-dependent properties of Ag + in the hosts NaI and KI may indicate that low-lying configurations can only occur when the impurity ion is smaller than the host ion it replaces (Na + < Ag + < K + < Rb+). This idea is supported by three different hydrostatic pressure measurements which have been used to transform off-center systems such as RbCI:Ag + (Holland and Luty 1979) and KCI:Li + (Devaty and Sievers 1980) to on-center ones. The almost discontinuous change observed in the impurity ion position at low temperatures and at finite temperatures would be a natural consequence if two different elastic configurations could coexist for a defect lattice system. The apparent transition would occur whenever the pressure-tuned ground states for the two configurations have approximately the same energy, so that direct configurational tunneling becomes possible. The detailed paraelectric resonance studies over a narrow hydrostatic pressure range for Ag + in the RbC1 host provide an experimental probe of the dynamics in this nearly degenerate two-configuration region. It has been

170

A.J. Sievers and J.B. Page

Ch. 3

shown that a modest hydrostatic pressure tunes the levels of what is interpreted as the on-center configuration into the same energy range as the off-center one, so that a microwave transition can be observed between these configurations (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). It is noteworthy that the integrated strength of this transition decreases much more rapidly with increasing temperature than Boltzmann population effects predict (Bridges and Jost 1988). 2.1.6. Microwave absorption measurements

Absorption measurements on KI:Ag + as a function of temperature from 5 K to 300 K have been made at three different frequencies in the microwave region, in order to explore the off-center Debye model in more detail. According to eq. (2.14) the model predicts a very simple result for the absorption coefficient in the limit w~- >> 1. These absorption measurements were made with an oversized nonresonant cavity technique (Hearon 1986) which was developed at the Max Planck Institute in Stuttgart (Kremer 1984; Poglitsch 1984). Backward-wave oscillators are used to span the frequency range from 40 GHz to 160 GHz. In order to approach an isotropic field in the cavity, mode stirrers are used to vary the effective shape of the cavity as a function of time, so that a large number of modes interact with the sample. The sample is surrounded by an unsilvered fused quartz double-walled cold finger which hangs in the cavity, and it is cooled to low temperatures by a continuous flow of cryogenic helium gas. The temperature dependence of the absorption coefficient for samples of KI +0.2 mole% AgI and KI +0.03 mole% AgI are shown in fig. 19. The impurity-induced absorption coefficient is given by the difference between the two temperature dependent curves at 160 GHz and 80 GHz. (The low concentration sample was destroyed before the 40 GHz data set could be completed.) To obtain these results, the detector signal for the cavity with the sample is divided by that without the sample at the same temperature to determine the absorption coefficient for a fixed frequency. The estimated uncertainty in the data is represented by the cross-hatched region around each set of data. The qualitative temperature dependence of the impurityinduced nonresonant absorption (the difference between the two curves in a given panel) for the limit a~- >> 1 can be seen most clearly for the highest frequency data shown in fig. 19 (a). These data, which appear to be temperature independent over much of the temperature interval studied, indicate that Nip2/T~ - is essentially a constant for temperatures large enough to favor the off-center configuration. One possible conclusion is that T~- = constant, just as was found earlier by comparing the far IR and the radio frequency data.

w

Unusual anharmonic local mode systems

0.4-

171

o) f=160

0.2-

0

IE 0.2

-

t

b) f: 80 GHz

I

c) f=40GHz

v

E ~0.1

g := O

a= ,,~

0

0. IO

,/ / ,~/ ,,,'/',, / /j...~

0.0S

0 I

0

I00 200 Temperoture (K)

300

Fig. 19. Temperature dependence of the impurity-induced microwave absorption coefficient of KI:Ag + at three different frequencies. (a) 160 GHz (5.33 cm-]), (b) 80 GHz (2.67 cm - ] ) and (c) 40 GHz (1.33 cm-1). The upper curve in each panel is for KI + 0.2 mole% AgI and the lower ones in frames (a) and (b) are for KI + 0.03 mole% AgI. The estimated errors for the oversized microwave cavity technique are identified by the cross hatched regions in each frame. (After Hearon 1986).

A.J. Sievers and J.B. Page

172

Ch. 3

The lower frequency data in fig. 19 (b) and (c) show additional temperaturedependent structure at low temperatures, which probably indicates that both resonant and nonresonant absorption are being observed for the off-center configuration. It now appears clear that given sufficient nonresonant absorption data versus temperature for different frequencies, both the applicability of the Debye model and the temperature dependent behavior of the different factors in Nip2/TT could be isolated and explored, identifying important aspects of the mysterious off-center configuration.

2.1.7. Rotational motion in the off-center configuration Perhaps the simplest way to produce two configurations, one of which has a much larger effective degeneracy than the other, is to assume that the off-center configuration of the Ag + ion contains a large number of free rotor states. In this model we assume that the crystal field splitting of such free rotor states is negligible. In addition, we assume that the rotational moment of inertia I of the Ag + ion about the normal lattice site is large enough so that the spacing between the rotational levels is small compared to kT at all temperatures for which the upper configuration is populated. In this case the rotational levels are dense and the rotational motion can be treated by classical statistical mechanics. The rotational partition function becomes Zr ~

2IkT h2

•

T Tr

.

(2.16)

If we assume that the vibrational degrees of freedom are roughly the same in both configurations, then the on-center population for the two-configuration model displayed in fig. 17 is simply

Pon(T) =

l+

T -~-~ Trr e

.

(2.17)

The surprising result is that with both the energy gap ~ and the moment of inertia Tr treated as free parameters, it is still not possible to fit the data shown in fig. 14. When Tr is made sufficiently small (generating a high density of states) so that at 25 K a small value for Pon(T) can be produced, then the abrupt knee at about 5 K is missed. Apparently, even this limiting case of free rotor states without the quenching produced by crystalline electric field effects does not produce a dense enough set of levels to match the measured temperature dependence. A better fit to the data of fig. 14 is obtained simply by replacing (T/Tr) in eq. (2.17) by a constant degeneracy factor g t> 100.

w

173

Unusual anharmonic local mode systems

2.1.8. Ag + in KI alloys

To test rotor-like models, disorder has been introduced into the lattice to see if the temperature-dependent properties of the resonant mode are changed. Early work on Li + ions in KBr alloys (Clayman et al. 1967) has shown that a resonant mode can be shifted both to smaller and larger frequencies, since an alloy of two alkali halides has an average lattice constant intermediate to those of the two constituents (Havighurst et al. 1925). The dependence of the mode frequency and linewidth on lattice constant was found to be linear in Clayman's work. Since the average lattice constant (a) is calculated from the measured alloy concentration by the Vegard relation, namely (a) = al + ( a 2 - al)z where al and a2 are the lattice constants of the two components and z is the molar concentration of component 2 (Vegard 1921), the centroid frequency of the resonant mode is expected to be a linear function of the molar alloy concentration. Since A V / V -- 3Aa/(a), the frequency shift can be expressed as -Aw = A ( A V / V ) cm -1. Because RbI has a larger lattice constant than KI, alloying some RbI with KI:Ag + would be expected to decrease the resonant mode frequency, making the defect system more unstable. The far IR temperature dependent spectrum of a KI + 1 mole% RbI + 0.2 mole% Ag + sample is presented in fig. 20

1.0

m

"T A

- o.s

i!

.J //

l 0

Z>v'"-7"~ C)"

I0

lyi'~" i

.

~

i 20

Frequency (cm "l)

30

40

Fig. 20. Temperature dependence of the Ag+-induced resonant mode absorption coefficient for a KI alloy. The sample is KI + 1 mole% RbI + 0.2 mole% AgI. The solid line is for T = 1.2 K, the dot-dashed line is for T = 10 K and the dashed line is for T = 20 K. The center frequency of the resonant mode is shifted slightly to the red and one half of the low temperature resonant mode area is missing when compared to an unalloyed sample grown at the same time. (After Hearon 1986).

174

A.J. Sievers and J.B. Page

Ch. 3

(Hearon 1986). At 1.2 K (solid line) the resonant mode centroid is shifted to a smaller frequency and has lost half of its strength when compared to a KI + 0.2 mole% Ag + sample, grown at the same time. The dot-dashed trace shows the same spectrum for I0 K. The temperature-induced area loss is very similar to that found for the unalloyed crystal: compare this result with the 9.3 K trace shown in fig. 9. At the highest temperature of 20 K (dashed curve), the resonant mode peak has nearly vanished. These temperature dependent results appear very similar to those presented earlier for the unalloyed samples. As the concentration of RbI is increased from 1 mole% to 5%, the frequency shift is found to obey the relation -Aw = 300(AV/V) cm -1. For each alloy, the strength of the mode appears to have the same temperaturedependent properties as described above. Unfortunately, the alloy-induced shift of the resonant mode to smaller frequencies is accompanied by a rapid decrease in the low temperature line strength. As a result, for 3 mole% RbI the strength is about 1/20 of that for the unalloyed crystal, and by 5 mole% RbI, the strength ratio is a barely observable 1/50. Is this rapid change in the low temperature strength with increasing alloy lattice constant a consequence of the off-center configuration coming into resonance with the on-center one? Probably not. For the different alloy concentrations, the observed strength most likely stems from the Ag + centers that have unperturbed neighbors. If the Ag + centers are in a random distribution and it is the next-neighbor Rb + ions that destroy the mode, then the absorption strength should vary as ,-~ (1 - : c ) 12 where z is the alloy mole concentration. The rapid strength change observed here with increasing z could be interpreted as evidence for some degree of nonuniformity in the Ag + :Rb + distribution. When the crystal is grown, the Ag + ions may prefer to remain in a substitutional lattice site near the Rb + ions in the crystal. A remarkably similar strength change with alloy composition has been found previously for the KBr:Li + resonant mode. In this case expanding the alloy lattice by 5 mole% KI also causes the mode strength to disappear (Clayman and Sievers 1968). Although the strength of the Ag + resonant mode is strongly influenced by lattice disorder, its temperature dependent properties appear to remain unchanged, within the experimental uncertainties generated by the large strength change.

2.1.9. Temperature dependence of the Ag + electronic transitions in KI Another way that the temperature dependence of the position of the Ag + ion can be monitored is by making use of the known optical properties of this ion in alkali halide crystals. This behavior can be understood in

w2

Unusual anharmonic local mode systems

175

terms of the 4d 1~ --+ 4d95s parity-forbidden electronic transitions of the Ag + defect (Fussg~inger 1969; Kojima et al. 1968; Holland and Luty 1979) being made allowed by dynamic or static symmetry breaking. When the ion is a substitutional defect in a cubic crystal these electronic transitions are made somewhat allowed by vibronic coupling to odd-parity vibrational modes, or if the defect is off center, by the static odd-parity lattice distortion. In each case the larger the odd-parity contribution, the larger the line strength. A schematic representation of the temperature dependence of the oscillator strength of the transition for the different possibilities is given in fig. 21. Four cases as a function of temperature are shown: (A) gives the optical signature for an on-center defect coupled to a single Tlu symmetry harmonic vibration; (B) is for an on-center defect coupled to a single Tlu symmetry anharmonic vibration; (C) is for an off-center impurity with a shallow barrier and (D) is for a frozen off-center impurity with a large potential barrier. Little optical work has been done on KI:Ag + because this particular system has the unusual property of "aging" with time at room temperature much faster than for the other alkali halide hosts (Fussg~inger 1969). A systematic way to reverse the aging process and hence to reactivate the samples was found during the radio frequency studies of KI:Ag + by Hearon (1986).

to-'

I D

I

,

f

~

[x,]

cons,

'

Id 2

td 3

0

,

I

|

200 Temperature (K) I00

Fig. 21. Schematic temperature-dependence signatures for the electronic oscillator strength of the Ag + ion coupled to different strength odd-parity distortions. (A) on-center ion coupled to a harmonic Ttu vibrational mode. (B) on-center ion coupled to an anharmonic Tlu vibrational mode. (C) off-center ion with a shallow potential barrier. (D) off-center ion with a large potential barrier. (After Holland and Luty 1979).

A.J. Sievers and J.B. Page

176

Ch. 3

o) 0.02 oo O O

O.OI

O O O

""

OO

? Jan.

O Oo

O

O

A& Aaat=a

21 Jan.

& "~' a a t=

aa~

A

,r

o ~ ~

.........................

b)

I

h-- 0 . 0 4

After ,txtxt•t="at=tbt= at=a ~ a

tx

0.02

o 0

t=

ooo

Before

oooooo,oooooooo 8 16 Temperature (K)

~ 24

Fig. 22. Aging and reactivation of isolated Ag + defect centers in KI. (a) The temperature dependent radio frequency dielectric constant versus temperature for an as-grown sample (circles) of KI + 0.2 mole% AgI and the aging observed in the same sample after it has been refrigerated for 14 days (triangles). The number of isolated defect centers has decreased by a factor two. (b) Radio frequency dielectric constant data (circles) for a KI + 0.5 mole% AgI sample maintained at room temperature for several months. The data for the same sample after heat treating at 200 C (triangles) shows that the isolated defect density is fully restored. (After Hearon 1986).

Such effects had been observed previously in the far IR, and samples were refrigerated between runs to slow the process. New crystals were always grown once previous samples had aged. Hearon (1986) found that the reactivation procedure for a Ag + doped KI crystal which has been stored at room temperature is first to heat it to 200 ~ in air and then cool it back down to room temperature in a few minutes. Examples of this aging and reactivation effect are shown in fig. 22. Here the temperature-dependent behavior of the dielectric constant is monitored to determine when the isolated Ag + defect state is present. Figure 22(a) shows the aging effect as measured via the radio frequency dielectric constant for a freshly grown sample

w

Unusual anharmonic local mode systems

177

of KI + 0.2 mole%AgI (circles) and for the same sample after it had been refrigerated for 14 days (triangles). The concentration of isolated centers has decreased by a factor 2. Figure 22(b) illustrates how the sample can be reactivated. A KI + 0.5 mole% AgI sample maintained at room temperature for several months is nearly devoid of isolated centers, as indicated by the circles. After cycling the sample to 200 ~ the temperature-dependent dielectric constant associated with isolated centers is fully restored (Hearon 1986). This aging effect is also quite pronounced in the UV absorption spectrum of the doped crystal. Figure 23 shows the spectrum of an aged KI:Ag + sample before (dashed) and after (dot-dashed) the heat treatment. Before heat treating, the UV spectrum has a strong edge near 420 nm that is in the same frequency interval as the band gap of pure AgI (Kleppmann 1976). After heat treatment, the AgI-like spectrum in the UV disappears, and the crystal becomes transparent. An Ag+-like absorption spectrum is now found at much higher frequencies. Apparently, at room temperature the Ag + ion is fairly mobile in the alkali halide host and tends to organize into silver 1.0

,

w

,..........

after before 0.8

.,,,..,,.v,

,,,-.--

9- - . ~ 9. , . ' . . ~ " ~- 9- . - ~ , . ~ . , ~ . .

/.r ,,

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! !

t

,..., 0 . 4 [.-,

/

i t

i i !

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/ /

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/

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'--

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/

/

/

,-. /

J

290

/

!

!

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330

370

410

450

Wavelength (nm) Fig. 23. Effect of heat treatment on the KI:Ag+ UV transmission spectrum. Dashed curve: aged crystal before heat treatment, showing the absorption produced by AgI aggregates. Dotdashed curve: after heating this sample to 200 C and quenching to room temperature, isolated Ag+ centers are produced. The crystal is now transparent beyond the 420 nm wavelength region. (After McWhirter and Sievers, unpublished).

178

A.J. Sievers and J.B. Page

Ch. 3

halide clusters which the modest heat treatment perfected by Hearon (1986) breaks up. Figure 24 shows the measured UV absorption spectrum of KI:Ag + for two different temperatures. Over this temperature range the center frequency of each of the three bands remains essentially fixed. A linear concentration dependence establishes that these three features labeled A, A' and C [using the nomenclature of Fussgiinger (1969)] in the figure are associated with isolated Ag + ions (Page et al. 1989). To display the three lines in one figure, three different Ag + concentrations are required, as described in the caption. The strength of the strong UV line C shown in fig. 24(a) is temperature independent, and is assigned to a delocalized charge transfer excitation in which an electron from a nearest neighbor anion occupies one of the empty Ag + excited states. The weak lines A and A' are assigned to the Alg(4d 10) -+ T2g(4d95s) and Alg(4d 10) --~ Eg(4d95s) transitions, respectively. They display nearly zero strength at 1.2 K but grow rapidly with increasing temperature. The change is much faster than can be obtained from the occupation number effect in the low-lying Tlu resonant mode, for example. This oscillator strength temperature dependent signature of A and A' does not match any of the cases displayed in fig. 21 but the increase in

Fig. 24. KI:Ag+ UV absorption spectra. Dashed line, T = 50 K; dot-dashed line, T = 1.2 K. Silver concentration: panel (1) 2 x 10-2 mole%, panel (2) 2 x 10-3 mole%, panel (3) 5 x 10-5 mole%. (After Page et al. 1989).

Unusual anharmonic local mode systems

w

1.0 .

i

179

,,,

o

m

It

-

Q_ 0

~. 0.5

~

r c-

\

(3

I

xU,..

tO

0.0 0

10 2O Temperature (K)

30

Fig. 25. Temperature dependence of Ag + on-center population, as determined by various measuring techniques. Dashed line, far IR resonant and gap mode strengths. Dotted line, dielectric constant data. Squares, UV absorption for the A' electronic state. Solid circles, Raman scattering intensity from the 16.1 cm -1 Eg mode. (After Page et al. 1989).

optical strength with temperature appears identical to that found earlier for the increase in the DC dielectric constant produced by the appearance of a permanent dipole moment with increasing temperature. This latter change was shown to be proportional to the population in the off-center configuration, Poff. By letting Pon 1 -Poff, both the UV and dielectric constant data can be compared directly with Pon determined from the temperature dependence of the Tlu resonant and gap mode strengths. These results are graphed in fig. 25. The three different experimental probes are seen to measure exactly the same rapid decrease in the population of the on-center population. -

-

2.1.10. Raman scattering from even-symmetry resonant modes

The polarized Raman intensities I((1,-1,0)(1,-1,0)),

I((1, 1,0)(1,-1,0)),

I(
for KI:Ag + have been reported for the 90 ~ scattering geometry (Page et al. 1989). The notation here identifies the polarization of the incident and scattered wave with respect to the crystal axes: ((incident)(scattered/). The results at three temperatures for the first two polarizations are shown in figs 26(a) and 26(b). No impurity-induced scattering was observed for the

180

Ch. 3

A.J. Sievers and J.B. Page 280

----T~T-~-'--'r----'-~T-------T---'--'T'-'-

I k

-

(o) N~

-

\x 160

,...

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160

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:

".

- ................' o... ........

40

12

Raman

~ ~:~,-A.

20 shift (cm - l )

t,

"

28

Fig. 26. Temperature dependence of the polarized Raman spectra of a { l, 1, 0}-cut KI + 0.3% AgI crystal. The spectra, taken with 488 nm excitation, are displayed for two polarization geometries: (a) ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ( A l g + ( 1 / 4 ) E g + ( 1 / 2 ) T 2 g ) a n d (b) (1,1, 0) (1, - 1 , 0) ((3/4) Eg). The three temperatures shown are 6.3 K (dotted line), 12.9 K (solid line), and 25.2 K (dashed line). No ( 1 , - 1, 0)(0, 0, 1) (1/2 T2g) polarized lines were observed at any temperature. The resolution is 2.5 cm -1. (Page et al. 1989).

third orientation. The data are classified according to the even-parity representations Alg, Eg and T2g of the cubic group Oh, which yields for the polarized intensities the expressions I ( ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ) - Alg + (1/4) Eg + (1/2) T2g,

I((1, 1,0)(1,-1,0))

- (3/4) Eg,

I((1, -1, 0) (0, 0, 1)) = (1/2) ZEg. The Raman measurements presented in fig. 26(b) support the earlier low temperature Eg mode frequency assignment by Kirby (1971). On the other hand, neither of the previously proposed Alg and T2g higher frequency resonances are found. In addition, no pronounced Raman features are found in the gap region. A new discovery is the strong temperature dependence of

w

Unusual anharmonic local mode systems

181

the Raman scattering intensity of this 16.1 cm -1 Eg mode, which is shown for all of the temperatures in fig. 25. As temperature is increased, the scattering strength of this Eg mode decreases rapidly (see the solid circles in fig. 25), yet its center frequency stays nearly fixed. The temperature dependence of this Raman line is remarkably similar to that of the far IR active resonant and gap modes and the optical features discussed above, but it takes on added importance because the Eg mode involves no motion of the Ag + ion. Unpolarized Raman spectra at a number of temperatures between 6.9 and 25.7 K are shown in fig. 27. With increasing temperature up to 17.1 K, the 16.1 cm -1 Eg resonant mode Raman peak decreases in strength and is finally replaced by a second excited state feature at 12.2 cm-1. This excited state peak shows both Eg and Alg components and is consistent with the appearance of an A1 symmetry mode in a Cnv off-center configuration. At still higher temperatures this feature also disappears (or broadens) into a pronounced central peak having the same polarization properties, and likely produced by the off-center tunneling levels themselves. This polarization character of the quasielastic central peak is preserved up to 60 K, while at higher temperatures T2g symmetry scattering appears (Fleurent et al. 1991).

25.7K

A i.0 6.9

9.2K

IlK

15.2K

r O Ir

o

>,0.5

B

t/3 t--

C

5

a~"S--

ZO 35 Roman shift (cm-I)

Fig. 27. Unpolarized Raman spectra of a {100}-cut KI:Ag+ crystal versus temperature. The six different temperatures are given in the figure. With increasing temperature the 16.1 cm-1 Eg resonant mode is replaced by a lower frequency A1 symmetry mode appropriate to a C4v symmetry defect site which subsequently disappears in a quasielastic scattering peak at the highest temperature. (After Fleurent et al. 1991).

A.J. Sievers and J.B. Page

182

Ch. 3

2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics

2.2.1. Local modes The Lifshitz harmonic Green's function method for computing general defect phonon properties has been described in detail (Maradudin et al. 1971; Bilz et al. 1984). For completeness, the application of this method to determine the frequencies, displacement patterns and IR spectra of localized modes arising from substitutional impurities is briefly reviewed here. A detailed discussion of the application of the Lifshitz method to the lowtemperature on-center dynamics of KI:Ag + is given by Sandusky et al. (1993a). The defect concentration is taken to be low enough so that the case of just a single impurity is required. For a system of N ions interacting via harmonic forces, the normal mode frequencies {w f} and displacements x ( f ) are determined by the 3N • 3N matrix eigenvalue equation

(4:' - w ~ M ) x ( f ) = O,

(2.18)

where ~ is the harmonic force constant matrix, M is the diagonal mass matrix and f = 1. . . . ,3N labels the modes. The force constant matrix is symmetric, and the modes are normalized such that the 0rthonormality relation is x ( f ) M x ( f ' ) = 5f f,. With a substitutional defect present, it is convenient to rewrite eq. (2.18) identically in terms of the host crystal harmonic Green's function matrix G0(w2) and the perturbing matrix C(w2), containing the impurity-induced force constant and mass changes. The result is +

x ( f ) = o.

(2.19)

This equation can be partitioned into two equations" one involving just components inside the "defect space", defined by the sites associated with nonzero elements of C, and a second equation which determines the mode displacements outside the defect space:

[III -k- GoiI(~)Cii(~o~)] xI(f) = 0

(2.20)

XR(f) = --GoRI(~)CII(OJ~)xI(f)"

(2.21)

and

w

Unusual anharmonic local mode systems

183

The subscripts I and R refer to components inside and outside the defect space, respectively. Equation (2.20) gives the determinental frequency condition: (2.22)

IIii + GoIi(w})Cii(w}) I - O.

For isoelectronic impurities, the defect space is usually small. Equation (2.22) then involves the determinant of a small matrix and is thus practicable, provided the defect-space elements of the unperturbed harmonic Green's function matrix can be computed. This is readily done for localized modes by direct summations involving the unperturbed host crystal phonons; for alkali-halides, these are well-known through phenomenological models (e.g. shell models) that account very well for the measured phonon dispersion curves. Once the local mode frequency is known, eq. (2.20) can be used to compute the defect-space displacement pattern, to within a normalization constant. The displacements outside the defect space may then be determined by eq. (2.21).

2.2.2. Infrared absorption In the long-wavelength limit, the interaction between an insulator obeying the Born-Oppenheimer approximation and an external field of monochromatic infrared radiation is

-M(u) . Ee -i~t,

(2.23)

where M(u) is the system's dipole moment for nuclear configuration u. If just the linear term MS(u) -- ~-,z# #~(l)u#(1) in the expansion of the ath component of the dipole moment in terms of the nuclear displacements is retained and a calculation similar to the one carried out by Klein (1968) is done, the resulting absorption coefficient a(w) for a cubic crystal with an impurity concentration Ci is given by 2

47rwCi ( n ~ + 2 ) ~(w)- cn(w) 3

~ I m [ G(W2+ ie)]/-~,

(2.24)

where n ~ is the high-frequency index of refraction, c is the speed of light in vacuum, G(w 2) is the harmonic Green's function matrix for the impurity crystal and the limit e --+ 0 + is understood. The quantity / ~ = {#~(l)} denotes the effective charges in this linear dipole moment approximation.

A.J. Sievers and J.B. Page

184

Ch. 3

In deriving eq. (2.24), it is assumed that the standard Lorentz local field correction holds and that the contribution from the electronic polarizability is adequately described by the high frequency dielectric constant coo. The index of refraction n(w) is taken to have its pure crystal value, owing to the assumption of low defect concentrations. Hence the integrated absorption strength Sf = f a(w)da~ for a single local mode f is n~+2

S.f = cn(w)

3

)

2

[~"~x(f)]2

(2.25)

Using the transformation properties of the dipole moment, it is straightforward to show that for an impurity crystal with Oh symmetry,/~ transforms under symmetry operations as the ath partner of irreducible representation Tlu. Hence, by a standard group-theoretic matrix element theorem, f ~ x ( f ) vanishes when x ( f ) is not an ath partner belonging to Tlu. In eq. (2.25), the sum implicit in ~ x ( f ) extends over the entire system, reflecting the fact that infrared radiation couples to all of the ions. For the case when the effective charges in the defect crystal are unperturbed from their host crystal values, Klein (1968) has shown how the absorption may be expressed in terms of just defect-space quantities.

2.2.3. Impurity model calculations The determination of the local mode (and resonant mode) displacement pattern and frequency requires knowledge of the perturbing matrix C(w 2) and the pure crystal Green's function elements. Two different methods have been employed to obtain the latter in the KI:Ag + defect mode calculations. The first is detailed in (Page 1974; Harley et al. 1971), and its present application is outlined here. The breathing shell model (Schrrder 1966) was used by Page et al. (1989) to compute the pure KI phonon frequencies {~ka} and complex plane wave displacement patterns {x(kj)} at 22,932 k vectors in the irreducible 1/48 element of the Brillouin zone, this being equivalent to one million vectors in the full zone. The imaginary parts of the unperturbed Green's function matrix Im Go(a;2) = 7r ~ x ( k j ) x +(kj) ~ (w~3 - to2) ka were approximated as histograms by dividing the pure KI phonon frequency range into 100 equally spaced "bins" and evaluating the sum over the modes

w

Unusual anharmonic local mode systems

185

whose frequencies fall within each bin. The real parts were then obtained by computing the Hilbert transforms of the imaginary parts. The defect model for a low-temperature on-center configuration of KI:Ag + consists of the mass change Am, longitudinal force constant changes 5 = - A ~ ( 0 0 0 , 100) between the defect and each of its six nearest neighbors, and longitudinal force constant changes 5' = - A ~ ( 1 0 0 , 200) between the defect's nearest neighbors and their adjacent fourth-nearest neighbors. Physically, 5 arises from the different binding of the impurity and is expected to be negative, consistent with the overall force-constant softening implied by the presence of the strong low-frequency IR impurity resonance at 17.3 cm -1. The force constant change 5' is postulated to arise from the expected defectinduced inward static relaxation of the nearest neighbors, also consistent with the overall force constant softening. Within this model, only the modes belonging to the Tlu, Eg and Alg irreducible representations of the Oh point group are perturbed. The necessary orthonormal symmetry basis vectors within the impurity space are shown in fig. 28. The two force constant changes (5, 5') were determined by requiring that the model reproduce the observed Tlu 17.3 cm -1 low-frequency resonant and 86.2 cm -1 gap mode IR frequencies. Both the resonant and gap mode frequencies are determined by the condition Re]l + Go(z)C(z)] = 0, where z = w2 + ie. For the gap mode, the Re in this equation is not necessary, since the imaginary parts of the Green's function vanish. Using this condition, 5 versus 5' curves are computed consistent with the 17.3 cm -1 and 86.2 cm -1 modes, with the results given in fig. 29. The crossing point of these two curves, denoted by a circle in the figure, gives the force constant changes for which the model reproduces both of these Tlu absorption peaks. The corresponding fractional force constant changes are d/kl - -0.563 and t~t/kl = --0.531, where kl -- 1.884 • 104 dyn/cm is the pure KI breathing shell model nearest-neighbor longitudinal overlap force constant. Having fixed the model's parameters, we next turn to its predictions. First, the model predicts the ratio of the IR absorption strengths for the 86.2 cm-1 and 17.3 cm -1 gap and resonant modes to be 1.4. This is in reasonable agreement with the observed value of ~ 3, considering that no local field or effective charge changes were included in the relative intensity calculation. Second, the mode predicts a low-frequency Eg resonance at 20.5 cm -1, in reasonable agreement with the observed Eg Raman peak at 16.1 cm -1. Figure 29 includes the 5 versus 5' curve computed for an Eg resonance at 16.1 cm -1, and one sees that this curve comes quite close to the circled (5, 5') point for this model. For comparison, fig. 29 also includes the (5, 5') curve computed for the case of a zero-frequency Tlu resonance (dashed curve). The force constant changes on this curve correspond to the defect lattice being unstable against Tlu displacements. As the force constants are

A.J. Sievers and J.B. Page

186

Ch. 3 2-1/2

~(Tlu x,1)

~(Tlu x,2) -

12-1/2

2 . 1 2 "1/2 0

o

~

~(Eg 1,1)

o

....

o

~(EQ2,1) z

6-1/2

~(Alg 1,1 ) Fig. 28. Symmetry-basis vectors on the defect and its nearest neighbors, used in the perturbed phonon model. For a basis vector ~(Fip, t), Fi labels the irreducible representation, p labels the partners within the representation, and t labels independent vectors for a given Fip. (After Sandusky et al. 1993a).

weakened, the resonant mode of Tlu symmetry is almost always the first to become unstable; hence only the portion of fig. 29 to the upper right of the dashed curve is physically reasonable. The close proximity of the fit point (circle) to the dashed curve means that this model corresponds to a nearly unstable defect/host system. Besides the Tlu (17.3 cm -1, fit) and Eg (20.5 cm -1, predicted) resonant modes, the model also predicts an Alg resonance at 37.3 cm -1. This mode is Raman-allowed, but has not been seen experimentally. However, since the Alg and Eg Raman strengths are determined by two independent electronic polarizability derivatives, the null Alg experiment could be used to place bounds on these derivatives.

w

Unusual anharmonic local mode systems - I .0

-0.5

..,

~~

o.o

187

0.0

I

I I jT, u zero freq.

T__!~_86.2 cm___[!

-

-0.5

-1.0 Fig. 29. Calculated fractional force constant changes for resonant modes and gap modes at fixed frequencies. The inset illustrates the force constant perturbations in our model; all other short range and Coulomb force constants are unperturbed. The fractional changes are given in units of the pure KI breathing shell model nearest neighbor longitudinal overlap force constant kl = 1.884 x 104 dyn/cm. The dashed curve is for a Tlu instability. (After Page et al. 1989).

A striking prediction of this model is the existence of three nearly degenerate gap modes" the three fold degenerate Tlu mode at 86.2 cm -1, a two fold degenerate Eg mode at 86.0 cm -1 and a nondegenerate Alg mode at 87.2 cm -1. These modes are found to have some very unusual properties. The first is the extent of the near-degeneracy- the frequencies are within about a wavenumber of each other. Second, they are essentially independent of the force constant change 6, as is evident from the curve for the 86.2 cm -1 gap mode in fig. 29. Third, ratios of the Raman strengths of the Alg and Eg gap modes to their low-frequency resonant mode counterparts are predicted to be negligible. Indeed, these modes have not been observed in Raman experiments. Detailed calculations of the mode displacement patterns reveals the origin of these unusual properties. All three gap modes are found to have displacement patterns which are strongly peaked on the impurity's fourthneighbor sites [(+200), (0 + 20), (00 + 2)], as shown in fig. 30(a)-(c). The displacements on these sites are a factor of 50 or more larger than those on the impurity or its six nearest neighbors. For comparison, Figure 30(d) also shows the displacement pattern for the 17.3 cm -1 Tlu resonant mode,

A.J. Sievers and J.B. Page

188

o

0

0

q

o

6

0

f

\

/

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0

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.o /

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0

o

o

0

o

0

o

0

o

0

o

0

~u

gap

(d)

0

Tlu

~

res.

Fig. 30. Calculated displacement patterns for different KI:Ag + modes. (a) 87.2 cm -1 Alg pocket gap mode. (b) 86.0 cm -1 Eg(1) pocket gap mode. (c) 86.2 cm -1 TluX pocket gap mode. (d) 17.3 cm -1 TluX resonant mode. Here Tlux denotes the Tlu partner which couples to z-polarized radiation, and Eg(1) denotes one of the two degenerate Eg partners. For our choice of partners (fig. 28), panels (a), (c), and (d) show displacements in the z - y plane, while (b) shows displacements in the y-z plane. Note that the displacement pattern for the resonant mode is peaked on the defect and its nearest neighbors, while the displacement patterns for the pocket gap modes, (a)-(c), are peaked on the fourth neighbor sites, away from the defect. The displacements for the different symmetries are not drawn to scale. (After Rosenberg et al. 1992).

w

Unusual anharmonic local mode systems

189

which is peaked at the impurity site and its nearest neighbors. The nearlydegenerate frequencies of these "pocket" gap modes reflect the fact that they are almost entirely determined by the local dynamics within each of the six pockets; the pockets are weakly coupled to produce the different symmetry modes. Moreover, the independence of these modes on the force constant change 5 is clear, since this force-constant change is seen to couple ions which have essentially no motion in these modes. Finally, the weak predicted Raman strengths for the Alg and Eg pocket modes is also explained by the negligible displacements on the impurity's nearest neighbors, since defect-induced first-order Raman scattering in alkali halides typically arises from the modulation of the electronic polarizability by the motion of the ions in the impurity's immediate vicinity. On the other hand, the Tlu pocket mode is readily observable with IR absorption, since this probe couples to all of the ions in the system. Unfortunately, the model is fit to the measured IR pocket mode frequency, so that the negligible Raman activity of the two even-parity pocket modes precludes a straightforward independent verification of the existence of these highly unusual modes.

2.2.4. Isotope mode splitting Owing to the negligible Raman activity of the Alg and Eg pocket gap modes, another test was needed to verify their existence. Fortunately, the naturally-occurring isotopic abundances of 7% 41K+ and 93% 39K+ in the host crystal provide just such a test. It has been demonstrated that the harmonic defect model predicts that the presence of one or more 41K+ ions at the impurity's fourth-neighbor sites strongly mixes the pocket gap modes of all three symmetry types, producing new IR-active "isotope" pocket gap modes which are experimentally observable (Sandusky et al. 1993a). To determine the frequencies and IR integrated absorption strengths for these isotope modes, nearly-degenerate perturbation theory is first applied to a single impurity/isotope combination (Sandusky et al. 1991, 1993a). This is done by expanding the isotope mode displacement patterns in terms of the six normalized unperturbed pocket gap mode patterns g,(i) = ~ , = 1 af'(i)X(ff) and substituting the expansion into eq. (2.18). Here i = 1,..., 6 labels the isotope modes. Multiplying the resulting equation on the left by ~(f) yields a 6 x 6 eigenvalue equation for the isotope-mode frequencies w~ and the expansion coefficients af(i)" 6

(W2f -- W2) af(i) = E w2f((f)Amx(f')aY '(i)" f'=l

(2.26)

190

A.J. Sievers and J.B. Page

Ch. 3

Here Am is the diagonal matrix containing the mass changes introduced by the 41K+ isotopes, and the {mos} are the six unperturbed pocket-gap-mode frequencies. A 41K isotope substitution changes the mass of the 41K+ ion, leaving its electronic structure and, hence the effective charges #~<~unchanged. Thus, the dipole moments for the isotope modes are produced exclusively by their Tlu components. When ff,(i) is substituted into eq. (2.25), the integrated z-polarized IR.absorption for the ith isotope mode, of frequency w~ produced by a single impurity/isotope combination in a crystal of volume V, is 2rr 2

2 (rzoo + 2 ) 2

si = cn(wi)V

3

[~=x(Tlux)]iaZ'ux(i)"

(2.27)

This expression is just the absorption for a single impurity with no isotopes present, multiplied by the Tlux fraction a2 uX(/) for the ith isotope mode and by an index of refraction correction n(w~ )ln(wO. Figure 31 shows the calculated splittlngs for a single Ag + impurity with a (200) 41K + isotopic substitution, which mixes the Alg, Eg2 and TluX pocket .

llu

No Isotope

87

Alg

-_

_

T E

" "

0

Tlu(x)

-

>'86 0

.

0

!

13"

-

L

_

Lt.

-

85

_

.

.

Eg(2) .

(200) Isotope S,

...~... . . . . \\ # \\ //

4.1%

\\11 V~\ 12 \

-(_1..3:_.

' \ A_,---\\\ \, \\

_. \ "

\

\\\ \\\

~\ %\ k\ It.

-

45z

....

\

.

.

.

.

52%

Fig. 31. Mixing of the three nearly degenerate Alg, Eg2 and Tlu x gap modes under a (200) 39K+ --+ 41K+ host-lattice isotopic substitution. The percentages give the strength sx of the z-polarized absorption per defect/(200) isotope combination, relative to a defect with no isotope substitution. Note that the percentages add up to slightly more than 100% because the absorption coefficient is not only proportional to the imaginary part of the dielectric function but is also inversely proportional to the host-crystal index of refraction, which is strongly frequency dependent in the gap mode region. (After Sandusky et al. 1991).

w

Unusual anharmonic local mode systems o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

II 0/-~ ) 0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

191

(a) 84.7 cm -1 isotope mode o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0/--o ) 0

II

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

0

o

9

0

(b) 86.1 cm -1 isotope m o d e Fig. 32. Computed displacements patterns for a (200) 39K+ -+ 41K+ host-lattice isotopic substitution. The impurity and the isotope are represented by a dark square and a dark circle, respectively. (a) 84.7 cm -1 isotope pocket gap mode and (b) 86.1 cm -1 isotope pocket gap mode. These are the isotope pocket gap modes having the largest and smallest frequency shifts relative to the unperturbed Tlu pocket gap mode. Note that the displacement pattern for the isotope mode with the largest frequency shift is strongly peaked on the isotope.

gap m o d e s to create three isotope modes. The calculated displacement patterns for the (200) isotope m o d e s with the largest and smallest frequency shifts from the unperturbed Tlu 86.2 cm -1 pocket gap m o d e are given in fig. 32. T h e s e isotope m o d e s are seen to be pocket gap modes with the displacements confined to a single pocket. It is not surprising that the isotope with a large displacement on the (200) 41K+ gives the largest frequency shift, while the isotope m o d e with a negligible displacement on this ion gives a very small frequency shift. As s h o w n in fig. 31, fourth-neighbor 41K+ substitutions can produce isotope m o d e s having frequency shifts larger than 1 cm -1 and IR absorption strengths c o m p a r a b l e to that p r o d u c e d by a defect with no isotopes present. In contrast, (110) (i.e. second-neighbor) and (400) 41K+ isotope substi-

A.J. Sievers and J.B. Page

192

Ch. 3

tutions are found to produce modes having negligible absorption strengths and/or frequency shifts of less than 0.1 cm -1, which would be very difficult to observe given the large width of the main mode. Since the displacements for the isotope mode are zero-order displacements in the perturbation approach, the Tlu fractions a2 ox(i) for a single isotope-impurity combination sum to unity. Hence, one sees from eq. (2.27) that provided the change with frequency of the index of refraction can be neglected, which is an excellent approximation for modes separated by less than 0.1 cm -1, the unperturbed mode strength obtained by including the weakly shifted modes is the same as if their contributions had been neglected from the outset. Similar arguments hold for the splitting of the pocket isotope modes, such as the 84.7 cm -1 (200) 41K+ mode, due to second-neighbor or (400) 41K+ isotopes- as long as the splitting produced by these substitutions is smaller than the experimental resolution, or these new modes have negligible absorption strengths compared with the fourth-neighbor pocket isotope modes, their contributions can be ignored. Accordingly, the second-neighbor and (400) 41K+ isotope substitutions were neglected in the isotope-induced IR absorption calculations (Sandusky et al. 1991, 1993a). However, in order to account for all 41K+ fourth-neighbor modes, the 41K+ fourth-neighbor isotope modes with frequency shifts of less than 0.1 cm -1 were included. The nearly-degenerate perturbation theory predicts the frequency shifts (eq. 2.26) and integrated IR absorption strengths (eq. 2.27) for isotope modes produced by a single impurity/isotope combination. However in a real KI:Ag + system, there is a distribution of impurity/isotope combinations. In a crystal with Nd impurities, the predicted x-polarized strength for the ith isotope mode produced by an impurity/isotope combination with 1 fourth neighbor isotopes and the Ni - 1 symmetry-equivalent modes produced by different isotope/impurity combinations, also with 1 41K+ fourth neighbors, is given by Si - NdNi(1 -

f)6-I fl si

(2.28)

where si is the x-polarized strength for the ith mode produced by a single isotope/impurity combination, as given by eq. (2.27). The computed shifts and strengths for 13 isotope/impurity configurations are tabulated in Sandusky et al. (1993a), which gives numerous additional details. The final result is that there are only two predicted isotope pocket gap mode IR peaks at frequencies resolvable from the main unperturbed line at 86.2 cm -1. These are an observably strong peak at 84.7 cm -1 and a much weaker one at 87.1 cm -1. The experimental isotope line is found at 84.5 cm -1 as discussed below.

w

Unusual anharmonic local mode systems

193

2.3. Pocket gap mode experiments

2.3.1. Isotope shift The absorption coefficient of KI+0.4 mole% AgI in the phonon gap region of KI for two different temperatures is shown in fig. 33 at a resolution of 0.1 cm -1. The strong impurity-induced feature at 86.2 cm -1 is the KI:Ag + gap mode corresponding to the low temperature on-center configuration of I

I

I

I

75

80

85

I 90

>

7 E 0

v t"-" 9 0

, m

q... 0 0 cO

o m

k.. 0

~0

..0


0

70

Frequency

(cm -1)

95

Fig. 33. Impurity-induced absorption coefficient of KI + 0.4 mole% AgI. The restricted frequency interval covers the phonon gap region of KI. The resolution is 0.1 cm -1 The temperature for the upper spectrum is 1.6 K, and that for the lower one is 8.8 K. The strong mode at 86.2 cm -1 is the on-center KI:Ag + gap mode. This mode has lost about half its strength in the higher-temperature spectrum. The doublet at 76.8 and 77.1 cm -1 is due to C1- and the single peak at 82.9 cm -1 is due to Cs +. The weak temperature-dependent peak at 84.5 cm -1 is the Ag + isotope mode. Note that the KI:Ag + modes, which have a FWHM of 0.5 cm -1, are significantly broader than other KI gap modes, whose FWHM is ,-~ 0.14 cm -1. Additional temperature-induced changes are increased broadband absorption due to difference-band processes in the host KI crystal and the appearance of a KI:Ag + gap mode at 78.6 cm -1, corresponding to population of the off-center configuration of the Ag + defect in the KI lattice. (After Sandusky et al. 1993a).

A.J. Sievers and J.B. Page

194

Ch. 3

the defect system. Most of the weaker spectral features seen here and identified in the figure caption are associated with other unwanted monatomic impurities, present in either the host or dopant starting materials. But the strength of the weak line at 84.5 cm -1 also varies linearly with the Ag + concentration and hence is not due to pair modes. When the temperature is increased from 1.6 K to 8.8 K, the strengths of the strong Ag + gap mode and the neighboring satellite line at lower frequency are reduced in strength by a factor of two while the weak but sharp modes due to the other impurities remain unchanged. At the highest frequencies shown in the figure, the temperature change produces an increase in the host absorption coefficient due to intrinsic difference band processes (Love et al. 1989). The interplay between the temperature dependence of the gap mode strength and the underlying difference band absorption can be seen more easily in the 3-dimensional plot of fig. 34, which displays the results obtained for several temperatures from 1.6 K to 19 K. At the highest tempera12"

0

0 ~

B~

~j

B7

~o

8 frocloeoCY

Fig. 34. Temperature dependence of the absorption coefficient in the region of the KI:Ag+ pocket gap modes between 1.6 and 19 K. The resolution is 0.1 cm-1. Note that the two KI:Ag+ modes have nearly disappeared in the high-temperature spectrum, where two weak Rb+ gap mode peaks at 86.3 and 86.9 cm-1 have become visible. Note also the increase in the difference-band absorption with increasing temperature. (After Sandusky et al. 1993a).

w

Unusual anharmonic local mode systems

195

ture shown, 19.0 K, the gap mode has nearly vanished; the remaining weak absorption peaks at 86.3 and 86.9 cm -I are due to a small concentration of naturally-occurring Rb + impurities. It is clear from these data that not only does the strength of the main mode disappear with increasing temperature, but also that the weak satellite line at 84.5 cm -~ disappears with a similar temperature dependence. At the same time that the difference band absorption increases in magnitude with increasing temperature, the weak features within and on the high frequency side of the main Ag + gap mode remain essentially temperature independent, but become easier to see with the disappearance of the Ag + gap mode. An identifiable property of the Ag + gap modes at 86.2 and 84.5 cm -1 is that their strengths become vanishingly small by ~ 25 K. This temperature dependence is associated with the depopulation of the Ag + on-center configuration. It is exhibited by all experimental probes of this defect system. In contrast, the KI:C1- doublet at 76.8 and 77.1 cm -1 and the KI:Cs + single peak at 82.9 cm -1, for example, have no temperature dependence in this restricted temperature region. There is another relevant feature generated by the Ag + defect in the 8.8 K spectrum. A weak broad gap mode is visible at 78.6 cm -1 which initially grows in strength with increasing temperature. Previously, this line was identified with a transition in the off-center Ag + configuration (Sievers and Greene 1984). The study by Sandusky et al. (1993a) shows that although the band initially grows with increasing temperature it then appears to stop growing for temperatures larger than about 12 K. It now is clear that the temperature dependence of the strength of this mode does not show the temperature dependence associated with the off-center configuration, which continues to grow in strength until about 25 K.

2.3.2. Temperature dependence of the isotope mode intensity To separate the temperature dependence of the gap mode spectrum from the two phonon difference band absorption a three step unfolding technique is used. First the data in the parts of the phonon gap region of KI where no impurity modes are present is used to fit a polynomial to the temperature dependent background absorption. Second, the background absorption is subtracted, leaving only the absorption peaks corresponding to the impurity modes. Third, the strengths of the KI:Ag + pocket gap modes are obtained by fitting the peaks to the sum of two Voigt functions corresponding to the unperturbed and isotope modes, respectively. Since the isotope mode is much weaker than the unperturbed mode (4% of the strength), it is necessary to limit the number of free parameters in the fits: the assumptions used are

196

A.J. Sievers and J.B. Page I

Ch. 3

'

Z

0 "0

0

I

I

'

I

10 Temperature (K)

20

Fig. 35. Temperature dependence of the strengths of the two ir-active modes produced by the Ag + center in the KI gap region. The solid circles are the data for the weak mode at 84.5 cm-1, produced by the 41K+ substitution on the fourth neighbor of the Ag + defect. The dashed line gives the temperature dependence of the unperturbed Ag + gap mode at 86.2 cm-1 (After Sandusky et al. 1991).

that the two Voigt functions have the same shape and that the separation of their center frequencies is temperature-independent. Thus, the remaining free parameters are the widths of the Gaussian and Lorentzian contributions to the lineshapes and the strengths of the two modes. The strength of the unperturbed gap mode obtained by this technique is in good agreement with the previous results, represented by the dashed line in fig. 35. The solid circles in this figure show the temperature dependence of the isotope mode as determined by this technique; within the experimental uncertainty, the temperature dependences of the unperturbed and isotope KI:Ag + pocket gap modes are clearly similar, and possibly identical. 2.3.3. Entire impurity induced spectrum at T = 0 K

It would be misleading to conclude from the success that of two-parameter defect model in describing the isotope effect that it can reproduce the complete impurity-induced far IR spectrum of KI:Ag + system. Figure 36 shows the measured low-temperature absorption spectrum of KI:Ag + versus frequency (curve A) over the frequency region below the reststrahl. In the same figure is presented the calculated impurity-induced absorption (curve B) in the acoustic spectrum generated by the two parameter model (8, St). The

w

Unusual anharmonic local mode systems

10 I

I

l

i

197

l

E o

1

c ~

o

0 o

c0. I o

Q_

L_

0 o3 _K3

0.01 0

25 50 75 Frequency (cm -1)

100

Fig. 36. Impurity-induced absorption coefficient of KI:Ag + vs frequency. Curve A: Far IR absorption coefficient of KI:Ag + below the optical phonon region, at 1.7 K. The dominant features are the KI:Ag + resonant and gap modes at 17.3 and 86.2 c m - l , respectively, and the gap modes due to KI:C1- at 76.8 and 77.1 cm -1 and KI:Cs + at 82.9 cm -1 (C1- and Cs + are present as natural impurities). Additional weak features due to KI:Ag + are at 30, 44, 55.8, 63.6 and 84.5 c m - 1 ; all are associated with the on-center configuration of the Ag + impurity. The instrumental resolution is 0.1 cm-1. Curve B: the calculated absorption coefficient according to the two-parameter harmonic (5, 5 ~) model. The gap mode delta-function is not shown. Note that the structure in the acoustic spectrum for curve A is not associated with density of states features in curve B. (After Sandusky et al. 1993a).

resonant mode peak height was matched to the one in curve A. For clarity, the gap mode delta-function is not shown. Although this perturbed harmonic model gives a good account of the positions and the strengths of the resonant and gap modes, as well as the pocket mode isotope effect, it cannot at the same time account for the frequency dependence or strength of the broadband acoustic absorption. This could be due to neglected anharmonic or electronic deformation effects involving the Ag + ion, or to the neglect of the static relaxation effects beyond the defect's fourth-nearest neighbors.

A.J. Sievers and J.B. Page

198

Ch. 3

2.3.4. Anharmonicity studied by uniaxial stress and electric field measurements Given the anomalous thermal behavior of KI:Ag +, which falls well outside the harmonic approximation, it is of interest to explore this system's anharmonic interactions. Accordingly, uniaxial stress shifts and electric field measurements have been made. Note that for the pocket modes, these measurements should provide information about host lattice anharmonicity near the (200) family of ions, whereas such measurements for the resonant modes should be sensitive to anharmonicity near the defect. Because the stress and E-field induced frequency shifts are small, a "global" analysis is used to obtain the experimental shifts. The method consists of overlaying the shifted line at some given non-zero perturbation onto the corresponding unshifted line and varying the position and width of the line at non-zero perturbation until the area between the two curves is minimized.

J

O~I(SCO)

~2(C0 -- A(co))

peak O~1

peak O~2

d~o

(2.29)

as a function of the two variables s and Aw, where the two absorption lines O~1(03) and a2(w), are normalized to unit height. The width of the first line relative to that of the second is then equal to s and its frequency shift is Table 1 Measured values of the stress coupling coefficients of the IR-active gap and resonant modes of the KI:Ag +, KI:CI-, KI:e-(F-center), KI:Cs +, and KI:Rb + defect systems. Note that B/A nearly vanishes for the first two modes, which are pocket modes, and for the 55.8 cm-1 KI:Ag + mode in the acoustic spectrum. On the other hand, the KI:Ag + resonant mode at 17.3 cm-1 and the last six "standard" gap modes have much larger values of B/A. The stress coupling coefficients are given in units of cm-1/unit strain. The measurements were made between 1.7 and 3.5 K (After Rosenberg et al. 1992). Mode

Frequency (cm -1)

A

B

KI:Ag + KI:Ag + KI:Ag + KI:Ag +

86.2 84.5 55.8 17.3

186 188 143 390

4- 21 4- 22 4- 32 4- 90

KI:C1KI:CIKI:eKI:Cs + KI:Rb + KI:Rb +

76.8 77.1 82.7 82.9 86.3 86.9

102 101 143 148 147 143

4- 36 4- 36 + 55 + 56 + 55 4- 55

C

9 4- 6 7-4- 6 2 4- 9 510 4- 70 53 455 439 480-l69 468 +

10 10 18 15 14 14

B/A

- 7 4- 6 0 4- 6 - 1 6 4- 9 - 1 5 4- 10

0.05 0.04 0.01 1.31

+ 0.03 + 0.03 + 0.06 4- 0.35

-19 -20 -20 -20 -21 -20

0.52 0.54 0.27 0.54 0.47 0.48

+ 0.21 + 0.22 4- 0.16 4- 0.23 -I- 0.20 + 0.21

4- 10 4- 10 4- 15 4- 15 4- 15 -i- 15

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Unusual anharmonic local mode systems

199

Table 2 Measured and QD model predicted electric-field-induced frequency shifts for the Tlu pocket gap mode. The values in parentheses give the predicted minimum and maximum shifts produced by anharmonic parameters consistent with the uncertainties in the measured Tlu gap and resonant mode uniaxial stress coefficients (After Sandusky et al. 1994). EDC

EIR

[100]

[100]

[100]

[010]

[110]

[110]

[110]

[1-10]

Aw/AE2[IO-6 cm-1/(kV/cm) 2] Exp. QD Exp. QD Exp. QD Exp. QD

1.70 -t- 0.09 1.13 (0.67/1.67) -0.93 4- 0.05 -0.42 (-0.28/- 0.58) 0.08 -t- 0.36 0.32 (0.12/0.54) 0.20 + 0.29 0.39 (0.27/0.55)

related to Aw. The stress and electric field results for the different crystal orientations and polarizations are given in tables 1 and 2, respectively. Stress measurements have been carried out on a variety of different gap modes in KI. The behavior of these gap modes is qualitatively different from that of the KI:Ag + pocket gap modes. Inspection of the coupling coefficients collected in table 1 reveals similar behavior for all impurity gap modes listed, except for Ag +. This suggests that a small B/A ratio is a stress behavior signature of the pocket gap modes, which sets them apart from the other gap modes. Note that the B/A difference is also large between the KI:Ag + pocket gap mode and the 17.3 cm -1 resonant mode (see table 1). Interestingly, the B/A ratio for the sharp band mode peak at 55.8 cm -1 for KI:Ag + is very small, comparable to that for this system's pocket modes. These examples demonstrate the unique nature of the stress behavior of the KI:Ag + pocket gap mode, the related isotope mode and the 55.8 cm -1 band mode.

2.4. The (6, 6', 6") and the quadrupolar deformability models As detailed in (Rosenberg et al. 1992; Sandusky et al. 1994), quasiharmonic extensions of the (6, 6') model prove inadequate to explain the anharmonic shifts revealed by the stress and E-field experiments. The original (6, 6') model assumed that defect-induced inward relaxation of the silver ion's six nearest neighbors produces the force constant change 6'. The magnitude of its fit value is roughly half the pure KI nearest-neighbor overlap force constant, implying substantial relaxation. This suggests in turn that relaxation-induced force constant changes 6" = -Aq~xx(200, 300) should also be included. These changes could have a strong effect on the pocket

200

A.J. Sievers and J.B. Page

Ch. 3

gap modes, since their displacement patterns are so strongly peaked on the (200) family of ions. Recent work has shown how 8" can be included without adding any free parameters. By assuming that the introduction of the defect into the unrelaxed crystal produces radial forces on just the defect's six nearest neighbors and working within a linearized theory (i.e., small relaxations), one can use the pure crystal harmonic shell model Green's functions to compute the static displacements throughout the lattice, relative to those on the defect's six nearest neighbors. If these relative static relaxations are combined with the assumption of cubic anharmonicity arising from nearest-neighbor central potentials, then the necessary force constant change ~;" can be computed uniquely in terms of 8'. For KI this procedure yields, 8" - 0 . 6 8 ' with no adjustable parameters added to the (8, 8') model. For this new "relaxation" model (8, 8~, 8"), the fits to the measured IR Tlu resonant and gap mode frequencies again give three nearly degenerate Alg, Eg and Tlu pocket gap modes, which are almost identical in frequencies and displacement patterns to the pocket gap modes predicted by the (8, 8') model. Nevertheless, despite improvements with some experimental results and the fact that the (~, St, 8,,) model is well-motivated physically, it cannot be complete: it predicts a Raman-active Eg symmetry resonant mode at 26.3 cm -1, 10 cm -1 above the experimental frequency and 6 cm -1 above the (8, 8') model prediction. Furthermore, like the (8, 8~) model, it cannot provide a basis for a consistent explanation for both the measured stress and Stark shifts. A consistent explanation of the measured stress and electric field pocket gap mode shifts has been obtained by introducing a Ag + electronic quadrupolar deformability-induced harmonic force constant change A4~xx(100,- 100) = 2A~bxu(100, -010) = t~QD (Sandusky et al. 1994). Quantum mechanically, such force constant changes arise from virtual s-d electronic transitions and have been argued to be important for the Ag + ion (Fischer et al. 1972; Fischer 1974; Dorner et al. 1976; Kleppmann and Weber 1979; Bilz 1985; Jacobs 1990; Corish 1990; Kleppmann 1976). This QD model adds but a single free parameter to those included in the original (~, ~') model; ~" is still determined uniquely in terms of ~' as described above. The force constant changes d; and ~' are obtained by fitting the measured IR resonant mode and pocket gap mode peaks at 17.3 and 86.2 cm -1, as before, and t~QD is adjusted to reproduce the observed 16.1 cm -1 Eg Raman peak. Overall, the resulting predicted harmonic properties are in substantially better agreement with experiment than for the (~, ~') model, and now the Eg resonant mode frequency is (necessarily) correct. Moreover, the QD model provides a consistent explanation for the gap mode E-field and uniaxial pressure induced shifts, plus the large E-field mixing seen for the Tlu and Eg resonant modes. The stress and E-field experiments probe the anharmonicity in the vicinity of the defect. To model the measured stress shifts, a quasiharmonic

w

Unusual anharmonic local mode systems

201

Applied Stress

Purecrystal,strains! Harmonic defect model

i_ [

i Defectcrystal !Anharm~ .........

t

strains Li ~

Force constant ! changes ~9 ......

1Frequency

Normalized gap mode ! displacement patterns .......................

shifts

Fig. 37. Schematic diagram illustrating the procedure for calculating the stress coupling coefficients of the KI:Ag + impurity modes. Note that both the defect-crystal strains and impurity-mode displacements are determined from the perturbed harmonic shell model. The anharmonicity is only needed to determine the force-constant changes produced by the harmonic defect-crystal local strains. The electric-field Stark shift calculations follow an analogous procedure. (After Rosenberg et al. 1992).

theory is employed as follows. First, the perturbed harmonic force-constant model is used to compute the local strains induced by the applied stress. These strains are then combined with the assumption of nearest-neighbor cubic anharmonicity to yield the local force-constant changes. Finally, these are used in a perturbation theory calculation of the stress-induced frequency shifts. This procedure is schematically outlined in fig. 37 and is detailed in the appendix of Rosenberg et al. (1992). The E-field Stark shift calculations follow an analogous procedure, with the different symmetry of this perturbation being taken into account. Figure 38 shows measured and predicted pocket mode difference spectra for an 87 kV/cm applied static field and EIRIIEDc[100] probe-field geometry. The predicted spectrum in (a) is for the (~;,d;', ~") model, with the anharmonicity parameters obtained from measured stress shifts, while the prediction in (b) is for the QD model, again based on stress-fit anharmonicities. It is seen that the QD model accounts well for the measured difference spectrum, whereas the (5, ~',~") model is in marked disagreement-

A.J. Sievers and J.B. Page

202

2.0 0.0 v

E

Ch. 3

(a)

""

f '\

\~~l, \

',.i .,]

-2.0

.,,.

!

~"-'"-'"

".9"~0.15

0.00

-0.15

,

s5

,

86

(0 (cm "1)

Fig. 38. Experimental Stark effect pocket gap mode difference spectrum, compared with the predictions of two different models. The measured spectrum, Aa = a(EDc = 87 k V / c m ) a(EDc = 0) for EIRIIEDc [100], is represented by the solid curves. (a) Comparison with the (6, 6', 6") model spectrum (dot-dash line), which is more than an order of magnitude too large. (b) Comparison with the QD model spectrum (dashed line). The theoretical Stark shifts of table 2 are computed using anharmonicities fit to measured stress shifts. The predicted difference spectra shown here are generated from the theoretical shifts by using a Voigt lineshape determined from the measured zero-field lineshape for the IR pocket gap mode. (After Sandusky et al. 1994).

it predicts shifts which are nearly two orders of magnitude larger than the experimental shifts. Table 2 compares the measured static E-field induced pocket mode frequency shifts predicted by the QD model using the QD model stress-fit anharmonicities. The computed shifts of table 2 are determined from strengthweighted frequency averages ~ - ~ i S i ~ i / ~ i Si as a function of applied field strength, and the uncertainties in the predicted shifts arise from uncertainties in the stress measurements. The predicted and observed shifts for the EIRtiEDC [100] p r o b e - field geometry are seen to overlap, while the predicted shift for EIRIEDc [100] is half the observed shift. The predicted and observed shifts for EDC [ 110] also overlap, but the uncertainty in the experimental shifts for this configuration precludes strong conclusions. Using the same stress-fit anharmonicities, the QD model also predicts a relatively

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Unusual anharmonic local mode systems

203

large E-field induced mixing for the low frequency Tlu and Eg resonant modes (frequency shifts ,-,, cm -1 at E = 100 kV/cm); these shifts are in excellent agreement with the experimental shifts (Kirby 1971). 2.5. Discussion and conclusions A comprehensive comparison of experiment and theory on KI:Ag + is collected in table 3. Since the theoretical work has necessarily focused on the T - 0 K results, much of the table is devoted to this limit. Ten different kinds of spectroscopic data are identified in the first column to compare with the three successively more refined harmonic shell models, labeled (a), (b) and (c). Column (a) identifies the two-parameter (3, 3') model; column (b) the two-parameter (3,/~, 3") model which includes relaxation without any additional parameters and column (c) the three-parameter QD model which includes electronic quadrupolar deformability in addition to (3, 3~, 3"). Although the simple (3, 3') model does a surprisingly good job in predicting the pocket modes and provides reasonable agreement with five of the ten experimental results listed, its quasiharmonic extension cannot account for the pocket mode stress and Stark electric field results listed as 11 through 14. The two-parameter (3, 3~, 3") model may be viewed as an intermediate model on the way to the QD model in that there is improved agreement with some of the eleven experimental items while at the same time the disagreement with the experimental Eg Raman resonant mode frequency increases. The QD model is a remarkable success. With its single additional parameter fit to the experimental Eg Raman resonant mode frequency, the QD model provides the same good overall agreement with the first ten experimental results as does the (3, 3~, 3") model, and its quasiharmonic extension describes both the stress and electric field measurements consistently. From the stress and Stark measurements (table 3, entries 11 through 15), there are 15 new pieces of spectroscopic data. When five of these from the stress measurements are used to fit the anharmonic QD model, the other 10 experimental results, which include the Stark electric field data, are predicted correctly. These successes of the QD model demonstrate that the silver ion possesses a significant electronic quadrupolar deformability in KI:Ag + and that this deformability plays an essential dynamical role. Conversely, the good agreement between the experimental data and the QD model also show that an anomalous potential energy anharmonicity does not appear to play a role in the description of the T = 0 K dynamics. Although the QD model provides an excellent description of most of the T = 0 K features, there are a few discrepancies that stand out and need to be considered. The most important ones are perhaps items 8 and 9 in table 3,

A.J. Sievers and J.B. Page

204

Ch. 3

Table 3 Qualitative Comparison of the KI:Ag + on-center experimental results with calculated results based on perturbed shell models. (a) the two parameter (6, 6t) model; (b) the two parameter (6, 6~, 6") model and (c) the three parameter (6, 8~, 6") + QD model. Experimental Results

SM Results

(a)

(b)

(c)

harmonic approx, harmonicapprox, harmonicapprox. fit to (6, 6') model fit to (6, 8', 6") 1. Tlu resonant and gap mode freqs (17.3 cm- l, model 86.2 cm- l)

fit to (6, 6', 6")+QD model

2. Relative Tlu resonant and gap mode strengths (,-~ 3)

fair agreement (1.4)

good agreement (3.0)

good agreement (3.0)

3. resonant mode isotope frequency shift (-0.14 4- 0.03)

poor agreement (-0.05)

good agreement (-0.12)

good agreement (-0.12)

4. gap mode isotope frequency shift ( - 1.7)

good agreement ( - 1.46)

good agreement (-1.63)

good agreement (-1.64)

5. relative gap mode isotope strength (0.04)

fair agreement (0.073)

fair agreement (0.074)

fair agreement (0.073)

6. Eg resonant mode (16.1 cm -1)

fair agreement (20.5 cm -1)

poor agreement (26.3 cm -1)

fit to QD model

7. Alg resonant mode not

Alg resonant mode Alg resonant mode Alg resonant mode (37.3 cm -1) (41.5 cm -1) (41.5 cm -1)

observed

poor agreement

poor agreement

poor agreement

9. weak absorption peaks poor agreement at 30, 44, 55.8, 63 cm -1

poor agreement

poor agreement

Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 cm-1, 86.0 cm -1)

Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 c m - 1, 86.0 cm -1)

8. magnitude of the broad acoustic Tlu absorption spectrum

10. Alg and Eg gap modes not observed in Raman.

Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.2 cm -1 , 86.0 cm -1)

anharmonic QD model 11. pocket mode and 17.3 cm-1 resonant mode stress effect

five parameter fit to the QD-based quasiharmonic model

12. isotope pocket mode stress effect

good agreement

w

Unusual anharmonic local mode systems

205

Table 3 (continued)

13. pocket mode Stark effect

Experimental Results

SM Results data agrees with predictions of the QDbased quasiharmonic model with no additional parameters

14. isotope pocket mode Stark effect

good agreement

15. resonant mode Stark effect

good agreement

16. Large gap mode line width in comparison to other gap mode systems (0.5 cm -1) r (0.14 cm -1)

outside of the model

Experimental temperature dependences (with increasing temperature)

Temperature dependent properties outside the framework of the QD-based quasiharmonic model

17. disappearance of the Tlu and Eg resonant and Tlu pocket mode strengths 18. weak broad IR gap mode (78.6 cm -1) appears and then levels off 19. Raman resonant mode (Alg + Eg) (12.2 cm -1) appears and then disappears into a broad central peak 20. Temperature dependent pocket mode A and B stress coefficients. which refer to the poor agreement between the model and the magnitude and weak structure in the measured impurity-induced absorption in the acoustic region, shown in fig. 36. It is found that this disagreement is preserved for both the (~, ~', d;") and the QD models; hence these experimental properties appears to be outside of the scope of the current models. The ultrasharp feature at 55.8 cm -1, listed in item 9, may not be unique to Ag + but may simply be an impurity-induced density of states peak, since a similar feature has been seen for Na + impurities in KI (Ward et al. 1975). However, the feature is produced by the Ag + ion since like the resonant mode, it disappears as the sample temperature is increased. The three models described here do not reproduce such an ultrasharp feature in the acoustic spectrum. Item 16, the T = 0 K pocket gap mode linewidth is another property that stands out. Except for Ag + which has a gap mode width of 0.5 cm -1, the linewidth for all other point defects in KI is about 0.14 cm -1. This factor

206

A.J. Sievers and J.B. Page

Ch. 3

of three difference may be significant. The smaller of these two values could be a consequence of random DC strains in the crystal producing an inhomogeneously broadened gap mode band but, according to table 1, the ratio of the hydrostatic coupling coefficients of Ag + to the Cs + gap mode, which is closest in frequency, is equal to 1.26: a value much too small to explain this linewidth variation. Since the resulting QD model parameters now indicate that only standard anharmonicity parameters are associated with pocket modes, the large linewidth observed for KI:Ag + does not appear to be driven by an unusual potential energy anharmonicity. The last section in table 3 lists a number of observed temperature dependences (17 through 20), all of which are outside of the QD model. We wish to make the point here that the temperature dependence results appear to be outside the scope of many other phenomenological models as well. For example, applying Boltzmann statistics to the two configuration model (fig. 17) shows that to account for the rapid temperature dependence of the resonant mode strength (fig. 14), the number of levels in the off-center C2 configuration state must be at least 100 times that of the on-center C~ ground state configuration. What physical process for a single point ion in a lattice could generate this much entropy by 25 K? One conclusion is that these temperature-dependent experimental results are not only outside of current lattice dynamics theory but are difficult to understand even in a more general phenomenological framework. One conclusion which appears to be independent of the details of the temperature dependence is that the quantitative temperature-dependences of the strengths of the IR resonant and pocket modes provide important complementary information on the participation of the Ag + impurity's surrounding ions in this system's anomalous thermally-driven on-to off-center transition of the Ag + ion. Measurements and analysis have demonstrated that both the low-frequency IR resonant mode and the strong pocket IR gap mode disappear at identical rates with temperature in the range 0 K to ~ 25 K, even though the dynamics of each, according to all three perturbed shell models, involve ion motion in different regions of the defect vicinity. Since the mode frequencies are nearly temperature-independent over this temperature interval, it appears that these two nearly-harmonic modes simply monitor, in different spatial regions, the population of the on-center configuration; hence, the thermally-driven instability in this system involves the entire coupled defect/host system in the impurity region.

3. Intrinsic localized modes in perfect anharmonic lattices We now tum from the previous, experimentally-driven study of a fascinating anharmonic solid-state defect system, to the still purely theoretical

w

Unusual anharmonic local mode systems

207

problem of intrinsic localized modes in anharmonic lattices. As noted in the Introduction, our focus is on simple descriptive aspects of some of the interesting phenomena. We will begin with the simpler case of monatomic lattices with harmonic plus hard quartic nearest-neighbor interactions and will later consider the important generalizations of adding cubic anharmonicity or treating full realistic potential functions V(r). When ILMs occur in these monatomic systems, their frequencies are above the maximum frequency of the harmonic lattice, and we will describe their existence, stability, motion and interactions with plane-wave phonons. We will then consider diatomic lattices for the same interactions; in this case one has the interesting additional possibility that ILMs can exist in the frequency gap between the acoustic and optic phonon branches.

3.1. One-dimensional monatomic lattices

3.1.1. Heuristic illustration of vibrational localization We begin with a qualitative example to illustrate why localized modes might be expected to appear in a perfect but anharmonic lattice. For a onedimensional monatomic harmonic lattice with particles of mass m connected by nearest neighbor spring constant k2, the normal modes are described by an orthogonal set of homogeneous plane waves which are confined to a band of frequencies with a high frequency cutoff at Wm. The amplitude pattern of this highest-frequency mode involves every atom vibrating 7r out of phase with its neighbor and is shown in fig. 39(a). To estimate the rms amplitude of a particular particle in a plane wave harmonic mode ~o for a chain of N particles, we use the virial theorem. For a harmonic system the mean energy in a mode is equal to two times the mean kinetic energy, e.g., hw/2 - N[mwe(u2)] hence the rms amplitude at each site n in this plane wave mode, (u2) 1/e ~ (N)-l/z(h/2mw) 1/e, is quite small for a long chain. Next we form a localized wave packet at t -- 0 at a particular site with the appropriate linear superposition of this set of plane wave modes. This localized excitation in which only a few atoms, Nd, are displaced from their equilibrium position is shown in fig. 39(b). If this were a normal mode, the central atoms would have relatively large rms amplitudes, (U2) 1/2 ~ (Nd)-l/2(h/2mw) 1/2, independent of the number of atoms N in the long chain. However, this is not a normal mode for the harmonic system, and for t > 0 lattice dispersion insures that this localized excitation would not remain confined for long. For a similarly constructed one-dimensional anharmonic lattice with N particles, the plane wave spectrum derived in the small oscillation limit is

208

A.J. Sievers and J.B. Page (o) 9

&

9

COm

9

9

Ch. 3

9

9

.... 9

9

(b)

......

_i

[

t

i

L_ r-

Nd

~uj

--J --I

(c)

3~

Wm

t

/////////////////~ ///1////////// .

.

.

.

.

.

.

.

[

I

Fig. 39. A schematic representation of the frequencies and eigenvectors of plane wave and localized modes. (a) The eigenvector of the highest frequency plane wave mode at Wm. (b) A localized mode with frequency w~ involving a subset Nd of the N lattice sites. (c) The plane wave vibrational spectrum located between 0 and win, the anharmonic localized mode with frequency w], and the third harmonic of this local mode (dashed line). The local mode eigenvector A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ) for the strongly localized (triatomic molecule) limit is also shown on the right.

necessarily similar to that found for the harmonic case. But when a wave packet is constructed at t = 0 to represent a localized disturbance in this chain, a different situation occurs from the harmonic example above. To be specific we assume that the potential energy anharmonicity is quartic with a positive coefficient, and we imagine that the wave packet ~is localized in stages. As the wave packet corresponding to fig. 39(b) is formed, the amplitude of the particle at the central site increases, but since the potential is no longer harmonic, so does its frequency of vibration. The quartic anharmonicity may be thought of as renormalizing the harmonic force constants within the packet to the higher values. According to Rayleigh's theorem the resulting frequency shift for each plane wave mode can be no larger than the frequency interval between the original lattice modes (Maradudin et al. 1971). As the amplitude pattern becomes more strongly localized around a particular site so that Nd --+ 1, the vibrational amplitude becomes

w

Unusual anharmonic local mode systems

209

larger, allowing the maximum harmonic lattice frequency win, which is not bounded from above, to rise above the plane wave spectrum, transforming to an inhomogeneous localized mode in the process. Given sufficient anharmonicity the end result is a true localized mode centered at the particle in question with frequency Wl as shown in fig. 39(c). The anharmonic system has evolved in such a way that both homogeneous and inhomogeneous modes exist simultaneously. Depending on the anharmonicity of the lattice, the eigenvectors shown in fig. 39 represent three different possibilities: Case (a) depicts the highest frequency ~m plane wave mode for the small anharmonicity limit. With increasing anharmonicity a localized solution with frequency a;~ > tom also exists above the plane wave spectrum, and its eigenvector is represented by (b). For this case there are roughly Na atoms involved in the localized vibration and since Na << N the amplitude of each atom in this mode is much larger than for the plane wave case. In the high frequency limit, Wl >> Wm, the local mode eigenvector takes on the particularly simple form of a vibrating triatomic molecule as shown in case (c). The spectrum shown in fig. 39(c) includes not only a plane wave spectrum of modes plus one localized mode but also an overtone of the local mode, represented by the dashed line. In principle, the anharmonic quartic potential produces a response at the third, fifth and higher harmonics, but because these frequencies are so much larger than both the plane wave and the local mode frequencies, it is reasonable to assume that the lattice cannot respond at these higher frequencies so that these nonlinear mixing terms can be ignored in the dynamical analysis. This approximation is called the rotating wave approximation (RWA). The original anharmonic lattice with the spectrum shown in fig. 39(c) is now described by a set of effective harmonic oscillators, both plane wave-like and localized.

3.1.2. Quantitative study of quadratic and hard quartic nearest-neighbor potentials (k2, k4) As noted in the Introduction, the prediction of Sievers and Takeno (1988) was based on this case, for which the potential energy function is given by ~2

v = -

2

(Un+l -n

k4 Z ( U n + l _

+ 7

Un)4

(3.1)

n

where k2 and k4 are the harmonic and quartic spring constants, respectively, and un is the longitudinal displacement of the nth particle from its equilibrium position. For a purely harmonic lattice, the solutions of the

210

A.J. Sievers and J.B. Page

Ch. 3

equations of motion are the familiar plane waves, with the dispersion relation w(k) = Odm sin(ka/2), where a:m =-- 2(k2/m) 1/2 is the maximum lattice frequency and a is the equilibrium lattice spacing. Writing the equation of motion for the nth particle, substituting the trial solution un(t) = A~,~ cos(a;t) and making the rotating wave approximation (RWA) by replacing the resulting cos3(wt) term by the first term (3/4)cos(cot) in its Fourier expansion, one obtains

--od2m~n

"-

k2(~n+l -- 2~n + ~,,+l) +

3k4A 2 4 [(~n+l -- ~n) 3 -- (~n -- ~n-1)3],

(3.2)

where m is the particle mass. This was the procedure followed by Sievers and Takeno (1988), and an approximate solution was then obtained via the use of lattice Green's function techniques. The result was that odd-parity ILMs with the approximate displacement pattern A(..., 0, - 1/2, 1, - 1/2, 0 . . . . ) given in w1.2 above can exist at any lattice site, with the frequency given by

(w) ~mm

3( ~

27A4 1+

16

'

(3.3)

where the anharmonicity parameter is A4 -- k 4 A 2 / k 2 9 This solution was argued to be a good approximation provided An >> 16/81. As we have noted, the molecular dynamics (MD) simulations of fig. 1 strikingly verify this solution. The value of A4 for the simulation in the lower panel was 1.63, well within the range of validity of eq. (3.3). A power spectrum of the MD displacements of fig. l(b) shows that the observed ILM frequency is within 2% of that predicted by this RWA equation. The theoretical arguments of Sievers and Takeno (1988) utilized a lattice Green's function formalism that is somewhat complicated to the uninitiated. Yet the ILM vibration appearing in fig. 1 is exceedingly simple. In the following subsection we give a simple and direct argument which shows that the tendency to localization in this case of strong quartic anharmonicity reflects a fundamental property of the underlying purely anharmonic system.

3.1.2.1. Asymptotic behavior; odd and even modes. To more simply understand the above ILM solution and its connection to the anharmonicity, we

w

Unusual anharmonic local mode systems

211

now focus directly on the equations of motion, for the hypothetical pure quartic case. Setting k2 - 0 in eq. (3.2) and rearranging, we have

w2 ---

3k4A 2

[(~n - ~n+l) 3 at- (~n -- ~n-1)3] 9

(3.4)

4m(n Let us now seek a localized odd-parity solution, centered at site n = 0. We thus require (0 = 1, ~-n = (n, and I~1 << I~11 for Inl > 1. Within these restrictions, the n = 0 and n = 1 versions of eq. (3.4) give

w2 =

3k4 A2

2(1 - (1)3,

n = 0,

(3.5)

4m and w2 ~

3k4A 2

[(~+(~1-1)3],

n=l.

(3.6)

4m~cl For a solution, these two frequencies should of course be the same, and we see that for ~1 = - 1 / 2 they become nearly equal"

w2k4A2(3) -

m

4 ~

,

n=0,

(3.7)

and k4A2 ( 3 ) 4 ( 60 2 ~

1+

1 )

n-

1

9

(3.8)

Next we proceed to the equation of motion for the particle at n = 2:

602

3k4 A2

[({2 -- ~3)3 -t- (~2 -- {1)3] 9

4m~2 In accord with our approximations, we set ~ 2 - (l ~,~ 1/2 and neglect ( 2 - (3 in comparison, obtaining w e ~ (3k4A2)/(32m(2). By equating this with the n = 0 expression for w 2 (eq. 3.7), we find 1

~2 ~ ~ . 54

(3.9)

A.J. Sievers and J.B. Page

212

Ch. 3

Thus 1~2] << I~l[ = 1/2. Finally, assuming I~n+ll << I ~ l for all n/> 2, we use eq. (3.4) to obtain a recursion relation for n/> 2

..~

.

(3.10)

Hence, all of the ~n's for In[/> 2 rapidly approach zero with increasing n, and it is seen that the odd-parity displacement A(..., 0, - 1/2, 1, - 1/2, 0,...) is indeed an approximate solution for the pure quartic case. The ILM frequency given by eq. (3.7) is the same as that given by the k2 = 0 version of eq. (3.3), and we find that this RWA frequency is within 2% of the exact frequency observed in MD simulations. When harmonic interactions are added, eq. (3.3) is recovered, and as noted earlier, it is a good approximation provided that the anharmonicity parameter does not become so weak that the homogeneous plane-wave harmonic solutions become dominant. Beyond the insight provided by this simple heuristic argument, it can easily be generalized to give an interesting exact result (Page 1990). Suppose that the nearest-neighbor pure quartic interaction is replaced by a nearestkr neighbor anharmonic interaction of arbitrary even-order: V = T ~,~(Un+lun) ~, where r = 4, 6 . . . . . The equations of motion are then

m~tn(~) ~ ]gr{ [Un+l(~)- Un(~)] r-1 -- [Un(~)- Un--l(~)] r-1 }.

(3.11)

Our focus here will be on the spatial behavior of the solutions, and for this purpose we could use the RWA for the time dependence, just as we did above for the pure quartic case. However, eqs (3.11) readily separate, giving rise to solutions periodic in time, for any even r (Kiselev 1990). Thus for the sake of generality, we briefly digress to bring in the exact, rather than the RWA frequency. For the trial solution un(t) = A~nf(t), it is straightforward to obtain an exact expression for the period T, from which the square of the frequency w = 27r/T is obtained as

W 2 B"k"A"-2[ rn~

(~ - ~+1

),.-1

+ (~n --

~-i

)r-l]

(3 12) ,

where the coefficient B~ is given by

)2

B,.-~

x/1-f ~

.

