Isotope Effects in Solid State Physics, Volume 68 (Semiconductors and Semimetals)

Isotope Effects in Solid State Physics SEMICONDUCTORS AND SEMIMETALS Volume 68 Semiconductors and Semimetals A Treati...

Author: Vladmir Plekhanov

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Isotope Effects in Solid State Physics SEMICONDUCTORS AND SEMIMETALS Volume 68

Semiconductors and Semimetals A Treatise

Edited by R. K. Willardson Consulting Physicist 12722 East 23rd Avenue Spokane, WA 99216-0327

Eicke R. Weber

Department of Materials Science and Mineral Engineering University of California at Berkeley Berkeley, CA 94720

Isotope Effects in Solid State Physics SEMICONDUCTORS AND SEMIMETALS Volume 68 Volume Editor VLADIMIR G. PLEKHANOV INSTITUTE FOR COMPUTER SCIENCE AND ENGINEERING TALLINN, ESTONIA

ACADEMIC PRESS San Diego San Francisco London Sydney Tokyo

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*

Copyright 2001 by Academic Press all rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777 The appearance of code at the bottom of the ﬁrst page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of speciﬁc clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-1999 chapters are as shown on the title pages; if no fee code appears on the title page, the copy fee is the same as for current chapters. 0080-8784/01 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two ﬁgures or tables from an Academic Press article in another scientiﬁc or research publication provided that the material has not been credited to another source and that full credit to the Academic Press article is given

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To my children

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Elastic Properties . . . . . . . . . . . . . . . . .

1

I. Theoretical Background of the Elastic Constant Measurements . . . . 1. Experimental Results and Interpretation . . . . . . . . . . . . . . . . . 2. Temperature Dependence of the Elastic Moduli in Diamond . . . . . . . . .

1 3 17

Chapter 2 Thermal Properties

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

I. Specific Heat and the Debye Temperature . . . . . . . . . . . . . II. Effect of the Isotopic Composition of a Crystal Lattice on the Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Dependence of the Thermal Conductivity of Diamond, Ge, and Si Crystals on Isotopic Composition . . . . . . . . . . . . . . . . . 1. Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . 2. T heoretical Models . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . IV. Dependence of the Lattice Constant on Temperature and Isotopic Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Vibrational Properties

23

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23

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27

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28 28 32 37

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46 46

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57

I. Theoretical Models of Lattice Dynamics . . . . . . . . . . . . . . . 1. Formal Force Constants . . . . . . . . . . . . . . . . . . . . . . 2. Rigid-Ion Model (RIM) . . . . . . . . . . . . . . . . . . . . . . 3. Dipole Models (DMs) . . . . . . . . . . . . . . . . . . . . . . . 4. Valence Force Field Model (VFFM) . . . . . . . . . . . . . . . . . II. Measurement of Phonon Dispersion by the Inelastic Neutron Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Raman Spectra and the Density of Phonon States . . . . . . . . . . .

vii

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57 57 59 60 62

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63 73

viii

Contents

Chapter 4 Raman Spectra of Isotopically Mixed Crystals . . . . . I. Low Concentrations: Localized, Resonant, and Gap Modes II. High Concentrations: Mixed Crystals . . . . . . . . . . 1. Introductory Remarks . . . . . . . . . . . . . . . . . 2. First-Order Raman Spectra . . . . . . . . . . . . . . 3. Second-Order Raman Spectra . . . . . . . . . . . . . . 4. Two-Mode Behavior of the L O Phonon: T he Case of L iH D V \V III. Disorder Effects in Raman Spectra of Mixed Crystals . . 1. Coherent-Potential-Approximation Formalism . . . . . . . 2. Disorder Shift and Broadening of the Raman Spectra . . . .

. . . . . . . . .

85 93 93 93 103 104 109 109 112

Chapter 5 Excitons in LiH Crystals . . . . . . . . . . . . . .

119

I. The Comparative Study of the Band-Edge Absorption in LiH, Li O, LiOH, and Li CO . . . . . . . . . . . . . . . . . . . II. Exciton Reflection Spectra of LiH Crystals . . . . . . . . III. Band Structure of LiH . . . . . . . . . . . . . . . . . . . IV. Dielectric—Metal Transition under External Pressure . .

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119 122 126 132

Chapter 6 Exciton--Phonon Interaction . . . . . . . . . . . . . .

135

I. Interaction between Excitons and Nonpolar Optical Phonons . II. Polarization Interaction of Free Excitons with Phonons . . . . III. Effects of Temperature and Pressure on Exciton States . . . . . 1. T heoretical Background . . . . . . . . . . . . . . . . . . . 2. Experimental Results . . . . . . . . . . . . . . . . . . . . . IV. Isotopic Effect on Electron Excitations . . . . . . . . . . . . 1. Renormalization of Energy of Band-to-Band Transitions in the Case of Isotopic Substitution in L iH Crystals . . . . . . . . . . . 2. T he Dependence of the Energy Gaps of A B and A B Semiconducting Crystals on Isotope Masses . . . . . . . . . . . . . . . . . . 3. Renormalization of Binding Energy of Wannier—Mott Excitons by the Isotopic Effect . . . . . . . . . . . . . . . . . . . . . . . 4. L uminescence of Free Excitons in L iH Crystals . . . . . . . . . .

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85

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135 136 139 139 146 156

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156

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158

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168 174

Chapter 7 Isotopic Effect in the Emission Spectrum of Polaritons

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181

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by Excitonic Polaritons in . . . . . . . . . . . . . . . . .

181 186

Chapter 8 Isotopic Disordering of Crystal Lattices . . . . . . . .

195

I. Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . II. Effects of a Disordered Lattice on the Energy of Interband Transitions in LiH D Crystals . . . . . . . . . . . . . . . . . . . . . . . . . V \V III. Broadening of Exciton Ground State Lines in Mirror Reflection Spectrum of LiH D Crystals . . . . . . . . . . . . . . . . . . . . V \V

195

I. Theory of Polaritons . . . . . . . . II. Experimental Results . . . . . . . . III. Resonance Light Scattering Mediated LiH (LiD) Crystals . . . . . . . . .

190

199 201

ix

Contents IV. Nonlinear Dependence of Binding Energy on Isotope Concentration . . . V. Effects of Disordering on Free Exciton Luminescence Linewidths . . . .

203 204

Chapter 9 Future Developments and Applications . . . . . . . .

211

I. II. III. IV. V. VI.

Isotopic Confinement of Light . . . . . . . . . . . . . Isotopic Information Storage . . . . . . . . . . . . . Neutron Transmutations . . . . . . . . . . . . . . . Isotopic Structuring for Fundamental Studies . . . . . Isotope Diffusion in Semiconductors . . . . . . . . . . Other Unexplored Applications of Isotopic Engineering

. . . . . .

. . . . . .

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. . . . . .

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211 214 215 216 216 221

Chapter 10 Conclusions . . . . . . . . . . . . . . . . . . .

225

References

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

229

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Volumes in This Series . . . . . . . . . . . . . . . . . .

241 247

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Preface The ﬁrst correct deﬁnition of the concept of the isotope was made by Soddy (1910) while investigating radioactivity. In 1912, using ion mass separation in the magnetic and electrical ﬁelds, Thomson found that Ne ions consist of two isotopes: Ne and Ne. According to the modern notion, the isotopes are almost identical atoms distinguished only by the numbers of neutrons, i.e., by their masses (weights). Therefore, the numbers of protons and electrons within different isotopes are identical. At present, about 300 stable and 1000 radioactive isotopes are known to exist. Some chemical elements are isotopically pure (e.g, Co), and others consist of a large number of isotopic modiﬁcations (e.g., Sn has 10 stable isotopes with masses from 112 to 124 and Xe has 23 isotopes, of which 9 are stable) see, for example, Firestone (1996). For a long time isotopes have been used in various areas of science, particularly in atomic (Bauche and Champean, 1976) and molecular (Muller, 1977) spectroscopy. It is interesting to note that the isotopic shift of the vibration band (0-0) of the oxygen ion was the ﬁrst direct observation of the presence of zero vibrations (Herzberg, 1945). In particular, the discovery of zero vibrations required the introduction of the half-integer in quantum mechanics. As is well known, zero-point vibration energy is different for ground and excited electron states and depends only on the isotope’s mass. It cannot be changed by any external action. The different meanings of zero-point vibration energy for different isotopes cause a shift of the pure electron transition in molecules with different isotopic compositions. Although the electron terms of the molecules do not depend directly on their isotopic composition, they are nevertheless variable for molecules containing different isotopes. This difference is initially determined by the molecular reconstruction of the electron—vibration interaction. The quantitative extent of this difference can serve as the nonadiabatic degree. The electron—lattice interaction coupling depends on the vibration frequency and thus on the isotopes. At the same time, it was noted that the harmonic approximation of the theory is insufﬁcient to describe the isotopic shift of pure electron transition in molecules (e.g., hydrides); see, e.g., xi

xii

Preface

Herzberg (1945). The successful study of the effect of isotopes in atomic spectroscopy demonstrated two contributions to the energy shift of the atomic levels. The ﬁrst is connected to the dependence on the nuclear mass and the second is due to the ﬁeld contribution (Bauche and Champeaun, 1976). Isotope effects become even more pronounced in the transition to solids. A prominent example is the experimental fact that the change in the electronic transition energy in a solid (for example, in LiH) on replacing H by D is two orders of magnitude greater than that in the hydrogen atom (Plekhanov, 1995a). As mentioned, the great number of stable and longlived isotopes currently available enables one to talk about the development of spectroscopy of solids with speciﬁed and controllable isotopic compositions. The study of crystals with various isotopic compositions has become possible because objects for studies are now available. Well-developed modern techniques for isotope separation (see, e.g., L aser Applications in Chemistry, 1986) provided the production of high-purity materials that possess virtually 100% uniform composition over different isotopes. In addition to LiH (Pretzel et al. 1960; Tyutyunnik et al., 1984), Cu O (Kreingol’d et al., 1976, 1977, Kreingol’d, 1985), ZnO (Kreingol’d, 1978; Kreingol’d and Kulinkin, 1986), and CdS (Kreingol’d et al., 1984; see also Zhang et al., 1998) crystals with different isotopic compositions of anionic and cationic sublattices, have long been grown; diamond (Chrenko, 1988; Collins et al., 1990; Ruf et al., 1998; Collins, 1998), GaN (Zhang et al., 1998), CuCl (Gobel et al., 1997), GaAs (Debernardi and Cardona, 1996), Ge (Agekyan et al., 1989; Itoh et al., 1993), Si (Capinsky et al., 1997), and -Sn (Wang et al., 1997) crystals, have recently been grown as well. Along with conventional applications, there are new opportunities in isotope engineering, such as the production of new media for data storage and ﬁber optics, UV, and visible lasers (Plekhanov and Altukhov, 1983; Takiyama et al., 1996); doping semiconductors by means of neutron transmutation; manufacturing of thermosensors from ultrapure materials; and many other promising applications (see also Chapter 9 and Berezin, 1989; Haller, 1995). Beginning with classical theory of the isotopic defect of Lifshitz (see, e.g., Lifshitz, 1987), interest in the dynamics of a defective lattice has not declined in more than ﬁve decades (see, for example, Maradudin et al., 1972; Dean, 1972; Bell, 1972; Taylor, 1975; Chang and Mitra, 1971; Elliott et al., 1974; Barker and Sievers, 1975; Thorpe, 1982; Dow et al., 1990). The volume of attention given to the problem of the simplest defect is explained in particular by the successful application of methods of the problem’s solution not only for mixed crystals but also for disordered systems (Maradudin et al., 1971; Bell, 1972; Ziman, 1979; Thorpe, 1982). According to the deﬁnition, the model of the isotopic defect assumes that only its mass changes, while

Preface

xiii

its force constant remains invariable. Dean (1972) and Bell (1972) showed by numerical calculations that the spectrum of the disordered structure exhibits two characteristic regions: the ﬁrst is continuous and related to the extended states, and the second is spiky and related to the localized states. The latter are commonly attributed to a single center, a pair of centers, and so on, with the subsequent clustering of defects (Chang and Mitra, 1971; Elliott et al., 1974; Taylor, 1982). In the limit of a high concentration of isotopes, when the interaction between defects becomes signiﬁcant, an isotopically mixed crystal is formed. Such mixed systems would most fully correspond to the model of a virtual crystal, because, at ﬁrst glance, the change in their effective charge and force constant should not take place. As a rule, the dependence of a lattice constant of isotopically mixed crystals on the concentration of isotopes is linear (Zimmerman, 1972; Holloway et al., 1991; Pavone and Baroni, 1994; see, however, Yamanaka et al., 1994). Therefore, the simplest isotopically mixed crystals could be used as intermediate model systems between isolated impurities (isotopic defects) and amorphous or totally disordered materials (Brodsky, 1979). In addition, because the isotope concentration can be varied over a wide range (0 x 100%), one can hope that this gives the rare opportunity to experimentally follow the percolation threshold, which separates localized states from extended states in the phonon spectrum (Kirkpatrick, 1973; Plekhanov, 1995c). As is well known, semiconductors differ from insulators primarily in the magnitude of the fundamental energy gap E . Clearly, it has a direct E inﬂuence on the possibility that signiﬁcant conductivity may occur as a result of thermal excitations of electrons across this energy gap. The equilibrium carrier density n resulting from such a process is strongly G temperature dependent (Ansel’m, 1978) n :2 G

m*kT E C exp $ , 2 kT

(1)

where k and are Boltzman’s and Plank’s constants and T is the absolute temperature. The Fermi level E is deﬁned relative to the conduction band $ edge, so that E 3kT E :9 E; . $ 2 4 ln(m* /m*) F C

(2)

The densities of thermally generated free electrons and holes n and h are C F equal to n for this intrinsically thermal excitation process, and the Fermi G level lies exactly midgap if the electron and hole effective masses m* and m* C F

xiv

List of Contributors

are equal. We shall see that m* is generally m* for direct gap semiconducC F tors, but these masses are much more similar for indirect semiconductors such as Si or GaP (see, e.g., Pankov, 1971). This temperature dependence of the carrier density, to which the bulk electrical conductivity is propor tional, gives a semiconductor its most characteristic property. If E is very E large, then n and : ne are clearly both very large. The large bandgap automatically gives relatively large effective masses m* and m* for free C F electrons and holes. The electron mass at the band edge is given by

1 2m p 3E ; 2 E :1; , m* 3 E (E ; ) C E E

(3)

and is therefore a strong function of E , being very small when E is small E E (for more details see, e.g., Pankov, 1971). In addition to traditional applications in nuclear power engineering, lithium hydride has also been considered a prospective material from the point of view of high-temperature conductivity (Ginsburg, 1977) in connection with the conduction band originating from metallic hydrogen (Weirs et al., 1996). The discovery of the linear luminescence of free excitons observed over a wide temperature range (Plekhanov, 1990a) has placed lithium hydride as well as diamond crystals (Takiyama et al., 1996) among the possible sources of coherent radiation in the UV spectral range. For LiH, isotopic tunning of this emission may also be possible (see also Plekhanov, 1981; Plekhanov and Altukhov, 1983). Besides this, new avenues have opened in isotope engineering. Note also the elements of ﬁberoptics using the different refractive index values of the different isotopes, from which it is easy to produce the core and cladding of the ﬁber. The instruments for recording and storing the information as well as radiation recorders and thermosensors based on the mixed-isotope compounds are easily realized from ultrapure materials. As has been shown (see, e.g., Haller, 1995) the basically uniform distribution of the neutral impurity over the crystal’s volume makes such crystals (e.g., germanium) the best IR recorders. The easiest current method to produce such semiconductor crystals is neutron transmutation (see, e.g., Magerle et al., 1995; Kuriyama and Sakai, 1996). Besides that, isotope substitution has opened new possibilities for the investigation of carrier scattering by neutral impurities (Erginsoy, 1950), a problem that has been apparent for more than half a century (Ansel’m, 1978). At the same time, we should note that one of the mechanisms causing the appearance of the double structure in the polariton emission spectra is polariton scattering by the neutral impurities (eg., donors; see also Koteles

Preface

xv

et al., 1985). Other prospective applications of materials with isotope composition (see also Berezin, 1989) require in-depth understanding of the fundamental physics of the electron and phonon states of these compounds. This is so because a large number of applied tasks are linked not only to the process of energy migration but also to electron excitation scattering by the phonons. One of the most successful tools to study these processes is the method of exciton spectroscopy used in this book. It has been more than six decades since the introduction of quasiparticle excitons by Frenkel (Frenkel, 1931a, 1931b), and the extreme fertility of this idea has been demonstrated most powerfully. According to Frenkel, the exciton is an electron excitation of one of the atoms (ions) of the crystal lattice resulting from the translation symmetry, which moves through the crystal in an electrically neutral formation. Since Frenkel, the concept of an exciton has been developed by Peierls (1932) and Slater and Schokley (1936). Problems concerning light absorption by the solid state have been considered somewhat differently (Wannier, 1937; Mott, 1938). According to the Wannier—Mott results, the exciton is the state of an electron and hole bonded by the Coulomb force. The electron and hole in the exciton state are spatially separated and their charges are screened. In the Frenkel papers, the excitations localized on the lattice site were described (after the Wannier—Mott papers) thus: The excitons became divided into the excitons on the Frenkel (small radius) excitons (for details see Davydov, 1968) and the Wannier—Mott (large radius) excitons (Knox, 1963). However, a description of the basic difference between these two models is absent (Davydov, 1968; Knox, 1963; Agranovich and Ginsburg, 1979). The experimental discovery (see, e.g., Gross, 1976) of the Wannier—Mott exciton (see Fig. 1 in Chapter 5) on the hydrogen-like absorption spectrum in the semiconducting crystals was the basis of a new subject — exciton physics (see also Agekyan, 1977; Permogorov, 1986). The inﬂuence of external perturbation (electrical and magnetic ﬁelds, one-axial and hydrostatic deformation) on the optical spectra of the Wannier—Mott excitons (see, e.g., Gross, 1976) and their energetic characteristics (see also Cardona, 1969) has been demonstrated repeatedly. These investigations enabled high-accuracy measurements not only of the exciton binding energy but also of their translational masses, the values of effective masses of the electrons and holes, their g factors, and so on. Moreover, the detailed account of the photon—exciton interaction has led to the concept of polaritons (Pekar, 1983). Since the experimental discovery of the Wannier—Mott exciton, the problem concerning the interaction of excitons and the crystal lattice has persisted for more than four decades (Haken, 1976). According to the modern concept, the dependence of the electron binding energy (Ryd) in the hydrogen atom on the nuclear mass is described by the

xvi

Preface

following expression: Ryd : 2

mc ,

c(1 ; m/M ) &

(4)

where m and M are electron and nuclear masses, respectively. In the & Wannier—Mott model, the Schro¨dinger equation for the exciton is solved using wave functions of hydrogen-like atoms, which have an effective charge Z : e/ (here is a medium permittivity); the exciton levels energy is C described by Knox (1963): e

k E (k) : E 9 ; . L E 2 n 2(m* ; m*) C F

(5)

Here k is the exciton’s quasi-impulse, and and n are its reduced mass and principal quantum number, respectively. The translational exciton mass (M) is equal to the sum of the m* and m* . The center of mass of the exciton can C F move through the crystal by diffusional or drift processes, just like the individual electronic particles. However, this exciton migration does not of itself produce electrical conductivity, since an exciton contains a pair of charges of opposite sign. The exciton binding energy E : (e)/(2 n) is @ analogous to Eq. (4) and does not depend on the nuclear mass. From the last formula (see also Fig. 1 of Chapter 5), we come to the natural conclusion that in the frozen crystal lattice, the isotope effect on the levels of the Wannier—Mott exciton is absent. This crude examination does not take into account the exciton—lattice interaction, which is characterized by the nonadiabatic degree (see also Pekar, 1951). The coupling of the exciton—phonon interaction depends on the phonon’s frequency, and thus on the mass of the vibrational atoms (ions); in other words, on the isotope’s mass. This dependence of the exciton binding energy in LiH (LiD) crystals was observed in Plekhanov et al. (1976), where the reﬂectance spectra at low temperatures were measured for the ﬁrst time (see also Klochikhin and Plekhanov, 1980). As seen later, the discovery of the dependence of E on @ the isotopic composition of the crystal lattice revealed the unique possibility of the experimental reconstruction of not only the value of the coupling of Coulomb and Fro¨hlich interactions, but also control of the effectivity of the exciton—phonon interaction. The problem of the electron excitation energy spectrum in condensed matter with a short-range interaction has a long history (Lifshitz et al., 1982). This history is shorter than the investigation of crystalline structures, although progress in the latter has been more rapid, and it is an exception to the rule. Although the kinetic phenomena in disordered media often have

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xvii

the same character as those in crystalline structures (see, e.g., Shklovsky and Efros, 1979), the energy spectra of electron excitations in noncrystalline media have a more complicated character. In addition to coherent states, whose movements obey the law of quasi-impulse conservation, there are localized states in noncrystalline media. At sufﬁcient degree of the disorder, the role of the latter may be large (so that in the one-dimensional case, all states are localized). In such cases, the localized states have an important effect on the picture of the kinetic phenomena (Mott and Davis, 1979). In modern physics, disordered systems are part of the wide and well-branched discipline of condensed matter physics. In addition to the monographs already cited, note the review papers by Elliott et al. (1974), Thorpe (1981), and Elliott and Ipatova (1988), which investigate the inﬂuence of different degrees of disorder in crystal lattices on the characteristics of electron excitations. Although Elliott and Ipatova (1988) present quite a detailed analysis of the inﬂuence of disorder in crystal lattices on the characteristics of free and bound excitons in chemically mixed crystals (mainly of A B and A B compounds), the effect of isotopes on the exciton energy levels was not considered. In this respect, this work appears as an addition to Elliott and Ipatova’s (1988) review on the optical properties of the chemically mixed crystals. There are three reasons for this addition. First, over the last two decades a large volume of information on the spectroscopy of large-radius excitons in crystals with isotopic effects has accumulated, which was not discussed in the earlier review. The second reason is due to the requirement for applied tasks in isotope engineering (for details, see Haller, 1995; Berezin, 1989). The last reason is the sporadically developing theory of Wannier— Mott excitons (see also Kanehisa and Elliott, 1987; Tanguy, 1995; Schwabe and Elliott, 1996), which continually requires knowledge of the static and dynamic characteristics of large-radius excitons embedded in various media and under various conditions. Let us now brieﬂy consider the effect of isotopes on the physical properties of a solid. The natural isotopic composition of a crystalline solid leads to a subtle state of disorder that affects several intrinsic physical properties. For example, the thermal conductivity, which has been determined as a function of temperature for many crystals, is rather sensitive to isotopic composition. The reduction or removal of the isotope disorder through isotopic enrichment was used by Geballe and Hull (1958) to demonstrate experimentally the increase in low-temperature thermal conductivity in Ge, which was predicted as early as 1942 (Pomeranchuk, 1942). The interest in the thermal conductivity of isotopically enriched diamond has led to some extreme speculations. Careful experimentation and a good theoretical understanding have produced record values of thermal conductivity, which are currently limited only by the size of the specimens and by the available experimental techniques. It is surprising that the 1% C in

xviii

Preface

natural diamond leads to a 30% reduction of the room-temperature thermal conductivity (Anthony et al., 1990; Banholzer and Anthony, 1992) as compared to an isotopically pure diamond. At low temperatures, these effects became very much larger, as is shown in Chapter 2. In this review, the effects of isotopic substitution on the elastic, thermal, and vibrational properties of crystals are discussed. The generality of results on the effects of isotopic substitution obtained for C, LiH, ZnO, ZnSe, CuCl, GaN, GaAs, CdS, Cu O, Ge, Si, and -Sn crystals suggests that a new line of investiga tion of solid state physics has appeared. Therefore, a brief review of the experimental data on isotopic effects in the lattice dyamics and on the electronic excitation seems to be timely. A consistent comparison of these data with existing theoretical models can enable us to discover the degree of their agreement (or discrepancy) and gives impetus to the development of new theoretical concepts and stimulates new experiments. A brief summary is presented in the conclusion. The difﬁcult and unsolvable problems of exciton physics and lattice dynamics in disordered media with isotopic effects are considered there. The main aim of this book is to familiarize readers with recent developments in isotope science and engineering in the hope that this will stimulate many creative ideas for studies and structures generally and for new scientiﬁc results and in due time new semiconductor and insulator devices.

