Dynamical Properties of Solids Volume 7
This Page Intentionally Left Blank
Dynamical Properties of Solids Volume 7 ...
71 downloads
1274 Views
20MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Dynamical Properties of Solids Volume 7
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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),
w1
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.
w
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
'
(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~ ,
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.
Lattice dynamical applications
135
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.
136
E.R. Cowley and G.K. Horton
Ch. 2
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
> 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
7 E 0
v t"-" 9 0
, m
q... 0 0 cO
o m
k.. 0
~0
..0
> 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 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] 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 / 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 = 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)
w
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)
w
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 > 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~,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 /,~(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)
w
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
[~
(5.1.13a-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
306
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
w
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 =
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
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, 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) - 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 '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.
422
J.D. D o w et al.
Ch. 5
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.
Phonons in semiconductor alloys
423
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.
424
J.D. D o w et al.
Velicky, B., S. Kirkpatrick and H. Ehrenreich (1968), Phys. Rev. 175, 747. Verleur, H.W. and A.S. Barker Jr. (1966), Phys. Rev. 149, 715. Vogl, P., H.P. Hjalmarson and J.D. Dow (1983), J. Phys. Chem. Solids 44, 365. von Barth, U. and L. Hedin (1972), J. Phys. C 5, 1629. Wang, Q., X.-W. Zhang and B.-L. Gu (1989), Phys. Status Solidi B: 153, 139. Waugh, J.L.T. and G. Dolling (1963), Phys. Rev. 132, 2410. Weber, W. (1977), Phys. Rev. B 15, 4789. Zhang, X.-W., Q. Wang and B.-L. Gu (1991), J. Amer. Chem. Soc. 74, 2846.
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
425
This Page Intentionally Left Blank
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
502
502
427
This Page Intentionally Left Blank
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
430
A.G. Eguiluz and A.A. Quong
Ch. 6
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-
432
A.G. Eguiluz and A.A. Quong
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
434
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
-
w
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
438
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
w
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)
w
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)
w
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).
w
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)
w
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