(3.13)

w3

Unusual anharmonic local mode systems

213

Notice that for the r = 4 pure quartic case, eq. (3.12) would go over to the RWA result eq. (3.4), provided/34 = 3/4. Indeed, evaluation of the elliptic integral appearing above gives B 4 = 0.718, and we again see that the RWA works very well for the pure quartic case. We now return to the question of the spatial behavior and focus on the (n's in the right-hand side of eq. (3.12). Following the preceding argument for eqs (3.5-3.10), we seek a localized odd-parity ILM, centered at site n - 0. Again neglecting (2 compared with (1, we find that the n = 0 and n = 1 versions of eqs (3.12) are nearly satisfied by (0 = 1 and (1 = - 1 / 2 :

co2__2krBrAr-2 ( 3 ) r-1 -

m

~

,

n-O,

(3.14)

and

w2 ~

2krBrAr-2(3)r-l[ (1) r-l] 1+

n-

1

(3.15)

These equations differ only by the factor 1 + (1/3) r - l , which approaches 1 in the asymptotic limit of large r. Turning to the n = 2 version of eq. (3.12), we again have ~2 - (1 .-~ 1/2 and neglect ~2 - (3 in comparison, obtaining w 2 ..~ (krB~A~-2)/(m(22~-l). Setting this equal to the n - 0 expression for w 2 (eq. 3.14), one finds (2 --~

1

2.3

,

(3.16)

r-1

which approaches zero in the limit of large r. As a final step, we again assume that I~n+xl << I~nl for all n >/ 2, and use eq. (3.4) to derive the general-r analog of the recursion relation eq. (3.10) for n t> 2:

,.+, ~n

..~

~n-1

.

(3.17)

Clearly, the (n's rapidly approach zero with increasing distance for fixed r, and they all vanish in the large r limit. Thus the odd parity ILM pattern A ( . . . , 0, - 1/2, 1, - 1/2, 0 , . . . ) is an asymptotically exact solution in the limit of increasing even anharmonic order (Page 1990). Even for the r = 4 "worst" case of a pure quartic system, this solution remains very a c c u r a t e - in the preceding reference, a more exact

A.J. Sievers and J.B. Page

214

Ch. 3

2.2 f

1.8

even ~

/i

odd

1.4

1.0

!

0

]

2

A4 Fig. 40. Computed odd- and even-parity ILM frequencies versus the quartic anharmonicity parameter A4. These curves were obtained by numerically solving the RWA equations of motion for all of the particles in a 40-particle monatomic linear chain, with periodic boundary conditions. Three of the corresponding odd-parity ILM displacement patterns are shown in the next figure.

calculation for the pure quartic case was found to correct the mode pattern only slightly, to A( . . . . 0, 0.02, -0.52, 1, -0.52, 0.02, 0 . . . . ). The simple odd-parity ILM pattern given by Sievers and Takeno (1988) therefore reflects a fundamental exact property of the underlying purely anharmonic system. This pattern is just that of a simple linear triatomic molecule of equal masses, and is in fact the most localized odd parity pattern which keeps the center of mass at rest. This leads naturally to the question of whether the most localized even parity displacement pattern, namely that of a linear diatomic molecule A( . . . . 0 , - 1, 1,0,...), might also be an asymptotically exact solution for the purely anharmonic system in the same limit of increasing evenorder anharmonicity. This was proven to be the case by Page (1990). For the pure quartic case this asymptotically exact mode pattern is corrected to A ( . . . , 0, 1/6, - 1, 1, - 1/6, 0 . . . . ), and the corresponding RWA frequency of the pure-quartic even-parity mode is given by w2 ~ (6kaA2/m)[1 +(7/12)3]. Again, this is readily verified by MD simulations. Even modes were discovered independently in numerical simulations by Burlakov et al. (1990a-d) and Bourbonnais and Maynard (1990). In the following section, we will see that the above asymptotic limit also gives simple insights into the stability properties of the odd and even ILMs. However, before moving to this topic, we return briefly to the harmonic plus quartic (k2, k4) case and note some additional simple aspects. As the quartic anharmonicity parameter An - kaA2/k2 decreases, the ILMs are expected to

w3

Unusual anharmonic local mode systems

215

(a) ~/O~m=1.79

(b) o)/O~m=l.

(c) o~/r

Fig. 41. Computed odd-parity ILM normalized displacement patterns {~n} as a function of the ILM frequency for a 40-particle (k2, k4) linear chain, with periodic boundary conditions. As for fig. 39, the RWA equations of motion were solved numerically for all 40 particles. In practice, one begins with an initial guess for the displacement pattern, and the routine converges to the correct pattern and frequency. The particle motion is longitudinal, but for clarity the displacements are plotted vertically. The displacement of the central particle is unity in each case. spatially broaden, and this is indeed found to be the case. For a fixed value of this parameter one can imagine moving out from the mode center until the displacements are so small that the anharmonic effects are negligible, and since the ILM frequency is necessarily above the maximum frequency Wm of the harmonic lattice, the amplitudes then decrease with distance just as for a localized impurity mode in the harmonic lattice. In one dimension this decrease is a simple exponential. This gives a straightforward numerical means for obtaining the mode displacement pattems: one simply solves the equations of motion (3.2) numerically for the particles having nonnegligible amplitudes and then applies the known harmonic-approximation analytic amplitude decrease for the particles beyond, as a boundary condition. A related approach is to apply periodic boundary conditions and simply solve the equations of motion numerically for all of the particles. These techniques have been used by a number of investigators (Bickham and Sievers 1991; Bickham et al. 1993; Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky and Page 1994).

A.J. Sievers and J.B. Page

216

Ch. 3

6.0

5.0

1D

4.0

I

I

2.0

1.0

0.0

/ 1.0

/

,I

2.0

,f

,,f./

3.0

, !

4.0

A4 Fig. 42. Local mode frequency versus A4 as calculated by two different rotating wave approximations. The dashed curve follows from the single frequency rotating wave approximation while the solid curve includes an additional contribution from the third harmonic term. The more exact two frequency calculation produces a slight lowering of the ILM frequency over that produced by the simple RWA. (After Bickham and Sievers 1991). Figure 40 plots the computed odd- and even-parity ILM frequencies for a (k2, k4) lattice versus the quartic anharmonicity parameter An, and fig. 41 shows odd-parity ILM displacement patterns for three different values of this quantity. The ILM spatial broadening with decreasing anharmonicity is clearly apparent; in the purely harmonic limit (k4 = 0), both the odd and even ILMs broaden into the zone boundary phonon mode A ( . . . , 1 , - 1 , 1 , - 1 , . . . ) . Interestingly, we will see in a later section that when cubic anharmonicity is added, this spreading is largely suppressed. In developing these analytic local mode solutions, it is assumed that the system only responds at the "fundamental" frequency in the assumed c0s(wt ) solutions. Because of the quartic potential, response at 3w, 5co, etc. should also be present. A straightforward approach to investigate the influence of the next higher order term is to generalize the rotating wave approximation to u,~ = (1 - 3)~n cos(wt) + 3~n cos(3wt). When this trial solution is inserted back into the equations of motion, the result is that the ILM frequency is corrected to a new lower frequency value. Figure 42 shows the magnitude

w3

Unusual anharmonic local mode systems

217

of the shift on the odd mode solution for one, two and three dimensions. As might be expected, the correction term grows with increasing anharmonicity parameter but it remains a small contribution over the entire parameter range. This figure confirms the idea that the simple rotating wave approximation is a valid approximation for identifying ILMs in anharmonic systems.

3.1.2.2. Stability. An important question concerns the stability of the ILMs against infinitesimal perturbations. By returning briefly to the asymptotic limit discussed in the previous section, we can easily determine the basic stability properties of these modes. Subsequently, we will sketch some of the quantitative aspects of these properties, followed by a discussion of their consequences for ILM motion. Much of the following stability material follows from the study by Sandusky, Page and Schmidt (1992), which should be consulted for details. The material on moving ILMs derives from that reference and the study by Bickham et al. (1992). It was seen above that the odd- and even-parity displacement patterns A(..., 0, - 1/2, 1, - 1/2, 0,...) and A(..., 0, - 1, 1,0,...), are asymptotically exact for a purely anharmonic lattice in the limit of increasing even-order anharmonicity. In this limit the interparticle potential becomes that of a square well. For point masses the "repulsive" side of the well occurs when the particle collide, at Un+l - - U n --" - - a , and because of the reflection symmetry possessed by an even-order potential, the "attractive" side of the well occurs at un+l - u n -- +a. Thus the square well has width 2a, and the particles move completely freely until either of two situations arise: 1) they collide elastically when their separation is zero, or 2) they attract impulsively ("snap back") when their separation reaches 2a. This limiting behavior for strong even-order anharmonicity is intuitively clear, and it has been derived rigorously by Sandusky et al. (1992). In this limit, the asymptotically exact even- and odd-parity ILM patterns above require that the amplitudes have the fixed values A - a/2 and A = 2a/3, respectively. This is easily seen in fig. 43. More importantly, one also sees clearly that for the even mode the collision and "snap" occur at different instants, whereas for the odd ILM the central particle simultaneously collides with one of its nearest neighbors and "snaps back" due to its attraction to the other nearest neighbor. It is then easy to see that any perturbation which destroys the simultaneity of the collision and snap will destroy the coherence of the odd-parity mode pattern, rendering this mode unstable. On the other hand, since the collision and snap in the even parity ILM are not simultaneous, this mode is stable. The above simple picture of ILM instability in the asymptotic limit of high even-order anharmonicity carries over to the case of even- and odd-parity modes in (k2, k4) systems; however, this case is more complicated than the

218

A.J. Sievers and J.B. Page O

L n

Ch. 3

_1 7

Collision (repulsion)

2a

L

_I

"Snap" (attraction)

A

-A

Even-parity mode A( .... 0,-1,1,0 .... )

Collision and snap simultaneous

21=1 I"

9

I_ I-

II

oi-.o -A/2

Odd-parity mode

-I _1 -I

~

A

-A/2

9

A( .... 0,-1/2,1,-1/2,0 .... )

Fig. 43. Collision (repulsion) and "snap" (attraction) for the even-parity ILM (top panel) and for the odd-parity ILM (bottom panel), in a monatomic lattice of point masses interacting via a nearest-neighbor anharmonic potential of even order, in the asymptotic limit of high order. In this limit, the potential becomes that of a square-well of width 2a and these mode patterns are exact. For the even-parity mode, the collision and snap do not occur at the same time, whereas for the odd-parity mode they do. This renders the odd-parity ILM unstable against any perturbation which destroys the simultaneity of the collision and snap. For the case of (k2, k4) lattices, the even- and odd-parity ILMs remain stable and unstable, respectively; moreover, the odd parity instability results in the ILM moving slowly from site to site, as discussed in the text. (After Sandusky et al. 1992).

above simple limiting case, and it requires careful analysis involving both analytic and numerical work. Briefly, one assumes a solution of the form

un(t) = A[~n + 5~ne~t] cos [wt + 6r

(3.18)

where 8~n and 8r are infinitesimal displacement and phase perturbations, respectively. Since M D simulations show that both the odd- and even-ILMs are generally stable over at least several periods, we assume that the above perturbations vary slowly in time with respect to a mode period. Performing an appropriate time-average (closely related to the RWA) and linearizing the equations resulting from the trial solution (eq. (3.18)), we arrive at a 4s • 4s eigenvalue problem to determine A, where s is the number of sites included in the unperturbed ILM displacement pattern {(n}. The ILM will

Unusual anharmonic local mode systems

w

219

0.18 o

0.14

= =,=,.

l-.

r 0.10 L_

i-.-

0.06

0

0.02 1.0

z~ I

2.0

I

3.0

I

4.0

5.0

(O/tOm Fig. 44. Instability growth rate vs. anharmonicity for odd-parity ILMs in a 21-particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is measured by the ratio of the ILM frequency to the maximum frequency of the harmonic lattice. The solid curve gives theoretical predictions obtained from the stability analysis sketched in the text, and the triangles are growth rates measured in MD simulations for various values of the amplitude and the ratio k4/k 2. The dashed line gives the predicted growth rate for the pure quartic lattice. (After Sandusky et al. 1992). be unstable if one finds a perturbation (determined from the eigenfunction) of the displacements or phases (i.e. velocities) which is associated with an eigenvalue A having a positive real part, since such perturbations will grow exponentially in time. Sandusky et al. (1992) should be consulted for details. Numerically applying this analysis, we find that the odd-parity pure quartic ILM is always unstable against even-parity displacement or phase perturbations [e.g. ( . . . , - d ; a , 0, d;a. . . . )], whereas the even-parity ILM is always stable. These results agree with the more intuitively clear asymptotic limiting behavior discussed above. With harmonic interactions (k2) included, the odd-parity ILM instability growth rates are predicted to decrease with decreasing anharmonicity, and the even-parity ILM is predicted to be stable. Figure 44 shows the predicted odd-parity ILM instability growth rates as a function of anharmonicity for the (k2, k4) lattice, and these are compared with growth rates measured in MD simulations. The dashed line gives the pure quartic limit. The predicted and MD results are seen to be in very good agreement; as discussed by Sandusky et al. (1992), the ILM spatial broadening with lowering anharmonicity was not included in the growth-rate predictions in this figure, and when they are included the minor discrepancies at low anharmonicities are removed.

A.J. Sievers and J.B. Page

220

Ch. 3

2.5 t-

1.5

. .O, .

o r (D

0.5

-~ -0.5 ...=

ca. -1.5 -2.5 19420

i

I

19430

19440

19450

time (units of 2~(Or.) Fig. 45. Even-parity mode stability in a 20-particle pure quartic lattice, as seen in MD simulations after more than 32,000 oscillations. The initial displacement pattern is the pure quartic even-parity pattern ( . . . . 0, 1 / 6 , - 1, 1 , - 1/6, 0 . . . . ), centered at sites (-0.5, 0.5). For this run k4 and the amplitude are chosen so that w = 1.7Wm, where Wm= 1.0 (eV/~, 2 amu) 1/2 is a convenient frequency unit. The displacements are magnified, for clarity, and the displacements on the particles not shown are negligible. This mode is exceedingly stable, as predicted by the perturbation theory analysis. (After Sandusky et al. 1992).

As noted above, no instability is predicted for the even-parity ILM, and MD simulations have found this mode to be extremely stable: runs for the pure quartic and for the harmonic plus quartic case found no changes in the even-parity mode over more than 32,000 oscillations, as is illustrated in fig. 45 for the pure quartic case. As pointed by Sandusky et al. (1992), the even-parity ILM stability is further manifested by the fact that even when given t = 0 perturbation seeds which cause the odd-parity ILM to move after just tens of oscillations, the even mode was still found to persist unchanged for more than 32,000 oscillations (the maximum extent of the MD runs). Given that the odd-parity ILMs are unstable, the question arises as to how the instability manifests itself. In MD runs, it is observed that the instability does not destroy the ILMs, but rather causes them to move.

3.1.2.3. Translational motion. To find a traveling ILM, Bickham et al. (1992) substituted the trial solution un = A~n(t)cos(wt- kna) into the equations of motion for the different particles, where A is the maximum amplitude of the moving ILM in a lattice of spacing a, (n(t) is a slowly varying envelope function, and k and w are the wave vector and frequency, respectively. The resulting set of equations are then numerically solved by assuming that the

w

Unusual anharmonic local mode systems

2.5

9

m

9

i .......

9

I

"

I

221

9

O

2.0 E

:3 :3

1.5

"

1.0

o.o

,

,

.

.

.

0.2

.

.

~

,

.

.

e

o.3 0.4 0.5 ka

Fig. 46. Dispersion curve of the t : 0 odd-symmetry traveling ILM for four different anharmonicity values. The values from top to bottom are A4 = 2.5, 1.6, 0.9 and 0.4. The dashed curves identify solutions of the equations of motion using the localization condition in the text. The solid lines indicate regions where simulations of moving modes are successful. The open circles are determined from the simulated displacements by assuming a Gaussian envelope function. Qualitatively similar results have been found for the t = 0 even-symmetry traveling ILM, but the odd modes cover a larger region of w(k) space. (After Bickham et al. 1992). t = 0 solution has the form: ~-n = ~,~ = (-1)nA~I e x p [ - ( n - 1)Ka] for positive n and K . For given values of the wave vector k and the anharmonicity parameter A4 = k4A2/k2, the time-dependent equations for sites 0, 1 and 2 are numerically solved to obtain the normalized frequency (W/Wm), the relative amplitude at the nearest neighbor site ~1, and the decay constant K . It is found that smoothly moving ILMs of the assumed type are produced only in a restricted region of w(k) space, with this region becoming more restricted as the anharmonicity and the ILM frequency increase. Figure 46 shows the dispersion curve that is found when the trial solution is used to fit numerically the displacements as the wave packet associated with a t - 0 odd-symmetry ILM travels across several lattice sites. The solid curves identify regions of w(k) space where the excitation moves with a constant envelope velocity for at least 15 lattice sites. This velocity is typically smaller than 15% of the harmonic lattice sound velocity. The absence

222

A.J. Sievers and J.B. Page

Ch. 3

Fig. 47. Displacement vs. time as the vibrational excitation passes through a fixed lattice site. The solid curve gives the results. The group velocity of the envelope is 7.2% of the harmonic lattice sound velocity. The dashed curve represents the best fit of the excitation envelope to a Gaussian envelope function. For comparison the dotted curve represents a hyperbolic-secant-function envelope. (After Bickham et al. 1992).

of a solid line at small k values identifies that region where the mode either remains stationary or only moves a few lattice sites before coming to a stop. At larger k values, on the other side of the solid line, the mode moves but decelerates. Uniform motion is observed over a larger region w(k) for the t = 0 odd mode than for the t = 0 even mode, consistent with the idea that the odd mode has an intrinsic translational instability. The solid line in fig. 47 shows a typical simulation trace of displacement versus time as the vibrational excitation passes through a particular site. The dashed line represents a gaussian function best fit to the pattern. Note that a hyperbolic-secant-function best fit represented by the dotted curve does not agree with the simulated amplitude in the wings, while the Gaussian envelope matches fairly well over the entire interval. One conclusion is that the shape is clearly different from the hyperbolic secant function previously found for solitons in continuous systems. The preceding discussion is for a traveling ILM described by the trial solution un = A~(t)cos(wt-$n), where the phase Sn is equal to kna; that

w

U n u s u a l a n h a r m o n i c local m o d e s y s t e m s

223

Fig. 48. MD simulation of a traveling ILM in a 21-particle harmonic plus strong quartic lattice, with periodic boundary conditions. The anharmonicity parameter is A 4 --- 1.63, and the t = 0 displacements are given by the translationally unstable odd-parity ILM pattern A(.... 0,-1/2, 1,-1/2,0 .... ), with A = 0.1a. The mode frequency is W/Wm = 1.67. The ILM motion is produced by seeding the MD run with a small initial velocity perturbation /q = -/~l = 0.00718, in units of wma, corresponding to a relative phase perturbation of 6r - 6r = 6r - 6r = -0.086 rad. The speed of this traveling mode is 0.053 in units of wma, well below the harmonic sound speed of 0.5. The displacements are magnified, for clarity. (After Sandusky et al. 1992).

is for a constant phase difference between adjacent sites. There exists another type o e smoothly traveling ILM, for which the relative phases between adjacent sites is not constant (Sandusky et al. 1992). This type of traveling mode can exist for large values of the anharmonicity, and an example is shown in fig. 48. This mode moves with a speed approximately 1/10 of the harmonic sound speed, and as it slowly moves from site to site, its mode pattern alternates approximately between the odd- and even-ILM patterns. Figure 49 plots the phase difference between adjacent relative coordinates d~ -- Un - Un-1 for the moving mode of fig. 48 as a function of time, as the mode passes a single site. The phase difference is seen to be nonconstant. Moreover for this mode, nonconstant phase differences of the same magnitude are obtained between the adjacent displacements {un} themselves, as well as between the adjacent relative displacements {dn}. Depending on the strength of the anharmonicity and the initial conditions, at least two other sorts of ILM motion have been reported: (1) The ILM becomes trapped at a site (Bickham et al. 1992) and converts to a stable even-parity ILM (Sandusky et al. 1992). (2) The ILM oscillates between

224

A.J. Sievers and J.B. Page

Ch. 3

Fig. 49. Phase difference between the relative displacements d_4 = u_4 - u - 3 and d_ 3 = u_3 - u_2 for the traveling mode of fig. 47. The triangles give the relative phase as the traveling mode passes the n = - 3 particle. This is a type of traveling ILM where the phase is nonconstant. (After Sandusky et al. 1992).

Fig. 50. ILM oscillations between lattice sites, observed in MD simulations for a 21particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is A4 = 1.25, and the initial displacement pattern is that of the odd-parity ILM for this anharmonicity, together with a small perturbation ( . . . . ~a, 0, - ~ a . . . . ), where da is 0.01% of the unperturbed mode amplitude. The dashed line is a guide that follows the mode's "center". Notice that this translationally unstable ILM does not move from its initial location until roughly 15 oscillations have occurred. The displacements are magnified, for clarity. (After Sandusky et al. 1992).

w

Unusual anharmonic local mode systems

225

Fig. 51. MD simulation of colliding ILMs in the 21-particle (k2,k4) lattice of fig. 48. The initial configuration produces two traveling ILMs of the same type as in that figure, and they are seen to reflect from the ends of the chain (for the symmetry of the initial configuration used here, the periodic boundary conditions are equivalent to free end boundary conditions). They then collide, producing ILMs traveling with much higher velocities. The displacements are magnified, for clarity. (After Sandusky and Page, unpublished).

adjacent sites (Sandusky et al. 1992). An example of this oscillatory behavior is shown in fig. 50. Finally, in passing we show in fig. 51 an MD simulation of two moving ILMs colliding. These are for a 21-particle (k2, k4) chain with periodic boundary conditions, and the symmetry of the initial conditions is such that the ILMs are symmetrically reflected inward from the ends. Upon collision it is seen that two ILMs emerge symmetrically, with much higher velocities. This is an isolated MD run, and it is not clear to what extent it is a special case. Nevertheless, this figure suffices to show clearly that traveling ILMs are not solitons in the usual sense.

3.1.2.4. The effect of a light mass defect. For the case when a light substitutional mass defect md < m is present at site n = 0, it is straightforward to generalize the arguments in w and show that in the asymptotic limit of increasing even-order anharmonicity, the exact odd-parity mode pattern A ( . . . , 0 , - 1/2, 1 , - 1 / 2 , 0,...) for the perfect lattice is replaced by

A.J. Sievers and J.B. Page

226

Ch. 3

(Kivshar 1991) A

md

md

)

. . . , 0 , - ~ m ' 1 , - ~,2m . . . .

(3.19)

Just as for the perfect lattice case, this remains a very good approximation for the case of pure quartic interactions. For the case of nearest-neighbor harmonic plus quartic potentials (k2, this mode pattern remains a good approximation provided that 16/81, just as for eq. (3.3), with the defect mode frequency given by

An>>

(Wd) 2 1 ( m ) ( ~

~

k4),

md)I l + ~ mm

3A4( r o d ) 2] 1+--~--l+2m

.

(3.20)

For md = m this reduces to eq. (3.3), and in the md --+ 0 limit of a very light mass, the defect mode pattern just becomes that of an Einstein oscillator A(..., 0, 1,0,...), as is intuitively clear, with the frequency going as

Bickhamet al. (1993) have studied the influence of a stationary light mass defect on the scattering of moving ILMs. Figure 52 shows the amplitude and frequency associated with the mass defect localized mode that is produced with the energy captured from a passing ILM. With defect masses in the range 0.33 < md/m < 0.5, the moving localized mode is partially captured and reflected. There is a peak in the amplitude of the mode localized at the defect when its frequency is near the vibrational frequency of the moving ILM, indicating a resonance effect. When the relative mass of the impurity increases to 0.5, the energy of the moving ILM is completely captured, except for some plane waves that are produced as the defect mode settles into its eigenvector. There is a dramatic increase of both the amplitude and frequency of the defect mode at this transition, as shown in the figure. As the mass increases further, the frequency of the defect mode decreases until it is nearly equal to the frequency of the moving localized mode at md/m = 0.91. Beyond this point, the impurity can support modes of the same frequency as the moving ILM and therefore becomes transparent. The key feature of this transfer process is evident by examining first the small defect mass limit. When the impurity is very light, it "adiabatically" follows the motion of its nearest neighbors and thus moves in phase with them at the incoming ILM frequency in the figure (dashed line). Therefore this defect region perfectly reflects moving ILMs in which adjacent particles vibrate 7r out of phase. Such reflection continues until the defect mass is sufficiently large that the impurity mode frequency approaches that of the moving ILM. The exact threshold at which the defect begins to capture energy depends on both the characteristic wave vector and anharmonicity of the moving ILM.

Unusual anharmonic local mode systems

w

0.30

9

I

"

I

.....

"

I

227

'

"13

,.,,0.20

E o0.10 (D

',I-,,

(D

"13

0.00

I

1.80

!

"

moving mode defect mode

......

1.60

'

E

a ~ a

.40 1.20 1.00

0.20

,

i

0.40

,

m

0.60

,

md/m

l

0.80

,

1.00

Fig. 52. Simulation results for the frequency and amplitude of the anharmonic mass defect localized mode produced with captured energy from a passing ILM. (After Bickham et al. 1993).

3.1.2.5. Relation to anharmonic zone boundary mode stability. As noted above, both the odd and even (k2, k4) ILMs spatially broaden and merge with the maximum frequency Wm of the harmonic lattice as the anharmonicity parameter A4 is decreased to zero. That is, the ILMs just become the harmonic lattice zone boundary phonon mode (ZBM), with displacement pattem A(..., 1 , - 1, 1 , - 1. . . . ). An interesting "inverse" of this connection between the ILMs and the ZBM has also been investigated, in recent analytic and numerical studies (Burlakov and Kiselev 1991; Burlakov et al. 1990d; Burlakov et al. 1990b; Kivshar 1993; Sandusky et al. 1993b; Sandusky and Page 1994); the last of these is quite extensive. The upshot of this work is that the anharmonic ZBM can evolve into one or more ILMs and that an instability of the ZBM is the first step in this decay. The question of the stability of such an extended lattice phonon mode against decay into ILMs is of interest, since it appears to provide a possible avenue for creating ILMs via the application of external perturbations.

A.J. Sievers and J.B. Page

228

Ch. 3

In the rotating wave approximation, the anharmonic ZBM frequency for a (k2, k4) lattice is easily shown to be given by O32

~=

1 + 3A4.

(3.21)

By applying a suitable version of the stability analysis sketched in the preceding section, we find that the anharmonic ZBM in this lattice is indeed unstable (Sandusky et al. 1993b; Sandusky and Page 1994). For a given value of the anharmonicity A4, it is convenient to decompose the instability perturbations into spatial Fourier components, and one can readily predict the ZBM instability growth rates as a function of the perturbation wavevector kp. Figure 53(a) shows such predictions, together with measured rates from MD simulations; these are seen to agree well with the predicted rates. For different values of the anharmonicity (as measured by 60/60m), the maximum ZBM instability growth rate is found to occur at different perturbation wavevectors (kp)max, as shown in fig. 53(b). One sees that as the anharmonicity increases, (kp)max increases towards its largest value, which is that for the pure quartic lattice. Thus, larger anharmonicity favors shorter wavelength instability perturbations - the ZBM instability has introduced a new, anharmonicity-dependent, length scale into the system, namely the wavelength 27r/(kp)max associated with the fastest-growing instability perturbation. This ZBM instability length scale turns out to correlate precisely with the spatial extent of the ILMs for each value of the anharmonicity. Furthermore, when one goes beyond the perturbation instability analysis and uses MD simulations to observe the nonlinear time-evolution of the ZBM instability over finite times, it is found that the ZBM indeed evolves into a periodic array of ILM-like localized displacement patterns, as is strikingly evident from fig. 54. The characteristic width of these ILM-like structures decreases with increasing anharmonicity; it is again just the wavelength associated with the fastest-growing ZBM instability perturbation. Analogous ILM-related zone boundary mode instabilities are found for the more realistic case of lattices with both quartic and cubic anharmonicity (Sandusky and Page 1994).

3.1.3. (k2, k3, k4) nearest-neighbor potentials While the (k2, k4) lattices discussed in the previous sections have been fruitful for establishing some fundamental aspects of localized dynamics in periodic anharmonic lattices, it is also clear that this model leaves out one of the most important properties of the interparticle anharmonicity in real

w

Unusual anharmonic local mode systems

229

0.10

~

0.05

0.00

t-tll "1o

0.00

0.50

o m

L _

0.25_

.-.

o

t-

/

• 0.00 E 1.0

%

0.25 0.50 kpa (units of n radians)

1.'5 ~/[.0m

Fig. 53. Zone boundary mode instability and its relation to ILMs. Upper panel: predicted and measured ZBM instability growth rates in units of the anharmonic ZBM frequency w, as a function of the perturbation wavenumber kp, for a 40-particle (k2, k4) lattice with lattice constant a and periodic boundary conditions. The anharmonicity parameter is An -- 0.068, corresponding to a ZBM frequency of W/Wm= 1.1. The solid curve gives the stability analysis prediction, and the circles are measured in MD simulations. Lower panel: ZBM stability analysis prediction of the perturbation wavenumber (kp)max associated with the largest instability growth rate, as a function of the anharmonicity (measured by the ZBM frequency w). The dashed line gives the value for a purely quartic lattice. For a given value of the anharmonicity, the wavelength associated with (kp)max correlates accurately with the spatial extent of the ILMs for the same anharmonicity. (After Sandusky et al. 1993a, b).

systems, namely cubic anharmonicity. Typical interatomic potentials such as Lennard-Jones, Morse, and Born-Mayer plus Coulomb are repulsivedominated, and their Taylor-series expansions lead to a strong negative cubic anharmonic term (k3). Our preceding restriction of the anharmonicity to just even orders disproportionately weights the attractive component of the potential, as is clearly evident in the "square-well" limit of increasing order discussed earlier, and we have seen that it is in this limit that the most localized ILMs are asymptotically exact. Therefore it is important to

230

A.J. Sievers and J.B. Page

Ch. 3

Fig. 54. MD simulation showing the finite-time evolution of the predicted fastest-growing ZBM instability perturbation in a 40-particle (k2, k4) lattice with periodic boundary conditions. Here, the anharmonicity parameter is A4 -- 0.0067, corresponding to a ZBM frequency of W/Wm = 1.01. The top panel shows the t = 0 unperturbed ZBM pattern that is used. The fastest-growing perturbation for this case occurs at (kpa)max = 0.107r, and its pattern is shown in the second panel. The bottom panel shows the displacement pattern after a finite time has elapsed, and one sees that the ZBM has evolved into a periodic array of localized ILMlike patterns. The rate at which this evolution occurs depends upon the magnitude of the perturbation seed. The particle displacements are plotted vertically, and the relative vertical scales are indicated by the tic marks. (After Sandusky et al. 1993b; Sandusky and Page 1994).

reconsider the problem, with cubic anharmonicity included. We will first discuss ILMs in monatomic (k2, k3, k4) lattices and then generalize to the case of the above realistic analytic interatomic potentials. With nearest-neighbor cubic anharmonicity included, the potential energy of eq. (3.1) is replaced by

k2

k3

V = --2 ~ ( u n + l - u,~)2 + -~- ~(u,~+l - un) 3 n

n

k4 y ~ ( U n + l n

-

Un)4.

(3.22)

w

Unusual anharmonic local mode systems

231

The main qualitative feature introduced by k3 is to make the interparticle potential asymmetric, resulting in the particles' time-average displacements being nonzero and amplitude-dependent. Hence, amplitude- and site-dependent static distortions have to be added to the RWA trial solutions: Un(t) = A[~n cos(oat) q- An] ,

(3.23)

where the {An } are the static distortions, relative to the overall amplitude A. The {(n} give the dynamical mode pattern, as before. Within the RWA, Bickham et al. (1993)carried out detailed numerical studies of ILMs for the (k2, k3, k4) case and verified their results using MD simulations. Briefly, their technique consisted of implementing a standard nonlinear equation solver to obtain numerical solutions to the RWA equations for a restricted number of sites near the ILM center, and then applying the harmonic-approximation boundary condition of exponential decay for the particles beyond. Free-end boundary conditions were used. They found that by locally distorting the perfect lattice with the sign of the distortion determined by the sign of k3, stable ILM's could be generated even for large cubic anharmonicities but that independent of the sign of the cubic anharmonicity its effect was to decrease the frequency of the ILM. As the cubic anharmonicity increases, the eigenvector also becomes more localized until it resembles a triatomic molecule, beyond which the mode becomes unstable and decays. Bickham et al. (1993) should be consulted for details. In recent studies focused on the question of ZBM-stability/ILM-existence in (k2, k3, k4) lattices with periodic boundary conditions, Sandusky and Page (1994) employed a similar but slightly more computational approach, in that the RWA equations of motion were solved numerically for all of the particles, the number of which was typically taken as 40. As detailed there, the presence of sufficient cubic anharmonicity is found to stabilize the ZBM and simultaneously prevent the formation of ILMs. For the case of isolated ILMs, their results are similar to those of Bickham et al. (1993), when allowance is made for the different boundary conditions used. Figure 55 shows the RWA calculations by Bickham et al. (1993) for stationary even-parity modes. The ILM frequencies are shown as a function of the cubic anharmonicity parameter A3 = k3A/k2, for three different values of the quartic anharmonicity An in a 512-particle lattice with free ends. Also shown are the results of MD simulations, and they are seen to be in good agreement with the RWA predictions. The slight discrepancy observed for larger values of An arises from the higher harmonics neglected in the RWA. Figure 56 shows the results of calculations by Sandusky and Page (1994) of the RWA dynamic (a) and static (b) displacement patterns for an oddparity ILM in a 40-particle periodic boundary condition lattice with

A.J. Sievers and J.B. Page

232 , 0

.....

9

I

'

!

'

I

"

'"

-o'~ ~.;..,

o --. o "-~ o o"-,.

1.8-

Ch. 3

I

'

'!'"

"

A,I.= 1.6 .,%=0.9 1,4.=0.4. /k4=O. 1

..... ---

0"\ 0"\ o'\.

E

1.6

3 3

o\

o~..

"o"o -..

~

o\

% o\

1.4-

~

\. o '

& '

~

ox

1.2

cr~ 1.0~ 0.0

, , 0.5 1.0 I

,

I

1.5

o~

................... I

2.0

,

I

2.5

A3-K~A/K2

~

1

3.0

9

3.5

Fig. 55. Frequencies of even-parity ILMs in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The dashed curves give the frequencies as a function of the cubic anharmonicity parameter A3, for four different values of the quartic parameter A4. The circles are the results measured in MD simulations. The results show that there is a limiting value of k 3 for a given value of k4. (After Bickham et al. 1993).

A3 = -2.1 and A4 = 1.1. The frequency is 1.1~m. Owing to the mode's symmetry, there is no static displacement on the central particle. However the static displacements on the remaining particles are nonzero, and they vary rapidly in the region where the dynamic displacements are large. Away from this region, the static displacements decrease linearly to zero at the boundaries, corresponding to a constant static strain away from the mode center. The magnitude of this strain decreases with increasing numbers of particles. The nature of the static displacements away from the mode center depends on the boundary conditions employed- for the free-end boundary conditions used by Bickham et al. (1993), these strains are in fact zero since the particles away from the mode center simply translate together, towards or away from the mode center, depending on the sign of k3. In the righthand column of fig. 57, we show the dynamical displacement patterns for odd-parity ILMs in a 40-particle periodic boundary condition lattice with fixed values of (k2, k3, k4), as the amplitude A is changed (Sandusky and

w

Unusual anharmonic local mode systems

233

(a)

....._ i,,,.--

Particle

(b)

b._ v

Particle Fig. 56. Predicted dynamical displacement pattern (a) and static displacement pattern (b) for an odd-parity ILM in a 40-particle (k2, k3, k4) lattice with periodic boundary conditions. The values of A3, A4 and the frequency are -2.1, 1.1 and 1.1 Wm, respectively. Notice that the static distortions away from the mode center decrease linearly to zero at the "supercell" boundaries, for this periodic boundary condition lattice. (After Sandusky and Page 1994). Page 1994). The lower pattern on the right corresponds to the same mode shown in fig. 56. The column on the left shows the results for ILMs of the same frequencies in a lattice that is identical, except that there is no cubic anharmonicity (k3 = 0). The interesting aspect here is that as the frequency is decreased, the presence of cubic anharmonicity is seen largely to quench the spreading out that occurs for the k3 -- 0 case. Eventually the mode rapidly broadens, merging with the zone boundary mode at a finite value of A3. It is straightforward to show for both monatomic (Sandusky and Page 1994) and diatomic lattices (Kiselev et al. 1994b) that the effect of the static displacements {An} is to renormalize the harmonic force constants k2 into site-dependent, effective harmonic force constants k~(n+l,n) -- k2 -k- 2k3A(An+l - A n ) + 3k4A2(An+l - An) 2

(3.24)

in one of the RWA equations of motion. As a result, the cubic anharmonicity is formally eliminated from the equation, and it is transformed into the pure

A.J. Sievers and J.B. Page

234

k =O

I m

Ch. 3

k ,0 "=

l

~

Particle

m=l

m---1

Fig. 57. Effect of cubic anharmonicity on ILM localization. The plots show the oddparity ILM dynamical displacement patterns for a 40-particle periodic boundary condition lattice having fixed values of (k2, k3, k4), as the ILM frequency is lowered by decreasing the amplitude A. The patterns on the left are for k3 = 0, while those on the right are all for the same nonzero value of k 3. The amplitude has been chosen to give the same frequencies for these cases. Notice that the k 3 # 0 ILM patterns remain localized as the frequencies decrease, while the k 3 = 0 ILM patterns spread out. (After Sandusky and Page 1994).

RWA eq. (3.20) above, except with k2 replaced by the site-dependent k~'s. This site-dependence is strong near the mode center, where the static displacements vary the most rapidly. Evidently, the dynamical displacements for the (k3 r 0) ILMs on the fight side of fig. 57 are formally like those of an impurity mode in a (kz, k4) lattice having defect-induced force constant weakening which enhances the mode's localization. (k2, k4)

3.1.3.1. ILM existence. Bickham et al. (1993) have generated an existence threshold curve for ILMs in the s a m e (k2,k3, k4) lattice used for fig. 55, with the results shown in fig. 58. The solid curve was obtained by fixing A3 and then finding the minimum value of An for which ILMs are supported- this is the point where the ILMs broaden into the zone boundary mode, and it occurs at the same point for the odd and even ILMs. The dashed curve above A4 ,~ 0.8 corresponds to the ILM dynamical displace-

w

Unusual anharmonic local mode systems 1.6

I

'

I

"

I

'

)

I

b cxi

1.2

...........

L'Xl <~-0.8

C

I

ble Re

I

I

I

I

I

I

I

235

I

II,d< 0.4 Unstable 0.0 -2.5

9

I

-1.5

~"~/k,~') ,

I

~

-0.5

I

0.5

A3-K3A/K2

Unstable ,

I

1.5

,

2.5

Fig. 58. ILM existence in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The solid curve is the existence threshold determined by the condition that the ILM spatially broadens into the zone boundary lattice phonon mode. The dashed curve corresponds to a high-localization triatomic-molecule limit. Area (d), lying between curves (b) and (c), designates the region of a continuum approximation. (After Bickham et al. 1993). ment patterns becoming more localized than the (k2, k4) asymptotic limit ( . . . , 0, - 1/2, 1, - 1/2, 0,...). At these points the ILMs were observed to be unstable in the MD runs of Bickham et al. (1993). The (A3, An) points along the dashed curve are very close to those where a double minimum appears in the (ke, k3, k4) interparticle potential (Sandusky and Page 1994).

3.1.3.2. ILM motion. Moving ILMs for (k2, k3, k4) lattices have been discussed by Bickham et al. (1993). The RWA was used within a moving envelope approximation characterized by a single wavevector, and numerical results were obtained for the same 512-particle free-end lattice considered in figs 55 and 58 above. For a fixed value of the quartic anharmonicity An, dispersion curves were predicted for several values of the cubic anharmonicity A3, as shown in fig. 59. Included in this figure are the corresponding results measured in MD simulations. The agreement is seen to be quite good, with the larger discrepancies occurring for larger values of the wavevector. The top curve is for the pure (k2, k4) lattice, and it reproduces the results given earlier by Bickham et al. (1992). Figure 60 shows the displacements as a function of time, for three particles in the above lattice as a moving ILM passes (Bickham et al. 1993). For

A.J. Sievers and J.B. Page

236

Ch. 3

1.5

9

1.4

........... b.~=-0.8 " - - - - - - - / ~ - - - 1.8 ..... .M=-1.2

I

9

I

9

I

o Simulation .

.

.

.

/~=0.0

.

9

i

9

i

Results -.~=-1.6

- -

E1.3 3 :3

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~

r. . . . .

0

1.2

.

.,.-

-

....

~-

"o--

.-,,

..........

0 ...........

g .......

" . . . . r_

o

0

1.1

0

o

0

~ 0 0

o0

,

0.0

I

0.1

,

I

0.2

'

i

0.3

.

~

0.4

.

,

0.5

9

0.6

ko Fig. 59. Dispersion curves for moving ILMs for the (k2, k3, k4) lattice of fig 58. The quartic anharmonicity is fixed at A 4 = 1.6, and the lines are the predictions for different values of the cubic anharmonicity A3. The circles are obtained from MD simulations. Note that the agreement between the simulation and analytic results is better at smaller wave vectors and higher frequencies because the effect of the second derivative that were removed from the equations of motion is then minimized. (After Bickham et al. 1993). this free-end boundary condition case, all of the particles on one side of the mode center and well away from it have the same static displacement, while the static displacements of the corresponding particles on the other side are equal and opposite. This is manifested in the figure by the change in the particle offsets from their equilibrium positions as the mode passes. The group velocity of this particular excitation is approximately 15% of the harmonic lattice sound velocity. Note that there is a small increase in the static displacement at the n = 10 site after a time corresponding to three periods of the m a x i m u m plane-wave frequency. This is caused by a small amplitude supersonic pulse that propagates in both directions from the ILM at the start of the simulation and eventually reaches the free ends of the lattice. This pulse may be a long wavelength acoustic kink soliton that is also a solution of these equations.

3.1.4. Realistic potentials The previous studies of ILMs in (k2, k3, k4) lattices show that for a fixed value of the quartic anharmonicity parameter A4, both the even and odd

w

Unusual anharmonic local mode systems

237

Fig. 60. Displacements versus time for a moving ILM in the (k2, k3, k4) free-end lattice of fig. 58. The total displacements (static plus dynamic) are shown for three different sites, as the moving ILM passes. The group velocity of this particular moving excitation is 15% of the harmonic lattice sound velocity. The anharmonicity parameters are A3 = 0.8 and A4 = 0.4, and the wavevector is given by ka = 0.3, where a is the lattice constant. The three curves have been displaced vertically, for clarity. (After Bickham et al. 1993). versions of these modes exist, provided the magnitude of the cubic anharmonicity A3 does not exceed a threshold value. Moreover, it is not difficult to show that these conditions can be satisfied easily for (k2, k3,/ca) values obtained from the Taylor-series expansions of realistic interparticle potentials V(r), such as Lennard-Jones, Morse and B o r n - M a y e r plus Coulomb, about their m i n i m u m points. The question then naturally arises as to whether or not ILMs can exist in lattices in which the particles interact via these full potentials, which of course include all of the higher-order terms in the Taylor-series expansions. One-dimensional lattices with realistic potentials have been studied recently by Kiselev et al. (1993, 1994a, b) for the monatomic and diatomic cases and by Sandusky and Page (1994) for the monatomic case. The diatomic case will be discussed in w3.2. The work of Sandusky and Page concerns general questions about the interrelations between zone boundary phonon stability and ILM formation, and it includes the use of the full potentials listed in the previous paragraph. In some of the calculations, these potentials are restricted to nearest neighbors, while in others they are allowed to act out to sixth neighbors. With respect to the question of ILM existence,

238

A.J. Sievers and J.B. Page

Ch. 3

the basic finding of both the monatomic and the diatomic studies is that ILMs do not exist in either type of lattice with these realistic potentials. The presence of anharmonic terms beyond fourth order in the full potentials acts to suppress the presence of ILMs. However, the diatomic case gives a new possibility, namely that ILMs could occur with frequencies in the gap between the acoustic and optic phonon branches. That such "gap" ILMs exist is shown by Kiselev et al. (1993, 1994b), as discussed w3.2.

3.1.5. Mass defect anharmonic mode with realistic potentials Although ILMs do not exist above the top of the plane wave spectrum of a monatofnic lattice with realistic potentials, high frequency anharmonic impurity modes are still a possibility. Kiselev et al. (1994a, b) have investigated the properties of mass and force constant defect modes as a function of amplitude for four nearest neighbor potentials, which in order of increasing anharmonicity are: Toda, Born-Mayer plus Coulomb (BMC), LennardJones, and Morse. These potentials were adjusted to have the same k2 and k4 terms in their Taylor's series expansion. The light impurity mass defect parameter is e = 1 - md/m -----0.95, where md and m are the defect and the host particle masses. The form of the interaction potential between particles has the same value for all nearest neighbors, except for the bonds involving the impurity. The force constant defect parameter is taken to have the value 77 = 1 - V~'/V" = 0.87, where V~' and V" are proportional to the nearest-neighbor force constants between the impurity and the host lattice and between the particles of the host lattice, respectively. The frequency of the harmonic impurity mode in such a chain has the value 60harm/60m -- 1.125, where tom is the frequency of the top of the harmonic phonon band. For small amplitude vibrations a light impurity supports a localized vibration with a frequency higher than the plane wave spectrum, but with increasing amplitude the anharmonicity of the interaction potential becomes important. The soft attractive part of the standard two-body potentials makes this frequency decrease as the amplitude of the local mode vibration increases and the excitation transforms from a local mode to an in-band resonant mode. Figure 61 shows the frequency of an anharmonic impurity mode in a lattice with a mass and force constant defect for three different two-body potentials as a function of normalized amplitude A/a, where a is the lattice constant. The odd-parity mode is centered at the impurity site. The solid curves show the analytic results, and the points give the simulation results" squares (Toda), diamonds (BMC), and circles (Morse). The results for the Lennard-Jones interaction potential (not shown) are between the BMC and the Morse results. The conclusion is that the more anharmonic the two-body potential, the lower the anharmonic impurity mode frequency.

Unusual anharmonic local mode systems

w 1.2

........... .

239 _

I

.

I

'

I

'

1.0 0.8 ~EO. 6 0.4 0.2 0

0.0

,

I

0.1

, .........

l

0.2

....................

i

I

0.3

Normalized amplitude

i

0.4

Fig. 61. The frequency of an anharmonic impurity mode in a monatomic lattice with a force constant and mass defect for three different two-body potentials, as a function of the normalized amplitude A/a. The odd-parity mode is centered at the light impurity (mass defect parameter e = 0.95), which is bonded with its neighbors through the perturbed potential having the forceconstant defect parameter r/ = 0.87. Analytic results are shown as solid curves, while the simulation results are represented by squares (Toda), diamonds (Born-Mayer plus Coulomb), and circles (Morse). For each of the anharmonic potentials, k2 and k4 are the same. (After Kiselev et al. 1994b).

3.2. One-dimensional diatomic lattices

The study by Kiselev et al. (1993, 1994b) of diatomic one-dimensional lattices with standard full potential functions establishes that ILMs can occur in the gap between the optic and acoustic phonon branches. The study was basically numerical and focused on a specific diatomic lattice, namely one with 257 particles, free ends, and having the masses of lithium iodide. Three full potential functions V(r) were emphasized, namely the Toda potential (Toda 1989), the Morse potential and the BMC potential, and all three were restricted to act between just nearest neighbors. In addition, a nearest-neighbor (k2, k3, k4) potential was studied, with the spring constants obtained from the expansion of the BMC potential for LiI. The parameters in the remaining two potential functions were adjusted to reproduce the BMC values of k2 and k4. To obtain the stationary ILMs, an RWA approach based upon the trial solution eq. (3.23) was made, and the resulting equations were solved numerically.

A.J. Sievers and J.B. Page

240

Ch. 3

1-(a) 0

to

- - - - ' - -

-1 "

5. E

(b)

o

<

I~

0

-1

-8

,

~

m

A

.... i

-4

0

i

4

Particle number

i

. .

1I 8

Fig. 62. Static and dynamic displacement patterns for ILMs in a one-dimensional diatomic lattice of 257 particles interacting via a nearest-neighbor Born-Mayer plus Coulomb potential. The masses and potential parameters are those of lithium iodide, and free-end boundary conditions are used. The circles and squares denote the light and heavy masses, respectively, with their open and solid versions giving the static and dynamic patterns. (a) Odd-parity gap ILM, with to = 0.74tom. This mode was found to be stable in MD runs. (b) Even-parity gap ILM, with to = 0.74tom. MD runs revealed this mode to be unstable. (c) Odd-parity ILM with to = 1.04tom, obtained by using just (k2, k3, k4), gotten by expanding the LiI potential used in (a) and (b). When the full LiI potential is used, no ILMs with to > tom are found. In all three panels, the overall amplitude is A/a = 0.233. (After Kiselev et al. 1993).

For the (k2, k3, k4) version of this diatomic lattice, an odd-parity ILM occurs, with a frequency W/Wm = 1.04, just above that of the maximum harmonic lattice frequency. Its static and dynamic displacement pattern are given in fig. 62 (c). Just as in the monatomic case discussed in w3.1.4, this high-frequency ILM disappears when any of the three full potentials is used instead of the (k2, k3, k4) potential. Again, no high-frequency ILMs have been found using the above full realistic potentials, for either the monatomic or diatomic cases; the higher-order anharmonic terms in the expansions of the full potentials suppress ILM solutions with frequencies above the phonon bands. However, both even- and odd-parity ILMs are found with frequencies in the phonon gap for either the (k2, k3, k4) case, or with the full potentials. (Note that for a diatomic periodic lattice, the midpoints between the particles' equilibrium sites are no longer symmetry centers, as they are for monatomic

w

Unusual anharmonic local mode systems

241

lattices. For the diatomic case, only the equilibrium sites are symmetry centers. Hence, even-parity modes for the diatomic case are qualitatively different than for the monatomic lattice; in particular they involve no motion of the central particle.) The odd-parity gap ILM is found to be stable, in the sense of persisting over many periods in MD simulations, whereas the even-parity gap ILM is found to be unstable. The displacements for the odd- and even-parity gap ILMs for the B M C potential are shown in panels (a) and (b) of fig. 62. Additional numerical tests show that odd-parity gap modes can be generated from the amplitude pattern for the nearest-neighbor potential cases even when second nearest-neighbor forces are included in the model. These findings suggest that anharmonic gap modes are also stable in the presence of long range forces (Kiselev et al. 1993). Figure 63 plots the frequencies of the odd-parity ILM gap mode for each of the three full potentials, as a function of the normalized amplitude of the 1.0 CO+ 0.8

0.6

~'~ 0.41-

~

I

0.2

0.0

o

| !

0.0

I 0.5

/f 1.0

Relative _.,

.

,

.

1.5

2.0

displacement .,

.

,

0.1 0.2 0.3 Normalized amplitude

,

0.4

.

Fig. 63. Frequencies of the odd-parity ILM gap mode in the diatomic lattice of fig. 62, for three different full potentials. The frequencies are plotted versus overall amplitude A/a, and w_ and w+ are the frequencies defining the gap between the acoustic and optic phonon branches in the harmonic crystal. The three frequency curves correspond to different nearest-neighbor potential functions: triangles (Toda), circles (Born-Mayer plus Coulomb), and squares (Morse). The inset shows these potentials, together with a Lennard-Jones and a (k2, k3, k4) potential. The Born-Mayer plus Coulomb potential is that appropriate to LiI, and the values of (k2, k 3, k4) were obtained by expanding this potential. The parameters for the remaining potentials were adjusted to reproduce these same values for k2 and kn. (After Kiselev et al. 1993).

A.J. Sievers and J.B. Page

242

Ch. 3

light particle A//a. The curves give the RWA results and the points give the results of MD simulations, and they are seen to be in good agreement with the RWA predictions for A/a < 0.2. When the normalized amplitude is increased to A/a ~ 0.05 the gap mode eigenvector becomes nearly that of a triatomic molecule. The discrepancies observed for larger amplitudes and "softer" potentials (e.g. Morse) indicate that the RWA is becoming less accurate. This is because the effective harmonic potential associated with the RWA looks very different from the effective anharmonic double well which forms at large ILM amplitudes (Kiselev et al. 1994b). The frequency spectrum for the displacement Of the light particle in the stable odd-parity ILM is given by the solid curve in fig. 64. It is obtained over a time interval of At = 1024(27r/o~m). In order of decreasing strength, the peaks are located at ~g, 3U)g, and 5Wg. This spectrum is remarkably similar to the dashed curve, obtained for the anharmonic ILM of the same amplitude in the same diatomic lattice with only the symmetric k2 and k4 potential terms, which is the form used in most earlier studies. Evidently, the

101

J ....

!

9

I

".,,

~....

9

!

!

ffl

m =

10 o

10 -1 E

"•10-2

%

.

/ /

0-3

~

0-4 0

!

2

4

6

8

O)I(JL)m

Fig. 64. Power spectra of gap and local modes in a diatomic chain with 257 particles. The solid curve corresponds to the gap mode (a~g = 0.74 OJm), with the Born-Mayer plus Coulomb potential for LiI, and the dashed curve shows the ILM spectrum (wi = 1.24~0m) in the same lattice with the (k2, k4) potential from the Taylor's series expansion of the Born-Mayer plus Coulomb potential. The ordinate of the dashed curve has been shifted by a factor 20, for clarity. In each case the amplitude is A/a = 0.233. The results are remarkably similar: the resonant features occur at the same first, third and fifth orders with about the same strengths. The "noise" that appears in the solid curve near these frequencies is due to weak coupling between the gap mode and the plane wave spectrum. (After Kiselev et al. 1993).

w

Unusual anharmonic local mode systems

243

main function of the asymmetric potential terms in the Taylor's expansion of the two body potential is to contribute to the static distortion in the displacement patterns shown in fig. 62. Inspection of fig. 62(a) reveals that the displacement pattern of the stable odd-parity gap ILM for the LiI BMC potential is very much like that of an isolated Einstein oscillator- the odd-order (asymmetric) terms in the interparticle potential have caused the equilibrium positions of the particles on each side of the central particle to uniformly shift outwards, and the dynamical displacements are ~, 0 except at the central particle. This Einstein-oscillator-like dynamical behavior is not surprising in view of the fact that the central mass is so much lighter than its neighbors. Indeed, this mode is behaving qualitatively like that of the light mass defect anharmonic mode discussed earlier for monatomic (k2, k4) systems, with the additional feature that the harmonic force constants k2 on either side of the central particle have been renormalized to k~, due to the local static distortion produced in the ILM by k3. Gap ILMs in pure (k2, k4) diatomic systems have recently been discussed (Aoki et al. 1993; Chubykalo et al. 1993). It is now evident that ILM solutions can be obtained numerically for a perfect diatomic 1D chain incorporating a variety of realistic two body potentials in the sense of persisting over many periods in MD runs, while simulations demonstrate that the odd-parity gap modes are stable, but the even-parity modes are not. The incorporation of a realistic two-body potential into the lattice has brought out several important new features of ILMs: (1) a localized mode will not exist above the top of the plane wave spectrum for realistic potentials; (2) a long-lived odd-parity localized mode can exist in the gap between the optic and acoustic branches; (3) in the large amplitude limit, the triatomic molecule-like eigenvector previously observed for local modes in the hard quartic case is recovered for gap modes with realistic potentials and (4) for each of these potentials the gap mode is accompanied by a localized DC expansion of the lattice.

4.

Speculation

In the preceding two sections, we have discussed two very different sorts of unusual localized vibrational behavior in strongly anharmonic systems. First, we have seen that detailed experimental and theoretical studies of the impurity system KI:Ag + reveal a nearly unstable defect/host system at low temperatures, which exhibits a rapid, thermally-driven on --+ off-center transition between T = 0 K and T = 20 K. As detailed above, this transition is highly anomalous, in that it cannot be explained within any sort of a

244

A.J. Sievers and J.B. Page

Ch. 3

harmonic or weakly perturbed anharmonic f r a m e w o r k - thus the system is strongly anharmonic in an as yet undetermined way. At the same time, the T = 0 K on-center configuration and its associate dynamical properties as probed by extensive stress, E-field and host lattice isotope effect experiments are found to be consistent with a perturbed harmonic phonon model. Even though the on-center configuration is therefore apparently harmonic (or quasiharmonic under applied stress or E-fields), its associated dynamics are themselves very unusual" the impurity-host system is strongly coupled but nearly unstable against off-center displacements and it possesses a new type of localized impurity mode, the pocket gap modes, whose displacements are confined to sites away from the defects. Second, we have discussed some of the basic properties of an unusual new class of localized vibrational excitations (ILMs) which can occur in periodic lattices, provided there is sufficiently strong local anharmonicity present. This anharmonicity could arise in two ways: 1) from large values of the anharmonic coefficients in the expansion of the potential energy about the equilibrium configuration, or 2) from the presence of large amplitudes in a localized region of the lattice. The study of ILMs has been purely theoretical up to the present and is still at an early stage. The focus has been on simplified models, but they are becoming more realistic, as exemplified by the recent studies involving realistic potentials (Kiselev et al. 1993, 1994b; Sandusky et al. 1994). As the properties of ILMs become better established, we anticipate that there will be significant activity in experimentally verifying their existence in real solids. In the following we briefly mention some of the speculative ideas which have been suggested previously. One of these involves a link between ILMs and KI:Ag +, and it will be discussed first, in a separate subsection. 4.1. ILMs and anomalous defect properties

4.1.1. KI:Ag + One of the early proposals was that perhaps one or more ILMs are trapped at the Ag + site at low temperatures only to be released into the lattice at some finite temperature. When these ILMs are released into the lattice at an elevated temperature, they would leave behind a defect space with a different vibrational pattem and hence account for the change in the defect induced spectrum. Since a free ILM could move in any direction in the lattice, a large number of accessible states would exist; hence, there is a great deal of entropy associated with a change from a bound to this "free" configuration. The release of ILMs would be expected to give a strong temperature dependent signature.

w

Unusual anharmonic local mode systems

245

1.0

O.B I I

0.6

\ I

\\

~

,

',,f ~

0.4

0.2

l

0.2

\

-

\

0.4

0.6

0.8

kT/~-Fig. 65. Temperature dependence of the degree of association pon(T) for three different concentrations. The reduced temperature is kT/( where ff is the Gibbs free energy of association. The three different concentrations are (a) c = 1.2 x 10 -3, (b) c = 1.2 x 10 - 2 , and (c) c = 1.2 x 10-1. (After Lidiard 1957).

The temperature dependence can be calculated by making use of the similarity with the corresponding impurity-vacancy-complex activation problem. (See, for example, Lidiard 1957). Assume that an ILM complexes with the impurity. If the degree of association Pon is such that Cpon is the molar fraction of complexes, then application of the law of mass action to the process (impurity-ILM complex r unassociated impurity + unassociated ILM) gives = c exp (1

--port) 2

(,) k-T

(4.1) '

where ff is the Gibbs free energy of association. Curves showing Pon(T) as a function of temperature for three different impurity concentrations are presented in fig. 65. The temperature dependence of the lowest concentration curve looks remarkably similar to the data shown in fig. 14(a). Since both c and ( could be taken to be free parameters at this stage, a reasonable fit to the data could be generated for a single concentration, but fig. 65 indicates that the exact temperature dependence depends strongly on the concentration. What cannot be fit by this model is the experimental observation that the line strength temperature dependence observed for KI:Ag + is independent of Ag + concentration over a range of at least a factor of three; hence, the correct model cannot depend on concentration.

A.J. Sievers and J.B. Page

246

Ch. 3

4.1.2. MOssbauer recoilless fraction for Sn in Pb There are completely different kinds of measurements on Sn impurities in metallic Pb which show unusual temperature dependent behavior in the lattice vibrational spectrum. Mt~ssbauer recoilless fraction measurements as a function of temperature provide another way to investigate the dynamical coupling between an impurity and the lattice. For a simple Debye solid this quantity reduces to ~D

f = exp

3R 1+ 2k69D

~

ex - 1

'

(4.2)

where R is the recoil energy of the free nucleus and 690 is the Debye temperature of the solid (Frauenfelder 1963; Ashcroft and Mermin 1976). At temperature large compared to 69D the temperature dependent factor in the exponent is proportional to the temperature. (Note the similarity between eq. (4.2) and eq. (2.5). The measured temperature dependence of the recoilless fraction of ll9Sn in Pb (Haskel et al. 1993) is shown in fig. 66. For temperatures comparable to or slightly larger than the Debye 690 of Pb (88 K), the data give a linear temperature dependence for a semilog plot, as expected from eq. (4.2). The anomalous behavior appears at somewhat higher temperatures. For the lowest concentration the recoilless fraction disappears at temperatures above 120 K. For the 1% 119Sn in Pb, the recoilless fraction can be followed to higher temperatures beyond the knee at about 145 K. Similar temperature dependent results are found for the 2% 119Sn in Pb sample with the knee now at about 170 K. Finally, for the highest concentration sample the knee appears at about 200 K but is now somewhat rounded. These results demonstrate that the characteristic knee temperature at which the recoilless fraction rapidly decreases moves to higher temperature with increasing Sn concentration, with the resulting temperature dependence being a strong function of concentration. These results have been interpreted as evidence for a large number of low-lying states, a number much larger than can be derived from hopping or tunneling of the defect. The large number of states is attributed to "local melting" of about 30 atoms in the lattice near the defect, even though the temperature is far below the melting temperature of the alloy. The remarkable concentration dependence is not a natural feature of such a model. From w4.1.1 a concentration dependence of the temperature dependence would be a natural consequence if one or more ILMs are complexed to the Sn impurity at low temperatures but are released as the temperature is raised into the 150 K region. (See fig. 65). During this transformation

Unusual anharmonic local mode systems

w

247

A

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t,._

v

o

>, 0.10

.u. (/)

e"

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150 Temperature (K)

200

Fig. 66. MOssbauer spectral intensity versus temperature for alloys of Sn in Pb. The different atomic % concentrations of l l9Sn for data taken with increasing temperature are as follows: crosses -0.5%; open squares -1%; solid diamonds -2%; open circles -3%. The z's show the data taken during cooling. The straight lines are fits to the experimental data using a standard anharmonic potential model for the Debye-Waller factor. (After Haskel et al. 1993). the Sn could move from an on-center site to an off-center site. The large entropy contribution would stem from the large number of states available to the ILMs and the effective knee activation temperature would increase with increasing Sn concentration in qualitative agreement with the results shown in fig. 66.

4.2. ILMs and other properties Section 3 focused on ILMs with frequencies outside the normal acoustic and optic phonon bands, since almost all of the ILM work up to now has concerned this case. However, Takeno and Sievers (1988) also theoretically discussed the possibility of low-frequency in-band ILMs for the case of monatomic (k2, k4) lattices having negative values of the quartic anharmonicity A4 - knA2/k2, arising from negative ka's. These "soft anharmonicity" in-band ILMs are reminiscent of defect-induced harmonic resonant modes, such as those occurring in the T = 0 K on-center configuration of KI:Ag +. Indeed, for the simple (k2, k4) lattice it was found that low-frequency resonant ILMs have frequencies and amplitude patterns behaving like those for a force constant defect in a harmonic lattice, with the role of the harmonic force-constant change parameter in the latter case being played by the anharmonicity parameter An. In the limit of low frequency, the resonant

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ILM amplitude pattern is confined to the central particle, in contrast to the behavior we have seen here for the high-frequency ILM in the hard quartic case, where the limiting displacement pattern is that of a simple triatomic molecule. On the other hand, the spectral width of the resonant ILM was found to be much greater than that for the corresponding harmonic forceconstant defect resonant mode, giving the resonant ILM a shorter lifetime against decay into the quasicontinuum of embedding band modes. All of the ILM work discussed here has been classical. Nevertheless, certain general aspects of the effects of ILMs on the equilibrium thermodynamics of lattices were discussed by Sievers and Takeno (1988) and by Takeno and Sievers (1988). The key point is that because ILMs can exist at any lattice site, they should give rise to a large configurational entropy and hence make an important contribution to the free energy for nonzero temperatures. Thus their equilibrium statistics should be like those for thermally-activated processes such as vacancy formation, being determined by the balance between the energy needed for ILM formation and the configurational entropy. For the case when the thermal equilibrium number n i l m of ILMs is much less than the number N of lattice sites, one has n i l m - - N exp(-/3f), where = 1/(kBT) and f is the work needed to create a single ILM in a lattice of fixed volume. These considerations apply to above-band ILMs, to gap ILMs and to in-band resonant ILMs, with the resonant case expected to have a smaller (but still positive) value of f (Takeno and Sievers 1988). Hence for each type of ILM, the picture which emerges is that the vibrational spectrum of a lattice at T = 0 K should be dominated by homogeneous plane-wave-like harmonic phonons (or anharmonically renormalized quasiharmonic phonons), but with increasing temperature ILMs are created and should be included in describing the dynamical properties. Interpreting the classical anharmonic potential functions as effective potentials for the quantum solids 3He and 4He, Sievers and Takeno (1988) and Takeno and Sievers (1988) speculated on the possible importance of ILMs for these highly anharmonic systems. 3He exhibits a low-temperature excess specific heat characteristic of thermally-activated processes, and this has been identified with vacancy production. X-ray measurements of the lattice parameter of both 3He and 4He in constant-volume cells have shown that there are indeed thermal vacancies produced. However, when these measurements and the specific heat results are interpreted in terms of localized vacancies, one obtains an unreasonably large energy for the production of a vacancy. An attractive mechanism in addition to thermal vacancies, is provided by thermally-activated ILM production. In this case the lowtemperature ILM formation energy should be relatively small, being the difference between zero-point energies of two types of vibrational modes,

w

Unusual anharmonic local mode systems

249

namely ILMs and phonons. Additional arguments, emphasizing the role of resonant ILMs, were advanced by Takeno and Sievers (1988). Another early application (Sievers and Takeno 1989) was to the problem of the observed anomalous low temperature specific heat of glasses. This application was based upon two main assumptions. 1) The disorder results in the presence of a substantial number of low-frequency resonant ILMs - these can be thought of as disorder-induced impurity ILMs. 2) The ILMs can move diffusively. Here in w3 we have seen that recent studies of moving ILMs in perfect lattices show that ILMs move very easily from site to site (Bickham et al. 1992; Sandusky et al. 1992); moreover, it is reasonable to assume that the glassy disorder would result in the motion being diffusive. Sievers and Takeno (1989) presented arguments leading from these assumptions to two main results. The first is that diffusive ILM motion produces a contribution to the low temperature specific heat which is linear in T, as is observed experimentally. The second is that the presence of stationary resonant ILMs contributes an excess specific heat term which is cubic in T, also as observed. For the simplified model calculations, the key parameter was the ratio of the resonant ILM frequency to the Debye frequency. Phenomenological fits of this parameter to the measured low-temperature linear specific heat term for 15 glassy solids yield similar values (0.063 to 0.12), even though the glasses range in type through covalent, van der Waals, ionic, metallic, inorganic and organic. Moreover, the resulting excess cubic specific heat in the model is of the correct order of magnitude. An obvious question is whether or not the atomic forces in real crystals are sufficiently strong to sustain ILMs at reasonable temperatures. Bickham and Sievers (1991) addressed this question in an approximate fashion in their work on high-frequency ILMs in (k2, k4) lattices having weak quartic force constants. As we have noted, strong anharmonicity An -- k4A2/k2 can still occur in such lattices, for sufficiently large amplitudes A. These authors obtained k4/k2 from lattice data for five alkali halides, with LiF being the most anharmonic (ka/k2 - - 5 . 5 0 ~-2). Then by combining the standard harmonic approximation expressions for the virial theorem and thermal mean square displacement, together with their results for the spatial width of the ILMs as a function of A4, they concluded that in thermal equilibrium, lattices with these force constants would have insufficient thermal energy to sustain ILMs at any temperature below the melting temperature. Despite the major approximations in these arguments, the results were sufficiently clear-cut that it was concluded to be very unlikely that high frequency ILMs could arise thermally in these (k2, k4) systems. This left open the possibility of thermally generated low-frequency resonant ILMs in such systems, but this question has not yet been addressed. Finally, these authors also suggested that crystals exhibiting soft-mode ferroelectric or anti-ferroelectric phase

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Ch. 3

transitions might be an appropriate candidate for ILMs - near such a phase transition the importance of the anharmonic potential energy terms should be strongly enhanced. Having shown that ILMs are not likely to be thermally generated in simple (k2, k4) lattices with force constants derived from alkali halide potentials, Bickham and Sievers (1991) then briefly considered the introduction of vibrational energy from a nonthermal source, e.g., a Mrssbauer recoil or an optical transition followed by local lattice relaxation. As an extreme case within their force constant parameters, they considered the vibrational energy relaxation of an F-center, which in LiF involves about 40 phonons of frequency hWm. The order-of-magnitude result was that for this case sufficient vibrational energy is available to produce ILMs. However, this would require a very rapid local relaxation of the optically excited center, and hence detailed model calculations are necessary before one can make definite predictions. The speculations summarized in the above paragraphs were carried out shortly after the ILM concept appeared in 1988, and they were therefore made in the context of simple (k2, k4) lattices. Owing to the novel possibility that perfect lattices could sustain localized vibrational excitations, and because of the inherent difficulties of dealing with nonlinear phenomena in general, most of the subsequent theoretical work has focused on detailed explorations of the properties of ILMs for this simple case. Although this work is still in a relatively early stage, we have seen in w3 that these studies have already revealed a rich variety of interesting phenomena. We have also pointed out, however, that the simple (k2, k4) case misses an important aspect of realistic potentials, namely cubic anharmonicity, whose implications for ILMs in real solids are only now being sorted out. Our recent analytic and numerical studies of the effects of including realistic values of k3 (Bickham et al. 1993; Sandusky et al. 1994) have shown that the qualitative features of ILMs in (k2, k4) lattices remain when k3 is included, with the primary additional feature being that the ILMs are accompanied by a strong, amplitude-dependent, local static distortion of the lattice. As discussed earlier, this distortion tends to enhance the ILM's spatial localization as the mode frequency is lowered. Moreover, recent studies using realistic interparticle full potentials show that ILMs above the maximum lattice frequency do not occur for the one-dimensional systems studied (Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky et al. 1994), whereas ILMs do occur in the phonon frequency gap in diatomic lattices (Kiselev et al. 1993). Again, these gap ILMs appear to retain most of the qualitative features of ILMs in simple (k2, k4) lattices, with the main effect of the odd-order terms in the potential function just being to produce the local static distortion.

Unusual anharmonic local mode systems

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Clearly, the study of ILMs is in its infancy, with a vast number of important open questions remaining. Nevertheless, the recent results using realistic interparticle potentials suggest that ILMs will be theoretically understood sufficiently well in the near future that sensible experimental investigations can be launched to establish their presence in real solids.

5.

Conclusion

The experimental and theoretical studies of the two strongly anharmonic systems discussed in this chapter have revealed a wealth of fascinating and unexpected new phenomena. While much understanding has been achieved, the important remaining unexplained questions should provide a fertile area for obtaining fundamental new insights into the dynamical behavior of condensed matter systems.

6. Acknowledgements During the preparation of this article A.J.S. was supported by NSF Grant DMR-9312381 and ARO Grant DAAL03-92-G-0369 and J. B. P. was supported by NSF Grant DMR-9014729 and the Alexander von Humboldt Foundation. We would also like to acknowledge the important contributions made by our many collaborators on this work, particularly R.W. Alexander, S.A. Bickham, B.P. Clayman, R.P. Devaty, H. Fleurent, L. Genzel, L.H. Greene, S.B. Hearon, A.M. Kahan, R.D. Kirby, S. Kiselev, J.T. McWhirter, C.M. Mungan, I.G. Nolt, M. Patterson, A.M. Rosenberg, T. R6ssler, K.W. Sandusky and S. Takeno.

Note added in proof Since the preparation of the manuscript for this chapter, the ILM-related literature has continued to grow. Listed below are some recent papers. 1. ILMs in perfect and/or defect (k2, k4) lattices are discussed in: Wallis, R.E, A. Franchini and V. Bortolani (1994),Phys. Rev. B 50, 9851; Flach, S. and C.R. Willis (1994), Phys. Rev. Lett. 72, 1777; Dusi, R. and M. Wagner (1995), Phys. Rev. B 51, 15847; Takeno, S. (1995), J. Phys. Soc. Jpn 64, 2380; Kovalev, S., F. Zhang and Y.S. Kivshar (1995), Phys. Rev. B 51, 3218. 2. Diatomic Toda lattices are discussed in: Aoki, M. amd S. Takeno (1995), J. Phys. Soc. Jpn 64, 809.

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3. The counting of ILMs is discussed in: Kiselev, S.A., S. Bickham and A.J. Sievers (1995), Commun. in Condens Mat. Phys. 17, 135.

4. Surface or edge ILMs are discussed in: Takeno, S., K. Hori, K. Ohtsuka and S. Homma (1994), J. Phys. Soc. Jpn 63, 1295; Watanabe, T. and S. Takeno (1994), Phys. Soc. Jpn 63, 2028; Bonart, D., A.P. Mayer and U. Schr~der (1995), Phys. Rev. Lett. 75, 870; Bonart, D., A.P. Mayer and U. Schr/Sder (1995), Phys. Rev. B 51, 13739.

5. Exact RWA solutions for ILMs driven by external fields are discussed in: Roessler, T. and J.B. Page (1995), Phys. Lett. A (accepted).

6. General nonlinear dynamics/soliton approaches" Huang, G. (1995), Phys. Rev. B 51, 12347; Neuper, A., EG. Mertens and N. Flytzanis (1994), Z. Phys. B 95, 397.

7. On-site potentials include: Flach, S. (1994), Phys. Rev. E 50, 3134; Flach, S., K. Kladko and C.R. Willis (1994), Phys. Rev. E 50, 2293; Flach, S. (1995), Phys. Rev. E 51, 3579.

8. Some quantum mechanical aspects related to ILMs are discussed in: Kitamura, T. and S. Takeno (1994), Phys. Lett. A 190, 327; Kitamura, T. and S. Takeno (1995), Physica A 213, 539; Roessler, T. and J.B. Page (1995), Phys. Rev. B 51, 11382.

9. The application to self-localized magnons is considered by: Takeno, S. and K. Kawasaki (1994), J. Phys. Soc. Jpn 63, 1928; Wallis, R.E, D.L. Mills and A.D. Boardman (1995), Phys. Rev. B 52, R3828.

10. An overview of some properties of ILMs is given by: Page, J.B. (1995), Physica B (accepted); Bickham, S.R., S.A. Kiselev and A.J. Sievers (1995), in: Spectroscopy and Dynamics of Collective Excitations in Solids, Ed. by B. DiBartolo (Plenum Press, New York).

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CHAPTER 4

Influence of Isotopic and Substitutional Atoms on the Propagation of Phonons in Anisotropic Media

TADEUSZ PASZKIEWICZ MAREK WILCZYI(ISKI Institute of Theoretical Physics University of Wroctaw pl. Maksa Borna 9 PL-50-204 Wroctaw Poland

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

9 Elsevier Science B.V., 1995

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Contents 1. Introduction

261

2. Scattering of phonons by substitutional and isotopic atoms

263

2.1. Interaction of phonons with substitutional and isotopic atoms 263 2.2. Probability density of transitions per unit time induced by mass difference scattering 268 2.3. Boltzmann kinetic equation for long wave length acoustic phonons scattered by substitutional and isotopic atoms 269 2.4. Properties of long wave-length acoustic phonons (LAP's) 273 3. Influence of the medium anisotropy on the ballistic propagation of phonons 279 3.1. 3.2. 3.3. 3.4.

Experiments with beams of phonons 279 Kinetic description of beams of ballistic phonons Density of energy and quasimomentum 283 Phonon focusing experiments 284

4. Relaxation of a gas of LAP's in an isotropic medium

282

287

4.1. Spectrum of the collision operator 287 4.2. Derivation of the diffusion equation 291 4.3. Fourier-Laplace transform of the distribution function (FLTDF) 295 4.4. Explicit dependence of the Fourier transform of the DF on time 300 4.5. Important conclusions 301 5. Spectral decomposition of the collision operator

302

5.1. Cubic media 302 5.2. Spectrum of the collision integral for transversely isotropic media 307 5.3. Relation of the spectral decomposition of the collision operator to heat conduction 310

259

260

Ch. 4

T. Paszkiewicz and M. Wilczyhski

6. Time dependence of the Fourier transform of the DF for media of arbitrary symmetry 311 6.1. 6.2. 6.3. 6.4.

Singularities of the FLTDF 311 Poles of the FLTDF 314 Symmetry properties of the scattering tensor S(~, ~) 316 Asymptotic diffusive behaviour of the phonon gas 319

7. Experimental observations of the diffusive propagation of phonon pulses 323 Acknowledgements

327

Appendix 1. Distribution function of impurities Appendix 2. Collision theory

327

329

A2.1. Formulation of the problem 329 A2.2. Probability density of transition per unit time 330 A2.3. Scattering of phonons by isotope impurities 334 Appendix 3. Time-reversal invariance in classical mechanics

335

Appendix 4. Time-reversal invariance in quantum mechanics

336

Appendix 5. Microreversibility

342

Appendix 6. Detailed balancing condition References

346

345

To the Memory

of DMITRH NIKOLAEVICH ZUBAREV

1.

Introduction

The effects of isotopic disorder on different properties of solids have recently received considerable interest, being triggered by the availability of nearly isotopically pure crystalline materials (cf. Fuchs et al. 1993). Here we restrict ourselves to one such phenomenon, namely the scattering of phonons by substitutional and isotopic atoms (SIA's for short). For reason of simplicity we consider perfect, macroscopic specimens of dielectrics, semiconductors and superconductors. Consequently, at temperatures T much lower than the Debye temperature 0D, phonons are mostly scattered by SIA's. This scattering mechanism of phonons was introduced by Pomeranchuk (1942) and Klemens (1955, 1958). Early experimental and theoretical studies of it were concerned with the stationary measurements of heat conduction. The results of this studies were summarized by Ziman (1960) and Carruthers (1961). This, generally, integral method is made, however, more selective by the use of crossing effects in phonon scattering (cf. Challis 1987). In 1964 von Gutfeld and Nethercot (cf. Gutfeld and Nethercot 1964) devised the very simple experimental technique of heat (phonon) pulses. For almost three decades this technique has been extended to several new kinds of solid state spectroscopies. One of them is called time-of-flight spectroscopy (cf. Northrop and Wolfe 1985). Wolfe and collaborators invented another integral type of spectroscopy termed phonon imaging (cf. Northrop and Wolfe 1985). Phonon imaging spectroscopy is related to the phenomenon of phonon focusing (or phonon enhancement), which was predicted and described by Maris (cf. Maris 1986) and Rrsch and Weis (1976a, b). They considered the phonon energy images and used geometrical arguments. The use of the Boltzmann kinetic equation (BKE) allows us to describe the phonon energy images and additionally the phonon quasimomentum images of crystals (cf. Jasiukiewicz et al. 1991). Strictly speaking, the geometrical description of phonon imaging is valid only for beams of ballistic (nonscattered) phonons. The kinetic approach allows us, in principle, to include their scattering. 261

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T. Paszkiewicz and M. Wilczyhski

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Obviously heat pulses (i.e. beams of phonons) are also scattered by SIA's. As it follows from the general theory (cf. Kubo et al. 1985) each scattering mechanism broadens the pulse width, changes its propagation velocity and obviously diminishes its amplitude. It also washes out any sharp structure from a phonon image, making any measurement method based on phonon images less sensitive. As an example of such a measurement technique we mention the phonon-mediated detection of radiation or elementary particles (cf. Sadoulet et al. 1990). In the majority of experiments with phonon beams, one studies long wavelength (i.e. nondispersive) acoustic phonons. However, beams of acoustic dispersive phonons have also been studied (cf. Northrop 1982; Tamura and Harada 1985; Hebboul et al. 1989; Jasiukiewicz et al. 1994). When a beam of high-energy dispersive phonons propagates in a macroscopic specimen one may restrict oneself to the dispersionless acoustic phonons. Simply, high-energy optical and acoustic phonons split gradually into low energy acoustic phonons (cf. Levinson 1986; Berke et al. 1988; Maris 1990). The group velocities of dispersive phonons are small in comparison to the group velocities of long wave-length acoustic phonons (LAP's). So a detector of phonons placed a macroscopic distance from their source registers mostly dispersionless acoustic phonons. Note that for LAP's crystals can be treated as anisotropic elastic contin-

uous media. Usually the scattering of phonon beams by isotopic and substitutional atoms is described in a very crude relaxation time approximation (cf. Tamura 1983). A much better approximation can be found in use of the Monte-Carlo simulation methods (cf. Shields et al. 1991). A natural tool for the study of the scattering of nonstationary phonon beams by SIA's is the Boltzmann kinetic equation (BKE). For this scattering mechanism the collision integral is a linear integral operator which for some important classes of elastic media can be spectrally decomposed. In most of the experiments with the transmission of heat pulses the scattering of phonons by the specimen boundaries can be neglected. Additionally, with a sufficiently slow repetition rate of the heat pulses, one can neglect a change of the state of the specimen. Therefore, the mathematical problem which we must solve becomes the relatively simple Cauchy problem for the BKE in an anisotropic half-space. In this chapter we describe the properties of the Boltzmann kinetic equation and its applications to the description of the propagation of long wavelength acoustic phonons in an anisotropic medium containing substitutional isotopic atoms. The basic principles of the phonon kinetics are introduced in w2 and in the Appendices.