Acknowledgments I would like to express thanks to my many students who have contributed to this study in various ways over the years. I would like to thank all of the authors who sent me their papers. I wish to thank Professor F. F. Gavrilov and Professor G. I. Pilipenko for many useful discussions. I am very grateful to my editors, Dr. Gregory Franklin, Professor Eicke Weber, and Ms. Peggy J. Flanagan. Vladimir G. Plekhanov

SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 1

Elastic Properties

I. Theoretical Background of the Elastic Constant Measurements . . . . 1. Experimental Results and Interpretation . . . . . . . . . . . . . . . . 2. Temperature Dependence of the Elastic Moduli in Diamond . . . . . . . .

1 3 17

I. Theoretical Background of the Elastic Constant Measurements The velocity v of an elastic wave in a crystalline solid depends on the density and the adiabatic elastic constant c . The velocities of the three GH modes of propagation are given by the solution of the Christoffel equation

9 v : 0, GI GI

where

(1)

:c n n , (2) GHIJ G J GI and n are the direction cosines of the direction of propagation (see G Musgrave, 1970). Materials with cubic symmetry have three independent elastic constants (see Nye, 1957), and for any of the directions 100, the solutions for the velocities of the three modes of propagation are v :

c ,

v :v :

(3)

c ,

(4)

where v is a longitudinal wave velocity and v is a transverse wave velocity. In the directions 100, the substitution of the appropriate into Eq. (2) GI leads to v : v : v :

[(c ; c ) ; c ] ,

(5)

c ,

(6)

c

9 c , 1

(7)

2

Vladimir G. Plekhanov

where v is a longitudinal wave velocity, v is a transverse wave velocity with particle motion in the direction 001, and v is a transverse wave velocity with particle motion in the direction 11 0. Provided that suitably oriented specimens are available, the elastic constants c , c , and c can be obtained by using Eqs. (3) to (7) together with the measured values of an appropriate set of velocities. For most directions of propagation in a crystal, the solution of Eq. (1) does not lead to a simple relation between the elastic constants and the velocities. Consequently, direct evaluation of the elastic constants is not possible for general orientations of specimens. However, Neighbors and Schacher (1967) have shown that a perturbation method can be used to determine the elastic constants from the measured velocities of ultrasonic propagation in arbitrary directions. Gluyas et al. (1975) used this method to evaluate the elastic constants of thallium chloride, where it was not possible to obtain the 100 and 110 faces with sufﬁcient accuracy to use Eqs. (3) to (7), according to the criteria of Watherman (1959), who has examined the errors introduced into the velocity due to misorientated specimens. Afterward, this method was used to determine the elastic moduli of LiH crystals and its isotope analog — LiD crystals (James and Kherandish, 1982). Another method for determining the elastic moduli in the solid state is the method of Brillouin scattering of light. There are many excellent accounts of the theory of Brillouin scattering (see, e.g., Born and Huang, 1968; Hayes and Loudon, 1978). In the context of our review, it is sufﬁcient to recall that the phonons, satisfying wave vector conservation in a given Brillouin scattering geometry in a cubic and hence optically isotropic material, have a wave vector given by q : <( k

G

sin , : <2 k 9 k ); q Q G 2

(8)

where q, k , and k are the wave vectors of the phonon and incident and G Q scattered light, respectively, and is the angle subtended by k and k . Note G Q that the k’s and are deﬁned inside the material so that refractive effects at the surfaces are not yet included in Eq. (8). In an isotropic medium, the velocity v of elastic wave is deduced from the frequency shift of the Brillouin line by the usual relation

v : 2n sin , c 2

(9)

where c is the velocity of light, n is the refractive index of the medium for the radiation of frequency , is the scattering angle inside the crystal, and is equal to the frequency of the hypersonic waves involved (see also C Hayes and Loudon, 1978).

1 Elastic Properties

3

1. Experimental Results and Interpretation a. L iH Crystals In this section we primarily follow the results of James and Kherandish (1982). Values of c and c were obtained from measured velocities of longitudinal and transverse wave propagation in the 100 direction. The elastic constants c and c were calculated from measured transit times by using Eqs. (3) and (4) together with values of specimen length and density corrected for thermal expansion. The densities of LiH and LiD at room temperature were taken to be 783 and 891 kgm\, respectively, as quoted by Guinan and Cline (1972) and by Gerlich and Smith (1974). The elastic constant c , corrected for thermal expansion, was obtained at 10 K intervals from values of v , v , and v using a computer program that carried out the perturbation calculations suggested by Neighbours and Schacher (1967). According to their results, the absolute uncertainties of c and c are due to the uncertainties in the measurements of length of the specimen (<0.1%), the transit time (<0.7%), and the density, estimated to be (<0.1%). However, the relative uncertainties arise only from measurement of transit time. The relative uncertainties in the values of c are much higher than those in c and c because of two velocities of almost the same value. Thus, together with uncertainties in the measurement of sample length, transit time and density result in absolute uncertainty in c of <5%. The room-temperature elastic constants of LiH and LiD have also been measured by Haussuhl and Skorczyk (1963) and Guinan and Cline (1972), and those of LiH by Gerlich and Smith (1974) and Laplaze and Boissier (1976). The values of elastic constants provided by different authors are given in Table I. The uncertainties quoted in this table are absolute uncertainties. The results for LiH are in good agreement. We should note the large discrepancy in the value of c of different authors. The reason for this discrepancy is not well understood. The data presented in Table I demonstrate a great difference in elastic constants c and c for both crystals. It is well known (see, e.g., Leibfried, 1963) that for crystals with the inversion center in the model of central forces, which act along the line connecting neighboring atoms, Cauchy’s relation should be valid, that is, c : c . In the microscopic description of lattice dynamics, Cauchy’s relation follows from the spherical distribution of charges of ions. Cauchy’s relation is quite satisfactorily obeyed in many ionic crystals (for details, see James and Kherandish, 1982). The greatest deviation from Cauchy’s relation is observed for LiF, where c /c : 1.53 and the effective Szigetti charge e* is 0.80 (see, e.g., Plekhanov, 1997a). One Q can see from Table I that for LiH (LiD) this ratio exceeds 3, which means that large deviations from Cauchy’s relation suggest that noncentral forces

TABLE I The Room-Temperature Values of the Elastic Constants of LiH [c] and LiD [c] (all in GN m\) G G Reference Haussuhl and Skorczyk (1963) Guinan and Cline (1972) Gerlich and Smith (1974) Laplaze et al. (1976) James and Kherandish (1982)

c

c

c

65.31<0.2

14.85<0.3 45<1.1

67.1<0.3 67.2<1.3 67.1 < 0.7 67.49<0.33

14.9<0.3 14.93<2.3 17.5<3.5 14.74<0.74

46.0<0.2 46.37<1.6 46.0<0.5 46.2<0.23

c

c

B 10\ dyn/c m Q

C

C

C

B 10\ dyn/c m Q

00.20 < 2 14.63 < 0.3 45.53 < 1.8 32.0 32.4 1.8

2.2—3.4

34.2(10 K)

67.8<0.2

14.2<0.2

47.8<0.2

31.7 33.5 35.6 (10 K)

5

1 Elastic Properties

play an important role in a LiH crystal and its isotopic analog (see also Wilson and Johnson, 1970, and details here). The isotope effect, which is manifested in the fact that elastic constants c and c in LiD are greater (while c is smaller) than in LiH is described here, later. Note also that direct calculations of elastic constants in LiH by Gerlich and Smith show that the c value strongly depends on the interaction between nearest neighbors in the second coordination sphere. Another important parameter of elastic properties of a lattice is the elastic anisotropy factor A : 2c /(c —c ). It is clear that for an elastic isotropic medium, A : 1. For LiH (LiD), the A value exceeds 1.7. Such a large value of this factor indicates a strong elastic anisotropy in lithium hydrate, which can be different for LiD. LiH and LiD have comparable values of the elastic anisotropy factor (A : 1.9) (Rastogi et al., 1974). The phenomenon of focusing of long-wavelength phonons is considered one of the manifestations of elastic anisotropy (see, e.g., Gutfeld, 1968). The adiabatic elastic modulus is calculated from elastic constants as B : (1/3)(c ; 2c ). As is known, this Q quantity characterizes the change in the lattice volume V with pressure p according to the relation B : 9V (dV /dp). One can see from Table I that Q values for LiH and LiD are small but different. The low accuracy of B Q values is probably explained by the low accuracy of measurement of elastic constants c . GI The Debye temperature can be calculated from the elastic constants (Leibfried, 1963; Born and Huang, 1968). The characteristic Debye temperature is, by deﬁnition, related to the boundary frequency of the spectrum of elastic continuum by the expression k : , K

(10)

where is Planck’s constant, and k is the Boltzmann constant. The boundary frequency is, in turn, determined by the normalization conditions of the distribution function g() of the frequency spectrum:

SK g() d : 3N,

(11) where 3N is the number of degrees of freedom for N discrete point masses. As mentioned, three types of different sound waves can propagate in a real crystal: a longitudinal wave, related to the expansion (compression), and two transversal waves, related to shear strains. Each of these waves has its own velocity, which depends in the general case on the crystallographic direction (Leibfried, 1963). For this reason, the total number of frequencies per spectral range d is equal to a sum of contributions from waves of each type averaged over all directions: g() d : 4V d ·

1 1 1 d ; ; , v (v ) (v ) 4 J R R

(12)

6

Vladimir G. Plekhanov

where V is the sample volume; d is the solid angle element; and v , v, and J R v are sound velocities. Taking into account Eqs. (10) and (12), expression R (11) becomes

: k

9N 2V

1 1 1 4 ; ; , v (v ) (v ) 4 J R R

(13)

The heat capacity is equal to C : T

12Nk 5

T

(14)

for temperatures lower than approximately /12. The integral in Eq. (13) is not yet taken in the general case. However, its value can be estimated by different methods (see, e.g., Allers, 1968). In the cited paper, six methods for estimating this integral are presented, starting from numerical integration and going to the expansion into a power series or over cubic harmonics. In addition to these methods, both graphic methods (Leibfried, 1963) and the calculation of the integral by means of tables are used (Allers, 1968). However, for an isotropic elastic medium, where sound velocities are independent of the crystallographic direction, the integration can be avoided (for details, see Plekhanov, 1997a). Because the isotropic medium is characterized by only two velocities v and v , which are related to the bulk J R modulus B and the shear modulus G by the equations Q B (4G/3) v : Q J

and

G v : , R

(15)

where is the sample density, expression (13) can be written in the form

9N 1 1 \ : . (16) ; k 2V v v J R Because polycrystals and glasses can be considered isotropic media, expression (16) is primarily valid for them. However, according to Anderson (1968), this expression can also be used for the approximate calculation of the Debye temperature in crystals, especially in the low-temperature limit. The Debye temperatures calculated in this low-temperature limit from elastic constants of LiH and LiD crystals are presented in Table 2 of the review by Plekhanov (1997a). Analysis of values obtained from the elastic constants and calorimetric measurements will be given in the next section. Here, we note that the isotopic substitution in LiH (substitution of D for H) results in an almost 3% increase in the elastic constant c in LiH GI

1 Elastic Properties

7

(Plekhanov, 1995b) and in a 0.5% increase in elastic constants in diamond (Ramdas et al., 1993, and see below). These changes in c can be reliably GI measured not only by the technique of Brillouin scattering of light but also by ultrasonic methods. Of great interest is the study of anharmonicity of vibrations in lithium hydride crystals. One can expect that the anharmonicity in these systems is quite large because of the small mass of hydrogen ions. The degree of anharmonicity of acoustic vibrations can be roughly estimated from the Gruneisen constant (Leibfried, 1963; Born and Huang, 1968), which is deﬁned as

:9

d In VB Q, :3 C d In V N

(17)

where is the linear expansion coefﬁcient, and V is the molar volume. Note here that, instead of B , the isothermal compressibility K : (I ; 3T )B\ Q Q is quite often used. The Gruneisen constant in many AHCs is close to 1.5 (see, e.g., Born and Huang, 1968). One can see that in LiH and LiD is of the same order of magnitude or even somewhat smaller than in AHCs. This means that LiH and LiD crystals are not strongly anharmonic with respect to the Gruneisen constant (see later). The degree of anharmonicity of acoustic vibrations is more exactly characterized by the values of baric derivatives (Table 1 of Plekhanov, 1997a), which were found from the dependence of the elastic constants on pressure (0 p 16 kbar). Gerlich and Smith (1974) also showed that the values of baric derivatives of lithium hydride can be adequately explained using the model of noncentral interaction between ions (see also Wilson and Johnson, 1970; Jex, 1974; Ho et al., 1997). The temperature dependences of the elastic constants of LiH crystals (Figs. 1 and 2) were studied both by ultrasonic method in the temperature range between 4.2 and 300 (James and Kherandish, 1982) and by Brillouin scattering in the temperature range between 100 and 300 (Vacher et al., 1981). It was shown that constants c and c increased with decreasing temperature, whereas c decreased (see also Table I). The temperature dependence of the bulk modulus proves to be very weak, which is most likely explained by the partial compensation of contributions from c and c . Leibfried and Ludvig (1961) have shown than the temperature dependence of the elastic constants may be represented approximately by c (T ) : c (0)(1 9 D ), GH GH

(18)

where D is a parameter that depends on the type of crystal or the model used and is the mean energy per oscillator. Several authors — for example,

8

Vladimir G. Plekhanov

Fig. 1. The elastic constants c and c of LiH and LiD (smoothed values: O, LiH; (, LiD; full curve extrapolated by means of Eq. (10). (After James and Kherandish, 1982.)

Varshni (1970), Lakhad (1971), and Plekhanov (1997a) — have used this expression in conjunction with either the Debye or the Einstein approximation for the frequency distribution in the solid to estimate the temperature variation of the elastic constants of the solid. Other authors — for example, Gluyas et al. (1975) — have used an extrapolation technique, based on the thermal energy obtained from the experimental values of the speciﬁc heat capacity to predict the temperature variation of the elastic constants for a number of ionic compounds. So far, however, no comparison of elastic constants observed experimentally at low temperatures (below 150 K) with extrapolated values have been reported. In James and Kherandish (1982) the methods of Leibfried and Ludvig were used to predict the temperature variation of elastic constants of LiH and LiD, which are compared with the corresponding values observed experimentally to evaluate the degree of success of this extrapolation technique. This technique has also been applied to calculate values of c

1 Elastic Properties

9

Fig. 2. The elastic constant c of LiH and LiD (smoothed values: O LiH; and ( LiD curve extrapolated by means of Eq. [10]; LiH and ▲ LiD from Guinan and Cline (1972); LiH and ▼ LiD from Haussuhl and Skorczyk [1969]; ■ LiH Gerlich and Smith [1974]. (After James and Kherandish, 1982.)

for LiH in the temperature range 0—160 K, where the measurements of the appropriate wave velocities were not possible. The method involved extrapolation proportional to the thermal energy so that the slopes of the curves of c , c , and c against temperature are each proportional to the heat capacity for both LiH and LiD. The elastic constants and the initial slopes used were the appropriate c and c /T at room temperature GH GH obtained for the experimental data. The heat capacity data used were from Yates et al. (1974). The temperature variation of the elastic constants of LiH and LiD calculated in this way from the room temperature values of the elastic constants are shown in Figs. 1 and 2. The extrapolated values of the elastic constants are in exellent agreement with the values observed experimentally down to approximately 120 K. Below 120 K, they are 0.6 to 0.9% lower than the observed values. Further comparisons need to be made to establish whether this method will, in general, provide reasonable values for the temperature variation of the elastic constants.

10

Vladimir G. Plekhanov

b. Diamond The strong, tetrahedrally coordinated, covalent bonds between nearest neighbors and the light mass of constituent atoms led to many striking and unique properties of diamond, for example, the largest elastic moduli (c ) GH are known for many materials and, correspondingly, the largest sound velocities (Grimsditch and Ramdas, 1975), and a very large Debye temperature, making it like LiH (Plekhanov, 1997a), a ‘‘quantum’’ crystal even at room temperature. Grimsditch and Ramdas made a comprehensive study of Brillouin scattering in natural diamond. On the basis of their investigations, they deduced the frequencies of the phonons with wave vectors corresponding to a large number of critical points of the phonon dispersion curves spanning the entire Brillouin zone of the elastic constants with high precision. Determination of ultrasonic velocities from the round trip transit time is, as is well known, an alternative technique that is equally precise. However, Brillouin scattering offers special advantages: it is a ‘‘contactless’’ technique; oriented specimens as small as a few millimeters in dimension are entirely satisfactory and can be conveniently placed in a high-temperature environment; and the incident light being brought to the samples and the scattered light can be collected with appropriate optical windows. In contrast, in the ultrasonic technique, for precision, large specimens and a suitable bond to the transducer are required. Finally, the Brillouin scattering geometries can be suitably devised, which obviates the need for the knowledge of the refractive index in deducing the elastic constants from the Brillouin shifts (see also Sandercock, 1975). The dependence of the elastic constants of diamond on the isotope effect was investigated in whole raw papers (Ramdas et al., 1993; Hurley et al., 1994; Vogelgesang et al., 1996; Zouboulis et al., 1998). In Ramdas et al., 1993; Vogelgesang et al., 1996; and Zouboulis et al., 1998, the method of Brillouin scattering of light was used, whereas in Hurley et al. (1994) the measurement of elastic moduli was performed using the method of ultrasound waves. The Brillouin experiments were performed in the backscattering geometry for phonon propagation vectors q along 001, 111, and 110. The incident beam was normal to the sample surface. Since the facets produced during growth for natural and synthetic diamonds were exactly normal to the desired direction, single measurements are sufﬁcient to determine the Brillouin shift for each particular wave vector (see, e.g., Fig. 3). For diamond surfaces that have been prepared by polishing, and hence possibly misoriented by a few degrees, the Brillouin shift was measured in Vogelgesang et al. for the various propagation directions close to the normal surface. As was noted in the cited paper, this does not change the scattering angle . This paper formulated the theory for the x dependence (CC ) of with lattice constant a and V \V elastic moduli c in terms of a lattice dynamical description for the zone GH center optical F phonon and the third-order bulk modulus, incorporating E anharmonicity as well as zero-point motion (see also Ramdas et al., 1993;

1 Elastic Properties

11

Fig. 3. Brillouin spectra of natural and isotopically enriched diamonds: (a) The spectra were recorded in the backscattering geometry for 5145-Å radiation incident along 111 and the backscattered light was analyzed with a (5 ; 4) tandem Fabry-Pe´rot interferometer. The phonon wave vector is along 111. Here L and T denote longitudinal and transverse, respectively. (b) Brillouin spectrum for C diamond in the same geometry as for (a) but analyzed with a ﬁve-pass interferometer with an FRS of 0.67067 cm\, and with an analyzer in the scattered beam to reduce the intensity of the longitudinal peaks. The longitudinal L and transverse T peaks are 8.4 and 5.4 orders from their parent laser line, as indicated by the arrows. (c) Same as (b) but for C diamond. The L and T peaks are now 8.7 and 5.7 orders from their parent line, respectively. (After Vogelgesang et al., 1996.)