'
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; ~)
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
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
Bachelet, G.B., D.R. Hamann and M. Schlilter (1982), Phys. Rev. B 26, 4199. Baroni, S., P. Giannozzi and A. Testa (1987), Phys. Rev. Lett. 58, 1861. Baym, G. and L.P. Kadanoff (1961), Phys. Rev. 124, 287. Baym, G. (1962), Phys. Rev. 127, 1391. Berk, N. and J.R. Schrieffer (1966), Phys. Rev. Lett. 17, 433. Bickers, N.E., D.J. Scalapino and S.R. White (1989), Phys. Rev. Lett. 62, 961. Bickers, N.E. and S.R. White (1991), Phys. Rev. B 43, 8044. Bohnen, K.-P. and K.-M. Ho (1988), Surf. Sci. 207, 105. Bornemann, T., J. Eickmans and A. Otto (1988), Solid State Commun. 65, 381. Brockhouse, B.N., T. Arase, G. Caglioti, K.R. Rau and A.D.B. Woods (1962), Phys. Rev. 28, 1099. Brosens, E, J.T. Devreese and L.F. Lemmens (1980), Phys. Rev. B 21, 1363. Brosens, F. and J.T. Devreese (1988), Phys. Status Solidi B: 147, 173. Chan, C.T., D. Vanderbilt and S.G. Louie (1986), Phys. Rev. B 33, 2455. Chen, X.M. and A.W. Overhauser (1989), Phys. Rev. B 39, 10570. Cooke, J.E (1973), Phys. Rev. B 7, 1108. Cooke, J.E, S.H. Liu and A.J. Liu (1988), Phys. Rev. B 37, 289. Dagotto, E. (1994), Rev. Mod. Phys. 66, 763. de Gennes, P.G. (1963), in: Magnetism, Vol. III, Ed. by G.T. Rado and H. Suhl (Academic Press, New York), p. 115. Deisz, J.J., A.G. Eguiluz and W. Hanke (1993), Phys. Rev. Lett. 71, 2793. Deisz, J.J., A. Fleszar and A.G. Eguiluz (1995), submitted. Doniach, S. and E.H. Sondheimer (1974), Green's Functions .for Solid State Physicists (Benjamin, Reading, MA). Doniach, S. and S. Engelsberg (1966), Phys. Rev. Lett. 77, 750. Dreizler, R.M. and E.K.U. Gross (1990), Density Functional Theory: An Approach to the Quantum Many-Body Problem (Springer, Berlin). Ehrenreich, H. and H.R. Philipp (1962), Phys. Rev. B 128, 1622. Eisenberger, P., P.M. Platzman and K.C. Pandy (1973), Phys. Rev. Lett. 31, 311. Eisenberger, P., P.M. Platzman and P. Schmidt (1974), Phys. Rev. Lett. 34, 18. Eisenberger, P. and P.M. Platzman (1976), Phys. Rev. Lett. B 13, 934. Eguiluz, A.G. (1983), Phys. Rev. Lett. 51, 3305. Eguiluz, A.G. (1985), Phys. Rev. B 31, 3303. Eguiluz, A.G. (1987a), Physica Scripta 36, 651. Eguiluz, A.G. (1987b), Phys. Rev. B 35, 5473. Eguiluz, A.G., A.A. Maradudin and R.F. Wallis (1988), Phys. Rev. Lett. 60, 309. Eguiluz, A.G., J.A. Gaspar, M. Gester, A. Lock and J.P. Toennies (1990), Superlatt. and Microstr. 7, 223. Eguiluz, A.G. and J.A. Gaspar (1991), in: Surface Science, Lectures in Basic Concepts and Applications, Ed. by F.A Ponce and M. Cardona (Springer, Heidelberg), p. 23. Eguiluz, A.G., M. Heinrichsmeier, A. Fleszar and W. Hanke (1992a), Phys. Rev. Lett. 68, 1359. Eguiluz, A.G., J.J. Deisz, M. Heinrichsmeier, A. Fleszar and W. Hanke (1992b), Int. J. Quantum Chem. Symp. 26, 837. Eguiluz, A.G., A. Fleszar and J.A. Gaspar (1995), Nucl. Inst. and Methods B 96, 550. Ehrenreich, H. and H.R. Philipp (1962), Phys. Rev. 128, 1622. Ekardt, W. (1984), Phys. Rev. Lett. 52, 1558. Ekardt, W. (1985), Phys. Rev. B 32, 1961. E1-Hanany, U., G.E Brennert and W.W. Warren Jr. (1983), Phys. Rev. Lett. 50, 540.
504
A.G. Eguiluz and A.A. Quong
Ch. 6