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Influence of isotopic and substitutional atoms

263

In w3 we describe the purely ballistic propagation of phonon beams and discuss briefly what kind of information is contained in the phonon images of crystals. Here the anisotropy of an elastic medium is the most important factor. For isotropic media, focusing phenomena are absent. However, the problem of the thermalization of an arbitrary phonon beam can be exactly solved (w4). Kozorezov and Krasilnikov studied the propagation of heat pulses in an isotropic medium containing SIA's in the relaxation time approximation (Kozorezov and Krasilnikov 1989). In w4 we discuss the main concepts of phonon kinetics. In particular the spectrum of the relaxation rates is obtained and the diffusion equation along with the diffusion coefficient are obtained. In particular we show that the diffusion constant determines the long-time asymptotics of a phonon pulse and can be measured experimentally. The suitable experimental data are presented in w7. In w5 we extend these results to the case of anisotropic elastic media of high symmetry (cubic and transversely isotropic). The general approach to the problem of thermalization of an initial state of a rarefied gas of dispersionless acoustic phonons based on the FourierLaplace transform (FLT) technique, valid also for elastic media of arbitrary symmetry, is presented in w6. The problem of propagation and scattering of beams of dispersionless acoustic phonons propagating in elastically anisotropic media is closely related to their elastic and acoustic properties. However, we pay less attention to them, otherwise this chapter would inevitably be much longer. The reader should be aware of this close relationship which, for example, makes the scattering a probe of the properties of the field of polarization vectors. These properties are described in a seminal paper by Al'shits et al. (cf. Al'shits et al. 1985). Their consequences were noted in experiments performed by the Wolfe group (cf. Ramsbey et al. 1988).

2. Scattering of phonons by substitutional and isotopic atoms 2.1. Interaction of phonons with substitutional and isotopic atoms The fact that fluctuations in the mass distribution throughout a crystal cause the scattering of phonons was first pointed out by Pomeranchuk (1942). This problem has been studied by Klemens (1955, 1958), Ziman (1960) and Carruthers (1961). They considered a crystalline dielectric or semiconducting specimen of volume V with a fluctuation in the mass distribution. Let us suppose that

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these fluctuations arise from the presence of SIA's and that the crystalline lattice contains no other lattice defects. Assume that the lattice of total volume V contains N unit cells, the equilibrium positions Xz of which are enumerated by integer numbers 1 = 1 , 2 , . . . , N. The volume of each cell is vo = V / N . For simplicity we confine ourselves to the case of lattices with one atom in each unit cell. The probability that an atom has the mass Mi of a substitutional atom is proportional to the concentration c. The dynamic state of the crystalline lattice is described by two sets of canonically conjugate variables, i.e. the momentap(1) and the displacements u(l). Since each site of the lattice can be occupied by a host or by a substitutional atom with masses Mh and Mi, respectively, we describe the state in the lth unit cell by the random variable r z (cf. Leibfried and Brauer 1978)

={0,

a host atom,

if in the/th unit cell there is a substitutional atom.

1,

We assume that the interatomic forces are unchanged when a host atom is replaced by its isotope. Then, only the kinetic energy part of the Hamiltonian is a function of the N-dimensional random variable r = {r 1, r 2 , . . . , rn}, namely

Hkin(7") = ~ l=l

1 } p2(l).

mhh (1 - "r,) + ~ii-r,

(2.1 1)

We separate from Hkin(r) the part which contains the isotope scattering effects, namely

lk{, l}

H'('r) = ~ /=1

Mi

Mh

(2.1.2)

rzp2(/)"

The total Hamiltonian of the lattice now contains the Hamiltonian of, a generally, anharmonic lattice without substitutional atoms N

~

p2(/) + /=1

~ n=2

(l)

~f~}~.2.:i2'~ I I u,~, (li),

(2.1.3)

i=1

where the summation over dummy indices is assumed and Y](O means summation over lattice sites, (1) - ll, 12... In and H ( r ) = Ho + H'(r).

(2.1.4)

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Influence of isotopic and substitutional atoms

265

Assume that the anharmonic part of H0 n

V= ~

I u~, (h) ~1 ~~,~2-..~,~1z,z~...,~

n=3

(l)

i=1

is small in comparison with the harmonic part . . . . z,h 2 U O ~ I (ll)u~2(12).

N 1 Hh-- E p2(/) 2M h + ~ E /=1

~0~10t

(2.1.5)

(/1,/2)

Then, the diagonalization of Hh provides us with a convenient basis, which allows us to expand the momentum and displacement operators 1

2

p~(l) : -~ ~ E kr

lhw(K)Mh

e~(K)exp(ik. Xz)(aK--a+K), (2.1.6a)

2N

j=o

21 2NMhw(K) h e~(K) exp (ik 9Xt) (a K +a +- K ) , (2.1.6b)

u,~(1) : E E kr

j=o

where K stands for the pair (k, j), in which k is the wave vector, j is the branch and polarization index and (-k, j) = - K . The frequencies w(K) and the polarization vectors e(K) can be found from the suitable eigenvalue problem (cf. Streitwolf 1967; Wallace 1972). For long-wavelength acoustic phonons (j = 0, 1,2) A

w(K) ~- c(K)k, A

(2.1.7) A

A

where k = Ikl, k = k/k, K = (k, j). The polarization vectors obey the normalization and the orthogonality conditions which, for acoustic phonons, read 2

(2.1.8a) j=O

3

(k, j) 5 (k, j') - 6,y. o~=1

(2.1.8b)

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Ch. 4

Additionally,

e ( - k , j) = e* (k, j),

(2.1.8c)

where e* is the complex conjugate vector to e. The creation, ag, + and annihilation, a K, operators are new variables obeying the boson commutation rules

[a K, aK, ] = O,

[a +, a+,] = O,

[a K, a+,] -- 5K,K,.

(2.1.9a--c)

In the new variables,

Hh=

Ehw(K) a + a K +

(2.1.10)

Kr

Introduce the set of eigenvectors In~-) of the phonon number operator +

n K :

aKa K

(n~c = 0 , 1 , 2 . . . . ).

(2.1.11)

They form an orthonormal basis (2.1.12) in which the operator a g diminishes while a g+ raises the number of phonons K by one

a K In~> = V/n'K In ~ - 1>,

a+[n'K> : ~/n'K +

l ln~ +

1>.(2.1.13)

Since in the harmonic approximation the subspaces of vectors In~-) with different K are independent, it is easy to find an eigenvector of Hn

I{n~)> : I-I In~>, |

where {n~}, is a set of occupation numbers of all 3N phonons. One has

(1)

~ 1 { - } } > : E ~(K) n~ + ~ I{-} } >. K

(2.1.14)

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Influence of isotopic and substitutional atoms

267

Two eigenvectors of Hh, I{n~-}) and I{n~.}) with two different sets {n~} and {n~.}, respectively, are orthogonal ({n~. } -r {n~.. }),

(2.1.15a)

and normalized to the unity =

1.

(2.1.15b)

Expressed in the terms of creation and annihilation operators, H ~ is not diagonal. Since the related processes do not satisfy the energy conservation law we drop in (2.1.2) the terms proportional to aKa K, and aKaK, + + , hence

H'(T) --+ HI(T) [ V . ( - K , K ' )aKa + K, + V~.*( - K , K ' )aK,aK] +

= E

(2.1.16)

K#K'

where V~-(-K,K')

(2.1.17) = ~A1(M_ 1).__~_hMhV/w(K)w(K,)e,(~)e~(~,)v( k, _ k),

A ( M -1) : (Mi -1 - m h l ) ,

and r(k) is the Fourier transform of the random variable r z 1

N

r(k) = -~ E rt exp(ik 9Xt). l=l

(2.1.18)

Functions of the random variable r may be averaged with a suitable distribution function P(r) (cf. Leibfried and Brauer (1978) and Appendix 1).

T. Paszkiewicz and M. WilczHtski

268

Ch. 4

2.2. Probability density of transitions per unit time induced by mass difference scattering

In the Appendix 2 we derive the Pauli master equation which defines the transition probability density per unit time. For our particular problem the probability density of a scattering event in which the initial number of phonons in states K and K', being unity and zero (ng = 1, n K, = 0), respectively, changes into the state with the corresponding phonon numbers n K = 0 and n K, = 1, is equal to w.(K,K')

IV.(-K,K')lZS[w(K) w(K')].

=

(2.2.1)

-

Since the system is spatially inhomogeneous the probability density of transition per unit time depends on both k and k", i.e. the quasimomentum is not conserved, so generally k ~ k'. The transition probability (2.2.1) is proportional to Ir(k k')l 2 -

_

1

N N

Ir(k - k')12 - N2 E

E

/=1 l'=l

rtrt' exp [i (k - k ' ) - ( X t - Xt,)].

Separating the terms for which 1 = l' one gets the expression [T(k _ k') l2

1

N

= N 2Erl

/=1

+

{ 1 N ~ErLexp[i(k-k')'X'] /=1

}

(2.2.2)

Generally the energy conservation law hw(K) = hw(K') does not imply the quasimomentum conservation law, therefore the second oscillating term on the rhs of eq. (2.2.2) vanishes. The degeneration points, where w(k,j) = w(k', j'), k = k', j ~ j', should be considered separately. So for k ~ k' eq. (2.2.2) reduces to

i

(k_k,)l = = --~

(w(k, j) - w(k', j'),

k r k').

(2.2.3)

Therefore, with exception of transitions at degeneration points, the probability density w~(K, K') does not depend on the particular impurity distribution and reads

w~.(K, K') - w(K, K') 7F

= 2N 9w(K)w(K')le(K)" e(ff2')125[w(K) - w(K')],

(2.2.4a)

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Influence of isotopic and substitutional atoms

269

where 9= c

Mh ) 2 ~ - 1

(2.2.4b)

and we take into account that for Bravais lattices the polarization vectors are real. Averaging the transition probability density w~(K,K') over all possible configurations (cf. Appendix 1) or over the volume (cf. Gurevich 1986) one restores the spatial homogeneity which manifests itself by the presence of the conservation law for quasimomentum hk. Generalization of our calculations to lattices with more than one atom per unit cell (i.e. lattices with basis) consists of introducing the fraction of ith isotope of the ~th atom of the unit cell, f~(~), the mass of the ith isotope atom being M~. If one considers only the scattering of long-wavelength acoustic phonons one obtains (cf. Tamura 1983) 71"

w(K, K') = - ~ w ( K ) w(K') 5 (w(K) - w(K'))

• E le(K' t~) "e(K~"~) 12g'(~)'

(2.2.4c)

t~

where 9'(~) is related to the fraction f~(~) of ith isotope of nth atom as

g'(~) = E fi(m) 1

mi(~)]2 _~(~)

i

Further we assume that 9, 9 t << 1. The transition probability obeys the microreversibility principle (cf. Appendix 5) w(K, K') = w(K', K).

(2.2.5)

2.3. Boltzmann kinetic equation for long wave length acoustic phonons scattered by substitutional and isotopic atoms 2.3.1. Boltzmann kinetic equation for phonons So far we have worked in a representation based on plane waves extending throughout the crystal. When dealing with an inhomogeneous system this

T. Paszkiewicz and M. Wilczyhski

270

Ch. 4

representation has disadvantages. Thus, we shall look for a more convenient representation. Such a representation is based on wave packets constructed as linear combinations of the above phonon states (cf. Peierls 1964; Kirczenow 1980). The wave packets will be localized in space to regions so small that an external force or local hydrodynamic parameters such as the temperature T(r, t) and drift velocity V(r, t) vary little over the region, but are large enough to contain many unit cells. Wave packets move with the group velocity v(K)

v(K) -

Ow(K) Ok '

(2.3.1)

transporting energy and quasimomentum (cf. Fedorov 1968, ch. 4). A wave packet is characterized by the polarization j, position r, a typical wave vector k and a typical frequency w(k, j) (cf. Elices and GarciaMoliner 1968). The state of a gas of such wave packets is described by the distribution function f(K;r, t) (DS). Multiplied by physically small volumes d3r and d3k, the distribution function gives the number of phonon wave packets of polarization j at an instant of time t in the volume d3r surrounding the point r with typical wave vectors belonging to the environment d3k of the wave vector k. In the absence of collisions, the distribution function is constant along the trajectories, so its complete time derivative vanishes

df(K;r,t) dt

--0.

Hence the distribution function obeys the Liouville equation

Of(K;r,t) -. ' =0, Ot + #" Vf(K', r, t) + hk. Of(K'olikr,t) where ~ is the group velocity v(K), hk is an external force F and the operator vector X7 is defined as

V

-

-

(V1, V2, V3)

(0 0 0) -

8rl' i~r2' 0r 3

-

and a . b is the scalar product of two vectors a and b. Collisions are characterized by a nonvanishing rate of change (df/dt)con (collision integral)

dt

-

d-t

con

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Influence of isotopic and substitutional atoms

271

or

Of(K;r,t) + v(K). Vf(K" r, t) + F . Of(K;r, t) ' Ohk Ot _ [ d f ( K ; r, t) L dt coll

(2.3.2)

The equation (2.3.2) is termed the Boltzmann kinetic equation (BKE). The microscopic derivation of the B KE has been given by many authors (cf. G6tze and Michel 1972; Enz 1974; Beck 1975). Further we assume that external forces are absent (i.e. F = 0), in particular we omit any coupling with the deformation field (cf. G6tze and Michel 1972), and we confine ourselves to the scattering of phonon wave packets by SINs only. Begining with w2.4, for the reason of simplicity, we shall consider only long-wavelength acoustic phonons. Under such conditions [dr~dr]coil is a linear functional of the distribution function, namely

df]

~

-

Bf(K;r,t)

coil

(2.3.3) (2703 j=0

d 3 k ' w ( K , K ' ) [ f ( K ' ; r , t ) - f(K;r,t)].

The kernel of this integral functional is the probability density of transitions per unit time (2.2.4).

2.3.2. General properties of the Boltzmann kinetic equation The B KE equation reveals several general properties, which we illustrate with the example of phonons scattered by substitutional or isotope atoms. According to Appendix 3, the BKE is not invariant under the operation of time reversal. The time reversed distribution function fx(K;r,t) obeys the equation

i3fI(K;r, t) + v(K). Vfi(K;r, t) + F . OfI(K;r, t) O(~k) i3t

-_ [dfi(K;r,

(2.3.4)

t) ] co.

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272

Ch. 4

Note that the collision term of eq. (2.3.4) has the opposite sign compared to that of eq. (2.3.2). Multiplying both sides of the BKE (2.3.2) by kB ln[(1 + f)/f], summing over j and integrating over k, we obtain the equation of the local balance

of entropy Os(r,t) 0t

(2.3.5)

+ div js(r, t) = S(r, t),

where s(r, t) is the entropy density 2

s(r,t)-j~ofd3ks(K;r,t) (2.3.6a)

2

d3kkB{ [1 + f(K;r,t)] In [1 + f(K;r,t)]

- f(K;r, t)In f(K;r, t)}, kB is the Boltzmann constant and js(r, t) is the current density of entropy

js(r't)-~ofd3kv(K)s(K;r't)'.= 2

(2.3.6b)

The entropy production density S(r, t) is related to scattering processes and can be written in the form

2

~r

t) =

v~ ~j, / d3k f d3kt w(K, K') (27r)3 , = 3

0~

(2.3.6c)

• [f(K';r, t) - f(K;r, t)] •

In l + f ( K ; r , t )

-In l + f ( K , , r , t )

'

which is obviously nonnegative. Therefore, when the net flow of entropy into and out of the system vanishes, the entropy may grow.

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Influence of isotopic and substitutional atoms

273

Note that the collision integral vanishes for an arbitrary function of w, r and t which does not depend on K. In particular the Planck function fo(w(K)) is a collision invariant and satisfies the BKE. Introduce the scalar product of two functions of K A

2

(2.3.7)

(f' 9) - E f d3k f(K)g(K). j=0

This scalar product defines the Euclidean space g. The collision integral is a linear and symmetric operator B acting in g, such that (f, B9) = - ~ E j,j'

d 3k

ffd3k 'w(K, K')

(2.3.8)

• I f ( K ) - f(K')] [ g ( K ) - 9(K')]. Thus, B is a symmetric, nonpositive linear operator (f, B9) -- (B f, 9) - (9, B f),

(f, B f) <~ 0.

(2.3.9a, b)

Mass difference scattering of dispersive phonons and the application to various lattice models has been considered by Tamura and collaborators (crystals of: Ge (Tamura 1983), GaAs and InSb (Tamura 1984), and for crystals of Si (Shields et al. 1991, 1989), (Tamura et al. 1991).

2.4. Properties of long wave-length acoustic phonons (LAP's)

2.4.1. Dynamics of continuous elastic media In the limit of long-wavelength (i.e. for small wave vectors), the microscopic details of the crystalline structure are not important and crystalline solids can be treated as anisotropic elastic media. The state of an elastic medium is described by the continuum deformation field u(r, t), which depends on the space variable r and time t. For a medium of density p and with the tensor of elastic constants C which has components C~u,~, (c~,/3, u, # = 1, 2, 3), the deformation field obeys the Christoffel equation (cf. Fedorov 1968)

p

i~2u~(r, t) i~t2 -- C~,~V~V,u~(r,t).

(2.4.1)

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T. Paszkiewicz and M. Wilczyhski

Ch. 4

The tensor of elastic constants C satisfies the following symmetry relations (2.4.2) Tensors fulfilling these relations have complete Voigt symmetry and, after Jahn, are referred to as [[V2] 2] type (cf. Sirotin and Shaskolskaya 1979). For a solution of eq. (2.4.1) having the form of a plane wave u(r, t) - exp [i(k. r - cot)] e

and propagating in an unbounded medium the polarization vector e must satisfy the algebraic equation (2.4.3)

co2e~ - A~z(k)ez,

where A~z(k) is an element of the propagation tensor A ~ ( k ) = p-1 kt~ Cc~t~,13uku"

There are three independent solutions of eq. (2.4.3) corresponding to one quasilongitudinal and two quasitransverse waves enumerated by the polarization index j = 0, 1,2. The polarization vectors are real, so from eq. (2.1.8c) A

A

(2.4.4)

e ( - K ) - e(K).

Equation (2.4.3) is equivalent to the long-wavelength limit of the lattice dynamical equation (cf. Wallace 1972). Since the wave vector k has length k - Ikl and direction k - k / k , eq. (2.4.3) implies eq. (2.1.7). This means that the phase velocity c(K) obeys the following algebraic equation c2e~ - A ~ ( k ) e ~ .

(2.4.5)

Thus, for LAP's both the phase velocity and polarization vectors depend only on the direction of the wave vector k and polarization j. Arthur Every (Every 1980) has obtained a closed form expression for components of the polarization vector and for the phase and group velocities of arbitrarily directed waves. A

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Influence of isotopic and substitutional atoms

275

2.4.2. Group velocity and surfaces of constant energy The equation

w(K) = coo,

(2.4.6)

defines in k-space a surface of constant frequency (w-surface for short). Each w-surface has the symmetry of the considered crystal. Having the phase velocity c(K) one can introduce the slowness, s, through s(K) - c-l(K).

(2.4.7)

A polar plot of the slowness gives the slowness surface. The equation (2.4.6) determining an w-surface is equivalent to A

k - s(K)~o. Thus, an w-surface, which is in fact a polar plot of k, has the same shape as the slowness surface for the LAP's. The ratio of the surface sizes is simply the frequency. The integral defining the operator B (2.3.3) can be written in a form allowing the explicit use of the law of energy conservation, namely

j d~k' - j dS" IX~:,~(K')I

-1

L

oo dw',

(2.4.8)

where dS~ is an element of the w-surface co(K ~) = const; hence,

Bf(K,r,t) =

vow 2

le(R), e(~')]

S d':

• [f(ff[';r,t)

-

(2.4.9)

f(ffU;r,t)].

The group velocity, being the gradient of the frequency (cf. eq. (2.3.1)), for any K is normal to the w-surface at the point indicated by the wave vector k, and, consequently, normal to the slowness surface at the corresponding point. In general, for anisotropic solids the group velocity v(K) of a phonon with wave vector k and polarization j is not parallel to k. This property is

T. Paszkiewicz and M. Wilczyhski

276

Ch. 4

related to the anisotropy of the medium. Its consequences will be discussed in w From eq. (2.1.7) it is seen that the group velocity of the LAP's depends only on K, i.e. A

v,~(/Y) = Oc(K)^ (a,~,.r _ ~,~.r ) + c(K)k~. ak.y

(2.4.10)

Therefore, the group velocity and the phase velocity obey the simple relation A

A

A

k,~v,~(K) = c(K).

(2.4.1 l)

When the tip of the wave vector k moves across the w-surface, the direction of the group velocity changes. The rate of this change depends on the curvature of the w-surface. In a fiat region the group velocity vectors are almost parallel. The local geometry of the w-surfaces is characterized by the Gaussian curvature F, which is the product of the two principal curvatures /-'1,/-' 2

V = F1/-'2.

(2.4.12)

The principal curvatures can be related to the derivatives of the group velocity in two perpendicular directions in the plane tangent to an w-surface at the given point (cf. Maris 1986; Lax and Narayanamurti 1980).

2.4.3. Collision integral for mass difference scattering of LAP's For long-wavelength acoustic phonons, the formula (2.4.9) can be simplified further. The surface element dS" when viewed from the F-point of the Brillouin zone subtends a solid angle dk' (cf. Kosevich 1988) where dk'=

1,) 2 dS" I~'" VK'W(K')[

(2.4.13)

A

Expressing the surface element dS" by the solid angle dk' and using eq. (2.4.11) we rewrite the integral over d3U in the following form A

(2.4.14)

Influence of isotopic and substitutional atoms

{}2

277

h

We introduce the mean value of a function A of K

lk/4

(A(K)}~ = ~

j=O

A

7r

~

A(K),

c(K') %

(2.4.15)

where co is the Debye velocity defined as dk ~-3(~). A

CD3 _._ 1 j=O

(2.4.16)

~r

The collision integral therefore can be written as

B f(ff~,r,t) - 3T -1 ([e(R). e(I~')[2[f(I~';r,t) - f(I~;r,t)])R,

(2.4.17)

where r -1 _= T--I(w)=

VogW4

(2.4.18)

4rrC3D " s

This form of the collision integral may be expressed in terms of a dyad

(8(2))~ = 8~ 0 = [e(K)| e(K)]~ e = e~(K)ee(k),

(2.4.19)

namely

B f(ff~,r, t) = -r-lv(ffS)f(Is

t) (2.4.20)

+ 3r -ls163

t))~,

where

v(R)- 3&e(R)(&e(R'))~,.

(2.4.21)

Thus the B KE equation for long-wavelength acoustic phonons scattered by SIA's is i3 + v ( K" ) . V-* + r - l u ( K ) ] f(/~; r, t) -~ (2.4.22)

= 3r-18~O(ff;)(s

T. Paszkiewicz and M. Wilczyhski

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Ch. 4

As we know for long-wavelength acoustic phonons a crystalline solid can be considered as an anisotropic continuous medium; therefore, the problem of calculating the polarization vectors and frequencies is relatively simple and model independent. Knowing them one can establish the spectrum of relaxation rates. Note that the number density of phonons of a given frequency w

N (w;r,t) - (f(w, ff2;r,t))~

(2.4.23)

is a collision invariant.

2.4.4. Mean value of the dyad s Consider the mean value of the polarization dyad (2.4.19). Since (g(2)(~))~ is a material tensor, the number of its independent elements de ~ends n p the symmetry of the medium. For isotropic and cubic elastic media the averaged polarization dyad simplifies to an isotropic tensor (2.4.24a) Due to the condition (2.1.8a) the trace of (s is equal to unity, so e = 1/3 and the expression (2.4.21) for any K satisfies A

A

u(K) = 1.

(2.4.24b)

For transversely isotropic media with the symmetry axis parallel to the z-axis of a Cartesian coordinate system, the components of the averaged polarization dyad expressed in that coordinate system read

(gc~/3(K))~. : El(~c~,x~/3,x + (~a,y(~13,y) -t- E3(~a,z~13,z,

(2.4.24c)

where (2.4.25a, b) and 2el + e3 = 1.

(2.4.25c)

Similarly, for hexagonal, tetragonal and trigonal systems the tensor (s has, in a suitable coordinate system, two independent diagonal elements. For media with the remaining symmetries there are three independent components of the diagonalized averaged polarization dyad (cf. Sirotin and Shaskolskaya 1979).

w .

Influence of isotopic and substitutional atoms

279

Influence of the medium anisotropy on the ballistic propagation of phonons

3.1. Experiments with beams of phonons Assume that we deal with a perfect macroscopic specimen at an ambient temperature much lower than its Debye temperature 0o (cf. (2.4.16)) h

OD -- kBa CD , -

-

where

a ~ v1/3. Then, the equilibrium density of high energy acoustic and optical phonons is exponentially small g(T) ,-~ Vo 1 exp

--~BT

(kBT << kBOD<

ha;),

(3.1.1a)

whereas for low energy acoustic phonons

~ ( T ) , . . , v o l (~DD T) 3

(T << 0O).

(3.1.1b)

The typical Debye temperature 0o is of the order of 102 K. Therefore at helium temperatures (T ~ 2K) there are several phonon wave packets per million unit cells. So at helium temperatures a phonon gas in a typical crystal is rarefied. In heat pulse transmission experiments one introduces long wave-length acoustic phonons in the form of wave packets (cf. Northrop and Wolfe 1985 and Maris 1986). Such phonons are usually generated at specimen boundaries. However, in experiments with the phonon mediated detectors of elementary particles (cf. Sadoulet et al. 1990) phonons are also generated inside the crystalline specimen. We shall confine ourselves to the former type of experiments. Due to the small dispersion (i.e. the weak dependency of the frequency on the wave vector) the group velocity of optical and acoustic dispersive

T. Paszkiewicz and M. Wilczyhski

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Ch. 4

1 q

r

z

/_ medium

X

vacuum Fig. 1. In transmission experiments a microscopic specimen can be treated as a spatially homogenous half space filled with an anisotropic medium.

phonons is small in comparison to the group velocity of LAP's. In a microscopic specimen on their way from the source towards the detector such phonons decay and their beams are never ballistic. From now we shall confine ourselves to beams of LAP's. Since in heat pulse transmission experiments one usually neglects the scattering of the propagating wave packets by specimen boundaries, a macroscopic specimen can be treated as a spatially homogeneous half-space filled with an anisotropic medium (fig. 1). Structural defects (point defects, dislocations etc.) violate the spatial homogeneity and scatter wave packets. Usually on one face of the specimen there is deposited a thin metallic film. A small region of this film is excited by a laser beam or by a beam of particles (e.g. electrons). This creates a highly spatially inhomogeneous transient distribution of energy and quasimomentum. The heated region cools down by radiating phonon wave packets. They move away with the group velocity v(K) (in spherical coordinate system v(K) = (v(K), O~(K), c~(K)) ) carrying energy and quasimomentum. In the case of a film in the normal A

A

A

A

A

w3

Influence of isotopic and substitutional atoms

281

nonsuperconducting state one assumes that the spectrum of radiated phonons is almost planckian. If one illuminates a superconducting film the spectrum of the generated phonons is close to monochromatic. Another plausible assumption concerns the directions of the wave vectors k = (k, 0, 4~) of the produced phonons. It is assumed that they are uniformly distributed over the body angle 27r. On the opposite face of the specimen there is a detector of phonons. It can be sensitive to energy (a bolometer) or quasimomentum (cf. Jasiukiewicz et al. 1991; Jasiukiewicz and Paszkiewicz 1990). In early experiments (cf. Gutfeld and Nethercot 1964) both source and detector (a superconducting bolometer) were immovable. In recent experiments usually the source is movable and the detector is fixed; however, several groups have used fixed sources and movable detectors. Changing the position of one of them, say, of the detector, and keeping the other one fixed, one varies the direction ~" of the line joining the source with the detector. This also means that the direction of propagation of a detected wave packets changes. Later on we shall assume that a source is located at the origin of a Cartesian coordinate system, the z axis of which is perpendicular to the boundary and is directed into the medium. For a given direction of propagation of nonequilibrium phonons the signal from the detector has the form of a number of well .separated sharp peaks (cf. Bron 1985) (fig. 2). Their number and the ratios of their amplitudes depend on the propagation direction. These peaks move with appropriate group velocities. Even for specimens of the highest quality the peaks have a finite width (the analogue of an inhomogeneous broadening of a spectral line). The scattering processes broaden them even more and enlarge the ambient background. ST

_J

Z

FT

oa0 =

_

~

_

_

TIME (,usec)

Fig. 2. Time-of-flight spectrum of a heat pulse injected into a perfect sample of A1203; after Bron (1985).

T. Paszkiewicz and M. Wilczyhski

282

Ch. 4

3.2. Kinetic description of beams of ballistic phonons Let us forget for a while about the scattering processes and consider freely propagating ballistic phonons in a crystal. To simplify the discussion we assume that the position of the source is fixed, say at r = 0 of the boundary. A movable detector is placed at an arbitrary point r of the medium. The vector r has the length r and direction F = (0~, r Consider an idealized experiment in which a real source of phonons is approximated by a point source producing infinitely short bursts of phonons. A nonequilibrium state of the phonon gas at ^an instant of time t is described by the deviation function 5 f ( K ; r , t ) = f ( K ; r , t ) - fo(w(K)). Under the described conditions the deviation function 5f(K;r, t) obeys the BKE with a source term and without the collision integral term Ia + v(K ^ ) . V ] ~ f ( K^ ; r , t ) = ~(K;r,t).

(3.2.1)

For an experiment with pulses of monochromatic phonons of frequency w0 the source term ~(K;r,t) has the form

~(K; r, t) - A(K)5(t)5(r)O [v=(~[)] ,

(3.2.2)

where 3(x) is the Dirac delta function, 5(r) = 3(x)6(y)~(z), O(x) is the unit step function and w0 is the frequency of generated phonons. The form of the source term of Planckian phonons is discussed elsewhere (cf. Jasiukiewicz and Paszkiewicz 1993). For a source of monochromatic phonons

A ( K ) = A 5[w(K)

-

wo].

(3.2.3a)

The source term is normalized in the following way +cx:~

2

oo

j=O

3+

(3.2.4)

where Noh is the number of injected phonons and R3+ is defined by the inequalities: - c ~ <~ x ~ +c~, - c ~ ~< y ~< +c~, 0 <~ z ~< +c~. The integral over the reciprocal space can be transformed in the same way as the collision integral (cf. w2.4.3). In this way we get A

A = 47rc(K) 3w2(K).

(3.2.3b)

w

Influence of isotopic and substitutional atoms

283

The solution of eq. (3.2.1) with the source term (3.2.2) which satisfies the boundary condition 3f(K;r, t ) = 0 in the infinite past, is (3.2.5)

5f(h'; t) = A(K)5 [r - v(K)t] 0(t)0 [vz(R')] 9

The scattering processes can be accounted for in the simplest way in the relaxation time approximation, which means that the collision integral C[~f] is replaced by (--T -1 ~f) and the solution (3.2.5) is multiplied by the factor exp(--t/T). However, in the example of scattering of phonons by isotope or substitutional atoms, we shall show (w167 and 5.1) that this approximation is very crude.

3.3. Density of energy and quasimomentum A

Having calculated the phonon deviation function ~f(K;r, t) we can calculate the components of the energy density currentjE(r, t) and the components of the quasimomentum density current 7r(r, t) 2

' /

j~(r, t) = Z

(2703

d3k hw(K)va(fi~) ~f(I~;r, t),

(3.3.1a)

j=0

2

7 r ~ ( r , t) =

-= (270 3

(3.3.1b)

For a bolometer placed at r with surface area AS and normal ~ the energy falling onto it per unit time is

In (r, t) = jE(r, t) . ~AS.

(3.3.2a)

Suppose that the detector of quasimomentum (i.e. a strip filled with a 2D electron gas or a cloud of excitons in the vicinity of the surface of a semiconductor (cf. Jasiukiewicz et al. 1991; Jasiukiewicz and Paszkiewicz 1990 and Paszkiewicz 1991) is lying in the plane with normal ~, has an elongated form, is directed along an axis ~ and its area is AS. Then, the suitable component of the quasimomentum density falling onto this strip per unit time is (3.3.2b)

T. Paszkiewicz and M. Wilczyhski

284

Ch. 4

Substituting the explicit form of 5f(K;r, t) (3.2.5) into (3.3.1a, b) we get the densities

e~(r,t) = O(t)hwol ~ f4~ ~dk

o[~(~)] (3.3.3a)

j=O

x [~. v(K)] 5 [r - v(K)t],

p~v(r, t) = O(t)~vo

0 [vz(R')] j=0

~r

(3.3.3b)

c(K) Note that the integrand of the integral defining the quasimomentum density contains an extra factor, namely the projection of the slowness vector s(K) = ks(K) = k c - l ( K ) (cf. (2.4.7)) onto ~.

3.4. Phonon focusing experiments In phonon focusing (imaging) experiments one measures respectively the density of total energy E and the density of total 0-component of quasimomentum falling onto the surface of the appropriate detector 2

2

c~

j =o

j =o

(3.4.1a)

2

p,~,(,')

2

p,~,( ) - ~

~ , c~

dt p~,)(,., t).

(3.4. lb)

j =0 j =0 d O Now we proceed to evaluate the densities en(r) and p~v(r). A phonon with the wave vector k = (k, 0, r falls onto the detector, if its group velocity direction ~(O~), r coincides with the direction of the radius vector r of the detector. Generally, due to the complicated nonspherical shape of surfaces of constant frequency (fig. 3) there is more than one solution to the vectorial equation

7(0~, r

~j(0~, r

(3.4.2a)

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Influence of isotopic and substitutional atoms

285

Fig. 3. The slowness surface of the slow transverse (ST) and fast transverse (FT) acoustic phonons for GaAs. Dash (parabolic) lines indicate vanishing Gaussian curvature. Parabolic lines separate convex, saddle-point and concave regions.

or to the two equations for angles 0~ - 0~)(0, r

r

- r

r

(j = 0, 1,2).

(3.4.2b)

Denote them by ,,jA(~)rA~,,,.,r

r

r

(i = 1,2, . . . , nj). For an arbitrary

function Fj of k we introduce the notation U '""' r

Fj

r

(0,. r

To evaluate the integrals defining the densities e~(r), pun(r) we shall use an identity valid for the Dirac delta function (cf. Jasiukiewicz et al. 1991) nj

5[r - v(K)t] - E x

[

5 t

r

[v~i)(O''

i=1

-

r ] 5 (0

.(i)~,~

-oj w,, r

x ~(.~)(o,, r

-

)(~ oj(i) (o,, r

6 (r

r

- r

r

(3.4.3)

sin O~J)(O,.,Cr) sin 0

Using eqs (3.3.3), (3.4.1) and (3.4.3) we obtain the densities (~.~) F~o 1 2 n~ en(r)

(3.4.4a) j=O i-1

T. Paszkiewicz and M. Wilczyhski

286

--

"

"3

a j

Ch. 4

.

(3.4.4b)

j=o i=1 The function Aj(O, r is the focusing (enhancement) factor (cf. Northrop and Wolfe 1985; Maris 1986). The focusing factor is related to the local geometric characteristics of the suitable w-surface at the point of intersection of this surface with the vector k. These local characteristics are the length of the radius vecl~or k, the angle between k and vj and the Gaussian curvature of the jth w-surface Fj (2.4.12)

Aj(0, r = I(;J

I(FJk ) - '

(3.4.5)

On parabolic lines and at flattenning points (e.g., the points of crossing of parabolic lines) of an w-surface the Gaussian curvature vanishes, hence

Fig. 4. The energy density distribution map for GaAs obtained by means of the computer Monte Carlo experiment (Gaficza and Paszkiewicz 1993). Lines (i.e. caustics) crossing the pattern center are related to the fast transverse acoustic phonons. The remaining lines and the "box" structure are produced by the slow transverse acoustic phonons. There are no caustics connected with the longitudinal acoustic phonons.

w

Influence of isotopic and substitutional atoms

287

at such points the densities of energy and quasimomentum are singular. Therefore, in the phonon imaging experiments one observes lines of points at which these densities are very large. In fig. 4 we show the pattern of energy density on the surface of a GaAs crystal perpendicular to a (001) direction, and for ~ = [1/x/~, l/v/2, 0]. It was obtained by means of a computer Monte Carlo experiment (Garicza and Paszkiewicz 1993, 1995). For each direction of propagation scattering events remove phonons from the beam, enlarging the number of phonons which move chaotically. With growing length of the path travelled by the quasiparticles of the beam its intensity diminishes and the intensity of the diffusive component grows. This effect has been observed by Ramsbey et al. (1988), (cf. also Shields et al. 1991; Tamura et al. 1991; Wolfe 1989; Held et al. 1989). This crossover from ballistic to diffusive motion can be studied in detail for an isotropic medium containing substitutional isotope atoms (cf. w4) for which phonon focusing is absent. The presence of anisotropy introduces phonon focusing and makes the problem of chaotization of the phonon beam much more complicated.

4.

Relaxation o f a gas o f LAP's in an isotropic medium

4.1. Spectrum of the collision operator In w2.3 we showed that the collision operator of the BKE is a linear integral operator B which acts on the distribution function f(K;r, t). As an linear integral operator it can be spectrally decomposed. In this section we shall illustrate this procedure and use the decomposed form of the collision integral to derive the diffusion equation for the number density of phonons (Jasiukiewicz and Paszkiewicz 1989). In our discussion we shall exploit two properties of an isotropic medium (Every 1980), namely (i) the phase velocities do not depend on the direction of propagation A

c(K) -= cj

(j = O, 1, 2).

(4.1.1 a)

(ii) one of the polarization vectors is parallel to k and the remaining two vectors form an arbitrarily oriented and mutually perpendicular pair of vectors lying in the plane perpendicular to k, i.e. A

e (k, j = 0) = k,

e (I~, j ) . e (K, j') = ~j,j,

(j, j' = 0, 1,2).

(4.1. l b)

T Paszkiewicz and M. Wilczyhski

288

Ch. 4

A

Since the phase velocity does not depend on k, according to eq. (2.4.10) the group velocity is parallel to k A

A

v(K) = cjk

(j = 0, 1,2).

(4.1.1 c)

In our considerations two fourth rank tensors play an important role, namely the tensor 3" with elements 1 3",~,-r6 = ~ ~,~s

(4.1.2a)

and and the tensor K: /C = Z - ,.7,

(4.1.2b)

where Z is the unit tensor of the fourth rank tensor algebra 1 Z,~,.~6 = ~(6,~,-y~,6 - ~o,,66,,.~).

(4.1.2c)

They bring about the decomposition of unity fl +/C = I,

(4.1.2d)

and make up the multiplication table ,j2 __ ,if-,

K~2 = K:,

f i e = K:3" = 0.

(4.1.2e-h)

For cubic and isotropic media the mean value of the dyad ~(2) is proportional to the unit tensor of second rank (cf. eq. (2.4.24a)). Therefore for these media A

u(K) = 1.

(4.1.3)

Consider the explicit form of the collision operator (2.4.20) acting on an arbitrary function of K, say A(K) A

TBA(K) = - A ( K ) + 3 C ~ ( K ) ( $ ~ ( K ' ) A ( K ' ) ) ~ , ,

w

Influence of isotopic and substitutional atoms

289

which can be rewritten as

TBA(K) - -A(K) + 3E~(K)ff~Z,.~e(g~(K')A(/~'))~, (4.1.4)

Note that the collision integral vanishes for any function which does not depend on K. Such a property has the distribution function of the equilibrium state. The distribution function of complete equilibrium is the Planck function. It depends on the frequency w and presents a special case. Generally, the phonon gas scattered by SIA's relaxes towards an incomplete equilibrium state fin, which depends on the initial condition. Let 7-/be the Hilbert space with the scalar product (4.1.5)

(A1, A2) - ( A I ( R ) A 2 ( R ) ) ~ . Now we introduce two projection operators acting in PuA(K)-

~ul ga~(I~),u

,,

(U = J , K ) , (4.1.6a)

where

,u

J',

,u

=/~.

The fourth rank tensor ~(4) of [V 4] type (cf. Sirotin and Shakolskaya 1979) with components (4.1.7) can be written in the tensorial basis ,,7, E as,

g(4) _ e j J

+

(4.1.8a)

eKK~.

The coefficients e v (U = J, K) are equal to

j - g~

l

- ~,

l( g ~

eK = ~

1

-- ~ g ~

)

=

9

(4.1.8b, c)

290

Ch. 4

T Paszkiewicz and M. Wilczyhski

The generalization of the expansion (4.1.8a) to an arbitrary tensor of the [[V2] 2] type (cf. Sirotin and Shaskolskaya 1979) for the thirty-two crystal classes has been done by Walpole (1984). The operators PJ, PK are real, symmetric and idempotent p2 _ Pu,

(PuA1, A2) - (A1, PuA2),

(4.1.9a, b)

and project onto mutually orthogonal subspaces qt'~J, qr'~K PJPK

-- PJPK

(4.1.9c)

= O.

Introduce also an operator Q projecting onto an orthogonal complement of qt'~J ~) "]~K Q=

Pu

I-

(4.1.6b)

U--J

Definitions (4.1.6) and (4.1.8) lead to the spectral decomposition of B K

B = -T-I(w)Q - E

(4.1.10)

TUI(w)Pu"

U=J

The eigenvalues r~ 1 are given by 7"ul(a) ) = T--I(w)(1

--

3ev).

(4.1.11) A

This means (cf. 4.1.8) that P j projects any variable of K onto the only collision invariant P j A ( K ) = (A(K)}~,

Tj 1 = 0

(or rj = oe),

(4.1.12a)

and _ 1 _ --.3 r g -- 5r

(4.1 .12b)

We see that for isotropic media the spectrum of the relaxation rates O,

-1 rg =

3 5r'

--r

-1

(4.1.13)

w

Influence of isotopic and substitutional atoms

291

is nonpositive; hence, in agreement with (2.3.9) the collision operator B is nonpositive. Note that TQB = - Q .

(4.1.14) ^

Operators Pu (U = J,K) project an arbitrary function A(K) onto the subspace of even functions of K, so the eigenvalue ( - r -1) is highly degenerate. Because the subspace 7-tQ contains all the odd functions of K, then, when the scattering by SIAs dominates, it is only the term -(1/T)Q of the collision integral which makes a contribution to the heat conductivity coefficient (cf. Gurevich 1986). This conclusion is also valid for media of lower symmetry (cf. w5.3). The subspace ~ j also plays a distinguished role for media of lower symmetry. For them ~ K splits into several mutually orthogonal subspaces and the eigenvalue TK1 separates into several different eigenvalues. This question will be discussed in detail in w5. Since the operators Pu (U = J, K) and Q project onto mutually orthogonal subspaces the BKE equation is equivalent to a set of three equations for the components of f (K; r, t) ^

O(Pjf)

^

..~

+ rPjcjk. VBf : 0,

(4.1.15a)

i~(Pgf) ^ + Pgcjk. V-~ Q f = - 5-'3 -~Pgf, ~t

(4.1.15b)

O(Qf____~)+ Qcjk. V Q f + Qcjk. V P j f + Qcjk-VPKf = _ l Q f . (4.1.15c) T ~t For initially spatially homogeneous systems the solution of the BKE can easily be found and we shall write it down in w5.1.2. Let us mention that for a model with three-phonon interaction the spectrum of collision rates was obtained by Claro and Wannier (1971) (cf. also J~ickle 1970; Buot 1972). 4.2. Derivation of the diffusion equation Now we shall demonstrate how the spectral decomposition (4.1.10) simplifies the derivation of the diffusion equation for the number density of

T. Paszkiewicz and M. Wilczyhski

292

Ch. 4

phonons N(r, t) (2.4.23). The diffusion coefficient is measured in experiments on the thermalization of heat pulses. Such measurements were performed by Rogers (1971) (cf. also Ivanov 1991; Sampat and Meissner 1993). Consider a gas of phonons close to the equilibrium state. Such a state is characterized by two characteristic lengths, namely the mean free path l

(4.2.1)

l ~ CD7",

and a length d characterizing the spatial inhomogeneity. We assume that d >> l.

(4.2.2)

The time dependence of the distribution function is marked by a characteristic time Tchar. In the considered situation the ratio d/(CDTchar) ~ 1, SO the ratio of terms on the lhs of the BKE (2.3.2) to terms on its rhs is proportional to a small parameter ~7 = l/d called the Knudsen number (Ferziger and Kaper 1972; Cercignani 1975). Therefore we write

(0I

77 - ~ + v .

Vf

)

=Bf

(77<<1).

(4.2.3)

Also the left hand sides of eqs (4.1.15) are proportional to the small parameter 77. A We shall introduce a new function ~(K;r, t) defined by ^

~

^

9(K;r, t) = -~ P j f ( K ; r , t).

(4.2.4)

Suppose that the functions Q f, P K f and 9 can be expanded into a series in ~7 CO

Qf = ~ l=0

CO

~TtQf(0,

P K f - ~ ~TtPKf(0, /=0

CO

~ - ~

71t~(t).

(4.2.5a-c)

/=0

Consider now the derivative of Q f and PKf. Following Chapman and Enskog (cf. Ferziger and Kaper 1972; Cercignani 1975) we assume that

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Influence of isotopic and substitutional atoms

293

the components P g f and Q f depend on time only via P s f and its space derivatives. Therefore, for example, 0(PKf) = # 0(PKf) 0t 0(P j f) oo

(4.2.6)

O(PKf)

+ E [VCtlVO:2"" "V~n~] O[V~,V,~z...V,~,,(pjf) ]" n=l

Expanding Pgf, Qf and 9 into a power series in r/we obtain

oo 0(PKf)0t = E

rlt+m OmPg

l,m=O

oo

O(Qf) = E 0t

l,m=O

(4.2.7a, b) r/Z+m i3mQf(t) 0t

where Om : ~(m) 0 0t 0(P j f)

(4.2.8)

i~(Pgf)

+ E [V0tlVs "''V~ s=l

O[VOllV~:7:Vozs(pjf)

]"

From (4.1.15a) we obtain

g,(n) = pjljk~ V.(Bf(n)),

(4.2.9)

where lj = cjr is the mean free path for acoustic phonons of polarization j. Equations (4.1.15a-c) are equivalent to two sets of equations for P g f (n) and

Qf(n) PK f(0) _ 0,

~ s=0

Qf(O) = 0,

OpKf(n-s-1) ^ Ot + PKcjk. ~Qf(n-1) = _

(4.2.10a, b)

pKf(n),

(4.2.10C)

T. Paszkiewicz and M. Wilczyhski

294 n-1

OQf(n-s-1) Ot

s=O

Ch. 4

+ Qcjk. ~Qf(n-1) + Qcjk. XTPrf(n-1) (4.2.10d)

+ Qc3k. VPdfSn,1 = _~.-1Qf(n). From eqs (4.2.10a, b) it follows that f(o) _ p j f.

(4.2.11)

This means that #(o) = 0,

0o/0t - 0.

(4.2.12a, b)

From eq. (4.2.10c) it is seen that P g f (1) vanishes P K f (1) = 0.

(4.2.13a)

Similarly, from (4.2.10d) (4.2.13b)

Q f(1) = - Q l j k 9x~pjf; therefore, accounting for (4.1.14) we obtain from (4.2.13b) B f (1) = Qcjk. V P j f , and we can find

~(1) _

~(1)

Pjljk.

VB/(1)= L Pgljk,~ljk'~V,~V~(Pgf) T

(4.2.13c)

-DA(Pjf). Here D is the diffusion coefficient. Since 3(2) - - 0

and

02/Ot - 0

with accuracy to terms proportional diffusion equation O(Pjf) = DA(P.jf) + O(r/3). 0t

to ?73

the Jth component of f obeys the

(4.2.14)

Influence of isotopic and substitutional atoms

w

295

The diffusion coefficient D depends on cD and cj (j = 0, 1,2) D = gl(p)DD,

(4.2.15a)

where DD is the Debyean diffusion coefficient

DD -

1 c~'r,

(4.2.15b)

5

91(39) is one of two functions of p = ct/c z

g n ( P ) - 3(n-3)/3

(pn + 2)

(393 -+- 2)n/3

(n -- 1,4)

(4.2.15c)

and 1 ( 2 1 ) c--~D-- c-~ + c-~ "

(4.2.15d)

Note that the ratio p obeys the inequality 0 < p < x/~/2 (cf. Landau and Lifshits 1987).

4.3. Fourier-Laplace transform of the distribution function (FLTDF) Solving the BKE with a given initial condition we can in principle determine how the distribution function f ( K ; r , t) depends on time. However, it is quite a complicated and generally unsolved problem. Therefore, we shall consider the much simpler question of the time dependence of the Fourier transform of the distribution function (FTDF). In particular we would like to show that it is an easy task to infer from the bTDF the long-time diffusion asymptotics. In w5 we extend the discussion to the case of cubic and transversely isotropic media. For media of lower symmetries this problem will be considered in w6. Because isotope scattering of phonons conserves both the energy and the total number of phonons, the coupled evolution of phonons of a given frequency w and of different polarizations does not interfere with the evolution of phonons of the remaining frequencies, so we can consider them separately. The distribution function of phonons of frequency co is determined by the BKE (2.4.22). It is assumed that the phonon gas fills the whole space R 3 and its distribution function obeys the initial condition A

A

f ( K , r , t = O) = h(K,r),

(4.3.1)

T. Paszkiewicz and M. Wilczyhski

296

Ch. 4

A

where h(K,r) is a given function of the space variable r. Introduce the Fourier-Laplace transform of the distribution function (FLTDF) r

q, z) =

/0

dt exp(-izt)

/:

d3r exp(iq 9r)f(l~,r; t).

(4.3.2)

3

The function r solves an operator equation

[ZT --iv(ff~)'qT + v(K)] r (4.3.3)

= 3E,~(ff~)(s162

+ Th'(I~,q),

A

where h'(K,q) is the Fourier transform (Fr) of the initial distribution of phonons

h'(ff[,q) - / R 3 d3r exp(iq, r)f(/~'; r; 0).

(4.3.4)

Introducing new variables and functions

= ZT, -

r

aT

e;, () = r

(4.3.5a) (4.3.5b)

= ~l'~l,

; -,

,

(4.3.5c)

T

h(K, e~) = h' (h', ~ ) ,

(4.3.5d)

we can transform eq. (4.3.3) into a more convenient form r

~, () = 3C~(K)(C~(~")r

~;, ~)>~,R(K, ~;, () (4.3.6)

A

A

+ Th(K, ro)R(K, to, (), where R is the resolvent

R(K, ~;, () -- [~ - i v ( K )

9~ + L,(K)]-1.

(4.3.7)

w

Influence of isotopic and substitutional atoms

297

Multiplying both sides of (4.3.6) by an element of $(2) (2.4.19) and averaging them over the polarization index and over the direction of k we get an equation for a tensorial function of the second rank 9 with elements A

(4.3.8a) namely

9.~(a, ~) -

S.~(~.

~)~.~(~, ~) +

~-7-t.~(a.

~),

(4.3.9a)

where S~Z~(tr r = (3C, z(K)E.~(K)R(K, a, ())~,

(4.3.8b)

7-/~f~(tr () = <E~(K)h(K, ~)R(K, ~, ()}~.

(4.3.8c)

We shall call the fourth rank tensorial function S(~, () (4.3.8b), the scattering tensor. The product b / o f two fourth rank tensors ]2, W is defined as follows

With the use of the tensor product, eq. (4.3.9a) may be written down in the compact form = S~b + 7-7-t,

(4.3.9b)

which allows us to solve it, viz. = 7-(77- S ) - I ~ .

(4.3.10)

Introducing (4.3.1 0) into (4.3.6) we obtain the explicit form of the FLTDF

r

~, () = 3 7 5 ~ ( K ) R ( K , ~, ()B,~c,,~,(~, () (4.3.11)

• (E~,,,(~')R(g', ~, r

m}~, + *R(g,,~, r

m,

298

T Paszkiewicz and M. Wilczy/tski

Ch. 4

where

: ( Z - S) -1.

(4.3.8d)

Analogous to ~(4) (eq. 4.1.8a) the tensors S and B can also be written as a linear combination of the basic tensors 3" and K;. The suitable coefficients are l

2

Sj(/~, r -- ~ ' ~ ,.),;4(icDl~[)-1 Q0(Ay), j=o

(4.3.12a)

2 sK(~, r - ~ s'j(~, r

(4.3.12b)

and B j ( ~ , r - [1-Sj(~, O] -~,

B K ( ~ , r = [1--SK(~, r

-~, (4.3.13a, b)

where 1 -t- r ~Xy- icjl~l'

c--Z ~/J- CD

(j = 0, 1, 2)

(4.3.14a, b)

and Qo(z) is the associated Legendre function of the second kind (cf. Abramowitz and Stegun 1970)

1/ 1 d# (z -

Oo(z)-

-~

1

1

#)- 1 = ~ In

- 1

= Arcth z.

(4.3.15)

This function is single valued and regular in the plane of complex ~ with a cut along the interval (-1 -iczl~ I, -1 + iczl~l). The points in this interval form a singular continuum Sc(~) (cf. w6.1). Consider the poles of the functions B u (U = J, K). They are located at the points flu resulting from the equations l

2

j=O

1 +~j] l(icDl~;l)-lQ0 icily] - 0,

(4.3.16a)

w

Influence of isotopic and substitutional atoms 2

l(icDItCl)-lQ0 j=0

1 +(K

= O.

icjl~l

299 (4.3.16b)

These poles form a set Sp(~;) (cf. w6.2). For small values of I~lc D (i.e. for large values of I~jl) ~-j

2 " --to 2 CD91 (19)

3 5

(K -~

or

or

Zg ~--

zj

_~

- D q 2,

3 5T"

(4.3 . 17a)

(4.3.17b)

One may check that the series for (a corresponds to the Chapmann-Enskog expansion (cf. Jasiukiewicz and Paszkiewicz 1989; Hauge 1974). If we describe a nonequilibrium state of the phonon gas with the set of Fourier harmonics (labelled by q) the characteristic length d is Aq = 2~r/Iql, so the Knudsen number r/is

l

r / - ~qq. The inequality I~lc D ~ 1 is equivalent to l << )~q,

(4.3.18)

where the mean free path l is l = CDT(co).

(4.3.19)

From the inequality (4.3.18) it follows that a phonon of frequency co passing a distance 1 expriences many collisions, so a local equilibrium state may be established. As it follows from w4.2 this local equilibrium state goes towards an incomplete equilibrium state via slow diffusion flows. When the inequality opposite to (4.3.18) holds

one collision occurs on the average over a distance of many wave-lengths. So, one speaks about the collisionless (kinetic) regime. For ~ = 0, i.e. for spatially homogeneous states, the positions of singularities (4.3.17) coincide with the eigenvalues (4.1.12) and the set Sc shrinks

T Paszkiewicz and M. Wilczyhski

300

Ch. 4

to a point ~ = - 1 (z -- - 1 IT). The general case of a spatially homogeneous state of a low-symmetry elastic medium is discussed in w6. With growing I~1, the poles {j and r move along the real axis towards the cut, approach it and vanish. The critical values of I~l, ~u) (u = J, K), such that for I~l >/,~(cU) the Uth pole vanishes, are obtained from eqs (4.3.16). Since

r

= lim Q ( - i e ) = 7ri lim Q 1 + r il~lc D ~-+o+ 2'

we get 71" ~J)(Jg) -- 2 ~ D g4(P),

/,~K)(.p) = ~

71"

(4.3.20a, b)

g4(P).

4.4. Explicit dependence of the Fourier transform of the DF on time

Introduce the deviation function from the state of incomplete equilibrium fin ^

A

(4.4.1)

5f(K;r, t) = f ( K ; r , t) - fin.

The inverse Laplace transform of r gives the time dependent Fourier transform of the distribution function (FTDF). To calculate it one should consider an integral along an axis parallel to the imaginary axis and crossing the real axis at an arbitrary point e > 0. All the singularities of r should be located to the left of the integration axis. Thus

5f""(K; ^ ~, t) -

1 ]f ~ + i ~ d~ exp (~t/T) r 27tit ~,e-ic~

~r ~)

(e > 0+). (4.4.2)

,..,,,

The integration contour defining 5f can be transformed to a set of contours encircling anticlockwise all of the singularities of r (fig. 5). Therefore, we should calculate an integral around the cut and around two (l~l < ~;(cg)), one (g~K) < [Nl < /'~(J)) or no (l~l > ~(J)) poles. Calculating them we get

R; +

1,1, t)

~ U=J,K

- fin(W) + exp

exp [@(p,

(--t/T)

q~c(W, K'; q,

I, 1, t)

I~l)t/~-] ~v(~o,g;~, I~l,t)O(~ v)- I~l).

(4.4.3)

Influence of isotopic and substitutional atoms

w

ic I

cj

c~ .1 /

icl~

cj

CK ,,,....-7

.)

301

Re ~

Re

-11

-iet~t~

.)

-ictx~t

-icl~l.t

) ...........

-i c ! ~t,tt

-ic I

5

-icier

Cc

Fig. 5. The contours of integration for the inverse Laplace transform of r a) For 0 < I~1 < m~cK)(P)it consists of C j, CK and Cc. For ~;~K)(p) < I~1 < m~:)(p) the contour CK should be missed. For I~1 > m~cd)(p) the contour of integration consists of Cc only. b) For I~;I ~ ,~K)(p) the pole at ffK(t~, p) lies very close to the cut. The same happens to the pole at ffj(~, p) for levi -~ ~cJ). The poles at ff = - 1 + il~lcjtz for (j = l, t) and # = k . ~ lie on the cut.

The exponential functions in eq. (4.4.3) are universal, i.e. they do not depend on the initial state. The preexponential functions r Cj, CK are non universal. These functions do not grow faster than the suitable exponentials. For a given initial condition the Fourier transform of the distribution function is a superposition of terms (4.4.3) with different vectors ~ and frequencies co. With passing time the terms with large I,~1 die out exponentially. This decay is characterized essentially by T(W) (cf. eq. (2.4.18)). After a long lapse of time only terms with CDl,~ I much smaller than unity survive and

5f ~ exp(-Dq2t). This means that for t >> T(W) the deviation function essentially obeys the diffusion equation (4.2.14). 4.5. Important conclusions In the simple case of isotropic media we can study the complete evolution of an initial state towards an incomplete equilibrium state (we do not consider

T. Paszkiewicz and M. Wilczyhski

302

Ch. 4

here the so-called initial slip). In particular we can study the crossover from the collisionless to the collision dominated regime. In the state of incomplete equilibrium the gas is spatially homogeneous and the distribution of wave vectors is isotropic. Due to polarization conversion processes the gas is a mixture of transverse and longitudinal phonons. Only collisions with thermalized quasiparticles and walls being in contact with a thermal bath lead to a state of true thermodynamic equilibrium characterized by the temperature. We checked that for ~ = 0 the set of singularities of the FLT of the distribution function coincide with the spectrum of the collision integral. For isotropic media we are able to fulfil the complete programme of studies of the kinetics of the phonon gas. Unfortunately, for anisotropic media we are unable to implement such a general scheme. For cubic media we can only derive the diffusion equation either via Chapman-Enskog scheme (cf. w5.1.2) or from the long-time asymptotics of the FLTDE For media of lower symmetry than cubic we have to use only the latter method.

5. Spectral decomposition of the collision operator 5.1. Cubic media

Now we shall show that the spectrum of the collision operator of a cubic medium resembles the spectrum of an isotropic m e d i u m - it contains several discrete eigenvalues (Paszkiewicz and Wilczyfiski 1992); however, it is not a general property of the collision operator. In the next subsection we shall show that for media of lower symmetry than cubic the spectrum of the collision operator usually contains also a continuous part (Paszkiewicz and Wilczyfiski 1990a, b).

5.1.1. Spectrum of the collision operator for a cubic medium Previously, (cf. w4.1), we noted that the eigenvalue 0 is related to the tensor 3" (4.1.2a). This eigenvalue is related to a collision invariant, an important and indispensable characteristic of the collision operator. This means that the remaining eigenvalues have to be related to the decomposition of the tensor E (4.1.2b). For cubic media (Walpole 1984) K: = s + M .

(5.1.1)

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Influence of isotopic and substitutional atoms

303

The building blocks of the fourth rank tensors 12 and 3,'l are the components of the unit vectors of the four-fold axes of a cubic crystalline structure ~,b and~

c-

1[

(~|174

+ (g|174174

(g|174

+ (~|174174

(~|174

x~ - ~ ( ~ | 1 7 4 1 7 4

[

.Ad-XM:~

A

A

(5.1.2a)

A

~..

(~@~-b@b)|174 + (b|174174

(b | 1 7 4

+ (~|174174

(~|174

(5.1.2b)

Walpole (Walpole 1984) established the multiplication table X 2 - A'v

(U - J, L, M),

Xv Xv - xv xu -o

( u r v ),

(5.1.3a) (5.1.3b)

and the decomposition of the fourth-rank unit tensor (4.1.2c) Z ~'

2 -

Xcr.

(5.1.3c)

U=J,L,M

An arbitrary fourth-rank tensor C of [[V2] 2] type (Sirotin and Shaskolskaya 1979) can be written as a linear combination of Xu (U - J, L, M )

C- ~

CuXU,

(5.1.4a)

u where

1

cj - -j C ~ Z ~ ,

1

cL - ~ s

1 C M -- -~ ./~c~13~/6Cc~13.y6.

(5.1.4b, c) (5.1.4d)

T. Paszkiewicz and M. Wilczyhski

304

Ch. 4

In the basis {if,/3, .M} the tensor g(4) (4.1.7) is decomposed as

g(4) _ ~ EU,S~U" U

(5.1.5)

The components e U (U = J, L, M) can be expressed in terms of a characteristic polynomial of direction cosines of the fourth degree (Paszkiewicz and Wilczyfiski 1992) P4(~)(K)= ~ e4(K), i=a,b,c

(5.1.6)

where, for example, ea(/Y) = e(K). ~'. We have 1 [ 1 - (P4(~'(~'))R]

EL - - ~

'

(5 1.7a)

and 1

[
(5.1.7b)

so the coefficients e L, e M are not independent. This is in agreement with a general rule. Because any permutation of indices does not change ~(4) it belongs to the class [V 4] (Sirotin and Shaskolskaya 1979) and has at most two independent components. Besides the operator P j (4.1.6a) we introduce three new operators PL, PM and Q. They are defined as PuA(K) - eu' E,~(ff[)x(~u~(C'Y~(ff[')A(ff['))R, (5.1.8a)

(U- J,L,M), and

Q-I-

Z U=J,L,M

Pu.

(5.1.8b)

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Influence o f isotopic and substitutional atoms

305

Pu (U = J, L, M) and Q are four idempotent operators, i.e. P2u=Pu '

Q2=Q,

(5.1.9)

that are mutually orthogonal P u P v - P v P u = QPu = P u Q = 0

(u, v - J, L, M).

(5.1.10a-d)

In the Hilbert space 7-/(7-/= 7/j ~]~~L ~ "]'~M @ 7"~Q) with the scalar product (4.1.5) these operators are real, symmetric and idempotent. Now we can write the spectral decomposition of the collision operator B B = -

E

T~Ipu-T-1Q,

(5.1.11)

U=J,L,M

where 7"{ 1 = r - ' ( 1 - 3eu).

(5.1.12)

Consequently Tj I = 0,

TL I = 7"-I ~

(5.1.13a-c)

3

and the eigenvalues TE l, r~r 1 obey the simple relation 1

3

rM

2TL

3 = ~.

(5.1.14)

2T

The collision rates r i 1, r ~ 1 are nonnegative, hence, the collision operator B is nonpositive, as it should be (cf. eq. (2.3.9b)). 5.1.2. Relaxation o f an initial state

Let us assume that the initial state of the phonon gas is spatially homogeneous, so that f ( K , t = 0) is a given function of K, which we denote by h ( K ) , i.e. A

A

f ( K , t - O) = h(K).

T. Paszkiewicz and M. Wilczyhski

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Ch. 4

At an arbitrary instant of time t # 0, the solution of the BKE A

af(K,t) = _ E~-~Ipuf(K,t)igt u

~--1Qf(K, t),

(5.1.15)

obeying the assumed initial condition, can be written as

f(K, t) - E exp (--t/Tu) Pub(K) + exp (--t/T) Qh(K),

(5.1.16)

u

where for an isotropic medium U - J, K, and for a cubic medium U --

J,L,M. For a cubic medium the components Puh(K) are quadratic forms of the polarization vectors the matrices of which depend on the initial state of the phonon gas (cf. Paszkiewicz and Wilczyfiski 1992). The number density of phonons at a given frequency w, N(w,r, t) (cf. (2.4.23)) is a collision invariant, i.e. BN(w,r,t)= 0. Therefore, Pjh(/~) = ( h ( K ) ) ~ = N(w), and the solution (5.1.16) automatically satisfies the conservation law

(f(K, t))~ = Pjf(K, t) = ( h ( R ) ) ~ = N(w).

(5.1.17)

The eigenvalues TE1,T~ 1 depend on the elastic constants Cll, 612, C44. This dependency has been studied by Paszkiewicz and Wilczyfiski (1992). The reader will find there a table summarizing the main elastic and scattering properties for a large number of compounds of cubic symmetry.

5.1.3. The diffusion coefficient The BKE (2.4.22) is equivalent to a set of four equations for the components Pvf(K;r,t), ( U - J,L,M) and Qf(K;r,t) A

OPjf(K;r,t) ~t 77

[ i~Puf i)t

+ ( V P j ) . vQf - 0,

-,

+ ( V P u ) . vQf

]

- -ru1puf

(5.1.18a)

(U = L, M),

(5.1.18b)

w

Influence of isotopic and substitutional atoms i3Qf ~ ot + ( V Q ) ' v Q f +

Z

~ ] (VQ).vPvf =-'r-lQf,

307 (5.1.18c)

U=J,L,M

where r/is the small parameter introduced in w4.2. Expanding Puf (U = L,M), Qf and ~b = OPjf/at into a power series in the Knudsen number 77, to the accuracy of terms proportional to r/3 we obtain the diffusion equation (4.2.14) with the diffusion coefficient equal to (Paszkiewicz and Wilczyfiski 1992)

D :

1

(5.1 19)

5

The coefficient r (2.4.18) depends on the experimental conditions, but the last factor of eq. (5.1.19) is universal. The values of it, for different cubic materials, are collected in a table enclosed in our paper (cf. Paszkiewicz and Wilczyfiski 1992). The general expression for the diffusion matrix valid for media of lower symmetries than cubic will be derived in w6.4.

5.2. Spectrum of the collision integral for transversely isotropic media To illustrate the general structure of the collision integral we consider

a transversely isotropic medium. As we already noted (cf. w2.4.4) for a Cartesian coordinate system with the z-axis parallel to the only symmetry axis ~ the t e n s o r (s is diagonal and has two different components (cf. (2.4.24c). Hence, for a transversely isotropic medium the nonintegral term of the collision integral r-1 u(K)f(K; r, t),

(5.2.1)

according to (2.4.25), is equal to u(K) = { 1 - [e(K). ~.]2} { 1 - ([e(K"). ~'] 2) ~, } (5.2.2) + [e(K) 9~'] 2 ([e(K") 9~'] 2) if, A

so it generally depends on K. The latter is valid for all intermediate and low symmetry media. We shall show that this property of v makes the spectrum

T. Paszkiewicz and M. Wilczyhski

308

Ch. 4

of the collision operator quite complicated as, in addition to a discrete part, it contains a continuous component, too. For transversely isotropic media the basis of all symmetric fourth rank tensors consists of six elements, i.e. g2(4), g~4), g(4) T., and G. All of these tensors are made up of two second rank tensors p(2), q(2) (cf. Walpole 1984)

"1"r

q(2) -- 1(2) _ ~- | ~-,

p(2) _ ~- | C_

where i(2) = ~c~,#.

The algebraic properties of the basic fourth-rank tensors follow from the multiplication table (table 1). The identity tensor 2- (4.1.2c) and the tensor K: (4.1.2b) can be decomposed as 2 - = 8 1 - + - ~2 + .fi" + ~,

(5.2.3a)

K~ = D + .T + G,

(5.2.3b)

where 2

v=

1

v~

(5.2.3c)

and (5.2.3d)

8, = (E3 + E4).

Table 1 Multiplication table for basic fourth-rank tensors for transversely isotropic media.

81 E2

81 0

0 E2

E3 0

0 E4

0 0

0 0

E3

o

E3

o

E~

o

o

~'4

E4

0

E2

0

0

0

.T G

0 o

0 o

0 o

0 o

.T o

0

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Influence of isotopic and substitutional atoms

309

Using table 1 we can check that

(u, v = j , z), y, 6),

X u X v = X v X u = X v ~u, v

(5.2.4)

where 2"a denotes 3", YD denotes 79, and so on. A general transversely isotropic tensor belonging to the class [[V2] 2] is constructed by a linear combination

.,4 = als

-k- a2E2

-k- a~,f,~ + a F f

+ acG,

(5.2.5)

where 1 a2 -- -~ q~zA~,y6q,,r6,

a l = Po,~.Ao,~.r6P..r6,

1

1 a C - ~9,~,y6A~76.

1 as = ~ po~.Ao~.-r6q,.r6, x/z,

(5.2.6a-e)

With the help of eqs (5.2.6) we can find the components eD, eF, e G of the tensor ~(4) (4.1.7) 2 1 x/~ eD = ~ el + ~ e2 -- - ~ e,, 1

1

/,

(5.2.7a) r .;'~..~] 2

e c - ( [e(K)c]2{ 1 - [e(/Y)~] 2 } } if,

2

\

(5.2.7b-d) (5.2.7f)

where

el -- ([e(K)c'] 4}ft.

(5.2.7g)

The projection operators Pu (U - 3", 79, .T', G) are defined analogously to the case of isotropic (cf. (4.1.6a)) and cubic media (cf. (5.1.8a)) P u A ( K ) - e~' s163

(5.2.8)

In the linear space 7-/with the scalar product (4.1.5), the operators Pu are symmetric, idempotent and project onto mutually orthogonal subspaces ~ v

310

T. Paszkiewicz and M. Wilczyhski

Ch. 4

(U = D, F, G, J) of 7-/. Adding and subtracting the term (-1 ~r f) we write the collision operator B (2.4.20) in the final form

B f(ff~;r, t) - --T -1 [1 -- u(ff~)] f(ff~;r, t) - v-lQf(ff~;r, t)

-

~

Tu1puf(K;r,t) 9

(5.2.9)

U--D,F,G,J

As previously (cf. (4.1.11) and (5.1.12)), the collision rates are r u 1 ~ r-1 (1 - 3eu)

(U - D, F, G, J),

(5.2.10)

and they are nonnegative; thus, the collision operator B is nonpositive. The spectrum of the collision operator for transversely isotropic media was previously studied (Paszkiewicz and Wilczyfiski 1990b). Numerically calculated characteristics of this spectrum for a number of compounds can be found there. The continuous part of the spectrum, related to the multiplication operator, that multiplies a function of K by the coefficient A

A

[1 - u(K)]

(5.2.11) A

disappears for u(K) - 1, i.e. for cubic and isotropic media. Eigenfunctions corresponding to the continuous part of the spectrum are generalized functions (cf. Case and Zweifel 1967; Ershov, Shikhov 1985), which are nonorthogonal to the eigenfunctions related to the discrete part of spectrum of B. This precludes the use of the previously applied method of derivation of the diffusion equation. This is why we shall derive it from the long-time asymptotics of FLTDF (cf. w6). 5.3. Relation of the spectral decomposition of the collision operator to heat conduction

We already noted that the operators Pv defined by expressions (4.1.6a), (5.1.8a) and (5.2.8) project any function of K onto the subspace of even functions of K. The coefficient (5.2.11) is also an even function of K (cf. (2.4.21)). The tensor of heat conductivity coefficients, "~, is related to the matrix elements of CR - the resistive (quasimomentum nonconserving) A

~.

w

Influence of isotopic and substitutional atoms

311

part of the collision integral C[f] of the BKE calculated between two odd functions of K (cf. Gurevich 1986; Beck 1975). So, for example, in the temperature region, where the part of CR related to B dominates, only the term ( - Q / T ) of (5.2.9), contributes to the heat conductivity coefficient ~. Consider an example: when isotope scattering is the dominant resistive mechanism A

~ar ~ Ts2(A-1)aB,

(5.3.1)

where s is the density of entropy of the phonon gas (2.3.6a) and

Ac~ = ~

j=O

d3k hkaB{fo(w(K)){ 1 + fo(w(K))}hk~}. (270 3

Hence, the tensor A reduces to the familiar form (Gurevich 1986)

dak

1 + fo(oJ(K))} hk~.

(5.3.2)

j=O Generally one should consider the matrix elements of the inverse of the linearized collision integral C~-1 containing the contribution of all the important resistive processes, e.g., phonon-phonon and phonon-isotope scattering processes. Thus the linear operator corresponding to such a CR does not have definite parity and a part of B related to the operators Pu can give rise to heat conduction.

6.

Time dependence of the Fourier transform of the DF for media of arbitrary symmetry

In w4 we discussed the properties of the FLTDF of an isotropic medium. Now we shall broaden this discussion to include a medium of arbitrary symmetry.

6.1. Singularities of the FLTDF In order to determine the time evolution of the amplitude of a Fourier harmonic tr (cf. eq. (4.3.5b)) of the DF of phonons with a certain K we

312

T. Paszkiewicz and M. Wilczyhski

Ch. 4

have to consider elements of the tensor ~ (4.3.8a) as functions of the complex variable ~ and to analyze their singularities. Because the tensorial functions S and 7/ and (4.3.8b, c) are defined by Cauchy-type integrals, they are holomorphic for all values of ~ outside the set A

A

Sc(~) = {~c: ~c = - u ( K ) + iv(K). ~

A

for any K}

(6.1.1)

and singular on it. Since the set Sc usually constitutes a continuum it will be called a singular continuum. Nevertheless, if for all values of K the condition u(K) = 1 is satisfied Sc(~ = 0) shrinks to a single point. The set Sc can constitute one compact set or can be composed of at most three compact subsets corresponding to each of the three polarizations. It is symmetric with respect to the real axis (cf. 2.3.1 and 2.4.4). For fixed K and ~, the only singular point of R is a pole at A

A

A

(6.1.2)

~R = - v ( K ) + iv(K). ~.

Since ~R ~ Sc we will not consider this pole separately. The only singularities of ~b (4.3.6) lying outside the singular continuum are the singularities of the tensorial function B (4.3.8d). Because a fourth rank tensorial function ,4 is singular iff in any chosen Cartesian coordinate system its matrix representation M(.A) (cf. Walpole 1984)

M(A) -

Al111 Al122 Al133 ~/'2.A1123 A2211 A2222 A2233 ~/~A2223 'A'3311 "A3322 r x/'2"A3323 v/-~.m2311 v/-~.A2322 v/-~.m2333 2.,4.2323 x/2,m1311 v/2.A1322 v/2.A1333 2,A1323 ~/2.A1211 vf2A1222 v@A1233 2.A1223

Vr2Al113 v/2A2213 V/2'A3313 2.,42313 2.,4.1313 2.,41213

v/-2A1112 V~A2212 x/~'A3312 (6.1.3) 2,,42312 2,,41312 2A1212

is singular, we shall examine the singularities of the matrix M(B). Let P(~, if) denote the matrix representation (6.1.3) of the tensorial function Z - S(~, if). According to Walpole (1984) P - M(Z) - M(S) = I - M(S),

(6.1.4)

where I is the unit matrix. Because all of the elements of M(S) are holomorphic in the plane of complex ff outside the singular continuum Sc(~),

w

Influence of isotopic and substitutional atoms

313

P is regular in that area, i.e. all of its elements are holomorphic. Since P = M(B-1) (cf. (4.3.8d)) we get M(B) = P-1,

(6.1.5)

M(B) - detP P'

(6.1.6)

or

where P is the matrix adjoint to P. From eq. (6.1.6) it follows that the matrix elements of M(B) are meromorphic outside the singular continuum So, and their only singularities in the ~-plane cut along Se are poles corresponding to the ~ values such that det P(~, ~) - 0.

(6.1.7)

Summing up, the set of singular points ~ of the FLTDF consists, for a given ~, of two parts" (i) a singular continuum S~(~), (ii) a countable set of poles Sp(~) Sp(~) = {~(~): d e t P ( ~ , ~ 0 r

(6.1.8)

h

To find f ( K , qr, t) - the time dependent Fourier harmonics of the D F one should calculate the inverse Laplace transform of r ~, if) (cf. (4.4.2)). All singularities of r ~, if) lie in the nonpositive half-plane (Re ff <~ 0). Analogous to what is shown in fig. 5 we can deform the integration contour defining the integral (4.4.2) to a set of contours enclosing anticlockwise all singularities of r if). Calculating the suitable contributions and assuming that we deal with s poles ffi(~), each with multiplicity c~i (i = 1,2 . . . . , s), we obtain A

j-1

f ( K , q, t) -

~ Cij(ff~, q) ~=1 j=l A

+ C(K, q, t),

exp (6.1.9)

T. Paszkiewicz and M. Wilczyhski

314

Ch. 4

where

^ C~j(K,q) = 3 g ~ ( K ) A

lim (a - j)! r162

• [((

-

ffi)~'

dO._j dr

(6.1.10)

R(K, qT, ()B,~.y6(qT, ()7-l.y6(qT, ()]

and

C(K, q, t) = ~-~

d( exp

qT, r

r

(6.1.11a)

In eq. (6.1.11a) the contour of integration C encloses anticlockwise the entire singular continuum Sc. The function C can be written in the form C(K, q, t) = exp

- ~

fc(K, q, t),

(6.1.1 lb)

where the modulus of fc increases in time slower than any exponential function exp(6t) (for any ~ > 0), A

Vm :

min v(K) = - max Re (

(

6

.

1

.

1

2

)

and T O I = Um.

(6.1.13)

T

Since all singular points ~ of the FLTDF satisfy the condition Re ((~;) <~ 0 there are no Fourier harmonics growing in time. Equation (6.1.1 lb) shows that the obligatory presence of the singular continuum leads to a nonexponential relaxation of the initial state. Besides, according to eq. (6.1.9) the poles with multiplicity greater then unity provide preexponential factors in the form of powers of (t/T). 6.2. Poles of the FLTDF

Equation (6.1.9) indicates that the processes and spatial structures the scattering rate of which is longer than r 0 eq. (6.1.13) are related to the poles of the FLTDF satisfying --Um <

Re (i,

1 <~ i ~< s.

(6.2.1)

w

Influence of isotopic and substitutional atoms

315

According to eq. (6.1.9), the pole contribution to the FTDF decays in time either purely exponentially or as an exponential function multiplied by a polynomial, the degree of which depends on the pole order. Because of eq. (6.1.6) the order of the pole ff~ is not greater than the order of the zero of detP(ff) at ff = ffi, it does not exceed the index of the lowest nonvanishing coefficient of the Taylor series det P(r

= al(r

-

r

+ a2(r

-

r

(6.2.2)

2 -+- . . . .

Consequently, for all poles r for which the polynomial p(r = det [P((:i) + P'(r162 - r

(6.2.3)

has a linear term the corresponding contribution to eq. (6.1.9) decays purely exponentially. In eq. (6.2.3), P'(ffi) is the derivative of M(ff) calculated at -ffi. In all the above formulae the parameter ~ = qT has been dropped. The explicit expression for the matrix P'(ff) can be easily found, when taking into account that P' = [ 2 - M(S)]' = - M ' ( S ) ,

(6.2.4)

where M(S) and M'(S) are the matrix representations (6.1.3) of the tensor S and its derivative S t. The derivatives of the matrix elements of S are related to the derivative of the resolvent R ' ( ~ ' , m, ( ) -- - [ (

- iv(K)

9tr + v ( ~ ' ) ] - 2

= _ R 2 ( ~ . ' m, ( )

(6.2.5)

with respect to r Taking into account eq. (4.3.8b) and eq. (6.2.5) we get

s',~(,~, r - -(3E~(R)E~(R)R2(~, ,~, r

(6.2.6)

The above formula and the procedure for checking the order of the zero of the detM(r can be easily extended beyond the first order. The calculation of the pole contributions to the F r D F (6.1.10) is very laborious and difficult to perform to a sufficient level of numerical accuracy. However, for the case of poles of the first order the formula (6.1.9) simplifies to

Cr(K, q) = ~

al(r

R(ff;, qr, r

r

r

(6.2.7)

316

T. Paszkiewicz and M. Wilczyhski

Ch. 4

where e and h are 6-dimensional vector representations of the second-rank tensors s (2.4.19) and 7-/ (4.3.8c) respectively (cf. Walpole 1984), for example (3 = (~11, ~22, ~33, V/2~23, V/2~13, V/2~12) 9

(6.2.8)

P(qr,(i) is the matrix adjunct to P(qr,(i), (6.1.6) and finally al((i) is the coefficient of the linear term of the Taylor series for detM(O in the vicinity of (i. In eq. (6.2.7) we used the summation convention over the repeated latin indices which take the values 1 , 2 . . . 6.