Plekhanov, 1995d). Lattice-dynamical theories for crystals with the diamond structure have been given by Musgrave and Pople (1962); by Keating (1966); and by Martin (1970) based on models in which the strain energy of a crystal is expressed in terms of the changes in the bond length and the bond angles. In this approach, called the valence-force-ﬁeld method (for details see also Chapter 3), the potential is manifestly invariant under both translations and rotations (Lax, 1965). Anharmonic forces were used by Keating in a model involving two harmonic and three anharmonic force constants. Vogelgesang et al. (1996) considered a simple model containing four parameters, two harmonic and two anharmonic. The harmonic part was identical to that employed by Musgrave (1970). Of the two anharmonic force constants, only one is of signiﬁcance, the bond stretching energy. Let rin be the change in distance between the central ion and its tetrahedral surrounding ions and let ijn be the change in the angle between the bonds of two adjacent tetrahedral ions. The energy required to

12

Vladimir G. Plekhanov

stretch (or contact) a bond by r is G 1 1 k (r ) 9 g (r ) ; · · · , 2 G 6 G

(19)

while that required to alter it at the indicated angle is 1 1 k (R ) 9 g (R ) ; · · · . GH GH 2 6

(20)

Here k and k are the force constants for the harmonic part of the potential, whereas g and g deﬁne the lowest-order anharmonic contributions. Neglecting interactions other than those between nearest-neighbor atoms, the deformation energy is 1 1 U : U : k (r in); k (R ijn) L 2 2 L L G GH 1 1 (21) 9 g (r in)9 g (R ) ; % . GHL 6 6 G GH Bulk modulus. The energy associated with the deformation in which all nearest-neighbor distances change from R to R ; u (u R) without change in symmetry, that is, rin : u and ijn : 0, is 2 U : N 2k u 9 g n ; % , (22) 3

where N is the number of primitive cells in the crystal. The change in volume is V : 3Vu/R, so that k (V ) g (V ) U : 9 ;%. 6aV 72V

(23)

Recalling the change in free energy F with volume

1 F F : (V ) 2 V

;%

(V )B , 2V

(24) where B is the bulk modulus, a comparison of Eqs. (23) and (24) at zero temperature yields k B: . 3a

(25)

Zone center optical mode. Consider a motion in which the two facecentered cubic (fcc) sublattices of the structure experience a relative displacement u with respect to each other as in the F zone center optical mode E and let u be along 111. Noting that the terms in g cancel exactly, the deformation energy due to this relative displacement is U : N

2 4g u (k ; 4k )u 9 ; % . 3 27

(26)

13

1 Elastic Properties

The kinetic energy per primitive cell associated with the relative motion of the fcc sublattices is P/M, where P is the momentum canonically conjugate to u, and M is the mass of the atom (M/2 is the reduced mass in the case of the diamond). Thus, the Hamiltonian per primitive cell associated with the motion is H:

P 2 4g u ; (k ; 4k )u 9 ; % . M 3 27

(27)

Neglecting the cubic term, the equation of motion for u is harmonic with angular frequency of the F zone center optical phonon E 8(k ; 4k ) . (28) : 3M

To estimate the effect of the zero-point motion in the cited paper the Hamiltonian in Eq. (27) was quantized and the nature of the ground state of such a system was investigated by a variational procedure. Choosing, after Vogelgesang et al. (1996), a displaced Gaussian distribution characterized by the normalized wave function, (u) :

(

exp 9

(u 9 ) , 2

(29)

where and are the variational parameters, one obtains the expectation value of H in state (u), E(, ) :

2 1 4g 3 ; (k ; 4k ) ; 9 ; ;%. 2M 3 2 27 2

For a given , the minimum of E(, ) occurs for :

M 16g 19 2

9M

(30)

(31)

and E( ) :

16 g 1 8 g

;% ; M 9 9 81M 9M 2 4

(32)

with

32 g . : 19 (33) 81M We note that the minimum of the total energy corresponds to a displacement : (8 g /9M) with respect to the classical value obtained neglect ing the kinetic energy arising from the zero-point motion of the atoms. As a result, the zone center optical phonon frequency is the renormalized value given by Eq. (33).

14

Vladimir G. Plekhanov

Now we brieﬂy discuss the effect of zero-point motion on the value of bulk modulus. Consider a uniform expansion of the crystal, without altering its symmetry, but including the effect of zero-point motion. By a procedure similar to that employed in the preceding for the zone center optical phonon, one can relate additional macroscopic parameters to the microscopic parameters k and g . For a uniform change in volume, the Hamil tonian per primitive cell is H:

P 2 ; 2k u 9 g u ; % . M 3

(34)

Taking a variational wave function of the form deﬁned in Eq. (29) and minimizing the expectation value of the Hamiltonian given in Eq. (34) with respect to the parameter , one obtains E( ) :

2k ; 2K ( 9 ) ; %, M

where K is the renormalized stiffness constant

g K :k 19 8(2 kM and

g (2 : . 8kM The renormalized bulk modulus T is then given by

K T: 3a

(35)

(36)

(37)

(38)

The lattice parameter was obtained from R : R ; , where R is the nearest-neighbor distance if the atomic mass was inﬁnitely large and hence the zero-point motion negligible. Taking it into account, we obtain

g a :a ; . (39) (6 kM The results of the Brillouin measurements (Vogelgesang et al., 1996) for the four directions of phonon wave vectors were allowed to obtain the elastic constants of diamond. The squares of the velocity of sound clearly show a decrease with increasing average mass (for details see Table 11 of the cited paper). The corresponding elastic moduli X were calculated according to X(x) :

c (x) (x), 4 n(x) *

(40)

15

1 Elastic Properties TABLE II Elastic Moduli c and Bulk Modulus T of Diamond, in Units of 10 dyn/cm GH (After Vogelgesang et al., 1996) x 0.0 0.01105 0.992

c

Sample D 29 D 1, D 2, and D 17 D 30

c

c

T

10.799(5) 10.804(5)

1.248(10) 1.270(10)

5.783(5) 5.766(5)

4.432(8) 4.448(8)

10.792(7)

1.248(14)

5.776(7)

4.429(12)

where C concentration dependent mass density is expressed by (x) : 8M /a(x) and a(x) was determined by Holloway et al. (1991) to be V a(x) : (3.56715 9 0.00053x).

(41)

The calculated (x ) : 3.5152 g/cm based on Eq. (41) compares very well with 3.5153 g/cm quoted by Mykolajewycz et al. (1964). From the data obtained by Vogelgesang et al. it was possible to determine c , c , and c separately for x : 0.0, x : 0.01105, and x : 0.992. The results are shown in Table II. From these elastic constants one can deduce the bulk modulus, the values being given in the last column of Table II. The theoretical prediction for T (x) is

T (x) a(0) M

g : 1; 19 M T (0) a(x) 8(2k M ) V

: 1 ; 0.0012x, (42)

where the values k and g equal, respectively, 4.76 · 10 dyn/cm and (4.5 < 0.4) · 10 erg/cm (see also Table III). The third-order bulk modulus T can be expressed in terms of anharmonicity parameter g . According to the results of Vogelgesang et al., we have T:

dT V T :9 dP T g

V g g a : : 4.9. : g 12(3 T 4(3 k

(43)

Assuming the bulk modulus to be a linear function of the pressure, one obtains the Mungham equation of state:

V T 2 : 1;P V T

(44)

where T is the derivative of the bulk modulus with respect to pressure,

16

Vladimir G. Plekhanov TABLE III The Main Parameters of Crystals LiH, LiD, LiT, LiF, Ge, C, and C

Parameter

LiH

LiD

LiT

LiF

Ge

C

Lattice constant (Å) Density (g · cm\) Binding energy (kcal/mol\) Melting temperature (K) Refractive index (K) E (eV) E

4.084

4.068

4.0633

4.017

5.658

3.567

0.775 9217.8

0.802 9218.8

2.640 9243.6

5.327

3.51

961

964

1115

1210

1.9847 3.61 12.9<0.5 1080 4.992 (2 K)

1.9856 3.63 14.0<0.5 1032 5.095 (2 K)

1.3916 1.96 9.01

4.0055

1120<10

880<9

(cm\) *-

C

2.417 5.7

16.5 0.85 (300 K)

1860 2114 5.48 (300 K) 1332 1280

(Data from Blistanov et al., 1982; Vavilov et al., 1985; and Plekhanov, 1997a)

evaluated at P : 0. With the experimentally determined value (McSkimin and Andreatch, 1972) T : 4.03, one can conclude that the molar volume of C diamond equals that of C diamond at zero pressure when P : 0.2 GPa. One can conclude that T (C) 9 T (C) : 1.8 ; 10\, T (C)

(45)

which is in reasonable agreement with the previous estimation using Eq. (42). Hurley et al. (1994) reported ultrasonic velocity determination in isotopically controlled diamonds with x ranging from 0 to 0.99. They measured the velocity of longitudinal and transverse sound waves for q along 001 as well as 111. Their data would yield c : 10.944;10 dyn/cm, when analyzed in the same manner as was done in Vogelgesang et al. For q !! 111 such an analysis does not appear reasonable in view of the essential constancy of v . Q In the paper of Hurley et al. the large c deduced by them for x : 0.99 in comparison to that for x : 0, namely, 2.379 · 10 dyn/cm versus 1.252 ; 10 dyn/cm, respectively, arises in the main from their v 111 for Q x : 0.99. The nearly factor of 2 increase in the value of c reported by Hurley et al., in contrast to the constancy with x observed by Vogelgesang et al., is very puzzling. Note here that the results of Vogelgesang et al. are consistent with the data obtained by Muinov et al. (1994). Based on experimental values of g a for LiH and LiD the value of g : 3.5 ; 10 erg/cm was obtained in the paper by Plekhanov (1997a). Inserting Eq. (36)

17

1 Elastic Properties

into Eq. (40) one can obtain the relation described by the elastic constant renormalized on isotopic substitution, which (see also Cerdeira and Cardona, 1972) is

k c : 8a

19

g 1 , 8K (K

(46)

where is the reduced mass of the unit cell. According to the experimental data (Gerlich and Smith, 1974), the relative change c in c on isotopic substitution amounts to 3.4—2.2% (see Table I), whereas the calculation based on Eq. (46) yields only 1.8% (see also Plekhanov, 1995f). Note in conclusion that the poor agreement between theoretical values and the experimental data of the elastic constant of LiH (LiD) could suggest that the consideration of the lowest order anharmonicity is insufﬁcient. In this connection, the reason for this discrepancy may be linked to the theoretical model used.

2. Temperature Dependence of the Elastic Moduli in Diamond In this section, we analyze the results of Zoubolis et al. (1998) based on the temperature dependence of the elastic constants of diamond in the temperature range 300 to 1600 K. Since the frequency ( ) of the F zone E center optical mode is determined by the bond stretching (k ) and bond bending (k ) force constants, which also determine the elastic constants (see earlier), a combination of the Raman (see later) and Brillouin results enable the temperature dependence of the microscopic force constants to be extracted. Figures 4—6 present all experimental data at different temperatures, different scattering geometries, and for different samples, according to the results of the cited paper. To ﬁt the data, these authors proceeded as follows: each one of the force elastic moduli was assumed to be described by a quadratic of the form c : c ; c (T 9 300) ; c (T 9 300), GH

(47)

thereby introducing nine ﬁtting parameters. Furthermore, because only the longitudinal backscattering data are free from possible systematic errors, the data from other scattering geometries were allowed an additional multiplicative factor. Since this factor must be the same for all phonons observed in a given experiment, it introduces ﬁve extra ﬁtting parameters. All the data in Figs. 4—6 were ‘‘least-squares ﬁt’’ to such a scheme and produced c values in agreement with literature values. Based on this agreement, the room temperature values of the ions in Eq. (47) were replaced with the more accurate values from the literature, namely c : 1080.4, c : 127.0, and

18

Vladimir G. Plekhanov

Fig. 4. Elastic moduli obtained in the backscattering geometry along different crystallographic directions. The lines are ﬁts described in the text. (After Zoubolis et al., 1998).

1 Elastic Properties

19

Fig. 5. Elastic moduli obtained in the 90° scattering geometry along different crystallographic directions. (After Zoubolis et al., 1998).

20

Vladimir G. Plekhanov

Fig. 6. Elastic moduli in the platelet scattering geometry along different crystallographic directions. (After Zoubolis et al., 1998.)

21

1 Elastic Properties TABLE IV Temperature Dependence of the Three Independent Elastic Stiffness Constants of Diamond (300—1600 K): c : c ; c (T 9 300) GH

c c c

c (GPa)

c (10\ GPa/k)

1080.4 127.0 576.6

929 < 8 93 < 18 922 < 7

The c values are taken from Vogelgesang et al. (1996), the linear temperature coefﬁcient is zero within experimental accuracy, and the c terms are obtained from ﬁts to the experimental results (after Zoubolis et al., 1998).

c : 576.6 GPa. The ﬁtting procedure was then repeated. Values obtained for constants c and c are presented in Table IV. The full lines in Figs. 4—6 were calculated using only the values given in Table IV, the dashed lines include the additional ﬁtting parameters that account for systematic errors (for details, see Zoubolis et al., 1998). The volume compressibility of diamond, or equivalently the inverse of its bulk modulus B, is an important parameter for experimenters using diamond anvil cells (DAC) or other applications involving high hydrostatic pressures at elevated temperatures. For a cubic crystal, the bulk modulus is equal to (c ; 2c )/3. Using the values in Table IV obtained from Zoubolis et al. in the temperature range from 300 to 1600 K, B(T ) : 444.8 9 0.000012(T 9 300).

(48)

The authors concluded that the bulk modulus and the elastic stiffness constants of diamond soften by only 7—9% when diamond is heated from ambient temperature to 1600 K. The temperature dependence of the elastic constants reported in the indicated paper together with that of the zone center optical phonons reported by Zoubolis et al. (1991) and by Herschen and Cappelli (1991) enables an estimate to be made of the temperature changes in the force constants at the atomic level. There are two well-known models that relate the Raman active F frequency ( ) and the elastic moduli to microscopic E interatomic force constants formulated by Keating (1966) and by Musgrave and Pople (1962). The assumptions in the two models are similar but lead to slightly different expressions for the various physical properties. Keating’s model had only two parameters, and , related to band stretching and bending, respectively; the other model includes four force constants. In the following analysis Zoubolis et al. (1998) kept only the two constants, which

22

Vladimir G. Plekhanov

described stretching and bending (k and k ). Following are the expressions and the values measured in Zoubolis et al. (1998) for the physical properties in terms of these atomic force constants: k ; 6k ; 3 : c : 1080.4 GPa : 3a a k 9 3k 9 : c : 127.0 GPa : 3a a

(49)

3k k : 4[a( ; )] c : 576.0 GPa : a(k ; 4k ) M k ; 4k ; : 440 GPa : : , 8a 3a a

where a is the lattice constant (3.567 Å) and M the mass of a carbon atom. Performing a least-squares ﬁt to the expressions and values in Eq. (49), these authors obtained k /a : 1090 GPa; k /a : 280 GPa; /a : 315 GPa, and /a : 240 GPa (see Table V). Finally, if we consider diamond at 1600 K as a ‘‘new substance,’’ we can estimate the decrease in its hardness compared with the hardness at ambient temperature. The hardness is H (1600 K) < 0.96 H (300 K). Although this is only a rough estimate, it nevertheless tends to indicate that diamond ‘‘conserves’’ a large portion of its reputed hardness even at a temperature as high as 1600 K. TABLE V Comparison of Experimental and Calculated Properties Model Experimental Keating Musgrave—Pople

c 1080.4 1035 923

c

127.0 75 83

c

M /8a

576.6 544 414

440 555 736

Values are based on Keating (1966) and Musgrave and Pople (1962) models. The force constants used are k /a : 1090 GPa, k /a : 280 GPa, /a : 315 GPA, and /a : 240 GPa (after Zoubolis et al., 1998).

SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 2

Thermal Properties

I. Specific Heat and the Debye Temperature . . . . . . . . . . . . . . II. Effect of the Isotopic Composition of a Crystal Lattice on the Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Dependence of the Thermal Conductivity of Diamond, Ge, and Si Crystals on Isotopic Composition . . . . . . . . . . . . . . . . . . 1. Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . 2. T heoretical Models . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . IV. Dependence of the Lattice Constant on Temperature and Isotopic Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . .

I.

.

23

.

27

. . . .

28 28 32 37

. .

46 46

Speciﬁc Heat and the Debye Temperature

The main features of thermal motion in a solid can be explained based on the temperature dependence of the heat capacity. The heat capacity per mole of substance is deﬁned as the energy that should be supplied to a mole of the substance to increase its temperature by 1 K. Therefore, the heat capacity at constant volume equals

E C : T T

,

(1)

T

that is, as the energy of a system changes by E, its temperature changes by T. From the point of view of classical statistics, for each degree of freedom of the system, there is an energy equal to k T /2. According to the law of uniform distribution of the energy over degrees of freedom, the average energy of such a system is equal to the production of the number of degrees of freedom by k T /2. As is known (see, for example, Ashcroft and Mermin, 1976), this result, which is valid for ideal gases, can also be applied to systems of interacting particles when the interaction forces are harmonic, that is, they obey Hooke’s law. In this case, we will consider the model crystalline lattice consisting of N atoms, which execute small vibrations near equilibrium positions in sites. Each atom vibrates independently of its 23

24

Vladimir G. Plekhanov

neighbors in three mutually perpendicular directions, that is, it has three independent vibrational degrees of freedom. In the harmonic approximation, such an atom can be represented by a set of three linear harmonic oscillators. As the oscillator vibrates, its kinetic energy periodically transforms to the potential energy and vice versa. Because the kinetic energy (which is exactly equal to the potential energy) equals k T /2, the average total energy of the oscillator, which is equal to a sum of the kinetic and potential energies, is k T. Therefore, the total energy of a crystal consisting of N atoms, is E : 3Nk T.

(2)

Thus, the molar heat, which is deﬁned as the energy required to increase the temperature by 1 K, is equal to C : 3Nk : 3R, T

(3)

where R is the molar gas constant (R : 8.314 J/mol K). Expression (3) is the well-known Dulong and Petit law. Note that this law is valid for many insulators and metals at sufﬁciently high temperatures. However, at sufﬁciently low temperatures, none of the substances obey this law. According to the experimental data, the heat capacity C at low temperatures is proporT tional to T . This dependence cannot be explained within the framework of the classical model, and one must use the concepts of quantum statistics to explain it. The dependence of C is explained in the Einstein model using T two assumptions: a solid is represented by a set of identical harmonic oscillators, which oscillate independently of each other with the same frequency in three mutually perpendicular directions, and their energy is quantized according to Planck. Thus, the problem is reduced to the calculation of the average vibrational energy of an atom along one of the three mutually perpendicular directions. The total thermal energy of the system is obtained by multiplying the result by 3N. According to the deﬁnition of the average value, n e9n /(k T )

E : L 9n /(k T ) , e L

(4)

where e9n /(k T ) is the Boltzmann factor, which determines the state of the system with energy ; and the number of oscillators that oscillate with energy is proportional to e9n /(k T ) . By introducing a new variable x : 9 /(k T ), after transformation of Eq. (4), we obtain

E :

d

ln(1 ; eV ; eV ; %) : . dx eV 9 1

(5)

25

2 Thermal Properties

By returning to the previous variable, we obtain

E :

. e /(k T ) 91

(6)

Thus, the total energy related to vibrations of N atoms in the lattice is E : 3N

. e /(k T ) 91

(7)

For kT 1, that is, at high temperatures, by expanding the exponential into a series e /(k

T ) 91

:1;

; 91 5 , k T k T

we obtain E : 3Nk T

and

C : 3Nk , T

that is, Dulong and Petit’s law. However, the result obtained in the low-temperature limit, when k T and e /(k T ) 1, is new. In this case, we have E : 3N e9 /(k T )

C 5 Nk T

and

9 /(k T ) e . k T

Therefore, C exponentially decreases with decreasing temperature. The heat T capacity calculated in the Einstein model decreases with decreasing temperature faster than in reality (the T law). As is well known, the discrepancy between experimental data and the Einstein theory results from the assumption that each individual atom executes harmonic oscillations with a frequency of independently of the rest of the atoms. Having retained the main idea of Einstein, Debye introduced an additional assumption that harmonic oscillators oscillate with different frequencies and their energy is quantized according to Planck. The total elastic energy at a temperature of T is equal to the integral over from the energy of the oscillator with a frequency of multiplied by the number of oscillators (phonon modes) per unit frequency interval: E:

9Nk T ( /T )

S" x dx eV 9 1

(8)

26

Vladimir G. Plekhanov

where x : /(k T ) and : /k . Expression (8) is called the interpola" tion Debye formula, and

S" x dx 3 D( /T ) : eV 9 1 ( /T )

(9)

is the Debye function. Expression (8) is interesting because the energy and, hence, the heat capacity [see Eq. (14) in Chapter 1] is expressed at any temperature via a single parameter , which is called the characteristic temperature of a solid or the Debye temperature. Its physical meaning, as is known, is that k T : represents a maximum quantum of vibrational " energy of a lattice. This quantity is the only parameter that takes into account the type of a substance, and its value varies from 100 to 2000 K for different substances (Reisland, 1973). The high value of the Debye temperature is well explained by the high rigidity of the interatomic bonds. The Debye function D( /T ) cannot be calculated explicitly; however, the analytic expressions for the energy and heat capacity can be obtained in the limiting cases of low and high temperatures. At high temperatures, that is, k T or x I in Eq. (8), the denominator in the integrand can be expanded into a series to obtain eV 9 I I ; x 9 I : x. Then, expression (8) will take the form

T S" E : E : 9Nk x dx : 3RT. ?

(10)

Therefore, the heat capacity C : 3R, that is, it is independent of temperaT ture and changes according to Dulong and Petit’s law. At low temperatures, that is, k T or x 1, the upper integration limit in Eq. (8) can be set to -. Then, the result of integration is independent of temperature, and the exact value of the integral is

x dx : eV 9 1 15

and

E:

3Nk T . 5

(11)

The latter expression is exact at low temperatures, and it adequately describes the temperature dependence of energy. In this case, the heat capacity C : T . This dependence agrees well with experimental data T within the narrow temperature range (near 0 K). Microscopic calculation based on the phonon formalism also gives Eq. (8) (Reisland, 1973). Therefore, one can conclude that the concept of phonons enables the use of mathematical ideas and procedures applicable to common real particles.