Enz, C.E (1992), A Course on Many-Body Theory Applied to Solid State Physics (World Scientific, Singapore). F~ildt, A. and J. Neve (1983), Solid State Commun. 45, 399. Feibelman, P.J. (1982), Progr. Surf. Sci. 12, 287. Feynman, R.P. (1939), Phys. Rev. 56, 340. Fetter, A.L. and J.D. Walecka (1971), Quantum Theory of Many-Particle Systems (McGraw-Hill, New York). Fleszar, A., A.A. Quong and A.G. Eguiluz (1995a), Phys. Rev. Lett. 75, 590. Fleszar, A., R. Stumpf and A.G. Eguiluz (1995b), to be submitted. Franchini, A., V. Bortolani, V. Celli, A.G. Eguiluz, J.A. Gaspar, M. Gester, A. Lock and J.P. Toennies (1993), Phys. Rev. B 47, 4691. Fritsch, J., P. Pavone and U. Schr0der (1993), Phys. Rev. Lett. 71, 4194. Fu, C.L. and K.-M. Ho (1983), Phys. Rev. B 28, 5480. Gaspar, J.A. and A.G. Eguiluz (1989), Phys. Rev. B 40, 11976. Gaspar, J.A., A.G. Eguiluz, M. Gester, A. Lock and J.P. Toennies (1991a), Phys. Rev. Lett. 66, 337. Gaspar, J.A., A.G. Eguiluz, K.-D. Tsuei and E.W. Plummer (1991b), Phys. Rev. Lett. 67, 2954. Gaspar, J.A. (1991c), PhD Thesis, Montana State University (unpublished). Gaspar, J.A., A. Fleszar and A.G. Eguiluz (1995), submitted. Gester, M. (1989), Diplomarbeit, Technical University of Munich, Bericht des Max-Planck-Institut fur StrOmungsforschung No. 107 (unpublished). Gester, M., D. Kelinhesselink, E Ruggerone and J.P. Toennies (1994), Phys. Rev. B 49, 5777. Giannozzi, P., S. de Gironcoli, P. Pavone and S. Baroni (1991), Phys. Rev. B 43, 7231. Gilat, G. and R.M. Nicklow (1966), Phys. Rev. 143, 487. Gonze, X. and J.-P. Vigneron (1989), Phys. Rev. B 39, 13120. Gonze, X., R. Stumpf and A. Scheffler (1991), Phys. Rev. B 44, 8503. Green, E, D. Neilson and J. Szymanski (1985), Phys. Rev. B 31, 2796. Green, E, D. Neilson and J. Szymanski (1987), Phys. Rev. B 35, 124. Gross, E.K.U. and W. Kohn (1985), Phys. Rev. Lett. 55, 2850. Gross, E.K.U. and W. Kohn (1990), in: Density Functional Theory of Many-Ferrnion Systems, Ed. by S.B. Trickey, Adv. Quant. Chem. 21, 255. Gross, E.K.U. and R.M. Dreizler, Eds (1994), Density Functional Theory (Plenum, New York). Gunnarsson, O. and B.I. Lundqvist (1976), Phys. Rev. B 13, 4274. Hall, B.M., D.L. Mills, M.H. Mohamed and L.L. Kesmodel (1988). Hamann, D.R., M. Schltiter and C. Chiang (1979), Phys. Rev. Lett. 43, 1494. Hamann, D.R. (1989), Phys. Rev. B 40, 2980. Hanke, W. and L.J. Sham (1980), Phys. Rev. B 21, 4656. Hannon, J.B. and E.W. Plummer (1995), Phys. Rev. B, to appear. Hedin, L. and S. Lundqvist (1969), in: Solid State Physics, Vol. 23, Ed. by E. Ehrenreich, F. Seitz, and D. Turnbull (Academic Press, New York), p. 1. Hellmann, H. (1937), Einfuhrung in die Quantenchemie (Deuticke, Leipzig). Hjelm, A. (1992), in: Physics of Transition Metals, Ed. by P.M. Openeer and J. Ktibler (World Scientific, Singapore), p. 275. Ho, K.M. and K.P. Bohnen (1986), Phys. Rev. Lett. 56, 934. Ho, K.M. and K.P. Bohnen (1988), Phys. Rev. B 38, 12897. Hohenberg, P. and W. Kohn (1964), Phys. Rev. 136, B864. Holas, A., P.K. Aravind and K.S. Singwi (1979), Phys. Rev. B 20, 4912.
Electronic screening in metals
505
Hubbard, J. (1957), Proc. R. Soc. London Ser. A 243, 336. Hubbard, J. (1963), Proc. R. Soc. London Ser. A 276, 238. Hubbard, J. (1964), Proc. Phys. Soc. 84, 455. Hybertsen, M.S. and S.G. Louie (1987), Phys. Rev. B 35, 5585. Hybertsen, M.S. and S.G. Louie (1988), Phys. Rev. B 38, 4033. Iwamoto, N., E. Krotscheck and D. Pines (1984), Phys. Rev. B 29, 3936. Janak, J.E (1977), Phys. Rev. B 16, 255. Jensen, E. and E.W. Plummer (1985), Phys. Rev. Lett. 55, 1912. Jepsen, D.W., P.M. Marcus and F. Jona (1972), Phys. Rev. B 5, 3933. Kittel, C. (1976), Introduction to Solid State Physics (Wiley, New York). Kleinman, L. and D.M. Bylander (1982), Phys. Rev. Lett. 48, 1425. Kohn, W. and L.J. Sham (1965), Phys. Rev. 140, Al133. Kunc, K. and R.M. Martin (1981), Phys. Rev. B 24, 2311. Kunc, K. and R.M. Martin (1983), in: Ab Initio Calculation of Phonon Spectra, Ed. by J.T. Devresee, V.E. van Doren and P.E. van Camp (Plenum, New York), p. 65. Lee, K.