6.3. Symmetry properties of the scattering tensor S(~, () The relations A

s

A

= C~(-K), A

(6.3.1)

A

R ( K , ~, () = R ( - K , -n,, (),

(6.3.2)

satisfied for any K, ~ and ( (cf. eqs (2.4.19) and (4.3.7)) allow us to write (6.3.3) Because in eq. (6.3.3) the average over K can be replaced by the average over ( - K ) we obtain the relation A

(6.3.4) Similarly we get A

A

R* (K, ~, () = R ( K , - ~ , (*),

(6.3.5)

sS~,~e(,~, r = s~:~e(-,~, r

(6.3.6)

SO

From eq. (6.3.4) and eq. (6.3.6) it follows that

s : ~ ( , ~ , ~) = S ~ ( , ~ ,

r

(6.3.7)

w

Influence of isotopic and substitutional atoms

317

It is obvious that eqs (6.3.4), (6.3.6) and (6.3.7) remain true if we replace the tensor S with ( Z - S), which implies that for the matrix P = M ( Z - S) the following statement is valid det P(~, ~) = 0

iff

det P(~, r

= 0.

(6.3.8)

The latter is equivalent to ff(~) 6 Sp(~)

iff

ff*(~) 6 Sp(~).

(6.3.9)

Relation (6.3.9) means that the poles of the FLTDF lie symmetrically with respect to the real axis of the complex plane, and we can confine ourselves to the half plane Im ~/> 0. Consider an element f of the crystal symmetry point group Fo. Let O(f) represent f in a three dimensional vector space and O(f) be the fourfold tensor product of O(f)

O(f) -- O(f) | O(f) | O(f) | O(f).

(6.3.10a)

For a given Cartesian coordinate system O(f) and O(f) can be identified with an orthogonal matrix O ~ ( f ) and with the eighth-rank matrix being the product of the components of O

O~,~,.r.r,66,(f ) = O~w(f)O~,(f)O.r.r,(f)O66,(f ),

(6.3. lOb)

respectively. According to well-known transformation rules, the Cartesian components of the transformed vector v and fourth-rank tensor S are

O(f)v~ = O(f)~.rv. Y,

(6.3.11 a)

[O(f)S]c~.r6 = Oaa,~,-r-r,66,(f)Sc~,~,-r,6,.

(6.3.1 lb)

Henceforth we will omit the symbol of the element f. Let us note, that if O is a matrix representing an element of the crystal symmetry group and e(K) is a polarization vector of a phonon K the following relations apply A

A

A

u ( O K ) - u(K),

(6.3.12a)

O r ( K ) - v(OK),

(6.3.12b)

318

T. Paszkiewicz and M. Wilczyhski Oe(K) - e(OK), A

A

Ch. 4 (6.3.12c)

A

A

where O K ( K = (k, j)) stands for the pair (Ok, j). As the transformation (6.3.11 a) conserves the scalar product and because of eqs (6.3.12a, b) we get, for any K , ~ and (, A

A

A

(6.3.13)

R ( K , ~, ~) = R ( O K , Otr ~). A

On the other hand, for any K, eq. (6.3.12c) implies

Og(4)(K) = ~(4)(OR),

(6.3.14)

where

s

) = g(2)(~.)|

s

(6.3.15)

Taking into account eqs (6.3.12b), (6.3.13) and (6.3.4) we derive the transformation rule for S(~, r

OS(~, () = (3g(4)(oR)R(R,~, ())~ : (3E(4)(OK)R(OA ",O~,

())~.

(6.3.16)

Because any change of the integration (averaging) variables related to an orthogonal transformation does not change the result of the integration we can replace the variable K by the variable K' = OK. Hence,

os(~, () : 3(g(4)(R')R(R ',0~, r

- s(o~, ().

(6.3.17)

Taking ~ = 0 we immediately get from eq. (6.3.17) os(,~

: o, ~) : s ( , r - o, r

(6.3.18)

Equation (6.3.18) means, that for all values of ~ and for ~ = 0 the tensor S has the complete symmetry properties of the considered crystal. We already used this important feature in w167 4.1, 5.1 and 5.2.

w

Influence of isotopic and substitutional atoms

319

6.4. Asymptotic diffusive behaviour of the phonon gas

Apart from very special cases, such as cubic media with two collision invariants (cf. Paszkiewicz and Wilczyfiski 1992), for almost all common materials the only set Sp(~) (6.1.8) which contains the pole r = 0, as a zero of the first order of the det P(~ = 0, r (cf. eq. (6.1.4)), is the set Sp(~ = 0). This means that the phonon gas tends steadily with time to a homogeneous and isotropic state with a uniform distribution of phonons in K and r. Although each Sp(~) may consist of several poles, for the long time asymptotics only the pole with the maximal real value should be considered. This is because the contribution to the DF originating from the other singular points decays in time much faster. Let Cj(e;) denote the element of Sp(N) with the greatest real component. Obviously A

Cj(0) = 0

(6.4.1)

and for any ~, Cj(~) satisfies the condition (6.1.7). Because a zero of the first order of a parametrized function is stable, it may move, but it neither vanishes nor splits up when the parameters are infinitesimally changed. So, in the vicinity of the point (~ = 0, Cj = 0), the equation det P(e~, Cj) = 0 unambiguously defines an implicit function, which coincides with (j(/'i;). From the above it follows that describing the evolution of the gas in a state close to equilibrium we can confine ourselves to the Fourier harmonics with small I~1. In other words, only the elements of Sp(~) for which ~ ~ 0 determine the long-time asymptotic behaviour of the phonon gas. Let us now calculate a few coefficients of the Taylor series for ( j ( n ) at the point n - 0. For this purpose let us fix the direction ~ = of the vector n and introduce a new function of the scalar variable r = ]~], r162 = r162

(6.4.2)

The function r162 is an implicit function determined by the equation F(r

Cy) - det P(r

Ca) = 0.

(6.4.3)

From a familiar theorem of mathematical analysis we know that the first derivative of r162 at a point r r is equal to

i~F(r r y) /ar (J(r

- i~F(r

(y)/a(g'

(6.4.4)

T Paszkiewicz and M. Wilczyhski

320

Ch. 4

whereas the partial derivatives of (6.4.4) are calculated at the point (r satisfying eq. (6.4.3). Let us recall that the derivative of a determinant F(x) all(X)

F(x) = det

anl(X))

""

. . . . . . al.(z)

n

F'(x) = __

k=l

(j)

...

(6.4.5) a..(z)

aik(z)

det (all(X) "'" anl(Z) "'' ...

""

aln(~))

oo,

(6.4.6)

ooo

alnk(X) ""

ann(X)

where a~j(x) is the derivative of the aij(x) element. Hence,

aF(r162 S~

1 -

6 =

....

S~

"

E det k--1

".

- S~ .

-S~I

. . . . .

.

.

. . . .

.

.

S~

S~

.

"

.

S~ .

.... . . . . .

.

...

S~

,

(6.4.7)

.

1 -

S~

where Si~ (i, j = 1. . . . ,6) are the components of the matrix representation of the fourth-rank tensor S0r ~) (4.3.8b) calculated for (~ = 0, ( = 0) s~o - 3(Eij(~')u(/~) -1 )ff

(6.4.8)

and S~~ are the partial derivatives of these elements with respect to the variable r calculated at the point (r = 0, ( = 0) O~

S~ - 08~(r

r162

)

__

3i~. (v(K)E~j(.K)v(K')-2)~.

(6.4.9a) A

Because the term being averaged in eq. (6.4.9a) is an odd function of K, all components of the average vanish. Consequently we get S'~j = 0 0t~

(i,j = 1 , . . . , 6 ) .

(6.4.9b)

w

Influence of isotopic and substitutional atoms

321

A

Above, E~(K) is an element of the matrix representation of the fourth rank tensor (6.3.15). From eqs (6.4.4), (6.4.7) and (6.4.9b) for any direction vector q we get (j(~)]r

-- i3(J(~q)/~3~lr

(6.4.10)

- O.

Let D denote the partial derivative of F(~b, (j) with respect to (a calculated at the point (~b = 0, ~j = 0). From eq. (6.4.6) we get D = OF(C, (j)/~(j ..

1 - S1~ 6

.

= Edet k=l

.

-S~

9 .

.

-.

O( --Slk .

.

0r

--Skk

... .

--S06 .

....

S~

(6.4.11)

. . . . . . --Sg 1

....

S60k~

...

1-

~'g6

where

sO~ -- ~ i j (~'q, ( ' j ) / 0 ( j l ( o , o )

= -3(E~3(K)u(K)-Z}ff.

(6.4.12)

To calculate the second derivative of ~j(//)) we recall another familiar formula -"

( aZF (~b' (j)

~J (r

= -

ar

i3:F (!b, (j) + 2(5(~b)

~ ~ j

(6.4.13)

+ (4(,)) Because of eq. (6.4.10) the latter simplifies ."-'t!

J

(r

~2F(r r162

(6.4.14)

T. Paszkiewicz and M. Wilczyhski

322

Ch. 4

Using eq. (6.4.6) we get 02F(r162 t' 1

sO 1

-

.

.

.

.

.

.

.

.

.

.

.

.

s~

..,

-S~

- 2 Z det

o,,

k=l

-so

. . . .

so%

. . . .

....

sO;

....

....

S~~

....

. . . .

sok6

sO~;

. . . .

so

S~

-..

.,,

\

--

so 61

1-sO 1 6

.

+2~det

.-.

.

.

-s ~

k=l

.

S O`` i k;ot/3 ~t/a ~t//3

--

.

.

.

.

-sO1

.

.

.

.

"..

. --

.

.

9 99

-sO 6

...

_ s ok6

.

S ko~ k ; a o q-a- -q-o

....

1 - Sg6 /

.

S 60~;'~ " ~ k;o~3qo~q3

(6.4.15)

'

. ""

1

S06

where j;o,3 = - 3 (Va(ff[)v~(I~)Eij(g)v(F, ) -3) p.

Si0/%/'i;

(6.4.16)

--

Because of eq. (6.4.9b) the first term of eq. (6.4.15) vanishes and we get 02

O@ 2 ( S ( ~ f q )

[

2D,~3

^

[

= -q'aq3 ~ ,

r

T

(6.4.17)

where 1

S~

-

.-.

_sOa*; 1k;oe/3

...

_sO~

. . . .

SO 16

oo,

6

D ~ - (r/D) Z det

-SOl

k=l

kk;ct~

"

..

_S O

k6

"

(6.4.18)

.**

-sO 1

9

-

S 0~

6k;oe~

""

1- SO

66

Ultimately, for small r we get ---

1

02

] r

--

-qo, q~ Do, z / r,

(6.4.19)

w

Influence of isotopic and substitutional atoms

323

or

zj(q) ~ -q~qzD~.

(6.4.20)

Thus we obtained a formula which generalizes eq. (4.3.17a). This means that, in agreement with the previous results, sufficiently close to the equilibrium state, after a sufficiently long lapse of time, for a medium of arbitrary symmetry, the DF and the number density N(w,r,t) (2.4.23) phonon gas approximately satisfies the diffusion equation

ON(w,r,t) O2N(w,r,t) ,.~ D,~ , i~t i~r~i~r~

(6.4.21)

with the diffusion tensor D (6.4.18) being a material tensor. For isotropic and cubic materials the tensor D simplifies to a single scalar coefficient (cf. eqs (4.2.15) and (5.1.19)).

7. Experimental observations of the diffusive propagation of phonon pulses The theoretical values of the diffusion constants (5.1.19) and (6.4.18) can be compared with experimentaly obtained diffusion constants. For such studies the yttrium-aluminium garnets containing rare earth substitution atoms of R type (R -- Dy, Gd, Lu, Yb) seem well suited. The ionic radii of yttrium and rare earth atoms are similar, so their presence does not distort the crystalline lattice. Besides, the elastic constants of YAG remain almost unaffected. For perfect massive specimens, containing substitutional atoms, at temperatures much lower than the Debye temperature the prevailing phonon scattering mechanism is the elastic scattering by point mass defects. Because the masses of Y and R atoms differ substantially the values of the scattering coefficient 9 are rather large (cf. w2.2). So, one can observe the diffusive motion of low energy nonequilibrium phonons almost undisturbed by inelastic phonon processes. In experiments performed by Ivanov, Khazanov and Taranov (cf. Ivanov et al. 1995) the specimen has the form of a slab, and its thickness L~ is much smaller than its lateral dimensions L~ and Ly. Nonequilibrium phonons are excited by heating a thin metallic film on one of the sample faces. Assuming that the film attains the local thermal equilibrium with the temperature TH slightly higher than the ambient temperature T, the distribution of the injected phonons is expected to be Planckian.

T. Paszkiewicz and M. Wilczyhski

324

Ch. 4

For small power density released in the heater mostly the long wavelength acoustic phonons (LAPs) are generated. Assume that the spectral distribution of the injected phonons attains the maximum for the frequency 60 - - WH ~ W m ( T H ) , where ~kBTH wm(TH) --

h

9

(7.1)

For the Planck distribution function ( = (P1 - 2.82. Because the ambient temperature T is in the experiment much lower then the Debye temperature 0D, the density of thermal phonons is very small (cf. w3.1) and an initially ballistic pulse of the injected phonons is gradually chaotized mostly due to scattering of phonons by substitutional atoms. Due to this scattering, the ballistic component of the pulse steadily diminishes, while the diffusive one grows. For long lapses of time and sufficiently large values of the coefficient g only the diffusive component of the phonon pulse is observed. In this case the number density N(w,r,t) of propagating phonons (2.4.23) obeys the diffusion eq. (6.4.21) The chemical structural formula for the yttrium-aluminum garnet can be written as Y3A12A13012. The unit described by this formula contains s = 20 atoms and there is eight (n = 8) of these formula units per unit cell. The yttrium atoms occupy 24 dodecahedral [C]-sites, the aluminium atoms occupy 16 octahedral ([A]-sites) and 24 dodecahedral sites. The remaining 96 sites are occupied by the oxygen atoms. The substitutional atoms can occupy dodecahedral [C]- as well as octahedral [A]-sites. Let us denote the abundance of the substitutional atoms in the [A]-sites by x (x = foct) and in the [C]-sites by y (y = fdod). Then, one can write the factor g for the yttrium aluminium garnet in the following form (cf. Kazakovtsev and Levinson 1986) 8x(1

-

x/3)(MR -

My) 2 +

8y(1

g=

-

y/2)(MR-

MA1) 2

(7.2) (8McFu)2

where 1

20

MCFtJ - ~-~ E M a .

(7.3)

ot=l

The Rayleigh characteristic time r is proportional to the factor 9 (cf. eq. (2.4.18)).

w

Influence of isotopic and substitutional atoms

325

Let us choose a Cartesian coordinate system so that the z-axis is perpendicular to the sample slab. Two remaining axes are lying in the surface of the slab and are arbitrarily oriented. Because of the chosen specimen geometry the phonon number density only weakly depends on the x and y coordinates and the diffusion equation simplifies. At the ambient temperature T much lower than the Debye temperature 0o one can neglect the presence of thermal phonons. As we previously noted (cf. eq. (7.1)), the frequencies of the dominating part of the injected phonons are proportional to the heater temperature TH. Assuming that the source produces short pulses of phonons of the frequency wH at one of the slab surfaces, one obtains the familiar expression

n(wH, Z , t ) , . , ~ l [ - - Z 2 J

Dv/-D-~exp

4D~wH)t

.

(7.4)

Assume that we generate a pulse of phonons at the instant t = 0. After a lapse of time t much longer than T(WH), at the opposite face of the slab, one observes a signal which attains its maximum at

tm --

4D (WH)A'

(7.5)

where A = 1/2. For a point source the coefficient A assumes the value of 3/2. Generally, the solution of the diffusion equation (6.4.21) has the form of a convolution of the initial function N(x, y, t = O) =_ h(x, y) with the function (7.4) depending now on r 2. However, since in the experiment the signal is averaged over the slab surface, when the dependency of h(x, y) on the spatial variables is weak one can expect that the formula (7.5) is still valid with 1/2 ~< A ~< 3/2. The dependence of the diffusion coefficient D on the heater temperature TH is shown for a number of different specimens in fig. 6. The upper line, labeled as 1 corresponds to the results of measurements of the arrival time t m for two pure Lu3A15Ol2 specimens. The diffusion coefficient was calculated with the use of eqs (7.2), (7.3), (2.4.18) and (7.5). We have assumed that phonons are scattered by the lutetium atoms substituting the aluminum atoms in the octahedral sites. The curve no 1 is consistent with the theoretically predicted D ,-~ TH 4 dependency. We also estimated the value of the coefficient ~ ~ 3.2 (cf. eq. (7.1)). The lower curve in fig. 6 corresponds to the experimentally measured arrival times for three samples of Y1LuzA15012 solid solutions (i.e. the matrix

326

T. Paszkiewicz and M. Wilczyhski

cm2/s)

I

I

I

Ch. 4

I

I

105

4

10

I

2

....

I

I

4

I

I

6 TIK)

Fig. 6. The diffusion coefficient vs the heater temperature TH (dots) for 1 - Lu3A15012, 2 Y1LuEA15012. The solid lines represent the results of calculation.

is a pure lutetium-aluminum garnet). For this matrix there are two scattering mechanisms. The residual one is related to some amount of foreign substitutional impurities as well as of oxygen vacancies. An additional scattering mechanism related to the substitutional Lu atoms placed in the octahedral aluminum sites. It was previously established that the concentration of the Lu atoms in the octahedral [A]-sites in pure Lu3A15012 matrices and in Y1LuEA15012 lattices are almost the same. So, the comparison of phonon scattering in Y1LuEA15012 and Lu3A15012 crystals allows us to establish the contribution to scattering made by the lutetium atoms placed in the dodecahedral [C]-sites. As previously, the measurements confirmed the the linear dependence of D on t h e TH 4 and gave the value of the ~ coefficient equal to 3.29. Similar considerations with the subtraction of the residual scattering term were done for results of measurements on other specimens (e.g., for Y2.65Luo.35A15012, Y2.TDyo.3A15012, Y2.65Gdo.15Ybo.sA15012). We accoun-

Influence of isotopic and substitutional atoms

327

ted for substitutional rare earth atoms in octahedral A-positions. The results confirmed the Rayleigh dependence of the diffusion coefficient on TH with not exceeding 4.1.

Acknowledgements This work and the works of the authors described herein were supported by a grant of National Committee for Scientific Research (KBN) through KBN contracts No 2 0096 91 01 and No 2 PO 3B 157 08. We thank Professor G.K. Horton for encouraging us to prepare this paper. We also thank Professors E.H. Hauge, O. Weis and V.L. Gurevich for hospitality and fruitful discussions.

Appendix 1. Distribution function of impurities In w2 we introduced the random variable r t which describes the state of a site that can be occupied by an isotope substitutional atom

_{0,

a host atom,

if in the lth unit cell there is

1,

a substitutional atom.

Each quantity depending on the set of variables r = {T1,..., TN } is a random variable and should be averaged with a suitable distribution function P(T). This distribution function should obey the normalization condition 1

1

1

EE

....

...... E

7"1 - ' 0 7"2 -~0

, TN)

-- 1.

(AI.1)

TN ---0

After Leibfried and Brauer (Leibfried and Brauer 1978) one can consider the grand canonical distribution function which depends on one macroscopic parameter, namely the concentration of impurities c N

N

P(T) ~ H p (TZ) -- H [CTZ+ ( I - C ) ( I - TZ)] . /=1

/=1

(A1.2)

T. Paszkiewicz and M. Wilczyhski

328

Ch. 4

It is easy to check that it obeys the normalization condition (AI.1) and to calculate the mean value of r l 1

: ~ r tier t + ( 1 - c ) ( 1 - r t ) r l

(A1.3)

] :c.

--0

To calculate the mean value of the product rllrz2 is a somewhat more complicated task C , 7"/1 "rl2 --

1,

(A1.4)

ll r 12,

ll

= 12.

Using (A1.4) one can calculate the mean value and the mean square deviation of the number of substitutional atoms

N~ = cN,

(A1.5)

N 2 - (N---~)2 _ c(1 - c)N.

(A1.6)

The mean value of Ir(k- k')12 contains two terms 1

N

[r(k - k') 12 = N2 ~

N

~

rt, rt2 exp [i (k - k ' ) . (X,, - Xt2)]

11=1 lz=l

1N N2

E

{ N 1 + c2

l=1

1 ~

exp [ i ( k - k ' ) . X,]

}2

.

l=l

In the thermodynamic limit (cf. eq. (A2.11)) the first term vanishes and we get I (k - k')I

: h

- hk')

This means that transition probability per unit time averaged over all allowed configurations of substitutional atoms satisfies both conservation laws, energy and quasimomentum. Let us mention, however, that the leading term of the transition probability w~(K, K') calculated for an arbitrary configuration of impurities r (cf. eq. (2.2.3)) is N/c times smaller in comparison to w~(K, K').

Influence of isotopic and substitutional atoms

329

Appendix 2. Collision theory A2.1. Formulation of the problem An incident beam of noninteracting particles or quasiparticles described by wave packets approaches the target. These wave packets must be spatially large, so they do not spread appreciably during the experiment and they must be large compared to the target particles, but small compared to the dimension of the laboratory, that is, they must not simultaneously overlap the target and detector. There follows interaction with the target and, finally, one sees two kinds of wave packets - one of them continues in the forward direction describing the unscattered particles and the other fly off at some angles and describe the scattered particles. The number of particles scattered into a given solid angle per time unit and unit incident flux is defined to be the differential cross-section. The Hamiltonian H of the composite system consists of three terms H = Ho + H + AH1, where H0 is the Hamiltonian of the beam particles, H is the Hamiltonian of the target and H1 is a weak (A << 1) interaction. Denote the eigenstates of target Hamiltonian by [a)

and the eigenstates of the Hamiltonian of the beam by IP) Ho[p) = E Ip).

The trace of the density matrix p of the total system over eigenstates of the target Hamiltonian gives the density matrix of the beam W(t) W(t)

-

Trap(t) - ~ (c~lp(t)lc~).

(A2.1)

O~

The diagonal matrix element of the operator W(t)

(p]W(t)lp) gives the probability density of the target particle to be in the state [P) at the time instant t. As the scattering particles do not change the state of the target its density matrix R(t) obeys Liouville equation with the Hamiltonian H

i~R(t)

i

it----~ + h [H, R(t)] - 0.

(A2.2)

T. Paszkiewicz and M. Wilczyhski

330

Ch. 4

A2.2. Probability density of transition per unit time

Since the particles of the beam and the target are statistically independent in the infinite past, we assume after Zubarev and Novikov (cf. J~drzejewski and Paszkiewicz 1976) the following boundary conditions valid for a macro-

scopic system

lim ex.

t~oo

[ i ,l]

~

p(t + tl)exp - ~

= t-~oolimexp[iH~~l]R ( t + t l )

(A2.3a)

P

The condition (A2.3a) can be written in a more convenient form (Zubarev 1980)

dtle ~tl exp

e

p(t + tl)exp -

h

oo

=

ef

dt 1 eSt' exp [iHtl ] R (t

~- t 1)

(A2.3b)

z

+

P

h

'

(e ~ 0 +). Calculating the integral on the rhs of eq. (A2.3b) we obtain the equation, the solution of which gives the retarded density matrix (Zubarev 1980)

p(t) = e / o _ _dtl

e et'

exp [ i Hh t l ] R ( t + t l )

oo

(A2.4) x ~ P

Wp(t + tl)IP)(Pl exp -

h

'

(e ~ 0+).

Influence of isotopic and substitutional atoms For e r 0 (and a finite system) eq. (A2.4) defines the operator obeys Liouville's equation with a source term

Opt(t) ~t

i

+ ~ [H, p~(t)] = - :

Ep~(t) - R(t) E

]

IP)(Pl 9

331

p~(t), which

(A2.5)

P

Since the target density matrix obeys eq. (A2.2)

exp[

h

h

1

'

we can rewrite the rhs of eq. (A2.4) in the form 0

p(t) -- : .~_

dr1 e st' U + (tl, O) R(t)

(X)

(A2.6)

~ Wp(t + tl)Ip>
(: -+ 0 +).

P

The operator

U(t, 0):

U(t,O) - exp [~ (H + Ho)] exp [--~it

HI,

(A2.7)

obeys the differential equation

OU(t, O) _ at

-

i AH[ (t)U(t, 0),

(A2.8)

h

where

H~(t) - exp [~ (H + Ho)] Hl exp [--~it (H +

H0)] .

Let us look for the solution of eq. (A2.8) in form of a power series in A. The first two terms of this series read U(~

O) = A~

u(1)(t, 0)- -A~if0t dtl H~ (t:).

332

T Paszkiewicz and M. Wilczyhski

Ch. 4

Taking the trace of both sides of eq. (A2.5) we obtain the equation for the operator Wq(t)

i~Wq(t)

i Tr. (q[ H1] lq) = ~ [p~(t),

i~t

(~ -+ 0 +)

Using eq. (A2.6) for p~(t) and integrating the rhs of the above equation by parts we obtain

OWq(t) ot

1

=

w

(t)

f~

dtl e

P

(~ --+ 0 +) 9 Above we assumed that Wv(t + tl) depends more weakly on time than the remaining terms of the integrand. Retaining the terms of the lowest order in A we obtain the Pauli master

equation owq(t) ot = ~ {wp (t)wpq(t)- wq(t)wqp(t)}.

(A2.9)

P

The function Wvq(t ) is the probability of transition from a state [p) to a state [q) per unit time at an instant of time t

Wpq(t)

-

d t l e etl

-~ -+

1 1 (tl) (HpqHqp

1 (tl)Hqp 1 + (Hpq

, (A2.10)

(3O

o+1

where, for example,

Hlpq (tl) = (plH~ (tl) Iq). The brackets (...)t mean the average with the target density matrix

(A) t - Tr{R(t)A}.

Influence o f isotopic and substitutional atoms

333

Since one can scatter either beams of particles or radiation on a target in a nonequilibrium state, its density matrix R(t) and the mean value (A) t may depend on time (cf. J~drzejewski and Paszkiewicz 1976). The explicit dependence on time of this average can be found in the frame of the Nonequilibrium Statistical Operator method (Zubarev 1974). Consider the thermodynamic limit of a mean value (A) t

lim

lim ( A ) t = - < A ~ - t ,

N--+ cx~ V--+cx~

(A2.11)

( N / V = pc = const).

In agreement with our remarks,

lim W~q (t) = - 1 ~ ~-,o+

0

f_ dtl eStl (A2.12a)

X {-~ glpqHlp

b-t nu -~ Hlpq ( t l ) H $ p b-t}.

(tl)

The order of the above limiting processes is important (cf. Zubarev 1974 and 1980). Let us suppose that the target is in an equilibrium state, i.e. R(t) - Po - e -~H / Z ,

Z - Tr e -c/H,

where/3 - (kB T ) - 1. In such a case the probability density of transition does not depend on time

1

Wpq = h--2 lim

e--+0+

f

cxz

dtl e etl

(A2.12b)

X {-~ HlqH~p(tl) b- -~- -~ Hlq(tl)Hlqp

For real systems the correlation function defining Wpq decays in time sufficiently fast, that, further on we put e - 0 in eq. (A2.12b) (Kubo et al. 1985; Zubarev 1974 and 1980).

334

Ch. 4

T Paszkiewicz and M. Wilczyhski

A2.3. Scattering of phonons by isotope impurities The term of the lattice Hamiltonian responsible for the scattering of phonons on substitutional and isotope atoms reads (cf. eq. (2.1.16))

+ H1 = E V (-K1, K2) aK, aK2 + h.c. K1,K2 Suppose that initially there are n K phonons K and n K, phonons K'. As a result of a scattering event there are (n K - 1) and (n K, + 1) phonons in the states K and K', respectively. Therefore

IP> =

I.--,

nK,

. . . , nK',

. . ->,

Iq> = I . . . , nK -- 1 , . . . , n K, + 1,...).

In the harmonic approximation the time dependent operators aK(t ), a+(t) oscillate in time

aK(t ) -- e-iw(K)taK,

a+(t)-

ei~(g)ta+"

Therefore, the matrix elements Hlq(t), Hip(t) oscillate too


IP) = 2 V r ( K , - K ' )

(A2.13a) x exp {i [ w ( K ' ) - co(K)] tl } i n k

(n K, + 1),

(pIH~ (tl) Iq) - 2 V r ( - K , K ' )

(A2.13b) x exp {i [ w ( K ' ) - co(K)] tl } i n K (n K, + 1). Now substitute expressions (A2.13a, b) into the integral (A2.12b) defining the probability density of transitions per unit time for a scattering event involving an initial state[..., n K . . . . , n g ..... ) and a final state I..., ( n g - 1), . . . , (n K, + 1),...}. We shall denote it by wn~'I)(n'+I)(T)'(n- In the thermodynamic limit, the energy spectrum of phonons is a continuous function of the wave vector. Using the definition of the Dirac delta distribution, we get the familiar expression !

W~nn-1)(n'-t-1)(T)

8~

= - ~ I V ( - K , K')I 2 n K (1 + n K,)5 [w(K) - w(K')].

(A2.14)

Influence of isotopic and substitutional atoms

335

This formula for the transition rate was named by Fermi the golden rule (cf. Baym 1974). The transition probability density for n = 1, n ' = 0

w~(K, K ' ) - wlO(T) defines the differential scattering cross-section (cf. Gurevich 1986).

A p p e n d i x 3. T i m e - r e v e r s a l i n v a r i a n c e in c l a s s i c a l m e c h a n i c s Consider a conservative system of N interacting particles, whose momenta Pk and coordinates rk(k =- 1 , . . . , N ) obey the Hamiltonian equations of motion

OH(p, r) /~ = - ~ ,

or~

§ =

OH(p, r)

(a - 1,2, 3).

ap~

(A3.1)

Let Pk0, rko (k = 1 , 2 , . . . , N ) be the position and velocity of the kth particle at t = to and allow them to proceed undisturbed for a time T, when their momenta and positions have become

Pk (to + T) = Pkl,

rk (to + T) = rkl.

Now, i.e. at t = to + T, another identical particle starts at rkl with the momentum (--Pkl)" The equations of motion (A3.1) are called invariant under time reversal operation when at a later time (to + 2T) we will find that the new position and momentum equal (rko,-Pko) (k = 1,2, 3 , . . . , N ) (Gottfried 1966). One can expect that random collisions may violate the time reversal invariance. Consider the set of new variables pIk(t),rI(t ) (k = 1 , . . . , N ) defined according to the relations p I ( t ) - - p k (to + T -

t),

?k(t) = rk (to + T - t).

(A3.2)

It is an easy task to write down the initial conditions for the new momenta and coordinates

p I ( t - to)

-

--Pkl,

rlk(t -- to) -- rkl

(k - 1,2 . . . . , N).

T. Paszkiewicz and M. Wilczyhski

336

Ch. 4

The familiar procedure (cf. Goldstein 1974) yields the equation of motion for the set of new momenta and coordinates ib~a(t) =

0H (-pI(t), rI(t)) -

Ori(t)

OH (-pI(t), rI(t)) '

§

OpI(t)

(A3.3) "

The classical equations of motion are said to be invariant under time reversal if the new momenta and new coordinates satisfy the equation of motion (A3.1), i.e. if

H ( - pI(t), rI(t)) - H (p(t), r(t)).

(A3.4)

To summarize, the equation of motion of classical mechanics is said to be invariant under time reversal if one can separate the manifold of all motions into two subsets {p(t), r(t)}, and {pI(t), rI(t)} between which at an arbitrary instant of time t there is a one-to-one correspondence defined by the relation (A3.2). Note that the initial instant of time to can be chosen arbitrarily, e.g., one can take to - 0 .

Appendix 4. Time-reversal invariance in quantum mechanics For the purposes of the lattice dynamics it is enough to consider quantum systems having classical analogues (e.g., systems of spinless particles), then the Hamiltonian H(p, r) of a classical system becomes the Hamiltonian opera t o r - a function of operators of particle momenta and coordinates. One can expect that for this class of systems one can easily generalize the discussion of time reversal operation to the case of quantum systems. In the Heisenberg picture the operators depend on time, e.g.,

pk~(t)=exp

(iHt) (iHt) ~ pk~exp----if-

(to = 0).

Define the set of new canonically conjugated variables

pI

(t) -

-pk~(-t),

rI.(t)

- rk~(-t)

(A4.1) (k = 1 , 2 , 3 , . . . , N ; a = 1,2,3).

Influence o f isotopic and substitutional atoms

337

There should exist a similarity operation relating two sets of variables {p(t), r(t)} and {pI(t), rI(t)}, namely for each of k and c~ p ~ ( t ) = Opk~(t)O -1,

(A4.2a, b)

rib(t) = Ork~(t)O -1.

Consider one of these relations, e.g., eq. (A4.2a). We have iHt)

pI~(t)-Oexp

--s

O- lOpk~O-lOexp

iHt)

----h-

0 -1.

(A4.3)

But according to eq. (A4.1) at the initial instant of time, the components of the momenta and coordinates satisfy the equations

Opk~O-1 _

--Pka,

Ol'kc~O-

1 --

(A4.4)

rk~.

Therefore, the following relations hold Oexp

(iHt) --~ O-l-exp

(iHt)

O--}z--O-1 - e x p

( --s -

There are two choices OiO-1 - i,

OHO -1 - - H ,

(A4.5a)

or

0i0-1 _ -i,

OHO- 1 _ H.

(A4.5b)

The example of the free particle is already enough to show that only the second possibility can be accepted, i.e. the operator 0 must be antilinear. For any complex number c~, 0~0 -1 - ~*.

(A4.6)

Consider a linear combination of two arbitrary state vectors q~l and t~2, namely c~4~1+ r From eq. (A4.6), it follows that 0(O~(/)1 -~-/~(/)2) -- O~*0(/)1 +/~* 0(/)2.

(A4.7)

T. Paszkiewicz and M. Wilczyhski

338

Ch. 4

This property of 0 was discovered by Wigner (cf. Wigner 1959). We postulate the conservation of the norm, i.e. for an arbitrary state vector q~ (0~b, 0~b) = (~b,~b),

(A4.8)

so the operator 0 is antilinear unitary operator, antiunitary operator for short. In the Schr6dinger picture (~b(0)= ~)

~2(t) = exp(-iHt/h)42.

(A4.9)

According to eq. (A4.5b)

O~2(t) - Oexp

iHt)

---K-

O- l O ~ = e x p

iHt)

--~

0~.

(A4.10)

Equation (A4.10) asserts that the configuration attained by permitting the system to evolve for a time t and then reversing all the momenta is indistinguishable from the configuration obtained by reversing momenta at the initial instant of time to = 0 and then evolving system backward for the time period t. Examine consequences of eqs (A4.7) and (A4.8) (Chem and Tubis 1967). Let ~b = ~31 = q~l nt- ~2, where ~bl and q~2 are two arbitrary states. Consider the norm of ~31 (~1, ~1) --- (q~l, q~l) -+- (q~2, q~2) -~- 2Re(~bl, q~2),

(A4.1 la)

( 0 ~ 1 , 0 ~ 1 ) = (0q~l, 0q~l) nt- (0~b2, 0~)2) -k- 2Re(0~bl, 0~b2),

(A4.1 lb)

or according to eq. (A4.8) (~1, ~1) -- (q~l, ~1) nt- (~b2, ~2) nt- 2Re(0~bl, 0q~2).

(A4.1 lc)

Comparing eqs (4.1 l a) and (4.1 l c) we conclude that Re(0q~l, 0~2) = Re(~bl, q~2).

(A4.12)

Now let ~b = ~2 --~1 + iq~2, then we conclude that Im(0~bl, 0~2) = -Im(~bl, q~2).

(A4.13)

Influence of isotopic and substitutional atoms

339

Hence, for the antiunitary operator 0 (0r162

= (r

(A4.14)

q~l) -- (q~l, r

The operator 0 does not change the probability of transition from state r to another state r (A4.15)

= l(r r

l(0q ~, 0r

Note that the similarity transformation (A4.2) preserves the canonical commutation relations (A4.16)

Iris, p~,~] - i fi~,Z~k,k,.

For lattice dynamics a natural basis in the Hilbert space is the coordinate representation. Let .~r,1..... r~v belong to the set of eigenvectors of the particle position operators, 1.e. rkc~

r

I

..... r N

' r

~-

7"k~

I

I

l ..... r N

.

The simplest antiunitary operator is the complex conjugation operator K. Consider a complete orthonormal basis {r then

r = Y'~(r r162 i

We define the complex conjugation operator K by Kr = Z ( r

r162 = ~ ( r

i

r162

(A4.17)

i

Note that K 2 = i, where I is the unit operator. So K -1 = K. Because of eq. (A4.4) the eigenfunctions of the coordinate operators are unaffected by 0. If r is a time reversed state of a system of spin 0 particles, its wave function is therefore given by

-

. ,,N ,,

, r ,2 , . . , 6 , ) .

T Paszkiewicz and M. Wilczyhski

340

Ch. 4

Hence ~bI (/~, r~ . . . . , r~v ) -- q~* (r], r ~ , . . . , r~v ) .

(A4.18)

From this argument we conclude that for spinless particles in the position representation the time-reversal operator 0 is simply the complex conjugation operator K. We shall now apply what we have learned about the transformation properties of the momentum and coordinate operators under time reversal to the phonon operators. Consider u~, p~, the elongation and the momentum operator of the ~th particle in /th unit cell of the lattice. According to eq. (A4.4)

0p/0 -1 : --p~.

0U/0 -1 -- U/,

(A4.19a, b)

The phonon creation (a +) and annihilation (aK) operators are defined by the Fourier series (cf. eqs (2.1.6a, b)) ~ i

h (A4.20a)

k#o j=o x exp(ik 9Xz)(a g + a +_K),

(A4.20b)

k#o j=0 x exp(ik. Xl)(a g - a+K),

where e(~, K) is the polarization vector, N is the number of unit cells (l = 1 , 2 , . . . , N). The index ~ enumerates s atoms contained in each unit cell (~ = 1,2 . . . . , s). The index j enumerates the 3Ns vibrational modes. Since 0=K,

Ouo~(l)O 1

Z

k#o j=o

2NMhw(K) e*~(a,K)

• exp(-ik. Xl)O(a K + a+K)O-1

(A4.21a)

Influence of isotopic and substitutional atoms

341

and

Opo,(1)O

1E

=

~

kr j=O

Jhw(K)Mhe*~(m'K) v

(A4.21b)

• e x p ( - i k . Xz)O(a K - a+K)O -'.

If we change K --+ - K in the sums (A4.21) and use eq. (A4.19) we obtain the transformation rules for the phonon operators

OaK 0-1 -- a_K ,

Oa+O-1 = a+K .

(A4.22a, b)

Now we can obtain the transformation rule for the.phonon number operator nK

OnK 0-1 -- OaKO-lOa+O -I -- n_K.

(A4.23)

Obviously the operation of time reversal leaves the total number of phonons unchanged, i.e.

ONO-1 - E OnK 0-1 : E n - K : E n K : N . Kr Kr Kr The eigenvectors of the occupation number operator are invariant under time reversal ol .... ,4-,...)

:

I .... ,~ .... ).

(A4.24)

The quasimomentum operator (cf. Kobussen and Paszkiewicz 198 l a,b) is an odd operator

OPO

- E K-C0

hkOnKO- 1 __. _p.

(A4.25)

342

T.

Paszkiewicz and M. Wilczyriski

Ch. 4

Appendix 5. Microreversibility Let us consider the scattering problem for the time reversed situation. According to Appendices 2 and 4 the system being in the infinite future (t --+ oo) in a time-reversed final state Olq) after collision and after an infinitely long lapse of time emerges in the time reversed initial state O]p). Therefore one should modify the boundary condition (A2.3a). Now the density matrix in the infinitely distant future factorizes

lira exp [i~~l] p(t + tl)exp Ii~'~ - h

t~-~oo

= lim exp tl--+oo

i~,~] ~ +,1,

,A,~,

x Z Wq(t + tl)lq~176 exp [-iHtl ] h ' q where

Iq~ - OIq}. As previously (cf. Appendix 2) the boundary condition (A5.1) can be written in an equivalent form e

/o~

[~~] p(t + [_iHtl ] [iHtl ] R(t +

dtl e -~tl exp i

= e

1

tl)exp

h

/0~

dtl e -st1 exp

tl)

h

x ZWq(t+tl)lq~176 q

[_iHtl] h '

where (e --+ 0 +). From this boundary condition we obtain the

p(al(t) - e

dtl

~0 ~176

(A5.2)

advanced density matrix

e-st'U+(tl,O)R(tl)

• ~ Wq(t + tl)lq ~ (q~ q

(A5.3) 0),

Influence of isotopic and substitutional atoms

343

which for e ~ 0 obeys Liouville equation with a source term

i

[

i~P(~a) ~- [H, p(~a)(t)] = e p(~a)(t) -- R(t) E

o--V- -h

]

IqO)(qOI

(A5.4)

q

Note that the source term has, in comparison to eq. (A2.5) for the retarded density matrix, the opposite sign. In the same way as for the retarded solution we obtain Pauli equation

o w (t) ~t

- ~ {Wq(t)Woqop(t)- Wp(t)wo~,oq(t)},

=

(A5.5)

q

where WOqOp = WqOpO. Note that, in comparison to Pauli equation (A2.9), the rhs of eq. (A5.5) has the opposite sign. The transition probability density per unit time WOqOp is related to the correlation function (cf. eq. (A2.10))

lim+ "1130qOp(t) - - - ~1 ~_..+0

dtl e-et~ f0 ~176

(A5.6)

where

1 __ (q~ n~qop

Ip~

(A5.7)

As previously (cf. Appendix 2) we confine ourselves to the case of an equilibrium state of the target. So R(t) = po and (A)0 = Tr(Ap0). Then it is possible to relate the probability densities wpq and Woqop. As previously (cf. Appendix 4), ~ denotes the state vector I~b}, the brackets (~b, ~b) denote the scalar product (~blq~) and 0~ denotes 01~b) = I~b~ Consider the correlation function (noqopnopoq(tl))o. 1 1 Since the eigenstates ~b~ of the Hamiltonian operator H form a complete set 1

1 Ot,'g

i

x exp { - ~ t l [(Ep - Eq) + (e~ - e,y)] } Z -1 exp ( - ~ e ~ ) ,

344

T. Paszkiewicz and M. Wilczyhski

Ch. 4

where

however, (A5.8)

0 + H~.yO - H.y~.

A little algebra with the use of eqs (A4.14) and (A5.8), which we do not reproduce, leads to relations 1

1

1 1 = (HpqHqp(tl

and

The relation (A5.6) defining the transition probability now reads 0

lim "1130qOp(t) - - - ~1 e-"~O+

/_

O0

dtl eet' { - ' < H q1v H p1q ( - t l ) >-o (A5.9)

1

1

}

+ -< H q p ( - t l ) H p q >'-o

so we have obtained the important relation (A5.10)

WOqOv = wvq.

When IP/, Iq} are the eigenstates of the momentum operator (A5.10) can be written in the familiar form (A5.11)

w_q,_p - wp,q,

called the principle of microreversibility. Now performing the change p ~ q we can rewrite eq. (A5.5) in the form of Pauli equation (A2.9)

owoq(t) ~t =

- ~

{Wop(t)wopoq(t) - Woq(t)woqop(t)}. P

(A5.12)

Influence of isotopic and substitutional atoms

345

Note that the signs of the rhs of eqs (2.9) and (5.12) differ. Let us check the relation (A5.10) for the scattering of a phonon K by isotopic impurities. Since phonon states are classified by the use of the quasimomentum/~ and linear polarization

- K = (-k, j) = OK,

- K ' = OK'.

Now a scattering event changes (nOK, + 1) phonons OK' and (nOK- 1) phonons OK into noK,, nOK phonons respectively. So from eq. (A2.14) we obtain

w~.(OK, OK') = w~.(K', K) = w~.(K, K').

(A5.13)

Appendix 6. Detailed balancing condition Until now the system of phonons scattered on impurities was in an arbitrary state. This resembles, for example, an optical experiment of scattering of a laser beam. Now we consider an equilibrium system of phonons, resembling black body radiation in a cavity of temperature T. We assume that the specimen containing isotopic or substitutional atoms is in the equilibrium state characterized by temperature T. The state of the gas of phonons is described by the phonon occupation r t K . The mean values of these occupation numbers are given by the Planck function fo(w(K)). Therefore, for an equilibrium system eq. (A2.14) reads 871"

w~q)_+K, -- h2 ]V(-K,K')]2 fo((w(K))

(A6.1)

• [1 + fo (w(K'))]5 [w(K) - w(K')]. Now note two identities

Io

+ So : [1 +

]

fo(w(K))]fo(w(K'))

(A6.2a)

(w(K) = w(K'))

and

IVr(-K, K')I 2 = IV,(K, -K')] 2.

(A6.2b)

T. Paszkiewicz and M. Wilczyhski

346

Ch. 4

M a k i n g use o f t h e m w e find

zo(eq) K__+K,(T) --

_ (eq) (7"). "WK,~ K

(A6.3)

It is a special case of the general relation called the balance. It is a direct c o n s e q u e n c e of the use of an ble. It states that scattering p r o c e s s e s c a n n o t drive the e q u i l i b r i u m state. T h e y can p r o d u c e small fluctuations

principle of detailed equilibrium ensems y s t e m out f r o m the only.

References Abramowitz, M. and I.A. Stegun (1970), Handbook of Mathematical Functions (Dover, New York), ch. 8. Al'shits, V.I., A.V. Sarychev and A.L. Shuvalov (1985), Sov. Phys. - JETP 62, 531. Baym, G. (1974), Lectures on Quantum Mechanics (Benjamin, Reading, MA), ch. 12. Beck, H. (1975), in: Dynamical Properties of Solids, Ed. by G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam), p. 205. Berke, A., A.P. Mayer and R.K. Wehner (1988), J. Phys. C: 21, 2305. Bron, W.E. (1985), in: Nonequilibrium Phonon Dynamics, Ed. by W.E. Bron (Plenum, New York), p. 227. Buot, F.A. (1972), J. Phys. C: 5, 5. Case, K.M. and P.E Zweifel (1967), Linear Transport Theory (Addison-Wesley, Reading, MA), ch. 4. Carruthers, P. (1961), Rev. Mod. Phys. 33, 92. Cercignani, C. (1975), Theory and Application of the Boltznumn Equation (Scottish Academic Press, Edinburgh-London), ch. 5. Challis, L.J. (1987), in: Physics of Phonons, Ed. by T. Paszkiewicz (Springer, Berlin), p. 264. Chern, B. and A. Tubis (1967), Amer. J. Phys. 35, 254. Claro, E and G.H. Wannier (1971), J. Math. Phys., 12, 92. Elices, M. and E Garcia-Moliner (1968), in: Physical Acoustics, Vol. 5, Ed. by W.P. Mason (Academic Press, New York), ch. 4. Enz, C.P. (1974), Rev. Mod. Phys. 46, 705. Ershov, Yu.I and S.B. Shikhov (1985), Mathematical Principles of Transport Theory (Moscow, Energoizdat), in Russian. Every, A.G. (1980), Phys. Rev. B 22, 1746. Fedorov, El. (1968), Theory of Elastic Waves in Crystals (Plenum, New York), ch. 4. Ferziger, J.H. and H.G. Kaper (1972), Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam), ch. 5. Fuchs, H.D., P. Etchegoin, M. Cardona, K. Itoh and E.E. Hailer (1993), Phys. Rev. Lett. 70, 1715. Gaficza, W. and T. Paszkiewicz (1993), in: Die Kunst of Phonons, Ed. by T. Paszkiewicz and K. Rapcewicz (Plenum, New York). Garicza, W. and T. Paszkiewicz (1995), Comput. Phys. Comm. 85, 423. Goldstein, H. (1974), Classical Mechanics (Addison-Wesley, Reading, MA). Gottfried, K. (1966), Quantum Mechanics, Vol. 1, Fundamentals (Benjamin, New York). GOtze, W. and K.H. Michel (1972), in: Dynamical Properties of Solids, Vol. 1, Ed. by G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam), p. 499.

Influence o f isotopic and substitutional atoms

347

Gurevich, V.L. (1986), Transport in Phonon Systems (North-Holland, Amsterdam). Gutfeld, R.J. and A.H. Nethercot, Jr. (1964), Phys Rev. Lett. 12, 641. Hauge, E.H. (1974), in: Lecture Notes in Physics, Vol. 31, Ed. by G. Kirczenow and J. Morro (Springer, Berlin). Hebboul, S.E. and J.P. Wolfe (1989), Z. Phys. B: Condens. Matter 73, 437. Held, E., W. Klein and R.P. Huebener (1989), Z. Phys. B: Condens. Matter 75, 17. Ivanov, S.N., E.N. Khazanov and A.V. Taranov (1991), Zh. Eksp. Teor. Fiz. 99, 1311. Ivanov, S.N., E.N. Khazanov, T. Paszkiewicz, A.V. Taranov and M. Wilczyfiski (1995), Z. Phys. B: Condens. Matter., to be published. Jasiukiewicz, Cz. and T. Paszkiewicz (1989), Z. Phys. B: Condens. Matter 77, 209. Jasiukiewicz, Cz. and T. Paszkiewicz (1990), in: Phonons '89, Vol. 2, Ed. by S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore), p. 1367. Jasiukiewicz, Cz. and T. Paszkiewicz (1993), in: Phonon Scattering in Condensed Matter, Ed. by M. Meissner and R.O. Pohl (Springer, Heidelberg), p. 96. Jasiukiewicz, Cz., D. Lehmann and T. Paszkiewicz (1991), Z. Phys. B: Condens. Matter 84, 73. Jasiukiewicz, Cz., T. Paszkiewicz and D. Lehmann (1994), Z. Phys. B: Condens. Matter 96, 213. Jfickle, J. (1970), Phys. Condens. Matter 11, 139. J~drzejewski, J. and T. Paszkiewicz (1976), J. Phys. C 9, 511. Kazakovtsev, D.V. and Y.B. Levinson (1986), Phys. Status Solidi B: 136, 425. Kirczenow, G. (1980), Ann. Phys. 125, 1. Klemens, P.G. (1955), Proc. Phys. Soc. A 86, 1113. Klemens, P.G. (1958), in: Solid-State Phys. 7 (1), Ed. by F. Seitz and D. Turnbull (Academic Press, New York). Kobussen, J.A. and T. Paszkiewicz (198 l a), Helv. Phys. Acta 54, 384. Kobussen, J.A. and T. Paszkiewicz (198 l b), Helv. Phys. Acta 54, 395 Kosevich, A.M. (1988), in: Theory of Crystalline Lattice (Vishtchaya Shkola, Kharkov), p. 95, in Russian. Kozorezov, A.G. and M.V. Krasilnikov (1989), Fiz. Tverd. Tela, 31, 109. Kubo, R., M. Toda and N. Hashitsume (1985), Statistical Physics H, Nonequilibrium Statistical Mechanics (Springer, Berlin). Landau, L.D. and I.M. Lifshitz (1987), in: Theory of Elasticity (Nauka, Moscow), ch. 3, in Russian. Lax, M. and V. Narayanamurti (1980), Phys. Rev. B 22, 4876. Leibfried, G. and N. Breuer (1978), Point Defects in Metals (Springer, Berlin), ch. 8. Levinson, Y.B. (1986), in: Nonequilibrium Phonons in Nometallic Crystals, Ed. by W. Eisenmenger and A.A. Kaplyanskii (North-Holland, Amsterdam), p. 91. Maris, H.J. (1986), in: Nonequilibrium Phonons in Nometallic Crystals, Ed. by W. Eisenmenger and A.A. Kaplyanskii (North-Holland, Amsterdam), p. 52. Maris, H.J. (1990), Phys. Rev. B 41, 9736. Northrop, G.A. (1982), Phys. Rev. 26, 903. Northrop, G.A. and J.P. Wolfe (1985), in: Nonequilibrium Phonons, Ed. by W.E. Bron (Plenum, New York), 165. Paszkiewicz, T. (1991), in: Ordering Phenomena in Condensed Matter, Ed. by Z.M. Galasiewicz and A. P~kalski (World Scientific, Singapore), p. 419. Paszkiewicz, T. and M. Wilczyfiski (1990a), Z. Phys. B: Condens. Matter 80, 287. Paszkiewicz, T. and M. Wilczyriski (1990b), Z. Phys. B: Condens. Matter 80, 365. Paszkiewicz, T. and M. Wilczyriski (1992), Z. Phys. B: Condens. Matter 88, 5. Peierls, R. (1964), Quantum Theory of Solids (Oxford University Press, London).

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Pomeranchuk, I.Ya. (1942), J. Phys. U.S.S.R. 6, 237 (Zh. Eksp. Teor. Fiz. 12 (1942), 245). Ramsbey, M.T., J.P. Wolfe and S. Tamura (1988), Z. Phys. B: Condens. Matter 73, 167. Rogers, S.J. (1971), Phys. Rev. B 3, 1440. R6sch, E and O. Weis (1976a), Z. Phys. 25, 101. R6sch, E and O. Weis (1976b), Z. Phys. 25, 115. Sadoulet, B. and B. Cabrera, H.J. Maris, J.P. Wolfe (1990), in: Phonons '89, Vol. 2, Ed. by S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore), p. 1383. Sampat, N. and M. Meissner (1993), in: Die Kunst of Phonons, Ed. by T. Paszkiewicz and K. Rapcewicz (Plenum, New York). Shields, J.A., J.P. Wolfe and S. Tamura (1989), Z. Phys. B: Condens. Matter 76, 295. Shields, J.A., S. Tamura and J.P. Wolfe (1991), Phys. Rev. B 43, 4966. Sirotin, Yu.I. and M.P. Shaskolskaya (1979), in: Principles of Crystallophysics (Nauka, Moscow), appendix D, in Russian. Streitwolf, H.-W. (1967), Gruppentheorie in der Festk6rperphysik (Akademische Verlagsgesellschaft, Geest & Portig, Leipzig), in German. Tamura, S. (1983), Phys. Rev. B 27, 858. Tamura, S. (1984), Phys. Rev. B 30, 849. Tamura, S., J.A. Shields and J.P. Wolfe (1991), Phys. Rev. B 44, 3001. Tamura, S. and T. Harada (1985), Phys. Rev. B 32, 1985. Wallace, D.C. (1972), Thermodynamics of Crystals (Willey, New York). Walpole, L.J. (1984), Proc. R. Soc. Lond. A 391, 149. Wigner, E. (1959), in: Group Theory (Academic Press, New York), ch. 26. Wolfe, J.P. (1989), in: Festk6rper Probleme/Advances in Solid State Physics, Vol. 29, Ed. by U. ROssler (Vieweg, Braunschweig), p. 75. Ziman, J.M. (1960), Electrons and Phonons (Clarendon, Oxford). Zubarev, D.N. (1974), Nonequilibrium Statistical Thermodynamics (Consultant Bureau, New York-London). Zubarev, D.N. (1980), in: Itogi Nauki i Tekhniki, Ed. by R.V. Gemkrelidze (Moscow), p. 131, in Russian.

CHAPTER 5

Phonons in Semiconductor Alloys JOHN D. DOW WILLIAM E. PACKARD

HOWARD A. BLACKSTEAD

Department of Physics Arizona State University Tempe, Arizona 85287-1504 USA

Physics Department University of Notre Dame Notre Dame, Indiana 46556 USA

DAVID W. JENKINS Institute for Postdoctoral Studies 1128 Almond Drive Aurora, Illinois 60506 USA

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

9 Elsevier Science B.V., 1995

349

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Contents 1. Introduction 353 2. Quantitative considerations 356 2.1. Classical equations of motion 356 2.2. Example: longitudinal vibrations of a perfect one-dimensional chain 359 2.3. Dynamical matrix 361 2.4. Example: one-dimensional perfect diatomic chain 362 2.5. Example: phonons of perfect GaAs and AlAs 363 2.6. Density of modes and additive functions of the mode frequencies 365 3. Theoretical approaches to alloys 365 3.1. One dimension 365 3.2. Three dimensions 370 4. Comparison with data 381 4.1. AI~Gal_~As 381 4.2. Gal_~In~As 385 4.3. Gal_~In~Sb 390 4.4. InAsl_~Sb~ 392 4.5. GaAsl_~Sb~ 393 5. Correlated alloys 394 5.1. General correlations 395 6. Combined treatment of thermodynamic, electronic, and vibrational properties 397 6.1. Order parameter 398 6.2. Spin Hamiltonian 400 6.3. Empirical tight-binding Hamiltonian 402 6.4. Force-constant model 405 7. Superlattices 418 8. Summary 419 Acknowledgements 420 References 421 351

This Page Intentionally Left Blank

1. Introduction Studies of the thermal properties of matter, and the lattice vibrations which determine those properties, have been central to the physics of condensed matter for almost a century. One of the classic textbooks on the matter, the work of Born and Huang (1954) was drafted over four decades ago, and yet is useful even today. One might ask why the physics of phonons is not completely understood, and how it can possibily be at the leading edge of condensed matter research today, after so many years. The answer, of course, is that the research has dealt with increasingly complex phenomena and has passed through as least three phases. Early work focussed on thermal and elastic phenomena, and the long-wavelength properties of matter, for example, elastic constants and the Debye theory of specific heats. The second phase dealt with all wavelengths of vibrational excitations in harmonic crystals, and began in the 1960's, when thermal-neutron scattering and infrared and Raman (light scattering) spectroscopy became widely used tools for studying the vibrational excitations of solids. The initial experimental studies with those tools were of the simplest crystals: monatomic or diatomic solids. Neutron and Raman spectroscopies focussed attention on the phonon dispersion relations S2(k, A), which could be determined directly from neutron spectroscopy and indirectly from infrared reflectivity or Raman data. (Here k is the wave-vector and A is the polarization index.) Optical spectroscopies had the disadvantage that they probed only states near k = 0 (because, on the scale of the Brillouin zone, namely ~ TriaL, where aL is the lattice constant, the wave-vector of light is nearly zero), but laser light is an inexpensive probe compared with a nuclear reactor, and so infrared and Raman measurements probably provided more information about the vibrational excitations of solids than neutron scattering data. Today we are in the third phase of phonon physics, which deals with the vibrational excitations of anharmonic crystals and harmonic non-crystalline solids, with the latter being the subject of this article. The phonon dispersion relation ~2(k,A) exists if the crystal is periodic and harmonic, and the harmonic approximation turned out to be a very useful model for the elementary vibrational excitations of crystals, despite the 353

354

J.D. D o w et al.

Ch. 5

facts that (i) all harmonic crystals are unstable (Leibfried and Ludwig 1961), and (ii) all anharmonic corrections to the harmonic crystal model (in powers of the ionic displacements from equilibrium) are necessarily divergent asymptotic series, because the perturbation operator is unbounded (Reed and Simon 1972). Indeed, some such expansions diverge in nth non-zero order as rapidly as e2n(n!) 2 - making perturbation theory almost useless for real solids, even those with anharmonic coupling constants e as small as ~,, 0.1. Nevertheless, the harmonic crystal approximation is the best model we have of vibrations in solids, and the classic book of Maradudin, Montroll, Weiss and Ipatova (1971) is the central reference for the physics of harmonic solids. The publication of its first edition shortly before the explosive growth of neutron-diffraction and Raman measurements of crystals set it apart as the classic reference on harmonic crystals. Early analyses of harmonic crystals were based primarily on the Bom-von Karman model describing the lattice vibrations in terms of empirical force constants associated with (mostly) short-ranged forces (Bom and Huang 1954). The long-ranged Coulomb interactions in semiconductors and insulators were incorporated using point-charge models (Ewald 1921). But even the point-charge models of long-ranged forces failed for materials as simple as NaCl and Ge, producing phonon dispersion relations in marked disagreement with the data, especially for wave-vectors near the edge of the Brillouin zone, where the transverse acoustic phonon dispersion relations were flatter than predicted. One of the most promising theoretical advances of those days of phonon dispersion physics involved going beyond the usual rigid-ion model (Kellerman 1940; Robinson and Dow 1968; Kunc 1973), in which the electronic clouds did not distort relative to the nuclei as the atoms vibrated, to the shell model (Dick and Overhauser 1958; Cochran 1959), which treated the electrons in atoms as spherical shells of charge which did not necessarily follow the atomic nuclei as they vibrated. The shell model was initially applied to alkali halides and then to Ge with great success, explaining, for example, the anomalously flat transverse-acoustic dispersion relations of Ge in terms of distortions of the Ge ion's electronic clouds, as the ions vibrated. However, the shell model eventually lost favor once it was recognized that the model's parameters could not be uniquely determined. Today's models of harmonic crystals rely on either empirical force constants extracted from measurements of phonon spectra (and include deformable-ion models such as the bond-charge model (Weber 1977) as well as the rigid-ion Bom-von Karman models), or on a p r i o r i force constants computed with methods such as the local-density approximation (Kohn and Sham 1965; von Barth and Hedin 1972; Martin and Galeener 1981). These a p r i o r i methods tend to be numerically cumbersome, and so we shall use

w1

Phonons in semiconductor alloys

355

empirical Bom-von Karman force constants in the present article. This is consistent with the spirit of the present work: to understand the essential physics of alloys and to determine how specific alloy clusters may manifest themselves in phonon spectra. The subject of the present article deals with collective atomic vibrations in one of the simplest types of solid that lies beyond the harmonic crystal approximation, being harmonic but non-crystalline: phonons in semiconductor alloys. Alloys are certainly not perfect crystals, and so one can only wonder if the harmonic crystal approximation has any relevance to real alloys. They nevertheless do have elementary excitations that are phonon-like, and so it behooves us to first understand the technologically-important simplest non-crystals, the substitutional zinc-blende crystalline semiconducting alloys, such as Al~Gal_xAs. These alloys would be zinc-blende crystals were it not for the disorder associated with occupancy by Ga vs. A1 of the cation sites. (For persons unfamilliar with the zinc-blende crystal structure, it is presented in fig. 21.) The present article deals with the physics of these nearly crystalline noncrystals, and their elementary vibrational excitations. Al~Gal_xAs is the prototypical material of this class of alloys, both because it has disorder on the cation site, and because Ga and A1 have essentially the same ionic radii - to a good (but not perfect) approximation GaAs and AlAs are identical except for the masses on the cation sites, which are occupied either by Ga with probability 1 - z or by A1 with probability z (when the site-occupancy is uncorrelated). For this particular material, the anion sites are invariably occupied by As atoms. The insights gained from studies of phonons in these simple non-crystals should guide us toward developing a comprehensive theory of phonon-like excitations in general alloys. Electronically the energy-band states of these AI~Gal_~As alloys in the vicinity of the fundamental electronic band-gap are well described by the virtual-crystal approximation: the cation is neither Ga-like nor Al-like, but an average cation, AI~Gal_~ (Parmenter 1955; Vogl et al. 1983). In other words, the electronic wavefunctions form an average over the crystal of Ga and A1, and so the resulting electronic material is a new hybrid material, and not just a mixture of GaAs and AlAs. However, we shall see that a virtualcrystal approximation to their phonons is completely invalid (in general), but that these materials have very interesting non-crystalline behavior over short distances, having spectral features found in GaAs, features of AlAs, and alloy modes found in neither compound. In all of the III-V semiconductor alloys, the long-ranged forces between atoms (Ewald 1921) are essentially the same, independent of the alloy disorder, and so the interesting physics is in the short-ranged disorder and how it affects the vibrational excitation spectra of the alloy materials. Thus our

J.D. D o w e t al.

356

Ch. 5

primary interest will be in understanding how this disorder displays itself, how to formulate and solve models of it, and how to interpret data to expract the physics of alloy clustering with an eye toward using the alloy physics to characterize the material and to determine its microscopic composition and local alloy configurations from Raman, neutron, or other spectra.

2.

Quantitative

considerations

Throughout most of this article, we shall use the Born-von Karman model of lattice vibrations in the harmonic approximation. This means that we shall obtain a set of empirical force constants from crystals such as AlAs and GaAs, assume that they can be transferred to the alloy AI~Gal_~As, and use these force constants to elucidate the physics of the alloy AI~Gal_~As. In particular, we shall use the second-nearest-neighbor model of Banerjee and Varshni (1969). Furthermore, in the first part of this article we shall assume that the occupancies of cation sites by Ga and A1 atoms are random and uncorrelated. In other words, the probability of finding an A1 atom on a given cation site is x. In w5 we shall discuss correlated alloys.

2.1. Classical equations of motion The equations of motion, in the harmonic approximation, for the atomic displacements from equilibrium I(fR) are

where we have taken the displacement from equilibrium to be

~n,b,i(t) = ~Tn,b,i(0) exp(-ig2t). Here we have

gn,b,~(t) -- Rn,b,~(t) -- R~,b,~(O), with Rn,b,i(O) being the ith component (i = x, y, or z) of the equilibrium (zinc-blende) lattice site of the bth atom (b - anion or cation) in the nth unit

w

Phonons in semiconductor alloys

357

cell. M is the mass matrix and 4i is the force-constant matrix, and we use Dirac matrix notation:

(n, b, ilSR) - ~Tn,b,~(O). In this notation, the mass matrix is diagonal M

In, b, i)Mn,b(n, b, i'[,

__ n,b,i,i ~

with the diagonal elements each being the mass of the atom on the site (n, b). We assume that the equilibrium position of the bth atom in the nth cell is !~n,b(0), which equals Rn(0) if b denotes an anion site, a n d / ~ ( 0 ) + ~7c if b denotes a cation site of the underlying zinc-blende lattice. Here we have Vc = (%/4)(1, 1, 1). The force-constant matrix is

(n,b, il~ln',b',i'), and its matrix elements are assumed to vanish whenever (n, b) and (n', b') denote atoms that are sufficiently distant from one another. In general the force-constant matrix is real and symmetric, and has on-site elements (n = n', b-- b') which are determined by its off-site elements and the sum-rule implied by infinitesimal translation invariance of the crystal:

(n, b, il~ln, b, i') = - ~ (n, b, il~[n', b', i'). nt,b '

Table 1 GaAs force-constants in the Banerjee and Varshni (1969) model, in units of kdyn/cm, and Ga, A1, and As masses in units of 10 -24 g. a /3 ,kAs AGa /Zas /~Ga VAs VGa MGa MAI MAs

-39.525 -34.000 4.500 4.500 -3.697 -4.467 -3.697 -4.467 116.43 45.06 125.12

J.D. Dow et al.

358

Ch. 5

Here, for the sake of being definite, we use the Banerjee-Varshni model because of its simplicity. The force constants of that model are presented in table 1; the nearest-neighbor force-constant matrix between an anion-site atom and its nearest neighbor on a cation site in the same cell is

(n, b = anion, i[4,Jn, b' = cation, i') =

fl

a

/~ ,

where the rows and columns range over i and i' = x, y, and z. The other nearest-neighbor atoms to an anion at Rn(0) are in cells other than the nth cell, and the corresponding force-constant matrices are obtained using group theory, as discussed in Kobayashi et al. (1985a), Kobayashi and Dow (unpublished), Kobayashi (1985). The choices of the force-constants a and ~ are determined by the atoms at either end of the bond between the atoms at sites (n, b) and (n', b'), e.g., aCaAs or aA1As, and are most often taken from the force constants fit to bulk GaAs or AlAs crystal data (e.g., table 1). The second-nearest-neighbor force-constant matrix connecting an anion at / ~ ( 0 ) and an anion at Rn,(0)=/~n(0) + (aL/2)(0, 1, 1) is

(n,b=anion, il#Jn',b'=anion, i')=

[

)~ 0

0 #a

0 ] va .

0

Va

~a

The cation second-neighbor force-constant matrix is similar. As with the first-neighbor matrices, the other second-neighbor matrices can be obtained using group theory 1) (Kobayashi et al. 1985a). l) These matrices can be written in more general form, using the direction cosines of the vector connecting two positions (n, b) and (n', b'), (l, m, n): for example ( - 1/x/~, 1/x/~, - 1/x/~) for (aL/4)(--1, 1,--1).

(n, b = anion,

iJ~Jn,b' =

cation, i I) =

[a 31mt3 3lnm:] 3lm~ a 3 [3 , 31hi3 3mn~

(n, b, iJ4)Jn',b', i')

[

)%(1 - 2/2) +

=

2lmvb 21nvb

2pb12

21mub ),b(1 -- 2m 2) +

2mnvb

2~bm2

21nvb 2mnv b )~b(1 -- 2n 2) +

] 2lZbn2

w

Phonons in semiconductor alloys

359

In this model we make two approximations which greatly simplify the calculations: (i) we assume a rigid-ion approximation to the force constants (Kellerman 1940; Robinson and Dow 1968), which makes the model especially addressable within the recursion method, and (ii) we omit the long-ranged forces in the interest of simplifying the calculations. These approximations introduce only minor theoretical uncertainties to the results. For example, the second-neighbor force constants simulate part of the longranged Coulomb force (Herscovici and Fibich 1980), and the remaining part, which causes the splitting between longitudinal and transverse optical modes at ~: = (), is small: the splitting is ~ 18 c m - 1 and ~ 38 cm -1 in GaAs and AlAs, small in comparison with the mode frequencies of ~ 270 cm -1 and 390 cm -]. The goal of most theories of alloys is to obtain a reasonable approximation to the eigensolutions O.y of the secular equation det [ M O 2 - ~] - O, and, with those approximate solutions, to predict various spectra of the alloy, including the density of modes or density of states per squared angular frequency -

2

3,

where N is the number of cells and S is the number of atoms per cell, i.e., S - 2 here.

2.2. Example: longitudinal vibrations of a perfect one-dimensional chain To illustrate these ideas, we show how they can be applied to a onedimensional monatomic chain with nearest-neighbor force constants, executing only longitudinal oscillations. The equations of motion are Mn ~2n = - r

un + ] - u n - ] + 2u n),

where i2 indicates a second time-derivative of the displacement from equilibrium u. Harmonically oscillating solutions solve - M n 2 l a / ~ ) - -~la/~),

J.D. D o w et al.

360

Ch. 5

w h e r e the mass matrix is (nlMIn') = M , ~ n , n , , and the force-constant matrix is expressed in terms o f K r o n e c k e r ' s delta-function and the nearest-neighbor force constant r

(nl~in') = --r

§ ~n',n-~

-

2~n,,n).

In the case that M n = M is a constant and there is no alloy disorder, the equations o f motion can be solved for the case of periodic b o u n d a r y I

.

.

.

9

.

.

.

........

.

.

.

.

.

.

,

o

"I

'

0

.

.

I

.

.

.

.

I

t

!

,

.

,

I

.

,

o~a~

.

.

.

.

.

.

I

.

.

.

.

2

I

~ 0.5

0 0

1

a/a~

2

3

Fig. 1. Dispersion relation 12(k) and density of modes D (per frequency) for a monatomic onedimensional crystal, with nearest-neighbor force constants r executing longitudinal vibrations. We have taken the force constant r to be I~1 of the Banerjee-Varshni model (Banerjee and Varshni 1969), and the mass to be the mass of As.

w

Phonons in semiconductor alloys

361

conditions, UN+n = un. In this case, the translational invariance of the lattice (Bloch's theorem) can be invoked to look for solutions of the form: (nla/~) - (lla/~; k ) N -112 exp (i1r where we have k in the first Brillouin zone: --(Tr/aL) ~< k < (rr/aL). Since it is the case that Rn(O) = na L, the dispersion relation is

s2(k) - 2(1r

sin (haLl2) l,

and the density of states or modes (per squared frequency) is:

D(a :) = [ . o ( o L Here O(z) is the unit step function and we have ~,~ = 2([d~[/M) 1/2 9 The dispersion relation and density of modes are plotted in fig. 1.

2.3. Dynamical matrix Since most alloys are not monatomic, it is often convenient to introduce the dynamical matrix Y - M - 1 / 2 ~ M -1/2 by executing a canonical transformation to normal coordinates IQ) - v/Ml~/~) 9This leads to the revised equations of motion

YIQ) = S?2IQ) 9 The main reason for introducing the dynamical matrix Y is to simplify the problem of finding the normal modes: we must diagonalize only Y, rather than simultaneously diagonalizing M and ~/i. The resulting secular equation is: det (~221 - Y) - O. Introducing the dynamical matrix's Green's function, G(~22) - (~221 - Y + iO)-', where i0 signifies a positive infinitesimal imaginary number, and specifies the boundary conditions to be observed when g22 corresponds to an eigenvalue

362

Ch. 5

J.D. D o w et al.

of O (and the Green's function would otherwise diverge), we find that the trace of its imaginary part is related to the density of modes" D(O 2) = ( - r r N S ) - I T r Im 17 = (NS) -1 ~

a(J? 2 - .(22). 7

Note that we shall use the term "density of states" to refer both to D(~2), the density per squared frequency, and to D(~), the density per frequency. The two are simply related: D(O) = 2 ~ D(a'22), and have units that differ by a factor of frequency.

2.4. Example: one-dimensional perfect diatomic chain The equations of motion for longitudinal vibrations of a diatomic linear chain are {2n,a "-" (r

+ Un--l,c -- 2Un,a)

?dn,c -- (r

+ U n - l , a - 2Un,c).

and

Invocation of translational invariance {n, b[Q) = {1, blQ; f c ) ( Y S ) - l M b l / 2 e x p

(i/~./~n(0) + ira. gb),

leads to a 2 • 2 secular equation, and the dispersion relations S'22(k) = (1r

4- { 1 - 2[1 - cos(kaL)] [MaMc/(Ma + Mc)2] }1/2),

where we have the reduced mass #, with #-1 = M a 1+ Me-1. The dispersion relations are plotted in fig. 2, along with the corresponding density of states, assuming r = [c~i, and Ma = MAs and Me = MGa. Note that the plus sign in the dispersion relation refers to the optical mode with ~2 __+ 2]r as k --+ 0, and the minus sign refers to the acoustical branch with speed of sound in the long-wavelength limit of c = [lr

+ Mc)] ,/2.

At the zone-boundary, k - 7r/aL, the modes reduce to an acoustical mode of frequency O - v/21Cl/Maand an optical mode of frequency O = ~v/21Cl/M~, where we have assumed Me < Ma. This simple diatomic model has an acoustical band of states separated by a gap from an optical band. While such a gap is to be expected in one-dimension, it may or may not occur in three dimensions, depending on the masses, the force constants, and the anisotropies.

w

Phonons in semiconductor alloys .

.

.

.

.

.

.

.

.

' .

.

.

.

,

.

0

1

.

.

.

,

363 !

,

,

v

9

I

2

a / a,.,

A

~' 0.5

0

1

a/a~

2

3

Fig. 2. Dispersion relations g2(k) and density of modes D (per frequency) for a diatomic onedimensional crystal, with nearest-neighbor force constants ~b, executing longitudinal vibrations. We have taken the force constant to be ~b = I~1 of the Banerjee-Varshni model (table 1), and the masses to be the masses of Ga and As.

2.5. Example: phonons of perfect GaAs and AlAs The Banerjee-Varshni model of GaAs can be similarly solved for the dispersion relations of the phonons in three dimensions. To do this, one introduces the transformation

(n,b, iIQ) - (1,b, ilQ;/~)Mbl/2

exp (i/~./~n(O) + i1r 9~Tb),

and directly diagonalizes the secular equation at each wave-vector k of the zinc-blende Brillouin zone. The resulting dispersion relations and the density

364

Ch. 5

J.D. D o w et al.

W a v e number ;~1 (crril)

r..

,~

0

Ik,,

I00

~//.~., , 2 GoAs

200

300

~'

~ ~ I . Z I"

~/'~ "

"'

/ GoA,

..

'

'

T I ~ : X - " ~ 2 TO'I~

,.o,,.11 I ~o=xl~_l_o.~

o.,~ ,,:~

AloaGao. As t.S~ Virtual crystal 1" approximation

= o.,

,,

0.0 t

~

~.e

!

"~

/

-~

~,:x

Y

~o=x~A ,lit

i~.__.,..~

~

0

i/

i

!

|'

!

|

'

~ 11"

'1\

"~

"/ I /

Or

9

I/

AI s

o?

L,~

~

400

'

,

I

,

0 I 0

20

40

4e60

Frequency f~ /lO"'rad/s) I0

20

30

Energy t~f~

40

eO i 50

(meV)

Fig. 3. From top to bottom, phonon dispersion curves of GaAs (solid) in comparison with the fit to the phonon dispersion from neutron scattering (Waugh and Dolling 1963) (dotted); density of states for GaAs obtained by the recursion (solid) and Lehmann-Taut (dashed) methods; virtual-crystal density of states; AlAs density of states; and AlAs dispersion curves, after Kobayashi et al. (1985b). The densities of phonon states (times 1012 rad/s) versus energy hO (in meV) of GaAs were calculated using the Banerjee-Varshni model. Note that the recursion method does not reproduce the van Hove singularities (van Hove 1953) of the Lehmann-Taut method, but that the differences are nevertheless not major. In the bottom panel, the phonon dispersion curves of AlAs are compared with infrared reflection data (circles) (Hass and Henvis 1962) and neutron diffraction data (dotted) (Farr et al. 1975). The symmetry points of the zinc-blende Brillouin zone are F = (0, 0, 0), L = (lr/%)(1, 1, 1),X = (lr/aL)(1, 0, 0), U = (27r/aL)(1,1/4, 1/4), and K = (37r/2%)(1, 1,0). Phonons such as the longitudinal optical phonons at k = X are denoted LO:X, etc.

w

Phonons in semiconductor alloys

365

of states (evaluated using the Lehmann-Taut method (Lehmann and Taut 1972)) are displayed in fig. 3 (Waugh and Dolling 1963; Kobayashi et al. 1985b; van Hove 1953; Hass and Henvis 1962; Farr et al. 1975).

2.6. Density of modes and additive functions of the mode frequencies In a harmonic crystal, most spectra and thermodynamic properties (such as the internal energy U, for which we have f(g2) = hOB(h~2), where B is the Bose-Einstein function) are additive functions f of the normal mode frequencies g2.~:

F

._,_

fo ~

3,

"1,

dO2f(O)NSD(~2).

= fo ~176

Since the eigenfrequencies g2.y have a continuous spectrum and do not necessarily have an obvious one-to-one correspondence with the normal modes of a perfect crystal, except in the very long wavelength (k --+ 0) limit, we focus instead on predicting the density of states D(~2). Thus the

primary goal of any alloy theory in the harmonic-crystal approximation is to predict the density of modes D(~2). 3.

Theoretical approaches to alloys

If we view the "alloy problem" theoretically as requiring the computation of the phonon density of states D(g22), then we must find numerical and analytic methods that can obtain a resonable approximation to the alloy density of states without excessive computer requirements. There are many different approaches to the alloy problem, but, unfortunately, the "best" choice can often depend on the dimension, the range of the forces, and the masses of the ions.

3.1. One dimension In one-dimension, for example, with short-ranged forces, the most efficient way to solve alloy problems is either by (1) directly solving the equations

366

Ch. 5

J.D. D o w et al.

of motion for a very large cluster of atoms (the brute-force method), or (2) using the negative-eigenvalue theorem (Dean 1961, 1972; Payton and Visscher 1967a, b, 1968; Myles and Dow 1979) for an extremely large cluster of atoms to obtain the density of states. The brute-force method becomes impractical for two and three dimensions, because the sizes of the clusters that can be considered become too small, and the level spacings vary as N -1/a where d is the dimension, often becoming too large for the clusters to responds in a physically correct way to a collision with an incoming wave-packet of lattice distortions. The negative-eigenvalue approach is still of practical use for some twodimensional problems with short-ranged forces, and so we review its essential elements here. Its application to three dimensions, however, is not practical even with very large contemporary computers. To illustrate the negative-eigenvalue method, we consider the simple case of a linear chain A~BI_~ with mass disorder and nearest-neighbor force constants executing vibrations. The secular determinant for the dynamical matrix Y is A1 X1,2 0 0 _

_

.

X1,2 0 A2 X2,3 X2,3 A3 0 X3,4

0 0 0 0 X3,4 0 A4 X4,5

0 0 0 0 0 0 0 0 X5,6 0 A6 X6,7 X6,7 A7

0 0 0 0

0 0 0 0

0

0

0 X7,8

0 0

... ... ... ... ... ... ...

0

0

0

X4,5

A5

0 0

0 0

0 0

0 0

X5,6 0

0

0

0

0

0

0

X7,8

A8

X8,9 ...

0

0

0

0

0

0

0

X8,9

A9

...

Here we have Xn,n' = r

1/2

and A~ = j?2 _ (2r with Mn being a random variable, taking on the values MA with probability c and MB with probability 1 - c . The way that the density of states is computed from this matrix is that the matrix is first brought into triangular form, with zero for all elements below

w

Phonons in semiconductor alloys

367

the diagonal, by elementary row operations. That is, (X1,2/A1) times the first row is substracted from the second row, making the second row's first element vanish. Substracting a constant times one row of a determinant from another row does not change the value of the determinant. This causes the first non-zero element of the second row to be Az(X2,z/A1). Then subtracting (X2,3/[A2- (X2,2/A1)]) times the second row from the third row, leaves the third row with two zeros as its first elements. Continuing this procedure brings the matrix to triangular form: its lower left triangle is all zeros. Then its determinant is the product of its diagonal elements only. But the determinant is an invariant of a matrix, and so this is also the product of the eigenvalue differences: det [ ~ 2 1 - Y] - 1~ ( ~ 2 _ g2~). 3/

The number of negative factors in this product is the number T'(122) of eigenvalues that are greater than 122 and so we have: T(122) = N -

~

0(122- 122).