2 Thermal Properties

27

II. Effect of the Isotopic Composition of a Crystal Lattice on the Speciﬁc Heat The ﬁrst detailed measurements of the speciﬁc heat of lithium hydride in a wide temperature range from 3.7 to 295 were performed by Kostryukov (1961). In this paper, the unusual behavior of speciﬁc heat was observed between 10 and 20 K. As shown earlier, at low temperatures, the heat capacity at a constant volume is C : T . Kostryukov observed strong T deviations from the T law, which were dependent on the sample prehistory, and explained them by the presence of free molecular hydrogen in lithium hydride. However, by using a special experimental procedure, Kostryukov managed to obtain more reproducible results that only slightly deviated from the T dependence. Based on these results, the cited author estimated the Debye temperature of lithium hydride to be : 860 K (Table 2 in Plekhanov, 1997a). Later, Yates et al. (1974) carefully studied the lowtemperature (5 T 320 K) speciﬁc heat of lithium hydride. In this paper, the effect of the isotopic substitution (H ; D) on the speciﬁc heat was studied for the ﬁrst time. Yates et al. investigated high-purity and highstoichiometric samples (the total content of impurities 2 ; 10\). The C (T ) dependence measured by Yates et al. at T 30 was used to calculate N the Debye temperature from the relation C : const. (T /) [see Eq. (14) in T Chapter 1]. The value of determined in this way depends on the temperature. It is known that this dependence is explained by the deviation of the postulated Debye spectrum from a real phonon spectrum. The values of found from the extrapolation of (T ) to T : 0 are equal to (LiH) : 1190 < 80 and (LiD) : 1030 < 50 K. One can see that the value of the Debye temperature for LiH obtained by Yates et al. is higher than that obtained by Kostryukov. In addition, data on the Debye temperature show that this temperature decreases with increasing isotope mass in accordance with the theory. Let us now brieﬂy discuss another unexpected result obtained by Yates et al. They observed the !-type anomaly of the heat capacity at temperatures of 11.1 < 0.2 K in LiH and 12.8 < 0.2 K in LiD crystals and concluded that this anomaly in the temperature dependence of C (T ) is related to some N phase transition. Indeed, in the early 1960s Schumacher (1962) had already predicted that lithium hydride should undergo a phase transition in the pressure range between 3 and 4 kbar. Later, using a simple qualitative model, Berggren (1969) found that the binding energy in LiH with a structure of the CsCl type is higher than with the NaCl structure. Based on these results, Yates et al. assumed that anomalies observed in LiH are related to the polymorphic transition from the NaCl structure to the CsCl structure. However, attempts to ﬁnd this transition, which were made in several papers (Voronov et al., 1966; Stephens and Lilley, 1969; Johnson et al., 1975; Kondo and Asaumi, 1988) using an external pressure of up to

28

Vladimir G. Plekhanov

330 kbar, failed. Note here that the CsCl ; NaCl phase transition was also not found in Chandehari et al. (1995), where the effect of the external pressure on CsH was investigated. Note also that numerous experimental studies on reﬂection spectra in the exciton region (see Plekhanov, 1998) and Raman spectra (Jaswal et al., 1974; Laplaze, 1976, 1977, 1979; Anderson and Luty, 1983; Tyutyunnik and Tyutyunnik, 1990; Plekhanov, 1994a) in these crystals did not reveal noticeable anomalies in this temperature range. Moreover, it should be pointed out that the electron gas model used by Berggren is not sufﬁciently substantiated, especially at normal pressures. In addition, more recent calculations made within the framework of the ion model (Ghosh et al., 1982) showed that the binding energy in lithium hydride with the NaCl structure is much higher than in the case of the CsCl structure (see also Perrot, 1976; Hama and Kawakami, 1988). In connection with the mentioned anomaly in C : f (T ), we should note that a weak N endothermic effect was observed in the temperature range between 12.2 and 14.8 K in LiH crystals irradiated by intense gamma rays. Catalano and Leider (1971) explained this effect by the crystallization of hydrogen in bubbles :10 to 10 in size (the triple point of hydrogen is observed at T : 13.96; Maksimov and Pankratov, 1975). Comparison of Debye temperatures calculated from elastic constants (: 1083 9 1135 K at 300 K) (Guinan and Cline, 1972; Gerlich and Smith, 1974) and calorimetric data ( : 1190 < 80 K at 0 K) (Yates et al., 1974) shows that they are in agreement, especially for LiH crystals. Note the strong temperature dependence of the Debye temperature of LiD crystals, which has not yet been adequately explained. It is likely that the Debye temperature of diamond also strongly depends on temperature.

III. Dependence of the Thermal Conductivity of Diamond, Ge, and Si Crystals on Isotopic Composition 1. Historical Remarks All solids — for better or for worse — are able to conduct heat. In isotropic solids the spreading of heat obeys the Fourier law, discovered in 1882: Q : 9k grad T : 9k

T , n

(12)

where Q is the surface density of the heat stream (e.g., it is a vector a module of which is equal to the heat ﬂow across the cross section, which is perpendicular to Q); T is the temperature; T /n is the gradient of the temperature along normal n to isotermic surface, and k is the heat conduc-

2 Thermal Properties

29

tivity. The minus sign to the right in expression (12) is connected with the fact that the heat is ﬂowing in the direction opposite to the gradient of the temperature, that is, from the hot side to the cold side. In any isotropic solids, the tensor of second rank and its form depends on the symmetry of the crystals. In the common case in solids, there are two main mechanisms of heat transfer: the transfer of the heat energy of free electrons and the transfer of the heat energy by the vibrations of the crystal-forming particles. In metals, the effect of both mechanisms act simultaneously, while in insulators, the heat energy transfer is carried out by atomic (ionic) vibrations. The kinetic energy of the vibrations is transferred from the hot strip to the cold strip. Macroscopically, the beam of the kinetic energy of atoms looks like the heat ﬂow. This process is similar to the process of the spread of elastic sound waves in solids. When explaining the heat conductivity phenomenon we can consider that the atoms perform the strong harmonic vibrations diffusing in the crystal lattice like a system of noninteracting elastic waves. Such waves would spread in the crystal freely without damping and, consequently, they would have an unlimited mean free path. Even at small temperature gradients, the heat ﬂow could exist indeﬁnitely long, before heat equilibrium would set in, and in such case, the heat conductivity would be endless. As has been shown in experiments (Berman, 1976), the heat of conductivity in real solids is ﬁnite. The ﬁnite value of the heat of conductivity is connected with the circumstance that in a real crystal the vibrations of the crystal-forming particles are not purely harmonic. The reason for this circumstance is that the forces interacting between atoms depend on the displacement of atoms in a nonlinear manner. The anharmonic character of the vibrations is taken into account by expanding the potential energy by the anharmonic term gx. By keeping the anharmonic term in the expansion of the potential energy, we take into account the presence of an interaction between vibration modes in real situations. This interaction is usually described as the mutual scattering of phonons. In the ﬁrst step, three phonon processes are considered: the decay of one phonon into two others, the sticking together of two phonons, and so on. Mutual scattering of phonons is accompanied by the creation and annihilation of phonons (see also Ashcroft and Mermin, 1976). As was shown by Peierls in 1929 (Peierls, 1955), the probability of the indicated transitions in three-phonon processes does not equal zero, if the following two conditions are fulﬁlled:

; : , ; k : k ;G .

k

(13) (14)

: 2H , and H : ha ; kb ; l c is the vector of the reciprocal lattice. Here G

30

Vladimir G. Plekhanov

Expression (14) is the energy conservation law for the three-phonon processes. The phonon with quasi-impulse k and frequency , generally speaking, does not possess an impulse like that of ordinary material particle. , called the quasi-impulse (wave vector), in the However, the value of k : 0 corresponds common case is equivalent to a pulse. Expression (14) at G to the wave vector conservation law. The interaction at which the condition : 0 is fulﬁlled is called a normal or N process. This term originated from G the analogy of the interaction of elementary particles (e.g., electrons) for which the law of conservation of energy and momentum is fulﬁlled. Unlike with ordinary particles, the number of phonons is not conserved in phonon interaction. Moreover, at phonon collisions, the wave vector could be conserved with a precision equal to the vector of the reciprocal lattice. This means that the crystal lattice in which the phonons move also takes part in : 2H . the collisions, removing part of the momentum equal to G " 0 the process Peierls called the interaction in expression (14) at which G of throwover (of the transfer) or U process, which is derived from the German word Umklapprozesse, the process of transfer. In the U process the energy must be conserved as in the normal process. The difference between is larger or smaller than the N and U processes is determined by whether G ) performed at the value of the wave vector of the third phonon (k the collision of the two phonons with the wave vectors k and k . (i.e., If k : k does not leave the border of the ﬁrst Brillouin k ; k G zone), all three vectors have a positive direction relative to k , which for V : 0. This picture is in them is justiﬁed in relations (13) and (14) at G accordance with an N-process. So far in this situation, k coincides with the direction in which the energy is effectively transferred by phonon modes with k and k , so that as was shown by Peierls, the N processes by themselves do not lead to the reconstruction of the equilibrium distribution of phonons. This means that the ﬁnished transfer of the energy can be conserved, and in the absence of a temperature gradient, for example, the heat conductivity is inﬁnitely large. After the U process, however, the heat energy is transferred in a direction that does not coincide with the direction of group velocities in the modes of k and k . Such essential changes to k always lead to the reconstruction of the phonon equilibrium distribution, and consequently to a ﬁnite value for the heat conductivity. In light of the described processes, we can analyze the dependence of heat conductivity on temperature. For this, we use the expression for the heat conductivity of phonons obtained from the kinetic theory of gases (see also Ashcroft and Mermin, 1976): 1 1 k : C v ! : C v ", 3 T 3 T

(15)

where C is the heat capacity of the single volume of the crystals connected T with the lattice vibrations; v is the average velocity of phonons,

2 Thermal Properties

31

approximately equal to the velocity of sound in crystal, which can be accounted as weakly dependent on the temperature; ! is the average mean free path of phonons and is equal to the average distance they pass between two sequence collisions; " : ! / v is the effective time of the relaxation, and the inverse meaning of "\ corresponds to the frequency of the phonon collisions. In Eq. (15) the main values that determine the dependence of the heat conductivity on the temperature are C and ! . At high temperatures T T , the speciﬁc heat is approaching the limited value, which is determined by the Dulong and Petit law [3Nk , see Eq. (3)]; that is, it becomes independent of the temperature. In this case, the dependence of the heat conductivity on the temperature is determined mainly by the temperature variations of the mean free path of the phonons. At these temperatures, the number of phonons is rather large and their change with temperature is linear:

n(k, s) :

1 e (k, s)/k

T \

<

k T .

(k, s)

(16)

The probability of the transfer process increases with rising temperatures and the frequency of the collisions "\ is expected to rise proportionally to T. In this case, the mean free path of the phonon changes by the opposite proportionality of the temperature; 1

! : . T

(17)

1 k : . T

(18)

Then at T we have

When the temperature decreases (T ), the average number of phonons able to take part in the transfer process, as follows from Eq. (16), is exponentially decreased:

n(k, s) :

1 e

, s)/k T

(k

\

e9/T.

(19)

The probability of the transfer process is also diminished, according to the exponential law, and this means that with decreasing temperature, the mean free path (at the time of the relaxation) of the phonon increases exponentially

! : eF2.

(20)

32

Vladimir G. Plekhanov

With the decrease of temperature, the heat capacity decreases in accordance with the Debye’s law, as :T (see also earlier), but the growth of the heat conductivity happens predominantly because of the sharp exponentially increased term for ! . Then we obtain k : T e/T. (21) When the temperature T approaches , and when the probability of the transfer process becomes small, ! is comparable to the dimensions of the specimen and does not depend on the temperature. Note that the defects (intrinsic and extrinsic) of the crystal lattice also inﬂuence ! . This inﬂuence diminishes with a decrease in defects, and the most signiﬁcant occurs for long-wavelength phonons, the length of whose waves achieve values :100 interatomic distances at 1 K. Defects with a size on the order of the average interatomic distances do not inﬂuence such waves, but they scatter on the crystal’s surface, therefore ! is mainly determined by the size of the specimen. The described change of heat conductivity with temperature has been proved by numerous experimental data (see, e.g., Klemens, 1959; Berman, 1976; as well as the following).

2. Theoretical Models Before analyzing the experimental results of the thermal conductivity of isotopically pure crystals, we brieﬂy discuss the main theoretical models of thermal conductivity (Berman, 1976; Klemens, 1959; Ziman, 1979). Klemens obtained a scattering rate (similar to the familiar Raleigh scattering of photons): "\ : A, ' gv A: 4v

(22) (23)

with the constant A containing the mass variance c M 9 ( c M ) G G G G , ( c M ) G G where c and M represent the concentration and the mass constituent G G isotopes, respectively. A mean free path obtained by Klemens is L : gT , G where, as already mentioned, g denotes the isotopic mass variance. In formulas (22) and (23) V is the volume per atom, and v is the average sound velocity. Equation (22) corresponds to a Debye-like phonon density of states D() : , which we show later to account for the global experimental results. g:

2 Thermal Properties

33

At low temperatures, boundary scattering leads to a T dependence of k, the prefactor being determined by the geometrical size of the sample and the details of the surface. The scattering rate can be written as "\ :

v L

(24) #

where L represents an effective phonon mean free path, which includes # effects resulting from sample size, geometry, aspect ratio, phonon focusing, specular (diffuse) reﬂection at the surface, and so on. In the following, we analyze the widely used scattering theory of k(T ) formulated by Callaway (1959) and the modiﬁcations introduced by Holland (1963). a. Callaway’s Model This model assumes 1. A Debye-like phonon spectrum with no anisotropies or particular structures in the phonon density of states, that is, no distinction of polarization (between longitudinal and transverse phonons); 2. One averaged sound velocity v ; 3. Diffuse scattering at the surface of the sample [see Eq. (24)]; 4. Normal three-phonon processes, including with a relaxation rate "\ : B T , which should be valid only for low-frequency longitudinal , phonons; 5. Three-phonon Umklapp processes assumed to have a relaxation rate like the N processes "\ : B T (see also Klemens, 1959); 3 6. That all phonon scattering processes can be represented by relaxation times depending on frequency and temperature; 7. The additivity of the reciprocal relaxation times for independent scattering processes. Then the total thermal conductivity k can be written as (see also, Berman, 1976): k:k ;k

(25)

where k and k are deﬁned by k : cT

F2

" (x)J(x) dx A

#$F2 [" (x)/" (x)]J(x) dx% A , k : cT : cT (I), $F2 [" (x)/" (x)" (x)]J(x) dx A , 0

(26) (27)

34

Vladimir G. Plekhanov

with :

$F2 [" (x)/" (x)]J(x) dx A , , $F2 [" (x)/" (x)" (x)]J(x) dx A , 0

" (x) I : $F2 A J(x) dx. " (x) ,

(28)

and 1 1 1 xeV , : ; , (eV 9 1) " (x) " (x) " (x) A , 0

k k m x: , m: , c: . k T

2v

J(x) :

(29)

In Eq. (29) k is Boltzmann’s constant, is Planck’s constant, and " (" ) , 0 denotes the relaxation time of N processes (resistive processes). The corresponding combined relaxation rate "\ can be written as the sum of the ! normal, nonresistive rate (N) and resistive rate R [Eq. (29)]. In the Callaway formulation in contrast to the earlier models of Klemens (1959) and Ziman (1979), all resistive scattering probabilities are taken to be additive (1/" ) : 0 (1/" ) [where " represents the isotopic (" ), the boundary (" ), and the G G G G Umklapp (" ) scattering times]; that is, the corresponding scattering mech3 anisms are assumed to be independent. The k term is not only a correction term to k (as sometimes stated in the literature; Callaway, 1959) but is essential to counteract the effect of treating N processes in " as if they were entirely resistive. Consequently, k ! is a nonnegligible part of Callaway’s theory. The magnitude of k is essentially controlled by the concentration of point defects. In the majority of cases of physical interest resistive scattering dominates (" " $ " " $ k k ) , 0 ! 0 and only k is important. Therefore, in the literature, very often only the k term is included. However, when N processes become comparable to the resistive processes (" " ), for example, in very pure, defect-free (i.e., , 0 isotopically pure) samples, the k integrals contribute signiﬁcantly to the total thermal conductivity (see also Berman, 1976). Thus, in isotopically pure samples, normal three-phonon scattering rather than Umklapp processes determine the phonon mean free path. b. Holland’s Model In the next step, we brieﬂy consider Holland’s (1963) theory, which extended the Callaway theory to explicitly include the thermal conductivity

35

2 Thermal Properties TABLE I Parameters for Debye and Callaway Models Fit for Diamond Sample A (cm) Debye model Natural type 11a Synthetic enriched C Callaway model Natural type 11a Synthetic enriched C

B (cm)

C (cm/K)

D (K)

0.1 0.1

4.0;10\ 0.6;10\

2.0;10\ 2.0;10\

550 550

0.055 0.055

1.5;10\ 0

1.4;10\ 1.4;10\

730 730

The Callaway model takes into account that N processes the phonon scattering.

by both transverse and longitudinal phonons, under the assumption k : 0. 1. Since the variation of the phonon relaxation times with frequency and temperature strongly depend on the actual phonon branch and its dispersion, the contributions to the thermal conductivity of the two kinds of differently polarized phonons (transverse and longitudinal) are considered separately, while normal processes are taken into account for the class of crystal at hand, as suggested by Herring (1954). 2. A more realistic representation of the very dispersive transverse acoustic modes of Ge is used (for details, see Nelin and Nilsson, 1972). It involves splitting the range of integration in two parts, a low and a high frequency range with different temperature and frequency dependences. The four scattering mechanisms assumed for the analysis with Holland’s model are chosen to have the following temperature and frequency dependence (see also Table I). "\ : A, ' v "\ : , L # "\ : B T for 0 , 22 "\ : B T for 0 , * * B "\ : 23 for , 23 sinh(x) "\ : 0 for , 23

(30) (31) (32) (33) (34)

36

Vladimir G. Plekhanov

where x : ( /k T ) and T (L) represent transverse (longitudinal) acoustic phonons. According to the Holland integral, k (Callaway’s notation) has been separated into TA and LA contributions k and k . The term k splits 2 * 2 up into the contribution of N processes k and that of U processes k : 223 k:k ;k :k ;k ;k , 2 * 223 *

(35)

with

2 F 2 : H T "2-(x)J(x) dx, ! 23

(36)

F 2 2 k : H T "23(x)J(x) dx, ! 23 3 23 F2

(37)

1 F 2 k : H T "*(x)J(x) dx, ! * 3 *

(38)

k

2-

where

v \ "2-(x) : , ; AmxT ; B mxT ! 2 L #

v B mxT \ , "23(x) : ; AmxT ; 23 ! L sinh(x) # v \ k m k "* (x) : and H : ; AmxT ; B mxT , m: . ! G * L

2v # G (39) In each of the three integrals the constant H contains the corresponding G sound velocity v . We should emphasize that U processes are neglected in G Eq. (36) because they should not contribute below ( : 101 K for Ge). The term for N processes was omitted from Eq. (37) since it should be relatively small above . Both assumptions have been checked to be quantitatively justiﬁed (Holland, 1963). Nevertheless, the integral formulation causes each of the scattering mechanisms to be operative over a large temperature interval. Thus, the effect of varying one of the coefﬁcients always induces modiﬁcations in the inﬂuence of the other coefﬁcients on the thermal conductivity.

2 Thermal Properties

37

3. Experimental Results a. Diamond Since the early work of Pomeranchuk (1942) demonstrating the role of isotopes as phonons scatter with a resulting inﬂuence on the thermal conductivity, and the work performed by Geballe and Hull (1958), it has been known that the maximum thermal conductivity k is strongly affected by the K isotopic composition (see Fig. 1). This fact has received considerable attention in recent years for the case of diamond: an <1% reduction of the C content in natural diamond enhances k by 50% (see, e.g., Anthony et al., 1990; K Banholzer and Anthony, 1992; Hass et al., 1992; Onn et al., 1992; Olson et al., 1993; Graebner et al., 1994). As we can see from results depicted in Fig. 1, Geballe and Hull observed an increase in the k of an enriched Ge sample K (with 95.8% of Ge) by a factor of 3 with respect to natural germanium. It has long been known (see, e.g., Wilks and Wilks, 1991) that diamond has an extremely high thermal conductivity (see also Berman, 1976). At room temperature, high-purity type-11a diamond has a thermal conductivity ﬁve times larger than that of copper and and indeed is the highest value

Fig. 1. Thermal conductivity of germanium. The solid lines are theoretical curves of Callaway’s model. The open circles represent experimental points read from the graph of Geballe and Hull (1958). (After Callaway, 1959.)

38

Vladimir G. Plekhanov

of any known material (Graebner et al., 1994). Diamond is thus very useful for heat dissipation, and a 50% enhancement of the conductivity at room temperature would therefore be of considerable practical interest. The magnitude of the enhancement is surprising, since in similar systems such as LiF (substitution of Li ; Li, and He ; He) the isotope effect is only 1—2% at the same temperature. On the experimental side, the enhancement in diamond has since been conﬁrmed in different papers (Banholzer and Anthony, 1992; Onn et al., 1992; Hass et al., 1992; Olson et al., 1993; Graebner et al., 1994) using the different experimental methods. The theoretical picture, however, remains rather cloudy, because the standard theories predicted only a few percent effect (see, e.g., Ziman, 1979). The thermal conductivity of diamond has been investigated in detail in the wide temperature region (10—1200 K). According to the results of Olson et al. (1993), the thermal conductivity of the two natural specimens is shown in Fig. 2. The measurements indicated

Fig. 2. Thermal conductivity of two type 11a diamonds. Also shown are the data by Berman (1976) as well as Burgmeister (1978) on type 11a diamonds. Burgmeister measured 30 different samples, all of which lie within the range shown. (After Olson et al., 1993.)