-H. and K.J. Chang (1994), Phys. Rev. B 49, 2362. Lehwald, S., F. Wolf, H. Ibach, B.M. Hall and D.L. Mills (1987), Surf. Sci. 192, 131. Liebsch, A. (1987), Phys. Rev. B 36, 7378. Liebsch, A. (1991), Phys. Rev. Lett. 67, 2858. Liu, A.Y. and A.A. Quong (1995), to be published. Liu, K.L., A.H. MacDonald, J.M. Daams and S.H. Vosko (1979), J. Magn. Magn. Mater. 12, 43. Louie, S.G., K.-M. Ho and M.L. Cohen (1979), Phys. Rev. B 19, 1774. Louie, S.G., S. Froyen and M.L. Cohen (1982), Phys. Rev. B 26, 1738. Lundqvist, S. and N.H. March, Eds (1983), Theory of the Inhomogeneous Electron Gas (Plenum, New York). Lynn, J.W., H.G. Smith and R.M. Nicklow (1973), Phys. Rev. B 8, 3493. Ma, S.-K. and K.W.-K. Shung (1994), Phys. Rev. B 49, 10617. Maddocks, N.E., R.W. Godby and R.J. Needs (1994a), Phys. Rev. B 49, 8502. Maddocks, N.E., R.W. Godby and R.J. Needs (1994b), Europhys. Lett. 27, 681. Mahan, G.D. (1980), Phys. Rev. A 22, 1780. Mahan, G.D. and B.E. Sernelius (1989), Phys. Rev. Lett. 62, 2718. Mahan, G.D. and K.R. Subbaswamy (1990), Local Density Theory of Polarizability (Plenum, New York). Mahan, G.D. (1990), Many-Particle Physics (Plenum, New York). Mahan, G.D. (1992), Int. J. Mod. Phys. B 6, 3381. Maradudin, A.A., E.W. Montroll, G.H. Weiss and I.P. Ipatova (1971), Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York). Mills, D.L. and P. Lederer (1966), J. Phys. Chem. Solids 27, 433. Mohamed, M.H. and L.L. Kesmodel (1988), Phys. Rev. B 37, 6519. Monkhorst, H.J. and J.D. Pack (1976), Phys. Rev. B 13, 5188. Mukhopadhyay, G., R.K. Kalia and K.S. Singwi (1975), Phys. Rev. Lett. 34, 950. Nachtegaele, H., F. Brosens and J.T. Devreese (1983), Phys. Rev. B 28, 6064. Needs, R.J. and M.J. Godfrey (1987), Physica Scripta T 19, 391. Ng, T.K. and B. Dabrowski (1986), Phys. Rev. Lett. 33, 5358. Niklasson, G., A. Sj/51ander and F. Yoshida (1983), J. Phys. Soc. Jpn 52, 2140. Nishijima, M., M. Jo, Y. Kuwahara and M. Onchi (1986), Solid State Commun. 58, 75. Northrup, J.E., M.S. Hybertsen and S.G. Louie (1987), Phys. Rev. Lett. 59, 819. Northrup, J.E., M.S. Hybertsen and S.G. Louie (1989), Phys. Rev. Lett. B 39, 8198. Nozi~res, P. (1964), Theory of Interacting Fermi Liquids (Benjamin, New York).
506
A. G. Eguiluz and A.A. Quong
Ch. 6
Overhauser, A.W. (1985), Phys. Rev. Lett. 55, 1916. Pacheco, J.M. and W. Ekardt (1992), Ann. Phys. (Leipzig) 1, 255. Papaconstantopoulos, D.A. (1986), Handbook of the Band Structure of Elemental Solids (Plenum, New York). Pick, R.M., M.H. Cohen, and R.M. Martin (1970), Phys. Rev. B 1, 910. Pickett, W.E. (1989), Comp. Phys. Rep. 9, 115. Pines, D. and P. Nozi~res (1966), The Theory of Quantum Liquids (Benjamin, New York), Vol. 1. Platzman, P.M. and P. Eisenberger (1974), Phys. Rev. Lett. 33, 12. Platzman, P.M., E.D. Isaacs, H. Williams, P. Zschack and G.E. Ice (1992), Phys. Rev. B 46, 12943. Plummer, E.W., G.M. Watson and K.