"7

Hence, by differentiating T(122), we obtain D(122). (Here we have used the fact that Dirac's d;-function is the derivative of 0). Calculating the density of states this way for a one-dimensional alloy of 100,000 atoms is rather easy on even a small personal computer. Some illustrative results are given in fig. 4 (Myles and Dow 1979; Soven 1967; Taylor 1967; Onodera and Toyozawa 1968; Velicky et al. 1968). To make contact with the physics of three-dimensional AlxGal_xAs, we have also computed the densities of phonon states for :r = 0.1,0.25, 0.5, 0.75, and 0.9 in a one-dimensional version of the Banerjee-Varshni model. The results are in fig. 5. They show that light-mass A1 produces a localized mode above the gap. As z increases, the isolated local Al-modes show satellites associated with clusters that eventually merge first into characteristic alloy features, and then into an AlAs band. The sort of approach used here for one-dimension can be extended into two-dimensions (Hu et al. 1984) when only short-ranged forces are involved, but it becomes computationally very cumbersome. With present-day computers, the method can be extended to simple models of three-dimensional crystalline alloys, but such models typically have so few atoms that they are unable to produce the desired small level spacings.

J.D. D o w et al.

368

(a)

Ch. 5

Ma

MA--

1.05

d:~

MA-

am

o.ol . , , , . . . . .

o.o I =~

LO

"

'

"

L*~

I (b) =

~,

I'OSl

I ....

'

1.40

, . . .1

2.0 a2/a "

'

I

'

3D

~'"~

z.o

I

'"~

"

'

:

/ i M'_.cMA+ J il.C)MB"

. a'f n=,.-.r---..~ (--'---,

9

4O

i.~,=i r ~

C-05 -. MN'MA=2

aoo . . . . . . . . . . . . . o.o ID

.-Jl

f.~!ni

.

,..

,

3D

n2/~ 2

,,cPA.~ 4.0

. . . . . . . . . . . . . . . . . . . . . . . . . I I I

1r 1.05

c=0.5 WIVIA =2

0.35 0.OG 12o ...........

1.0

2.o

3D

4.0

Fig. 4. Vibrational density of states for a perfect one-dimensional linear chain A c B l - c with nearest-neighbor force constants and random mass disorder: M = MA, with probability c = 0.5 and M = MB with probability 1 - c = 0.5. These results are from Myles and Dow (1979). We have MB = 2MA, and O A - - (q~/MA)1/2 and I2B = (c~/MB)1/2. Panel (a) shows the perfect-crystal results. Panel (c) shows the results obtained using the negative-eigenvalue theorem (histogram) and using the embedded cluster method of Myles and Dow (1979). Panel (b) shows the virtual crystal approximation, and the coherent potential approximation (CPA) (Soven 1967; Taylor 1967; Onodera and Toyozawa 1968; Velicky et al. 1968). Note that the "exact" negative-eigenvalue result is well-reproduced only by the embedded cluster method, and even that method has problems near the band gap.

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Phonons in semiconductor alloys 1

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ta "/ta,, ~

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a z/caaJ Fig. 5. Computed densities of phonon states in one-dimensional A l x G a l _ x A s , using the masses and the nearest-neighbor force constant 4~ = Ic~l of table 1. Note the localized mode associated with vibrations of the light-mass impurity that emerges from the top of the optical band when A1 is introduced. Increasing z leads to clustering effects, and ultimately to AlAslike structures.

370

J.D. D o w et al.

Ch. 5

3.2. Three dimensions

The physics of three-dimensional "real" alloys, even in the harmonic approximation, requires development of well thought-out models. Since our primary goal is to evaluate the density of states, we are seeking a method that models the phonons well on a short length scale, or k ~ TriaL. The reason for this is that the density of states comes mostly from the surface of the Brillouin zone, with very little coming from the center. In the limit of d --+ ~ dimensions, the surface-to-volume ratio approaches unity, and the Brillouin-zone surface provides all of the states. Even for d = 3, the major contribution comes from near the zone surface. Furthermore, g22(k, )~) is a periodic function of k in the Brillouin zone, and so has van 2 Hove singularities g2vn, at which the density of states is singular, behaving as 1~"22 - ,Q2H[1/2 on one side of the singularity and a constant on the other (van Hove 1953). These singularities, for topological reasons, tend to occur near the Brillouin-zone surface. Therefore, a good theory of alloys should pay particular attention to modes whose frequencies correspond to the zone-boundary frequencies of the perfect crystals, which often lie near the surface of the Brillouin zone, at k ~ TriaL, with short wavelengths. This requirement, that the theory reproduce short-wavelength features which are sensitive to specific local clusters of atoms, makes developing a successful theory of alloys difficult. 3.2.1. Qualitative considerations

Before discussing the vibrational physics of three-dimensional alloys and AI~Gal_~As in particular, we first clarify some important qualitative points that restrict how the vibrational states of successful alloy theories must behave. Quite a few of the quasi-analytic theories of alloys do not reproduce all of these qualitative features. 1) The zinc-blende substitutional crystalline semiconductor alloys all have two atoms per unit cell and N unit cells, where N is of the order 6.02 • 1023. They are not periodic in the usual sense, but do have unit cells whose cation occupancy varies from cell to cell, being, e.g., either Al-occupancy or Gaoccupancy. Each atom can vibrate in any of three directions: x, y, and z. Consequently, the alloy will have 6N vibrational normal modes. 2) These materials can be displaced over macroscopic distances in three independent directions,.x, y, and z, without exciting internal vibrations. This implies that three of the normal modes, in the long-wavelength or k = 0 limit, have zero frequency: there are acoustic modes in the alloy. 3) The crystalline materials AlAs and GaAs, having two atoms per unit cell, consequently have one set of three long-wavelength or k = 0 optical

w

Phonons in semiconductor alloys

371

phonon modes (that approach finite frequency in the long-wavelength limit, or at k --+ 0). These correspond to modes in which the anion and cation vibrate out of phase, either stretching and contracting the bond between them (for the longitudinal vibrations) or executing transverse motions that are in opposite directions. 4) The crystalline materials AlAs and GaAs each have a dipole moment per unit cell, and so there is a splitting between "transverse" and "longitudinal" fr --+ (~ modes that is proportional to the square of the dipole moment. The transverse modes correspond, in the long-wavelength limit, to atomic vibrations perpendicular to the wave-vector k of the modes; similarly the longitudinal modes vibrate parallel to k. Although the wave-vector k is not a good quantum number in the alloy for general values of k, it is still a valid concept in the alloy in the long-wavelength k ~ 0 limit, and so this splitting observed in crystals is expected to persist in alloys. However, since most of the interesting physics of III-V alloys lies in clustering and shortranged phenomena, and since the long-ranged forces are all similar, we shall concentrate on the short-ranged forces. (A discussion of how to incorporate the long-ranged forces is included in Kobayashi and Dow (unpublished) and Ren et al. (1987, 1988a, b), Chu et al. (1988), Ren and Chang (1991) .) 5) The low concentration limits of the alloy AlxGal_xAs are defined by z --+ 0 and z --+ 1. In these limits the basic physics of the alloy is the same as for the perfect crystal containing isolated defects. For example, in the limit z --+ 0, AlxGal_~As behaves as GaAs with only a few A1 defects on cation sites; similarly z --+ 1 corresponds to AlAs with dilute concentrations of Ga. Similarly, up to concentrations of order 3% or so (i.e., z < 0.03 or z > 0.97), the alloy is essentially a crystal with small numbers of isolated and paired defects. Therefore the physics of low concentration alloys is identical to the physics of perfect crystals with isolated defects, pairs of defects, and small clusters. The "alloy problem" corresponds to the regime 0.03 < z <~ 0.5. (For z > 0.5, we merely replace z by 1 - z . ) A successful theory of alloys must reduce to the theory of perfect crystals with defects and pairs of defects, to order z 2.

3.2.2. Amalgamation versus persistence One of the classic papers on the theory of alloys was presented by Onodera and Toyozawa (1968), who clarified the concept of amalgamated versus persistent spectra of an alloy. Although the Onodera-Toyozawa paper was written with the spectra of excitons in alloys in mind, it can nevertheless be applied to phonon problems in alloys. The basic idea is that there is a criterion which determines whether the spectra of an alloy are almost

J.D. Dow et al.

372

Ch. 5

superpositions of spectra of the components (the persistent limit) or are an amalgam or average of the spectra of its components (the virtual-crystal limit (Parmenter 1955)). For example, if the two components of the alloy, such as AlAs and GaAs, each had a peak as a principal spectral feature, in the amalgamated limit there would be one peak at the average of the AlAs and GaAs frequencies, but in the persistent limit there would be two peaks: one at the GaAs position and the other at the AlAs position. The criterion for having amalgamated spectra is that the transfer or hopping integral between sites, 14~1,is much larger than the variation of on-site matrix elements, IMA1- MGaIf22. In terms of our phonon problems, this usually amounts to the masses of A1 and Ga being equal to within several percent, for the zone-boundary phonons that dominate the density of states. This criterion is rarely met for short-wavelength vibrations in semiconductor alloys such as Al~Gal_~As. (Long-wavelength excitations, such as sound waves with f2 --+ 0, can have amalgamated spectra while the short-wavelength modes are nearly persistent.) Thus we have the ironic situation that the substitutional crystalline alloys electronically are invariably amalgamated and well-described by a virtual-crystal approximation, but vibrationally these alloys exhibit persistent spectra, and so are rather difficult to describe when they have any character significantly different from the vibrations of small clusters. In the extreme persistent-limit, the density of states is a superposition of those for AlAs and GaAs: D(f22; AI~Gal_~As) = zD(~22; AlAs) + (1 - z)D(~22; GaAs). Thus the goal of the theory of alloys is to understand and correctly predict the deviations from this persistent-limit: the alloy modes.

3.2.3. Unsuccessful theories It is worth while briefly reviewing several unsuccessful theories of alloys, and pointing out why they fail to describe the phonons in materials such as Al~Gal_~As.

3.2.3. I. Virtual-crystal approximation. In the virtual-crystal approximation to an alloy with mass disorder, such as A~BI_~, one would replace the masses MA and MB with an average mass M* = cMA +(1--c)MB (Parmenter 1955; Elliott et al. 1974). This clearly yields a poor approximation to the density of states in one-dimension (fig. 4), and is a comparably poor approximation in three dimensions (fig. 3). The virtual-crystal omits the clustering and fluctuations of alloy composition that produce local configurations whose vibrations yield the characteristic peaks of D(~22).

w

Phonons in semiconductor alloys

373

3.2.3.2. Random-element isodisplacement model. The random-element isodisplacement model (Chen et al. 1966; Cheng and Mitra 1968, 1970, 1971; Verleur and Barker 1966) of crystalline alloys such as AlxGal_xAs is designed to treat long-wavelength optical phonons, and assumes that both a rigid A1 and a rigid Ga sub-lattice vibrate against a rigid As sub-lattice. It also makes a virtual-crystal-like approximation to average force constants. This approach can be valid as k --+ 0, but clearly breaks down for zoneboundary phonons, k ~ 7r/aL, for which the displacements of atoms in adjacent cells have different phases. Therefore, while the model has utility for describing how optical phonon frequencies change in alloys, it lacks the general features necessary for correctly predicting densities of states. 3.2.3.3. Elementary clusters. In a substitutional alloy such as AlxGal_xAs, the As ions each have one of five elementary environments: (1) four Ga atoms, (2) three Ga atoms and one A1 atom, (3) two each of Ga and A1, (4) one Ga atom and three A1 atoms, and (5) four A1 atoms. If the alloy is random, these clusters occur with probabilities P given by a binomial distribution, viz., P = 4z3(1 - z) for three A1 atoms and one Ga. Therefore, one could construct a statistical theory of vibrations in the solid, based on suitable averages of the vibrational properties of elementary clusters. While such small clusters execute vibrations characteristic of the elementary units that make up the alloy, the collisions of those vibrations with adjacent clusters are poorly modeled if larger clusters are not also treated comparably. Of course, if sufficiently large clusters are included as "elementary", the elementary cluster model will yield the correct density of states for the alloy, being equivalent to the brute-force approach. However such inclusion is not easy and undercuts the central simplicity (and value) of the elementary cluster model. Talwar et al. (1980, 1981) have used many of the ideas of the elementary cluster model, together with the Green's function method, to make good progress on the alloy problem. As we shall see, however, the recursion method seems to present advantages over all elementary cluster methods for computing short-wavelength features of alloy spectra.

3.2.3.4. Coherent potential approximation. The coherent potential approximation (CPA) (Soven 1967; Taylor 1967; Onodera and Toyozawa 1968; Elliott et al. 1974) is an attempt to rigorously treat the alloy problem by

374

J.D. Dow et al.

Ch. 5

finding the "best" normal modes of a statistically averaged alloy. Unlike the virtual-crystal approximation, which would assign a single average mass to the cations of A1,Gal_~As, the CPA seeks an average self-consistent propagator or Green's function (Velicky et al. 1968; Shen et al. 1987; Bonneville 1984). The requirements that determine this Green's function are (i) the "best" normal modes produced in this method must scatter from each atomic site minimally: the single-site effective-medium transition matrix (or effective interaction) must vanish when averaged over all alloy configurations; and (ii) the Green's function must assume the form related to the Green's function for the perfect crystal with dynamical matrix Ieb, but with an additional self-energy 27: G(~22) = [O21 - l~b- 27(02)] -1, where s162 is the self-energy matrix, and provides a single frequencydependent (but not spatially-dependent) lifetime and level shift for all of the effective normal modes of the alloy. By construction (Onodera and Toyozawa 1968), the CPA reproduces the isolated-defect limits a: -+ 0 and a: -+ 1, but the method fails in order a:2 or (1 - z ) 2 to produce the pair spectra appropriate to those limits. Therefore it makes more sense to compute the isolated-defect limit directly (which is less difficult than executing the coherent potential approximation) when that limit applies, and otherwise to ignore the CPA - the answers provided by the CPA are physically incorrect in general when the alloy's constituents are not isolated. The mathematics of the CPA is reviewed in Myles and Dow (1979) and in Onodera and Toyozawa (1968), but it is perhaps better to recast the CPA approach to alloys in terms of a story about boating in a mountain lake, in the coherent potential approximation. This story is somewhat exaggerated, but nevertheless graphically illustrates the weaknesses of the CPA. A young scientist was given the task of rowing east across a mountain lake every day. Unfortunately, the lake was in the mountains and filled with many very small and rocky islands. Furthermore the lake was invariably shrouded in mist and fog. So every day he set out toward the east in the fog, rowing until he bounced off one island, and then another, and another, finally arriving at his destination on the other side of the lake after several scattering events. Having been educated in the CPA, he decided to replace his task of rowing across the lake with an equivalent problem. He hired a bulldozer to remove all of the rocky islands, so that he did not collide with them any more. Then he dumped some gelatin into the lake to make its water more viscous. Finally he took a sledgehammer and beat on his boat so that it exhibited the effects of the collisions he once had experienced. He

w

Phonons in semiconductor alloys

375

even punctured the boat, so that it leaked, and would sink if the distance across the lake was too great. But he did this all in such a way that the time to get across the bulldozed but viscous lake in his leaky CPA boat was exactly the same as the time to cross the real lake in the real boat. The leaky boat was his Green's function or propagator, and introducing leaks into the boat gave it a lifetime or (complex) self-energy. While the leaky boat leaked, and the leaks slowed it down, it did not collide with any islands, because they had been removed by a bulldozer. (The lowest-order scattering had been replaced by a viscous medium.) So the leaky boat went straight across the lake, but slower than the real boat had, which had collided with real islands, and followed a complicated path as it had repeatedly scattered against islands on its way across the real lake. Clearly, so long as he only concerned himself with crossing the lake toward the east, the leaky CPA boat solution to the young scientist's problem was satisfactory, because he had been careful when creating leaks in the boat and adding gelatin to the water to do both in such a way that he preserved his schedule for crossing the lake. A different problem, such as crossing the lake to the north, where he would have to pass through a canal, would cause the time of crossing for the CPA boat to be different from the time of crossing for the real boat, and the coherent potential approximation would fail. The moral of this somewhat over-simplified story is that the CPA is a good approximation only for selected problems, and should not be expected to produce a realistic model of the short wavelength modes of an alloy. There have been a number of attempts to develop improved versions of the coherent potential approximation, which account for multiple-site scattering (the boat bouncing off several islands simultaneously), discussed in Myles and Dow (1979). These multi-site CPA theories have self-energies with spatial dispersion, S(fr S22), tend to be mathematically complicated, often have difficulties achieving self-consistency, and at first were plagued with non-analyticities or negative state densities (Elliott et al. 1 9 7 4 ) - a problem that Kaplan et al. (1980) showed can be solved, but with considerable computational effort.

3.2.3.5. Average t-matrix approximation. The t-matrix or transition matrix is the effective interaction in an interacting system. For the mathematics of the average t-matrix approximation, the reader is referred to Myles and Dow (1979), Elliott et al. (1974) and Bemasconi et al. (1991). In the average t-matrix approximation, the self-energy S is evaluated using RayleighSchr6dinger perturbation theory, rather than Brillouin-Wigner perturbation theory, as in the CPA. Thus the exact Green's function is approximated by the Green's function of the unperturbed reference lattice, say GaAs. As a result, the average t-matrix approximation lacks the self-consistency of the CPA, but has virtually all of the defects of the CPA.

376

J.D. Dow et al.

Ch. 5

3.2.3.6. Embedded cluster approximations. One way to introduce local vibrational modes, while limiting the computational requirements, is to embed a cluster of modest size in some sort of statistical medium. The formalism for this is worked out in Myles and Dow (1979) and in Gonis and Garland (1977). The density of states, however, is primarily dependent on the asymptotic behavior of the normal modes at large distances. As a result, the density of states of a statistical medium plus a cluster embedded in it, will normally be close to that of the medium, not that of the cluster. This problem can be circumvented by heavily weighting the cluster in evaluating the density of states, but there is still the problem that the medium will have forbidden bands, and when an excitation of the cluster propagates out to the medium, it will be reflected if its frequency corresponds to a forbidden frequency of the statistical medium. Therefore, embedded cluster methods, while holding some promise for computing vibrational properties of modest-sized clusters and for accounting for their interaction with a statistical medium, will be only marginally better than computing the densities of states of the clusters themselves, especially in the spectral vicinities of the gaps in the medium's density of states. As we shall see, the recursion method invariably provides a better approximation than most embedded cluster schemes. 3.2.4. A solvable limit: low-concentration alloys

In the low-concentration limit, for example in Al~Gal_~As with z < 0.005 or z > 0.995, the alloy reduces to a perfect crystal with an ensemble of isolated defects: As z ~ 0 we have GaAs with a small number of isolated A1 atoms replacing Ga atoms. In this isolated-defect limit, the density of states of the alloy can be evaluated numerically. The resulting A1 defects alter the mass matrix M only on the defect sites and perhaps alter the forceconstant matrix ~, but only for sites (n, b) in the vicinity of each d e f e c t and so the method of localized perturbations applies (Dawber and Elliott 1963). There are some subtle issues with respect to the differences between perturbed and unperturbed basis sets which are handled properly by these authors. In this method, one first constructs the Green's function for the perfect crystal, namely the matrix Go(~22) - (~21 - Yo + iO) -1 9 We also define the defect matrix A = Mol/2[(M

- Mo)g) 2 - (~ - ~o)] M o 1/2,

w

377

Phonons in semiconductor alloys

where M0 and 4i0 are the mass and force-constant matrices of the perfect crystal. Then we have the Dyson equation G = Go - GoG,

where G is the Green's function for the perturbed crystal, and the secular equation reduces to det(1 + G0) = 0. An important feature of this secular determinant is that it is not 6N • 6N in size, but rather the size of the defect matrix A, which is limited to the sites on which the Ga mass is replaced by an A1 mass, and to the sites whose bonds are altered. For example, for a single mass defect in a linear chain at the origin, the secular equation is algebraic: (0[G010)<01zal0> - 1 , where 10) is the vibrational basis state at the origin. Moreover, the Green's function for the perfect crystal can be evaluated (numerically) rather easily if the phonon dispersion relations are known (even numerically), following techniques discussed in Dowber and Elliott (1963) and Maradudin et al. (1971). -

3.2.5. The approach o f choice: the recursion method

The recursion method (Haydock 1980; Nex 1978, 1984; Kelly 1980; Heine 1980) is currently the theoretical scheme of choice for studying substitutional crystalline alloys such as AlxGal_xAs. It takes into account the local environment of a particular atom in a cluster, and also correctly incorporates the recoil of the cluster of atoms surrounding the central atom, for any desired cluster size. In the recursion method, one starts with a dynamical matrix Y expressed for a very large but finite cluster of atoms, in a local basis In, b, i) = IP), with p = 0, 1 , 2 , 3 , . . .

(p[Ylq) =

Yo,o Yl,o Y2,0 Y3,0

Yo,1 Yl,1 Y2,1 Y3,1

Yo,2 Yl,2 Y2,2 Y3,2

Yo,3 " "" Yl,3 "'" Y2,3 "'" Y3,3

J.D. D o w e t al.

378

Ch. 5

Here the basis states are 10), I1), 12), etc. The method then generates a new basis set IP'), which transforms the matrix Y to tridiagonal form:

(p'lYlq') =

a0 b1 0 0

b1 a1 b2 0

0 b2 a2 b3

0 0 b3 a3

... ... ... ...

with the basis states 10'), I1'), 12'), etc. The method is designed to compute matrix elements of the Green's function between "starting states" such as 10)"

, For example, if 10) is a state localized at a particular site, then we have

(-1/TrNS)Im(OIGIO) - - ( N S ) - I ~ - "

I(OlE, A)Ie6(s2 e - s~(fc, A)2),

k,)~

the local density of states. Or alternatively, by selecting 10) to be a uniform linear combination of states localized at each site of a large cluster, viz., states that transform according to k = 0, one can obtain the density of states projected onto k = 0. The new basis set IP') can be generated from any chosen intial state 10) = [0') by employing the recursion relations, which are the foundation of the method: YI0') = a010') + bill'), YI 1') = b 110') + a 111') + b212'), and

YIP') - bp[(p - 1)') + aplp') + bp+ll(p + 1)'). The requirement that the new states LP') obey orthogonality relations determines the coefficients ap and bp in terms of the matrix elements involving the basis set IP) of the original dynamical matrix. For example, we have (O'IY[O' / = %(0'10') + bl (0'11').

w

Phonons in semiconductor alloys

379

Orthonormality leads to (0'10') = 1 and (0'l 1') = 0, and an expression for a0:

a 0 = (0'[YI0') - (01YI0) = Y0,0" From the recursion relations we have I1')

-

bll ( Y - ao)lO')

= b l l ( Y - ao)lO )

-- bl 1{[Yo,olO) + Yl,Ol1) + Y2,ol2) + ' " "1- Yo,o[O)}, or

= b~-l{yl,ol 1) -I- Y2,ol2) §

I0)},

where b l is determined by the normalization conditions ( l'l 1') -- 1. Hence the state I1') is a linear combination of the states connected to the starting state ]0) through the off-diagonal matrix elements of the original dynamical matrix Y in the basis Ip). Vibrations initially localized at the site corresponding to 10) propagate outward: the new states IP') correspond to regions increasingly remote from the starting state's site 10). With increasing p, the coefficients ap and bp become increasingly less important in determining the physical properties of the starting state 10). The "starting-state" matrix element of the Green's function



= (o1[y221

_ y]-i

IO>

can be calculated rather easily because the dynamical matrix is tridiagonal in the IP') basis. Hence the leading element of the inverse matrix [g221- y ] - l is the cofactor divided by the determinant, and so we have

-

2 -

ao

-

where we have NO -- b 2 [ ~ 2 -

a 1 - ,~1] -1,

and

'~1 -- b2

[j?2 a2_/~2]-1,

J.D. D o w e t al.

380

Ch. 5

and so on. Thus we have a continued-fraction representation of the Green's function matrix element (01GI0). The continued-fraction can be terminated at any level, say at L. Then we have the approximate local density of states projected onto 10): d(0; J'22) - ( - 1 / T r N S ) I m ( 0 1 c

(s22) 10).

Here the "0" denotes the site or sites onto which the state-density is projected. The level L of continued fraction is normally determined by a convergence criterion: either the requirement that the Lth and ( L - 1)st level produce the same results (within some specified uncertainty), or the Lth level for a large cluster does not involve any states at the surface of the cluster. A computer code that executes the recursion method is available from Cambridge University, and is reproduced in Kobayashi (1985). This code generates the coefficients a n and bp, while verifying upper and lower bounds on the relevant densities of states. The density of states spectra of this paper were computed using the recursion method, typically for 1000-atom clusters, with L - 51. A slight drawback of the recursion method is that it computes a local density of states at site /~, d(/~; J22), not the global density of states. To overcome this, we divide the density of states into anion and cation parts, where we have: Dcation ( n 2) - (1 - x) d(Ga; 0 2) + xd(A1; 02). Here the (approximate) Ga- and Al-site local densities of states are computed for an ensemble of various clusters with Ga and A1 atoms at the center. The anion density of states is obtained from an average over mini-clusters of anion-site local densities of states: Danion(J?2) -- ~"~p•

j?2).

x

At the center of a 1000-atom cluster, a specific five-atom mini-cluster is generated, such as a central As atom with one Ga and three A1 neighbors. Then the probability p , ( x ) of this mini-cluster occurring is determined. This probability is a binomial distribution if the mini-cluster is randomly formed. The remaining 995 atoms of the 1000-atom cluster are added, with the probability of A1 occupancy of a cation site being x in the random alloy. Then the local density of states at the central site is determined, for each mini-cluster and cluster combination, using the recursion method, with the process being repeated for all possible mini-clusters, each embedded in a 995-atom cluster. The (approximate) anion-site density of states is then obtained, and the total density of states is the sum of the anion and cation contributions.

w 4.

Phonons in semiconductor alloys

381

Comparison with data

4.1. AI~Gal_~As The theoretical densities of states computed with the recursion method can be compared with Raman scattering data of Tsu et al. (1972), Kawamura et al. (1972), Kim and Spitzer (1979), Saint-Cricq et al. (1981), Caries et al. (1982), and Jusserand and Sapriel (1981). Since these alloys are in the persistent limit, a first approximation to their densities of states is D(g22; AI~Ga,_~As) - zD(j22; AlAs) + (1 - x)D(j22; GaAs). It is the deviations from this persistent limit that are to be determined by an alloy theory - the "alloy modes". These alloy modes are associated with small clusters in alloys not present in either GaAs or AlAs. Experimentally, there are two types of phenomena in the Raman spectra of these alloys which require explanation; (i) "two-mode behavior" and (ii) "disorder activated alloy modes". The fact that the spectra are almost in the persistent limit implies that the optical band will be two bands, one GaAs-like and the other AlAs-like. This two-mode behavior is observed (Cheng and Mitra 1968, 1970, 1971; Illegems and Pearson 1970). The spectra reveal three frequency regions where there are significant deviations from "persistent" behavior: (i) near 370 c m -1, (ii) near 250 cm -1, and (iii) near 75 c m -1. These are the "alloy modes". By dissecting the calculations, we can identify specific peaks with specific clusters of atoms. Figure 6 gives the computed densities of states of AI~Gal_~As, as z varies, and shows how the features that change with z originate with certain atoms or clusters of atoms. (i) The persistent spectrum (fig. 6; dashed line) has ,-~ 345 cm -1 and ,,~ 390 cm -1 features due to vibrations of a central A1 atom with A1 atoms at its second-nearest-neighbors. The deviations from the persistent-limit of these modes in this vicinity is assigned to alloy modes associated with a central A1 atom and with Ga atoms on the second-neighbor sites. (ii) The deviation from the persistent-limit around ,,~ 250 cm -1, namely the shoulder at ,-~ 250 c m -1 is due to a combination of a central Ga atom with A1 atoms on the nearest cation sites (second-nearest-neighbor sites), and (ii) a central As atom with neighboring A1 atoms. (iii) For z ~ 0.5, the transverse and longitudinal acoustical models are composed mostly of As atoms vibrating while surrounded by nearest-neighbor Ga and A1 atoms. The deviation from the persistent-limit spectrum

J.D. D o w e t al.

382

Ch. 5

Wave number ~1 (crdl)

0

I00

200

I

X

o.6~-

!

300

I "AIIG~|.xAs I 0.9

400

X = O. !

/

AI

I

O0

":,

0.61

X=0.3

0.6

X

/

t...

m -

0

l

0.5

o., I

(Go,All .,, (A,,AII~;

__

(AI,Ga)

,,,9 ..~_

C a

O.

0.6

X = 0.9

Ga ( As ,Go )

o.3 ~ 0

0

I

I0

20

30

40

50

60

Frequency Fl (lfl=rad/p) I0

20

Energy fl~

30

40

(meV)

70

,

80

50

Fig. 6. Densities of phonon states D for AlzGa]_xAs alloys, computed using the recursion method (solid lines) for various alloy compositions z, and with the persistent approximation for z = 0.5: D(AlxGa]_=As) = zD(A1As) + (1 - z)D(GaAs). Labels such as "As" denote clusters with central As atoms; (As, A1) denotes central As atoms and neighboring A1 atoms; and (Ga, A1) denotes a central Ga and second-neighbor A1. After Kobayashi et al. (1985b).

near ~ 75 c m -1 is due to clusters with 25% of the nearest-neighbor sites containing either G a or A1. In m a k i n g c o m p a r i s o n s with R a m a n or infrared data, one must m a k e allowances for the fact that the theory does not have R a m a n or infrared matrix elements, that long-ranged forces have been omitted, and that longitudinaltransverse splittings have been set to zero. Thus one must be prepared to split and shift the theory by an amount of order ~ 20 cm -1 to bring it into coincidence with the data.

w

Phonons in semiconductor alloys

383

Wave number ~-1(crril) 200.

I00 I

AIo.7, GOo.,4 As

R omon spectrum

, m

c

~

300

I

TA:L

400

I

~ x -~

TO o , A ,

LA:L

AL}rLOo.,.

I

TO,,,.

n

E~ I

m ~..o

6 -

I

I

I

I

I

I

60

70

AI o.Te Go o.=4 AS

(b) ~ -1~ 0 . 3 c a

0.0

o l

0

I0

20

30

40

12

50

Frequenc~ f4 (10 t rad/s) I0

20

Energy .l'tQ

30

I

40

80 I

50

(meV)

Fig. 7. (a) Raman spectrum of AlxGa]_xAs with z = 0.76, after Tsu et al. (1972) and Kawamura et al. (1972) (solid) and resonant Raman spectrum, after Jusserand and Sapriel (1981) (dashed); (b) calculated density of phonon states. AL denotes an acoustic local mode. The mode assignments are those of Tsu et al. (1972), Kawamura et al. (1972), and Jusserand and Sapriel (1981). After Kobayashi et al. (1985b).

Figure 7 shows Raman scattering data of AlxGal_xAs for z - 0.76 (Tsu et al. 1972; Kawamura et al. 1972) and resonant Raman data (Jusserand and Sapriel 1981) for z - 0.75, in comparison with the calculated density of states of Kobayashi et al. (1985b). The experimental selection-rule conditions for the data of fig. 7, in a perfect crystal, would yield only the LO:F (longitudinal optical at F) modes, but the disorder of the alloy activates some other non-longitudinal optical modes. The theory confirms the assignment of the LOA1As peak, while also suggesting that the disorder-activated TO:L (transverse optical at the L-point of the Brillouin zone) and TO:X modes contribute to the width and asymmetry of the ,-,, 390 cm -1 feature. The peak assigned to TOAIAs in the data for ~ 360 cm -1, is reassigned to an alloy mode associated with A1 vibrations, with some Ga atoms on about three of the twelve second-neighbor sites. Other assignments are confirmed (Tsu et al. 1972; Kawamura et al. 1972): the GaAs-like LO modes, the

384

Ch. 5

J.D. D o w et al.

disorder-activated TO modes correspond to the persistent peak in the theory at ,,- 270 cm -1 (fig. 3). The shoulder of the GaAs optic mode, denoted AL (an acoustic local mode), is associated with the theoretical structure at ,-, 250 cm-1 _ a vibration of As with A1 atoms instead of Ga atoms at three of its four nearest-neighbor sites. The disorder-activated LA (longitudinal acoustic) and TA (transverse acoustic) modes are the two lowest bands of Jusserand and Sapriel (1981), displayed in Fig. 7a. The acoustic region is AlAs-like with LA:X(A1As), LA:L(A1As), TA:X(A1As), and TA:L(A1As) features. LA:U, K(A1As) and TA:L(A1As) should also contribute to those LA peaks. The peak labeled TA:L(A1As) is at least partly due to an alloy mode, with As surrounded by three A1 atoms and one Ga atom. Figure 8 shows the data of Kim and Spitzer (1979) for z = 0.54. The AlAs-like LO mode and the GaAs-like LO mode are sharp features, and the theory confirms the DALA, LOCaAs, and LOAIAs assignments. The shoulders on the main peaks are from disorder-activated alloy modes. The LOGaAs

Wave number Z'l(cm 1 ) 0

I00

200

AIo.s4Go0.4e AS

~A c ~

300

I

i

400

I

LOG,A,-I

1

--

LOAtA,-

Romon spectrum

(o)

i

DALA ~ ] ~

E ~ m I

,

!

i

I

,, =

I I

_ AIo.s4Goo.4sAs

r

I

(b) C

o.o

J

I

j~

o

t 0

i

20

!. I0

_

40

-1-2

! 20

Energy "h~

I 50

80

6O

Frequency fZ (10 rad/s)

! 40

! 50

(meV)

Fig. 8. (a) Raman spectrum and (b) calculated densities of states for A l x G a l _ x A s with z = 0.54. The disorder-activated longitudinal acoustic mode is denoted DALA. After Kobayashi et al. (1985b).

w

Phonons in semiconductor alloys

%.1(cnil)

Wave number 200

IOO T

400

I

,Roman

2TADALA " ~

DATA

(o)

C ffl 19,*-'_

500

I

AIo.2 Gao.e As

385

1"-'-

spectrum

DATO I

OA O o:r OAO

,

E~

m ee

I

(/) 19

0.9

~

0.6

o~

i

I

,,

i

I

I

i

60

70

- AIo. 2 Gao. o As (c)

"~ ~ 0.3 C 19

o.o

0

tO

20

30

40

50

t

! Freqt~ency f21 (1012ra~s)

o

I0

20

Energy ~

30

40

80

t 50

(meV)

Fig. 9. (a) Raman spectrum of AlzGal_zAs for z = 0.2, after Saint-Cricq et al. (1981) and Caries et al. (1982). (b) The same spectrum with two-phonon contributions, the background and the DAO (disorder-activated optical mode) removed. (c) Computed density of states. After Kobayashi et al. (1985b).

vibration has side-bands due to vibrations of a central Ga atom with some A1 atoms as second neighbors and to clusters with As having some nearestneighbor A1. The low-energy tail of the LOA1As peak is caused by vibrations of A1 atoms with some Ga second-neighbors. The Raman spectrum of Fig. 9 was measured under conditions that, in a perfect crystal, forbid the LO:F and TO:F modes. The main peaks are disorder-activated and correspond to broken selection rules, and the experimental assignments and theory are consistent with one another. 4.2. G a l _ = I n = A s

Following the same procedures as for AI=Gal_=As, and using the InAs phonon spectrum (fig. 10) (Kobayashi and Dow unpublished; Hass and Henvis 1962; Orlova 1979) obtained from the GaAs force constants (Banerjee

386

J.D. D o w e t al.

Ch. 5

Wave number ~1 (cn~l) 0

I00

I

20O

!

0.9[-

-,-

/

.

lnAs

I

TO:X

LO:X

I -TO:L

~~" 0"6F TA:X LO:L~ I/~ LA:X~, }/ w6:~ L TA'L ITA:U,K.~LA:L~ ~

I~

~

I

~

~

O:Y

~

i

!

"

~

-'~

.~.

,

" _~

'

I

0

20

I

4O

12

Frequency a 110 rad/s) 0

I0 20 30 Energy ~ (meV)

Fig. 10. Calculated densities of states (solid for recursion and dashed for Lehmann-Taut) and phonon dispersion curves of InAs, after Kobayashi and Dow (unpublished). The circles are infrared reflection data of Hass and Henvis (1962) and the dotted dispersion relations are extracted from X-ray diffuse scattering (Orlova 1979).

and Varshni 1969) and the mass of In, one can obtain the density of states spectra for Ga]_=In=As alloys (fig. 11). Once again the densities of states approximate the persistent limit. The higher frequency optical band near ,-~ 270 cm -1 is associated with GaAs, while the ,,~ 210 cm -] band is InAslike. The Ga-As-like optical band has alloy modes which create a shoulder

w

Phonons in semiconductor alloys

Wave 0

%'1(cm1)

number

I00

,,,

200

0"9- Ga"xInxAs '

o.,:

,i

/ 1

!

(As,In)

'

/

~i

0.5

~o,t x=o5 ==

"

'~

o.3-

"6

oo-1

ii

O.e

e~

1 i'

i

x =o.t

0

>,

300

'

o.~

1,1

387

(As,Go)=

in

I,,,,~

J

Ih

J

(G,,~.l~n

II1

1/

...,~,,..z. C-. t . X =0.7

t

p

A

Q

!

0

20

40

Frequency i o

Energy

f2 t

Io

2o

~

60

(101=rad/s) I ~0

(meV)

Fig. 11. Calculated densities of phonon states D for Gal_zlnxAs alloys, for various x. The dashed line is the persistent-limit for x = 0.5. After Kobayashi and Dow (unpublished).

388

Ch. 5

J.D. Dow et al.

near ,,~ 250 cm -1, due to vibrations of As, with some In atoms as neighbors, and vibrations of Ga atoms with In second-neighbors. The InAs-like optical band is associated primarily with As atomic vibrations, with alloy modes attributable to neighboring sites being occupied by Ga instead of In. The acoustic bands, which change their positions and shapes as z varies, are dominated by the motions of the cations, with second-neighbor disorder leading to alloy modes. For small values of z, an In impurity mode appears at-,~ 210 cm -1, corresponding to vibrations of As atoms neighboring isolated In atoms in mostly Ga environments. For large z, an impurity mode appears, associated with isolated Ga atoms in an InAs environment and As atoms bonded to such isolated Ga atoms. 4.2.1. One-two mode behavior

Gal_=In~As is a two-mode sets of optical in Gal_~In=As

exhibits one-two mode behavior, unlike AI=Gal_~As, which system for all compositions z. By this we mean that two modes are apparent for z > 0.2, but only one for z < 0.2 (fig. 12). The second, InAs-like, mode is actually present, ::500 1

o i

i

0

o

280 A '7

E U

'7""

i

i

LO

0 o

)-

260

~

0

0

-~

-

0 Oo

TO

O

" " ' o " - ~ o..

i_ .Q

E

240

9

r

e-- L

4)

..

>

220

LO

"0

ee ee

9e

T0

""~176 e

one - m o d e 200

two:mode

0.0

t

GaAs

,I 0.2

I 0.4

I

0.6

I 0.8

1.0

x Ga~. z In x A s

InAs

Fig. 12. Dependence on alloy composition z of the LO and TO modes of Ga]_zlnzAs alloys, obtained from Lucovsky and Chen (1970), after Kobayashi and Dow (unpublished).

w

Phonons in semiconductor alloys

Wave number ~.l(cm'l)

0

I O0

200

l

Gao.47 Ino.53 As

l

300

LO n

( a ) g,//~',

389

i

rt~

>, f/IA m

-=~

I

1 I ro~ (b) "~, .L "~,

.=~ E~

LO ,

ffl

,

,

o.s~ Go o47Ino.~3 As o~

.~ ~,o 5

O'Ot 0

I 0

I0

20

:50

4.0

50

Frequency ~1 (1012rad/s) I I0

I 20

Energy "h~

60

I 30

(meV)

Fig. 13. Raman data (Pearsall et al. 1983) with incident and scattered light polarization vectors (a) parallel and (b) perpendicular to each other, compared with (c) the computed density of states for Gao.47In0.53As. After Kobayashi and Dow (unpublished).

but is not visible because it is resonant with the continuum of GaAs states. (Recall that there is no gap between the acoustical and optical bands of GaAs.) A comparison of Raman data (Pearsall et al. 1983) for Gao.n7In0.53As with theory (fig. 13 (Pearsall et al. 1983)), reveals features with TO:F and LO:F character. The ~ 270 cm -1 peak is an LO:F GaAs mode. A disorder-activated GaAs-like mode (mostly TO:X and TO:L) at 254 cm -1 corresponds to the density of states feature at ~ 265 cm -1, the differences in frequency being attributable to uncertainties in the theory. The experimental feature at ~ 224 cm-1 is a disorder-activated feature associated with As atoms surrounded by In atoms, and Ga atoms with second-neighbor In atoms. Finally the lowest energy experimentalpeak, at 226 cm -1, has TO:F character, due to InAs-like TO:F phonons and InAs-like disorder-activated TO:X and TO:L vibrations.

390

J.D. D o w et al.

Ch. 5

4.3. G a l _ x l n ~ S b Gal_~In~Sb has persistent features in its density of states, characteristic of both GaSb (fig. 14) and InSb (fig. 15) (Price et al. 1971), and behaves much as Gal_~In~As (fig. 16), exhibiting "one-two-mode" behavior (Brodsky et al. 1970) because of overlapping optical and acoustical bands. Ga

Fig. 14. Densities of states (top) calculated with recursion (solid) and Lehmann-Taut (dashed) methods, and phonon dispersion curves (bottom) for GaSb, in the Banerjee-Varshni model. The circles are infrared data (Hass and Henvis 1962) and the dots are from neutron scattering data (Farr et al. 1975). After Kobayashi et al. (1985a).

w

Phonons in semiconductor alloys

391

Wave number ~1 (cml) o

| 8F ~

. . . . .

IOO

2oo

:

I

i

I

1.5

TA:X

|TA:L

0.6F

|

LO'L

Jl

Loix

!',

LA:L

',i

LA:XI

| II

fl

O0

'-

I

1-

I l/

1

t._.

0

x

$

-J

0 20 40 12 Frequency Q (10 tad/s) I,,

I0

Energy fi~

,

I

20

(meV)

Fig. 15. Densities of states (top) calculated with recursion (solid) and Lehmann-Taut (dashed) methods, and phonon dispersion curves (bottom) for InSb, in the Banerjee-Varshni model. The neutron scattering data (Price et al. 1971) are represented as dots. After Kobayashi and Dow (unpublished).

vibrations dominate the GaSb-like optical band, with second-neighbor In atoms producing alloy modes. An impurity mode at ,-~ 165 cm -1 appears for z - 0.1, associated with isolated In atoms surrounded by mostly GaSb, and with Sb atoms bonded to such isolated In atoms. The InSb optical band overlaps the longitudinal acoustical band for all values of z, while the GaSb-like optical band at ~ 240 cm -1 is separate and distinct.

392

Ch. 5

J.D. D o w e t al.

Wave number ~.'l(cm'l)

0

,,,

I00

200

I

I

Go I-xlnxSb X=O.I

o.s 0.6 -

In

( Sb

0.3.

/

,fin )

"1

l

I

7 x=o. ~

"

O.

-

Zr,

| ~'

:~(Ga,ln}l~

.... , ,~'~ U, \

=: (9

X=O. 0.3 O.

x=o. O.

0

i/oo

I0

20

30

~

40

50

Frequency ~ (101=rad/s) I.

0

i I0

I 20

t 30

Energy "hf~ (meV) Fig. 16. Densities of states for Gal_=InxSb, as calculated from the recursion method (solid). The dashed line is the persistent-limit result for z = 0.5. After Kobayashi and Dow (unpublished).

4.4. InAsl_xSb~ Figure 17 displays the calculated results for InAsl_xSbx. The InAs-like optical band (near 210 cm -1) features mostly As vibrations, and the alloy mode corresponds to a central As atom surrounded by some second-neighbor Sb. The alloy modes in the vicinity of 170 cm -1, correspond to In or Sb atoms with nearby As atoms.

w

Phonons in semiconductor alloys

393

Fig. 17. Computed densities of states for InAsl_xSbx alloys (solid lines). The persistent-limit approximation for z = 0.5 is dashed. After Kobayashi and Dow (unpublished).

InAsl_xSb~ should exhibit two-mode behavior, although there was once a suggestion that it is a one-two-mode system (Lucovsky and Chen 1970).

4.5. GaAsl_xSb~ Figure 18 shows the computed densities of states of GaAsl_xSbx for selected values of Sb-content z. The slight deviations from the persistentlimit at ~ 170 cm -1, ,-~ 230 cm -1, and ~ 255 cm -1 are attributable to alloy modes associated with As and Sb vibrations. This system also displays

J.D. Dow et al.

394

Ch. 5

Fig. 18. Computed densities of phonon states for GaAsl_xSbx alloys. The dashed curve is the persistent-limit for x = 0.5. After Kobayashi and Dow (unpublished).

"One-two-mode" behavior (Lucovsky and Chen 1970), again because there are two modes, but one is obscured by an overlapping acoustical band.

5.

Correlated alloys

Up to this point, we have assumed that alloys such as AI~Gal_~As are random and uncorrelated, and that the probability of finding an A1 atom on a cation site is x. However, all of these alloys are products of chemistry,

w

Phonons in semiconductor alloys

395

and in chemistry, certain atoms prefer to stick together. In other words, the norm is that alloys are correlated, not that they are uncorrelated. As far as the recursion method is concerned, it makes little difference if the alloy is correlated or n o t - the very large (1000-atom) cluster that is fed into the computation must have the correlations built into it, and then the method proceeds. Therefore, with this method, the problem of correlations in the alloys is separated from the problem of determining the spectra of the (correlated or uncorrelated) alloys. 5.1. General correlations

It is often the case that we wish to predict the phonon density of states for a correlated alloy such as Inl_~Ga~AsuSbl_ u, but do not know how to generate the large cluster that initiates the recursion method, because we do not know how to construct that cluster with the appropriate correlations. For an uncorrelated alloy, we merely use random number generators to assure ourselves that the cation site is occupied with In a fraction 1 - x of the time, and with Ga with probability x; we similarly distribute As and Sb on anion sites. Constructing a cluster with desired nearest-neighbor correlations is more difficult than creating an uncorrelated cluster. The random-numbergeneration procedure cannot be used for a correlated cluster, because it will generate different correlations in different parts of the cluster: the central atom is deposited without any knowledge of its neighbors; the second atom is deposited with knowledge of only the first atom, which may or may not be a nearest-neighbor of the second; similarly the third atom's deposition senses the previous two atoms, but not subsequent ones. For the last atom, all of the neighbors are known. As a result, the cluster has higher-order correlations that depend on the deposition sequence. An easy way to circumvent this problem has been suggested by Redfield and Dow (1987), who proposed application of Monte Carlo procedures to a four-component Ising-like model of the cluster (Metropolis et al. 1953; Binder 1979). With this method, one can find an alloy configuration with the desired average alloy composition and nearest-neighbor correlations. This configuration can then be fed into the recursion method, and a phonon spectrum for the correlated crystal can be generated. The starting point is the energy for a particular alloy configuration, with the atoms occupying sites R = (n, b):

E/(kBT) = ~ j (R, R') + ~ h(R ), ~,h,

J.D. D o w e t al.

396

Ch. 5

where_._. we have h(R) = H~ if an atom of type v occupies site R, and j(R, R') = J~,~ if an atom of type # is at site R while an atom of type u is at R t, with R and R t being nearest-neighbors. The numbers H~ and J~,~ are adjustable parameters, which are independently varied until the desired correlations are produced. The probability of a particular alloy configuration (of, say, 1000 atoms) is proportional to exp(-E/kBT), where kB is Boltzmann's constant and T is the processing temperature for the cluster, which is normally not known. (Only E/kBT need be known.) Changing one atom of the cluster changes the probability of the configuration by altering one value of H~ and the values of J~,,~ for its four neighbors. We first choose values for Hv and J~,,~ and then solve this Ising-like model for an equilibrium configuration, using Monte Carlo techniques (Binder 1979). Then we adjust the parameters H~ and J~,~ by trial and error until we find an equilibrium configuration with the desired nearest-neighbor correlations, while executing enough iterations of the Monte Carlo method to obtain convergent values of z and y (the average alloy compositions) and N~,~, (the nearest-neighbor correlations). This procedure leads to a single

-

-

-

In,_~Ga= Asy Sb,_y NGo,As = 0 . 2 5 (uncorreloted) NGo,As = 0 . 3 4 II

GaAs

t

" "

"" "

N

Go,As

Il

=0.16

I

InAs

,

+

t

GaSb

K

I

I I

X o~

,

InS

t

L,' , el

m

L ....

t. . . . .

30

I"

,I!, 7 Ie

i

40

I

5O

(~ ( T H z ) Fig. 19. Theoretical Raman spectrum ixu,xu at room temperature for the substitutional crystalline alloy In0.sGao.sAso.sSb0.5. NGa,As is 0.25 (uncorrelated), 0.34, and 0.16 for the chained, solid, and dashed curves, respectively. Note how the GaAs and InSb peaks vary with correlation. After Redfield and Dow (1987).

w

Phonons in semiconductor alloys

397

cluster configuration with the required average composition and nearestneighbor correlations; this cluster (or an ensemble of such clusters) can then be fed into the recursion method to generate the phonon spectrum of the correlated cluster. Using this approach, Redfield computed the Raman spectrum of Inl_xGa~AsuSbl_ u, for NGa,As taking on the values 0.25 (uncorrelated), 0.34, and 0.16, obtaining the results of fig. 19.

0

Combined treatment of thermodynamic, electronic, and vibrational properties

The different theoretical machinery for thermodynamic, electronic, and vibrational properties of semiconductors can be merged to treat alloy phase transitions and their consequences. Particularly interesting examples are the substitutional crystalline alloys (GaSb)l_xGe2x and (GaAs)l_xGe2~, alloys that are actually metastable thin films, but from a practical viewpoint are stable because their lifetimes are ~ 10 29 years at room-temperature. These alloys necessarily undergo a transition from the zinc-blende crystal structure for small z (-+ 0) to the diamond structure for larger z (-+ 1) as the alloy composition varies, and this order-disorder transition has been described using a combination of a spin Hamiltonian (for the thermodynamic properties), an empirical tight-binding Hamiltonian (for electronic properties), and a Born-von Karman force constant model (for harmonic vibrational properties). The relationships among these three Hamiltonians are instructive, and provide a prescription for treating most of the physical properties of alloys using convenient Hamiltonians, all of which are inter-related, and each of which is well-tailored for the type of problem it is best suited to handle. The combination of the three Hamiltons, with the spin Hamiltonian treated in a mean-field approximation, is known as the Newman model (Newman and Dow 1983a, b; Newman et al. 1985, 1989a, b; Bowen et al. 1983; Barnet et al. 1984; Shah and Greene unpublished; Shah et al. 1986). The Newman model was actually developed in parallel with measurements of the optical absorption data for (GaAs)l_~Ge2~, and was completed before the data were completely analyzed. As a result it had a strong predictive element, as well as the normal descriptive character of a theory which explains new data. The experiments were by Greene and co-workers (Newman et al. 1983), and eventually showed that the fundamental band gaps of (GaAs)l_~Ge2~ thin films exhibit a characteristic "V"-shaped bowing as a function of alloy composition (fig. 20 (Newman and Dow 1983a, b; Newman et al. 1983)). The minimum of the "V" occurred at z ..~ Zc = 0.3. Data for other semiconductor alloys, prior to those measurements, had invariably

398

Ch. 5

J.D. D o w et al. 1.6

I

I

1.4

I

I

I

9 ~

,

%

\

L2 .

~

0 W W

I

\

~

\%% %N

1.0,

-

0s

9

-

0.6-

(Go As)

Composition, x

(Ge)

Fig. 20. Data for the direct-gap of (GaAs)l_xGe2x compared with mean-field theory (solid) and the virtual-crystal approximation (dashed). After Newman and Dow 1983a, b; Newman et al. 1983.

produced parabolic bowing: a band gap whose dependence on alloy composition z was a parabola. This "V" shape suggested an unanticipated phase transition, whose nature was novel at the time. 6.1. Order parameter The first parameter to be determined when studying a phase transition is the order parameter M, a quantity which is non-zero on one side of the phase-transition, and zero on the other. GaAs and Ge, to an excellent approximation, have the same lattice constant and the same crystal structure, with each having two atoms per unit cell (fig. 21), except that the cations and anions are the same, viz., Ge atoms, in the diamond structure of Ge, while they are different (Ga cations, As anions) in the zinc-blende structure of GaAs. When viewed along the [1,1,1] direction, GaAs has alternating rows of Ga and As atoms. Ge impurities introduced at low concentrations into GaAs occupy both Ga and As sites. At high Ge concentrations, z --+ 1, there is so little Ga or As that a Ge atom is unable to discern which row is supposedly a Ga row, and which row is

w

Phonons in semiconductor alloys

399

Fig. 21. A schematic model of the zinc-blende crystal structure with the (a) GaAs "ordered" structure, (b) the Ge "disordered" structure, (c) the GaAs-rich "ordered" structure, and (d) the Ge-rich "disordered" structure, after Bowen et al. (1983).

purportedly an As row. At low Ge concentrations the crystal structure has a non-zero average electric dipole moment per unit cell, whereas for z near unity this dipole moment is zero. At some intermediate Ge-concentration Zc, presumably at the bottom of the "V" in the optical absorption data, the alloy loses m e m o r y of its zinc-blende crystal structure, the dipole moment per unit cell vanishes, and for z > Zc Ge atoms occupy nominal "Ga" and "As" sites with equal probabilities, being unable to determine from the local environment the difference between nominal "Ga" and "As" sites. To describe the

J.D. Dow et al.

400

Ch. 5

order-disorder phase transition from the zinc-blende to the diamond crystal structure, we first define the order parameter M which characterizes the order in the zinc-blende structure and vanishes for the diamond crystal structure. The relevant parameter is proportional to the average electric dipole moment per unit cell, namely,

1

M = - { (PGa)"Ga"- (PGa)"As" + (PAs)"As"- (PAs)"Ga"}, 2 where, for example, (PGa)"As" is the probability of finding a Ga atom on a nominal "As" site. Thus M = 0 corresponds to a diamond structure, M = 1 - x is the ordinary random alloy with all Ga atoms on nominal "Ga" sites and all As atoms on "As" sites. M = x - 1 corresponds to the random alloy as well, but with the nominal sites incorrectly labeled: Ga is on "As" sites and As is on "Ga" sites.

6.2. Spin Hamiltonian With this order parameter, it is possible to model the order in the alloy with a spin Hamiltonian: on a particular site (the R-th site, where R stands for both the unit cell index n and the site index b), the spin Sh is up (+ 1), down ( - 1 ) , or zero, respectively, representing occupancy of that site by Ga, As, or Ge. In this picture, GaAs is a zinc-blende "antiferromagnet" diluted by "non-magnetic" Ge - a dilutional phase transition similar to the "ferromagnetic" 3He-nile superfluid to normal-fluid transition successfully treated by Blume, Emery, and Griffiths (1971). The resulting Hamiltonian reflects facts such as Ga's preference for bonding to As rather than Ga, and is derived in Newman and Dow (1983a). It has the form

H-JESf~S

h, - K E

Sf~S~, 2 2

2 h,,h

2 ~

h

where J, K, L, h, and A are parameters. While the spin-Hamiltonian formalism is not absolutely necessary for treating phase transitions, it is nevertheless the most widely studied and understood formalism for phase transitions, and so in practice, the first step after determining the order parameter in studying any phase transition is to

w

Phonons in semiconductor alloys

401

construct the appropriate spin Hamiltonian. Note that this Hamiltonian is designed to correctly model (only) the long-ranged correlations that are so important for scaling behavior and critical p h e n o m e n a - and so is normally relatively independent of the details of the physical system and does not normally model the atomic scale properties at all well, such as electronic or vibrational structure. While the spin Hamiltonian will provide information on the site-occupancies in semiconductor alloys, we shall have to construct different Hamiltonians whose parameters depend on those site-occupancies, if we are to model the electronic and vibrational properties of the alloys. Thus we shall require three different Hamiltonians to describe the physics of alloys: (1) the spin Hamiltonian to determine the equilibrium occupancies of the various sites; (2) a virtual-crystal empirical tight-binding theory of the electronic structure, and (3) a Born-von Karman empirical force-constant model of the lattice vibrations. Here we show how all three Hamiltonians can be constructed for a single alloy theory in a complementary manner. An equation for the order parameter M(x, T) can be obtained from the spin Hamiltonian, using standard variational techniques, and employing a mean-field approximation (Newman and Dow 1983a)

( J z M / k e T ) - tanh-1 [M/(1 - x)], where d is positive and is the "antiferromagnetic exchange" coupling, z is the number of nearest-neighbors to a site (z - 4 in our tetrahedral structures),

'~I

Order P a r o m e t e r M ( x ) = i .... I

0.8

0.6

u

v

~E

m

0.4

0.2

O.C

0.0

0.1

O.2

0.5

Composition x

I, 0.4

0.5

Fig. 22. Absolute value of the order parameter M(z) for (GaSb)l_~Ge2x or (GaAs)l_xGe2x, assuming Xc = 0.3. After Newman and Dow (1983b).

402

J.D. Dow et al.

Ch. 5

kB is Boltzmann's constant, and T is the effective "equilibrium" samplepreparation temperature. Neither the coupling constant J nor the effective equilibrium sample-preparation temperature T is known (in part because the sample-preparation is invariably not an equilibrium scheme), but the critical composition Zc ~ 0.3 at the bottom of the "V" in the optical data is known. In terms of this known quantity zc, the order parameter M(z; Zc) solves the equation [M/(1 - z)] = tanh[M/(1 - Zc)], and has the "traditional" form of fig. 22.

6.3. Empirical fight-binding Hamiltonian Once the order parameter M is known, the electronic structure can be evaluated in a virtual-crystal approximation (Parmenter 1955), using the empirical tight-binding method (Vogl et al. 1983). The virtual-crystal approximation is valid because electronically almost all of the technologically important semiconductors are in the amalgamated limit of Onodera and Toyozawa (1968), having band-widths of order 10 eV, with nearest-neighbor transfer matrix elements of the tight-binding Hamiltonian being about the same for all semiconductors with comparable lattice constants. In contrast, the variation of on-site matrix elements in most semiconductor alloys is almost an order of magnitude smaller than the conduction-band or valence-band width - implying that the electronic structures of these materials (especially for energies near the fundamental band gap) are appropriately described by the amalgamated limit of alloys or the virtual-crystal approximation. The empirical tight-binding method is especially well-suited for treating semiconductor alloys, because a single nearest-neighbor model Hamiltonian has been found that describes the electronic structures of all semiconductors rather well (Vogl et al. 1983). This model reproduces valence bands accurately, and provides an adequate description of the lowest conduction band of each semiconductor, while having manifest chemical trends in its matrix elements: off-diagonal matrix elements scale inversely with the square of the bond-length, according to Harrison's Rule (Harrison 1980), and on-site matrix elements vary as the electronegativity or as atomic energies in the solid (Vogl et al. 1983). This means that there are simple rules for adapting this Hamiltonian to treat any tetrahedral semiconductor, rules which can be used to treat alloys. The basis states lub,/~(o)) for this model are an sp3s * basis: one s-orbital, three p-orbitals, and one excited s-orbital termed s* on each of the two sites

w

Phonons in semiconductor alloys

403

per unit cell. The fight-binding basis states constructed from these orbitals partially diagonalize the Hamiltonian, reducing it to a 10 x 10 matrix. Those tight-binding states are

I~,b, re)

=

N -112Z lub,/~)) exp [i/~./~) + ik. gb}, n

where we have u = s, p~, Pu, P~ or s*, b(= a or c) refers to the anion or cation site, a n d / ~ ) specifies the nth unit cell. Then the empirical tight-binding Hamiltonian, at a wave-vector k in the first Brillouin zone, has the form

.

H0(k) =

[.s .sHp p] ' H~p

where the matrices Hs, ns,p, and Hp are: Is*a)

Hs =

i Vs,,s* s-,a9~ 0 0

Ipxa)

Is*c)

Es*,a 0 0

lpya)

isa)

Isc)

o

o I 's'a

0 0 Es,a K,s9O Vs,sg(~ Es,a

IS*C) [sa) Isc)

Ipza)

Ipuc)

Ipxc)

Ipzc)

I 0 0 0 gs*a,pcgl Vs*a,pcg2 Vs*a,pc g30 1 Hs,p ----- -- Vpa,s*cg~ - Vpa,s*cg~ - Vpa,s*cg~ 0 0 0 0 0 Vsa,pcgl Vsa,pcg2 gsa,pcg3 0 0 0 -- Vpa,scg 19 - Vpa,scg2* -- V,pa,scg3*

and [p~:a)

Sp

--

Ep,a 0 0

[pua)

0 Ep,a 0

IPza)Ipxc)

0 0 Ep,a

]pyC) IPzC)

Vx xgo Vx,yg3 Vx,yg2 Vx',yg3 Vx,xgo Vx,ygl Vx,yg2 Vx,ygl Vx go

Vx,xg~ Vx,yg~ Vx,ygl Ep,c v~,~g; v~,~9~ v~,~g; o Vx,yg~ Vx,x9~ Vx,xg~ 0

Ip~c)

0

Ep,r 0

Ip~a) [pya) [pza)

0

[puc)

Ep,c

Ip~c)

IS* a)

Is*c) Isa) Isc)

404

J.D. D o w e t

Ch. 5

al.

Here the basis state Lpuc) corresponds to the tight-binding state at wavevector k, with a p-orbital polarized along the y-direction, and centered on a cation (c) site. The functions g~ are 490 = exp(ik 9:~0) + exp(ik 9~l) + exp(ik 9~2) + exp(ik 9~3), 491 = exp(ik 9:to) + exp(ik 9:~1) -- exp(ik 9s

- exp(ik 9:~3),

492 -- exp(ik 9~0) - exp(ik 9:~1) + exp(ik- s

- exp(ik 9X3),

and 493 = exp(ik 9:~o) - exp(ik 9:~1) -- e x p ( i k . :~2) + exp(ik 9~3).

Here the :~i's are the relative coordinates of the nearest-neighbor atoms" x-'o - (aL/4)(1, 1, 1), Xl = ( a L / 4 ) ( 1 ,

-- 1, -- 1),

x~2 = ( a L / 4 ) ( - - 1, 1, -- 1),

and x-'3 = ( a L / 4 ) ( - - 1, - 1, 1), where aL is the lattice constant. The parameters of this Hamiltonian (Vogl et al. 1983) for GaAs and Ge are given in table 2. The virtual-crystal approximation for these alloys is achieved by averaging the matrix elements as follows" Es,"Ga" = ( P G a ) " G a " E s , G a ( G a A s ) + ( P G e ) " G a " E s , G e ( G e )

+ ( PAs )"Ga" Es,As (GaAs), where we have (PGa)"Ga" = (1 -- x + M ) / 2 , (PGa)"As" = (1 - x -

M)/2,

(PAs)"As" = (1 -- x + M ) / 2 ,

(PAs)"Ga"- (1 -- X - M ) / 2 ,

w

Phonons in semiconductor alloys

405

Table 2. Empirical tight-binding parameters of the nearest-neighbor sp3s * model Hamiltonian, in eV, after Vogl et al. (1983). For additional details, see Newman and Dow (1983a), which also goes beyond the present model, changes some of these matrix elements slightly, and incorporates some second-nearest-neighbor matrix elements assumed to be zero here. GaAs

Es,a Es,c

Ep,a Ep,c Es* ,a Es*,c Vs,s Vx,x Vx,y

Vsa,pc Vsc,pa

Vs*a,pc Vs*c,pa

-8.3431 -2.6569 1.0414 3.6686 8.5914 6.7386 -6.4513 1.9546 5.0779 4.4800 5.7839 4.8422 4.8077

Ge -5.8800 -5.8800 1.6100 1.6100 6.3900 6.3900 -6.7800 1.6100 4.9000 5.4649 5.4649 5.2191 5.2191

and ( V G e ) " G a " - ( P G e ) " A s " - X.

These relations connect the order parameter M of the spin Hamiltonian to the tight-binding matrix elements of the electronic Hamiltonian. Note that in the random-alloy limit we have M = ( 1 - x), and the usual virtual crystal approximation is recovered (because no Ga atoms are allowed to occupy "As" sites). The tight-binding Hamiltonian matrix can then be diagonalized, yielding its eigenvalues (band structure) and eigenvectors. The band edges obtained by Newman, using a Hamiltonian of this general type, are given in fig. 23 (Newman and Dow 1983a, b), and show characteristic splitting of the X minima on the zinc-blende side of the phase transition. 6.4. Force-constant model

Even though the phonons are invariably close to the persistent limit, it is possible to use the order parameter in a manner similar to the electronicproperties case to construct a Born-von Karman force-constant model of the Banerjee-Varshni type. Here we consider (GaSb)l_~Ge2~ as our prototypical alloy material, rather than (GaAs)l_~Ge2~, which has the masses

406

Ch. 5

J.D. D o w et al. 2.5

Theoreticol Bond Gop of (Go.~s)l.xGe2r

2.01-\

..C

C

uJ 1.0

o.sL 0.0

o.0

(GoAs)

-

I

0.2

r,c t

0.4

,

l

0.6

Composilion x

0.8

~.o

(Ge)

Fig. 23. Conduction band edges at F, L, and X, with respect to the valence band maximum, for (GaAs)l_zGe2z calculated using the mean-field model and a virtual-crystal approximation to the band structure. The assumed value of Zc is 0.3. The empirical tight-binding theory used here is a generalization of that in Vogl et al. (1983) that obtains a slightly better energy for the L minima of the conduction band (Newman and Dow 1983a). of its three constituent atoms all nearly equal, and so is atypically in the amalgamation regime of phonon alloy theory. The central idea behind constructing a Born-von Karman model of the alloys is that the occupancies by Ga, Sb, and Ge of nominal "Ga" or "Sb" sites in (GaSb)l_=Ge2= are given in terms of the order parameter M, as in the electronic case. Then the first-neighbor force constants c~ and/3 of the Banerjee-Varshni model of the alloy (GaSb)l_=Ge2= are obtained from those of the crystals GaSb and Ge for the various bonds, using the following averages: o~[(GaSb)l_zGe2z; Ga-Ga] = o~[GaSb; Ga-Ga], c~[(GaSb)l_=Ge2=; Sb-Sb] = o~[GaSb; Sb-Sb], o~[(GaSb)l_~Ge2~; Ga-Sb] = o~[GaSb; Ga-Sb], cz[(GaSb)l_=Ge2=; Ge-Ge] = c~[Ge], c~[(GaSb)l_=Ge2=; Ga-Ge] = (1 - z)o~[GaSb; Ga-Ga] + zoo[Gel, c~[(GaSb)l_=Ge2=; Sb-Ge] = (1 - z)c~[GaSb; Sb-Sb] + zoo[Gel.

w

Phonons in semiconductor alloys

407

Wave number ;~1(cn~l) 0 m

.

0.9

t

I00

200

300

!

|.

TO:X~ J

Ge

~o.6~-

r.:x

Lo:x(LA:X) Lo:rl

~ 9 O.O/

n

~iTO: L1

C-,

I

I

1

i

.I...~ I ...."'~

~x >

._10 i, 0

20

4.0

12

60

30 (meV)

40

F~requencYl fl (10 ,n~d/s) I0 20 Energy "ritZ

1

Fig. 24. (Top) Densities of states of Ge calculated using the recursion method (solid) and the Lehmann-Taut method (dashed), and (bottom panel) phonon dispersion curves (solid) compared with curves obtained from neutron scattering data (Nelin and Nilson 1972) (dotted). After Kobayashi et al. (1985a).

Similarly the second-neighbor force constants in the alloy, ~, #, and u, are given by the following averages

~[(GaSb)l_~Ge2~; Ga-Ga] = ~c[GaSb; Ga-Ga], )~[(GaSb)l_xGe2x; Sb-Sb] = )~a[GaSb; Sb-Sb], ~[(GaSb)l_~Ge2~; Ge-Ge] = 1[Ge], 1[(GaSb)l_~Ge2~; Ga-Sb] = {~c[GaSb; Ga--Ga] + ~a[GaSb; Sb-Sb]}/2, ~[(GaSb)l_xGe2x; G a G e ] = (1 - Z)~c[GaSb; Ga-Ga] + z)~[Ge],

408

J.D. D o w e t al.

Ch. 5

and )~[(GaSb)l_~Ge2~; Sb-Ge] = (1 - z))~a[GaSb; Sb-Sb] + z)~[Ge]. The phonon dispersion curves and density of states of Ge are given in fig. 24 (Nelin and Nilson 1972). With these prescriptions for constructing the force-constant matrices of the alloy, one need only (1) select a mini-cluster of five central atoms and note the probability p(z) with which that cluster appears (see p. 165 and p. 188 of Kobayashi 1985), (2) generate the remaining 995 atoms of the 1000-atom cluster using random number generators, (3) compute the density of states with the recursion method, and (4) average over all mini-clusters. 6.4.1. Results: (GaSb)l_zGe2~

Figure 25 shows the densities of phonon states of (GaSb)l_~Gee~ computed with the recursion method for the random alloy ( M = 1 - z), and in the virtual-crystal approximation. Clearly the virtual-crystal approximation is not appropriate for these materials. The main spectral features of the random alloy can be associated with vibrations of specific bonds (by dissecting the calculations), which are indicated on the figure. The acoustic bands evolve from GaSb-like to Ge-like as z increases. The top of the optical band in the alloy is Ge-like, while the bottom is GaSb-like. For small z, the highest-frequency peak of the optical band corresponds to zone-boundary transverse optical phonons, mostly from near L and X points of the Brillouin zone. The zone-boundary longitudinal optical phonons, mostly from near U and K, produce the shoulder at ~ 220 cm -1 for z = 0.1. An impurity local mode associated with light-mass Ge emerges at ~ 270 cm -1 for z ..~ 0.1, due to Ga-Ge bond vibrations, and as z increases combines with Ge-Ge features to form a Ge-like optical band. Comparable results for the mean-field theory value of M, together with the persistent limit, are given in fig. 26. Note that the persistent limit is not an extremely poor approximation, but the there are significant spectral features absent from that limit, associated with alloy modes. The main new features in the mean-field theory are associated with antisite disorder, namely Sb-Sb bonds vibrating (near ,-~ 195 cm-1 for x = 0.1) and Ga-Ga bonds (near ~ 270 cm -1, and near Ga-Ge vibrations). A rather narrow feature associated with Ga-Sb and Ge-Sb vibrations is apparent for 0.3 <~ z <~ 0.5. Acoustic mode intensities and positions vary with z, becoming Ge-like as z ~ 1. The vibrations of Ga-Ge bonds produce a local mode at ~ 270 cm -1 for small z < 0.1. The top of the optical band is Ge-like, while the bottom is GaSb-like.

w

Phonons in semiconductor alloys Wave number ~ "l(cm'l)

0

IO0

9| X=O.I

•

- o~

200

(M=0.9)

o.31-

~

II

Go-Ge

II I

~ , : : ~ l i i r~*~'

o~

II

!il~~.-G,

r/ x

= o,

"

300

CM=o.T~ ~i

~

"

409

x=o5

I o31

(M=O~) G,-sb G,-s~

i!

~,-o,

!i r~o-G,

-'Aii~

.~. oo = O. e"

x=o7

(M=031

. A too=fly Ge I~

[

oi ! / .

.Fl

0

0.6t X=0"9

O,

0

(M'O.I)

20

~

I

40

60

t Frequency ,~. (lOl=rad/s),

0

I0

20

Energy "h~

30

(meV)

40

Fig. 25. Computed densities of states for phonons in (GaSb)l_zGe2x in the virtual-crystal approximation for z = 0.5 (dashed) and for the random alloy approximation, M = 1 - z. The bonds responsible for spectral features are indicated on the figure. After Kobayashi et al. (1985a, b).

6.4.1.1. Comparison with (GaSb)l_xGe2~ data. (GaSb)l_~Ge2~ Raman spectra have been reported by Beserman et al. (1985) and by Krabach et al. (1983). These data are compared with both the mean-field theory (with order parameter M ) and the random alloy theory ( M = 1 - z) in figs 27, 28, 29, 30, and 31. The Raman data contain matrix elements (and selection rules (Hayes and Loudon 1978)) which weigh different parts of the density of states differently, and the theory lacks long-ranged forces that can shift the mode frequencies, but a comparison between the two imprecise theories and the data is highly instructive. Two major issues are: (i) is there anti-site

410

J.D. D o w e t al.

Ch. 5

Wave number ~.'l(crn'l)

0

I00

I~-

I

200

o.9~(GoSb)t.x

300

I

I

Gezx r---G,-Sb

/ M =mean f i e l d theory

O.S~-X=O.I

(M=0.667)

l/

Ga-G,

o.s]- X = 0.3 ( M = 0.003 )

Go- Sb.--~ Ge-Sb A

|

,'-

/

Ga-G,

I--Go-Ge --

O

Ge-

os~- X=0.5 "/ m m I

" "

o.3k

Ga-Ge

(M-O) Go-Go G.-Sb._,~, I r-S.-G,' Ga-Sb :t~ I , i!

9

"6 o.o~

-~== 9 o6}. x=o.7 ( M - O )

a

o3

x'~ ("'~

A

Oi!o I

0

Frequency 9 (lO'~ad/s), I .... I0

20

Energy "fif~

30

(meV)

....

40

Fig. 26. Computed densities of states for phonons in (GaSb)]_xGe2x in the persistent approximation for z = 0.5 (dashed) and for the mean-field approximation, with the values of the order parameter M indicated. The bonds responsible for spectral features are indicated on the figure. After Kobayashi et al. (1985a, b). disorder? and (ii) is such disorder well-described by the mean-field theory? A positive answer to the first question would validate the Newman model's central qualitative feature, namely a zinc-blende to diamond phase transition, while a positive answer to the second would confirm the model quantitatively. An examination of figs 25 and 26 reveals that the most prominent characteristic of the mean-field theory is the spectral feature associated with vibration of Sb-Sb bonds near ,,~ 195 cm -1. This feature is the signature of anti-site disorder, namely Sb on nominal Ga sites. We shall argue that a careful comparison of the two theories, the random alloy theory (M = 1 - z) and the mean-field theory, with the data indicates that the Sb-Sb vibrations

w

Phonons in semiconductor alloys Wive number k"(cni 1 ) IOO 200

300

'A

,

~" ,

T'-

i

i

,

l

,

i

~

i

(b

0.3

!o.o r

,_.

: o.s

,,

I[

X=0.13 M=I-x

!

'(o,q

i

X=O.13 o.s M = mean field theory

411

(c)

II

"

II

F?q...=y ,~ (~,"r.f.) ,o ,o ,o Energy f ~

"o

(meV)

Fig. 27. (a) Raman spectrum (Beserman et al. 1985; McGlinn et al. 1988; Krabach et al. 1983) and calculated density of states (b) in the mean-field approximation and (c) in the random alloy approximation (M = 1 - z) for (GaSb)]_xGe2x with z = 0.13. Note the gap in the random alloy approximation spectrum around ,-~ 195 cm -1, which corresponds to the absence of Sb-Sb vibrations. After Kobayashi et al. (1985a).

Wive number k'l(cnl 1 ) 0

I00

200

i

I

I

300

!

X =0.24

I

|

(a)

I

l

i

l

j ~o.3F

/I A

H

' ~

II

I

o.o

~)

FlmquencYIfl I0

Energy

20 fit3

(lO'=rald/=) 30 (meV)

, 40

Fig. 28. (a) Raman spectrum (Beserman et al. 1985; McGlinn et al. 1988; Krabach et al. 1983) and calculated density of states (b) in the mean-field approximation and (c) in the random alloy approximation (M = 1 - z) for (GaSb)]_xGe2x with z = 0.24. Note the gap in the random alloy approximation spectrum around ,,~ 195 cm -1 , which corresponds to the absence of Sb-Sb vibrations. After Kobayashi et al. (1985a).

412

Ch. 5

J.D. Dow et al. Wavenumber~.'l(cm'l)

o.

=~' 9

,,oo

~,oo

3o0

I Romon spectrum

I

I

I

I

I

I

o.s~ M=meonfield

I

I

I

]

theory

l |

I/

'0 00

|

20 40 60 F~equencyis (101=rald/s) I I0 2o 3o 40 Energy 'h~ (meV)

l 0

Fig. 29. (a) Raman spectrum (Beserman et al. 1985; McGlinn et al. 1988; Krabach et al. 1983) and calculated density of states (b) in the mean-field approximation and (c) in the random alloy approximation (M = 1 - z) for (GaSb)l_zGe2x with z = 0.34. Note the gap in the random alloy approximation spectrum around ~ 195 cm -1, which corresponds to the absence of Sb-Sb vibrations. After Kobayashi et al. (1985a).

o | ~

|...

Wave number~'l(cm'l) IOO 200 300 , i , X=0.56 (a)l R a m a n ~ I

I

I

I

I

/l

-~ ,r

"k

.

.

"~= 1" X= 0.56 ~ ~ - M=I-x

0

.

.

.

.

I

t.

1 J,

(c)1'

20 40 60 F?quency ~ (1012rald/a) ,o Energy

=o

fl~

3o

(meV)

,'o

Fig. 30. (a) Raman spectrum (Beserman et al. 1985; McGlinn et al. 1988; Krabach et al. 1983) and calculated density of states (b) in the mean-field approximation and (c) in the random alloy approximation (M = 1 - z) for (GaSb)l_=Ge2= with z = 0.56. After Kobayashi et al. (1985a).

w

Phonons in semiconductor alloys

413

are present (confirming the qualitative physics of the Newman model while leaving unanswered the issue of whether those data quantitatively support the numerical values of the mean-field order parameter M). z ~ 0.13. The main features of the Raman data displayed in fig. 27 are also present in both the mean-field (order parameter M) and random alloy (M = 1 - z) theories: (i) the Ga-Ga and Ga-Ge local modes at 270 cm -1, (ii) a Ge-like longitudinal optical model at ~,, 260 cm -1 which is unusually broad for semiconductor alloys - most likely due to Ge atoms in Ge-Ge and Ga-Ge bonds forming sidebands to the Ge-Ge vibrations, and (iii) an asymmetry in the lineshape of the GaSb-like longitudinal optical mode which dominates the spectrum. Beserman et al. (1985) assigned this asymmetry to a Brillouin-zone center GaSblike transverse optical mode which appears in the Raman data due to a slight breakdown of a selection rule. We confirm this identification while suggesting that zone-boundary longitudinal optical phonons of Wave number ~.'l(crn'l) I00

200

500

'"

' A'(a

X=0.8'

Raman s p e c t r u m ~ ~ .

_=

)1I

E n,, |

I

!

I

I

i

X=O.8

0.6

I

I

(~))

o.3 I

~ c ~

X=O.8

0.6

.( )

0.5 0.0

20

0

40

60

Frequency a (lO~adis) 0

,

1

I0

,,,

I

20

Energy flQ

I

50

(meV)

I

40

Fig. 31. (a) Raman spectrum (Beserman et al. 1985; McGlinn et al. 1988; Krabach et al. 1983) and calculated density of states (b) in the mean-field approximation and (c) in the random alloy approximation (M = 1 - z) for (GaSb)l_xGe2z with z = 0.8. After Kobayashi et al. (1985a).

414

Ch. 5

J.D. Dow et al.

GaSb at ~ 220 cm-] in the density of states may also contribute to the line's asymmetry, by being "disorder-activated" (a breakdown of the crystalline selection rules caused by alloying). In the spectral region near the Sb-Sb vibration at ~ 195 cm-1, the data lack the gap of random alloy theory, and so seem to favor the mean-field interpretation with its anti-site disorder. ii) 0.24 ~< z ~ 0.56. The Raman data for 0.24 <~ z <~ 0.56 (figs 28, 29, and 30) have two broad, asymmetric main peaks (the higher one associated with Ge longitudinal optical vibrations, and the lower with GaSb longitudinal optical phonons), and a weaker disorder-activated longitudinal acoustical mode at ,-~ 150 cm -1. The breadth of the Ge longitudinal optical band is attributed to sidebands (inhomogeneous broadening) associated with Ge-Ge, Ga-Ge, and Ga-Ga bonds. The long tail on the GaSb-like longitudinal optical phonon band near 195 cm -1 is probably due to Sb-Sb vibrations and the anti-site disorder of the mean-field model (which is absent in the random alloy model with M = 1 - z). Wave number k'l(cm'l)

o

IOO

200

3oo

'oo

Lt

lU O.

o.s

X=0.5

~.o.~

i~!

t ,.o,

ft l

~

I

0

20

4O

6O

Frequency f2 (1012rad/s)

~1,

I I0

I 20

Energy "hfl

310

I 40

(meV)

Fig. 32. Computed densities of states for (GaAs)l_zGe2x in the virtual-crystal approximation (solid) and persistent limit (dashed), after Kobayashi and Dow (unpublished).

w

Phonons in semiconductor alloys

415

iii) z ,,~ 0.8. For z ~ 0.8 the Raman data show an asymmetric Ge-like band associated with longitudinal optical phonons at the L and X points of the Brillouin zone and longitudinal acoustic modes at X. The low frequency shoulder appears to be due to vibrations of G a G e and Ge-Ge bonds.