2 Thermal Properties

39

here at and below room temperature agree very well with these previously reported values. The authors of the cited papers noted that there is an :10% difference between the two samples, which is due to the different quality of these samples. All indicated data are in close agreement. One is confronted with a situation where small concentrations of isotopic impurities lead to an expectably large reduction in the thermal conductivity. In an attempt to quantify the phonon scattering resulting from isotopic impurities in diamond, the experimental data (see Fig. 2) have been analyzed using the Debye model of thermal conductivity

T F"G2 xeV &(T ) : Nk v l(x) dx, G (eV 9 1) G

(40)

where N is the number density of atoms (for diamond N : 1.762; 10 cm\) and the sum over i denotes a sum over the one longitudinal and the two transverse phonon modes, v is the sound velocity for that mode [for G diamond (Novikov, 1987) v : 1.75 ; 10 cm/s and v : 1.28 ; 10 cm/s]. J R Note is the effective Debye temperature for mode i, given by "G : 2.997 ; 10\v NsK, "G G

(41)

l(x) is the phonon mean free path, and x:

v G , k T!

(42)

where ! is the phonon wavelength. As earlier [see formula (29)], the assumption is made that the resistive scattering rates add, so that ! l(x) : (l \ ; l\ ; l\ )\ ; . 2

(43)

The terms l , l , and l are the phonon mean free path associated with the sample boundaries, points defects, and Umklapp processes, respectively. The term !/2, half the phonon wavelength, is included to avoid the nonphysical case where the mean free path becomes short compared to the phonon wavelength. Although the Debye model is sometimes referred to as the Klemens— Callaway model, the authors Olson et al., (1993) felt that the Debye model is more appropriate. Klemens made the assumption that the scattering rates add and ignored the N processes, while Callaway took N processes into account. At the same time we note that when Callaway applied his theory to Ge, he ignored N processes (see earlier) and simply used the Debye

40

Vladimir G. Plekhanov

model. The data of Berman (1976) and one set of high-temperature data (sample N1 in Olson et al., 1993) have been ﬁtted, using phonon mean free path s of the form 1 l\ : , A

(44)

B l\ : , !

(45)

T l\ : C e\"2. !

(46)

Formulas (45) and (46) were performed by Klemens (1959) and Peierls (1955). A good ﬁt (see Fig. 3) is found for the following parameters: A : 0.1 cm,

B : 4.0 ; 10\ cm,

C : 2.0 ; 10\ cm/K,

D : 550 K.

Fig. 3. Thermal conductivity of natural type 11a diamond and synthetic diamond with 0.07% C isotope concentration. The dashed line is the ﬁt calculated from the Debye model. The solid line is the ﬁt for the nearly isotopically pure diamond, where the point defect scattering (Raleigh term) was the only parameter changed. (After Olson et al., 1993).

2 Thermal Properties

41

These parameters are listed in Table I and are close to those published (see also Hass et al., 1992; Onn et al., 1992; Graebner et al., 1994). In Olson et al., a reasonable ﬁt was found for B : 0.6 ; 10\ cm (see Table I). Thus it was necessary to reduce B by 3.4 ; 10\; this is the Rayleigh term caused by the isotope scattering. Klemens calculated the Rayleigh term taking into account not only the mass difference of the impurity but also the difference in volume occupied. It was found that

M 2 l\ : , ; 2 4 M !

(47)

where is the volume per atom equal to 1/N : 5.68 ; 10\ cm, is the atomic fraction of the impurity (assumed 1), M/M is the fractional mass difference for the impurity, is the Gruneisen parameter equal to 1.1 for diamond (Novikov et al., 1987), and is the fractional volume difference of the impurity. For 1.1% C in C, where M/M : 1/12 and : 0.0005 (Olson et al., 1993), l calculated from Eq. (47) is 5.38 ; 10\ cm. This is 6.3 times smaller than the scattering rate obtained from experiment. Moreover the same conclusion was obtained by Onn et al. (1992), who found the calculated scattering rate to be 5.4 times smaller than the measured scattering rate. The main reason for this observation is that in the Debye model, the nonresistive three phonon N processes are ignored (see also Hass et al., 1992). These scattering processes do not themselves lead to a degradation of the heat ﬂow, but instead restore the phonon distribution to a displaced Planck distribution compatible with the heat ﬂow. Suppose there were a defect that strongly scattered phonons at a particular frequency f . In the Debye model, phonons with frequency f would not contribute to the thermal conductivity, but phonons of other frequencies would be unaffected. N processes would act to channel other phonon modes into that frequency, effectively reducing the thermal conductivity contribution from the other modes as well. The N process scattering rates in diamond are largely unknown, so a quantitative computation is not possible. In Olson et al. (1993) it was assumed that the N process scattering rate is much larger than the resistive scattering rate, and both the Callaway model and the Ziman variational method led to the same result (Berman, 1992):

T [$F"G2xeV(eV 9 1)\] &(T ) : Nk v dx, G $F"G2l\(x)xeV(eV 9 1)\ "G G

(48)

where l\(x) is given by the inverse of Eq. (43). The data of natural type 11a diamond have been ﬁtted using the phonon mean free paths as given in Eqs. (41) to (46). A good ﬁt has been found for the following parameters: A : 0.055 cm; B : 1.5;10\ cm; C : 1.4;10\ cm/K, and D : 730 K.

42

Vladimir G. Plekhanov

These values are listed in Table I (Callaway’s model). Not surprisingly, these parameters are somewhat different from those found using the Debye model. The Rayleigh scattering term was found l\ : 0 in this case. Thus 1.5 ; 10\!\ cm had to be removed. This corresponds to 3.6 times less scattering than predicted by Eq. (47). In this case, making the assumption that the N processes dominate enhances the isotope scattering by a factor of 23 over the case where the N processes are ignored. The same conclusion was reached by Hass et al. (1992). Thus, the observed isotope effect can be explained by existing theory by taking into account the effect of N processes, but the assumption that the N processes dominate leads to an incorrect temperature dependence of thermal conductivity. b. Germanium and Silicon Few papers have been devoted to investigating the dependence of thermal conductivity on the isotope compositions of germanium and silicon crystals. In view of the high purity ( N 9 N 10 /cm\) and the perfection of B ? used samples, in the following we rely mainly on the experimental results of Asen-Palmer et al. (1997), where only four scattering mechanisms were considered: 1. 2. 3. 4.

Normal (N) three-phonon scattering; Three-phonon Umklapp (U) processes; Boundary (B) scattering; Isotopic (I) mass ﬂuctuations (point defects).

The parameters for the two latter mechanisms are taken ﬁxed by theory and are therefore not adjustable. Dislocation scattering was also to be considered. The thermal conductivity versus temperature measured for various isotopic compositions are displayed as log-log plots in Figs. 4 and 5. The data for the isotopically purest sample Ge (99.99%) are shown together with the results for the less pure Ge (86%), natural Ge, and the most isotopically disordered sample containing 43% Ge and 48% Ge. The maximum of k(T ) amounts to 10.5 kW/mK near 16.5 K for Ge (99.99%), which is the highest value of k measured for Ge, higher than the thermal conductivity maximum of sapphire (6 kW/mK near 35) and comparable to that of silver (11 kW/vK near 8 K, Asen-Palmer et al., 1997). The isotopically most disordered sample shows, as expected, the lowest thermal conductivity for undoped Ge (0.75 kW/mK near 15.4 K). In this sample, the isotopic scattering is dominant, in contrast to the pure Ge (99.99%) where it becomes negligible. As pointed out by Asen-Palmer et al., with the exception of sample Ge (95.6%), all samples had a similar geometry

2 Thermal Properties

43

Fig. 4. Thermal conductivity versus temperature of ﬁve Ge samples with different isotopic compositions: Ge (99.99%), Ge (96.3%), Ge (86%), Ge 1, Ge. Two of the samples, Ge (99.99%) and Ge 1, have been measured with two different experimental setups, in Stuttgart (S) and in Moscow (M). The dot-dashed line represents simply a T law, expected for pure boundary scattering, while the dashed line shows a 1/T dependence expected for phonon at high temperatures. (After Asen-Palmer et al., 1997.)

(within 10% equal cross-sectional dimensions and identically prepared surfaces). They were cut with a diamond saw and the surfaces lapped with a 20 pm diamond powder slurry (see details in Asen-Palmer et al., 1997). The overall features of the k(T ) curves displayed in Figs. 4 and 5 are those found for defect-free insulators: The T behavior at sufﬁciently low temperatures, due to boundary scattering, and a maximum resulting of normal and Umklapp phonon processes, which lead to a 1/T dependence above 100 K. Comparing the results depicted in Fig. 4 one can see that the maximum thermal conductivity k of Ge is 14 times smaller than that of Ge K (91.91%). The value of k for natural Ge is increased by a factor of :8 in K the Ge (99.99%) sample. The increase of k, however, is only 30% at 300 K. Note also that (1) the maximum of k(T ) shifts slightly to higher temperatures with increasing isotopic purity in accordance with theory (Jackson and Walker, 1971) and (2) the strong inﬂuence of isotopic disorder on k is clearly displayed over the entire temperature range in which k(T ) was measured. Point defect scattering from isolated atoms of different isotopes or different elements with very similar force constants is one of the rare cases for which phonons can be calculated analytically without adjustable parameters. Using the Callaway model in its original form — that is, keeping both k and k [see formulas (26) and (27)] — two free parameters (B , B ) were adjusted in the combination of the four scattering mechanisms considered,

44

Vladimir G. Plekhanov

Fig. 5. Thermal conductivity of various 100 oriented natural Ge bars. The dashed line represents the thermal conductivity calculated with the full Callaway model (k ; k ), where the parameters B : B : 2.6;10\ s/k was used. The continuous line represents the heat conductivity calculated with the model of Holland (only k ) using the single set of parameters for all samples: B : 1.5;10\ 1/k; B : 4.5;10\ s; B : 9.0;10\ s/k and L : 3.8 2 23 * # mm; g : 58.7;10\ for Ge. (After Asen-Palmer et al., 1997).

with the scattering rates for isotopic and boundary scattering ﬁxed to the values given in Eqs. (22) to (24) (v : 22.6;10\ m; v : 3500 m/s; and 0.01 g 58.7;10\; 3.6 L 4.8). In this manner, an acceptable rep# resentation of all the data was achieved but only below about 30 K. Moreover, the adjustable coefﬁcients obtained were not the same for the various samples (for details see Asen-Palmer et al., 1997). As a typical example, the ﬁt to the data for natural Ge with Callaway’s theory was shown in Fig. 5 (dashed line). The convexity of the calculated thermal conductivity above maximum, describing a steeper decrease of k(T ) with increasing temperature than found experimentally, cannot be removed by changing the parameters. The reason can be traced to an underestimation of the U processes in that model. Instead of using an exponential function for the Umklapp scattering probability, as proposed in the literature (Peierls, 1955; Klemens, 1959), the N processes, as well as the U processes, are represented by the same temperature and frequency dependencies B T . The prefactors B and B are thus indistinguishable in Callaway’s theory. This analysis points out that in isotopically pure samples, normal three-

45

2 Thermal Properties TABLE II Comparison of the Effect of Isotopic Scattering on the Thermal Conductivity of Natural Ge, Si, and Diamond at 300 K. (After Asen-Palmer et al., 1997, and Capinski et al., 1997)

Ge Si Diamond

Percentage increase in k

; 10\

30 60 50

376 658 1860

5.80 2.01 0.76

The percentage increase in k is the increase of the thermal conductivity of the nearly isotopically pure sample compared to the natural sample. The isotope is deﬁned as : f (M /M). G G G

phonon scattering rather than Umklapp processes determine the phonon mean free path. Further, we should brieﬂy analyze the experimental results with Holland’s theory. It was Holland who extended the Callaway theory to explicitly include the thermal conductivity by both transverse and longitudinal phonons, under the assumption that k : 0. The four scattering mechanisms assumed for the analysis with Holland’s model were chosen by Asen-Palmer et al. (1997) to have the temperature and frequency dependence described by formulas (30) to (34). In this model, the three free adjustable coefﬁcients — B , B , and B (equal to 1.0 ; 10\ 1/k, 6.9 ; 10\ s/k, and 2 * 23 5.0 ; 10 s) — were obtained by linear regression. The isotopic and boundary scattering rates were ﬁxed by Eqs. (22) to (24), the corresponding mass variance g, the effective mean free path L , and the sound velocity v , # respectively. With this model it has been possible to obtain a good representation of the thermal conductivity of all samples studied in AsenPalmer et al. (1997). Using a unique set of parameters in the temperature range 2—200 K, the agreement between experimental data and ﬁtted curves is rather good (<5%), as exempliﬁed by the solid line in Fig. 5 for natural Ge. Note also that Capinski et al. (1997) showed that at the isotopical substitution in Si the thermal conductivity was increased by a factor of 7 in the k (at T 25 K) with respect to natural Si (see also Table II). Thus, the analysis of the thermal conductivity of Ge and Si samples with several isotopic compositions, using a modiﬁed Callaway—Holland formalism, works well below 200 K. To conclude this part we should mention that the crystals of diamond and Ge (Si) at the present time include a few unique samples where much detailed investigation of the inﬂuence of the isotope effects on thermal conductivity was done. In this connection, we must note that analogous investigation of the isotope effects of the thermal conductivity in LiH D crystals will be very intriguing, taking into account the V \V

46

Vladimir G. Plekhanov

presence of the local vibrations at low concentration x and two-mode behavior of LO phonons at large concentrations (Plekhanov, 1995b).

IV. Dependence of the Lattice Constant on Temperature and Isotopic Composition 1. Background The lattice parameter at any given temperature is determined by three different contributions. First and most obvious is the size of the atomic radii and the nature of the chemical bonding between them, which are most important in determining interatomic spacings and crystal structure. Second is the effect of temperature on the distance between atoms, which normally produces a volume expansion with increasing temperature. Finally, there is the effect of the zero-point displacement, which is a purely quantum effect with no classical analog. This last contribution results from the fact that the lowest-energy state of the system, the zero-point energy, generally corresponds in an anharmonic potential to an atomic displacement somewhat larger than that associated with the potential minimum. Since the zero-point displacement is usually a small contribution to the lattice parameter at 0 K, its contribution is often ignored, particularly since its magnitude is difﬁcult to determine experimentally. Important exceptions (see later), however, are the crystals that are isotopic variants of lithium hydride. Since they are chemically identical, the contribution to the lattice parameters due to the atomic radii and chemical bonding may be taken as constant in all of them. Consequently, the differences in the lattice parameters at various temperatures may be attributed solely to differences in the thermal expansion and the zero-point displacement. As happens, these differences are relatively large in these crystals, mostly because of the large relative differences in the atomic masses in the three isotopic forms of hydrogen, and the relatively large changes that these produce in the anion— cation reduced masses (see also Plekhanov, 1997a). The ﬁrst paper devoted to the calculation of a change in the molecular volume upon isotopic substitution was that of London (1958), which has now become a classic. He started with an expression for free energy using the Einstein free energy function to reach the expression

M dV d ln : [U 9 E 9 T C ] , T V dM V d ln M

(49)

where V is the molecular volume, M is the atomic mass, is the Gruneisen constant : (V /C ), is the volume expansion coefﬁcient, is compresT

47

2 Thermal Properties

sibility, C is the molar speciﬁc heat, E is the potential energy, is the T phonon frequency, and U is the total energy. For monoatomic solids d ln 1 :9 d ln M 2

(50)

and dV /dM can be expressed through a Debye function with characteristic temperature . At high temperatures, one can express the Debye function " by a power series in ( /T ). Then Eq. (49) becomes (London, 1958, 1964) "

M dV 1 11 : 9 T ( /T ) 1 ; ( /T ) ; % , " V dM 20 420 "

(51)

and at absolute zero M dV 9 :9 R . V dM 16 V "

(52)

Here, R is the gas constant. For diatomic cubic crystals with atomic masses M and m, we have (see also Born and Huang, 1968) :

1 1 ; . M m

(53)

Therefore d ln 1 1 :9 , d ln M 2 (1 ; M/m)

(54)

which leads to

M dV 1 1 : 9 T " . T V dM 1 ; M/m 20

(55)

As pointed out by London, to obtain more accurate results one should have a detailed knowledge of the frequency spectrum, since does not depend on M in a simple fashion. The values predicted by London’s analysis are in reasonable agreement with the experimental ﬁndings (for details see London, 1958).

48

Vladimir G. Plekhanov

a. Lithium Hydride Various investigators (Anderson et al., 1970; Mel’nikova, 1980; Ruffa, 1983; Tyutyunnik, 1992, 1994; Zimmerman, 1972) experimentally and theoretically examined the effect of isotope substitution on the lattice parameter of the LiH crystals (see also reviews by Shpil’rain et al., 1983; Berezin and Ibrahim, 1988). These investigations demonstrate that the isotope effect on the lattice parameter and coefﬁcient of thermal expansion shows a deﬁnite trend in which the lighter isotopes produce larger lattice parameters and smaller coefﬁcients of thermal expansion than the heavier isotopes. The effects are more pronounced when a lighter element is substituted and at low temperatures. Closely related to the molar volume is the thermal expansion (see, e.g., Kogan, 1963). Table III gives values of lattice constants and thermal expansion coefﬁcients for isotopic LiH (Smith and Leider, 1968; Anderson et al., 1970; Zimmerman, 1972). The data indicate that the isotope effect — that is, lighter isotopes having larger lattice constants — is reduced at higher temperatures. Heavier isotopes have larger thermal expansion coefﬁcients. Similar ﬁndings were also reported by other workers (see, e.g., Shpil’rain et al., 1983; Tyutyunnik, 1992; Plekhanov, 1997). These ﬁndings are expected, since the Debye theory predicts a larger heat capacity (see also earlier) for heavier isotopes, and then from the Gruneisen relation, : C /3V, one T can obtain . TABLE III Lattice Constants and Thermal Coefficients for Isotopic LiH (After Anderson et al., 1970) Material lattice constants a (Å)

LiH LiH LiD LiD LiT

9190°C

25°C

140°C

240°C

4.066 4.0657 4.0499 4.0477 4.0403

4.0851 4.0829 4.0708 4.0693 4.0633

4.1013 4.1005 4.0888 4.0893 —

4.1218 4.1224 4.1110 4.1119 —

Material thermal expansion coefﬁcient ; 10

LiH LiH LiD LiD LiT

9190—25°C

25—140°C

140—240°C

21 < 0.3 19.8 < 0.4 24.0 < 1.0 24.8 < 0.4 26.4 < 0.5

34.3 < 0.8 37.4 < 0.4 38.4 < 1.8 42.9 < 1.0 —

50 < 1.0 53.3 < 0.6 54.3 < 1.0 55.0 < 1.4 —

2 Thermal Properties

49

Fig. 6. Temperature dependence of the lattice constant of (1) LiH, (2) LiD, and (3) LiT crystals. Experimental data are taken from Smith and Leider (1968) and Anderson and co-workers (1970). The solid line is theoretically calculated. (After Plekhanov, 1997a.)

After London’s classical paper (London, 1958), Anderson et al. (1970) suggested a simple empirical expression that related changes in the reduced mass of the unit cell and in the lattice constant on isotopic substitution: a : A ; B,

(56)

where A and B are constants, which are, however, dependent on temperature. This relation can be readily obtained taking into account a linear temperature dependence of the lattice constant, which is typical for high temperatures. The nonlinear temperature dependence of the lattice constant of LiH and LiD crystals (Smith and Leider, 1968; Anderson et al., 1970) observed in experiments (Fig. 6) can be described by the second-degree polynomial a ( 9 ( * & [A ; B(T 9 T ) ; C(T 9 T )], * " : a ( * & * "

(57)

where a : a 9 a , I/ : 1/M ; 1/M , T : 25°C, and : 1080 K * & * " * & & * is the Debye temperature of a LiH crystal. The values of constants A, B, and C, determined by the method of least-squares, are presented in Table IV. For comparison, the values of these constants, calculated in a similar way for diamond, silicon, and germanium (Pavone and Baroni, 1994), are also given. The value of temperature T : 810 K, at which the lattice constant is the same for LiH and LiD crystals, was found from theoretical calculations. The lattice constant is equal to 4.165 Å. This means that the temperature dependence a(T ) in LiD crystals (a heavy isotope), which have a smaller Debye temperature, is stronger than in LiH crystals. This general conclusion is valid for a broad class of compounds, from an ionic LiH crystal to a

50

Vladimir G. Plekhanov TABLE IV Values of Coefficients of Polynomials [Eq. (57)] Describing the Temperature Dependence of the Lattice Constant on Isotopic Substitution of a Mass (After Plekhanov, 1997) Substance LiH C Si Ge

A ; 10

B ; 10

C ; 10

55.4 95.48 91.60 90.72

955.54 3.55 3.94 2.27

9102.8 8.21 96.90 96.40

covalent germanium crystal. It follows from theoretical calculations (Fig. 6) that for T 810, the lattice constant of LiD crystals is larger than that of LiH crystals. This agrees qualitatively with the results of microscopic calculations of the temperature dependence of a change in the unit cell on isotopic substitution, according to which a : aLiD at 9900—950 * & (Shpil’rain et al., 1983). The change in the lattice constant on isotopic substitution is mainly determined by the anharmonicity of vibrations, which results in the dependence of the distance between atoms on the vibration amplitude, that is, on the mean vibrational energy. It is well known that the vibrational energy depends not only on temperature, but on the isotopic composition as well. Therefore, to take into account the thermal expansion of a crystal lattice, one should consider the effect of anharmonic terms in the expression for potential energy of pair interaction between atoms at temperature T. b. Germanium Among semiconducting crystals, the Ge crystal was the ﬁrst for which the dependence of the lattice constant on an isotope effect was investigated both theoretically and experimentally (Buschert et al., 1988; Pavone and Baroni, 1994; Noya et al., 1997). In the very ﬁrst paper (Buschert et al., 1988), this dependence was experimentally studied with a highly perfect crystal of natural isotopic composition (average M : 72.59) and a second crystal was isotopically enriched, containing 95.8% of Ge (average M : 73.93). Using the following equation [analogous to Eq. (55)] for the relative changes in the lattice constant a with isotopic mass at low temperature (e 9 /k T 1)

a C M 3 :9 ; k , 4 ? a a M

(58)

where : 1.12 and : 0.40 are the Gruneisen parameters for optical and ?