-D. Tsuei (1991), in: Su~.ace Science, Lectures in Basic Concepts and Applications, Ed. by F.A. Ponce and M. Cardona (Springer, Heidelberg), p. 49. Putz, R., R. Preuss, A. Muramatsu and W. Hanke (1995), submitted to Phys. Rev. Lett. Quong, A.A., A.A. Maradudin, R.F. Wallis, J.A. Gaspar, A.G. Eguiluz and G.P. Alldredge (1991), Phys. Rev. Lett. 66, 743. Quong, A.A. and B.K. Klein (1992), Phys. Rev. B 46, 10734. Quong, A.A. and A.G. Eguiluz (1993), Phys. Rev. Lett. 70, 3955. Quong, A.A., A.Y. Liu and B.M. Klein (1993), in: Materials Theory and Modeling, Vol. 291, Ed. by J. Broughton, P. Bristowe and J. Newsom (Material Research Society, Pittsburgh), p. 43. Quong, A.A. (1994), Phys. Rev. B 49, 3226. Quong, A.A. (1995), to be submitted. Rahman, T.S. and G. Vignale (1984), Phys. Rev. B 30, 6951. Rappe, A., K. Rabe, E. Kaxiras and J.D. Joannopoulos (1990), Phys. Rev. B 41, 1227. Sakurai, J.J. (1994), Modern Quantum Mechanics (Addison-Wesley, New York). SchOne, W.-D., W. Ekardt and J.M. Pacheco (1994), Phys. Rev. B 50, 11079. Schtflke, W., U. Bonse, H. Nagasawa, S. Mourikis and A. Kaprolat (1987), Phys. Rev. Lett. 59, 1362. Schialke, W., H. Nagasawa, S. Mourikis and A. Kaprolat (1989), Phys. Rev. B 40, 12215. Schtilke, W., H. Schulke-Schrepping and J.R. Schmitz (1993), Phys. Rev. B 47, 12426. Serene, J.W. and D.W. Hess (1991), Phys. Rev. B 44, 3391. Sham, L.J. (1969), Phys. Rev. 188, 1431. Shung, K.W.-K. and G.D. Mahan (1986), Phys. Rev. Lett. 57, 1076. Shung, K.W.-K. and G.D. Mahan (1988), Phys. Rev. B 38, 3856. Sigalas, M.M. and D.A. Papaconstantopoulos (1994), Phys. Rev. B 50, 7255. Sinha, S.K. (1980), in: Dynamical Properties of Solids, Vol. 3, Ed. by G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam). Smith, N.V. (1970), Phys. Rev. B 2, 2840. Spr~3sser-Prou, J., A. vom Felde and J. Fink (1989), Phys. Rev. B 40, 5799. Stentzel, E. and H. Winter (1986), J. Phys. F. Met. Phys. 16, 1789. Stott, M.J. and E. Zaremba (1980), Phys. Rev. A 21, 12. Streight, S.R. and D.L. Mills (1989), Phys. Rev. B 40, 10488. Stumpf, R. and M. Scheffler (1994), Phys. Rev. Lett. 72, 254. Sturm, K. (1982), Adv. Phys. 31, 1. Sturm, K., E. Zaremba and K. Nuroh (1990), Phys. Rev. B 42, 6973. Toennies, J.P. (1988), in: Solvay Con.:. on Surface Science, Ed. by EW. de Wette (Springer, Heidelberg). Toennies, J.P. (1990), Superlatt. and Microstr. 7, 193.
Electronic screening in metals
507
Tom~inek, D., Z. Sun and S.G. Louie (1991), Phys. Rev. B 43, 4699. Tsuei, K.-D., E.W. Plummer, A. Liebsch, K. Kempa and P. Bakshi (1990), Phys. Rev. Lett. 64,44. Tsuei, K.-D., E.W. Plummer, A. Liebsch, E. Pehlke, K. Kempa and P. Bakshi (1991), Surf. Sci. 247, 302. Trickey, S.B., Ed. (1990), Density Functional Theory of Many-Fermion Systems (Academic Press, New York). Troullier, N. and J.L. Martins (1991), Phys. Rev. B 43, 1993. Vanderbilt, D. (1990), Phys. Rev. B 41, 7892. vom Felde, A., J. Fink, Th. Bueche, B. Scheerer and N. Nticker (1987), Europhys. Lett. 4, 1037. vom Felde, A., J. Sprt~sser-Prou and J. Fink (1989), Phys. Rev. B 40, 10181. von Barth, U. and L. Hedin (1972), J. Phys. C 5, 1629. vonder Linden, W. (1992), Phys. Rep. 220, 53. Vosko, S.H. and J.P. Perdew (1975), J. Phys. 53, 1385. Wallis, R.E, A.A. Maradudin, V. Bortolani, A.G. Eguiluz, A.A. Quong, A. Franchini and G. Santoro (1993), Phys. Rev. B 48, 60543. Wang, Y.R., M. Ashraf and A.W. Overhauser (1984), Phys. Rev. B 30, 5580. Whang, U.S., E.T. Arakawa and T.A. Callcott, (1972), Phys. Rev. B 6, 2109. Wimmer, E., H. Krakauer, M. Weinert and A.J. Freeman (1981), Phys. Rev. B 24, 864. Wuttig, M., R. Franchy and H. Ibach (1986a), Z. Phys. B 65, 71. Wuttig, M., R. Franchy and H. Ibach (1986b), Solid State Commun. 57, 455. Zangwill, A. and P. Soven (1980), Phys. Rev. A 21, 1561. Zeller, R. (1992), in: Unoccupied Electron States, Ed. by J.C. Fuggler and J.E. Inglesfield (Springer, Berlin), p. 25. Zhu, X. and A.W. Overhauser (1986), Phys. Rev. B 33, 925.
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
This Page Intentionally Left Blank