6.4.1.2. Comparison with (GaAs)l_xGe2x data. The masses of Ga, Ge, and As are all nearly equal, and so (GaAs)l_~Ge2~ is one of the few semiconductor alloys that can be described reasonably well by the virtual-crystal approximation (Parmenter 1955). Indeed, fig. 32 shows that the density of states spectra predicted by the virtual-crystal approximation and the persistent limit are quite similar. Figures 33 and 34 are the corresponding predictions for the random-alloy and mean-field models, and are so close to one another (in comparison with uncertainties in the theory) that one could not reliably discriminate between the two theories by comparing them with spectra.

Wave number ;L'l(cm"1)

0

I00

200

300.....

o.,I ,oo,,i,., o,,'.

'7, 06 t

X=0.3 (M=O.71

{o

= o.,t •

_ ~

N 9

(,.o.~)

ii

i

0,3

0.0 o.6

C

ol

o

0.6t

X"0.9

(M=O.I)

A

t Frequency ,CZ,, (101=rad/s) I I , I

~ 0

I0

20

Energy flC4

30

(meV)

40

Fig. 33. Computed densities of states for phonons in (GaAs)]_zGe2x in the virtual-crystal approximation for z = 0.5 (dashed) and for the random alloy approximation, M = 1 - z, for various compositions z. After Kobayashi and Dow (unpublished).

416

J.D. Dow et al.

Ch. 5

Wave number ~.'l(cm'l)

o

200

I00

300

O.g

0.6 0.3 0.0 "7

O.S 0.3

~0.0

X-O.5 (M-O)

' ~ O.a W ~ 0.3 ffl ~ 0.0 o

-_>'o=g,, x.o7 0.6

(M.o)

"

A

9

0.3 O.

0

20

40

I 60

, Frequency,a (lO'=rad/s,) o |o 20 30 40 Energy ~ (meV)

Fig. 34. Computeddensities of states for phonons in (GaAs)l_xGe2x in the persistent approximation for x = 0.5 (dashed) and for the mean-field approximation for various compositions x. After Kobayashi and Dow (unpublished).

6.4.1.3. Discussion of (GaSb)l_zGe2x and (GaAs)l_xGe2x. We have shown that the theory of alloys can be broken up into three complementary Hamiltonians, each with predictive power. By comparing the data for (GaSb)l_~Ge2~ with Raman spectra, we have confirmed that there is an order-disorder transition in these materials, and that there is anti-site disorder. The Sb-Sb vibrations provide a spectral feature which is a signature of such disorder. X-ray data, which show the characteristic (200) zinc-blende reflection disappearing as the alloy makes the transition from the zinc-blende structure to the diamond structure confirm this viewpoint (Newman et al. 1989b), providing additional quantitative support for the phase transition and qualitative but not necessarily quantitative confirmation of the Newman model (fig. 35). The issue of the quantitative nature of these alloys remains open. Holloway and Davis (1984), and Davis and Holloway (1987) have argued that the Newman treatment, which assumes an effective growth temperature and

w 1.0 0.8 C ,eJ C

0 o N

~

- ',,

Phonons in semiconductor alloys

417

G o A s ) , . , (Gez) , / Go P

\

I. ..........

0.6

0.4

0 N .ip

"~ o . z

9

E

,.~

0

z

0.0 0.0

O.Z

0.4 Alloy

0.6

Composition

0.8

1.0

x

Fig. 35. The intensity of the (200) x-ray diffracted beam of (GaAs)i_xGe2x (on a GaP substrate) and (GaSb)l_xGe2x (on a GaAs substrate), normalized to the intensity of the (400) diffraction spot, versus alloy composition x, after Newman et al. (1989b), Barnett et al. (1984), Shah and Greene (unpublished). This intensity should vanish as the zinc-blende phase vanishes. some aspects of equilibrium growth, should be replaced with a kinetic model of growth. Stem et al. (1985) have reported intricate extended z-ray absorption fine-structure data analyses, which lead them to conclude that there is a zinc-blende to diamond transition on a large length scale, but not on the ,-~ 3 A scale of the Newman model. (This implies Sb-Sb bonds, but at the surfaces of large clusters.) They have constructed a kinetic model to go along with their hypothesis. Gu and co-workers have tried to include even more complicated correlations (Gu et al. 1992; Wang et al. 1989; Zhang et al. 1991). It is clear that these materials are non-equilibrium alloys whose growth is governed by both kinetic and thermal effects, and whose physics remains interesting, and not fully understood. 6.4.1.4. Disorder and entropy. One of the very nice results to emerge from studies of (GaAs)l_~Ge2~, (GaSb)l_~Ge2~, and other (III-V)I_~(IV)2~ alloys (Jenkins et al. 1984, 1985) is the fact that the entropy of disorder has been measured optically (Newman et al. 1985). The Ge-like LO phonon line of (GaSb)l_~Ge2~ is inhomogeneously broadened. The simplest way to understand this broadening is to realize that the broadening is a measure of the alloy disorder, and the disorder is a measure of the entropy S(z) - which

418

J.D. Dow et al. T

=

E u 40.0

~'

=

Ch. 5 "

i

1.4

Phase-transition

a"="--- - ~ x , =

model-

90.3

1 , 2 . wm

i.o .-~

"i 30.0

0.8 ~ ==m

-- zo.o

0 . 6 a. o

o E n., I 0 . 0 o

I'~

0.4 = W 0 . 2 ~ '~-

I,' ~/

0.0

( Go Sb)

0

~

O.Z

I

0.4

I

X

0.6

I

0.8

( GoSb)l.x Get,t

.o

(Ge)

Fig. 36. Total Raman line-width (left-hand axis) and entropy (fight-hand axis) of the Ge-like LO mode in (GaSb)]_=Ge2=, after (Newman et al. 1985). The dashed line is the random alloy or on-site model of the entropy, while the solid line in the mean-field theory.

can be computed from the spin Hamiltonian in the mean-field approximation, and is (Newman et al. 1985). 1

S(x)/kB = : {[(1 - x + M)ln[(1 - x + M)/2] + 2x ln[x] Z

+ [(1 - x - M)] ln[(1 - x - M)/2]}. The theoretical entropy is presented in fig. 36, where it is compared with the data for the line-width (Newman et al. 1985), producing a strikingly direct measurement of the entropy of alloy disorder.

7. Superlattices A few words about alloy superlattices, such as GaAs/AI~Gal_~As superlattices, are in order. What we have learned about bulk semiconductor alloys is that their density-of-states spectra are not virtual-crystal-like, but are best described by a theory which contains the main persistent features of the constituents' densities of states, plus alloy modes. The main spectral features can be assigned to vibrations of specific clusters or bonds, and these assignments reflect the local order on a scale of ~ 3 ,~.

w

Phonons in semiconductor alloys

419

The periods of most superlattices (excluding the extremely short-period superlattices) are much larger than 3 A, and so we expect the density of states spectrum of a superlattice such as GaAs/Al~Gal_xAs to exhibit the features of GaAs, AlxGal_~As, and some peaks associated with "zonefolding". That is, an NGaAs • NAICaAs superlattice with periodically repeated NGaAs bilayers of GaAs and NAIGaAs bilayers of AI~Gal_~As has a new Brillouin zone that is smaller in the growth direction, and can be obtained by "folding" the GaAs or Al~Gal_~As zone to reflect the larger true lattice constant in the growth direction: (NGaAs+ NAIGaAs)agrowth instead of agrowth, where t~growth, is the lattice constant of GaAs or AI~Gal_~As in the growth direction. The phonon dispersion curves are then "folded" into the new Brillouin zone of the superlattice, and, for example, acoustic modes of the GaAs and AI~Gal_~As constituents at k ~ 0 can be folded back to near k = 0 in the superlattice's Brillouin zone, and become infrared or Raman active (with their activity perhaps insured by the relaxation of selection rules broken by the alloy). These folded modes are the primary new spectral features of alloy superlattices, and are discussed thoroughly by Jusserand and Sapriel (1981), Sapriel (1989) and others (Ren et al. 1987, 1988a, b; Ren and Chang 1991; Chu et al. 1988). One of the especially interesting topics of the last few years has been the growth of spontaneously ordered superlattices instead of alloys. Attempts to grow alloy semiconductors result in unexpected superlattices, because of spontaneous ordering (Gu et al. 1987, 1992; Wang et al. 1989; Zhang et al. 1991; Kuang et al. 1985; Jen et al. 1986; Gomyo et al. 1994; Mascarenhas et al. 1989; Homer et al. 1993, 1994; Sinha et al. 1993). The techniques developed here for treating correlated alloys should prove especially applicable to this new field.

8. Summary Until theory and experiment reach new levels of precision, the physics of semiconductor alloys, whether superlattices or not, appears to be adequately treated by the recursion method and a Bom-von Karman model such as those described here. But light-scattering studies such as resonant Raman scattering spectroscopy are beginning to provide new and precise information about vibrations in alloys, and, when analyzed with improved models of long-ranged forces, will no doubt allow very detailed comparisons of theory with data. A better understanding of correlations in alloys should follow, and will involve the theory of phase transitions as well. In summary, the recursion method provides a very good way to understand the phonon spectra of the semiconductor alloys discussed here, as well as

420

J.D. Dow et al.

Ch. 5

the spectra of other III-V and II-VI alloys (Amirthara et al. 1985; Fu and Dow 1987; Lucovsky et al. 1975, 1976). While this method provides a satisfactory description of the density of states per squared frequency in an alloy, its primary limitation is that it normally does not provide matrix elements (which can be calculated for large clusters (Ren and Dow 1992), but not for clusters so large as those amenable to treatment with the recursion method). The discussion of one-mode versus two-mode k = 0 optical phonon behavior is seen to be largely moot: unless the masses of the atoms are virtually equal, one invariably finds two-mode behavior, but one of the modes may be resonant with a phonon continuum, and therefore may not be visible. The phonon spectra of the substitutional crystalline semiconducting alloys are nearly "persistent", but the deviations from this persistent behavior are the alloy modes, and those modes are the subject of investigations of alloy physics. (One should not claim success for an elegant theory that does little more than reproduce the persistent limit!) The advantage of the recursion method as applied to the alloy physics of the semiconductor alloys is that it can be dissected to associate specific features of a spectrum with particular local configurations. With the Redfield method of generating correlated clusters, and with the separation of the alloy problem into the following parts: (i) determination of the cluster (correlated or not), (ii) evaluation of the forces, and (iii) computation of the density of states, it is now possible to predict the main properties of the substitutional crystalline alloy semiconductors. The present work assumed that all of the atoms of the alloy occupied crystalline sites on a zinc-blende lattice. Of course, with strains in the materials, and with diffusion at finite temperatures, this assumption will not be 100% valid. Therefore, one of the next major problems to be addressed is the physics of substitutional nearly-crystalline alloys whose crystal structures deviate from the perfect geometry of the zinc-blende structure, due to strain, for example. This will be interesting from the viewpoint of pure physics, and it has important technological implications: interfaces between alloy semiconductors exhibit interdiffusion, and this interdiffusion is one of the elements scattering electrons and limiting the electronic mobility- and hence limiting the performance of high-speed optoelectronic devices based on these materials.

Acknowledgements

We are grateful to the U.S. Office of Naval Research, the U.S. Air Force Office of Scientific Research, the U.S. Department of Energy for their gen-

Phonons in semiconductor alloys

421

erous support of this work (Contract Nos. N00014-92-J- 1425, AFOSR-910418, and DE-FG02-90ER45427), to D. Pulling for his assistance, and to P. Leath for a stimulating discussion of the CPA. Finally we wish to remember E.P. O'Reilly, who introduced us to the Cambridge recursion routines, and A. Kobayashi, whose computational industry produced many of the results reported here. References Amirtharaj, P.M., K.K. Tiong, P. Parayanthal, F.H. Pollack and J.K. Furdyna (1985), J. Vac. Sci. Technol. A3, 226. Banerjee, R. and Y.P. Varshni (1969), Can. J. Phys. 47, 451. Barnett, S.A., B. Kramer, L.T. Romano, S.I. Shah, M.A. Ray, S. Fang and J.E. Greene (1984), in: Lauered Structures, Epitaxy, and Inter.aces, Ed. by J.M. Gibson and L.R. Dawson (North-Holland, Amsterdam). Bernasconi, M., L. Columbo, L. Miglio and G. Benedek (1991), Phys. Rev. B 43, 14447. Beserman, R., J.E. Greene, M.V. Klein, T.N. Krabach, T.C. McGlinn, L.T. Romano and S.I. Shah (1985), in: Proc. 17th Int. Conf. on the Physics of Semiconductors, San Francisco, Ed. by D.J. Chadi and W.A. Harrison (Springer, New York), p. 961. Binder, K., Ed. (1979), Monte Carlo Methods in Statistical Physics (Springer, New York). Blume, M., V.J. Emery and R.B. Griffiths (1971), Phys. Rev. A 4, 1071. Bonneville, R. (1984), Phys. Rev. B29, 907. Born, M. and K. Huang (1954), Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford). Bowen, M.A., A.C. Redfield, D.V. Froelich, K.E. Newman, R.E. Allen and J.D. Dow (1983), J. Vac. Sci. Technol. B 1, 747. Brodsky, M.H., G. Lucovsky, M.F. Chen and T.S. Plaskett (1970), Phys. Rev. B 2, 3303. Carles, R., A. Zwick, M.A. Renucci and J.B. Renucci (1982), Solid State Commun. 41, 557. Chen, Y.S., W. Shockley and G.L. Pearson (1966), Phys. Rev. 151, 648. Cheng, I.F. and S.S. Mitra (1968), Phys. Rev. 172, 924. Cheng, I.F. and S.S. Mitra (1970), Phys. Rev. B 2, 1215. Cheng, I.F. and S.S. Mitra (1971), Adv. Phys. 20, 359. Chu, H., S.-F. Ren and Y.-C. Chang (1988), Phys. Rev. B 37, 10746. Cochran, W. (1959), Proc. R. Soc. London, Ser. A: 253, 260. Davis, L.C. and H. Holloway (1987), Phys. Rev. B 35, 2767. Dawber, P.G. and R.J. Elliott (1963), Proc. R. Soc. London, Set. A: 273, 222. Dean, P. (1961), Proc. R. Soc. London, Ser. A: 260, 263. Dean, P. (1972), Rev. Mod. Phys. 44, 127. Dick, B.G. Jr. and A.W. Overhauser (1958), Phys. Rev. 112, 90. Elliott, R.J., J.A. Krumhansl and P.L. Leath (1974), Rev. Mod. Phys. 46, 465. Ewald, P.P. (1921), Ann. Phys. 64(4), 253. Farr, M.K., J.G. Traylor and S.K. Sinha (1975), Phys. Rev. B 11, 1587. Fu, Z.-W. and J.D. Dow (1987), Phys. Rev. B 36, 7625. Gomyo, A., K. Makita, I. Hino and T. Suzuki (1994), Phys. Rev. Lett. 72, 673 and references therein. Gonis, A. and J.W. Garland (1977), Phys. Rev. B 16, 2424. Gu, B.-L., K.E. Newman and P.A. Fedders (1987), Phys. Rev. B 35, 9135. Gu, B.-L., L. Ni and J.-L. Zhu (1992), Phys. Rev. B 45, 4071.

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Harrison, W.A. (1980), Electronic Structure and the Properties of Solids, Ed. by W.H. Freeman (San Francisco). Hass, M. and B.W. Henvis (1962), J. Phys. Chem. Solids 23, 1099. Haydock, R. (1980), in: Solid State Physics, Vol. 35, Ed. by H. Ehrenreich, E Seitz and D. Turnbull (Academic Press, New York), p. 215. Hayes, W. and R. Loudon (1978), Scattering of Light by Crystals (Wiley, New York). Heine, V. (1980), in: Solid State Physics, Vol. 35, Ed. by H. Ehrenreich, E Seitz and D. Turnbull (Academic Press, New York), p. 1. Herscovici, C. and M. Fibich (1980), J. Phys. C 13, 1635. Holloway, H. and L.C. Davis (1984), Phys. Rev. Lett. 53, 830. Horner, G.S., A. Mascarenhas and R.G. Alonso (1994), Phys. Rev. B 49, 1727. Horner, G.S., A. Mascarenhas and S. Froyen (1993), Phys. Rev. B 47, 4041. Hu, W.M., J.D. Dow and C.W. Myles (1984), Phys. Rev. B 30, 1720. Illegems, M. and G.L. Pearson (1970), Phys. Rev. B 1, 1576. Jen, H.R., M.J. Cherng and G.B. Stringfellow (1986), Appl. Phys. Lett. 48, 1603. Jenkins, D.W., K.E. Newman and J.D. Dow (1984), J. Appl. Phys. 55, 3871. Jenkins, D.W., K.E. Newman and J.D. Dow (1985), Phys. Rev. B 32, 4034. Jusserand, B. and J. Sapriel (1981), Phys. Rev. B 24, 7194. Kaplan, T., EL. Leath, L.J. Gray and H.W. Diekl (1980), Phys. Rev. B 21, 4230. Kawamura, H., R. Tsu and L. Esaki (1972), Phys. Rev. Lett. 29, 1397. Kellerman, E.W. (1940), Philos. Trans. R. Soc. London, Ser. A: 238, 513. Kelly, M.J. (1980), in: Solid State Physics, Vol. 35, Ed. by H. Ehrenreich, E Seitz and D. Turnbull (Academic Press, New York), p. 296. Kim, O.K. and W.G. Spitzer (1979), J. Appl. Phys. 50, 4362. Kobayashi, A. and J.D. Dow, unpublished. Many of the details of the approach discussed here can be found in (Kobayashi 1985). Kobayashi, A. K.E. Newman and J.D. Dow (1985a), Phys. Rev. B 32, 5312. Kobayashi, A., J.D. Dow and E.E O'Reilly (1985b), Superlatt. Microstruct. 1, 471. Kobayashi, A. (1985), PhD Thesis, University of Illinois at Urbana-Champaign, Department of Physics. Kohn, W. and L.J. Sham (1965), Phys. Rev. 140, 1133. Krabach, T.N., N. Wada, M.V. Klein, K.C. Cadien and J.E. Greene (1983), Solid State Commun. 45, 895. Kuan, T.S., T.E Kuech, W.I. Wang and E.L. Wilkie (1985), Phys. Rev. Lett. 54, 201. Kunc, K. (1973), Ann. Phys. 8, 319. Leburton, J.-P., J. Pascual and C. Sotomayor Torres, Ed. (1993), Phonons in Semiconductor Nanostructures, Proc. NATO Advanced Research Workship on Phonons in Semiconductor Nanostructures, St. Feliu de Guixols, Spain, September 15-18, 1992 (Kluwer, Dordrecht). Lehmann, G. and M. Taut (1972), Phys. Status Solidi B: 54, 469. Leibfried, G. and W. Ludwig (1961), in: Solid State Physics, Vol. 21, Ed. by E Seitz and D. Turnbull (Academic Press, New York), p. 275 and references therein. Lucovsky, G. and M.E Chen (1970), Solid State Commun. 8, 1397. Lucovsky, G., K.Y. Cheng and G.L. Pearson (1975), Phys. Rev. B 12, 4135. Lucovsky, G., R.D. Burnham and A.S. Alimonda (1976), Phys. Rev. B 14, 2503. Maradudin, A.A., E.W. Montroll, G.H. Weiss and I.P. Ipatova (1971), in: Solid State Physics, Supplement 3, 2nd edn, Ed. by H. Ehrenreich, E Steitz and D. Turnbull (Academic Press, New York). The first edition was co-authored by Maradudin, Montroll and Weiss only. Martin, R.M. and EL. Galeener (1981), Phys. Rev. B 23, 3071. Mascarenhas, A., S.R. Kurtz, A. Kibbit and J.M. Olson (1989), Phys. Rev. Lett. 63, 2108. McGlinn, T.C., M.V. Klein, L.T. Romano and J.E. Greene (1988), Phys. Rev. B 38, 3362.

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Metropolis, N., A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller (1953), J. Chem. Phys. 12, 1087. Myles, C.W. and J.D. Dow (1979), Phys. Rev. B 19, 4939. Nelin, G. and G. Nilsson (1972), Phys. Rev. B 5, 3151. Newman, K.E. and J.D. Dow (1983a), Phys. Rev. B 27, 7495. Newman, K.E. and J.D. Dow (1983b), J. Vac. Sci. Technol. B 1, 243. Newman, K.E., A. Lastras-Martinez, B. Kramer, S.A. Barnett, M.A. Ray, J.D. Dow, J.E. Greene and P.M. Raccah (1983), Phys. Rev. Lett. 50, 1466. Newman, K.E., J.D. Dow, A. Kobayashi and R. Beserman (1985), Solid State Commun. 56, 553. Newman, K.E., J.D. Dow, B.A. Bunker, L.L. Abels, P.M. Raccah, S. Ugur, D.Z. Xue and A. Kobayashi (1989a), Phys. Rev. 39, 657. Newman, K.E., J.D. Dow, B.A. Bunker, L.L. Abels, P.M. Raccah, S. Ugur, D.Z. Xue and A. Kobayashi (1989b), Phys. Rev. 39, 657. Nex, C.M.M. (1978), J. Phys. A 11, 653. Nex, C.M.M. (1984), Computer Phys. Commun. 34, 101. Onodera, Y. and Y. Toyozawa (1968), J. Phys. Soc. Jpn 24, 341. Orlova, N.S. (1979), Phys. Status Solidi B: 93, 503. Payton, D.N. and W.M. Visscher (1967a), Phys. Rev. 154, 802. Payton, D.N. and W.M. Visscher (1967b), Phys. Rev. 156, 1032. Payton, D.N. and W.M. Visscher (1968), Phys. Rev. 175, 1201. Pearsall, T.P., R. Caries and J.C. Portal (1983), Appl. Phys. Lett. 42, 436. Parmenter, R.H. (1955), Phys. Rev. 97, 587. Price, D.L., J.M. Rowe and R.M. Nicklow (1971), Phys. Rev. B 3, 1268. Ray, M.A. and J.E. Greene, unpublished. Redfield, A.C. and J.D. Dow (1987), Solid State Commun. 64, 431. Reed, M. and B. Simon (1972), Methods of Modern Mathematical Physics (Academic Press, New York). Ren, S.-E and Y.-C. Chang (1991), Phys. Rev. B 43, 11857. Ren, S.-F., H. Chu and Y.-C. Chang (1987), Phys. Rev. Lett. 59, 1841. Ren, S.-E, H. Chu and Y.-C. Chang (1988a), Superlatt. Microstruct. 4, 303. Ren, S.-F., H. Chu and Y.-C. Chang (1988b), Phys. Rev. B 37, 8899. Ren, S.Y. and J.D. Dow (1992), Phys. Rev. B 45, 6492. Robinson, J.E. and J.D. Dow (1968), Phys. Rev. 171, 815. Saint-Cricq, N., R. Caries, J.B. Renucci, A. Zwick and M.A. Renucci (1981), Solid State Commun. 39, 1137. Sapriel, J. (1989), Surf. Sci. Rep. 10, 189. Shah, S.I. and J.E. Greene, unpublished. Shah, S.I., B. Kramer, S.A. Barnett and J.E. Greene (1986), J. Appl. Phys. 59, 1482. Shen, J., C.W. Myles and J.R. Gregg (1987), J. Phys. Chem. Solids 48, 329. Sinha, K., A. Mascarenhas and G.S. Horner (1993), Phys. Rev. B 48, 17591. Soven, P. (1967), Phys. Rev. 156, 809. Stern, E.A., F. Ellis, K. Kim, L. Romano, S.I. Shah and J.E. Greene (1985), Phys. Rev. Lett. 54, 905. Talwar, D.N., M. Vandeyver and M. Zogone (1980), J. Phys. C 13, 3775. Talwar, D.N., M. Vandeyver and M. Zogone (1981), Phys. Rev. B 23, 1743. Taylor, D.W. (1967), Phys. Rev. 156, 1017. Tsu, R., H. Kawamura and L. Esaki (1972), in: Proc. Int. Conf. Phys. of Semiconductors, Vol. 2 (Elsevier, Amsterdam), p. 1135. van Hove, L. (1953), Phys. Rev. 89, 1189.

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Ch. 5

CHAPTER 6

Electronic Screening in Metals" from Phonons to Plasmons

ADOLFO G. EGUILUZ

ANDREW A. QUONG

Department of Physics and Astronomy The University of Tennessee Knoxville, TN 37996-1200 and Solid State Division Oak Ridge National Laboratory Oak Ridge, TN 37831-6032 USA

Computational Materials Sciences (8341) Sandia National Laboratory Livermore, CA 94551-0969 USA

Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin

9 Elsevier Science B.V., 1995

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Contents 1. Introduction

429

2. Electronic screening in metals

434

3. Density response and lattice dynamics

439

3.1. Interatomic force constants as a screening problem 439 3.2. Force constants for non-local pseudopotentials 445 3.3. First-principles calculations of phonon dispersion curves in bulk metals 450 3.4. Surface force constants and surface phonons in A1 463 4. Dynamical electronic response in bulk metals

474

4.1. Plasmon dispersion relation and dynamical structure factor of A1 477 4.2. Elementary excitations in Cs 483 4.3. Spectrum of charge fluctuations in Pd 492 4.4. Spectrum of spin fluctuations in paramagnetic Pd 496 Acknowledgements References

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502

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1.

Introduction

Most physical properties of a metal are determined by the valence (or conduction) electrons. In particular, the rapid readjustment of the valenceelectron density that results from the presence of a local charge imbalancethe phenomenon of screening- is a key characteristic of metallic behavior. It has been known for many years that a great variety of physical phenomena related to screening can be concisely described in terms of response functions (or correlation functions) appropriate to the physical situation at hand (Pines and Nozi~res 1966). For example, the density-response function determines the cross-section for inelastic scattering of fast electrons and X rays; thus, it provides a direct link between the space-time correlations which govern the physics of the electron liquid and experiment. The analytical structure of this response function is significant: the energy and lifetime of the collective excitations of the conduction electrons (plasmons) whose existence is a signature of the presence of long-range correlations - are determined by the poles of the response function in the lower half of the complex frequency plane. The same response function (its static limit) is a crucial element in the microscopic theory of phonon dynamics. The analytical structure of another correlation function - the transverse magnetic susceptibility - provides a criterion for the onset of the magnetic instability of the conduction electrons in metals. Other response functions are related to light scattering, etc. The key physical ingredients which must be included in an ab initio theory of correlation functions in metals are: a full representation of the frequency dependence of the response, a realistic description of the effects of the oneelectron band structure, and an explicit treatment of the electron-electron Coulomb correlations. It is the combination of these physical requirements which is responsible for the relatively-slow pace of the progress which has been made over the years in the evaluation of response functions for realistic models of the correlated electron liquid in metals. By contrast, major progress has been reported over the last two decades in the evaluation of observables pertaining to the ground state of a manyelectron system such as a metal. The most important role in that progress is to be assigned to the development of density functional theory into a -

429

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powerful computational tool. Density functional theory has provided a fundamental framework for the treatment of electron correlation in the ground state (Hohenberg and Kohn 1964) and a formally exact one-electron-like scheme (Kohn and Sham 1965) that considerably simplifies the implementation of the method. Furthermore, the simplest possible non-trivial ansatz (Kohn and Sham 1965) for the exchange and correlation energy functional, the so-called local-density approximation (LDA), has turned out to be extremely successful in practice. The conceptual simplicity of calculations performed within the LDA resides in the fact that the intricacy of the correlation problem is really not an issue - in this approximation one sets up the one-electron potential for the crystal environment starting from the knowledge of the (approximate) solution of the correlation problem for electrons in jellium. (For extensive monographs on and reviews of density functional theory see, e.g., Lundqvist and March 1983; Trickey 1990; Dreizler and Gross 1990; Gross and Dreizler 1994.) The development of accurate and efficient methods for the self-consistent solution of the Kohn-Sham one-electron equation in the presence of a periodic lattice constitutes the second major building block responsible for the impressive achievements which have been reported in ground-state studies. (For a brief review, see Zeller 1992.) In particular, the linearized all-electron schemes (Andersen 1975; Wimmer et al. 1981) and the modem ab initio pseudopotentials (Hamann et al. 1979; Bachelet et al. 1982) have considerably simplified the computations- while, at the same time, sufficient accuracy is retained, for many purposes. The above two theoretical advances, combined with the enormous recent improvements in the power of the available computational resources, have opened up the field of first-principles investigations of total energies and related physical quantities for realistic models of metallic systems. The most sophisticated calculations performed to date in this general area deal with systems composed of many (500 or more) atoms per unit cell; see, e.g., Stumpf and Scheffler (1994). In this chapter we outline some of the progress which has been made very recently in the first-principles evaluation of electronic response in metals. Our aim is to convey a flavor as to where the state of the art is, at the time of this writing, in this broad area of condensed matter physics. As hinted at above, this area has until recently remained relatively "underdeveloped" by comparison with the voluminous research which has been published on the ab initio evaluation of total energies and related condensed-matter observables. We hope to make a case to the effect that this situation is beginning to be reversed. For the most part, our discussion centers on the densityresponse function and its impact on physical observables- such as the loss spectrum for high-energy electrons and X rays, and dispersion relations of

w1

Electronic screening in metals

431

phonons and plasmons. For completeness, we also touch on the dynamical "screening" of a local spin imbalance, and discuss the itinerant-spin response in the paramagnetic phase. In the static-screening case, problem which is within the realm of validity of density-functional theory as originally formulated, the quality of the calculated phonon dispersion curves benefits from the same factors which, as noted above, have had an impact on the ground-state problem - densityfunctional theory within the LDA, ab initio, norm-conserving pseudopotentials, and access to modem high-performance computers. The net result is that accurate bulk phonon dispersion curves have now been obtained for several metals for arbitrary wave vectors in the first Brillouin zone (Quong and Klein 1992; Quong 1994). The calculations of dynamical electronic response in metals discussed in this chapter represent successful applications of non-local, norm-conserving pseudopotentials for the study of electronic process outside the ground-state problem for which they have been used almost exclusively so far. Indeed, our pseudopotential-based method leads us to novel results of rather general significance in condensed matter physics. For example, we demonstrate the crucial importance of band-structure effects in the dynamical response of prototype sp-bonded metals such as A1 (Quong and Eguiluz 1993; Fleszar et al. 1995a) and the "anomalous" heavy alkalis (Fleszar et al. 1995b). Our first-principles results for a transition metal (Pd) display a rich spectrum of elementary excitations which is in good agreement with the available experimental data (Gaspar et al. 1995). Thus, the calculations reported herein, together with concurrent work by other authors (e.g., Maddocks et al. 1994a, b; Aryasetiawan and Karlsson 1994; Aryasetiawan and Gunnarsson 1994) help bring the field of ab initio investigations of electronic excitations in metals to a new level of sophistication. We would like to note at the outset that, for the most part, our presentation refers to work in which we have been directly involved, either by ourselves or with close collaborators. While a brief comparison with some of the most pertinent recent results by other workers is included, we have not engaged in a discussion of the details of such work. References are given to all closely related theoretical/experimental work that we are aware of, particularly in the emerging area of first-principles calculations of dynamical electronic response in metals. In the presence of the actual band structure of the metal, dynamical correlations have so far been treated in the random-phase approximation (RPA), in which short-range correlations between the electrons involved in the screening process are ignored altogether, and via simplified descriptions of the electron-hole attraction. We consider the so-called time-dependent localdensity approximation- and its generalization for the case of the finite-

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Ch. 6

frequency spin response in the paramagnetic p h a s e - and other simple vertex corrections which are in the spirit of Hubbard's original approximation for the effects of exchange. Thus, the treatment of dynamical correlations contained in our calculations is certainly susceptible to improvement; in particular, it would be of interest to address the frequency dependence of the electron-hole vertex. The study of the screening response of metal surfaces lags behind its counterpart for the bulk, a reflection of the increased computational demands posed by the reduced symmetry of the surface environment. As a consequence, most of our survey refers to bulk metals. Our only incursion into the metal surface problem refers to density-response-based calculations of the phonon spectrum at low-index surfaces of A1. Now, the total-energy approach to the evaluation of surface phonon dispersion curves (Ho and Bohnen 1986, 1988) was developed before the ab initio screening method presented herein came to fruition- a reflection of the factors which, as noted earlier on in this Introduction, have shaped the progress of work in the area of ground-state calculations. An advantage of the density-response method, relative to total-energy methods, is that it is well suited for a detailed analysis of the surface-induced modifications of all the interatomic force constants. Furthermore, phonon eigenvectors- whose knowledge is required in, e.g., the evaluation of the reflection coefficient for He atom-surface scatteringare automatically generated as part of the solution of the surface vibrational problem. The outline of this c h a p t e r - whose flavor is largely pedagogical- is as follows. We start out in w2 with a sketch of some of the fundamental concepts related to screening. We then proceed to discuss in detail the interconnection between the response method and experiment through the evaluation of physical observables. In w3 we deal with the static-response case. We begin with an overview of the microscopic theory of lattice dynamics, which we adapt for its implementation in the presence of non-local pseudopotentials. We then discuss the evaluation of bulk phonon dispersion curves. We present results for A1 and Au, elements which serve as prototypes of nearly-free electron metals (A1), and metals in which d-electrons participate in the bonding (Au), respectively. The calculated phonon dispersion curves agree with the neutron-scattering data extremely well (w3.3). We explicitly show that the inclusion of the many-body effects of exchange and correlation in the effective electronelectron interaction is a significant element in this success of the theory. We also discuss the case of Pb, whose phonon spectrum contains prominent anomalies for large wave vectors. The metal surface problem is considered in w3.4, where we present results for sp-bonded A1. We address this problem at two levels. First, we present results for the surface phonon dispersion

w1

Electronic screening in metals

433

curves for low-index surfaces of A1, obtained via the ab initio evaluation of the surface force constants with full inclusion of the effects of the band structure (Quong 1995). Next, we discuss a simplified- yet self-consistent treatment of the surface force constants of A1 (Gaspar et al. 1991 a), based on the use of pseudopotential perturbation theory. Starting from the solution of the surface-screening problem for the electron-gas (jellium) surface, this work proceeds all the way to the evaluation of the reflection coefficient for He atom-surface scattering (for a review, see Toennies 1988). Agreement with high-resolution time-of-flight measurements on A1 is satisfactory (Franchini et al. 1993). In w4 we deal with the dynamical-response problem. Following a brief outline of the theoretical framework, we present a summary of a series of recent ab initio calculations for several metals. In the case of A1 (w4.1), we obtain an overall satisfactory picture of its dynamical response for all wave vectors. The theoretical plasmon dispersion curve is in quantitative agreement with experiment; this agreement is traced to the combined effects of the one-electron band structure and of the electron-hole vertex evaluated in the LDA (Quong and Eguiluz 1993). From the calculated dynamical structure factor we extract new insight into a long-standing controversy involving the physics of short-range correlations in the electron liquid in metals. We show that a much-discussed double peak, whose existence is the most intriguing feature of the measured inelastic X-ray scattering spectrum for large wave vectors (Platzman et al. 1992; Schtilke et al. 1993) is, in fact, an inherent property of the response of non-interacting electron-hole pairs propagating in the actual band structure of A1 (Maddocks et al. 1994b; Fleszar et al. 1995a). Coulomb correlations are shown to play a quantitative role. Inclusion of a vertex correction improves the quality of the calculated intensities on the low-frequency side of the double peak (Fleszar et al. 1995a); this quality is assessed by detailed comparison with the measured X-ray intensities. The dynamical density-response of Cs is considered in w This is a very interesting system, as the latest high-resolution electron energy-loss experiments (vom Felde et al. 1989) have been interpreted as signaling drastic departures from available theories of the response of the interacting electron liquid. We show that band structure effects - rather than short-range Coulomb correlations- are the crucial physical ingredient behind the anomalous nature of the measured plasmon dispersion relation (Aryasetiawan and Karlsson 1994; Fleszar et al. 1995b). Indeed, the R P A - when implemented for band electrons - turns out to provide a good first approximation for all wave vectors (Eguiluz et al. 1995). Finally, we consider the dynamical response of metals for which the d-electron bands straddle the Fermi surface; thus, in addition to partaking in the bonding, the relatively-localized d-electrons play an important role in -

A.G. Eguiluz and A.A. Quong

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Ch. 6

the physics of the response, for all energies. As a prototype, we consider the transition metal Pd. The calculated spectrum of elementary excitations includes loss peaks which show some similarities, but also significant differences, with, e.g., the A1 plasmon. All the measured excitations up to about 40 eV (Bornemann et al. 1988) are accounted for by our calculations (w4.3). We also discuss the spectrum of spin-density fluctuations (w4.4), whose nature is directly related to the large density of states at the Fermi level which characterizes the physics of Pd. The exchange-correlation-enhanced dynamical spin susceptibility exhibits a prominent paramagnon mode (Gaspar et al. 1995); the same has been studied theoretically in the past mostly in the context of lattice models.

2. Electronic screening in metals In this section we outline the theoretical framework required for the study of dynamical screening in real metals, where by "real" we mean that the electron band structure is fully incorporated in the calculations. As hinted at in w1, the physical process of electronic screening is a fundamental feature of the metallic-state of condensed matter. Screening has a bearing on the nature of the ground state (its energy, details of the bonding, geometry of the equilibrium crystal structure, etc.) and also on the nature of the available elementary excitations. We emphasize that these excitations are directly probed by various spectroscopic techniques- thus the quality of the theoretical description of the electronic response can be subjected to an immediate "reality check." The screening process is embodied in the Kubo linear-response formula (see, e.g., Pines and Nozi~res 1966) 6nind(:~, t) --

F l dt t

d3x t Xnn (~, :~tlt - t t) 6Uext(:~tltt),

(2.1)

0o

which relates the induced electron number density ~nind to the external potential (energy) 3Uext which polarizes the medium. Equation (2.1) serves as the definition of the (retarded) density-response function Xnn(X,~ ' l t - t'). This function contains the essential physics of two complementary manifestations of screening: (i) response of a many-electron system to an external potential - due to, for example, a fast electron scattered off a s o l i d - and (ii) formation of an exchange and correlation hole in the neighborhood of a Fermi-sea electron by virtue of, respectively, the antisymmetry of the manyelectron wave function and the electron-electron Coulomb interaction.

w

Electronic screening in metals

435

In the many-body theory of interacting electrons one also introduces an "irreducible polarizability" ~,(:~,~'[t- t'). This function incorporates the elementary processes- such as electron-hole pair excitation, electron-hole ladders, etc. - from which the response of the ensemble of electrons is built up. The irreducible polarizability serves as the kernel of a Dyson-like integral equation for Xnn (Fetter and Walecka 1971; Mahan 1990), namely

Xn.

xt[ 60) =

+/

tt~

n[to)v(:~n -- ~nt)xnn (~,tn, :~t[60),

(2.2)

where u is the bare Coulomb interaction. In eq. (2.2) we have made use of the fact that the Hamiltonian of the isolated solid is time-independent, and have introduced the frequency-Fourier transforms of Xnn and ~,, defined in the usual way. (Note that eq. (2.2) holds in the general case for the timeordered counterparts of the retarded response functions introduced above.) For the benefit of the reader who may be more familiar with the concept of the dielectric function e(~, :~~lw), we note that the same is directly related to the irreducible polarizability by the equation e = 1 - u~. From eq. (2.2) we have that the inverse dielectric function e - l ( ~ , ~ l w ) is obtained from the density-response function Xnn(~,:Ulw) according to the equation e-1 _ 1 + UXnn. Note that all elements of this equation are to be understood as "matrices" in ~, ~ space; thus the term VXnn involves a matrix product. Because of the computational demands of the full problem of interacting electrons in the presence of a periodic potential, the study of the many-body effects associated with the screening process in metals has traditionally been carried out within the homogeneous electron-gas model. In this model one concentrates on the effects of the electron-electron interaction and ignores the effects of the lattice of ions in which the conduction electrons are embedd e d - the lattice is smeared into a uniform background. This is the so-called jellium model. Clear evidence will be presented in w4 to the effect that, even in the so-called simple metals, this model leaves out important physical processes which result from, or are strongly influenced by, the one-electron band structure. The simplest treatment of screening corresponds to the RPA, in which we set ~ - X(~ where X(~ is the free-electron "bubble," which corresponds to excitation of non-interacting electron-hole pairs (Pines and Nozi~res 1966). This ansatz ignores self-energy effects (interaction between the electron and hole and the other particles in the Fermi sea) and vertex corrections (such as the ladders for the interaction between the electron and the hole).

436

A.G. Eguiluz and A.A. Quong

Ch. 6

The structure of eq. (2.2) is such that long-range correlations between electron-hole pairs automatically build up in the process of setting up the physical response X n n , which leads to the appearance of a plasmon "pole" in the solution of eq. (2.2) for the full response X n n . The formation of such pole follows directly from the behavior of the RPA bubble for small wave vectors, X(~ ,-~ q2/w2 for q --+ 0, together with the long-range nature of the Coulomb interaction, which gives v ,,~ q - 2 . Clearly, then, for zero wave vector the equation Re(1 - vX~~ - 0 has a root at a finite frequency the plasma frequency. (This condition is the same as the vanishing of the "denominator" of the solution for X n n . ) Note that this simple argument strictly applies for the homogeneous electron gas; the band structure of an actual metal affects its validity to a greater or lesser extent, depending on the metal in question. Explicit examples are given in w4. Coulomb correlations are treated in the RPA in a mean-field sense. Shortrange correlations originate from higher-order diagrams for ~, the RPA ansatz ~. = X~~ being of zeroth order in u. In the diagrammatic approach, the key difficulty encountered in going beyond the RPA in a systematic way stems from the requirement that self-energy effects and vertex corrections must be incorporated in a "balanced way". This can be achieved in principle through the use of "conserving approximations" (Baym and Kadanoff 1961; Baym 1962) - which provide a prescription for generating self-energies and vertex functions self-consistently- or, equivalently, by the use of Ward identities (see, e.g., Mahan 1992). Such calculations are notoriously difficult. Significant progress has been made recently for discrete lattice models of the cuprate high-Tc superconductors (Bickers et al. 1989; Serene and Hess 1991; Putz et al. 1995). Actually, a conserving approximation, as defined by Baym and Kadanoff, only ensures that the correct mixture of self-energy effects and vertex corrections is introduced in the evaluation of the one-electron Green's function. It turns out that for the evaluation of two-particle correlation functions an example of which is the dynamical density-response function Xnn - one must in principle impose further constraints in order to have a controlled many-body "mixture" (Bickers and White 1991). The implementation of such stringent theories for realistic models of a metal appears to be virtually non-existent. In recent years, and driven largely by the well-documented explosion of interest in the high-Tc superconductors, a "numerical" approach to the treatment of electron correlations has been developed rather extensively. The compromise made in these studies is the use of highly-simplified band structures, such as contained in the Hubbard model (Hubbard 1963, 1964) - in which, furthermore, the assumption is usually made that metallic screening is so extreme that two electrons in opposite spin states only interact when they

-

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Electronic screening in metals

437

encounter each other at the same atomic site. With these simplifications, the still-formidable problem of the on-site Coulomb interaction is treated without appeal to perturbation theory by, e.g., exact numerical diagonalization of the Hamiltonian of a finite s y s t e m - in practice, a rather small cluster - (for a review, see Dagotto 1994), or by "exact" numerical evaluation of correlation functions via quantum Monte Carlo techniques (for a review, see v o n d e r Linden 1992). Again, these techniques have yet to be employed for the study of correlation functions for realistic models of, e.g., transition metals. In the work reviewed in this chapter we explore the consequences of a more modest approach to the many-body p r o b l e m - which, however, is implemented with full use of the actual band structure of the metal. The investigation of the feasibility of some of the powerful techniques just alluded to for the study of dynamical correlation functions for realistic models of metals is left for future work. The density-functional method provides a "natural" way of going beyond the RPA. Most importantly, this scheme allows for the treatment of manybody correlations self-consistently with the effects of the electronic structure. In practice, this appealing notion has been carried out almost exclusively within the LDA. In this approximation we have a simple prescription for the contribution from exchange and correlation to the effective electron-electron interaction, namely

v(e, e ' ) = . ( ~ - ~') +

dVxc(:~) dn(~)

6 ( e - i'),

(2.3)

where Vxc(:~) is the self-consistent exchange-correlation potential for the electron number density n(~). Equation (2.3), which corresponds to a local picture of the electron-hole attraction, follows directly from the Kohn-Sham equation in the LDA (Kohn and Sham 1965), upon linearizing the change in the electron density brought about by a change in the external potential - produced by, for example, the presence of a phonon. In the general case (that is, if we do not invoke the LDA), the second term in eq. (2.3) becomes a non-local function of s and i ' ; this function (or "vertex") is given formally by the second functional derivative of the exchange-correlation energy functional with respect to the density. Not much is known explicitly about this non-local vertex for a strongly inhomogeneous electron system, such as a metal surface, for which the importance of many-body correlations is further enhanced (Eguiluz et al. 1992a, b; Deisz et al. 1993, 1995). The use of eq. (2.3) for the study of dynamical screening defines the so-called time-dependent "extension" of local-density-functional theory

A.G. Eguiluz and A.A. Quong

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Ch. 6

(TDLDA). Noting that in the Kohn-Sham scheme the diagram for the irreducible polarizability is of the same form as the RPA bubble, we readily conclude that in the TDLDA the response function is of the symbolic form X - X(~ 1 - u(1

-G)X(0)) -1,

(2.4)

where we have introduced the "local-field factor"

a(q)--u(q)-If

. . . dVxc(s

d3xe -~q'x

dn(~)

= GTDLDA(qT).

(2.5)

Alternatively, fxc(q)= -u(q)GTDLDA(q~) is said to introduce a "vertex correction", its presence in eq. (2.4) reflects the existence of an exchange and correlation hole associated with each electron participating in the responsewhich leads to a "weakening" of the screening. Note that G = 0 in the RPA, which thus ignores the short-range aspects of the screening. It should be noted that the propagators which define X(~ in the density-functional context are automatically "dressed," i.e., they include exchange and correlation effects via the eigensolutions of the Kohn-Sham ground state. The LDA description of the exchange-correlation process is that the electron system behaves as if it were locally homogeneous- which of course it is not in many cases of interest, such as metals with localized electron orbitals, and metal surfaces. Thus, in essence, the TDLDA is an uncontrolled approximation with regard to both its treatment of electron dynamics and its neglect of spatial non-locality in the electron-hole interaction. Nevertheless, TDLDA-based calculations have produced good results for polarizabilities of atoms (Zangwill and Soven 1980; Stott and Zaremba 1980; Mahan 1980; Mahan and Subbaswamy 1990), clusters (Ekardt 1984, 1985; Pacheco and Ekardt 1992; ScNSne et al. 1994), bulk metals (Quong and Eguiluz 1993), semiconductor surfaces (Streight and Mills 1989) and metal surfaces (Eguiluz 1987a; Gaspar et al. 1991b; Eguiluz and Gaspar 1991; Tsuei et al. 1990, 1991; Liebsch 1987, 1991). On the other hand, it has been demonstrated that the TDLDA does not account properly for the continuumexciton effect in the optical response of Si (Hanke and Sham 1980). It is noteworthy that eq. (2.4) is of the form originally proposed by Hubbard as a physically-motivated approximation for the ladder diagrams for the electron-hole interaction (Hubbard 1957). In fact, it has become rather customary to write down the response function beyond the RPA in the Hubbard form. Of course, the actual expression for the local-field factor G(0") depends on the details of the treatment of correlation. We will consider another approximation for G(q) (additional to TDLDA) in w4.1, in the course

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Electronic screening in metals

439

of our discussion of the dynamical structure factor of A1 for large wave vectors. A first-principles investigation of the frequency dependence of the local-field f a c t o r - in particular, for band electrons- remains an outstanding theoretical challenge; progress has been reported by Gross and Kohn (1985, 1990) for electrons in jellium. For a detailed discussion of the many-body problem in a variety of condensed-matter situations see, e.g., Hedin and Lundqvist (1969), Mahan (1990), and Enz (1992). In the bulk of this chapter - constituted by w3 and w4 - we provide an account of some of the progress which has been made very recently in the computation of Xnn, and related physical observables, with full inclusion of the effects of the band structure of the metal. Our discussion focuses on two distinct physical limits: static electronic response and electronic response in the optical and ultraviolet spectral regions, respectively. (A subtle interconnection between both cases exists, since, for example, the ground-state energy can be obtained from the knowledge of the density-response function for all frequencies.) We also discuss the finite-frequency spin response of a metal in the paramagnetic phase.

3. Density response and lattice dynamics When an ion vibrates it gives rise to a longitudinal potential to which the electrons respond. This response, in the form of a change in the electron density, is "sensed" by a nearby ion. The conduction electrons are thus the intermediary of an indirect ion-ion interaction- which is the origin of an electronic contribution to the interatomic (or interionic) force constants providing the restoring force for the lattice vibrations. This simple physical picture is elaborated on in w3.1, where we review the microscopic theory of lattice dynamics. In w3.2 we obtain the interatomic force constants for the general case in which the pseudopotential is nonlocal. In w3.3 we discuss the computation of phonon dispersion curves in bulk metals from first principles. In w3.4 we discuss the surface force constants of A1 and associated physical quantities. 3.1. Interatomic force constants as a screening problem

Of course, at the formal level the microscopic theory of interatomic force constants is a mature subject, whose origins are traced back to articles by Sham (1969) and Pick et al. (1970). In practice, the first-principles implementation of the theory has proved to be non-trivial; this topic is discussed in w3.3 and w3.4. The following demonstration- presented in the interest of

A. G. Eguiluz and A.A. Quong

440

Ch. 6

making this section self-contained- is designed to emphasize the fact that the key physical element of the problem is the electronic screening process, which we frame in the context of the response theory outlined in w2. Let us denote the equilibrium position of the sites of a Bravais lattice by {:g (l)}. In the case of a perfect bulk crystal, the index 1 is a collective symbol for a set of three integers (1 (11,12,13)) such that :g(1) spans the Bravais lattice; we have in mind monatomic crystals, which elemental metals are. A simple redefinition of our labeling convention makes the present discussion applicable to the more general case of a metal with a surface (w3.4). We introduce a set of ionic displacements {z7(l)}, defined with respect to the equilibrium configuration of the lattice. In the presence of this displacement field the Hamiltonian for the electron-ion interaction is given by the equation =

(3 1)

/~ei --/~(Q) + (~Ve i, ,

e,l

,

where we have made the definitions

/~(Q)e,, -"

f

d3 x ~(:~) ~

v,(Z - Z (1)),

(3.2)

l

and

(~Ve,i = f d3xn(x)(~(z),

(3.3)

where ~(:g) - g,t(:g)g,(:g) is the operator of the electron number density, given in terms of the usual field operators. The Hamiltonian --e,,D(Q)clearly corresponds to the electron-ion interaction appropriate for a system of conduction electrons embedded in a lattice in which all the ions are at rest at their equilibrium positions (rigid lattice). The interaction ~Ve,i accounts for the change in energy of the electronic system brought about by the ionic displacements. The polarization potential 6r generated by the displaced ions is defined by the equation h

(3.4) where ~ , ( ~ - ~ (z))

l

cz

ax,~(l)

u~(~)

(3.5)

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Electronic screening in metals

441

is of first order in the displacements, and 1

ar

- 2

i~2~,(~" - ~ (1)) l

c~,~

Ox (Z)Ox (Z)

u~(l)um(1)

(3.6)

is of second order. The neglect in eq. (3.4) of higher order terms in the ionic displacements constitutes the harmonic approximation. The small parameter which controls the validity of this approximation is the ratio lul/ao, where a0 is the lattice constant. If the displacements are thought of as being due to the lattice vibrations, [u[ is the temperature-dependent root-mean-square displacement from equilibrium. It is useful to think of the electron-ion interaction as being built up continuously from the rigid-lattice case (for which ~Ve,i is absent) to the full physical interaction Hamiltonian given by eq. (3.1). This is achieved by defining the electron-ion interaction for arbitrary values of a "coupling constant" A (0 ~< A ~< 1) according to the equation

/~e,i(/~)- /~(Q) e,1 -+- A ~Vei,

9

(3.7)

The convenience of this approach is that it allows us to invoke the so-called Feynman theorem of elementary quantum mechanic.s (this results is also attributed to Pauli). The same provides us with a simple prescription for the evaluation of the change in the ground-state energy of the conduction electrons, ~Eel, due to the ionic displacements, namely 1

~Eel =

L

^ d/~ (~Ve,i) I ,

(3.8)

where the quantum mechanical mean value refers to the ground state of the total electronic Hamiltonian A

gel(A)

A

A

-- ge,e + ge,i(A),

(3.9)

whose first term is the many-body Hamiltonian for the interacting electrons by themselves. Substituting eq. (3.3) into eq. (3.8) we have that 1

442

A. G. Eguiluz and A.A. Quong

Ch. 6

where (~,(~))a is the electron density for the ground state of /~el(~) for arbitrary )~. We evaluate this density by setting (n(:~)) ~ = no(~) + ~n(~; ~),

(3.11)

where n0(~) -- (~(~))~=0 is the ground-state density for the rigid lattice, and 3n(~; ,~) is the screening density induced by the ionic displacements. To first order in ,~ 3r the latter density is given by the equation

(3.12)

6n(~; )Q = f d3x ' Xnn(~, ~'lw - 0 +) ~ ~r

which is nothing but a static version of the Kubo formula given by eq. (2.1). We note that higher order terms are not required in eq. (3.12) because 3Eel already contains a factor of 3r in eq. (3.10). We also note that implicit in the present argument is the assumption that the electrons adjust themselves quickly, and without dissipation, into the ground state of the Hamiltonian Hel(A) for the instantaneous configuration of the lattice. This is the adiabatic or Born-Oppenheimer approximation. Because of the simple dependence of 3n(:~; )~) on )~, the coupling-constant integration required in eq. (3.10) is performed without difficulty. This leads us to the result that A

~Eel -- f d3xno(x) ~r (3.13)

-k-~l f d3x f d3x, t~r

Xnn(:~,~tlw __ 0+ ) t~r

)

in which there is no term of first order because the displacements {g(l)} have been defined with respect to the equilibrium configuration. Making use of eqs (3.5) and (3.6) in eq. (3.13) we readily obtain 3Eel in the canonical "harmonic-oscillator" form

t~Eel : ~1 ~~--~ el~)o~,fl(1,l')uo~(l)u~(l'), ll' ot~

(3.14)

w

Electronic screening in metals

443

where el

02V(X - X(1))

d3x n o ( s

, l') -- 5z,t,

+

f

d 3x

Ou(s

f

d 3 x'

~x~(1) i~x~(1)

Oxa(1) o~,(~eq))

Xnn

(~, ~'1~ -- o+)

(3.15)

- s (l'))

Oz~(l') is the electronic contribution to the interatomic force constants. The presence of the (static) density-response function Xnn(~, ~']w - 0 +) in the second term in eq. (3.15) reflects the fact that the central physical process in the above derivation is the response of the conduction electrons to the polarization field generated by the ionic displacements. It is instructive to rederive eq. (3.15) from a different perspective. This method proves very fruitful in the study of structural properties, and in the implementation of the frozen phonon approach to lattice dynamics (Kunc and Martin 1981, 1983). In this derivation we are not concerned with "building up" the electron-ion interaction via a coupling-constant integration. Rather, we give the ion 1 a virtual displacement ~(l) from its equilibrium site; at this point the other ions may or may not be in their equilibrium positions. The electronic Hamiltonian is now given by A

A

Hel(U(/))-

A

He,e -~- He,i(u(/)),

(3.16)

where the dependence of the electron-ion interaction on the displacement g(1) has been made explicit. Denoting by Eo(g(1)) the energy of the ground state of Hel(U(/)), the force on ion / due to the electron liquid in which it is immersed is obtained as

F~(1) -

(OEo(~(Z)) ) o~(l) ~(z)=o'

(3.17)

in the limit of vanishing ~(1). Now the right hand side of eq. (3.17) can be expressed in a manner suitable for computation by appealing to the HellmannFeynman formula (Hellmann 1937; Feynman 1939) A

OEo(~(z)) o~.(z)

(3.18)

A. G. Eguiluz and A.A. Quong

444

Ch. 6

where the average is taken with respect to the ground state of Hel(~(/)). This average is formally evaluated without difficulty, with the result that Ou(~. - 2(1)) F~(1) - f d 3x ~,o(x)

oz.(l)

(3.19)

where ~ o ( e ) = ( E 0 ( ~ q ) = 0 ) l ~ ( e ) l E o ( ~ q ) = 0)),

(3.20)

is the electron number density for ~(l) = 0. Note that the difference between ~o(:~) and the ground-state density no(~) of the rigid lattice (see eq. (3.11)) is that in the configuration represented by Hel(~(1) = 0) we may have the other ions in arbitrary positions, not necessarily the sites ~ (1) of the Bravais lattice in stable equilibrium. All that is required in eq. (3.19) is that ~0(~) be the self-consistent electron density for the assumed ionic configuration consistent with the ground state of Hel(~(l) = 0). In fact, if ~0(:~) is interpreted as the "distorted" density which results from our displacing another ion (say, ion l') by ~(l') relative to what would otherwise be a perfect Bravais lattice, from the Kubo formula (2.1) we have that, to first order in ~(l'), A

no(x) - no(~') - ~

f d3x ' Xnn (~, :~'lw = 0 +) (3.21)

Ou(~,' - Z (l')) ~(l').

Ox~(l') The result of substituting the second term of eq. (3.21) into F~(1) given by eq. (3.19) is quickly recognized to be exactly the same as 8Eel/SU~(1) obtained from eq. (3.13). We thus arrive again at the result for the electronic contribution to the interatomic force constants for 1 ~ l' given by eq. (3.15). (A slightly modified argument produces the diagonal force constants, 1 = l'.) Finally, we note that the total interatomic force constants ~b~Z(1,l') are given by the equation

9,~z(l,l') - #~(l, i

el l') + ~b~;~(1, l'),

(3.22)

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Electronic screening in metals

445

i (1, l' ) are the direct force constants for ions coupled by the bare where ~b~n Coulomb interaction. From the force constants we obtain the dynamical matrix D ~ ( q ) , defined by the equation (Maradudin et al. 1971)

'(m(z)-m(z'))~=~(l l'), Dc~g(0*)- ~1 ~ e -lq" l'

(3.23)

where M is the mass of the ions, and q' is a wave vector in the first Brillouin zone. The (square root of the) eigenvalues of the dynamical matrix as a function of wave vector define the phonon dispersion relations. The eigenvectors of the dynamical matrix specify the phonon displacement patterns. 3.2. Force constants for non-local pseudopotentials

As it stands, the result for the electronic contribution to the interatomic force constants given by eq. (3.15) is strictly applicable in an all-electron calculation, in which the potential u(:g-s (1)) coupling an electron at :g and an ion at :g (1) is the bare Coulomb potential - a local function of the coordinate of the electron. However, all-electron calculations of phonon dynamics based on eq. (3.15) are rare (Sinha 1980). For the most part, eq. (3.15) has so far been used in conjunction with local, empirical pseudopotentials. (An example of this approach, in the context of surface phonons, is given in w Now, the modem theory of pseudopotentials has evolved into an extremely successful technique (for a review, see Pickett 1989). In this method one eliminates the core states from explicit consideration and concentrates on the electrons which are responsible for most condensed-matter-properties- the valence electrons. The condition of norm conservation (Hamann et al. 1979) with which these ionic pseudopotentials are constructed has proved capable of ensuring a high degree of transferability into the condensed-matter environment. Thus the ab initio pseudopotentials lend themselves to firstprinciples calculations of condensed-matter observables. (Vanderbilt 1990 has developed an alternative scheme which is not based on norm conservation.) However, these pseudopotentials are inherently non-local functions of the electron's position coordinate. For example, the Hamann et al. (1979) pseudopotential operator is of the form ~ps - ~ Ilm)Vz(f)(lml, lm

(3.24)

A.G. Eguiluz and A.A. Quong

446

Ch. 6

where the {llm)} are eigenkets of the angular momentum operators ~2 and Lz, and ~" denotes the operator for the radial coordinate of an electron. The matrix elements of eq. (3.24) in the position representation are of the form A

;7

- r')

0,

~

(3.25)

lm

where the amplitude Yzm(0, r = (0, r is a spherical harmonic (Sakurai 1994). Clearly, eq. (3.25) is non-local in the angular coordinates. Other popular forms for the ab initio pseudopotentials are also non-local (and separable) in the radial coordinate (Kleinman and Bylander 1982; Gonze et al. 1991). It is then necessary to reformulate the all-electron result (3.15) for the interatomic force constants in order to make it suitable for use with ab initio pseudopotentials. We show next how this can be done within density-functional theory. To this end it is useful to go back to the expression for the change in the groundstate energy of the electrons due to the ionic displacements, 5Eel, given by eq. (3.13). Noting that the ground-state electron density for the equilibrium configuration of the lattice is given by

no(~) - ~ fi, lr

2,

(3.26)

V

where the {r are Kohn-Sham one-electron wave functions and the {fi,} are fermion occupation numbers, we rewrite the first term in eq. (3.13) as follows:

d x no(x) 5r

(x)

= / d3x E f~(v[x)5r

(3.27)

V

1

= ~ E E E fi'

0 2 ( u ] u ( ~ - ~, (/))]u)

Ox,~(1)i~x~(1)

u,~(1)u~(1),

where :g denotes the position operator for an electron. In arriving at eq. (3.27) we have made use of the closure relation for the eigenkets of this operator and of the definition of &b(2) given by eq. (3.6).

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Electronic screening in metals

447

As for the second term in 5Eel (cf. eq. (3.13)), consistent with the physics behind the Kohn-Sham one-electron equation we have that

/

d3z ' Xnn (:f:f'lw - 0 +) 6q~(1)(:~t) = 5n(1)(:~)

/

(3.28)

d3x ' X(~

= 0 +) 6q~eff(:~t),

where X(~ is the RPA irreducible polarizability introduced in w (As noted below eq. (2.5), the X(~ one computes in density functional theory is automatically dressed, in that it includes effects of exchange and correlation which are in the nature of self-energy effects.) The interpretation of eq. (3.28) is clear: The density induced by the potential (~q~(1)(:~) due to the ionic displacements can be obtained not only from the Kubo f o r m u l a - in which case 3qr163 plays the role of an "external" potential, and the relevant response function is the full density-response function Xnn -- but also as the response of non-interacting "Kohn-Sham electrons" which are acted on by an effective, self-consistent potential, &beff(:g). This effective potential is defined by the equation

6q~eff(~')- 6~(1)(X)-+-

f

d3x ' V(:~, :f') 5n(1)(~'),

(3.29)

where V ( s 1 6 3~) is an effective electron-electron interaction. We note that if we set V = u we have Hartree response theory, or RPA. It is quite straightforward to incorporate a contribution to V(s s from exchange and correlation as long as we work within the L D A - this was done in eq. (2.3). We will show in w3.3 that this many-body contribution has a significant influence on the phonon frequencies. Making use of eqs (3.28) and (3.29), together with the spectral representation of X(~ in terms of one-electron energy eigenfunctions and eigenvalues

fu -- fu, x(~

2")- ~

E ~ , - EL,,

+;(e)~, (e)~;, ~(e'),

(3.30)

A.G. Eguiluz and A.A. Quong

448

Ch. 6

we rewrite the second term in 5Eel as

E~, _- Ev S.'

= _~ E VV

f

f d3x(u[~)5r

!

(3.31) d3m'(v' le') 5r

(e'lv)

VV !

Now, 5r163 ~) is given explicitly in terms of the ionic displacements by eq. (3.5). Furthermore, 5r - and thus eq. (3.31)- is also expressible in terms of the {us(l)} upon noting that the electron density induced by the ionic displacements may be written as

5n(:)(2) u~(1). l

(3.32)

a

Collecting together the results just derived for the two terms in the expansion of 5Eel to second order in the displacements, we arrive at the following result for the electronic contribution to the interatomic force constants el ( / , l ' ) - 51,l' E f~' #~,~ A

+ E

VV !

fv -- fv, E~ - E~,

(3.33)

Ox~q)

[ o(~'l~(e)- e(z')l~) A

Ox~(l')

+

f

5n(1)(:~,) ]

d3x'(v'[V(~,:U)lv)-~u-(iTi

9

Of course, the physical content of eq. (3.33) is exactly the same as that of eq. (3.15). However, eq. (3.33) lends itself to our "pseudizing" the force constants, which we do via the replacement (3.34)

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Electronic screening in metals

449

The idea behind this replacement is that, since the pseudopotential operator ~ps(l) for the ion l appears inside a matrix element, eq. ( 3 . 3 4 ) - and thus eq. (3.33) as well - is suited whether the pseudopotential is local or not. We note that the above result is consistent with the formalism put forth by Gonze and Vigneron (1989) starting directly from the Kohn-Sham equation in the presence of a non-local pseudopotential. (Since density-functional theory holds rigorously for interacting electrons in the presence a local external potential, that formalism must also invoke an ansatz in the spirit of eq. (3.34).) We thus have an alternative version for the all-electron result (3.15) in a form which is designed for use with non-local pseudopotentials. Now, at first sight, the appearance of the variational derivative 6n(1)(s ') on the right hand side of eq. (3.33) seems to suggest that the evaluation of the force constants is to be performed as part of the electronic self-consistency procedure from which the induced density ~n(1)(:f) is to be obtained. It is important to note that this is not the case. In effect, substituting eq. (3.29) into eq. (3.28), and making use of eqs (3.5) and (3.30), we are led to the result that

6n(1)(1)-f d3x'f

= o +)

'')

x V(Y', s

(3.35)

f v -- f v ' l

a

0z,,(l)

uv'

From eq. (3.35) we quickly conclude that the variational derivative required in eq. (3.33) is a functional of X(~ - which itself is a functional of the ground state density in the absence of the phonons. In fact, we can proceed from the above argument and rewrite eq. (3.33) in a form which is more suitable for its evaluation. First, we introduce the definition

6n11)(:~) : 6n(1)(:~)/6u,~(1),

(3.36)

and rewrite the integral eq. (3.35) as (1) (:~)

(Snl, a

,---(1) (:~)

= onl, a

(3.37)

+fd3x'fd3x"x(~

=

o+)v(e', e") 0"l~l,~(a,

),

450

Ch. 6

A. G. Eguiluz and A.A. Quong

where

~)

-

f~ - f~,

t'~(~) = ~///,/p EL,- E~,

(~,le)(el~,')

(3.38)

Ox.q)

can be thought of as the electron density induced by a displacement of the ion 1 in the absence of screening. In the inhomogeneous term in eq. (3.37) we recognize that

~V/(1)(s - f d3x ' V(s :~') ~nl,o~(x(1) -.,),

(3.39)

J

is the variational derivative of the screening potential (cf. eq. (3.29)). A few mathematical manipulations lead us then to a more compact version of eq. (3.33), namely

~

el

~a,~O(1, I t) -- t~l,l' ~

f v OXa(1) OXl3(l ) -t-

f

--(1) x ,~(--') d 3X t~n!1)l,t~(X) t~Vl'

1,1

(3.40) + ~121,, I E . - E~..

Ox.~(1)

Ox~(l')

'

where we have incorporated (3.34) throughout. Equation (3.40) is further simplified in w3.3 for the specific case of a periodic crystal.

3.3. First-principles calculations of phonon dispersion curves in bulk metals In this subsection we describe the numerical procedure that we have developed for the evaluation of the electronic contribution to the interatomic force-constant. We subsequently present results of the implementation of the method for the evaluation of phonon dispersion curves for A1, Au and Pb. As a prelude, it seems pertinent to note that the traditional method for the calculation of bulk phonons from first principles is the frozen-phonon method (Kunc and Martin 1981, 1983). In this approach, a distortion corresponding to a chosen phonon displacement pattern is frozen into the lattice. The change in total energy of the crystal is calculated and assigned to the phonon. Alternatively, the Hellmann-Feynman forces are determined, and the elements of the dynamical matrix are calculated. Diagonalization of the

w

Electronic screening in metals

451

dynamical matrix yields the phonon energies. This method has produced accurate results for the phonon dispersion curves along high-symmetry directions in the Brillouin zone for a variety of materials, ranging from metals and semiconductors to the high-temperature cuprate superconductors. However, the frozen-phonon method has its drawbacks. Small wave vector phonons require very large supercells to describe the phonon distortion. This is inconvenient, since the computational requirements grow as the third power of the size of the system. Additionally, the displacement pattern corresponding to a phonon mode for arbitrary wave vectors in the Brillouin zone cannot be established a priori on the basis of symmetry - as is the case for high symmetry directions. Thus, it is not possible to "freeze in" the phonon distortion. As a result, the determination of the dynamical matrices over a fine mesh of wave vectors is very difficult, and, therefore, the interatomic force constants (3.22) cannot be obtained with sufficient accuracy. Incidentally, a similar complication arises in the surface vibrational problem. In effect, the frozen-phonon method cannot be directly implemented at a surface, since the change in the amplitude of the phonon displacement pattern in the outer atomic layers is a "dynamical" issue not answered by symmetry - rather, it is determined from the self-consistent solution of the surface vibrational problem. A total-energy method involving a variation on the frozen-phonon method has been developed by Ho and Bohnen (1986, 1988) for the study of surface phonons. Another approach is the method introduced by Baroni et al. (1987) for the case of bulk semiconductors. Although this method has its roots in linear response theory, it circumvents the explicit evaluation of the dielectric matrix. Instead, eqs (3.29) and (3.35) are solved iteratively, and this yields the screened potential produced by the presence of a phonon; the dynamical matrix and phonon energies are determined from the knowledge of this potential. Baroni et al. (1987) eliminate the need to evaluate the sums over the unoccupied states required in eq. (3.35) via mathematical manipulations which start out from the introduction of the one-electron Green's function. This method has been successfully applied for the study of phonons in bulk semiconductors and insulators (Gianozzi et al. 1991), and most recently for the case of the (100) surface of GaAs (Fritsch et al. 1993). In the present approach, we use eq. (3.40) in conjunction with the selfconsistent solution of the screening problem built into eq. (3.37). We begin by making use of the assumed perfect periodicity of the bulk crystal, and introduce the Fourier transform of all quantities in the usual way. For example,

A.G. Eguiluz and A.A. Quong

452

Ch. 6

for the static RPA irreducible polarizability we set BZ

x(~

~')

~---~--~ei(r162

1

=

~?N

~,

-. -.-.

q

(3.41)

G,G ~

x x (~ (r + O, r + 6'), where ON denotes the normalization volume, the sum over the wave vector q runs over the first Brillouin zone, and G is a vector of the reciprocal lattice. The Fourier coefficients X(~ (~, ~'+ (~) are given explicitly by the w - 0 limit of eq. (4.7) - we refer the reader to that equation for the explanation of all the symbols related to the band structure. A key simplification comes about by noting that from Bloch's theorem we have that

O(k + r n'lPps(Z)lk, n)

i~(k + r n'l~ps(O)lk, n) -- e - i 4 ' ~ (/)

0z~q)

0z~(o)

'

(3.42)

where :~ (0) is the coordinate for the pseudo-ion at the (arbitrarily chosen) origin. We can then define/-independent response quantities according to the equation

6n (') (r 1,0r

G) = e-ir

(qT+ G)

(3.43)

and write down an /-independent version of the Fourier transform of the integral equation (3.37), namely (~n~) ((1 -at- 6 )

+ E

~-(1) (r 6) = on~

X(~ r

6, r

6')V(r

6', r

6")~n~)(r

6"),

(3.44)

~,,~,, where BZ f g n - f g+r l E E ' ~N g n,n' E~:,n - Eg+r A

• (~, nle-i(~§ ~ I~ + r n' / 0<~ + ~,n'l~ps(O)l~,n>

0z~(o)

(3.45)

w

453

Electronic screening in metals

With the above results at hand, we proceed to rewrite eq. (3.40) for the case of a perfectly periodic crystal as el ~ (l,l' ) - Eei0"(:~(l)-~(/')) E ~ ) ( ~ q ~Sa, r d

+ E eir O"

G) *~v(l)(~q- G)

(t)-~ (z'))

BZ

O(k, nl~ps(O)lk + q; n') Ox.(O) g n,n' Eg,n - Eg+~,n' k n -- kq--~',n'

(3.46)

O(k + q, n'l~ps(O)lk, n)

Oxn(O) where for clarity we have only written down the force constants for 1 r l'. In eq. (3.46) we have introduced the Fourier coefficients of the screening potential defined by eq. (3.39) according to

~V~I) (q'-t- 0) - ~--~ V(q'-ff- G, q'+ 0 p) 5n~)(r

0').

(3.47)

d, The simplicity of eq. (3.46) resides in the fact that the dependence of the force constants on the location of the sites l and 1p is only through exponentials in the nature of Fourier transforms - which leads us to the following explicit result for the dynamical matrix D~Z(0") defined by eq. (3.23): 1 O2(k, nl~ps(0)lk, n ) De~ (~') - ~ E fg,n i ~ ( O ) i ~ ( O ) k,n 1

+ -M E

6) *sg~l) (r

--(1) ( r

) (3.48)

1 BZ

.fg n

-- k+~',n'

re n,n' E k , n -

Eg+r

O(k + q, n'lUps(O)lk,n)

Oxn(O)

O(k, nl~ps(O)lk + ~, n') axe(O)

454

A. G. Eguiluz and A.A. Quong

Ch. 6

A remarkable feature of eq. (3.48) is that its evaluation requires that we displace just one pseudo-ion - the one at the origin. The self-consistent screening response to such displacement is handled via eq. (3.44), whose solution enters eq. (3.48) through the potential ~V~(1)(q~+ (~). All other elements of eq. (3.48) are obtained from the knowledge of the pseudopotential, the band structure, and associated wave functions. The starting point of our phonon calculations is the solution of the groundstate problem in the LDA. In the work reported in this subsection we have chosen to expand the Kohn-Sham wave functions in a plane wave basis. The sum over the Brillouin zone required in the evaluation of the ground state density is carried out by the special-points method, with meshes determined by the scheme of Monkhorst and Pack (1976). The partial occupancy of the levels near the Fermi surface is taken into account by broadening those levels with a Gaussian (Fu and Ho 1983) whose width is typically 0.1 eV. From the knowledge of the Kohn-Sham one-electron wave functions and eigenvalues, together with the crystal potential and charge density, we have the necessary input for the calculations of phonons and interatomic force constants. The matrix elements entering the evaluation of ~(1) as defined by eq. (3.45), and of X(~ as defined by the static limit of eq. (4.7), are straightforwardly evaluated from the known expansion coefficients of the Bloch states in the plane wave basis; we use the same energy (or wave vector) cutoff utilized for the ground-state of each metal. The sum over the Brillouin zone required in the evaluation of ~(1) and X(~ are performed on the same ~r used in the ground-state p r o b l e m with one proviso. In effect, for cubic lattices the Monkhorst-Pack (1976) mesh defines an evenly spaced grid that avoids the point at the origin, k - 0, by shifting the "conventional" mesh by one half of the grid spacing in each Cartesion direction. In our discussion below we refer to this mesh as the "odd" m e s h - this is the only mesh needed in the ground-state calculations. Now, because wave functions at both k and k + q are required in eq. (3.45) and in the evaluation of X(~ the wave vector ~' has to be chosen in such a way that k + q"also belongs in the mesh. As it turns out, for cubic lattices the 0-points satisfying this constraint define an evenly spaced mesh that includes the point ~ = 0; we refer to this "undisplaced" mesh as the "even" mesh. The key ingredient of the phonon calculations is the solution of the selfconsistent screening problem posed by eq. (3.44), which is a linear matrix equation for the coefficients n~(1)(0"+ d~). We solve this matrix equation (whose size is determined by the plane wave cutoff) using a standard linearalgebra package such a LINPACK. Diagonalization of the 3 x 3 dynamical matrix yields the phonon energies and eigenvectors for each propagation wave vector q" for the phonon modes.

w

Electronic screening in metals

455

Before presenting some representative results for selected metals, let us briefly consider some of the technical aspects of our scheme in the light of the linear response method developed recently by Baroni et al. (1987) - who published the first linear response calculations (for Si) which used non-local ab initio pseudopotentials. The main difference between the present method and that of Baroni et al. (1987) lies in the procedure for the determination of the screening density ~n(1)(:~). Baroni et al. solve eqs (3.29) and (3.35) iteratively, by a procedure similar to what is done in the ground state calculation. By contrast, the above scheme relies on the solution of a matrix equation for the screening density. One advantage of the iterative solution is that no large matrices have to be "inverted" and stored; on the other hand, there is the drawback that the screening density has to be calculated as many times as required for convergence (self-consistency). In the present calculations, we find that the evaluation of the kernel of the matrix equation from which we determine 3r~~1) typically takes as little as twice the time required for the evaluation of the inhomogeneous term, i.e., the "unscreened" density 3~1); for systems with a large number of plane waves this factor is at most five. On this basis, our method appears to be at least as efficient as the iterative m e t h o d - in fact in most cases it may be more efficient, as more than a few iterations are needed for self-consistency in the latter method. The price to be paid in the present scheme is that we need large amounts of memory to store the kernel of the matrix equation; however, we have not found this to be prohibitive. We note that eq. (3.48) involves a single sum over the Brillouin zone - by contrast, eq. (3.46) involves a double such sum. Thus it turns out to be more efficient to base the analysis of the interatomic force constants on a prior evaluation of the dynamical matrix and subsequent "inversion." In other words, we calculate the dynamical matrix D ~ ( q ) at the points of the even mesh defined above, and then take the inverse Fourier transform numerically. This procedure yields the full force-constant tensor ~b~,~(1, l'). Now it should be noted that both the odd and even meshes are relatively coarse - there are on the order of 8-10 points between the zone center and a zone comer. Nonetheless, it turns out that the mesh contains enough kpoints that the force constants are obtained accurately. We have verified this by checking the convergence of phonon energies along high symmetry directions with respect to the number of k-points used in the computation of the dynamical matrix. The evaluation of the interatomic force constants serves two purposes. First, their knowledge provides a useful way of interpolating the calculated dynamical matrices. By this we mean that if we were to plot the phonon dispersion curves on the basis of the knowledge of the dynamical matrix on the even mesh, we would not obtain smooth curves. However, after the

A. G. Eguiluz and A.A. Quong

456

Ch. 6

interatomic force constants have been obtained, the dynamical m a t r i x - and thus the phonon dispersion curves as w e l l - can be evaluated on a fine mesh via eq. (3.23). Second, and most important, the knowledge of the force constants provides information about the range of the interatomic forces and their nature - e.g., whether they are central or not. Of course, with the force constants ~b,~,~(1,l') at hand we can also evaluate the dynamical matrix according to eq. (3.46) by summing over shells of neighbors in real space. In fact, comparison with the "all-neighbor" dynamical matrix obtained directly from eq. (3.48) proves very instructive. For example, by examining the convergence of the former calculation to the allneighbor result we can assess the range of the interactions for the particular metal under study. Let us consider first the phonons of sp-bonded A1. In this case, it is expected that a plane wave expansion of the pseudo wave functions would be rapidly convergent. Using a pseudopotential generated by the method of Troullier and Martins (1991) we find that approximately 80 plane waves are required to represent both the Kohn-Sham wave functions and the response functions. We employ a Monkhorst-Pack mesh with 60 points in the irreducible-element of the Brillouin zone and find very good agreement with experiment for both the lattice constant and the bulk modulus - see table 1. Use of the same mesh for the determination of the dynamical matrix at the X-point of the Brillouin zone yields phonons which agree with experiment very well, as seen in table 2. Table 1 Calculated parameters for the ground state of metals for which phonon dispersion curves are discussed in the text, and their comparison with experiment (Kittel 1976).

A1 Au Pb

Lattice constant (,~)

Bulk modulus (GPa)

Theory

Experiment

Theory

Experiment

4.01 4.10 4.95

4.02 4.08 4.95

74 189 49

72 173 43

Table 2 Calculated (Quong and Klein 1992) and experimental (Gilat and Nicklow 1966) values of the phonon energies of A1 at the X-point (in meV).

Longitudinal phonon Transverse phonon

Linear response

Frozen phonon

Experiment

39.43 24.16

39.43 23.76

40.02 24.02

w

457

Electronic screening in metals

"• >.

40.0

"

30.0

~,~ 20.0 10.0 o.o

F

X

K

F

_

Fig. 1. Comparisonof ab initio bulk phonon dispersion curves for A1 (Quong and Klein 1992) with experiment (Gilat and Nicklow 1966). In fig. 1 the phonon dispersion curves are plotted along with the experimental data, given by the solid circles (Gilat and Nicklow 1966). The dynamical matrix was originally evaluated with use of an even mesh containing 89 points in the irreducible element. The interatomic force constants were subsequently evaluated as outlined above, and the phonon dispersion curves were obtained upon diagonalizing the dynamical matrix on a dense k-point grid. The comparison with experiment is clearly extremely good throughout the entire Brillouin zone. In particular, the calculated dispersion relation for the transverse mode is quite fiat in the neighborhood of the L-point, in agreement with experiment. As is well known, this feature of the phonon spectrum is indicative of the existence of long-range interatomic interactions; furthermore, we conclude that such long-range forces are accurately determined with the present method. We find that eleven shells of neighbors are required for full convergence of the dynamical matrix evaluated by the shell-by-shell method discussed above. It is useful at this point to compare the linear-response results with those of the total-energy frozen-phonon method, which we have also implemented for the purpose of the present comparison. In table 2 we have listed phonon frequencies obtained with both methods; the two sets of values are in agreement, as expected. However, we find that the frozen-phonon calculations require a much larger number of k-points than the response method. For example, using as benchmarks phonon frequencies converged to ,,~ 2% (table 2), we have that, while the frozen-phonon approach requires 550 points in the irreducible element (which corresponds to 8,000 points in the full Brillouin zone), the linear response scheme requires only 60 (or 2048 points in the full zone). This result is traced to the fact that in the response method we basically compute an expression for the force constants which has been reduced to explicit analytical form, whereas in the frozen-phonon calculations one must evaluate numerically differences between total e n e r g i e s these energies are large and differ by a small amount, the phonon energy.