2 Thermal Properties

51

acoustical phonon modes in Ge, : 374 K is the Debye temperature, and

: 37.3 meV. Buschert et al. evaluated Eq. (58) for a 95.8% enriched Ge crystal in comparison with a natural crystal. Equation (58) predicts 12 ; 10\ and 6 ; 10\ reduction in a for T : 0 and T : 300 K, respectively. Buschert et al. experimentally found reductions of 14.9 and 6.3 ppm at 77 K and T : 300 K, respectively. The agreement between calculated and measured values is very good, considering the uncertainties of the Gruneisen parameter values used in the theory. After Buschert et al. (1988), two theoretical papers were published (Pavone and Baroni, 1994; Noya et al., 1997) where the isotope effect and its temperature dependence were studied. In Pavone and Baroni (1994), the dependence of the lattice constant of C, Si, and Ge on their isotopic purity using ﬁrst-principles calculations were performed by treating nuclear vibrations by density-functional perturbation theory. The main results of this paper are depicted in Fig 7. The values of the constants A, B, and C [see Eq. (57)], as ﬁtted to theoretical data for the three materials studied in the paper of Pavone and Baroni, as well as for LiH (Plekhanov, 1997a), are reported in Table IV. Noya et al. (1997) studied the dependence of the lattice parameter upon the isotope mass for ﬁve isotopically pure Ge crystals by quantum path-integral Monte Carlo simulations. The interatomic interactions in the solid were described by an empirical of the Stillinger—Weber type. At 50 K, the isotopic effect leads to an increase of 2.3 ; 10\ Å in the lattice parameter of Ge with respect to Ge. Comparison of the simulation results with available experimental data for Ge (see also Buschert et al., 1988) shows that the employed model provides a realistic description of this anharmonic effect. Noya et al. (1997) showed that the calculated fractional change of the lattice parameter of Ge with respect to a crystal whose atoms have the average mass of natural Ge amounts to a/a : 99.2 ; 10\ at T : 0 K. This compares well with the results of Buschert et al. c. Diamond The isotopic dependence of the lattice constant of diamond has also attracted interest in connection with the preceding thermal properties (see the earlier section on thermal conductivity). Banholzer et al. (1991) reported the lattice constant of natural and C diamond by x-ray diffraction using the powder samples and single crystals. Holloway and co-workers (1991, 1992) examined the inﬂuence of the isotope ratio on the lattice constant of mixed crystals of CC by single-crystal x-ray diffractometry. AccordV \V ing to their results, the lattice constant, as in the case of LiH, decreases linearly with C content according to the expression a(x) : 3.56714 9 5.4 ; 10\ x.

(59)

52

Vladimir G. Plekhanov

Fig. 7. Dependence of the equilibrium lattice constant of C, Si, and Ge on temperature for different isotopic masses. The temperature is given in units of T * [T *(C) : 1941 K, T *(Si) : 744 K, T *(Ge) : 440 K]. The arrows indicate room temperature (25°C). The lattice constants are in units of the zero-temperature lattice constants at the natural isotopic compositions (a : 6.71, a : 10.23, and a : 10.61 a.u.). Note the different units in three ! 1 % panels, which are indicated by the vertical bars. (After Pavone et al., 1994).

The fractional difference a/a between both end compositions is 91.5; 10\. The lattice constants of the ﬁve samples of diamond mixed with different isotopic compositions were studied in Yamanaka et al. (1994). In this paper it was shown that the standard deviations of the lattice constant were in the range of 5 9 9 ; 10\ Å. The lattice constant is varied (see also Fig. 8) with the isotope ratio and it can be expressed in quadratic form as a(x) : 3.56712 9 9.0 ; 10\ x ; 3.7 ; 10\ x,

(60)

2 Thermal Properties

53

Fig. 8. Isotope dependence of the lattice constant of diamond. The curved line shows the quadratic [Eq. (60)], which ﬁts to solid circles obtained by Yamanaka et al. The straight line ﬁts to open squares those obtained by Holloway et al. (1991). (After Yamanaka et al., 1994.)

where x : [C/C ; C]. This expression contrasts with the linear relation reported by Holloway and co-workers (1991). A linear relation between the lattice constant and the isotope ratio would be somewhat puzzling, because compressibility and Gruneisen parameter are not the same for different isotopes (see also Plekhanov, 1997). In concluding this section we should stress the premier role of the anharmonic effect in the dependence of a on the isotopic effect. Really, as was pointed out by Vogelgesang and co-workers (1996) the concentration dependent lattice parameter incorporating zero-point motion in combination with anharmonicity, deduced from Eq. (39) of Chapter 1,

a(x) : a

9

g M 19 (6k M ) M V

(61)

with M : (1 9 x)M ; xM . A comparison of Eq. (61) with the data of V Holloway et al. for a(x) and k : 3Ba : 4.76 ; 10 dyn/cm for natural diamond yields g : (4.5 < 0.4) ; 10 erg/cm. The Yamanaka et al. results analyzed in the same manner yield g : (4.7 < 0.4) ; 10 erg/cm, that is, very close to Holloway’s data.

54

Vladimir G. Plekhanov

d. Compound Semiconductors GaAs and ZnSe These compounds were studied by Garro et al. (1996) and Debernardi and Cardona (1996). In binary semiconductors, the calculations of the dependence of volume (or lattice parameter) on isotopic masses is more complicated. It is not possible to write the relative variation of the crystal volume as a simple function of the relative variation of the mass. Phonon frequency depends differently on the two masses, and this dependence has to be known, together with the corresponding Gruneisen parameter for all phonon modes, to calculate the dependence of the lattice constant on the isotopic masses. For ZnSe Garro et al. have employed an 11-parameter rigid-ion model (RIM, Talwar et al., 1981) at two different unit-cell volumes to obtain the Gruneisen parameters. Due to the absence of a similar dynamical model for GaAs (note that the only stable As isotope is As), an estimation of the variation of its lattice parameter with Ga and As masses has been performed by interpolating the ZnSe and Ge results. To connect the change in volume with phonon parameters Garro and co-workers used the Helmholtz free energy F, which is related for a system of independent oscillators (phonons) through a partition function (Kubo, 1990). In terms of the energy of individual oscillators, F can be written as F: , q

1

( q ) ; kT ln[1 9 ( ( q )/kT )] . J 2 J

(62)

The volume of a sample is related to the bulk modulus through (V /V ) : 9p/B, while p can be written as p : (F/V ) . Using these 2 2 expressions, we can write

1 1 V : V ; ( q ) ( q ) n [ ( q )] ; , B , q J J J 2

(63)

where it has introduced the mode Gruneisen parameters ( q ) deﬁned as J (q ) : ( ln )/( ln V ) and n [ ( q )] is the Bose—Einstein factor. In the J J J last equation, V represents the crystal in the limit of inﬁnitive masses. In terms of the lattice constant of the conventional unit cell (a in the limit of inﬁnitive masses) for zinc-blende-type materials, the last equation can be written as

a(M , M ) 9 a 4

1 : n ( q ) ( q )[ ( q )] ; , J J J a 3Ba , q 2

(64)

where a(M , M ) is the lattice constant for a ﬁnite mass of atoms 1 and 2 in the primitive cell at a given temperature. Here we are interested in the

2 Thermal Properties

55

change of the lattice parameter when one of the atomic masses changes ( ln a/M ) and in the low temperature limit in which n 0. If we change I the mass of atom k (k : 1, 2) from M to M ; M , the relative change in I I I the lattice parameter is a(M ; M ) [a(M ; M ) 9 a ] 9 [a(M ) 9 a ] I I < I I I a(M ) a I 2

< [ ( q ) ( q )], J 3Ba , q I J

(65)

q )] is the mean difference of the quantity in brackets where [ ( q ) ( J I J evaluated at two different isotopic masses. As was shown by Garro et al., this term is usually negative for an increase in either of the masses. Thus it can be understood as an ‘‘isotopic contraction’’ of the lattice parameter. The calculation of Eq. (65) requires an integration over the whole Brillouin zone. For this reason, it is convenient to deﬁne the ‘‘lattice spectral function’’: 2

'(M , M , ) : ( q ) [ 9 ( q )], J 3Ba , q J

(66)

which represents the spectral dependence of the changes in lattice parameter induced by a mass conﬁguration M and M . In terms of Eq. (66), Eq. (65) becomes

a I : d '(M , M , ). I a

(67)

Garro et al. calculated the spectral functions to two different isotopic masses of Zn. These authors compared the results for ZnSe with those for Ge obtained by Pavone and Baroni (1994), where it was noted that the effect of changing both masses in the unit cell is nearly the same for both materials. However, whereas for Ge the two atoms contribute equally, for ZnSe, the contributions of the anion and the cation are rather asymmetric. The dependence of the GaAs lattice parameter on the Ga and As masses was obtained by linear interpolation of the values found for Ge and ZnSe because of the less extensive knowledge of Gruneisen parameters for GaAs than for ZnSe. Debernardi and Cardona (1996) described an efﬁcient way to compute the derivatives of the lattice constant with respect to the mass in polar semiconductors.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 3

Vibrational Properties

I. Theoretical Models of Lattice Dynamics . . . . . . . . . . . . . . . 1. Formal Force Constants . . . . . . . . . . . . . . . . . . . . . . . 2. Rigid-Ion Model (RIM) . . . . . . . . . . . . . . . . . . . . . . 3. Dipole Models (DMs) . . . . . . . . . . . . . . . . . . . . . . . 4. Valence Force Field Model (VFFM) . . . . . . . . . . . . . . . . . II. Measurement of Phonon Dispersion by the Inelastic Neutron Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Raman Spectra and the Density of Phonon States . . . . . . . . . . .

57 57 59 60 62 63 73

I. Theoretical Models of Lattice Dynamics 1. Formal Force Constants The objective here is to present and discuss the models that have been developed to date with particular emphasis on the dynamics of ionic crystals (LiH); we also describe some of the work that has been done on the lattice dynamics of covalently bonded crystals (e.g., C, Ge, and Si). However, the number of models that have been developed to date is very large and growing rapidly (Cochran, 1973; Hardy, 1974; Bilz and Kress, 1979; Klein, 1990). Thus it is impossible to discuss all of them in great detail. The main reason for concentrating the discussion in this part of the book on the subject of ionic and to a lesser extent on covalently bonded materials is that these are the materials that phenomenologically have been extensively used. The lattice potential ' of a crystal in thermal equilibrium can be derived from the free energy F(T, V ). In the classical harmonic approximation (' : ' ), ' is 1 ' : u(L )'(L L )u(L ), 2 **Y

(1)

where L : (l, k) denotes the kth particle in the lth cell. Here, u (l, k) denotes 57

58

Vladimir G. Plekhanov

the displacement vector of the ions with equilibrium positions x (l, k) : x (l) ; x (k), where x (l) is pointing to the lth cell of the crystal, and x (k) describes the relative position of the kth ion in the cell. The 3 ; 3 two-ion (L , L ) are subject to the conservation laws of force constant matrices ' energy, momentum, and angular momentum and to the symmetry restrictions of the space group (see, e.g., Birman, 1974). We note that this treatment is based on the assumption of crystal stability and on the adiabatic approximation (Born and Huang, 1968). Furthermore, the force constants being derived from a series expansion of the free energy F(T, V ) are pseudoharmonic entities that contain a quasi-harmonic correction due to thermal expansion (Leibfried and Ludvig, 1961) as well as self-energy renormalization arising from anharmonicity (see, e.g., Cochran and Cowley, 1967). From the vector equation of motion, ' (L L )u (L ), : 9 ' M u¨ (L ) : 9 I u (L ) *Y

(2)

using, for example, periodic boundary conditions, one obtains the eigenvalues and eigenvectors of the system as the solution : ( q, j) H and e : e ( q, j) of the secular equation H (q I : 0, det D ) 9 M

(3)

is deﬁned as where the dynamical matrix D (kk q (L L ) exp#9iq D ) : ' [x (L ) 9 x (L )]%. J

(4)

: (M ), that is, it is a diagonal mass matrix, while I is a unit Here M I matrix, I : ( ), where and are Cartesian indices. The index j labels ?@ IIY the different branches belonging to every wave vector q. The most important entity to be known, in addition to the dispersion relation ( q, j), is the phonon density of states (DOS)

D() : n (), H H

(5)

with the contribution from branch j, n () : [ 9 (q, j)] : V H ? q

dSq

q (q, j)

,

(6)

where dS q denotes a surface element in q space. Note that the frequencies

3 Vibrational Properties

59

do not uniquely determine the eigenvectors and force constants of a crystal (Bilz and Kress, 1979). Therefore, in a calculation of dynamical properties that contains phonon eigenvectors (such as Raman scattering) additional arguments from microscopic theory and the like are required to demonstrate the physical signiﬁcance of a set of (formal or model) (see also Plekhanov, 1997) force constants (Kellerman, 1940; Taylor, 1975). 2. Rigid-Ion Model (RIM) The lattice energy of ionic crystals is mainly given by two-ion Coulomb interactions of pointlike charges. In diatomic cubic crystals with ionic charges, Z :
x(lk) 9 x(0k)

JIIY

(7)

with the Madelung constants and the nearest-neighbor distance r . At K equilibrium, the resulting attractive forces are balanced by repulsive shortrange electronic overlap forces. They are conveniently described by twobody central Born—Mayer potentials: V (r) : A exp(9r/ ). IIY IIY IIY

(8)

In this Born model the total lattice energy per cell is given by U(r) : 9 K

(Ze) ; V (r). IIY r IIY

(9)

Kellerman (1940) used this model for a calculation of dispersion curves in alkali halides. Here the force matrix can be split into a long-range Coulomb :' A;' 0. and a short-range repulsive part as ' The Coulomb force matrix is given by A:Z C Z , Z : (Z ), ' I

(10)

. The Fourier transform of ' (refer to the with the ‘‘Coulomb matrix’’ C corresponding transform by Leibfried and Ludwig, 1961) is the rigid-ion dynamical matrix. 0'(q ) : Z C (q )Z ;R (q ). D

(11)

(q ) is the Fourier transform of C and has a nonanalytical part at Here C

60

Vladimir G. Plekhanov

q : 0, which leads to the splitting of the longitudinal-optic from transverseoptic modes in the nonpolariton regime (Lyddane—Sachs—Teller relation, Born and Huang, 1968). A speciﬁc technique originated by Ewald has been (q ) (see also Maradudin et al., 1971). In developed for the calculation of C the simplest nearest-neighbor approximation, this rigid-ion model reduces with the help of the equilibrium condition, U(r) : 0, to a one-parameter model that gives a surprisingly good ﬁt to the acoustic and transverse optic modes. The calculated longitudinal optic mode frequency is, of course, too high, since no electronic screening has been considered. In other words, : 1 in the RIM. 3. Dipole Models (DMs) The difﬁculties in properly describing the longitudinal optic modes and related entities can be overcome by an explicit consideration of the electronic charge distortion induced at a displaced ion (Cochran, 1973; Bilz and Kress, 1979). Several models developed during the last 4 decades treat this distortion in the dipole approximation (see also Klein, 1990). The most pictorial of these models is the shell model (SM). a. Shell Model This model, originally due to Dick and Overhauser (1958) and Cochran (1971), introduces an electronic polarization coordinate w (L ) in addition to the ion vector displacement u (L ). The potential depends now on a bilinear form on all w (L ) and u (L ) of the lattice. The equations of motion are ' , M u¨(L ) : 9 I u (L )

(12)

' M w¨(L ) : 9 : 0. (13) CJ w (L ) The second equation is the adiabatic condition for the shell model. One now introduces Coulomb and short-range forces between electrons as well as between electrons and ions and proceeds similarly to the case of the RIM; then one obtains then the dynamical matrix (q ) : D 0'(q ) ; D "'.(q ), D

(14)

0' is the dynamical matrix of the RIM and D "'. that of the induced where D dipolar forces: "'. : 9(T ; Z C Y )(S ; YC Y )\(T ; Y C Z ). D

(15)

3 Vibrational Properties

61

represent the short-range electron—ion and electron— Here T and S electron coupling, and Y is an electronic charge matrix analogous to the . The matrix S consists of the short-range electron— ionic charge matrix Z electron coupling S and of the local shell—core coupling k : (k ). In the I simplest version of the shell model (SSM) one assumes that short-range : T : S ) and, furthermore, one forces act through the shells only (R neglects the polarizability of the positive ions as compared to that of the negative ions (k ; -). In the nearest-neighbor approximation there are > , as compared with Kellerman’s then two additional parameters, Y and K RIM. They can be ﬁtted to the optical constants in the following equations (Bilz and Kress, 1979):

4 ; 2 e : R 9 Z , 2 QV 3 3 ? 8 ; 2 e : R ; Z . * QV 3 3 ?

(16) (17)

Here is the reduced mass of the lattice cell. The effective Szigetti charge Z is given (Born and Huang, 1968) by Q Y \ Z :Z ; . Q > 1 ; k /R \

(18)

Here R deﬁnes the center frequency 1 : (2 ; ), A 3 2*-

(19)

R . : R : A 1 ; R /k \

(20)

This three-parameter SM (A, Y , K ) improves the agreement between \ \ experiment and theory remarkably as compared with the RIM. Here A is a longitudinal force constant to nearest neighbors. The extended shell model (ESM) was suggested by Cochran. ESM is, of course, able to describe every type of phonon dispersion relation by introducing a sufﬁcient number of ,S , T (for details see Cochran, 1973). parameters in the matrices R b. Deformation Dipole Model (DDM) The shell model, in its extended version, is equivalent to the deformation dipole model (DDM) originally developed by Hardy (see, e.g., Hardy, 1974). The equivalence can be shown in the DDM and is completed with a

62

Vladimir G. Plekhanov

polarization self-energy that was missing in the original version of Jaswal (1975). The interrelation between these models has been illuminated in a microscopic calculation by Zeyher (1975). The reader can ﬁnd other types of shell models in Bilz and Kress (1979). 4. Valence Force Field Model (VFFM) This approach can be regarded as an extrapolation of the theory of normal vibrations of molecules. Here, the potential energy of the molecule is often given in terms of internal displacement coordinates, such as changes of bond distances, bond angels, and the like (see also Wilson et al., 1955). The interatomic forces stem from distortions of the electronic charge density, which is strongly concentrated along the interionic connecting lines due to hybridization of the atomic orbitals. Since the covalent crystals behave in many respects like very large molecules, one should be able to describe them by an appropriate extension of molecular methods to almost inﬁnitive systems. The valence force ﬁeld constants in these crystals can be obtained from analogous organic molecules with similar covalent bonds. For example, Schacht-Schneider and Snyder (1963) performed a leastsquares determination of valence forces in branched parafﬁns that has been used successfully to determine the phonons in diamond, silicon, germanium, and gray tin (Tubino et al., 1972). Note that the valence force ﬁeld potential does not contain any Coulomb interactions, which means that the interacting electronic and ionic charges are well screened. On the other hand, the VFFM lacks a clear relation to the dielectric properties of the crystals, which gives the shell models their strong appeal. In conclusion, note that in microscopic theories of phonons one begins with Coulomb electron—electron and electron—ion interactions and eventually derives the phonon dispersion relations by some approximation to them, using Hamiltonian. The essential assumption is that the deﬁnition of ‘‘ions’’ includes the tightly bound electrons. The Coulomb potential of the ions must then be replaced by a pseudopotential that takes into account the exchange and correlation effects of the valence electrons with the core electrons. As an example, a successful sequence of investigations (Zeyher, 1975) led to the ﬁrst parameter-free calculation of phonon dispersion curves in LiD (see also Baroni et al., 1987). The shell model has been successfully applied to describe the dynamics of various ionic compounds. Comparison with experimental data reveals a general disadvantage of this model, resulting from the inadequate description of phonons at the L point of the Brillouin zone (see, for example, Bilz et al., 1984). In NaCI-type structures, all displacements of ions vanish at the L point. This means that the sublattices move independently at one another; that is, three phonon branches from the six correspond exactly to the

3 Vibrational Properties

63

displacements of cations, and the other three phonon branches correspond to the displacements of anions. For this reason, the polarizability of ions does not play a role; that is at the L point, the shell model and the rigid-ion model yield the same results (Hardy, 1974). At the L point the largest discrepancy between the theoretical and experimental data is observed (Laplaze, 1979; Plekhanov, 1997a). Another problem of the shell model is that it is a model of central forces, and hence, it requires the fulﬁllment of Cauchy’s relation for elastic constants. Because, as mentioned, this relation is often violated, one should overcome the discrepancy by matching force constants without taking into account the equilibrium of the lattice. For this reason, the shell model is often considered to be the interpolation scheme for the description of dispersion curves. These problems resulted in numerous attempts to reﬁne the shell model (Hardy, 1974; Bilz et al., 1984). In this connection, note also that, as Cochran (1971) showed, the elements of the dynamic matrix are similar for dipole—dipole and shell models, although polarization effects in these models are considered differently.

II.