458

A.G. Eguiluz and A.A. Quong

Ch. 6

Table 3 Calculated values of selected interatomic force constants for A1 (103 dyn/cm). Czz(O, 1) = Cuu(O,1) Czz(O, 1) Czu(0, 1)

-9.93 1.51 - 10.95

Cxx(O,2) Cuu(O,2) = Czz(O,2)

-2.14 -0.17

Cxx(o, 3) Cuu(o, 3) = r r 3) = r Cyz(o, 3)

3) 3)

0.33 0.25 0.35 0.12

We consider next the nature of the force constants in bulk A1. The nonzero elements of the force constant tensor for the first two shells of neighbors are given in table 3. We recall that for force constants which derive from a pair potential which is strictly central we have that (Maradudin et al. 1971)

4,,~,~(0, l) - - (c~(1)5.,~ + (~(l) - c~(l)) x,~(1)x~(1) ) i (Z)l 2

(3.49) '

where a(1) and fl(1) are, respectively, the tangential and radial force constant. Now, the radial and tangential force constants can be obtained from appropriate linear combinations of the interatomic force-constant tensor. However, such linear combinations are not unique. If the force constants were strictly central, all such combinations would yield precisely the same values for c~(1) and/3(l). For first nearest neighbors we find that different linear combinations yield values of a(1) and/3(1) with a "dispersion" of about 5%, as can be verified by using the results of table 3. Second neighbors can be described as strictly central because symmetry restricts the number of nonzero elements of the force-constant tensor, so that there is a unique linear combination. On the other hand, for third neighbors the dispersion can be as large as 50%. We then conclude that while the leading force constants are nearly central - and thus the overall bulk phonon spectrum of A1 is not affected much by the deviations from central forces - the force constants for third and more distant neighbors are quite non-central (Wallis et al. 1993); however, as just noted, their impact on the dispersion curves is not large. We have also examined the influence of the many-body effects of exchange and correlation on the calculated phonon spectra. To this end, we performed numerical "experiments" in which we switched off the contribution from exchange and correlation to the effective electron-electron interaction V(~, s entering the screening potential defined by eq. (3.39). (All

w

Electronic screening in metals

459

60.0 .....

,~ rY t.~ Z 0 Z 0

!

....... I

!

"'"'t

-t

&5.0

P

30.0 D

15.0 0.0

s

_

Fig. 2. Calculated bulk phonon dispersion curves for A1. The solid curves correspond to a full LDA calculation. The dashed curves correspond to a calculation without exchange and correlation in the effective electron-electron interaction (dashed curves). other phonon calculations reported in this chapter treat this interaction in the LDA, according to the prescription given by eq. (2.3).) Figure 2 illustrates the results of such calculations for the case of the A1 phonons propagating along the (111) direction; similar results were obtained for Au. Clearly, in the absence of the exchange-correlation contribution to V(s s the quality of the phonon dispersion curves is very poor. We can visualize this effect as follows. We first note that, if screening is completely neglected, the eigenvalues of the dynamical matrix become negative; the phonons have imaginary frequencies, and the crystal is unstable. Allowing for screening via the second term of eq. (3.48), including the contribution to eq. (2.2) and (3.39) from the exchange-correlation hole, yields stable phonons - the corresponding dispersion curves are the solid lines in fig. 2. Next, if we tum off the exchange-correlation hole contribution to eq. (2.2) - i.e., if screening is treated in the R P A - we have the dashed curves in fig. 2. The higher phonon frequencies which obtain in this case are a result of the fact that in the absence of an exchange-correlation hole the efficiency of the electronic screening is magnified. We conclude that the many-body contribution to the electronic response plays a significant role in the calculated phonon frequencies- and in their agreement with experiment. In the above argument we have ignored the fact (already alluded to below eq. (2.5)) that in the LDA the bubble X(~ entering eq. (3.37) is automatically dressed. The effect of the exchange-correlation hole on the screening response is also discussed in w4.1 in the context of the plasmon dispersion in A1. In that subsection we also argue that, at least in the case of A1, the loss spectrum which obtains upon undressing the RPA irreducible polarizability agrees with the one obtained from the LDA bubble. We turn next to the phonons of the noble metal Au. Now, it may perhaps be said that the above plane-wave implementation of our method was bound

A. G. Eguiluz and A.A. Quong

460

Ch. 6

to work well for an sp-bonded metal such as A1; similarly it can expected that it should also work for an elemental semiconductor such as silicon. A more demanding test of the method is its implementation for a transition or noble metal. In such systems the localized d-electrons are in close proximity to the Fermi level and thus contribute to the bonding in the system. A related difficulty is that the pseudopotentials are no longer weak, and thus a much larger number of plane waves may be required to represent the localized d-functions. However, recent advances in techniques to generate "ultra soft" pseudopotentials (Vanderbilt 1990; Rappe et al. 1990) allow such systems to be studied with a relatively modest number of plane waves. The calculations for Au were performed within the same framework as the calculations for A 1 - the only changes required to describe the noble-metal bonding having to do with the details of the construction of the pseudopotential and the larger number of plane waves required in the ground-state and response calculations. The pseudopotentials for Au were generated by the method of Rappe et al. (1990), which allows us to use the relativelysmall energy cutoff of 30 Ry, the associated number of plane waves being approximately 350. The meshes used for the Brillouin zone summations are the same as for A1. The phonon dispersion curves (Quong 1994) are plotted in fig. 3 along with the experimental data, given by the full circles (Lynn et al. 1973). The solid curves were obtained by computing the dynamical matrix by the shell-by-shell method described above, with inclusion of interatomic forceconstant to four neighbors. The agreement between theory and experiment is excellent across the Brillouin zone. The largest errors occur near the K-point, and are of the order 5%. An interesting feature of the results of fig. 3 is the positive curvature of the lowest transverse branch for phonon propagation along the (110) (or FK) 25.0 >,,.

L~ C

o

C 0

20.0 15.o

I0.0 5.0

0.0

F

X

K

F

L

Fig. 3. Comparison of calculated phonon dispersion curves for Au (Quong 1994) with experiment (Lynn et al. 1973). The solid (dashed) curves correspond to a calculation in which the range of the interatomic force constants was taken to fourth (second) neighbors.

w

Electronic screening in metals

461

direction; indeed, even at the K-point the phonon frequency is above the extrapolated velocity of sound line. That this subtle f e a t u r e - which has an impact on the measured specific heat of Au - is reproduced by the theoretical curves demonstrates the high accuracy with which phonon dispersion curves can be determined from first principles. The interatomic force constants of Au show a dramatic drop off with distance, as shown in table 4. In fact, the third neighbor force constants are smaller than the nearest neighbor force constants by two orders of magnit u d e - by comparison, in the case of A1 this ratio is ~ 30. The dashed curves in fig. 3 correspond to phonon dispersion curves calculated keeping force-constant up to second neighbors. The overall agreement with the full calculation is quite good, but the positive curvature of the dispersion relation for the transverse mode in the (110) direction is lost, and the overall discrepancy between theory and experiment is slightly larger. Including interactions past the fourth neighbor does not shift the curves on the scale of fig. 3. Table 4 Calculated values of selected interatomic force constants for Au (103 dyn/cm) (Quong 1994). Model calculations M1 and M2 are from Lynn et al. (1973). Response method

M1

M2

Czz(0, 1) = CVV(0,1) Czz(0, 1) CxU(0, 1)

18.05 -6.62 21.84

16.43 -6.54 19.93

16.61 -6.65 19.93

Czz(0, 2) Cyy(O, 2) = Czz(O,2)

4.66 -0.68

4.04 - 1.27

3.95 - 1.13

Czz(0, 3) Cyy(0, 3) = Czz(0, 3) r 3) = Cz~(0, 3) Cyz(0, 3)

0.19 -0.02 0.06 0.16

0.80 0.39 0.16 0.54

1.00 0.28 0.24 0.48

Cxz(O, 4) = Cuu(O,4) Czz(O, 4) Czy(0, 4)

-0.84 0.02 -0.90

-0.75 -0.14 -0.36

-0.57 -0.21 -0.36

r 5) Cyy(o, 5) Czz(O, 5) Cxu(O, 5)

0.01 0.05 -0.02 0.08

Czz(O, 6) = Cuu(O,6) = Czz(O,6) r 6) = r 6) = Cuz(O,6)

0.06 0.05

Cxz(O, 7) Cuu(O, 7) r 7)

0.04 0.02 0.00

-0.17 -0.02 0.00 -0.06

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462

Ch. 6

Table 5 Calculated values of the tangential (c~(/)) and radial (fl(/)) force constants for Au (in 103 dyn/cm). Also listed are the values of X2 (in dyn/cm) 2) obtained from the fits described in the text. Neighbor

cz(l)

/3(/)

1 2 3 4 5 6 7

5.21 0.68 0.05 0.04 0.01 0.01 0.02

39.90 4.66 0.25 1.74 0.06 0.15 0.09

X2

400000 0 3447 105 1368 0 176

Additionally, we find that the force constants are highly non-central in nature, a reflection of the fact that for this noble metal d-electrons participate in the bonding. When we try to obtain the tangential and radial force constants as in the case of A1 discussed above, we find a large dispersion of the values obtained from different linear combinations of the force-constant tensor. In order to better quantify the deviations of the force constants from central form, we performed a least-squares fit of our interatomic force constants to the form given in eq. (3.49). The quality of the fit is reflected in the value of X2 obtained, which in this case is a measure of the central nature of the force constants - if the forces were purely central, X2 would be zero. The best-fit values for a(l) and/3(1) are listed in table 5. With the exception of the symmetry-restricted force constants coupling second and fifth neighbors along the (100) direction, the force constants are in general clearly non-central. Even for the seventh neighbor X2 is fairly large, considering the smallness of the corresponding force-constant. This analysis suggests that any force-constant model for Au must be general in n a t u r e i.e., non-central - and extend to at least four shells of neighbors. Finally, we briefly consider the case of lead, element which has been the focus of much attention, since its phonon spectrum includes prominent anomalies (Chen and Overhauser 1989). We proceed from the premise that in order to gain a reliable physical picture of the nature of these anomalies, one must start out with an accurate description of the ground state and vibrational properties of the crystal, with as few approximations as possible. Thus, we have performed calculations of the phonon dispersion curves for Pb using the method outlined above. In the present case the pseudopotential was generated by the Troullier-Martins (1991) method. The calculated dispersion curves for phonons propagating along the (100) direction are plotted in fig. 4, together with the experimental data, extracted from neutron scattering experiments (Brockhouse et al. 1962). We observe

w

Electronic screening in metals

463

10

II I > E

v

r

i11 cO tO cO.

5

//

F • Fig. 4. Comparison of calculated phonon dispersion curves for Pb (Quong et al. 1993) with experiment (Brockhouse et al. 1962). that the difference between theory and experiment is larger than for the cases of A1 or Au. However, the qualitative agreement between theory and experiment is quite good. Most importantly, the dispersion curve develops its anomalous shape in the same place in the Brillouin zone as is found experimentally. The largest deviations from experiment occur near the zone edge. Now lead has a fairly large spin-orbit interaction, and the same has been neglected in the present study. Preliminary calculations suggest that inclusion of the spin-orbit interaction lowers the phonon energies near the zone edge, yielding better agreement with experiment; furthermore, it has a negligible effect at the point (1/2, 0, 0) where the anomaly develops (Liu and Quong 1995). The exact nature of the dip in the dispersion is still to be determined, and more calculations are required to obtain the force constants. Nonetheless from the present results we can already state that the unique character of the phonon dispersion of this metal follows from first principles from a "conventional" LDA ground state and related vibrational spectrum. More exotic explanations, such as the family of charge density waves proposed in the phenomenological calculations of Chen and Overhauser (1989), do not appear warranted by the ab initio method reported herein.

3.4. Surface force constants and surface phonons in AI The presence of a surface gives rise to new phenomena, such as the appearance of surface collective modes (surface plasmons, surface phonons, etc.), and to surface-induced modifications of various quantities, such as the interatomic force constants for atoms in the vicinity of a surface. Of

464

A. G. Eguiluz and A.A. Quong

Ch. 6

course, these modifications reflect in part the change in coordination number of surface atoms relative to those in the bulk. At a deeper level, qualitative changes in the surface force constants may also reflect the distinct features of the surface screening process (Gaspar and Eguiluz 1989; Gaspar et al. 199 lb; Hannon and Plummer 1995). Knowledge of these surface-induced changes is an important issue in surface physics. Now, surface screening is considerably more difficult to address at the first-principles level than bulk screening, a consequence of the breakdown in translational invariance in the direction normal to the surface. This broken symmetry has the effect that there is a new scale of length associated with the screening response of the surface to, e.g., a long-wavelength probe. In effect, in addition to slowly-varying fluctuations on the scale of length of the probe, there are density fluctuations on the microscopic scale of length over which the electron density drops to zero in the vacuum outside the surface (Feibelman 1982). (Of course, one must also contend with the microscopic density fluctuations brought about by the crystal lattice - but this is also a feature of the response in the bulk of the metal.) As a consequence, the state-of-the-art in the evaluation of screening response at metal surfaces has lagged behind its counterpart in the bulk. We model the surface via a slab of approximately 11-15 layers. Actually, we introduce a periodic repetition of the physical slab in the direction normal to the surface, consecutive slabs being separated by six to eight layers of vacuum. In this supercell geometry we have periodic boundary conditions in all three directions - we can then treat the system as a bulk-like system containing many atoms per unit cell. For a monatomic metal the number of atoms in our basis is equal to the number of atomic layers composing the physical slab. The formalism of w3.2 and w3.3 can be applied for the present case once we redefine our notation as follows. We denote the position of an atom by the pair of indices (l, x), where l identifies the unit cell and x labels a particular atom in the unit cell. In the metal surface case considered here, x labels the atomic layers in the direction normal to the surface. The dynamical matrix defined by eq. (3.48) is labeled by the combined index c~x - thus the rank of this matrix is now 3NL, where NL is the number of layers in the physical slab. The derivatives required in eqs (3.46) and (3.48) are taken according to the prescription i~Ups(0)//i~Xc~(0) ---} OUps(0; X)//OX~(0; X). So far, we have only implemented the method for low-index surfaces of A1 (Quong 1995). We have used the same cutoffs as for the bulk case; the 12 Ry cutoff for both plane waves and response functions corresponds to approximately 1200 plane waves for a system composed of an elevenlayer slab with six vacuum layers. For the Brillouin zone summations on the odd mesh, we have used 21 Monkhorst-Pack (1976) special points in

w

Electronic screening in metals

465

the irreducible element of the square surface Brillouin zone. The dynamical matrix was calculated on an even mesh containing 28 points in the irreducible element of the surface Brillouin zone. Consider the case of the (001) surface of A1. First, we determined the equilibrium geometry of the aluminum slab. We computed the total energy of the slab as a function of the interlayer spacing normal to the surface, and obtained the configuration of minimum energy. We found a small outward relaxation of the surface layer of about 0.5%, relative to the lattice constant a0 - 4.02 A determined in our calculations in the bulk. This value of the relaxation is in good agreement with experiment (Jepsen 1972) and previous theoretical work (Eguiluz 1987b; Bohnen and Ho 1988). Due to the smallness of the relaxation for this surface, we chose to neglect it in the calculation of surface force constants and phonon dispersion curves. In table 6 we list a few of the force constants obtained for surface layers. We have also listed the corresponding bulk force constants for comparison. For nearest-neighbor coupling in the surface layer, we find that the intralayer force constants are s o f t e r - which tends to shift the Rayleigh mode upwards from the value obtained with use of bulk force constants. Additionally, we find that the zx component of the force-constant tensor is no longer zero at the surface as it is in the bulk, which is a consequence of the broken cubic symmetry at the surface. For the other nearest neighbor force c o n s t a n t an interlayer force c o n s t a n t - we find a very different effect. There is a stiffening in components that are along the bond. For second neighbor coupling, we find large changes in force constants; in particular, we note that the intralayer force constant stiffens along the bond. This is consistent with the presence of surface stress in this surface, but it is not the dominant contribution to the shift in the Rayleigh mode. These results agree rather well with the earlier findings of Gaspar and Eguiluz (1989). Interestingly, the substantial changes in surface constants are not restricted to the top layer. We find that even in the third layer there are sizable changes in the surface force constants, but the magnitude of these changes decreases. We also find an alternating pattern to the c h a n g e s - e.g. the first and third layer force constants exhibit similar behavior as the second and fourth. The long-ranged nature of the force constants is consistent with the results we have for the bulk, and the larger oscillations are due to the enhanced Friedel oscillations at the surface. Note that while some force constants change by 90%, the change in the force constants is only about 1000 dyn/cm which corresponds to a shift of about 10% of the maximum value. In fig. 5 we present the dispersion curves for the surface phonons of A1 (001) along high-symmetry directions of the surface Brillouin zone. The dense background of surface-projected bulk modes is achieved from our use of a 51 layer slab. The dynamical matrix for this thick slab is obtained by

466

A. G. Eguiluz

and

A.A.

Ch. 6

Quong

Table 6 Calculated interatomic surface force constants for A1 (001) (in 103 dyn/cm). Intraplanar force constants Nearest neighbors x = X'

1 2 3 Bulk

R1 - R2 -- (0.5, 0.5,0.0) xx = yy

8904 9706 9148 9929

Next nearest neighbors X = X'

1 2 3 Bulk

zz

133 1704 1567 1510

xy

10419 10478 10751 10948

ZX

734 781 0 0

R1 - R2 = (1.0, 0.0,0.0) xx

2286 2032 2702 2141

yy

zz

ZX

572 323 417 168

157 141 57 168

472 132 341 0

Interplanar force constants Nearest neighbors + 1 = x'

1 2 3 Bulk

1 2 3 Bulk

xx

10538 9390 10120 9929

Next nearest neighbors x+

R1 - R2 - (0.5, 0.0,0.5)

2 = x'

yy

1479 1297 1619 1510

ZZ

10377 10367 9906 9929

ZX

12564 10635 10637 10948

R1 - R2 = (0.0, 0.0,1.0) xx = yy

48 148 2 168

zz

1046 1754 1752 2141

using calculated bulk force constants for the interior layers; the a b i n i t i o surface force constants were used for the five surface layers on either side of the midplane of the physical slab. The energy obtained for the Rayleigh mode at X is 15.84 meV in good agreement with the zero-temperature experimental result of 16.1 meV (Gester 1994), and with the theoretical value of Gaspar and Eguiluz (1989) 16.3 meV. The value reported by Bohnen and Ho (1988) is 14.88 meV. In addition to the Rayleigh wave there is a lower-frequency mode for all directions of propagation. This mode has shear-horizontal polarization along FX, and changes character to shear-vertical (SV) polarization as it approaches the SV-polarized Rayleigh wave at a point between X and M. We also find a SV mode, which occurs above the Rayleigh wave near X, and very close to the bottom edge of the continuum of bulk modes. This mode has displacements primarily in the second and fourth layers. A similar mode was found

w

Electronic screening in metals

467

Fig. 5. Calculated surface phonon dispersion curves for a 51 layer A1(001) slab (Quong 1995). for Na (001) (Quong et al. 1991) and a relaxed Lennard-Jones film (Allen et al. 1971). This mode was not obtained in the perturbation theoretic calculations of Gaspar and Eguiluz (1989). Thus, while the present results are overall very similar to the ones by these authors, there are some differences, which are ultimately related to the more elaborate treatment of the surface screening response contained in the present non-perturbative calculations. An additional feature of the linear-response method is that the electron density induced by the presence of selected surface modes can be easily examined. This capability of the method is of great interest in the interpretation of the He-atom scattering data. In effect, under the assumption that the scattering can be approximated by the value of the electron density at the turning point of the atom, it becomes relevant to ask what modes will generate significant modulations of the electron density. In fig. 6 we show a series of induced density contours for selected surface phonons of the A1 (110) surface. In fig. 6a (fig. 6b) the mode has polarization normal to the surface, and the induced density has been plotted for the surface located at 0.8 * (1.6 *) into the vacuum. We notice that closer to the surface, the charge densities oscillate up and down, in correlation with the up and down oscillations of the surface atoms; as one moves away from the surface, this behavior persists, but the magnitude of the changes is smaller. The calculations are repeated in fig. 7, but for a mode with polarization in the plane of the surface. Surprisingly, the nature of the results is similar to

A.G. Eguiluz and A.A. Quong

468

Ch. 6

1.4142

0.7071 -

....

0 -0.5

0 ~

I

0

0.5

Fig. 6a. Contours of electron density induced at the A1 (110) surface by the presence of a phonon with polarization normal to the surface plane; a) in the plane lying 0.8 ,/~ above the surface (Quong 1995).

the case of the mode with polarization normal to the surface. We interpret this finding as follows. As surface atoms move towards each other they will push the charge away from the interstitial between them, and the easiest place for the charge to move is out of the plane. The results for both polarizations differ mostly in the magnitude of the charge buildups and in the position along the surface of the charge oscillations. For the mode polarized out of the plane, the charge builds on the atomic sites, but for the polarization in the plane, this takes place between the atomic sites. Finally, we note that for the case farther from the surface shown in figs 6b and 7b the mode with polarization normal to the surface induces changes in the electron density that are almost an order of magnitude larger than for the

w

Electronic screening in metals

469

1.4142

0.7071

-

-

0 -0.5

0-'-'--

0

0

0.5

Fig. 6b. Contours of electron density induced at the A1 (110) surface by the presence of a

phonon with polarization normal to the surface plane; b) in the plane lying 1.6 A above the surface (Quong 1995).

in-plane polarization. Nonetheless these calculations suggest that it is indeed possible for He atoms to scatter off modes which are polarized in the plane of the surface; this could explain the observation of several near-resonances on noble and transition metal surfaces. We close this subsection by summarizing the main conclusions of an earlier series of calculations performed for A1 surfaces (Gaspar and Eguiluz 1989; Gaspar et al. 1991a; Franchini et al. 1993). Such calculations were performed by treating the pseudopotential to second order; for details of the method see Eguiluz et al. (1988), Franchini et al. (1993). This perturbationtheoretic procedure corresponds to evaluating the density-response func-

A. G. Eguiluz and A.A. Quong

470 1.4142

0

0.7071 - --------0

o

-0.5

Ch. 6 0

-~

/

!

0

0-

0.5

Fig. 7a. Contours of electron density induced at the AI (110) surface by the presence of a

phonon with polarization in the plane of the surface; a) in the plane lying 0.8 ,~ above the surface (Quong 1995).

tion X for the electron-gas model of a metal surface (Eguiluz 1983, 1985, 1987a, b). The results of such simplified- yet self-consistent- response calculations bring out additional key features of the physics of the surface force constants and surface phonon spectrum. For A1 (100) and A1 (111) (Gaspar and Eguiluz 1989; Eguiluz et al. 1990; Gaspar et al. 1991 a) the calculated dispersion curve for the Rayleigh wave lies appreciably higher than the curve obtained with the use of force constants calculated for bulk A1. It is important to note that this behavior is not due to a stiffening of the force constant coupling nearest neighbors in the outermost two layers (Mohamed and Kesmodel 1988), as it appears to be the case for other fcc metals (Wutting et al. 1986a, b; Lehwald et al. 1987). In fact, this force constant actually softens, a consequence of the fact that, as noted above, the calculated relaxation amounts to a small (~ 0.7%) expan-

Electronic screening in metals

w

471

o

r

f

, N,

/ ~

~

~,..-----'"~0

''

-0.5

0.2

,,~

.

1

0

0.5

Fig. 7b. Contours of electron density induced at the A1 (110) surface by the presence of a phonon with polarization in the plane of the surface; b) in the plane lying 1.6 A above the surface (Quong 1995).

sion of the first interlayer spacing (Eguiluz 1987b). Instead, the upward shift of the Rayleigh wave frequency is found to be determined by the surface value of the tangential force constant coupling first nearest neighbors in the outermost layer for displacements normal to the bond and to the surface. For a simple nearest-neighbors model, the tangential force constant just alluded to is proportional to the radial derivative of the (central) pair potential at the equilibrium position, which, in this model, is directly related to surface stress (Wuttig et al. 1986; Hall et al. 1988). However, it is the derivative of this pair potential normal to the surface the one which impacts the Rayleigh wave frequency at the zone boundary, and not its lateral derivative, which is the one that contributes to the surface stress. We then have that the presence of surface stress - which is certainly finite for this surface (Needs

472

A. G. Eguiluz and A.A. Quong

Ch. 6

and Godfrey 1987)- does not find its way into the Rayleigh wave dispersion curve near the zone boundary, as it does in the simple models. The physics behind the above result is traced to the fact that the pair potential at the surface is far from central. Now, the perturbative method yields a pair potential which is strictly central in the bulk (fig. 8(a)); as noted above, in reality this is not a good approximation for neighbors beyond the second, but it is a fair approximation for the leading force constants, namely the ones coupling an atom with its first two shells of neighbors. By contrast, the calculated pair potential is found to be highly non-central near the surface, as illustrated in fig. 8(b) for A1 (100). This result is an inherent feature of the surface vibrational problem, and is a direct manifestation of the nature of surface screening. The subject of resonances in the surface phonon spectra, first observed for the (111) surfaces of the noble metals, has now a rather long history (for a brief review, see Toennies 1990). Part of the original excitement brought about by this problem was the expectation that the resonances may provide a signature of changes in the interatomic force constants at the surface. Without venturing much into such a large topic, we briefly touch on this issue for the case of sp-bonded A1, for which the resonances are rather weak. Figure 9 shows phonon spectral densities for all three low-index surfaces of A1, projected onto a scan curve, together with the experimental time-offlight spectra for He atoms backscattered from the surface (Gester 1989). (The scan curve is the relation between the momentum transfer hq-~l and the energy transfer No which results from conservation of energy and momentum parallel to the surface, for given values of the energy of the incident He atom E~ and of the angle of incidence O~.) The relatively sharper resonance observed for the (110) surface has been shown to be a consequence of the substantial multilayer relaxation associated with this open surface (Gaspar et al. 1991a); the changes in force constants are in this case of the order of (15-25%), and correspond mostly to relaxation-induced stiffening of the force constants. The reduced spectral weight of the resonance for the (100) and (110) surfaces is attributed to the fact that these more densely-packed surfaces relax very little. It is noteworthy that the "lateral" surface force constants play no role in the calculated resonances. For the test case of A1, we proposed that the observed resonance originates from changes in the entire surface force field, and not to a softening or other change of a single force constant. This means that, contrary to previous suggestions, the resonance cannot be used as a simple diagnostic tool for individual surface force constants. A detailed calculation of the reflection coefficient for He atom-surface scattering in the distorted wave Born approximation has also carried out

w

Electronic screening in metals

473

Fig. 8. Equipotential lines for the pair potential (in Ry) at the A1 (100) surface, drawn on a plane normal to the surface (Gaspar 1991c). The dashed line identifies the intersection of the plane parallel to the surface, containing the reference atom, and the plane for which the pair potential is drawn. The arrows indicate the position of the first and second nearest neighbors, respectively, a) Neighborhood of the third atomic plane, which basically corresponds to the case of the b u l k - note that the equipotential lines are circles; b) Neighborhood of the outermost atomic plane, negative displacements being directed toward the vacuum outside the surface note that the equipotential lines are not circles, i.e., the pair potential is not central.

A. G. Eguiluz and A.A. Quong

474 ..~

Ai(UO)

"~Z L

AI0")

AlO00)

E~=33.1meV

.,4,,,,

Et=26.9 meV

Et=34.1 meV

0~=37.1"

Ch. 6

0~=40.6"

~

& _J z t~ " -

9

.

a z

~

z

o

I

0 .,.I

----

9

r~ -16

-12

-8

-4

-16

-12

-8

-4

-16

ENERGY TRANSFER ( meV )

-12

-8

-4

Fig. 9. Experimental time of flight spectra for A1 (Gester 1989), and calculated phonon spectral densities (Gaspar et al. 1991a) along the scan curve for modes polarized normal to the surface (solid lines) and longitudinal modes (dashed lines). Each panel is labeled by the incident angle Eki. The incident beam energy Ei is also given. The arrows show the location of the resonances that were resolved experimentally.

(Franchini et al. 1993) - this was the first such calculation to start out from the self-consistent solution of the underlying surface screening problem. Overall, the agreement with the He atom time-of-flight data for all three low index surfaces of A1 is very good. From the calculated reflection coefficient we have been led to characterize the resonances for A1 as weak resonances of the Fano-Wigner type. The resonances observed for noble metals are in this scheme characterized as strong resonances.

4. Dynamical electronic response in bulk metals The spectrum of charge-density fluctuations is directly accessible to experimental observation. For example, from first-order time-dependent perturbation theory we have that the differential cross section for a process in which a fast electron is scattered from an initial plane-wave state Ik ) into a final plane-wave state Ik '), while the target (solid) undergoes a transition

w

Electronic screening in metals

475

between many-body states [i} and If) is given by the equation d2o"

dg2 dw

(I -+ F ) =

27r mf2N

h

hk

[(FIVlI)I25(EF- El),

(4.1)

where II) = [i) | Ik ) and IF) = If) | [k') are the initial and final states for the combined projectile-target system, E1 and EF being the corresponding energies. The operator for the interaction between the electron and the solid is given by the equation

V" - - / d 3x ~(:g) Uel(:g),

(4.2)

where ~(:g) was introduced in eq. (3.3) and Uel(:g) is the potential due to the incoming electron. In order to make contact with experiment, eq. (4.1) - in which we have already accounted for the particle flux hk/m~2g associated with the incident beam - must be summed over all final states available to the scattered electron (the density of such states is ($?N/27r3)(mk'/h2)) and over all possible initial and final states of the solid (the former are weighted with an appropriate statistical factor) consistent with the energy-conserving delta function. Standard quantum mechanical manipulations lead us to the Van Hove formula

d2cr dS? dw

=

171,2

kI

-- ~2(q)S'nn(0', co), (27r3)h5 k

(4.3)

where ,-,qnn(q',cO) is the frequency- and wave vector-Fourier transform of the auto-correlation function for density fluctuations induced in the solid, i.e.,

d3zf x

f+oo

d3x'e -iq'(g-z') (4.4)

dt e iwt <~(~., t) ~(~', 0)).

oo

In eq. (4.3) ~-" is the momentum transferred by the projectile to the solid and ha; is the energy transfer. The differential cross section for inelastic X-ray scattering is governed by an equation of the same form as eq. (4.3);

A.G. Eguiluz and A.A. Quong

476

Ch. 6

see, e.g., Platzman et al. 1992. The fluctuation-dissipation theorem, which at zero temperature adopts the form

d3zf d3xte-iq'(~'-~")ImXnn(X,Xt;W),

Snn(q';w)--2hf

(4.5)

establishes a direct link between the experimental cross section and the density-response function Xnn defined in w2 via the Kubo linear-response formula (2.1). For a periodic crystal we introduce the spatial Fourier transform of the density-response function Xnn(~', :~t; 6o) according to the equation

BZ Xnn(~,~t;(a.)) -- "~NNE_. ~_._. ei(q'+G)'~e-i(q'+G')'~" XO,O,(~'; w ), 1

q

(4.6)

G,G'

where we have adopted the matrix notation XO,O,(q;w)

= Xnn(q' + G,

q + (~';w). For the RPA bubble X(~ the expansion coefficients defined according to eq. (4.6) are given explicitly by (o) .-.

1

'

fr~,n - fr~+r

BZ ~e

n,n'

E;,n - Es

+

+ it/)

(4.7)

A

A

•

+ q, n'lei( +o"

n ),

where the sums over n and n ~ run over the band structure for each wave vector tic in the first Brillouin zone, and 77- 0+. Note that in the evaluation of the matrix elements entering eq. (4.7) the Block states may be expressed in an arbitrary basis, not necessarily a plane-wave basis. We emphasize that a complete evaluation of X(~ entails a full account of its frequency dependence and the inclusion of all the unoccupied bands that may contribute to the final result for the physical observable under considera t i o n - in particular, for energies comparable with, e.g., the plasmon energy. This has traditionally proved to be a major roadblock for the realization of ab initio investigations of dynamical electronic response in metals. It is only in the last two years that such calculations have become available (Quong

w

Electronic screening in metals

477

and Eguiluz 1993; Maddocks et al. 1994a, b; Lee and Chang 1994; Aryasetiawan and Karlsson 1994; Aryasetiawan and Gunnarsson 1994; Fleszar et al. 1995a, b; Gaspar et al. 1995). In the above Fourier representation the integral equation for Xnn (eq. (2.2)) is readily transformed into a matrix equation for the Fourier coefficients Xd,d,. From the numerical solution of that equation the dynamical structure factor is immediately obtained according to the equation S n n ( q , C.d) :

--2h~NIm X~=6,C'=6(q; w),

(4.8)

where the wave vector transfer q"can take on arbitrary values - in particular it may lie outside the first Brillouin zone. Equation (4.8) is used repeatedly in our discussion of the dynamical density-response of selected metals presented in w167 4.1-4.3.

4.1. Plasmon dispersion relation and dynamical structure factor of AI The input to the entire calculation of the loss spectrum based on eq. (4.8) is well-converged, self-consistent solution for the ground state of the crystal in the LDA (Kohn and Sham 1965). Such solution was constructed by expanding the Kohn-Sham one-electron Bloch states in a plane-wave basis, for which we used a kinetic-energy cutoff of 12 Ry. In our calculations the electron-ion interaction is described by a non-local, norm-conserving ionic pseudopotential. Pseudopotentials constructed using several schemes (Bachelet et al. 1982; Hamann 1989; Troullier and Martins 1991) gave essentially the same results for the loss spectrum of A1. In the evaluation of the RPA bubble according to eq. (4.7), convergence was achieved by including on the order of 40 bands. Now, in the collisionless regime (Pines and Nozi~res 1966) in which we work, in principle in eq. (4.7) we must set rl = 0 +. However, in practice, the sampling of the Brillouin zone required in that equation was done in conjunction with a choice of a finite value of r / - for A1 we typically used r/,-~ 0.5 eV. We have checked that this small numerical broadening does not affect our results in any significant way. Wave vector sampling of the Brillouin zone was implemented on a dense mesh, defined according to conventional algorithms (Monkhorst and Pack 1976). For the determination of the plasmon dispersion curve we used 182 k-points in the irreducible element of the Brillouin zone; for the accurate representation of the large wave vector response discussed below we used 408 k-points in the irreducible element. Figure 10 illustrates the frequency dependence of Re xo=o, ~'=0 and Im X~=0, ~'=0 for a small wave vector transfer. The fulfillment of the

478

Ch. 6

A. G. Eguiluz and A.A. Quong

Dynamical density-response of AI '

0.4

I

'

'i"

'

I

'

I

I

'

/

1

TDLDA

=,,.,,=

t--

..... Re ~, G=O.G...o(q;(O)

::3

.=5 II,..

v

r

E

m

0.2

_

0.0

~;e,o.~...o(q;~)

=

I

,, i | ',

(9

n-

Im

0

_l

I

5

I

I

,

,

!"

q=O.07 k F

-0.2

a'

f

I

I

I

10 15 20 Energy (eV)

25

J

30

Fig. 10. Typical result for the dynamical density-response function of A1 for a small wave vector transfer (Eguiluz et al. 1995). Note that Imx~=0, ~,=0(q';w) is proportional to the dynamical structure factor Snn(q; w) according to eq. (4.8).

Kramers-Kronig equation relating both functions is visually apparent. Figure 10 looks very "uneventful," in that the shape of Im XO=0, ~'=0, which is clearly dominated by a well-defined plasmon, is the type of loss function one expects from the jellium model. This result for A1 is to kept in mind when we discuss the case of Cs, which contains structure absent in the loss spectrum for electrons in jellium, and that of Pd - which, as expected, looks very different from the loss spectrum for the simple homogeneousbackground model. The plasmon dispersion relation was obtained numerically by recording the energy position of the peak in Snn(~';to) as function of the wave vector transfer ~ (Quong and Eguiluz 1993). In fig. 11 we present results of three such calculations, which we compare with the electron energy-loss data of Spr6sser-Prou et al. (1989). (We note that for ~"s larger than those in fig. 11 the plasmon ceases to be a well-defined excitation due to Landau damping; for A1, 27r/ao ~ 0.89 kF.) First, we have the RPA for electrons in jellium with the average density of A1 (rs = 2.07). Figure 11 makes clear that this is a rather poor approximation, particularly for large wave vectors. Next, we have the RPA implemented for band electrons according to the outline given above; the quality of the theoretical dispersion curve improves substantially for all wave vectors. Note, in particular, that for large q's the effect of the band structure

w

Electronic screening in metals

479

Plasmon dispersion in A! !

> v

27

m

25

o~

23

r tO

E u~ IX.

'

I

'

I

'

I

'

I

'

9 Jellium - - R P A o

I

A _

C r y s t a l -- R P A

9 Crystal --TDLDA -§

,t

-

Experiment

o ,t O

21

o

19

&

e..§

O

.~"§

e.q~"

17 15 I

0.0

i

I

0.2

I,

I

0.4 2

I

i

I

0.6

2

I

0.8

i

I

1.0

q (2~/a o)

Fig. 11. Plasmon dispersion relation in A1 (Quong and Eguiluz 1993). Shown are calculated RPA and TDLDA dispersion curves for A1 crystal, and the RPA curve for jellium with rs = 2.07. The experimental data are from SprOsser-Prou et al. (1989). Plasmon propagation is along the (100) direction.

on the plasmon frequency is quite large (about 4 eV). Finally, we have the TDLDA for band electrons, which yields extremely good agreement with experiment for all wave vectors. The downward shift of the TDLDA plasmon dispersion curve for finite wave vectors relative to the RPA result can be understood as follows. When we turn on the exchange-correlation contribution to eq. (2.3) we turn on an exchange-correlation hole about each electron. This dressing of the electrons translates into a less efficient screening response than would be the case for fictitious electrons without an exchange-correlation hole - thus the downward shift of the plasmon dispersion curve for finite q (the local-field factor vanishes for zero wave vector; see eq. 2.4). The main conclusion to be drawn from fig. 11 is quite apparent: in order to reproduce the measured plasmon dispersion relation in A1, both physical features included in our calculations, namely a realistic treatment of the one-electron band structure and the many-body effect of the electron-hole vertex in TDLDA, are required. Remarkably, the crystal effect turns out to be the larger of the two - the electrons-in-jellium shape of the loss spectrum shown in fig. 10 notwithstanding. It was been argued that, plotted versus the square of the wave vector transfer, the downward shift of the experimental dispersion curve, com-

480

A. G. Eguiluz and A.A. Quong

Ch. 6

pared to jellium theory, is nearly q-independent (Spr6sser-Prou et al. 1989; Sturm 1982). Our ab initio results do not support such conclusion, as can be easily visualized from the RPA dispersion curves shown in fig. 11 for jellium and for A1 crystal. Now, the inelastic scattering of X rays is currently the method of choice for the study of short-range correlations in metals, as large momentum transfers are in principle easily accessible to experimental observation. Early inelastic X-ray scattering experiments performed on A 1 - and other systems as well - revealed a then-unexpected two-peak structure in the loss spectrum for large wave vectors (Eisenberger et al. 1973; Platzman and Eisenberger 1974; Eisenberger et al. 1974; Eisenberger and Platzman 1976). From the premise that the band structure is unlikely to determine the observed physics for this "free-electron" metal for the large frequencies of interest (w/> 2WF, NOF being the Fermi energy), it was concluded that the double peak must be a direct manifestation of fundamental physics of the correlated electron liquid (Platzman and Eisenberger 1974; Platzman et al. 1992). It was this conclusion what ignited the large amount of theoretical and experimental work that has been carried out on this problem over the last twenty years. See, among other papers, those by Schtilke et al. (1987, 1989, 1993), Mukhopadhyay et al. (1975), Holas et al. (1979), Aravind et al. (1982), Awa et al. (1982a, b), Niklasson et al. (1983), Iwamoto et al. (1984), Rahman and Vignale (1984), Ng and Dabrowski (1986), Green et al. (1985, 1987), Maddocks et al. (1994a, b). In fig. 12 we show the dynamical structure factor Snn(0";w) evaluated for fictitious non-interacting electron-hole pairs (Fleszar et al. 1995a). Let us call the result of such calculation S(n~ The same was obtained by replacing in eq. (4.8) the density-response function Xnn by the RPA bubble X(~ 9 It is clear that S (nn~ (g; w) reproduces the main features of the experimental data quite well. Not only is there a prominent double peak in the theoretical spectrum, but its energy position is rather accurately given, and so are the intensities of the main features. Thus, fig. 12, in which we also show the result for Sn(~ w) obtained from the jellium counterpart of X(~ i.e., the well-known Lindhard function - makes it unequivocally clear that the overall two-peak nature of the loss spectrum is an inherent property of X(~ for A1 crystal. In other words, the existence of the double peak is not a consequence of the electron-electron correlations - which have been switched off in the calculations underlying fig. 12. A variety of explanations of the double peak, which were proposed over the years on the basis of many-body mechanisms (Platzman et al. 1992), are thus ruled out. [As noted below eq. (2.5), the propagators from which X(~ is evaluated in the TDLDA are automatically dressed. We have performed numerical tests

w

Electronic screening in metals

481

Non-interacting electrons in AI vs. X-ray data

2.4 .,=, .~P~_

2.0 1.6

~

1.2

8- 9

0.8

."

,,

,,,

"

u) = 0.4

0,0

q=l .7k F

..... jellium 9 0

I

X-ray data

9 I

'

I

I

I

I

I

'

I

I

~=.=i I

I

I~li

10 20 30 40 50 60 70 80

Energy (eV)

Fig. 12. Dynamical structure factor Onn~176w) for non-interacting electrons-hole pairs, and its

comparison with the X-ray data of Platzman et al. (1992) (Fleszar et al. 1995a). The solid curve incorporates the actual band structure of A1; the dashed curve is for jellium (rs = 2.07).The wave vector transfer ~' is along the (1/2, 3/2, 0) direction; its magnitude equals 1.7kF. in which we undressed the TDLDA bubble, i.e., we computed it with use of the eigenfunctions and eigenenergies which obtain from a ground state calculation in the Hartree approximation. The ensuing Sn(~ w) is basically indistinguishable from the one shown in fig. 12; this finding corroborates the physical mechanism just put forth. (Since exchange and correlation are needed in order to achieve stability, we performed these tests by "clamping" the crystal at the experimental value of the lattice constant.)] We note that fig. 12 corresponds to exactly the same wave vector, both in magnitude and direction (the direction is (1/2, 3/2,0)), as the wave vector transfer for which the high-resolution X-ray data of Platzman et al. (1992) display the double peak. This is an important point for a conclusive comparison between theory and experiment, as we have found that the details of the fine structure in finn(q; w) are strongly wave vector dependent. The effects of the Coulomb interaction are investigated in fig. 13, in which our theoretical spectra are once again plotted on the same absolute scale as the experimental intensities. At the simplest level we have the RPA; the corresponding Snn(q';a)) agrees rather poorly with the X-ray data Platzman et al. (1992), particularly on the small-frequency side of the double peak. Clearly, the RPA worsens the quality of the spectrum obtained in fig. 13 for strictly non-interacting electrons.

A. G. Eguiluz and A.A. Quong

482

Ch. 6

Dynamical structure factor of AI Wave vector transfer a/ong (1/2,3/2,0)

2.4

o.,,~ ~~,

2.0

1.6

De,,j ,/

1.2

:/" /:

8- 9

0.8

~

0.4

0.0

:-, ~'/I'F-~d'~t';

I : .:" "::'

://"

r ""

j,..: ~.~' "' 0

q=1.7k F

- TDHF

~,",,~.' ',

- r(.,',~,

~',

.... TDLDA ~,~,., -.... RPA , T ,X'ray, d,ata,

10 20 30 40 50 60 70 80

Energy (eV) Fig. 13. Comparison of the calculated dynamical structure factor Snn(~;w) of A1 and the X-ray data of Platzman et al. (1992) (Fleszar et al. 1995a). The wave vector ~' is the same as in fig. 12. The theoretical curves correspond to three different choices for the vertex or local field factor G(~') - see text for details.

Next, in fig. 13 we present the results of two calculations representative of efforts to go beyond the RPA. First we have the TDLDA, which clearly brings about a substantial improvement over the RPA. Physically, this improvement reflects the approximate TDLDA inclusion of the electron-hole attraction. Now, the TDLDA vertex f~)LDa = --UGTDLDA (see eq. (2.5)) is only exact for the homogeneous electron liquid for q ~ 0 (and for w = 0). Since this vertex is q-independent- a consequence of the local approximation - it ignores the details of the structure of the exchange-correlation hole. The trend apparent in fig. 13 suggests that the q-dependence of the vertex is important, and that, furthermore, a stronger vertex is needed for 10"1 1.7 kF. From the many published G(q)'s we have found that in the present case the one obtained by Brosens and Devreese (1988) (see also Brosens et al. 1980; Nachtegaele et al. 1983) via a numerical solution of the timedependent Hartree-Fock (TDHF) equation for jellium ("dynamical exchange decoupling" method) works best. In fact, agreement with the X-ray data of Platzman et al. (1992) is now quantitative, for all energies. (For ha; ~ 70 eV the X-ray data of Platzman et al. 1992 show the onset of core excitations; these are absent in our pseudopotential calculations.) Numerically, this result

w

Electronic screening in metals

483

is traced to the fact that, for the large wave vector under consideration, we have that G ~ 1 and X -'~ X(~ (cf. eq. ( 2 . 4 ) - thus the loss spectrum approaches that of fig. 12 for non-interacting electron-hole pairs, which resembled the experimental spectrum rather closely. It should be noted that the TDHF vertex, with its prominent spike for q = 2kF, remains a controversial theoretical concept (Wang et al. 1984; Zhu and Overhauser 1986). In particular, this vertex ignores the screening of the ladders for the electron-hole attraction. The remarkable quantitative agreement with experiment which we have just highlighted may be due to the fact that for the high frequencies involved such screening may be ineffective (in A1, hwr, ..~ 15 eV). In any event, our results strongly suggest that the electron-hole vertex manifests itself rather directly in the X-ray data. It would be extremely interesting for a systematic experimental investigation of Snn(0~;co) to be performed for A1 and other "simple" metals in the neighborhood of 2kF. Such study may help elucidate the physics of short-range correlations in metals in a more direct way than the phonon probe suggested by Zhu and Overhauser (1986). Additionally, our results suggest the need for a more elaborate treatment of dynamical correlations. In particular, it appears desirable to carry out a full numerical solution of the integral equation for the irreducible electron-hole vertex in the presence of the actual band structure of A1. In summary, we have demonstrated that the one-electron band structure plays a significant role in the dynamical response of simple-metal A1, for all wave vectors. In addition, many-body effects manifest themselves in the X-ray intensities for large wave vectors, and in the plasmon dispersion relation measured via inelastic electron scattering.

4.2. Elementary excitations in Cs The physics of the alkali metals has attracted renewed attention in recent years. Several experiments performed during the past decade reported findings that were in conflict with the belief that the alkalis were well understood. There is of course the much-discussed issue of the possibility of the formation of a charge-density wave state (Jensen and Plummer 1985; Overhauser 1985; Shung and Mahan 1986, 1988; Northrup et al. 1987, 1989; Ma and Shung 1994). Furthermore, the anomalous enhancement of the magnetic susceptibility of expanded Cs (E1-Hanany et al. 1983), and the anomalies in the experimentally-determined plasmon dispersion relation of the heavy alkalis (vom Felde et al. 1987), have led to the conjecture that the physics of these metals is dominated by strong electron-electron correlations, bandstructure effects playing a smaller role (vom Felde et al. 1989). Interestingly,

484

A. G. Eguiluz and A.A. Quong

Ch. 6

this was also the premise behind the X-ray investigations of the dynamical structure factor of A1 discussed above. Our calculations for Cs were carried out in a plane wave basis, with a kinetic-energy cutoff of 8 Ry (Fleszar et al. 1995b). In the construction of the pseudopotential (we used the scheme of Troullier and Martins 1991) we included a "partial-core correction" (Louie et al. 1982). This correction is designed to account for the spatial overlap between the large Cs core and the valence orbitals, and its impact on the non-linear dependence of the exchange-correlation potential on the total electron density. Twenty bands were included in the computation of the Fourier transform of X(~ according to eq. (4.7). The Brillouin-zone sampling required in eq. (4.7) was performed with use of 440 wave vectors in the irreducible element, the numerical-broadening parameter we used being typically r / = 0.1 eV. This small value of 77, and the rather large number of k-points employed in the sampling, were dictated by the need to resolve the fine structure present in the theoretical spectra in an unequivocal way. Figure 14 shows a loss spectrum, obtained in the RPA, which is typical of our results for small wave vectors. Now, the local field factor G(q) ~ 0 for q ~ 0, i.e., the RPA is exact in this l i m i t - which is why fig. 14 is indistinguishable from the spectrum obtained in, e.g., the TDLDA. The dominant spectral feature in fig. 14 is a prominent plasmon at ~ 3 eV, additional fine structure being clearly visible (cf. fig. 10 for A1). We assign the fine structure to interband transitions. With reference to fig. 15, which shows our calculated density of states, it seems natural to attribute the loss features at 1, 3.5, 4, and 5 eV, to one-electron transitions into empty final states giving rise to the peaks present in the density of states at about the same energies. We note that, while for small q the spectral weight of the fine structure in Snn(~'; w) is rather small, this is not necessarily the case for large q's. We make further comments on the fine structure below. The main object of interest in this subsection is the plasmon dispersion relation of Cs. Some of our key findings are contained in fig. 16. In it we display our RPA results for plasmon propagation along the (110) direction, together with the electron energy-loss data of vom Felde et al. (1989), and the very recent RPA results of Aryasetiawan and Karlsson ( 1 9 9 4 ) - w h o presented results for the (110) direction only. We first note that the RPA dispersion curve for electrons in jellium with the average density of Cs crystal, rs = 5.6, is in drastic disagreement with experiment. We stress that the value of the plasmon energy for ~' = 0 is not the issue, since in this limit most of the difference between the jellium result and experiment can be assigned to the effect of the core polarizability. The key point is that the experimental curve shows a - small- negative dispersion for small wave vectors; moreover, the measured plasmon dispersion relation,

w

Electronic screening in metals

485

Cesium

Wave vector along the (111) direction

0.5 '>

'

i

'

~ '

I

'

i

'

,

,

i

,

q=O.05a.u.t

0.4

0.3

.~ 0.2 cO= 0.1 0.0 0

i

I

1

l

I

2

I

I

3

Energy

J

I

4

i

I

5

a

I

6

7

(eV)

Fig. 14. Representative result for the dynamical structure factor Snn(q"w) of Cs for small q (Eguiluz et al. 1995). The figure corresponds to an RPA calculation, for ~" directed along the (111) direction. For small wave vectors the spectrum looks the same in other directions; inclusion of the effects of exchange and correlation via a local field factor G(~') (e.g., TDLDA) does not alter it, either.

plotted vs. q2, is quite flat for large wave vectors, while the RPA curve for jellium shows a pronounced dispersion. Inclusion of a many-body localfield correction (e.g., TDLDA) on top of the RPA-jellium result does not resolve the qualitative failure of jellium response theory. It was precisely this situation, coupled with the expectation that the effects of the band structure must be small, that prompted the conclusion that in Cs (and Rb as well) there is a fundamental breakdown of the existing theories of the response of the interacting electron liquid (vom Felde et al. 1989). Next, in fig. 16 we consider the RPA for band electrons. We performed two such calculations, both yielding dispersion curves endowed of the crucial features of negative dispersion for small wave vectors, and flat dispersion for large wave vectors. The calculation performed strictly within our ab initio pseudopotential framework ignores the effect of the core polarizability completely - thus it is not particularly accurate for q --+ 0. At the next level of approximation, we have taken into account the effect of the core in a simplified fashion by adding to the real part of the dielectric function the experimental value of the core polarizability reported by Whang et al. (1972). The overall agreement with experiment is now extremely good, for all wave

A. G. Eguiluz and A.A. Quong

486

Ch. 6

Cesium A

>

3.0

r

c: 2.5 0

t,.,

0

2.0 1.5

"~ 1.0

"6

~9 0.5 (9

CI 0.0.2

-1

0

1

2

3

I, 4

I, 5

I , 6 7

Energy (eV) Fig. 15. Calculated density of states of Cs. The pseudopotential was generated according to the scheme of Troullier and Martins (1991), with inclusion of a partial core correction (Louie et al. 1982).

vectors, except for the slightly exaggerated negative dispersion obtained for q --+ 0. It is intriguing that the RPA may turn out to work so well for large wave vectors, since the formal development of diagrammatic response theory leads one to except that the bubble should be the dominant polarization diagram for q --+ 0. In fig. 16 we also show the RPA dispersion curve obtained by Aryasetiawan and Karlsson (1994), who performed all-electron calculations in a generalized linear-combination-of-muffin-tin-orbitals (LMTO) basis. It is puzzling why these authors' result for the plasmon energy for q --+ 0 would basically equal the value for jellium, since their LMTO method seemingly includes the polarization of the Cs 5p orbitals accurately (in our pseudopotential calculations the Cs 5p states was assigned to the core; additional work is in progress, in which we treat these orbitals as a valence states). Similarly, it would be interesting to understand why the dispersion curve of Aryasetiawan and Karlsson differs from our own RPA dispersion curve (and experiment) for large wave vectors. We stress that, although we have so far introduced the effect of the core polarizability in a phenomenological way, this approximation does not determine the nature of our results. In effect, the dispersion of the plasmon for small wave vectors is negative with or without the core polarization; the fiat dispersion for large wave vectors does not depend on the effect of the

w

Electronic screening in metals

487

Plasmon dispersion in Cs

Wave vector along (11 O) direction 6.0

5.5 -e- Present work

5.0 >

4.5

>,

4.0

0

0

m c UJ

3.5

3.0 2.5

2.

Without c.p, Experiment A-K --Theory o Jellium

I_=_"'~--=----~ i-~.,.d~_,_i,__.i.__~__,__=+

Ooio

I

0.1

,

I

0.2

,

I

0.3

,

I

0.4

q2 (A-2)

i

I

0.5

,

0.6

Fig. 16. Plasmon dispersion relation in Cs, within the RPA, for plasmon propagation along the (110) direction (Fleszar et al. 1995b). Shown are the dispersion curves calculated with use of the band structure of fig. 15 and the corresponding result for electrons in jellium with rs = 5.6. The curve labeled "present work" includes the effect of the core polarizability as discussed in the text; the curve labeled "without c.p." does not. The RPA results of Aryasetiawan and Karlsson (1994) are also given, as are the experimental data of vom Felde et al. (1989). core. (A scheme for the introduction of the effects of the core polarizability has been given by Sturm et al. 1990.) Summarizing the results of fig. 16 we have that, contrary to expectation arising from the interpretation of state-of-the-art experiments (vom Felde et al. 1989), there is no fundamental breakdown of mean-field response theory in Cs - the metal with the lowest-available electron density. The RPA does provide a good overall account of the dynamical response of Cs for all wave vectors, provided that we implement it for band electrons, as opposed to electrons in jellium. The anomalous plasmon dispersion must then be attributed to band-structure effects. In order to elucidate what it is about the band structure of Cs that may explain this "by default" conclusion, we find it convenient to discuss the impact of the band structure on the dielectric function (Eguiluz et al. 1995). The Fourier transform of the definition given in the second paragraph below eq. (2.2) reads eO,O,(~';~o) = ~ d , O ' - u ( q + G)~o,O,(q;w); we recall that in the RPA, on which our present discussion is based, ~, = X~~

For a

A. G. Eguiluz and A.A. Quong

488

Ch. 6

Dielectric function of Cs -- R P A Wave vector along (I 11) direction '

20

I

'

0

'

/! Jellium I

0

I

I

'

I

Im~:G=0,G,=0(q;(0)

Crystal , I , 1

""--r-,---..~...- .... 2 3 4

Energy (eV) Fig. 17. Imaginary part of the dielectric function of Cs in the RPA, for a small wave vector transfer (Eguiluz et al. 1995). The dashed curve shows the corresponding result for electrons in jellium with rs = 5 . 6 . given wave vector q, the condition for plasmon formation corresponds to the existence of a root to the equation Re eo=o,O,=0(~';~) - 0 such that Im eO=0,O,=o(q; ~) << 1. The calculated dielectric function of Cs is shown in figs 17 and 18. The figures also show the dielectric function for electrons in jellium with rs = 5.6. For the small wave vector transfer under consideration the dominant effects of the band structure of Cs are: (i) the one-electron absorption spectrum (Im ~) has low-frequency structure which reflects the spiky nature of the density of unoccupied states shown in fig. 15. A low-lying p-d complex gives rise to the peaks observed in the density of states at about ~0.6 eV and 1 eV above the Fermi energy; the peaks in the range 1 eV <~ ha) ~< 5 eV are due to d-like states. (Papaconstantopoulos 1986, page 50, gives a partial-wave decomposition of the density of states of Cs.) (ii) the absorption spectrum has a "high-frequency" tail, which, in addition, includes a weak fine structure stemming from interband transitions into dlike empty states lying at energies higher than 3 eV above the Fermi level (such absorption structure gains spectral weight for larger wave vectors). Figure 19 highlights the difference between the dielectric function for electrons in Cs and for electrons in jellium with rs = 5.6. The prominent

w

Electronic screening in metals

Dielectric function of Cs

489 RPA

Wave vector along (111) direction

20

\~

i

'

I

\

....

'

i

'

I

ReF-,G=O,G,=O(q;0))

10

Crystal

/ /

-1 0

q=0.14a.u. \ j

0

I

1

1

dellium

/

i'

I,

I

f

2 3 Energy (eV)

i

i

4

Fig. 18. Real part of the dielectric function of Cs in the RPA, for a small wave vector transfer (Eguiluz et al. 1995). The dashed curve shows the corresponding result for electrons in jellium with rs = 5.6.

low-frequency spike in the difference curve for Re e is clearly the leading spectral feature in this figure. On first inspection, it may then seem that the strong low-frequency interband transitions may be responsible for the anomalous plasmon dispersion obtained in fig. 16. Note that, if that were the case, the anomaly would have a similar origin as the classic case of the 3.8 eV plasmon-like loss in Ag - in that case the localized d-states providing the polarizability which shifts the plasmon (by about 5 eV) are occupied (Ehrenreich and Philipp 1962). However, this turns out not to be the case. We reached this conclusion via "numerical experiments" designed to visualize the contribution to the total (or physical) dielectric function from selected empty states. For example, we show in fig. 19 the dielectric function which obtains upon eliminating from the sums over bands required in eq. (4.7) all states lying higher in energy than 2.5 eV above the Fermi level. We refer the reader to fig. 15 for the significance of the 2.5 eV "threshold" we have chosen. The result of this evaluation is also given in figs 17 and 18 - in fact, the present argument could already be made from the latter figure. It is apparent that, in the absence of those interband transitions, the zero of Re ed=0,d,=0(q;o~ ) -- 0 occurs at the same frequency as in the case of electrons in jellium.

A.G. Eguiluz and A.A. Quong

490

Ch. 6

Real part of dielectric function of Cs Crystal vs. jellium

10

'

,\l

....

J ....

j , ,j

,-~176 1

.

/

Crystal

/

-1

-5

1

'

P'

,t I , , , ,

=

2 3 Energy (eV)

. . . .

4

-2

,,

i /~,,,,;/

2

,

' .i~ ,

,

,

3 Energy (eV)

,m

i

,

'

4

Fig. 19. Illustration of the effect of the band structure of Cs on the position of the zero of Re e (Eguiluz et al. 1995). The curves labeled "Difference" correspond to the difference between Re e for Cs and its jellium counterpart. The dashed lines correspond to a calculation in which only bands up to 2.5 eV above the Fermi level are included in the computation of X(~ according to eq. (4.7).

This finding reflects two features of the dielectric function and the underlying band structure: the "high-energy" structure in question may be weak (for small wave vectors) but the plasmon happens to lie in the frequency region spanned by this structure; in addition, Re eo=o,O,=0(q; w) reaches its zero very gently, and thus the precise location of this zero is quite susceptible to even small "perturbations," such as produced by the bands lying just above the plasmon energy. It would be extremely interesting to repeat these calculations by including the effect of the core polarizability on the same footing with the polarizability of the "conventional" valence electrons. (We have already briefly commented on the all-electron work of Aryasetiawan and Karlsson (1994).) Since the effect of the core is to lower the energy of the plasmon for q = 0 by about 0.5 eV (see fig. 16), and since the energy location and strength of the absorption fine structure is open to debate- see b e l o w - the full answer to the question of the relative influence on the plasmon dispersion of the states lying below and above the nominal plasmon energy may require the explicit self-consistent inclusion in the RPA bubble of, e.g., the 5p semicore state; that calculation is in progress. Our present results clearly suggest that, for the reasons just given, the weak higher-energy interband transitions play

w

Electronic screening in metals

491

an important role in the plasmon anomaly for small q's. As alluded to above, the spectral weight of those transitions grows with q, and this is consistent with the negative plasmon dispersion obtained in fig. 16. Now, the fine structure present in our calculated loss spectra has not been observed experimentally (vom Felde et al. 1987, 1989). We have further tested this point by computing the optical conductivity ~r = (co/47r)Ime. Our results reproduce the energy position of the low-frequency absorption peak (co ,-~ 1 eW) measured by Smith (1970) and F~ildt and Neve (1983), and, upon subtraction of the Drude background, the intensity measured by the latter authors as well. Similarly, our calculations reproduce the energy position of the high-energy peak measured by Whang et al. (1972). On the other hand, our calculated or(co) contains structure in the range 3.5-4 eV which was not observed in the optical absorption experiments of Whang et al.; this structure is of the same origin as the fine structure present in our Snn(q; co); interestingly, the calculation of Aryasetiawan and Karlsson (1994) gives rise to similar fine structure (see their fig. 3). We expect that the reality - or not - of the fine structure, and the details of its impact on the plasmon dispersion, is tied up to the quality of our calculated band structure, whose associated density of states is shown in fig. 15. It is well known that the construction of a reliable pseudopotential for Cs is a subtle issue, due to the semi-core nature of the 5p orbital. Additional demands are posed by the fact that it is the unoccupied band structure in which we are particularly interested. This topic is under investigation, as is the effect of many-body corrections to the RPA. Finally, we note a related feature of our results, namely, the anisotropy of the dynamical response of Cs. This is illustrated in fig. 20, which shows the RPA dispersion curves for the three high-symmetry directions. Of course, the value of the plasmon energy for q = 0 does not depend on the direction in which this limiting value is approached. However, the dispersion curves for larger wave vectors differ appreciably. Note, in particular, that the dispersion curve for the (100) direction terminates at a finite, and rather small, wave vector- the fine structure in S'nn(q*; co) becomes so prominent for larger q's that it becomes impossible to identify the position of the plasmon peak. In this context we should note that the experiments of vom Felde et al. (1989) were performed on polycrystalline samples. The tantalizing feature contained in fig. 20 is that, while our dispersion curve for the (110) direction agrees extremely well with the measurements of these authors, the dispersion curve for the (111) direction lies below the experimental data, and that for the (100) direction is indicative of plasmon energies above the data. Clearly, it would be extremely interesting for the electron energy-loss experiments to be performed on single crystals.

A. G. Eguiluz and A.A. Quong

492

Ch. 6

Plasmon dispersion in Cs 6.0

'

I

'

I

'

I

'

I

'

I

5.5

'

I

o 1

RPA o

5.0

A (111)

0

A>

4.5

>, 4.0

~oo~ o

13 L..

r-

* 9.... (I I 0)

0

Q

3.5

0

0

o

(1001

§

Experiment

o Jellium

UJ

s.0

o

"~ ,.r--~~.§

:':t, 'o.o

o.1

......" - ;

,,,,,,,

0.2

o.a

0.4

9.......,.1

'-"

o.s

1

,I

0.6

q2 ( A "2)

Fig. 20. Illustration of the anisotropy of the plasmon dispersion relation in Cs. Shown are RPA dispersion curves along the three high-symmetrydirections. In summary, we have shown that one-electron transitions into empty states just above the plasmon energy are the root of the anomalous negative plasmon dispersion of Cs. Further work is in progress in which the 5p semicore states are treated on the same footing with the valence states. The many body effects beyond RPA, which may be expected to become more important in the presence of these spatially localized states are also under investigation.

4.3. Spectrum of charge fluctuations in Pd The spectrum of charge fluctuations of transition metal Pd is expected to show qualitative differences from the response of the sp-bonded metals discussed above. This expectation originates immediately from a consideration of the density of one-electron states depicted in fig. 21. Clearly, the initial states for polarizability calculations are dominated by the d-band complex, which, as is the case in all transition metals, straddles the Fermi level. Palladium is particularly remarkable for the sharp change with energy exhibited by its density of states right at the Fermi l e v e l - which lies very close to the upper edge of the d-band manifold. The dynamical density-response calculations for Pd were performed in a mixed basis (Louie et al. 1979) consisting of plane waves and Gaussians localized at the atomic sites. Of course, the same basis was used in the

w

Electronic screening in metals

493

Palladium 5

0

4

33

2

(1)

0

3(!)

t't

-6

-4

-2

0

2

4

6

8

Energy (eV)

Fig. 21. Calculated density of states of Pd. The pseudopotential was generated according to the scheme of Bachelet et al. (1982). solution of the ground state problem in LDA. The motivation for the use of this basis is physical" the plane waves are well suited for the description of the response of the delocalized sp-electrons, and the Gaussians are designed to represent the more localized d-electrons. We used a plane-wave cutoff of 12 Ry; five d-type Gaussians were placed at each atomic site. Pseudopotentials constructed using the schemes of Bachelet et al. (1982) and Hamann (1989) gave basically the same results. Figure 22 shows a TDLDA spectrum which is typical of our results for small wave vectors. It is interesting to note that the figure actually depicts the results of two evaluations of the dynamical structure factor Snn(q';a)) (which, according to eq. (4.8), is proportional to the G - G' = 0 matrix element of the Fourier transform of the response function Xnn)" (i) eq._(2.2) was solved as a matrix equation from the knowledge of the full (G, G') matrix for the bubble X(~ and (ii) we only retained the (2 - (2' - 0 element of the bubble, and solved eq. (2.2) for a scalar response function. It is apparent that the "crystal local fields," which by definition correspond to the effect of the non-diagonal matrix elements of the bubble, have a small effect on the energy position of the loss peaks. Moreover, for energies less than about 40 eV the loss intensities are modified by a rather modest amount by these local fields. This somewhat surprising result, which runs contrary to the expectation that charge localization (such as due to the d-electrons) would enhance the importance of the non-diagonal elements of the dielectric

A. G. Eguiluz and A.A. Quong

494

D e n s i t y r e s p o n s e of Pd --

.-~ "~-

c:

Ch. 6

T D L D A

Wave vector along (100) direction 1.5

, . . . .

~

'

. . . .

'

. . . .

Full response

'

. . . .

'

. . . .

....... Diagonal response

,~.

-1

| |

1.0

o~- 0.5 II

P,r

E --

0.0

0

10

20

30

Energy (eV)

40

50

Fig. 22. Im X~=0, ~'=0(~;w) for Pd for a small wave vector transfer (Gaspar et al. 1995). The solid (dashed) curve refers to the case in which X6=0, ~ , =0(q'; w) is obtained via eq. (2.2) from the full matrix x- (0) Q,Q, (from its (~ = ~ = 0 element only).

Table 7 Energy position (in eV) of the loss peaks in Pd for q --+ 0, Note: the energy assignments for the highest three experimental peaks were made directly from the published spectra. Theory

Experiment

Present work

Bornemann et al. (1988)

Nishijima et al. (1986)

7.6 17.5 25.5 33.4

7.2 18 26 33

7.5

matrix, has been discussed recently by Aryasetiawan and Gunnarsson (1994) for the case of Ni. The main features of the spectrum shown in fig. 22 are loss peaks lying at 7.6, 17.5, 25.5, and 33.4 eV. The comparison with the electron energyloss experiments of Bornemann et al. (1988) is very encouraging: all the measured excitations are accounted for by our results (Gaspar et al. 1995). Indeed, there is quite good quantitative agreement with the energies of the losses measured for small wave vector transfers, as illustrated on table 7. The physics behind the loss spectrum of fig. 22 is elucidated with reference to fig. 23, in which we show the calculated dielectric function of Pd. Consider first the loss which occurs at about 7.6 eV. The same is assigned to a plasmon-like mode, in that the energy position of the loss clearly cor-

w

Electronic screening in metals

495

Dielectric function of Pd -- TDLDA 15

Wave vector along (I 00) direction I

I -I I

10 -i

I

'i'

'

t I

'

'

. . . .

i

. . . .

i

I

. . . .

q-O.07 a.u.

~lm %_O,G,.o(q;co) \

-

\

\

\

1 I

\"'"---

~" - ' - - J " ~ ~

.J. .................................................................................................

-5 ~/,I

0

,

,

,

I

10

' mi

I

I

I

I

J

t

20

J

I

30

~

,

~

40

Energy (eV) Fig. 23. Calculated dielectric function of Pd for a small wave vector transfer.

responds to a zero of Re ~, and, furthermore, Im e reaches a minimum at a nearby energy. This low-energy loss has been observed, and its physics discussed, by Nishijima et al. (1986) and Bornemann et al. (1988). Interestingly, both experimental groups have reached different conclusions with regard to the question of which electrons are responsible for the measured loss. From the fact that the energy position of the loss shows a small dispersion with wave vector, Bornemann et al. (1988) (who conducted electron energy-loss experiments in the transmission mode) concluded by default that only d-electrons partake in the plasmon. Nishijima et al. (1986) (who performed electron energy-loss experiments in a reflection geometry) concluded that the loss is due to s-electrons by arguing that the effective number of electrons per atom participating in the collective motion is ~ 0.6. Our theoretical results do not support either conclusion. First, we note that the plasmon dispersion relation of Cs discussed above provides an obvious counter-example to the argument of Bornemann et al. (1988). Next, we note that we have estimated the effective number of electrons participating in the 7.6 eV loss, and we obtained, in fact, a value on the order of 0.6. (Our estimate is based on evaluating the integral f o dc~176Im e(cot), where energies are measured from the bottom of the occupied bands.) However, this finding does not necessarily support the conclusion of Nishijima et al. (1986), since for energies comparable to that of the loss in question there

496

A. G. Eguiluz and A.A. Quong

Ch. 6

is a significant contribution to the dielectric function from band structure features which are not s-like. This result is not particularly apparent in fig. 23, but we have performed explicit tests which bear it out. For example, we have computed the loss spectrum with and without the contribution to the RPA bubble from a (presumably d-like) flat band located at ~ 8 eV above the Fermi level; our results show that such band influences the position of the loss appreciably. Thus, the 7.6 eV peak is not quite the same "pure" plasmon as it exists in A1. The two high-energy peaks present in the loss spectrum of fig. 22 at about 25.5 eV and 33.4 eV are in the nature of plasmons. This conclusion follows from the observation that the peaks occur for energies which are approximate roots of Re e = 0. Furthermore, the energy positions of these losses are relatively insensitive to changes we introduced in the band structure via computer experiments designed to test this point, and this is consistent with the plasmon interpretation of these losses. The small-but-finite value of Im e in the respective neighborhoods accounts for the width of the peaks in the loss function- these collective modes are rather efficiently damped by interband transitions. The peak in S'nn(q*;w) lying at ~ 17.5 eV is more of a hybrid mode. Not only does the dielectric function not show a clear-cut collective-mode behavior for this energy (fig. 23), but additional numerical tests we have performed indicate that this loss is strongly affected by the details of the one-electron band structure for energies in the vicinity of the peak.

4.4. Spectrum of spin fluctuations in paramagnetic Pd The large value of the density of states at the Fermi level (see fig. 21) is intimately connected with the special magnetic properties of Pd. These properties have intrigued the condensed matter physics community for many years (see, e.g., Doniach and Engelsberg 1966; Mills and Lederer 1966; Berk and Schrieffer 1966; Liu et al. 1979; Stentzel and Winter 1986; Cooke et al. 1988; Hjelm 1992). In particular, the static magnetic susceptibility of Pd is strongly enhanced by exchange and correlation- indeed, Pd is close to fulfilling the Stoner criterion for the onset of the ferromagnetic instability. Before discussing the physics of Pd further, a brief presentation of our method of evaluation of the spectrum of spin-density fluctuations is in order. In the first Born approximation, the differential cross section for inelastic scattering of slow neutrons by virtue of their magnetic coupling with the spin of the conduction electrons is given by the equation (see, e.g., de Gennes 1963; Cooke 1973) d2o- _ [ g N r O ] 2 k ' / \ 1 S+_((;w), d ~ dw k,, , ] P B k 27rh

(4.9)

w

Electronic screening in metals

497

where r0 is the classical electron radius, 9N is the gyromagnetic ratio for the neutron (= 1.91), and #B is the Bohr magneton. In writing down eq. (4.9) we have assumed that the metal is in the paramagnetic phase. The dynamical structure factor entering eq. (4.9), S+_(q';w), is the frequency- and wave vector-Fourier transform of the "transverse" correlation function involving space-time fluctuations in the magnetic moment density, i.e.,

e -iq'(~-~')

F

oo

dt

oo

(4.10)

x e i~t (~+(~, t ) ~ _ (:~', 0)), where ff~+(s = ff~x(s i~u(s t). The Cartesian components of the magnetic moment density operator in a plane orthogonal to an assumed external magnetic field (conventionally taken to be along the z-axis), are given by the equations ff~x -- --#B(ff'~ff'+ + ff,+tff,~) and ff~u - --i#B(--ff'~ff'+ + ff'+t~1-), respectively, where ff,,t(s t)is the field operator which creates a spinup electron at position s and time t, etc. The fluctuation-dissipation theorem relates S+_(q';w) to the "transverse" spin susceptibility X + - ( s 1 6 3 ;w) via an equation of the same form as eq. (4.5). We evaluate the susceptibility as follows. First we note that the full susceptibility tensor X~# obeys the Kubo linear-response formula

6mo,(:F,;w)-- ~ f d3x ' Xc,~(:E, :~'; w) B~Xt(s w),

(4.11)

fl which gives the magnetic moment density induced by an external magnetic field /~ext. In addition, we define a mean-field susceptibility tensor (0) ,-. ~., ;w) such that X~#~x, ama(:~; w) -- ~ / fl

d 3x'-x~et (0) ,,~, i ,; w) B}ff(e'; w),

(4.12)

where the effective magnetic field /~eff differs from /~ext because of the electron-electron interaction- this difference is the source of the muchstudied exchange-correlation enhancement of the spin susceptibility; cf. eq. (3.29) for the charge-density case. Of course, the precise identification of/~eff via the solution of the many-body problem is the key issue. We

A.G. Eguiluz and A.A. Quong

498

Ch. 6

adopt a time-dependent "extension" of the local spin-density approximation (let us call it TDLSDA) of spin-polarized density-functional theory (von Barth and Hedin 1972), which for the paramagnetic state yields the simple result that Beff(:~; w) a

next,-.. ~ X , W) -- J(n(~)) amc~(:~; w),

(4.13)

---- / J a

where the exchange-correlation contribution to the inverse susceptibility, J(n(:~)), is given by the equation

J(n(s )) - - n(:~) ( - i~2 -

e~(n,

i~m2

m)

)

,

(4.14)

n---n(:~); m=O

where e~(n, m) is the exchange-correlation energy per electron in an electron gas with number density n and magnetic moment m. (Note that X-1 = (X(~ - 1 - J.) Utilizing eqs (4.11), (4.12) and (4.13) we readily establish an integral equation, valid in the TDLSDA, from which the full tensor X~fi is to be obtained. Now, from the spin isotropy of the paramagnetic state we have that

1

X ~ - Xzz3~ = -~ X+-3~, Z

(4.15)

which allows us to write down an integral equation for the transverse susceptibility,

(4.16) - /

d3x"

X~)-(:~,x,";w)J(n(x")) X+ _(~'", :~'; w),

which is the equation that we actually solve. The diagrammatic interpretation of eq. (4.16) corresponds to a summation to all orders of the ladder diagrams for repeated exchange scattering of a pair of electrons. Equation (4.16) was first obtained by Vosko and Perdew (1975) for the ground state (w = 0) susceptibility- in that case spin-polarized densityfunctional theory provides a formally exact theory of the spin response. The local approximation (eq. (4.13)) was invoked by these authors only after setting up a more general integral equation for the static susceptibility.

w

Electronic screening in metals

499

As was the case for the density-response method, we make use of the assumed perfect periodicity of the crystal and work with the spatial Fourier transform of the susceptibility - we shall denote it Xd,d, + - (q; ~o) - defined according to eq. (4.6). Finally, we note that in the paramagnetic state we have that X~_(:~, : ~ ' ; ~ ) - 2X~)(~, :~'; w ) - 2#2X(~

:~'; w),

(4.17)

where the last response function is precisely the RPA bubble defined above. Thus, the non-interacting transverse susceptibility X(+~ which is the central element in the computation of X+-, is already available from our solution of the density-response problem. We solve eq. (4.16) by turning it into a matrix equation for the Fourier coefficients Xd,d,(0"; +co); from its solution we obtain S+-(0"; w) via the direct counterpart of eq. (4.8). Janak (1977) has computed the static spin susceptibility of Pd from first principles. Within the local spin density approximation, he obtained a result of the Stoner form

X+- =

,

(4.18)

where we have used our own notation in order to simplify the discussion. It is important to note that eq. (4.18) is of the same symbolic form as the solution of the more general integral e q u a t i o n - eq. ( 4 . 1 6 ) - which is the basis of our dynamical calculations. Now, in Janak's case, which is restricted to the q - - 0 susceptibility, one has that X(+~ - N(EF), where N(EF) is the density of states at the Fermi level. (A multiplicative constant in this equation takes care of the units.) Janak obtained I - N ( E F ) J - 0.77, which translates into an enhancement of the static susceptibility given by X+_/X(~ ~ 4.35. Recent ab initio work by Sigalas and Papaconstantopoulos (1994) reported a somewhat larger value for I (~ 0.85). Note that if I were unitary - which nature would have had to "arrange" by further enhancing the importance of correlation, or by increasing the value of N(EF) - the denominator of eq. (4.18) would vanish, and Pd would go ferromagnetic. This is the Stoner criterion for the ferromagnetic transition. Our calculations of the dynamical spin susceptibility were performed in a linear-combination-of-atomic-orbitals (LCAO) basis, the atomic orbitals being replaced by self-consistently chosen Gaussians (Chan et al. 1986; Tom~inek et al. 1991). There is no definitive physical reason to pick this D

A. G. Eguiluz and A.A. Quong

500

Ch. 6

Spin susceptibility of Pd 0.25

'

I

'

I

'

I

'

I

'

"7" A

r

<:9 0.20 >

XC-enhanced --

II1

~

Non-interacting

0.15

.~.

o.lo

$ P'r 0.05

E

~

q-O.08 a.u. / =

I 0.2

i

I 0.4

I

I

0.6

I

I

0.8

I

1.0

Energy (eV)

Fig. 24. Imaginary part of the spin susceptibility of Pd (Gaspar et al. 1995). The full curve corresponds to the TDLSDA calculation. The dashed curve is for non-interacting electrons. Note that Im X+ 0(~'; ~) is proportional to the dynamical structure factor S+ (~';~o) by =0,~' = the spin counterpart of eq. (4.8).

basis over the mixed basis used for the study of charge-density fluctuations in Pd. The motivation for this switch in basis is that we are developing computational schemes for the study of electron dynamics in metals, and we are exploring alternative approaches. The LCAO basis has the advantage of a relatively small size, and an appealing "chemical" interpretation. Moreover, for the small excitation energies typical of the magnetic-response problem, we feel that it is as accurate as the more "expensive" mixed basis. In the specific case of Pd, our basis is similar to the one employed by Tom~inek et al. ( 1 9 9 1 ) - we use 40 Gaussians per site. We have utilized slightly different choices for the parameters of the pseudopotential and have subsequently readjusted the parameters contained in the Gaussians. We have explored a parameter space for the Gaussian orbitals and we have convinced ourselves that our results for the spin susceptibility are converged from the basis-expansion standpoint. The spin-response problem was solved as outlined above. We used the Gunnarsson-Lundqvist (1976) parametrization of d(n(~,)) defined by eq. (4.14). In figs 24 and 25 we present typical results for Im X + - and Re X + - , respectively, which we compare with the corresponding results for the non-interacting susceptibility. The most obvious feature of our results is the presence of a strong exchange-correlation enhanced low-energy peak in the susceptibility, which is referred to as the paramagnon mode. (It may be worth noting that a similar calculation of the spin susceptibility of A1 gave

w

Electronic screening in metals

,'7"

501

Spin susceptibility of Pd

0.5 0.4

~0.3 ~

Non-interacting

O.2

"~0.1[~\ii,, rr 0

q=O.08a.u.

.

0.0

0

0.2

~

0.4 0.6 Energy (eV)

0.8

1.0

Fig. 25. Real part of the spin susceptibility of Pd (Gaspar et al. 1995). The full curve corresponds to the TDLSDA calculation. The dashed curve is for non-interacting electrons. a spectrum without a paramagnon mode, which agrees with the fact that the calculated Stoner enhancement of A1 is small.) The possibility of the existence of this mode in Pd has been discussed in the past almost exclusively in the context of discrete models; see, e.g., Doniach and Engelsberg (1966), Berk and Schrieffer (1966). Results for band electrons have been reported by Stentzel and Winter (1986). The physics behind the paramagnon can be visualized as follows. In the absence of electron-electron interactions, the frequency range of the spin response is of the order of the Fermi energy. This implies that if a given spin is flipped, it will relax to its equilibrium configuration in a time scale on the order of h/EF. However, in the exchange-correlation enhanced case shown in fig. 24, the response has become much more prominent for small energies, which in effect means that the energy range controlling the relaxation has narrowed considerably. This translates into a much longer relaxation time, which can be viewed as signaling a tendency towards the formation of local magnetic ordering (as opposed to long-range ferromagnetic ordering). In fact, the enhanced susceptibility displayed in fig. 24 looks rather similar to what comes out of Hubbard-type models (Doniach and Engelsberg 1966; Doniach and Sondheimer 1974) - although for larger wave vectors our results show the presence of significant fine structure due to the band structure of actual Pd. It would be of interest to explore the interconnection between ab initio response methods such as the one described here and physicallymotivated models, particularly with regard to many-body effects due to the d-electrons.

502

A.G. Eguiluz and A.A. Quong

Ch. 6

Now, the paramagnon mode in Pd has not been observed experimentally. We hope that our ab initio results will stimulate further experiments, since detection of this model would serve as a powerful test of the physics of electron correlations in transition metals. We should point out that the dynamical enhancement we have obtained for the spin response is physically related to the enhancement of the static susceptibility. Indeed, the interconnection between the dynamic and static problems serves as a guideline for our numerical results. Specifically, we have that for q = 0 the ratio [Re X+ _(w)/Re X~~ ~_~0 must reproduce the value of the calculated static enhancement. At the time of this writing the value we have obtained for this ratio from the calculations underlying fig. 25 is ~ 3.6, corresponding to the use of 1300 k-points in the irreducible element of the Brillouin zone, with a numerical broadening given by 7/= 0.0125 eV (Gaspar et al. 1995). In summary, we have outlined calculations of dynamical charge- and spinresponse in Pd which incorporate the ingredients which are believed to be crucial for the description of this type of electronic system. These ingredients are the many-body effects giving rise to, e.g., the large enhancement of the spin susceptibility, and the band structure effects associated with the fact that the density of states at the Fermi level is large, and has significant structure in its neighborhood.

Acknowledgements The calculations outlined in w4 are the result of collaborative work with Andrzej Fleszar and Jorge A. Gaspar. Their help is greatly appreciated. Additional collaboration with Osvaldo Cappannini and David Tom~inek is acknowledged with thanks. A.G.E. acknowledges support from NSF Grant No. DMR-9207747, the San Diego Supercomputer Center, and the National Energy Research Supercomputer Center. A.A.Q. acknowledges support from DOE, Office of Basic Energy Sciences, the National Research Council, and the Pittsburg Supercomputer Center. References Allen, R.E., G.E Alldredge and E W. de Wette (1971), Phys. Rev. B 4, 1661. Andersen, O.K. (1975), Phys. Rev. B 12, 3060. Aravind, EK., A. Holas and K.S. Singwi (1982), Phys. Rev. B 25, 561. Aryasetiawan, E and K. Karlsson (1994), Phys. Rev. Lett. 73, 1679. Aryasetiawan, E and O. Gunnarsson (1994), Phys. Rev. B 49, 16214. Awa, K., H. Yasuhara and T. Asahi (1982a), Phys. Rev. B 25, 3670. Awa, K., H. Yasuhara and T. Asahi (1982b), Phys. Rev. B 25, 3687.

Electronic screening in metals

503

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504

A.G. Eguiluz and A.A. Quong

Ch. 6

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This Page Intentionally Left Blank

Author Index

Abels, L.L., see Newman, K.E. 397, 416, 417 Abrahams, E., see Zhang, X.Y. 7-9 Abramowitz, M. 298 Acocella, D. 84, 120 Al'shits, V.I. 263 Alder, B., see Ceperley, D. 6 Alexander, R.W. 159, 162 Alimonda, A.S., see Lucovsky, G. 420 Alldredge, G.P., see Allen, R.E. 467 Alldredge, G.P., see Quong, A.A. 467 Allen, M.P. 145 Allen, R.E. 467 Allen, R.E., see Bowen, M.A. 397, 399 Alonso, R.G., see Horner, G.S. 419 Ambrose, W.P., see Love, S.P. 148, 149, 156, 194 Amirtharaj, P.M. 420 Andersen, O.K. 430 Anderson, J.B. 6 Anderson, P.W. 8 Aoki, M. 146, 243 Arakawa, E.T., see Whang, U.S. 485, 491 Arase, T., see Brockhouse, B.N. 462, 463 Aravind, P.K. 480 Aravind, P.K., see Holas, A. 480 Aryasetiawan, E 431, 433, 477, 484, 486, 487, 490, 491,494 Asahi, T., see Awa, K. 480 Ashcroft, N.W. 246 Ashraf, M., see Wang, Y.R. 483 Aubry, S. 122 Awa, K. 480

Barker, A.S. 141, 142, 151, 154 Barker, Jr., A.S., see Verleur, H.W. 373 Barma, M. 7,8 Barnes, T. 7 Barnett, S.A. 397, 417 Barnett, S.A., see Newman, K.E. 397, 398 Barnett, S.A., see Shah, S.I. 397 Baroni, S. 451,455 Baroni, S., see Giannozzi, P. 451 Baroni, S., see Sorella, S. 7, 8 Baskaran, G., see Anderson, P.W. 8 Baym, G. 335, 436 Beck, H. 311 Beck, T.L., see Doll, J.D. 5, 6, 53 Beck, T.L., see Freeman, D.L. 5, 6, 29, 42, 53 Behre, J. 7 Benedek, G., see Bernasconi, M. 375 Berk, N. 496, 501 Berke, A. 262 Bernasconi, M. 375 Berne, B.J. 7, 9, 53 Beserman, R. 409, 411-413 Beserman, R., see Newman, K.E. 397, 417, 418 Bickers, N.E. 436 Bickham, S.R. 146, 147, 215, 217, 220223, 226, 227, 231, 232, 234-237, 249, 250 Bickham, S.R., see Kiselev, S.A. 146, 147, 215, 233, 237-242, 244, 250 Bilz, H. 141, 152, 182, 200 B ilz, H., see Fischer, K. 200 Binder, K. 5, 10, 16, 395, 396 Bishop, A.R., see Cai, D. 146 Blake, EC., see Havighurst, R.J. 173 B lankenbeclker, R. 7, 8

Bachelet, G.B. 430, 477, 493 Bakshi, P., see Tsuei, K.-D. 438 Banerjee, R. 356, 357, 360, 385

509

510

Author index

Blume, M. 400 Bohnen, K.P. 465, 466 Bohnen, K.P., see Ho, K.M. 432, 451 Boninsegni, M. 6 Bonneville, R. 374 Bonse, U., see Schtilke, W. 480 Born, M. 5, 9, 19, 104, 353, 354 Bornemann, T. 434, 494, 495 Bortolani, V., see Franchini, A. 433, 469, 474 Bortolani, V., see Wallis, R.E 458 Bourbonnais, R. 147, 214 Bouwen, A., see Fleurent, H. 181 Bouwen, A., see Page, J.B. 178-180, 184, 187 Bowen, M.A. 397, 399 Brennert, G.E, see E1-Hanany, U. 483 Breuer, N., see Leibfried, G. 264, 267, 327 Bridges, E 141, 142, 170 Bridges, E, see Wong, X. 141 Brockhouse, B.N. 462, 463 Brodsky, M.H. 390 Bron, W.E. 281 Brosens, F. 482 Brosens, E, see Nachtegaele, H. 482 Brout, R. 141 Bueche, Th., see vom Felde, A. 483, 491 B tihrer, W., see Dorner, B. 200 Bunker, B.A., see Newman, K.E. 397, 416, 417 Buot, EA. 291 Burlakov, V.M. 146, 214, 227 Burnham, R.D., see Lucovsky, G. 420 Bylander, D.M., see Kleinman, L. 446 Cabrera, B., see Sadoulet, B. 262, 279 Cadien, K.C., see Krabach, T.N. 409, 411413 Caglioti, G., see Brockhouse, B.N. 462, 463 Cai, D. 146 Callcott, T.A., see Whang, U.S. 485, 491 Campbell, D.K., see Kivshar, Y.S. 146 Car, R., see Sorella, S. 7, 8 Cardona, M., see Fuchs, H.D. 261 Caries, R. 381,385 Caries, R., see Pearsall, T.P. 389 Caries, R., see Saint-Cricq, N. 381, 385 Carruthers, P. 261,263 Case, K.M. 310 Celli, V., see Franchini, A. 433, 469, 474

Ceperley, D.M. 6 Ceperley, D.M., see Pollock, E.L. 7-9, 51 Ceperley, D.M., see Schmidt, K.E. 7-9 Cercignani, C. 292 Challis, L.J. 261 Chan, C.T. 499 Chang, K.J., see Lee, K.-H. 477 Chang, Y.-C., see Chu, H. 371,419 Chang, Y.-C., see Ren, S.-E 371,419 Chen, C.X., see Schtittler, H.B. 5, 6 Chen, M.E, see Brodsky, M.H. 390 Chen, M.E, see Lucovsky, G. 388, 393, 394 Chen, X.M. 462, 463 Chen, Y.S. 373 Cheng, B.K., see Janke, W. 85 Cheng, I.F. 373, 381 Cheng, K.Y., see Lucovsky, G. 420 Chern, B. 338 Cherng, M.J., see Jen, H.R. 419 Chiang, C., see Hamann, D.R. 430, 445 Choquard, P.F. 92, 93, 102, 132 Chow, D., see Bridges, E 142, 170 Chu, H. 371,419 Chu, H., see Ren, S.-E 371,419 Chubykalo, O.A. 146, 243 Claro, E 291 Claude, C. 146 Clayman, B.P. 173, 174 Clayman, B.P., see Sandusky, K.W. 143, 199, 200, 202, 244, 250 Clayman, B.P., see Ward, R.W. 205 Coalson, R.D., see Doll, J.D. 132 Coalson, R.D., see Freeman, D.L. 7-9 Cochran, W. 358, 390, 407, 409--413 Cohen, M., see Feynman, R.P. 6 Cohen, M H., see Pick, R.M. 439 Cohen, M.L., see Louie, S.G. 484, 486, 492 Columbo, L., see Bernasconi, M. 375 Cooke, J.F. 496 Corish, J. 200 Cowley, E.R. 10, 16, 17, 84, 93, 111, 120, 124, 125 Cowley, E.R., see Acocella, D. 84, 120 Cowley, E.R., see Liu, S. 5-7, 9, 19, 49, 51, 52, 67--69, 84, 111, 116 Cowley, E.R., see Shukla, R.C. 111, 113, 116 Cowley, E.R., see Zhu, Z. 84, 110 Cox, D.L., see Deisz, J. 7

Author index Cuccoli, A. 5-7, 9, 17, 19, 49-53, 59, 60, 62, 68, 69, 71, 84, 98, 99, 108-111, 118120, 122, 123, 128 Cullen, J.J. 7 Daams, J.M., see Liu, K.L. 496 Dabrowski, B., see Ng, T.K. 480 Dagotto, E. 437 Dagotto, E., see Moreo, A. 7 Dahler, J.S., see Jhon, M.S. 121 Dauxois, T. 146 Davis, L.C. 416 Davis, L.C., see Holloway, H. 416 Daw, M.S., see Maradudin, A.A. 5-7, 19, 28, 37, 44, 47-49, 51 Dawber, P.G. 141,376, 377 Day, M.A. 10, 16, 17 de Gennes, P.G. 496 de Gironcoli, S., see Giannozzi, P. 451 de Jongh, L.J. 8 de Raedt, B., see de Raedt, H. 19, 20, 26, 27, 36 de Raedt, H. 5, 6, 19, 20, 26, 27, 36 de Wette, EW., see Allen, R.E. 467 Dean, P. 366 Deisz, J.J. 7, 437 Deisz, J.J., see Eguiluz, A.G. 437 Devaty, R.P. 141, 169 Devreese, J.T., see Brosens, F. 482 Devreese, J.T., see Nachtegaele, H. 482 Dick, B.G. Jr. 354 Diekl, H.W., see Kaplan, T. 375 Diep, H.T., see Nagai, O. 7 Dolgov, A.S. 144 Doll, J.D. 5-7, 53, 132 Doll, J.D., see Freeman, D.L. 5-9, 29, 42, 53 Dolling, G., see Waugh, J.L.T. 364, 365 Doniach, S. 496, 501 Dorner, B. 200 Doucot, B., see Anderson, P.W. 8 Dow, J.D., see Bowen, M.A. 397, 399 Dow, J.D., see Fu, Z.-W. 420 Dow, J.D., see Hu, W.M. 367 Dow, J.D., see Jenkins, D.W. 417 Dow, J.D., see Kobayashi, A. 358, 364, 365, 371, 382-389, 391-394, 409, 410, 414416 Dow, J.D., see Myles, C.W. 366-368, 374376

511

Dow, J.D., see Newman, K.E. 397, 398, 400, 401,405, 406, 416--418 Dow, J.D., see Redfield, A.C. 395, 396 Dow, J.D., see Ren, S.Y. 420 Dow, J.D., see Robinson, J.E. 354, 359 Dow, J.D., see Vogl, P. 355, 402, 404--406 Dransfeld, K., see Stolen, R. 147 Dreizler, R.M. 430 Dreizler, R.M., see Gross, E.K.U. 430 Eguiluz, A.G. 433, 437, 438, 465, 469--471, 478, 485, 487-490 Eguiluz, A.G., see Deisz, J.J. 437 Eguiluz, A.G., see Fleszar, A. 431, 433, 477, 480-482, 484, 487 Eguiluz, A.G., see Franchini, A. 433, 469, 474 Eguiluz, A.G., see Gaspar, J.A. 431, 433, 434, 438, 464-467, 469, 470, 472, 474, 477, 494, 500-502 Eguiluz, A.G., see Quong, A.A. 431, 433, 438, 467, 476, 478, 479 Eguiluz, A.G., see Wallis, R.E 458 Ehrenreich, H. 489 Ehrenreich, H., see Velicky, B. 367, 368, 374 Eickmans, J., see Bornemann, T. 434, 494, 495 Eisenberger, P. 480 Eisenberger, P., see Platzman, P.M. 480 Ekardt, W. 438 Ekardt, W., see Pacheco, J.M. 438 Ekardt, W., see Sch/Sne, W.-D. 438 E1-Hanany, U. 483 Eldridge, J.E. 147 Elices, M. 270 Elliott, R.J. 372, 373, 375 Elliott, R.J., see Dawber, P.G. 141,376, 377 Ellis, F., see Stern, E.A. 417 Emery, V.J., see Blume, M. 400 Engelsberg, S., see Doniach, S. 496, 501 Enz, C.P. 271,439 Ershov, Yu.I. 310 Esaki, L., see Kawamura, H. 381,383 Esaki, L., see Tsu, R. 381,383 Etchegoin, P., see Fuchs, H.D. 261 Every, A.G. 274, 287 Ewald, P.P. 354, 355 Faldt, ,~. 491 Fang, S., see Barnett, S.A. 397, 417

512

Author index

Farr, M.K. 364, 365, 390 Fedders, EA., see Gu, B.-L. 419 Fedorov, EI. 270, 273 Fedro, A.J., see Schiattler, H.B. 5, 6 Feibelman, EJ. 464 Fermi, E. 146 Ferziger, J.H. 292 Fetter, A.L. 435 Feynman, R.P. 5, 6, 9, 43, 66, 67, 84-87, 90, 93, 119, 131,443 Fibich, M., see Herscovici, C. 359 Fink, J., see Spr6sser-Prou, J. 478-480 Fink, J., see vom Felde, A. 433, 483-485, 487, 491 Fischer, E 146 Fischer, K. 200 Flach, E. 146 Flach, S. 146 Fleszar, A. 431, 433, 477, 480-482, 484, 487 Fleszar, A., see Deisz, J.J. 437 Fleszar, A., see Eguiluz, A.G. 437, 478, 485, 487-490 Fleszar, A., see Gaspar, J.A. 431,434, 477, 494, 500-502 Fleurent, H. 181 Fleurent, H., see Page,. J.B. 178-180, 184, 187 Flory, EJ. 24 Flytzanis, N. 146 Flytzanis, N., see Pnevmatikos, S. 146 Fradkin, E. 8 Franchini, A. 433, 469, 474 Franchini, A., see Wallis, R.E 458 Franchy, R., see Wuttig, M. 470 Frauenfelder, H. 246 Freeman, A.J., see Wimmer, E. 430 Freeman, D.L. 5-9, 29, 42, 53 Freeman, D.L., see Doll, J.D. 5, 6, 53, 132 Fritsch, J. 451 Froelich, D.V., see Bowen, M.A. 397, 399 Froyen, S., see Horner, G.S. 419 Froyen, S., see Louie, S.G. 484, 486 Fu, C.L. 454 Fu, Z.-W. 420 Fuchs, H.D. 261 Furdyna, J.K., see Amirtharaj, EM. 420 Fussg~inger, K. 175, 178 Fye, R.M. 7, 8

Galeener, EL., see Martin, R.M. 354 Gaficza, W. 286, 287 Garcia-Moliner, E, see Elices, M. 270 Garland, J.W., see Gonis, A. 376 Gaspar, J.A. 431,433, 434, 438, 464 467, 469, 470, 472--474, 477, 494, 500-502 Gaspar, J.A., see Eguiluz, A.G. 438, 470, 478, 485, 487--490 Gaspar, J.A., see Franchini, A. 433, 469, 474 Gaspar, J.A., see Quong, A.A. 467 Gester, M. 466, 472, 474 Gester, M., see Eguiluz, A.G. 470 Gester, M., see Franchini, A. 433, 469, 474 Gester, M., see Gaspar, J.A. 433, 469, 470, 472, 474 Giachetti, R. 5-7, 67, 84, 90, 93, 98, 109, 131 Giannozzi, E 451 Giannozzi, E, see Baroni, S. 451,455 Gilat, G. 456, 457 Gillis, N.S. 92, 124-127 Glyde, H.R., see Samathiyakanit, V. 5, 6, 9 Godby, R.W., see Maddocks, N.E. 431,433, 477, 480 Godfrey, M.J., see Needs, R.J. 471 Goldman, V.V. 67, 111, 132 Goldstein, H. 336 Gomyo, A. 419 Gonis, A. 376 Gonze, X. 446, 449 Gottfried, K. 335 G0tze, W. 271 Grant, EM., see Schiatler, H.B. 7 Gray, L.J., see Kaplan, T. 375 Green, E 480 Greene, J.E., see Barnett, S.A. 397, 417 Greene, J.E., see Beserman, R. 409, 411413 Greene, J.E., see Krabach, T.N. 409, 411413 Greene, J.E., see McGlinn, T.C. 411-413 Greene, J.E., see Newman, K.E. 397, 398 Greene, J.E., see Shah, S.I. 397, 417 Greene, J.E., see Stern, E.A. 417 Greene, L.H. 142, 152, 153, 157-160, 162, 164 Greene, L.H., see Sievers, A.J. 142, 155, 156, 160, 161, 163-167, 195 Gregg, J.R., see Shen, J. 374 Griffiths, R.B., see Blume, M. 400

Author index Grc~nbech-Jensen, N., see Cai, D. 146 Gross, E.K.U. 430, 439 Gross, E.K.U., see Dreizler, R.M. 430 Gross, M. 7 Gu, B.-L. 417, 419 Gu, B.-L., see Wang, Q. 417, 419 Gu, B.-L., see Zhang, X.-W. 417, 419 Gubernatis, J.E. 5-7 Gubernatis, J.E., see Doll, J.D. 5-7 Gubernatis, J.E., see Loh, E.Y. 7-9 Gubernatis, J.E., see Silver, R.N. 5, 6 Gubernatis, J.E., see White, S.R. 7, 8 Gunnarsson, O. 500 Gunnarsson, O., see Aryasetiawan, F. 431, 477 Gurevich, V.L. 269, 291,311,335 Gursey, E 19, 109, 110 Gutfeld, R.J. 261, 281 Haberkorn, R., see Fischer, K. 200 Hader, M. 111 Haldane, F.D. 8 Hall, B.M. 471 Hall, B.M., see Lehwald, S. 470 Hailer, E.E., see Fuchs, H.D. 261 Hamann, D.R. 430, 445, 477, 493 Hamann, D.R., see Bachelet, G.B. 430, 477, 493 Hammersley, J.M. 10 Handscomb, D.C., see Hammersley, J.M. 10 Hanke, W. 438 Hanke, W., see Deisz, J.J. 437 Hanke, W., see Eguiluz, A.G. 437 Hanke, W., see Putz, R. 436 Hannon, J.B. 464 Hansen, J.P., see Levesque, D. 16 Harada, T., see Tamura, S. 262 Hardy, J.R. 147 Hardy, R.J., see Day, M.A. 10, 16, 17 Harley, R.T. 184 Harrison, W.A. 402 Hashitsume, N., see Kubo, R. 262, 333 Haskel, D. 246, 247 Hass, M. 364, 365, 385, 386, 390 Hauge, E.H. 299 Havighurst, R.J. 173 Haydock, R. 377 Hayes, W. 150, 409 Hearon, S.B. 165, 167, 168, 170, 171, 173178

513

Hearon, S.B., see Sievers, A.J. 164, 165, 167 Hebboul, S.E. 262 Hedin, L. 439 Hedin, L., see von Barth, U. 354, 498 Heine, V. 377 Heinrichsmeier, M., see Eguiluz, A.G. 437 Held, E. 287 Hellmann, H. 443 Henvis, B.W., see Hass, M. 364, 365, 385, 386, 390 Herscovici, C. 359 Hess, D.W., see Serene, J.W. 436 Hibbs, A.R., see Feynman, R.P. 84, 87, 131 Hino, I., see Gomyo, A. 419 Hirsch, J.E. 7, 8 Hirsch, J.E., see Scalapino, D.J. 8 Hjalmarson, H.P., see Vogl, P. 355, 402, 40'tl ~06 Hjelm, A. 496 Ho, K.-M. 432, 451 Ho, K.-M., see Bohnen, K.-P. 465, 466 Ho, K.-M., see Fu, C.L. 454 Ho, K.-M., see Louie, S.G. 492 Hoffman, G.G. 8 Hohenberg, P. 430 Holas, A. 480 Holas, A., see Aravind, P.K. 480 Holland, U. 166, 168, 169, 175 Holloway, H. 416 Holloway, H., see Davis, L.C. 416 Holt, A.C., see Squire, D.R. 10, 16, 17 Homma, S., see Takeno, S. 147 Hoover, W.G., see Klein, M.L. 10, 16, 17 Hoover, W.G., see Squire, D.R. 10, 16, 17 Hori, K. 146 Hori, K., see Takeno, S. 146 Horner, G.S. 419 Horner, G.S., see Sinha, K. 419 Horner, H. 93, 132 Horton, G.K. 111 Horton, G.K., see Acocella, D. 84, 120 Horton, G.K., see Cowley, E.R. 84, 93, 120, 124, 125 Horton, G.K., see Goldman, V.V. 67, 111, 132 Horton, G.K., see Kanney, L.B. 132 Horton, G.K., see Klein, M.L. 92 Horton, G.K., see Liu, S. 5-7, 9, 19, 49, 51, 52, 67-69, 84, 111, 116 Horton, G.K., see Zhu, Z. 84, 110

514

Author index

Hsu, T., see Anderson, E W. 8 Hu, W.M. 367 Huang, K., see Born, M. 5, 9, 19, 104, 353, 354 Hubbard, J. 436, 438 Huebener, R.P., see Held, E. 287 Hughes, A.E., see Alexander, R.W. 159, 162 Hughes, A.E., see Kirby, R.D. 141, 151, 166 Hybertsen, M.S., see Northrup, J.E. 483 Ibach, H., see Lehwald, S. 470 Ibach, H., see Wuttig, M. 470 Ice, G.E., see Platzman, P.M. 433,476, 480482 Illegems, M. 381 Imada, M. 8, 9, 45 Imada, M., see Takahashi, M. 9, 19, 27, 29, 31, 36 Ipatova, I.P., see Maradudin, A.A. 182, 208, 354, 377, 445, 458 Isaacs, E.D., see Platzman, P.M. 433, 476, 480--482 Itoh, K., see Fuchs, H.D. 261 Ivanov, S.N. 292, 323 Iwamoto, N. 480 J~ickle, J. 291 Jacobs, EW.M. 200 Janak, J.E 499 Janke, W. 85 Jarrel, M., see Deisz, J. 7 Jasiukiewicz, Cz. 261,262, 281-283, 285, 287, 299 Jaswal, S.S., see Sievers, A.J. 141 Jen, H.R. 419 Jenkins, D.W. 417 Jensen, E. 483 Jepsen, D.W. 465 J~drzejewski, J. 330, 333 Jhon, M.S. 121 Jo, M., see Nishijima, M. 494, 495 Joannopoulos, J.D. 7 Joannopoulos, J.D., see Rappe, A. 460 Jona, E, see Jepsen, D.W. 465 Joosen, W., see Fleurent, H. 181 Jost, M., see Bridges, E 142, 170 Jusserand, B. 381,383, 384, 419

Kadanoff, L.E, see Baym, G. 436 Kahan, A.M. 142, 154 Kalia, R.K., see Mukhopadhyay, G. 480 Kalos, M.H. 6 Kanney, L.B. 132 Kaper, H.G., see Ferziger, J.H. 292 Kaplan, T. 375 Kapphan, S. 166 Kaprolat, A., see SchUlke, W. 480 Karlsson, K., see Aryasetiawan, E 431,433, 477, 484, 486, 487, 490, 491,494 Karo, A.M., see Hardy, J.R. 147 Kawamura, H. 381,383 Kawamura, H., see Tsu, R. 381,383 Kaxiras, E., see Rappe, A. 460 Kazakovtsev, D.V. 324 Kelinhesselink, D., see Gester, M. 466 Kellerman, E.W. 354, 359 Kelly, M.J. 377 Kembry, K.A., see Eldridge, J.E. 147 Kempa, K., see Tsuei, K.-D. 438 Kesmodel, L.L., see Hall, B.M. 471 Kesmodel, L.L., see Mohamed, M.H. 470 Khazanov, E.N., see Ivanov, S.N. 292, 323 Kibbit, A., see Mascarenhas, A. 419 Kikuchi, M., see Okabe, Y. 7 Kim, K., see Stern, E.A. 417 Kim, O.K. 381, 384 Kirby, R.D. 141, 151-153, 159, 160, 166, 180, 203 Kirczenow, G. 270 Kirkpatrick, S., see Velicky, B. 367, 368, 374 Kiselev, S.A. 146, 147, 212, 215, 233, 237242, 244, 250 Kiselev, S.A., see Bickham, S.R. 146, 215, 220--223, 249 Kiselev, S.A., see Burlakov, V.M. 146, 214, 227 Kisoda, K., see Takeno, S. 146 Kittel, C. 456 Kivshar, Y.S. 146, 147, 226, 227 Kivshar, Y.S., see Chubykalo, O.A. 146 Kivshar, Y.S., see Claude, C. 146 Klein, B.K., see Quong, A.A. 431, 456, 457, 463 Klein, M.L. 10, 16, 17, 19, 40, 92 Klein, M.L., see Goldman, V.V. 67, 111, 132 Klein, M.V. 141, 154, 183, 184

Author index Klein, M.V., see Beserman, R. 409, 411413 Klein, M.V., see Krabach, T.N. 409, 411413 Klein, M.V., see McGlinn, T.C. 411-413 Klein, W., see Held, E. 287 Kleinert, H. 85, 87, 131 Kleinert, H., see Feynman, R.P. 5, 6, 67, 84, 93, 119, 131 Kleinert, H., see Janke, W. 85 Kleinman, L. 446 Klemens, P.G. 261,263 Kleppmann, W.G. 177, 200 Kluth, O., see Claude, C. 146 Kobayashi, A. 358, 364, 365, 371, 380, 382-389, 391-394, 408-410, 414--416 Kobayashi, A., see Newman, K.E. 397, 416418 Kobussen, J.A. 341 Koehler, T.R., see Gillis, N.S. 92, 124-127 Kohn, W. 354, 437, 477 Kohn, W., see Gross, E.K.U. 439 Kohn, W., see Hohenberg, P. 430 Kojima, K. 175 Kojima, T., see Kojima, K. 175 Kooin, S.E., see Sugiyama, G. 7 Kosevich, A.M. 146, 276 Kosevich, Y.A. 146 Kotliar, G., see Zhang, X.Y. 7-9 Kovalev, A.S., see Chubykalo, O.A. 146, 243 Kovalev, A.S., see Kosevich, A.M. 146 Kozorezov, A.G. 263 Krabach, T.N. 409, 411-413 Krabach, T.N., see Beserman, R. 409, 411413 Krakauer, H., see Wimmer, E. 430 Kramer, B., see Barnett, S.A. 397, 417 Kramer, B., see Newman, K.E. 397, 398 Kramer, B., see Shah, S.I. 397 Krasilnikov, M.V., see Kozorezov, A.G. 263 Kremer, F. 170 Krotscheck, E., see Iwamoto, N. 480 Krumhansl, J.A., see Elliott, R.J. 372, 373, 375 Kuan, T.S. 419 Kubo, R. 262, 333 Kuech, T.E, see Kuan, T.S. 419 Kunc, K. 354, 443, 450 Kuroda, A., see Suzuki, M. 5-8

515

Kurtz, S.R., see Mascarenhas, A. 419 Kuwahara, Y., see Nishijima, M. 494, 495 Ladd, A.J.C., see Maradudin, A.A. 5-7, 19, 28, 37, 44, 47-49, 51 Lagendijk, A., see de Raedt, H. 5, 6 Landau, D.P., see Cullen, J.J. 7 Landau, L.D. 11, 14, 20, 29, 295 Lastras-Martinez, A., see Newman, K.E. 397, 398 Lax, M. 276 Leath, P.L., see Elliott, R.J. 372, 373, 375 Leath, P.L., see Kaplan, T. 375 Lederer, P., see Mills, D.L. 496 Lee, K.-H. 477 Lehmann, D., see Jasiukiewicz, Cz. 261, 262, 281,283, 285 Lehmann, G. 365 Lehwald, S. 470 Leibfried, G. 264, 267, 327, 354 Lemmens, L.E, see Brosens, E 482 Levesque, D. 16 Levinson, Y.B. 262 Levinson, Y.B., see Kazakovtsev, D.V. 324 Liang, S.D., see Anderson, P.W. 8 Lidiard, A.B. 245 Liebsch, A. 438 Liebsch, A., see Tsuei, K.-D. 438 Lifschitz, E.M., see Landau, L.D. 11, 14, 20, 29 Lifshitz, I.M. 142 Lifshitz, I.M., see Landau, L.D. 295 Lindenberg, K. 53, 54 Liu, A.J., see Cooke, J.F. 496 Liu, A.Y. 463 Liu, A.Y., see Quong, A.A. 463 Liu, K.L. 496 Liu, S. 5-7, 9, 19, 49, 51, 52, 67-69, 84, 85, 87, 111, 116, 131 Liu, S., see Zhu, Z. 84, 110 Liu, S.H., see Cooke, J.E 496 Lock, A., see Eguiluz, A.G. 470 Lock, A., see Franchini, A. 433, 469, 474 Lock, A., see Gaspar, J.A. 433, 469, 470, 472, 474 Loh, E.Y. 7-9 Loh, E.Y., see Scalapino, D.J. 8 Loh, E.Y., see White, S.R. 7, 8 Loudon, R., see Hayes, W. 409 Louie, S.G. 6, 484, 486, 492 Louie, S.G., see Chan, C.T. 499

516

Author index

Louie, S.G., see Northrup, J.E. 483 Louie, S.G., see Tom~inek, D. 499, 500 Love, S.P. 148, 149, 156, 194 Lovesey, S.W. 53, 54, 57, 120 Lucovsky, G. 388, 393, 394, 420 Lucovsky, G., see Brodsky, M.H. 390 Ludwig, W., see Leibfried, G. 354 Lundqvist, B.I., see Gunnarsson, O. 500 Lundqvist, S. 430 Lundqvist, S., see Hedin, L. 439 Ltity, E, see Holland, U. 166, 168, 169, 175 Ltity, E, see Kapphan, S. 166 Liitze, A., see Zavt, G.S. 147 Lynn, J.W. 460, 461 Ma, S.-K. 483 Macchi, A., see Cuccoli, A. 5-7, 9, 19, 49, 50, 52, 68, 69, 71, 84, 108, 118, 119 MacDonald, A.H., see Liu, K.L. 496 Mack, E., see Havighurst, R.J. 173 Maddocks, N.E. 431,433, 477, 480 Mahan, G.D. 435, 436, 438, 439, 461 Mahan, G.D., see Shung, K.W.-K. 483 Makita, K., see Gomyo, A. 419 Manousakis, E. 7 Manousakis, E., see Boninsegni, M. 6 Maradudin, A.A. 5-7, 9, 19, 28, 37, 44, 47-49, 51, 182, 208, 354, 377, 445, 458 Maradudin, A.A., see Cuccoli, A. 5-7, 9, 17, 19, 51-53, 59, 60, 62, 68, 69, 71, 84, 120, 122, 123, 128 Maradudin, A.A., see Eguiluz, A.G. 469 Maradudin, A.A., see Liu, S. 5-7, 9, 19, 49, 51, 52, 67, 69, 84, 116 Maradudin, A.A., see McGurn, A.R. 5-7, 19, 28, 37, 39, 40, 52, 53, 55, 63, 64, 110 Maradudin, A~A., see Quong, A.A. 467 Maradudin, A.A., see Sievers, A.J. 141 Maradudin, A.A., see Wallis, R.E 458 March, N.H., see Lundqvist, S. 430 Marcu, M. 7, 8 Marcus, P.M., see Jepsen, D.W. 465 Maris, H.J. 261,262, 276, 279, 286 Maris, H.J., see Sadoulet, B. 262, 279 Martin, R.M. 354 Martin, R.M., see Kunc, K. 443, 450 Martins, J.L., see Troullier, N. 456, 462, 477, 484, 486 Mascarenhas, A. 419

Mascarenhas, A., see Horner, G.S. 419 Mascarenhas, A., see Sinha, K. 419 Mashiyama, H., see Tomita, H. 53, 54, 57, 58 Mayer, A.P., see Berke, A. 262 Maynard, R., see Bourbonnais, R. 147, 214 McGlinn, T.C. 411-413 McGlinn, T.C., see Beserman, R. 409, 411413 McGurn, A.R. 5-7, 19, 28, 37, 39, 40, 52, 53, 55, 63, 64, 110 McGurn, A.R., see Cuccoli, A. 5-7, 9, 17, 19, 51-53, 59, 60, 62, 68, 69, 71, 84, 120, 122, 123, 128 McGurn, A.R., see Liu, S. 5-7, 9, 19, 49, 51, 52, 67, 69, 84, 116 McGurn, A.R., see Maradudin, A.A. 5-7, 19, 28, 37, 44, 47-49, 51 McWhirter, J.T., see Fleurent, H. 181 McWhirter, J.T., see Page, J.B. 178-180, 184, 187 Meissner, M., see Sampat, N. 292 Mermin, N.D., see Ashcroft, N.W. 246 Mertens, EG., see Hader, M. 111 Meserve, R.A., see Lovesey, S.W. 53, 54, 120 Metropolis, N. 8, 10, 14, 15, 395 Michel, K.H., see GOtze, W. 271 Miedema, A.R., see de Jongh, L.J. 8 Miglio, L., see Bernasconi, M. 375 Mijashita, S., see Suzuki, M. 7 Mikeska, H.J., see Behre, J. 7 Mills, D.L. 496 Mills, D.L., see Hall, B.M. 471 Mills, D.L., see Lehwald, S. 470 Mills, D.L., see Streight, S.R. 438 Mitra, S.S., see Cheng, I.E 373, 381 Miyake, J. 7, 8 Miyashita, S. 7, 8 Miyashita, S., see Suzuki, M. 5-8 Miyatake, Y., see Nagai, O. 7 Mohamed, M.H. 470 Mohamed, M.H., see Hall, B.M. 471 Monkhorst, H.J. 454, 464, 477 Montroll, E.W., see Maradudin, A.A. 182, 208, 354, 377, 445, 458 Moreo, A. 7 Morgan, M., see Bridges, E 142, 170 Morgenstern, I. 7 Mori, H. 29, 42, 53, 54, 56, 84, 120 Mourikis, S., see Schiilke, W. 480

Author index Mujashita, S., see Behre, J. 7 Mukhopadhyay, G. 480 MUller, J., see Marcu, M. 7 Mungan, C.E., see Rosenberg, A. 143, 188, 198, 199, 201 Mungan, C.E., see Sandusky, K.W. 143, 189, 190, 192, 196 Muramatsu, A., see Putz, R. 436 Myles, C.W. 366-368, 374-376 Myles, C.W., see Hu, W.M. 367 Myles, C.W., see Shen, J. 374 Nachtegaele, H. 482 Nagai, O. 7 Nagasawa, H., see Schtilke, W. 480 Narayanamurti, V. 141 Narayanamurti, V., see Lax, M. 276 Needs, R.J. 471 Needs, R.J., see Maddocks, N.E. 431,433, 477, 480 Negele, J.W. 5-7, 10, 13, 14 Negele, J.W., see Joannopoulos, J.D. 7 Neilson, D., see Green, F. 480 Nelin, G. 407, 408 Nethercot, Jr., A.H., see Gutfeld, R.J. 261, 281 Nette, P., see Weber, R. 156 Neumann, M., see Cuccoli, A. 5-7, 9, 19, 49, 50, 52, 68, 69, 84, 108, 118, 119 Neve, J., see F~ildt, ,~. 491 Newman, K.E. 397, 398, 400, 401, 405, 406, 416--418 Newman, K.E., see Bowen, M.A. 397, 399 Newman, K.E., see Gu, B.-L. 419 Newman, K.E., see Jenkins, D.W. 417 Newville, M., see Haskel, D. 246, 247 Nex, C.M.M. 377 Ng, T.K. 480 Ni, L., see Gu, B.-L. 417, 419 Nicklow, R.M., see Gilat, G. 456, 457 Nicklow, R.M., see Lynn, J.W. 460, 461 Nicklow, R.M., see Price, D.L. 390, 391 Niklasson, G. 480 Nilsson, G., see Nelin, G. 407, 408 Nishijima, M. 494, 495 Nishino, K., see Nagai, O. 7 Nolt, I.G. 151 Nolt, I.G., see Clayman, B.P. 173 Nomura, K. 7 Northrop, G.A. 261,262, 279, 286

517

Northrup, J.E. 483 Nozi~res, P., see Pines, D. 429, 434, 435, 477 Nticker (1987), N., see vom Felde, A. 483, 491 Nuroh, K., see Sturm, K. 487 Ogata, M. 7, 8 Okabe, Y. 7 Olbrich, E., see Each, E. 146 Olson, J.M., see Mascarenhas, A. 419 Onchi, M., see Nishijima, M. 494, 495 Onodera, Y. 367, 368, 371,373, 374, 402 O'Reilly, E.P., see Kobayashi, A. 364, 365, 382-385, 409, 410 Orland, H., see Negele, J.W. 5-7, 10, 13, 14 Orlova, N.S. 385, 386 Otto, A., see Bornemann, T. 434, 494, 495 Overhauser, A.W. 483 Overhauser, A.W., see Chen, X.M. 462, 463 Overhauser, A.W., see Dick, B.G., Jr. 354 Overhauser, A.W., see Wang, Y.R. 483 Overhauser, A.W., see Zhu, X. 483 Pacheco, J.M. 438 Pacheco, J.M., see SchOne, W.-D. 438 Pack, J.D., see Monkhorst, H.J. 454, 464, 477 Page, J.B. 142, 144, 178-180, 184, 187, 212-214 Page, J.B., see Fleurent, H. 181 Page, J.B., see Harley, R.T. 184 Page, J.B., see Rosenberg, A. 143, 188, 198, 199, 201 Page, J.B., see Sandusky, K.W. 143, 146, 147, 182, 186, 189, 190, 192-197, 199, 200, 202, 215, 217-220, 223-225, 227235, 237, 244, 249, 250 Pandy, K.C., see Eisenberger, P. 480 Papaconstantopoulos, D.A. 488 Papaconstantopoulos, D.A., see Sigalas, M.M. 499 Parayanthal, P., see Amirtharaj, P.M. 420 Parmenter, R.H. 355, 372, 402, 415 Parrinello, M., see Sorella, S. 7, 8 Pasta, J.R., see Fermi, E. 146 Paszkiewicz, T. 6, 9, 19, 283, 302, 304, 306, 307, 310, 319 Paszkiewicz, T., see Garicza, W. 286, 287 Paszkiewicz, T., see Ivanov, S.N. 323

518

Author index

Paszkiewicz, T., see Jasiukiewicz, Cz. 261, 262, 281-283, 285, 287, 299 Paszkiewicz, T., see J~drzejewski, J. 330, 333 Paszkiewicz, T., see Kobussen, J.A. 341 Patterson, M. 154, 155 Patterson, M., see Kahan, A.M. 142, 154 Pavone, P., see Fritsch, J. 451 Pavone, P., see Giannozzi, P. 451 Payton, D.N. 366 Pearsall, T.P. 389 Pearson, G.L., see Chen, Y.S. 373 Pearson, G.L., see Illegems, M. 381 Pearson, G.L., see Lucovsky, G. 420 Pehlke, E., see Tsuei, K.-D. 438 Peierls, R. 87, 270 Perdew, J.P., see Vosko, S.H. 498 Pettifor, D.G. 57 Peyrard, M., see Dauxois, T. 146 Philipp, H.R., see Ehrenreich, H. 489 Pick, R.M. 439 Pickett, W.E. 445 Pines, D. 429, 434, 435, 477 Pines, D., see Iwamoto, N. 480 Plaskett, T.S., see Brodsky, M.H. 390 Platzman, P.M. 433, 476, 480-482 Platzman, P.M., see Eisenberger, P. 480 Plummer, E.W., see Gaspar, J.A. 438, 464 Plummer, E.W., see Hannon, J.B. 464 Plummer, E.W., see Jensen, E. 483 Plummer, E.W., see Tsuei, K.-D. 438 Pnevmatikos, S. 146 Pnevmatikos, S., see Flytzanis, N. 146 Poglitsch, A. 170 Pohl, R.O., see Narayanamurti, V. 141 Pollack, EH., see Amirtharaj, P.M. 420 Pollock, E.L. 7-9, 51 Pollock, R. 7-9 Pomeranchuk, I.Ya. 261,263 Portal, J.C., see Pearsall, T.P. 389 Pratt, L.R., see Hoffman, G.G. 8 Preuss, R., see Putz, R. 436 Price, D.L. 390, 391 Putz, R. 436 Pyrkov, V.N., see Burlakov, V.M. 146, 214, 227 Quong, A.A. 431,433, 438, 456, 457, 460, 463, 464, 467-471,476, 478, 479

Quong, A.A., see Fleszar, A. 431,433,477, 480-482 Quong, A.A., see Liu, A.Y. 463 Quong, A.A., see Wallis, R.E 458 Rabe, K., see Rappe, A. 460 Raccah, P.M., see Newman, K.E. 397, 398, 416, 417 Rahman, T.S. 480 Ramsbey, M.T. 263, 287 Rappe, A. 460 Rau, K.R., see Brockhouse, B.N. 462, 463 Ray, M.A., see Barnett, S.A. 397, 417 Ray, M.A., see Newman, K.E. 397, 398 Recce, M., see Bridges, E 142, 170 Redfield, A.C. 395, 396 Redfield, A.C., see Bowen, M.A. 397, 399 Reed, M. 354 Reger, J.D. 7, 8 Remoissenet, M., see Flytzanis, N. 146 Remoissenet, M., see Pnevmatikos, S. 146 Ren, S.-E 371, 419 Ren, S.-E, see Chu, H. 371,419 Ren, S.Y. 420 Renucci, J.B., see Caries, R. 381,385 Renucci, J.B., see Saint-Cricq, N. 381,385 Renucci, M.A., see Caries, R. 381,385 Renucci, M.A., see Saint-Cricq, N. 381,385 Robinson, J.E. 354, 359 Rogers, S.J. 292 Romano, L.T., see Barnett, S.A. 397, 417 Romano, L.T., see Beserman, R. 409, 411413 Romano, L.T., see McGlinn, T.C. 411-413 Romano, L.T., see Stern, E.A. 417 ROsch, E 261 Rosenberg, A. 143, 188, 198, 199, 201 Rosenberg, A., see Sandusky, K.W. 143, 146, 186, 189, 190, 192-194, 196, 197, 199, 200, 202, 216-220, 223-225, 244, 249, 250 Rosenbluth, A.W., see Metropolis, N. 8, 10, 14, 15, 395 Rosenbluth, M.N., see Metropolis, N. 8, 10, 14, 15, 395 Rowe, J.M., see Price, D.L. 390, 391 Rubinstein, R.Y. 5, 10 Ruggerone, P., see Gester, M. 466 Rupasov, V.I., see Burlakov, V.M. 146, 214, 227 Ryan, P., see McGurn, A.R. 5-7, 19, 28, 37, 39, 40, 110

Author index Sadoulet, B. 262, 279 Saint-Cricq, N. 381,385 Sakurai, J.J. 446 Sakurai, M., see Kojima, K. 175 Salje, E.K.H. 124 Salvador, R., see Manousakis, E. 7 Samathiyakanit, V. 5, 6, 9 Sampat, N. 292 Sanchez-Velasco, E., see Gross, M. 7 Sandusky, K.W. 143, 146, 147, 182, 186, 189, 190, 192-197, 199, 200, 202, 215, 217-220, 223-225, 227-235, 237, 244, 249, 250 Sandusky, K.W., see Rosenberg, A. 143, 188, 198, 199, 201 Santoro, G., see Wallis, R.E 458 Sapriel, J. 419 Sapriel, J., see Jusserand, B. 381,383, 384, 419 Sarychev, A.V., see Al'shits, V.I. 263 Scalapino, D.J. 7, 8 Scalapino, D.J., see Bickers, N.E. 436 Scalapino, D.J., see B lankenbecker, R. 7, 8 Scalapino, D.J., see Fye, R.M. 8 Scalapino, D.J., see Moreo, A. 7 Scalapino, D.J., see Schtittler, H.B. 7 Scalapino, D.J., see White, S.R. 7, 8 Scalettar, R.T., see White, S.R. 7, 8 Sch~ifer, G. 141 Scheerer, B., see vom Felde, A. 483, 491 Schemer, A., see Gonze, X. 446 Schemer, M., see Stumpf, R. 430 Schltiter, M., see Bachelet, G.B. 430, 477, 493 SchlUter, M., see Hamann, D.R. 430, 445 Schmatzer, F.K., see Marcu, M. 7 Schmidt, K.E. 7-9 Schmidt, K.E., see Sandusky, K.W. 143, 146, 147, 182, 186, 189, 192, 195, 217, 227230 Schmidt, P., see Eisenberger, P. 480 Schmidt-Rink, S., see Miyake, J. 7, 8 Schmitz, J.R., see Schtilke, W. 433, 480 Schneider, T. 109, 110 Schoemaker, D., see Fleurent, H. 181 Schoemaker, D., see Page, J.B. 178-180, 184, 187 Scht~ne, W.-D. 438 Schrieffer, J.R. 8 Schrieffer, J.R., see Berk, N. 496, 501

519

Schrt~der, U. 184 Schr6der, U., see Fritsch, J. 451 SchUlke, W. 433, 480 SchtJlke-Schrepping, H., see Sch01ke, W. 433, 480 SchUttler, H.B. 5-7 Serene, J.W. 436 Shah, S.I. 397, 417 Shah, S.I., see Barnett, S.A. 397, 417 Shah, S.I., see Beserman, R. 409, 411-413 Shah, S.I., see Stern, E.A. 417 Sham, L.J. 439 Sham, L.J., see Hanke, W. 438 Sham, L.J., see Kohn, W. 354, 437, 477 Shaskolskaya, M.P., see Sirotin, Yu.I. 274, 278, 289, 290, 303, 304 Shastry, B., see Anderson, P.W. 8 Shastry, B., see Barma, M. 7, 8 Shechter, H., see Haskel, D. 246, 247 Shen, J. 374 Shiba, H., see Ogata, M. 7, 8 Shields, J.A. 262, 273, 287 Shields, J.A., see Tamura, S. 273, 287 Shikhov, S.B., see Ershov, Yu.I. 310 Shockley, W., see Chen, Y.S. 373 Shukla, R.C. 111, 113, 116 Shung, K.W.-K. 483 Shung, K.W.-K., see Ma, S.-K. 483 Shuvalov, A.L., see Al'shits, V.I. 263 Sievers, A.J. 141-144, 149, 150, 152, 155, 156, 160, 161, 163-167, 195, 209, 210, 214, 248, 249 Sievers, A.J., see Alexander, R.W. 159, 162 Sievers, A.J., see Aoki, M. 146, 243 Sievers, A.J., see Barker, A.S. 141, 142, 151, 154 Sievers, A.J., see Bickham, S.R. 146, 147, 215, 217, 220-223, 226, 227, 231, 232, 234-237, 249, 250 Sievers, A.J., see Clayman, B.P. 173, 174 Sievers, A.J., see Devaty, R.P. 141, 169 Sievers, A.J., see Fleurent, H. 181 Sievers, A.J., see Greene, L.H. 142 Sievers, A.J., see Hearon, S.B. 165, 167, 168 Sievers, A.J., see Kahan, A.M. 142, 154 Sievers, A.J., see Kirby, R.D. 141, 151, 166 Sievers, A.J., see Kiselev, S.A. 146, 147, 215, 233, 237-242, 244, 250 Sievers, A.J., see Love, S.P. 148, 149, 156, 194

520

Author index

Sievers, A.J., see Nolt, I.G. 151 Sievers, A.J., see Page, J.B. 178-180, 184, 187 Sievers, A.J., see Rosenberg, A. 143, 188, 198, 199, 201 Sievers, A.J., see Sandusky, K.W. 143, 146, 186, 189, 190, 192-194, 196, 197, 199, 200, 202, 216-220, 223-225, 244, 249, 250 Sievers, A.J., see Takeno, S. 143, 146, 157, 247-249 Sigalas, M.M. 499 Siggia, E., see Gross, M. 7 Silver, R.N. 5, 6 Simon, B., see Reed, M. 354 Singh, A. 7 Singwi, K.S., see Aravind, P.K. 480 Singwi, K.S., see Holas, A. 480 Singwi, K.S., see Mukhopadhyay, G. 480 Sinha, K. 419 Sinha, S.K. 445 Sinha, S.K., see Farr, M.K. 364, 365, 390 Sirotin, Yu.I. 274, 278, 289, 290, 303, 304 Sivia, D.S., see Silver, R.N. 5, 6 SjOlander, A., see Niklasson, G. 480 Smith, D.Y. 152 Smith, H.G., see Lynn, J.W. 460, 461 Smith, N.V. 491 Sondheimer, E.H., see Doniach, S. 501 Sorella, S. 7, 8 Soven, P. 367, 368, 373 Soven, P., see Zangwill, A. 438 Spatschek, K.H., see Claude, C. 146 Spicci, M., see Cuccoli, A. 84, 111 Spitzer, W.G., see Kim, O.K. 381,384 SprOsser-Prou, J. 478-480 Spr0sser-Prou, J., see vom Felde, A. 433, 483-485, 487, 491 Squire, D.R. 10, 16, 17 Srivastava, S. 85 Staal, P.R., see Eldridge, J.E. 147 Stegun, I.A., see Abramowitz, M. 298 Stentzel, E. 496, 501 Stern, E.A. 417 Stern, E.A., see Haskel, D. 246, 247 Stolen, R. 147 Stoll, E., see Schneider, T. 109, 110 Stoneham, A.M. 141 Stoneham, A.M., see Hayes, W. 150 Stott, M.J. 438

Strauch, D., see Bilz, H. 141, 152, 182 Streight, S.R. 438 Streitwolf, H.-W. 265 Stringfellow, G.B., see Jen, H.R. 419 Stumpf, R. 430 Stumpf, R., see Fleszar, A. 431,433, 477, 484, 487 Stumpf, R., see Gonze, X. 446 Sturm, K. 480, 487 Subbaswamy, K.R., see Mahan, G.D. 438 Sugar, R.L., see Blankenbecker, R. 7, 8 Sugar, R.L., see Scalapino, D.J. 7, 8 Sugar, R.L., see White, S.R. 7, 8 Sugiyama, G. 7 Suhm, M.A. 6 Sun, Z., see Tom~.nek, D. 499, 500 Suzuki, M. 5-8, 19, 21, 25, 36 Suzuki, T., see Gomyo, A. 419 Swanson, A.S., see Barnes, T. 7 Szymanski, J., see Green, E 480 Takahashi, H. 109, 110 Takahashi, M. 7, 9, 19, 27, 29, 31, 36 Takahashi, M., see Imada, M. 8, 9, 45 Takasu, M., see Suzuki, M. 7 Takeno, S. 143, 146, 147, 157, 247-249 Takeno, S., see Aoki, M. 146, 243 Takeno, S., see Bickham, S.R. 146, 214, 217, 235, 249, 250 Takeno, S., see Hori, K. 146 Takeno, S., see Sievers, A.J. 142-144, 209, 210, 214, 248, 249 Talwar, D.N. 373 Tamura, S. 262, 269, 273, 287 Tamura, S., see Ramsbey, M.T. 263, 287 Tamura, S., see Shields, J.A. 262, 273, 287 Taranov, A.V., see Ivanov, S.N. 292, 323 Taut, M., see Lehmann, G. 365 Taylor, D.W. 367, 368, 373 Teller, A.H., see Metropolis, N. 8, 10, 14, 15, 395 Teller, E., see Metropolis, N. 8, 10, 14, 15, 395 Tesanovic, Z., see Singh, A. 7 Testa, A., see Baroni, S. 451,455 Thomas, H., see Salje, E.K.H. 124 Tildesley, D.J., see Allen, M.P. 145 Timusk, T., see Ward, R.W. 205 Tiong, K.K., see Amirtharaj, P.M. 420 Toda, M. 110, 239 Toda, M., see Kubo, R. 262, 333

Author index Toennies, J.P. 433, 472 Toennies, J.P., see Eguiluz, A.G. 470 Toennies, J.P., see Franchini, A. 433, 469, 474 Toennies, J.P., see Gaspar, J.A. 433, 469, 470, 472, 474 Toennies, J.P., see Gester, M. 466 Tognetti, V., see Cuccoli, A. 5-7, 9, 17, 19, 49-53, 59, 60, 62, 68, 69, 71, 84, 98, 99, 108-111, 118-120, 122, 123, 128 Tognetti, V., see Giachetti, R. 5-7, 67, 84, 90, 93, 98, 109, 131 Tom~inek, D. 499, 500 Tomita, H. 53, 54, 57, 58 Toyozawa, Y., see Onodera, Y. 367, 368, 371,373, 374, 402 Traylor, J.G., see Farr, M.K. 364, 365, 390 Trickey, S.B. 430 Trotter, H.E 6, 7, 21 Troullier, N. 456, 462, 477, 484, 486 Tsu, R. 381,383 Tsu, R., see Kawamura, H. 381,383 Tsuei, K.-D. 438 Tsuei, K.-D., see Gaspar, J.A. 438, 464 Tubis, A., see Chern, B. 338 Ugur, S., see Newman, K.E. 397, 416, 417 Ulam, S.M., see Fermi, E. 146 Usatenko, O.V., see Chubykalo, O.A. 146, 243 Vaia, R., see Cuccoli, A. 5-7, 9, 17, 19, 49-53, 59, 60, 62, 68, 69, 71, 84, 98, 99, 108-111, 118-120, 122, 123, 128 Vaia, R., see Giachetti, R. 5-7, 67, 84, 98, 109 Vanderbilt, D. 445, 460 Vanderbilt, D., see Chan, C.T. 499 v.d. Osten, W., see Dorner, B. 200 Vandeyver, M., see Talwar, D.N. 373 van Hove, L. 121,364, 365, 370 Varma, C., see Miyake, J. 7, 8 Varshni, Y.P., see Banerjee, R. 356, 357, 360, 385 Vegard, L. 173 Velicky, B. 367, 368, 374 Venables, J.A., see Klein, M.L. 16, 19, 40 Verleur, H.W. 373 Verucchi, P., see Cuccoli, A. 84, 109 Vignale, G., see Rahman, S. 480

521

Vigneron, J.-E, see Gonze, X. 449 Vishwamittar, see Srivastava, S. 85 Visscher, W.M. 141 Visscher, W.M., see Brout, R. 141 Visscher, W.M., see Payton, D.N. 366 Vogl, P. 355, 402, 404 406 vom Felde, A. 433, 483-485, 487, 491 vom Felde, A., see Spr0sser-Prou, J. 478480 von Barth, U. 354, 498 vonder Linden, W. 437 Vosko, S.H. 498 Vosko, S.H., see Liu, K.L. 496 Wada, N., see Krabach, T.N. 409, 411-413 Wagner, M., see Zavt, G.S. 147 Walecka, J.D., see Fetter, A.L. 435 Walker, C.T., see Harley, R.T. 184 Wallace, D.C. 265, 274 Wallis, R.F. 458 Wallis, R.E, see Eguiluz, A.G. 469 Wallis, R.E, see Liu, S. 5-7, 9, 19, 49, 51, 52, 67, 69, 84, 116 Wallis, R.E, see Maradudin, A.A. 5-7, 19, 28, 37, 44, 47-49, 51 Wallis, R.E, see McGurn, A.R. 5-7, 19, 28, 37, 39, 40, 52, 53, 55, 63, 64, 110 Wallis, R.F., see Quong, A.A. 467 Walpole, L.J. 290, 302, 303, 308, 312, 316 Wang, Q. 417, 419 Wang, Q., see Zhang, X.-W. 417, 419 Wang, W.I., see Kuan, T.S. 419 Wang, Y.R. 483 Wannier, G.H., see Claro, E 291 Ward, R.W. 205 Warren Jr., W.W., see E1-Hanany, U. 483 Watanabe, S., see Yoshimura, K. 146 Watts, R.O., see Suhm, M.A. 6 Waugh, J.L.T. 364, 365 Weaire, D.L., see Pettifor, D.G. 57 Weber, R. 141, 149, 156 Weber, W. 354 Weber, W., see Fischer, K. 200 Weber, W., see Kleppmann, W.G. 200 Wehner, R.K., see Berke, A. 262 Wehner, R.K., see Bilz, H. 141, 152, 182 Weinert, M., see Wimmer, E. 430 Weis, J.J., see Levesque, D. 16 Weis, O., see R~sch, E 261 Weiss, G.H., see Maradudin, A.A. 182, 208, 354, 377, 445, 458

522

Author index

Wen, X.G., see Schrieffer, J.R. 8 Werthamer, N.R. 124 Werthamer, N.R., see Gillis, N.S. 92 West, B.J., see Lindenberg, K. 53, 54 Whang, U.S. 485, 491 Wheatley, J., see Anderson, P.W. 8 White, S.R. 7, 8 White, S.R., see Bickers, N.E. 436 Wiesler, A. 7 Wiesler, A., see Marcu, M. 7 Wigner, E. 338 Wilczyr~ski, M., see Ivanov, S.N. 323 Wilczyriski, M., see Paszkiewicz, T. 302, 304, 306, 307, 310, 319 Wilkie, E.L., see Kuan, T.S. 419 Williams, H., see Platzman, P.M. 433, 476, 480--482 Willis, C.R., see Flach, E. 146 Willis, C.R., see Flach, S. 146 Wimmer, E. 430 Winter, H., see Stentzel, E. 496, 501 Wolf, F., see Lehwald, S. 470 Wolfe, J.P. 287 Wolfe, J.P., see Hebboul, S.E. 262 Wolfe, J.P., see Northrop, G.A. 261, 279, 286 Wolfe, J.P., see Ramsbey, M.T. 263, 287 Wolfe, J.P., see Sadoulet, B. 262, 279 Wolfe, J.P., see Shields, J.A. 262, 273, 287 Wolfe, J.P., see Tamura, S. 273, 287 Wong, X. 141 Wood, W.W. 16 Woods, A.D.B., see Brockhouse, B.N. 462, 463

Wruck, B., see Salje, E.K.H. 124 Wuttig, M. 470

Xia, J. 5, 9 Xue, D.Z., see Newman, K.E. 397, 416, 417 Yacoby, Y., see Haskel, D. 246, 247 Yamada, Y., see Nagai, O. 7 Yasuhara, H., see Awa, K. 480 Yoshida, E, see Niklasson, G. 480 Yoshimura, K. 146 Young, A.P., see Reger, J.D. 7, 8 Zangwill, A. 438 Zaremba, E., see Stott, M.J. 438 Zaremba, E., see Sturm, K. 487 Zavt, G.S. 147 Zeller, R. 430 Zhang, S.C., see Schrieffer, J.R. 8 Zhang, X.-W. 417, 419 Zhang, X.-W., see Wang, Q. 417, 419 Zhang, X.Y. 7-9 Zhu, J.-L., see Gu, B.-L. 417, 419 Zhu, X. 483 Zhu, Z. 84, 110 Ziman, J.M. 261,263 Zogone, M., see Talwar, D.N. 373 Zow, Z., see Anderson, P.W. 8 Zschack, P., see Platzman, P.M. 433, 476, 480-482 Zubarev, D.N. 330, 333 Zweifel, P.E, see Case, K.M. 310 Zwick, A., see Caries, R. 381,385 Zwick, A., see Saint-Cricq, N. 381,385

Subject Index

Bethe ansatz 111 Boltzmann equation 291,292, 295 Chapman-Enskog approximation 302 collision invariants 306 collision operator 287 -spectrally decomposed 287 relaxation time approximation 283 solution for cubic and elastic media 306 source term 282 Boltzmann factor 107 Boltzmann kinetic equation 271 Boltzmann population 162 Born-Mayer plus Coulomb (BMC) potential 237, 238 Bosons 9 Branches longitudinal 119 Branches transverse 119 Bravais lattices 269 Brillouin zone 105, 108 first 104 Broadening of a spectral line inhomogeneous 281 Bulk modulus 108

Absorption coefficient 148 Absorption strength 184 Ag + 142 Ag + electronic quadrupolar deformability 200 Ag + electronic transitions in KI: Ag + optical transitions 174 Aging 175 Aging and reactivation of isolated Ag + defect centers in KI 176 AlxGal_xAs 355, 356, 367, 369, 370-372, 374, 377, 381-385, 388, 394 Alkali halides 141 Analytical treatments 6 Anharmonic effects 124 Anharmonic perturbation theory 111-113, 115 Anharmonic ZBM frequency 228 Anharmonicity 95, 109 Anisotropic media 302 Anisotropic solids 275 Ar 97, 110, 112, 115, 119 Associated Legendre function 298 Asymptotic ILM behavior 210 Atom host 264 Atom substitutional 264 distribution function 327 impurities 326 number of 328 random variable 264, 327

Christoffel equation: plane waves solution 274 Classical mechanics time reversal invariance 335 -violation by collisions 335 Classical Monte Carlo methods (CMC) 10-18, 111, 112, 115, 119 Classical partition function 11 Clausius-Mossotti equation 167 Coherent potential approximation 373

Backward-wave oscillators 170 Ballistic phonons 282 Beams of phonons 279 523

524

Subject index

Collision integral 273, 276, 277, 290 collision invariant 290 decomposed form 287 general structure -continuous part 307 -discrete part 307 nonpositive 290 spectral decomposition 290 spectrum of relaxation rates 278 Collision invariant 278, 302, 273 Collision operator 302, 305, 310 nonpositive 305 spectrum -continuous 302 -discrete 302 -eigenvalues related to collision invariants 302 Collision rates 305 Collision term 272 Collision theory detector 329 differential cross-section 329 incident beam 329 incident flux 329 target 329 wave packets 329 Complex dielectric constant 167 Continued fraction 53-58, 63, 64, 69-71, 122 expansion 53 representation 53, 120 Continuum deformation field 273 Coulomb potentials 106 Crystalline lattice long-wavelength limit 273, 274 energy transport in 270 fluctuations in mass distribution 263 -scattering on 263 quasimomentum transport in 270 spatially homogenous 269 spatially inhomogeneous 268, 269, 280 structural defects 280 with basis 269 Crystals Lu3A15012 325 of Ge, GaAs, InSb, Si 273 solid solutions 325 Cubic anharmonicity 147 Cubic materials 307

Cubic media 302 Cumulant expansion 132 Curie point 129 Curie-Weiss law 128 (6, 6', 6") model 199 4d 1~ -+ 4d95s parity-forbidden eleet,-onic transitions 175 DC lattice distortions 147 Debye frequency 156 Debye model 167 Debye temperature 115, 279, 323, 325 Debye-Waller factor 121, 158 Decomposition of the fourth-rank unit tensor 303 Density matrix 85, 132 Detailed balance 14 Diatomic chain 362 Diffusion coefficient 292, 294 equation 301 flows 299 matrix 307 Dipole moment 183 Dispersion curve 221 Dispersion curves for moving ILMs 236 Dispersive phonons 273 Displacement patterns 187 for different KI:Ag + modes 188 Displacive model 126 Displacive model ferroelectric 127, 129 Distribution function 295, 296, 313 Chapmann-Enskog expansion 299 deviation from state of incomplete equilibrium 300 Fourier harmonics 299 Fourier transform 295, 300, 301 -time dependent 295 Fourier-Laplace transform 296, 300, 313 -set of poles 313, 317 -singular continuum 312 -singularities 300 incomplete equilibrium state 301 initial condition 295 long-time diffusion asymptotics 295 time reversed 271 true thermodynamic equilibrium 302 Double-well order-disorder model 126

Subject index Effective potential 87, 89, 90, 92, 96-98, 107, 108, 117, 126, 127, 131 formalism 125 formulation 67-71 method 130 Monte Carlo method (EPMC) 107, 108, 110-112, 115-117, 121 theory improved 132 E-field dependence of the resonant mode 160 Elastic constants 306 symmetry relations 274 Voigt symmetry 274 Elastic media anisotropic 273, 276 basic fourth rank tensors 308 Christoffel equation 273 cubic 278, 295, 310 -diffusion coefficient 307 -elastic properties 306 -scattering properties 306 -transversely isotropic 295 density 273 hexagonal 278 isotropic 278, 290, 301,310 of arbitrary symmetry 311 of lower symmetry 291 tensor of elastic constants 273, 274 tetragonal 278 transversely isotropic 278, 307, 308 -tensor of [[V2]2] class 309 trigonal 278 Electric field measurements 152, 198 Stark effect 143 Electron gas: 2D electron gas 283 Energy density current 283 Entropy 272 current density 272 density 272 local balance 272 production 272 Envelope solitons 146 Equal time correlation functions 122 Equilibrium statistical mechanics 121 Euclidean space 273 Even ILM stability 219 Even moments 121 Even parity ILM 213 Even-symmetry resonant modes 179

525

Exact ILM frequency 212 Excitons: cloud of 283 FCC Lennard-Jones crystal 16-18, 44-52, 69-71 Fermi golden rule 335 Ferroelectric model 96 Ferroelectrics 124, 128, 131, 144 Filtering transformations 53 First-order effective potential 116 First-order self-consistent phonon approximation 67-69 First-order transition 126 Fluctuations 90, 93 Focusing (enhancement) factor 286 Force-constant matrix 111 Four-fold axes of a cubic crystalline structure 302 Fourth rank tensor decomposition 302 Fourth-neighbor pocket isotope modes 192 Free energy 86, 87, 101, 106 Free-end boundary condition case 236 Free rotor states 172 Frequency moments 53-64, 69-71 f sum rule 152 GaAs/AlxGal_xAs 418 (GaAs)l_xGe2x 405, 406, 414, 415, 417 GaAsl_xSbx 393, 394 Gal_xlnxAs 385-388, 390 Gal_zlnxSb 390, 392 (GaSb)l_xGe2x 405, 406, 408-413, 416--418 Gap ILMs 238 Gap modes 141 Gas of phonons 279 collisionless regime 299 incomplete equilibrium state 299 local equilibrium state 299 rarefied 279 Gaussian 126 Gaussian approximation 57, 58, 63, 64 Gaussian pair distribution 93 Gaussian sampling 17 Gaussian smearing 88 Ginzburg expansion 111, 116, 119, 120 Ginzburg parameters 96, 97, 116 Gram-Schmidt procedure 54 Green's function Monte Carlo method 6

526

Subject index

Green's harmonic function 182 Ground state tunneling systems 168 Group velocity 222, 270 Grtineisen parameter 17 Haldane conjecture 8 Hamiltonian equations 335 Harmonic oscillator 86, 89-91, 125, 132 He 97 Heat capacity 107, 108, 112, 115 Heat conductivity coefficient 291, 310 Heat pulse ballistic component 324 diffusive component 324 transmiss/ion experiments 279 Heisenberg antiferromagnet 8 Heisenberg spin systems 7 Hubbard model 8, 43 Hydrostatic pressure 154 measurements 169 Identical particles 42-44 ILM (intrinsic localized mode) and static distortion 231 ILM and zone boundary mode stability 227 ILM collisions 225 with a mass defect 227 ILM dynamic and static displacement pattern 231 ILM in glass 249 ILM induced force constant renormalization 233 ILM instability growth rates 219 ILM motion 235 ILM power spectra 242 ILM stability 215 ILMs and diatomic lattices 239 ILMs and (k2, k3, k4) potentials 228 ILMs and realistic potentials 236 ILMs even-parity 144 ILMs stability 146 ILMs traveling 146 Improved numerical convergence 25-27, 35-39, 46, 47 Improved self-consistent phonon theory (ISC) 116, 117, 118 Impurity modes 141 Impurity-induced dielectric constant 167

InAsl_xSbx 392, 393 Inert gas 115 Inert-gas crystals 97, 112 Infinitesimal displacement 218 Inl_xGaxAsuSbl_y 395, 397 Initial state, spatially homogeneous 305 Internal energy 107, 108, 110, 114, 117 Intrinsic localized modes (ILM) 143, 206, 207 IR-active "isotope" pocket gap modes 189 Ising model 8 Isotope effect 143 Isotope mode intensity temperature dependence 195 Isotope mode splitting 189 Isotropic approximation 116, 117 media 302 tensor 278 trial function 118 KI 173 KI alloys 173 KI phonon gap 160 KI + 1 mole% RbI + 0.2 mole% Ag + 173 KI:Ag+ aging 175 KI:Ag+ (~, ~') model 185 KI:Ag+ Debye spectrum 167 KI:Ag+ impurity 142 KI:Ag+ pocket mode experiments 193 39K+ __+ 41K+ host-lattice isotopic substitution 191 (k2, k4) lattice 209 Knudsen number 292, 307 Lattice constant 47-49 disorder 173 dynamics 104 sound velocity 221 spacing 116 unit cell 340 Lattice Hamiltonian anharmonic part 264, 265 harmonic approximation 266 harmonic part 265 part containing isotope scattering 264, 334 Law of mass action 245

Subject index Lennard-Jones chain 123 interaction 122 parameters 118 potential 18, 93, 111, 115, 237, 238 solid 18 Lifshitz method 182 Lifshitz theory 142 Linear chain 90 Linear coupling between a resonant mode and a Debye spectrum 159 Liouville equation 270 Liquid helium 9 Local hydrodynamic parameters 270 Local modes 141 Long-wavelength acoustic phonons 265, 269, 276-279 Low coupling approximation (LCA) 98, 99, 104, 105, 107, 108, 110, 111, 119, 120 Many fermion systems 9 Markovian generation process 13 Mass defect 225 Mass difference scattering 273 Material tensor 278 fourth rank tensor algebra 288 number of independent elements 278 MD simulations 235 Metropolis sampling 9, 10, 14-17 Microreversibility 342 Microreversibility principle 269 Microwave absorption 170 Microwave transition 170 Molecular dynamics (MD) 123, 131, 144, 210 simulations 122 Molecular field 124 Molecular field ansatz 124 Mollwo-Ivey rule 150 Moment expansion 128, 131 Moment expansion method 123 Moments 121, 128 Monte Carlo simulation 107-109, 115, 131 Morse potential 237, 238 MOssbauer recoilless fraction 246 Nearest neighbor distance 114 Nearest-neighbor cubic anharmonicity 230

527

Ne 97, 110, 115, 120 22Ne 117 Non-crystalline solids 353 harmonic 353 Non-identical particles 9 Nonresonant absorption 160, 167 Normal coordinates 101 Normal modes 101 Normal mode frequencies 106 Number density of phonons 306 Odd parity ILM 210 Odd parity ILM gap mode 241 in the diatomic lattice 241 Odd- and even-ILM patterns 223 Off-center configuration 142, 172 Off-center population 169 On-center configuration 142 One-dimensional anharmonic potential 20 chain 28--44, 52-64, 69, 70, 359 lattice 110 models 111 system 124 One-phonon approximation 121 One-phonon spectral function 128 Operator antilinear 337 antiunitary 338, 339 antiunitary conjugation 339 complex conjugation 339 Order parameter 125-127 Order-disorder model ferroelectric 126, 127, 129 Oscillator strength temperature dependent signature 178 Oversized nonresonant cavity technique 170 Paraelectric resonance 169 Parallel computer: Connection Machine 120 Partition function 20-23, 26, 30, 31, 36, 43, 44, 46, 66, 67, 85, 86, 89, 106, 108, 109, 111, 120, 124, 130, 131 Path integral 85, 87 Path integral theory 131 Path integral methodology 7

528

Subject index

Path integral techniques 6 Pauli equation 343 Pauli master equation 268, 332 Peierls inequality 87, 131 Periodic boundary conditions 214 Phase perturbations 218 Phase transitions 131 displacive 124 of 1st order 125 order-disorder 124 Phonon acoustic dispersive 279 acoustic: mean free path 293 annihilation operator of 266 ballistic: Monte Carlo computer experiment 286, 287 ballistic motion 287 ballistic beams 280 collision integral 270, 289, 303 collision operator 288, 291 coupling with deformation field 271 creation operator 266 Debye velocity 277 densities 285 densities of energy 285 densities of quasimomentum 285 detector 281,282 -fixed 281 -movable 281 -sensitive to energy 281,283 -sensitive to quasimomentum 281, 283 diffusion coefficient 325 diffusion equations 291,294 diffusive motion 287 dispersion law degeneration points 268 dispersion 279 -distribution function 270, 287, 289, 292 -diffusive behaviour 319, 323 -incomplete equilibrium state 289 -of complete equilibrium 289 distribution transient 280 equilibrium system of phonons 345 focusing 287 focusing (imaging) experiments 284 frequencies 132, 278 gas entropy 272 gradient of frequency 275

Phonon group velocity 274-276, 281,288 high energy acoustic 279 injected Planckian 323 isotope scattering 295, 311 -contribution to heat conductivity coefficient 311 local thermal equilibrium 323 long-wavelength acoustic 271 low energy acoustic 279 mediated detectors of elementary particles 279 number density 278, 287, 291 number operator eigenvectors 266, 267 -normalization condition 267 -orthogonality condition 267 occupation number 345 occupation number's Planck function 345 operator 341 -annihilation of 340, 341 -creation of 340, 341 -number of 341 -total number of 341 optical 279 phase velocity 274-276, 288 plane waves representation 269 polarization -conversion processes 302 -quasilongitudinal 274 -quasitransverse 274 polarization vectors 265, 278, 287 -normalization condition 265 -orthogonality condition 265 pulse transmission experiments 280 pulses 323, 325 -diffusive propagation 323 quasimomentum density 283, 284 resistive processes 311 scattering 287 -mean free path 292 -processes 281 scattering by isotope impurities 334, 345 scattering by point mass defects 302, 277 -BKE (Boltzmann kinetic equation) 277 -distribution function: rate of change 270

Subject index Phonon scattering by point mass defects -energy conservation law 268 -transition probability density 269 -violation of quasimomentum conservation law 268 slowness 275 slowness surface 275 source 281,282 -monochromatic 282 -Planekian 282 -point 282 spectral function 128 spectrum of relaxation rates 290 states time reversed 342 surface of constant frequency 275, 284, 286 -curvature of it 276 -principal curvatures 276 -solid angle 276 -surface element 276 thermal 325 thermalization of heat pulses 292 three-phonon interaction spectrum of collision rates 291 wave packets 279-281 -representation of 270 Phonons 110 generated at boundaries 279 generated inside crystalline specimen 279 Planck function 273, 289 Planckian phonons 282 Pocket gap modes 143, 189 Pocket mode stress coupling coefficients 201 Point mass defects: random variable distribution function of 267 Fourier transform of 267 Polarization dyad 278 Polarization vector 274 Probability density of transitions per unit time 271 Propagation tensor 274

QD (quadrupolar deformability) model 200 Quadratic variational function 93, 131 Quantum fluctuations 95

529

Quantum mechanics canonically conjugated variables 336 time reversal operation 336 time-reversal invariance 336 Quantum Monte Carlo method (QMC) 87, 110, 116-121, 132 Quantum relaxation functions 123 Quantum renormalization parameter 91 Quantum solids 248 Quasi harmonic frequencies 118 Quasielastic central peak 181 Quasiharmonic lattice dynamics 131 Quasimomentum conservation law 269 density current 283 Radio frequency dielectric constant measurements in KI:Ag+ 168 signature 167 spectra 168 Raman polarization selection rules 180 Raman scattering 179 Raman spectra polarized 180 Random element isodisplacement model 373 Random variable canonical distribution function 327 Rayleigh characteristic time 324 Rayleigh's theorem 208 RbCI:Ag+ 168 RbI 173 Reactivation of samples 175 Realistic potentials 147 Recursion method 377 Reflectivity coefficient 148 Regime of relaxation collision dominated 302 collisionless 302 Reorientational relaxation time 169 Resolvent 296 Resonant and gap mode strengths 165 Resonant modes 141 Rotating wave approximation (RWA) 143, 209, 210 Rotational motion in the off-center configuration 172

530

Subject index

Scattering advanced density matrix 342 by SIAs 291 detailed balancing condition 345 differential cross-section 335 function 121 of phonon wave packets by SIA's 271 of phonons 263 principle of detailed balance 346 principle of microreversibility 344 probability density of transitions 334 retarded density matrix 343 tensor 297, 316, 318 -symmetry properties 316, 318 transition probability density per unit time 343, 344 vector 121 Scattering theory density matrix of composite systems 329 density matrix of the beam 329 density matrix 329 Liouville equation 329 probability density 329 probability of transition 332 retarded density matrix 330 thermodynamic limit 333 Second-order theory 98 Self-consistent phonon theory 93, 131 improved 111, 132 -free energy 96 of 1st order (SC1) 9, 92, 95-97, 110, 111, 116, 125, 130, 132 Shell model 142 Short-range correlations 132 Silver halide clusters 178 Sine-Gordon model 109 Sine-Gordon potential 90 Single particle model 20-25 Single particle anharmonic oscillator 19 Slowness vector 284 Smeared force constants 96, 98, 101, 106 Smeared potentials 93, 111, 132 Soft mode behaviour 128 Solid argon 16 Solid He 144 Solid xenon 17 Solid-liquid-gas interfaces 16 Soliton 145

Sources of phonons fixed 281 monochromatic 281,282 movable 281 Planckian 281 Specific heat 110 Specimen boundaries 280 Spectral density 52 Spectral function 120, 122, 130, 131 Splittings isotope 190 Stark effect pocket gap mode difference spectrum 202 Stark shifts 200 Static correlation functions 120 Static distortions 231 Static lattice contributions 118 Stress coupling coefficients 151 of the IR-active gap and resonant modes 198 Stress shifts 143 Superlattices 418, 419 Surface of constant frequency flattenning points 286 Gaussian curvature 286 local geometric characteristics 286 parabolic point 286 Susceptibility 126, 128, 129 Tensor of fourth rank 289 basic 298 material 289 Tensorial basis 289 Termination 57, 58, 63, 64, 69-71 Termination procedure 128 Thermal excitations of ILMs 249 Three-body forces 106 t-matrix approximation 375 Time reversal operation 271 Time-dependent correlation functions 120 Time-dependent quantum Monte Carlo method 52-64, 69-71 Toda lattice 110 Toda potential 238 Transfer matrix 109, 110 Transition probability 269 density per unit time 268 per unit time 328 Translational motion 220 Transmission coefficient 148 Trial action 87, 99

Subject index Trial free energy 88, 92 Trial partition function 88 Trial potential energy 100 Triatomic molecule 231 Trotter identity 6-8, 21, 25-27, 30-32, 36, 45, 66 Two different elastic configurations 169 Two-elastic-configuration model 165 Two-configuration arrangement 166 Two-configuration model 172 Two-phonon difference band processes 160 Two-phonon difference process 147

531

Variational procedure 87 Variational techniques 6 Vegard relation 173 Vertex model 8 Virial theorem 108 Virtual crystal approximation 355, 372 Voigt functions 195 Wave packets of phonons 279 Wigner expansion 93, 130 Wigner solid 9, 43 Xe 112

Unfolding technique 195 Uniaxial stress 151, 198 UV absorption spectrum 177

Yttrium-aluminium garnets 323 containing rare earth atoms 323

Variational approximations 65-69 Variational function 88, 90, 126

ZBM instability 228 Zero phonon line 159

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