Measurement of Phonon Dispersion by the Inelastic Neutron Scattering Method

The phonon spectrum is characterized by the dispersion law ( q ) and the frequency distribution function g() (Krivoglaz, 1967). Both these functions are commonly determined from experiments on the scattering of thermal neutrons, if coherent and incoherent scattering can be separated. Thermal neutrons play an important role in the study of lattice dynamics, because their energy (k T : 10\ to 10\ eV) is of the same order of magnitude as the phonon energy. At the same time, their de Broglie wavelength is comparable with the interatomic distance in a crystal. In this respect, neutrons have an advantage over electromagnetic waves, for which matching can only be obtained in energy (in the IR range) or the wavelength (gamma rays). The features of scattering of thermal neutrons by a lattice are determined by the following main factors: 1. Because the neutron wavelength greatly exceeds the size of the nucleus, the scattering is isotropic and independent of the neutron energy. 2. Scattering of neutrons by the lth nucleus is described with good accuracy by the Fermi pseudopotential, which is proportional to the delta function ), U ( r ) : (2 b /m ) ( r 9 R J J L

(21)

where m is the neutron mass and b is the so-called scattering length. L J

64

Vladimir G. Plekhanov

3. The total scattering cross section by an ensemble of nuclei forming a crystal is determined by the summation over individual nuclei, taking properly into account the phase relations for scattered waves. If nuclei are vibrating, the neutron scattering can be both elastic and inelastic; that is, it can be accompanied by creation and annihilation of a phonon or several phonons; and 4. The scattering length b can greatly differ for different isotopes of the J same element. In addition, when the nucleus spin is not zero, b J depends on the mutual orientation of the neutron and nucleus spins. These are the factors that cause incoherent scattering; that is, the scattering during which the law of conservation of momentum is not satisﬁed because of violation of the translational symmetry (Dolling, 1974). The principle of measuring phonon dispersion by means of one-phonon coherent scattering can be understood from the laws of conservation of energy and momentum:

E 9E : (k 9 k ) : ( q ), (22) Q G Q 2m G L : , Q k 9 (23) k : q ; G G Q where E and E are energies of the incident and scattered neutrons, G Q is the reciprocal lattice vector; q are their momenta; G respectively; k is the phonon wave vector; and j is the number of the phonon spectrum branch. The frequency of the phonon involved in scattering is found from Eq. (22) by measuring neutron velocities before and after the interaction. If the scattering angle is also measured, the phonon wave vector q can be found from Eq. (23). Thus, the dispersion law in a point of the Brillouinn zone is totally determined. What actually happens is that the situation is complicated by the presence of several frequency values for each q ; in other words, , there are several groups of for each value of the transmitted momentum Q neutrons scattered by each branch (Dolling, 1974). In the case of incoherent scattering, only the law of conservation of energy, Eq. (22), is fulﬁlled, so that the scattering spectrum proves to be continuous within E < . The G

scattering cross section is proportional to the frequency distribution function g() (see earlier) and also contains a frequency factor that depends on the polarization vectors of phonons e Q(q) (Krivoglaz, 1967). For this reason, function g() as reconstructed from experiments reﬂects to some degree the model that was used to determine the polarization vectors. In the case of lithium hydride crystals, neutron scattering strongly depends on the isotopic composition. The nonzero spin of the hydrogen nucleus results in a very strong incoherent scattering. For this reason in lithium hydride, only the frequency distribution function can be measured, but not the dispersion law. In contrast, in LiD (containing 100% D), only coherent scattering takes

3 Vibrational Properties

65

Fig. 1. Dispersion of phonons in a LiD crystal: (1) and (2) measurements of inelastic neutron scattering by TO and LO phonons, respectively (after Verble et al., 1968); (3) IR measurements (after Brodsky and Burstein, 1967; Laplaze, 1977); and (4) data on Raman scattering (after Anderson and Luty, 1983; Tyutyunnik and Tyutyunnik, 1990; Plekhanov, 1994a). Solid lines correspond to calculations according to the shell model of Verble et al.

place. However, an addition of a small amount of hydrogen enables one to observe incoherent scattering as well. Scattering of thermal neutrons by LiH (LiD) crystals was investigated in papers of Zemlianov et al. (1965) and Verble et al. (1968). Zemlianov et al. studied the incoherent scattering. Figure 1 presents dispersion curves for lithium deuteride, measured by Veble et al. The accuracy of these data, according to the authors, is 3—4%. Despite great efforts, Verble et al. failed to observe scattering by longitudinal optical phonons. They explained this fact by the very short lifetime of LO phonons (see also Plekhanov, 1987). Verble et al. described their experimental results by means of the shell model with seven parameters: constants A and B >\ >\ of the short-range potential for the interaction between nearest neighbors, constants A and B of the same potential for the interaction between \\ \\ nearest anions; the effective charge Z; and electrical and ‘‘mechanical’’ polarizabilities, and d, respectively, of the deuterium ion (cations were assumed to be nonpolarizable). All these parameters were found from the best ﬁt with experimental data. They are presented in Table 4 of Plekhanov (1997a). Figure 1 shows the dispersion curves of LiD calculated with these parameters. One can see that the greatest discrepancy, as expected, is observed in the vicinity of the L point of the Brillouin zone. The values of the parameters obtained can be used to calculate elastic and dielectric constants, which can be compared with experimental data. The latter represent the average values of data presented earlier. One can see from Table I that calculated values agree satisfactorily with experimental data except for the c and constants. As for the ﬁrst constant, it is not

66

Vladimir G. Plekhanov TABLE I

Comparison of Calculated and Measured Parameters of LiD (After Verble et al., 1968) c Theory Experiment

(GPa)

69.3 67.7

c

(GPa)

3.0 14.8

c

(GPa)

47.1 47.0

(cm\) *819 880

e*/e Q

2.89 3.61

0.51 0.56

surprising if one bears in mind that noncentral forces are treated incorrectly in the shell model, as was noted above. Note also that the ionicity Z : 0.88e obtained in the shell model is in good agreement with the preceding estimates. As mentioned, the dispersion curves for LiH crystals cannot be measured. However, it is reasonable, as suggested by Verble et al., that all parameters of the model are, with good accuracy, the same for LiH and LiD crystals except, of course, for the anion mass. The dispersion curves calculated by Verble et al. under this assumption are presented in Fig. 2. The shell model correctly predicts frequencies of not only TO and LO phonons shown in Fig. 2, but also of other phonons found from absorption spectra of color centers in LiH crystals (Anderson and Luty, 1983; Pilipenko et al., 1985). Very accurate experiments of the dispersion relation were performed for diamond (Fig. 3) by Warren et al. (1967), for silicon (Fig. 4) by Dolling and Woods (1965), and for germanium by Nilsson and Nelin (1971). The numerous calculations (Warren et al., 1967; Tubino et al., 1972; Bilz et al., 1984; Baroni et al., 1987) of the dispersion relation for these semiconductors

Fig. 2. Dispersion of phonons in a LiH crystal: (1) IR measurements (after Brodsky and Burstein, 1967; Laplaze, 1977) and (2) data on Raman scattering (after Anderson and Luty, 1983; Tyutyunnik and Tyutyunnik, 1990; Plekhanov, 1994a). Solid lines correspond to calculations according to the shell model of Verble et al.

67 Fig. 3. The dispersion relation for the normal modes of vibration of diamond in the principal symmetry directions at 296 K. (After Warren et al., 1967.) The full curves represent a shell-model ﬁt to the data points. (After Bilz and Kress, 1979.)

68 Fig. 4. Single-phonon dispersion curves of Si deduced from neutron spectroscopic data at RT (the data of Dolling, 1965). (After Bilz and Kress, 1979.)

3 Vibrational Properties

69

are shown in rather good accordance with the experimental ones (see also Dolling and Cowley, 1966), excluding diamond. Diamond has unusual static properties when compared to other group IV tetrahedral semiconductors characterized by a small lattice parameter, a large bulk modulus, and a large cohesive energy (Pavone et al., 1994). Lattice dynamical characteristics, such as phonon dispersion and thermal expansion are also distinctive. Additionally, the occurence of a maximum in the phonon dispersion of the most energetic phonons away from the Brillouin zone center is peculiar to diamond. A necessary condition to have such an overbending is to have sufﬁciently large second-order nearestneighbor force constants (Tubino and Birman, 1975; Klein et al., 1992; Pavone et al., 1994). Interest in the lattice dynamics of diamond has been especially strong since the report of a sharp peak in the second-order Raman spectrum at a frequency slightly higher than twice the largest single phonon frequency; that is, at 9 2 : 2 cm\ (0.25 meV) at room temperature [ : E(), Solin and Ramdas, 1970, and references therein]. This situation is unlike that of Si (Uchinokura et al., 1974), and a ﬁrst explanation invoked the occurence of phonon—phonon interactions to produce a two-phonon bound state (Cohen and Ruvalds, 1969). However, later theoretical work led to the conclusion that anharmonic coupling constants have the wrong sign to enable formation of a bound state (Vanderbilt et al., 1984; Wang et al., 1990). An alternate explanation was suggested involving an unusual LO phonon dispersion with an energy maximum away from the Brillouin zone center (Musgrave and Pople, 1962; Vanderbilt et al., 1984; Hass et al., 1992). The occurence of this overbending for the branch was supported by the valence force ﬁeld model (Tubino and Birman, 1975). They obtained a sharp peak in the overtone density of states (see later) at 2 (i.e., without a shift) due to a van Hove singularity at . Later, Hass et al. (1992) reported that the model used by Tubino and Birman does not yield the peak in the overtone density of states. However, their Raman scattering studies performed on various isotopic compositions of diamond support the occurrence of such a peak (see also later). Hass et al. added ad hoc a peak to the DOS to achieve agreement with their two-phonon Raman spectra. The ﬁrst ab initio calculations by Vanderbilt and co-workers (1984) found overbending along . More recently, Windl and co-workers (1993) obtained such dispersion in an ab initio calculation along all three directions for diamond having a natural isotopic abundance. However, they obtained a two-phonon shift of 25 cm\ (3 meV), an order of magnitude larger than the observed value. Although measurements obtained for diamond dispersion by inelastic neutron scattering thirty years ago were not focused on subtle features in LO phonon dispersion (Warren et al., 1967), overbending along has been reported for recent neutron scattering measurements (Kulda et al., 1996). The results of Schwoerer-Bohning and co-workers (1998) of phonon dispersion of diamond measured by inelastic x-ray scattering are

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Vladimir G. Plekhanov

Fig. 5. Dispersion of the high-energy branches as obtained by HRIXS (data points) together with ab initio results (lines) from Windl and co-workers (1993). The error of the data is their symbol size (except along ). The horizontal lines indicate the phonon energy at the zone center . (After Schroerer-Bohning et al., 1998).

depicted in Fig. 5. Their data agree quite well with the raw data from Kulda et al. Although the data of Schwoerer-Bohning et al. deviate from the neutron results toward the zone boundary, they show the same magnitude and position of the maximum in the phonon dispersion. The maximum obtained by Schwoerer-Bohning et al. is 1.2 meV above and appears at the phonon wave vector q : 0.4(2/a), which is closer to the results of Kulda et al. [maximum at 1.5 meV above and q : 0.35(2/a)]. However, this is in contradiction to the ab initio calculation presented by Kulda et al., which predicts a different magnitude (3 meV) and position [0.48(2/a)]. Thus, the momentum-resolved inelastic x-ray scattering experiment yields directly the dispersion of optical phonons. The Schwoerer-Bohning results demonstrate overbending along , and possibly along the directions (for details see Schwoerer-Bohning et al., 1998). In Fig. 6 we show the LO optical phonon branch from to X obtained for a Ge enriched crystal and natural one according to the results of Etchegoin and co-workers (1993). Measurements of the TO branch are shown in Fig 6b. The latter include the mapping of the branches along two

3 Vibrational Properties

71

Fig. 6. Inelastic neutron scattering results. (a) The LO phonon branch. The frequencies of the isotopically enriched sample Ge were multiplied by 1.0208 so as to eliminate the trivial mass dependence of the phonon frequencies. Differences between the two measurements are due to isotopic disorder. (b) Results for the TO phonon branch. Two sets of measurements are displayed, for an incident focused or defocused neutron beam. (After Etchegoin et al., 1993).

72

Vladimir G. Plekhanov TABLE II Disorder-Induced Changes in the Frequencies of TO Phonons in Natural Ge at Different Points of the Brillouin Zone Obtained from Inelastic Neutron Scattering (After Etchegoin et al. 1993) Phonon TO TO * TO 5 TO 6

(cm\) N

Re[] (cm\)

303 290 278 275

0.4 < 0.2 90.2 < 0.07 0.1 < 0.15 90.2 < 0.07

different lines in the reciprocal space with a different orientation of the resolution ellipsoid with respect to the dispersion (focusing and defocusing conditions). The frequencies of the Ge crystal have been multiplied by 1.0208, thus eliminating the trivial proportionality to 1/(m (where m is the average atom mass). The differences that are left should be solely due to disorder (Etchegoin et al., 1993). Although these differences are small, it is possible to discern systematic deviations between the experimental values of the enriched and natural samples. The change in the broadening of the peaks can also be evaluated. The numerical values for the energy shift of the TO phonons at the critical points are summarized in Table II. These results show that the shift of the LO phonons is larger than that of the TO phonons, and the LO phonons (as is shown later) are not isotopically broadened in the Ge crystal (see also Agekyan et al., 1989; Fuchs et al., 1991). In conclusion, note also that comparison of the dispersive curves for lithium hydride and other ionic crystals (for example, alkali-halide crystals) shows (Bilz et al., 1984) that lithium hydride and lithium deuteride as a whole exhibit ‘‘normal’’ dispersion of phonons. The only qualitative difference consists of the behavior of the transverse optical branch in the —L direction. In most alkali-halide crystals, () (L ). In addition, 22in crystals with strongly different masses of ions, (L ) commonly proves 2to be the lowest frequency in the optical band, that is, this frequency determines the top of the energy gap (Karo and Hardy, 1963). In LiH, the dispersion of the TO branch is the reverse [ () (L )], and 22for this reason, the top of the energy gap corresponds to (); that 2is, the frequency of the IR-active phonon. We see later that this fact substantially determines an unusual behavior of the structure of the absorption spectra of mixed crystals in the IR spectral region.

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73

III. Raman Spectra and the Density of Phonon States The phonon density of states g() of a crystalline solid is deﬁned as follows. The function g() d signiﬁes the fraction of the total number of phonon states in the frequency interval (, ; d) if $ g() d is normalized to 1. Knowledge of g() is essential for understanding the thermodynamic properties, as well as for extraction of the electron—phonon coupling and the like. It is difﬁcult to make good direct measurements of the density of states, and few spectra (e.g., LiH) have actually been studied. Usually, measurements of incoherent scattering of neutrons reveal only the main peaks. Theoreticians have produced mathematically accurate g() functions for lattice models in about 4 decades, and one of the most striking results of these calculations is the existence of kinks in g() or discontinuitues in dg/d. A direct comparison between Raman spectra and overtone density of states cannot be accomplished with great precision because the Raman scattering efﬁciency is the combination and overtone phonon density of states weighted by electron—phonon matrix elements and energy denominators (see, e.g., Cardona, 1975). For tetrahedrally coordinated semiconductors, these matrix elements in the second-order spectra involving phonon overtones dominate the scattering efﬁciency and, away from resonance, vary weakly with Raman frequency shift; thus, second-order spectra are in good measure of the overtone density of states. It must be kept in mind, however, that resonant intermediate states will generally distort the shape of Raman spectra away from of the density of states. Figure 7 shows frequency distribution functions g() calculated in the shell model for lithium hydride and lithium deuteride crystals (see, e.g., Plekhanov, 1997a). One can see that spectral distributions of phonons in LiH and LiD are virtually identical in the region of acoustic vibrations (450 cm\). At the same time, the optical band in LiH is broader by the factor of (2 than in LiD. This is explained by the fact that in a crystal lattice containing ions with substantially different masses, heavy ions primarily take part in acoustic vibrations, and light ions are involved in optical vibrations. In addition, there is a narrow gap in g() between 583 and 608 cm\ in LiH crystals, whereas in LiD, such a gap is absent. Note also that the frequency of a transverse optical phonon in LiH lies at the band boundary, where the density of states is very low. Figure 8 compares the frequency distribution function g() calculated within the framework of the shell model (Verble et al., 1968) with the function obtained in experiments on incoherent neutron scattering (Zemlianov et al., 1965). Note several interesting facts that follow from this comparison. For LiD crystals, the agreement is good, except for the region of LO phonons. However, in LiH,

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Vladimir G. Plekhanov

Fig. 7. Density of phonon states in (1) LiH and (2) LiD crystals. (After Plekhanov, 1997a.)

there is a great discrepancy between the theory and experimental data (Fig. 8). This can be due to several reasons. First, in the determination of the frequency distribution function by Verble et al. for some polarization vectors, a rough model of rigid ions was used that took into account the interaction between nearest neighbors. Second, incoherent scattering in LiH can contain a considerable contribution from multiphonon processes, which can severely distort the true frequency distribution function. This is obviously the main reason for the strong overstating of the phonon frequency in LiH crystals, especially in the region of LO phonons (see also Fig. 9). As noted, from the appropriate models for different materials it is relatively straightforward to compare the frequency distribution function g() for the normal modes of vibrations. As we can see (Figs. 10 and 11), the curves clearly display sharp critical points arising from regions of the dispersion curves having zero gradient. Many of these can be correlated with features of ( q ) for wave vectors along the principal symmetry directions, although some must result from non-symmetry-direction modes. The twophonon absorption spectra (Raman spectra) of semiconducting crystals (C, Ge, Si, -Sn) were compared with the density of the combined vibrational states in the paper of Tubino and co-workers (1972), where the comparison was shown to hold only concerning the peak position and not the intensity ratios, since the ‘‘joint density of states’’ has not been modiﬁed by the thermal factor and the frequency dependence of the coupling Hamiltonian has not been taken into account because of the intrinsic difﬁculty of a priori knowledge of the second-order dipole moment (see also Lax and Burstein, 1955). In studying the Raman scattering due to phonons, one is concerned with the changes in the electronic polarizability of the crystal due to the lattice vibrations (Loudon, 1964). For small oscillations, the polarizability tensor

3 Vibrational Properties

75

Fig. 8. Density of phonon states in (a) LiH and (b) LiD crystals. Dashed curves correspond to data on incoherent scattering of thermal neutrons. (After Zemlianov et al., 1965.)

can be expanded in terms of the ionic displacement about the equilibrium position of the ions. Since a cubic crystal (with NaCl structure; e.g., LiH) has inversion symmetry about every ion, the ﬁrst-order terms in the polarizability expansion are zero, and hence this crystal does not produce any one-phonon Raman scattering. Note that the diamondlike crystals (C, Si, Ge, etc.) do not have inversion symmetry and possess the ﬁrst-order Raman scattering. The second-order terms in the expansion give rise to the continuous two-phonon Raman scattering due to all phonon pairs, with wave vectors ;q and 9q being allowed by the symmetry of the crystal. The intensity of the Raman scattered light from the system of phonons is given by Born and Huang (1968): I( ) : G n I n I i ()E\ E>, (24) ? @ ?@AB A H D 2c I ?@AB with * v v P v% , (25) i () : # v P ?A ?@AB @? ?T TY

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Vladimir G. Plekhanov

Fig. 9. Raman spectra (A , E , T , and AET) at 78 K and a histogram of temperatureE E E weighted density of two-phonon states of a LiH crystal. (After Jaswal et al., 1974.)

where : 9 , with and being the frequencies of incident and G D G D scattered light, respectively; n and n are two mutually perpendicular unit vectors in a plane perpendicular to the direction of scattering; E\ and E> A H are the components of the amplitudes of the incident electric ﬁeld; v and v, respectively, refer to the initial and ﬁnal vibrational states of the system; c is the velocity of light; #% means thermal average over the initial states; P ?T ?@ is the electronic polarizability tensor; the subscripts , , and so on refer to the three axes of our coordinate system; and is the vibrational energy of T the state v of the system. In Eq. (25) actually should be but the small G D change in corresponding to the change in the vibrational energy of the G system is negligible. By choosing polarization of the exciting light parallel to one of the axes (x, y, z), one can avoid the summation in Eq. (25). Then the intensity of the scattered light is essentially determined by i (). ?@AB The polarizability tenzor is expanded in terms of the displacement of the ions about their equilibrium positions. The zeroth-order term gives rise to the Rayleigh scattering and the ﬁrst term is zero because of the inversion symmetry about every ion (excluding the crystals with diamond-like struc-

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77

Fig. 10. (a) Two-phonon and overtone density of states of diamond C and (b) second-order Raman spectrum of diamond C in the high-frequency region. The solid lines represent the calculated spectra, and the dashed lines are experimental spectra from Solin and Ramdas (1970). (After Windl et al., 1993.)

ture). When the Fourier displacements are analyzed in terms of the normal coordinates, the second-order term in the polarizability expansion becomes

9q q q 9q 1 : P P Q Q , ?@ 2 ?@ jj j j EHHY

(26)

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Vladimir G. Plekhanov

Fig. 11. (a) Two-phonon and overtone density of states of silicon and (b) second-order Raman spectrum of silicon. The solid lines represent calculated spectra; the dashed lines are experimental spectra from Temple and Hathaway (1973). (After Windl et al., 1993).

with

P ?@

9q q l 9l : P ?@AB jj k k JYIIY

l 9l ; (k q j)(k 9q j)(M M )\ exp 2iq a A I IY k k

,

(27)

3 Vibrational Properties

79

where Q( qj ) is the normal coordinate corresponding to wave vector q and branch j of the phonon dispersion curves, and

l 9l P ?@ P : ?@AB k k u (lk)u (lk)

A B

(28)

is the coefﬁcient of the second-order term in the polarizability expansion with u ( J ) being the component of the kth ion in the lth cell; (k q j) is ? I the eigenvector and M is the mass of the kth type of ion; a ( J \JY) is the I IY I position vector of ion (lk) with respect to ion (lk). When Eq. (26) is substituted in Eq. (25) and the thermal average is taken, one gets

i () : [ (q )HY (9q)]\ ?@AB 8 q jj H ;P ?A

9q q 9q q P f [ ( q )] · f [ (9q)] # < (9 q)% HY H HY @B jj jj (29)

where ( q ) is the phonon frequency, H f [ ( q )] : n[ ( q )] ; 1, H H for the creation of a phonon, and f [ ( q )] : n[ ( q )] H H when a phonon is destroyed with n[ ( q )] : [exp# ( q )/kT % 9 1]\ H H being the usual Bose factor; the ; and 9 signs in the function correspond to the destruction and creation of a phonon respectively. One can see from Eq. (29) that the second-order Raman spectrum is a weighted two-phonon DOS. For positive and negative values of , the spectra are called Stokes and anti-Stokes, respectively. In this book we consider only the Stokes spectra. If we assume the polarizability tensor to be symmetric, then the tensor i () has the same transformation properties as the elastic constant tensor ?@AB (see Chapter 1). Cubic LiH (LiD) crystals possess three different components of scattering tensor i : i ; i :i ; i :i . It is known from the group theory (see, e.g., Birman, 1974) that components P of the ?@ second rank tensor are transformed over three irreducible representations

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Vladimir G. Plekhanov

A , E , and T of the O group. In nonpolarized scattering, all three F E E E components are present with certain weights. However, by choosing the experimental geometry of scattering in a proper way, one can separate the E and T components of the scattering tensor. The intensities of these E E components are described by the expressions (see also Plekhanov, 1997a) I(A ) : E I(E ) : E

A ;A ;A VV WW XX ; (3

A ; A 9 2A A ;A VV WW XX ; VV WW (6 (2

(30)

A ; A ; A WX XV . I(T ) : VW E 3 In this case i () :

I(A ) ; I(E )/4 ; 3I(T ) E E E : AET, 3

I(E ) E :E , i () : E 4 i () : I(T ) : T , E E I(A ) ; I(E ) E E : AE. i () : 3

(31)

One can see from Eq. (31) that theoretical calculation of the Raman spectrum consists of the calculation of three tensor components i , i , and i . Such calculations were done in the paper of Jaswal et al. (1974). In this paper, the lattice dynamical model used for LiD and LiH was a deformation dipole model. This model is based on the neutron scattering data for LiD (Verble et al., 1968) and the experimental second-order Raman spectra of LiD. Jaswal and co-workers (1974) showed that if the contributions to polarizability from nearest and next-to-nearest-neighbor are taken into that enable one account, one can select the values of tensor components P ?A to reproduce Raman spectra with good accuracy for different polarizations (see also Laplaze, 1977). Figure 9 shows a comparison of experimental Raman spectra (A , T , E , and AET) recorded at liquid nitrogen temperaE E E ture (LNT) with a histogram of the density of two-phonon states of a LiH crystal, obtained by Jaswal et al. To improve agreement between the theory and experimental data, the interaction between next-to-nearest neighbors was taken into account. To obtain agreement with experimental data, 6 and 13 parameters were introduced for the T and E (or A ) components, E E E respectively. It was also noted (Jaswal et al., 1974) that high polarization of

81

3 Vibrational Properties

hydrogen (deuterium) ions requires the consideration of the ;-; interaction between next-to-nearest neighbors. Similar conclusions were made later by Dyck and Jex (1981), where good agreement was achieved between experimental Raman spectra and spectra calculated based on the model of deformable dipoles with thirteen parameters. However, although agreement between theory and experimental data in the region of 2LO() phonons (see also Plekhanov, 1997a) obtained by Dyck and Jex is better than in the paper of Jaswal et al., the values of frequencies at the L point calculated by Dyck and Jex are nevertheless in poor agreement with experimental values. Laplaze (1979) managed to substantially improve agreement between calculated and experimental elastic constants, but he failed to obtain agreement with experimental data in the region of the TO ; LO and LO() phonons in the Raman spectra of LiH crystals (for details, see Plekhanov, 1997a). The DOS of diamond and Si was calculated in papers by Pavone et al. (1994) and Windl et al. (1993). The eigenvectors and eigenvalues that are [Eq. (27)] have been calculated in the cited papers necessary to evaluate P ?@ using the local-density-approximation (LDA) plane-wave pseudopotential method. The results for the second-order Raman spectra for diamond C and silicon are shown together with experimental curves in Figs. 10 and 11. As shown in Table IV and Fig. 10, for diamond C, Pavone et al. found

TABLE III Critical Point Analysis of the Second-Order Raman Spectra of Silicon (After Windl et al., 1993)

Overtones and combinations 2O() 2L (X) 2TO(X) 2TA(X) L (X) ; TA(X) L (X) ; TO(X) TO(X) ; TA(X) 2O (W ) : 2A (W ) 2O (W ) 2A (W ) O (W ) ; O (W ) A (W ) ; A (W ) 2LO(L ) 2TO(L ) 2LA(L ) 2TA(L ) LO(L ) ; TO(L ) LA(L ) ; TA(L )

Active in >; > ; >; > ; >; > ; >; > ; > > > >; > ; >; > ; >; > ; > ; > >; > ; >; > >; > ; >; > >; > ; > ; > > ; >

> > > >

> > > > > >

Theory (cm\)

Experiment (cm\)

1034 828 932 290 559 880 611 715 945 409 830 677 838 988 755 220 912 488

1038 825 923 299 562 874 611 743 948 434 845 691 841 983 760 226 912 493

82

Vladimir G. Plekhanov TABLE IV Critical-Point Analysis of the Second-Order Raman Spectra of Diamond C (After Windl et al., 1993)

Overtones and combinations ‘‘Sharp peak’’ 2O() 2L (X) 2TO(X) 2TA(X) L (X) ; TA(X) L (X) ; TO(X) TO(X) ; TA(X) 2O (W ) : 2A (W ) 2O (W ) 2A (W ) O (W ) ; O (W ) A (W ) ; A (W ) O (W ) ; A (W ) 2LO(L ) 2TO(L ) 2LA(L ) 2TA(L ) LO(L ) ; TO(L ) LA(L ) ; TA(L )

Active in >; > ; >; > ; >; > ; >; > ; >; > ; > > > >; > ; >; > ; >; > ; > ; > > ; > >; > ; >; > >; > ; >; > >; > ; > ; > > ; >

> > > > >

> > > > > >

Theory (cm\)

Experiment (cm\)

2671 2646 2453 2186 1596 2024 2319 1891 2035 2375 1864 2205 1949 2119 2548 2459 2147 1117 2503 1632

2670 2667 2370 2138 1614 1992 2254 1864 1998 2358 1817 2177 1907 2178 2504 2422 2011 1126 2458 1569

(especially in comparison with previous works by Cowley, 1965; Go et al., 1975) very good agreement between calculation and experiment (peaks position generally deviate by less than 2%). From Fig. 10 it can also be seen that within the harmonic approximation by Pavone et al., the sharp peak at the frequency cutoff in all representations is well reproduced, as it should be from the symmetry considerations (Solin and Ramdas, 1970; Birman, 1974). The origin of this sharp peak is connected with the overbending of the LO branch dispersion (see earlier). A detailed analysis (Windl et al., 1993) of the force constants shows that an overbending of the LO branch in the phonon dispersion can be obtained only for signiﬁcantly large values of the force constants between second neighbors (see also Hass et al., 1992). To show that the origin of the sharp peak at the high-frequency cutoff is not an effect of the matrix elements, Windl et al. also ﬁtted the second-order Raman spectrum in its three representations for silicon. The results of these calculations are shown in Fig. 11. Again Pavone et al. ﬁnd good agreement between calculation and experiment for all representations. The positions of

3 Vibrational Properties

83

the calculated peaks with experimental values are generally within less than 2% (see also Table III). As expected from phonon dispersion (see Fig. 4) and the high-frequency density of states, there is no sharp peak at the highfrequency cutoff. Thus, these results provide strong evidence that the long-standing controversy seems to be resolved.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 4

Raman Spectra of Isotopically Mixed Crystals

I. Low Concentrations: Localized, Resonant, and Gap Modes II. High Concentrations: Mixed Crystals . . . . . . . . . . 1. Introductory Remarks . . . . . . . . . . . . . . . . . . 2. First-Order Raman Spectra . . . . . . . . . . . . . . 3. Second-Order Raman Spectra . . . . . . . . . . . . . . 4. Two-Mode Behavior of the L O Phonon: T he Case of L iH D V \V III. Disorder Effects in Raman Spectra of Mixed Crystals . . 1. Coherent-Potential-Approximation Formalism . . . . . . . 2. Disorder Shift and Broadening of the Raman Spectra . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

85 93 93 93 103 104 109 109 112

I. Low Concentrations: Localized, Resonant, and Gap Modes Studies of vibrational properties of crystals containing impurities (defects of various types) were described in detail in a number of reviews (Maradudin, 1966; Maradudin et al., 1971; Dean, 1972; Bell, 1972; Taylor, 1975; Barker and Sievers, 1975; Chang and Mitra, 1971; Elliott et al., 1974; Taylor, 1982; Lifshitz, 1987a). These studies are traditionally classiﬁed under two regions corresponding to low (x 10%) and high (x ( 10%) concentrations of defects (impurities). The latter region, which corresponds to the so-called mixed crystals, is considered in the next section. Here, we analyze the results obtained in the region of low concentrations, when the interaction between defects (impurities) can be neglected in the ﬁrst approximation (Barker and Sievers, 1975). The simplest defects in a crystalline lattice that distort its translational symmetry are isotopes of the elements forming a crystal. In most cases, the effect of isotopes on the vibrational spectrum is small and can be neglected in calculations of the dispersion curves. However, in the LiH—LiD system, the situation is different. The great change in a mass on replacement of H by D results in a shift of the entire optical vibrational band by :250 cm\. This means that even low concentrations of isotopes can produce substantial effects (Maradudin et al., 1972), especially in phenomena related to the motion of defects themselves (Lifshitz, 1987a; Andreev, 1976). In the case of low concentrations, the main effects consist 85

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Vladimir G. Plekhanov

of the appearance of various processes induced by defects. For example, in the harmonic approximation, the IR absorption spectrum of an ideal lattice consists of a single delta-shaped peak at a frequency of (), and the 2introduction of defects lifts the selection rule over the wave vector, which results in the appearance of additional absorption in the entire frequency region (with the intensity proportional to the concentration of defects). Lifshitz (1956) developed the theory for the case of ultimately low concentrations, which do not contain free parameters in the case of an isotopic defect. The method developed by Lifshitz is based on the technique of Green’s functions. This method is especially convenient, because it enables one to uniformly write the solution of the vibrational equation in the presence of point defects of different types. A change in the atomic mass results in the appearance of three features in the phonon spectrum (Lifshitz, 1956; Lifshitz and Kosevich, 1966): localized, gap, and quasi-localized (resonant) modes. Localized modes appear on a sufﬁciently strong decrease in the atomic (ion) mass, and their frequencies lie above the maximum frequency of the phonon spectrum. Frequencies of gap modes lie in the gap between optical and acoustic bands. These modes can appear on both decreasing and increasing atomic mass. Resonant modes appear if the defect mass greatly exceeds the mass of the substituted ion. These modes are manifested as sharp maxima of the frequency distribution function in the region. Frequencies of all these modes are determined by the

equation (Lifshitz, 1956) 1 : ! Re GQ (). Q

(1)

Here, ! : I 9 M /M (M is the mass of a defect in the sth sublattice), and Q Q Q Q 1

eQ(q) G () : 3N q j 9 ; i0 q j >

eQ(q) 1 1 : ; i eQ(q j ) ( 9 q j). 3N q j 3N q j [ 9 ] OH N

(2)

Here, P denotes the principal value. For lithium deuteride, a condition of the appearance of a localized mode upon replacement of D\ ions by H\ ions has the form M/M : 0.66 (Plekhanov, 1997a) (for the phonon spectrum in the shell model). Therefore, the introduction of H\ ions into a LiD crystal should result in the appearance of the localized mode. Figure 1 shows the second-order Raman spectrum of a pure LiD crystal at room temperature (Plekhanov and Veltri, 1991). Note ﬁrst that this spectrum agrees well with the spectra measured earlier (Jaswal et al., 1972; Anderson and Luty, 1983). Although according to a nomogram of excitonic states (Plekhanov, 1990d), this crystal is a pure LiD crystal, its Raman spectrum

4 Raman Spectra of Isotopically Mixed Crystals

87

Fig. 1. Second-order Raman spectrum of a LiD crystal excited at ! : 532 nm at room temperature. (After Plekhanov and Veltri, 1991.)

nevertheless exhibits a high-frequency peak at 1850 cm\. This peak has no analog in the Raman spectra of a pure LiH crystal. It was observed earlier in second-order Raman spectra (Wolfram et al., 1972; Anderson and Luty, 1983) and was interpreted (Wolfram et al., 1972) as the localized vibration of the hydrogen ion, in LiD crystals. The calculated value of lies between 917 (Wolfram et al., 1972), 948 cm\ (Elliott and Taylor, 1967), and 995 cm\ (Jaswal and Hardy, 1968). The frequency : 1850 cm\ of the peak observed in Raman spectra (Wolfram et al., 1972; Plekhanov and Veltri, 1991) is in good agreement with the double value of the most consistently calculated frequency ( : 917 cm\). Although the concen tration of H\ in LiD crystals under study is very low (according to Plekhanov, 1990d, the position of the n : lS line in the reﬂection and luminescence spectra corresponds to a pure LiD crystal), it is nevertheless sufﬁcient for observation of the localized mode in the second-order Raman spectra (Fig. 1). The intensity of the peak of the localized mode increases with increasing concentration of hydrogen ions (the concentration was estimated by Plekhanov and Veltri from the position of the ground-state level of excitons in the reﬂection spectra), and it shifts slightly (Plekhanov, 1997a) to the blue (Fig. 2). This behavior is retained up to a concentration of hydrogen ions x10%. Note here that the study of the concentration dependence of the shape of this peak enables one to examine the percolation threshold, which separates the local states from continuous phonon states (Anderson, 1955; Kirkpatrick, 1973; Plekhanov, 1995b). Figure 3 shows the dependence of frequency of the localized mode on the concentration of hydrogen ions obtained from the Raman spectra presented in Figs. 1 and 2 and the spectra of other crystals (Plekhanov and Veltri, 1991; Plekhanov, 1995b). Figure 3 also shows the theoretical dependence : f (x) cal culated in literature (Elliott and Taylor, 1967; Jaswal and Hardy, 1968; Wolfram et al., 1972). In Elliott and Taylor (1967), the concentration dependences of the frequency and intensity of the localized mode in LiH D were calculated V \V

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Vladimir G. Plekhanov

Fig. 2. Second-order Raman spectra of LiH D crystals at room temperature: x : 0 (1), V \V 0.05 (2), 0.12 (3), and 0.35 (4). (After Plekhanov, 1997a.)

by means of Green’s thermal function. These authors used the model phonon spectrum consisting of two Gaussian bands and bounded by the region. This approximation turned out not to be quite *2adequate, as was shown by Verble et al. (1968). Jaswal and Hardy repeated calculations of the concentration dependence of the frequency and intensity of the localized mode using the model of deformable dipoles and the real phonon spectrum of a LiD crystal. As mentioned, the values of calculated in these two papers are close to each other, but somewhat overestimated compared to the reﬁned calculation performed by Wolfram et al. (1972), where : 917 cm\. More serious discrepancies between the : f (x) dependences were obtained by Elliott and Taylor and Jaswal and Hardy, because, according to Elliott and Taylor, decreases with increasing x, whereas, according to Jaswal and Hardy, increases with increasing x. Despite this discrepancy, note that concentration dependences of the intensity of the localized mode found in these papers were in close agreement (see also Elliott et al., 1974). The experimental concentration

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Fig. 3. Dependence of the localized mode frequency on the concentration of hydrogen ions. Data from Elliott and Taylor (1967) (1); Jaswal and Hardy (1968) (2); Wolfram et al., (1972) (3); and Anderson and Luty (1983) (4). Experimental data from Plekhanov and Veltri (1991) and Plekhanov (1995b).

dependence of the intensity of the local mode (Plekhanov and Veltri, 1991) qualitatively agrees with the calculations performed by Elliott and Taylor as well as Jaswal and Hardy. In addition, experimental data (Plekhanov and Veltri, 1991) agree better with calculations based on the model of deformable dipoles (Jaswal and Hardy, 1968) than with calculations performed using Green’s functions (Elliott and Taylor, 1967). The experimental dependence : f (x) is almost parallel to the theoretical dependence (Jaswal and Hardy, 1968), being however shifted to lower frequencies (Fig. 3). Note that this dependence agrees with the value calculated for x : 0 in the paper of Wolfram et al. (1972). These data show that the model of deformable dipoles quite adequately describes the behavior of the localized mode, which is related to vibrations of hydrogen ions in lithium deuteride crystals. On introduction of deuterium ions into a LiH crystal, the gap mode appears, whose frequency in the model of deformable dipoles is : 576 E cm\ (Jaswal and Hardy, 1968). The shell model yields almost the same value of . One can see from Table I that this frequency is close to the E IR-active frequency in LiH. For this reason, it is unlikely that the gap 2mode can be observed at very low concentrations of deuterium ions. However, the gap mode has a number of interesting properties that should be manifested with increasing concentration of deuterium ions. At low concentrations of defects (impurity deuterium ions), the induced IR absorption should appear at the renormalized frequency (Maradudin, 1966)

M 2> : 1 9 c , E E " M ; M 9 E > \ 2-

(3)

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Vladimir G. Plekhanov TABLE I

Frequencies of Optical and Acoustic Phonons (in cm\) Calculated at Different Points of the Brillouin Zone (Verble et al., 1968; Dyck and Jex, 1981; Plekhanov, 1997a) LiH Substances phonon branches LO TO LA TA

LiD

X

W

L

K

X

W

L

K

1085 608 0 0

944 825 399 291

889 847 366 336

968 749 580 248

913 808 387 279

880 444 0 0

672 589 285 285

633 604 338 338

689 535 251 251

951 578 278 278

where c is the concentration of deuterium ions, rather than at the frequency " . Expression (3) shows that the peak in the induced IR spectrum should E be strongly shifted, even in the case of very low c , because the frequency " of the gap mode is close to in LiH. The effect of the isotopic 2composition on the IR absorption spectra of lithium hydride was ﬁrst studied by Montgomery and Yeung (1962). These authors measured IR spectra of thin ﬁlms made of materials containing pure lithium or hydrogen isotopes (LiH, LiD, LiH, and LiD) or of mixed materials in a broad range of concentrations (LiH D , 0 x 1). The transmission spectrum V \V of ﬁlms of pure isotopic composition consists of a single broadband whose maximum corresponds to the TO phonon frequency. According to the results of the paper of Montgomery and Yeung (1962), as the concentration of deuterium in LiH increases, the frequency of the IR-active phonon ﬁrst changes slightly (584 589 cm\) at c 0.03%, but at c : 5%, it " " sharply decreases to the value that is typical for a pure LiD crystal and then changes insigniﬁcantly with increasing c (Fig. 4). Such a behavior of the IR " absorption has still not been adequately explained (Behera and Tripathi, 1974), despite a number of attempts (Elliott et al., 1974; Elliott and Taylor, 1967; Jaswal and Hardy, 1968; Maradudin et al., 1971; Montgomery and Hardy, 1965). For low c , it is likely that the IR peak consists of two peaks " [which are probably not resolved (Plekhanov, 1994a)] related to the transverse optical and gap modes. As the concentration increases, the frequency of the gap mode rapidly decreases, according to Eq. (3), and its relative intensity increases. However, this shift should occur smoothly, which contradicts the experimental data (Fig. 4). This problem has attracted attention for many years. As mentioned above, all attempts to describe experimental results (Mongomery and Yeung, 1962) within the framework of the isotope effect (Lifshitz, 1987a) have failed. In this connection, the remark of Maradudin (Maradudin, 1966) that the replacement of the hydrogen ion by the deuterium ion, which is smaller in size, requires

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Fig. 4. Concentration dependence of the infrared dispersion frequency obtained from transmission spectra of thin LiH D ﬁlms at room temperature (according to Montgomery and V \V Yeung, 1962). The dashed curve corresponds to the dependence : \, where is an average mass of the unit cell (after Montgomery and Hardy, 1965).

consideration not only of the change in a mass (isotope effect) but also of the change in the force constant in calculations of U centers in alkali-halide crystals (Barkers and Sievers, 1975) deserves special attention. Taking into account the decrease in the force constant upon replacement of H by D results in a substantial improvement of agreement between the theory and experimental data for U centers in alkali-halide crystals (Barker and Sievers, 1975). Later, Taylor (1975, 1982) considered the change in the force constant of the isotopic defect in calculations of localized modes of isotopic impurities in semiconductors. The consideration of a change in the force constant in the calculation of the localized mode on introducing D into LiH allowed Behera and Tripathi (1974) to achieve agreement with experimental data (Montgomery and Yeung, 1962) in the region of low concentrations (c 0.1). For this purpose, the renormalization of was performed 2" according to the equation

C 4!" : 1 ; , 222 [" 9 !]

(4)

where ! : (M 9 M/M) is the mass defect parameter, and " : (' 9 ' )/ ' : '/' is the parameter describing the change in the force constant between the defect and lattice atoms. The agreement between the theory and

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Fig. 5. The calculated concentration dependence (full lines) of , , and together with * 2- E the experimental data (chain lines) for LiH—LiD systems. (After Behera and Tripathi, 1974.)

observation in the case of low c (LiH : D) was obtained for ! : 0.5 and " " : 0.21 (Fig. 5), while in the case of LiH : D, these parameters were 1.0 and 0.21, respectively. This shows that an actual change in the force constant on replacement of the hydrogen ion by the deuterium ion, which is smaller in size, is quite large. Note, in conclusion, that although the ﬁrst comprehensive papers on lattice dynamics of isotopic mixed semiconductor germanium (Agekyan et al., 1989; Fuchs et al., 1991a, 1991b, 1992, 1993a, 1993b; Cardona et al., 1992, 1993; Etchegoin et al., 1993; Spitzer et al., 1994; Cardona, 1994; Zhang et al., 1998) and diamond (Chrenko, 1988; Hass et al., 1991, 1992; Anthony and Banholzer, 1992; Ramdas et al., 1993; Spitzer et al., 1993; Muinov et al., 1994; Haller, 1995; Hanzawa et al., 1996; Vogelgesang et al., 1996; Zoubolis et al., 1998; Ruf et al., 1998) crystals have already appeared, lattice dynamics in the region of low concentrations has not been investigated in these papers. In addition, the absence of quasilocalized (localized) modes in Raman spectra of isotopically mixed germanium crystals was explained by a weak phonon scattering potential (Etchegoin et al., 1993), which enabled Etchegoin et al., (1993) and Hass et al., (1992) to use the CPA (Elliott et al., 1974) of lattice dynamics for the description of their experimental results.

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93

II. High Concentrations: Mixed Crystals 1. Introductory Remarks If the impurity concentration in a crystal is high enough that the interaction between impurity atoms (ions) plays an important role, such a system is called a mixed crystal with a various degree of disorder. There are two types of disordered systems: disordered alloys (isotopic mixtures) and mixed crystals and glassy substances, which possess a more pronounced spatial disorder than conﬁgurational disorder. The ﬁrst theoretical dynamic model of mixed crystals was a linear chain, which represented the development of the virtual model (Nordheim, 1931; Pant and Joshi, 1969). Despite its simplicity, this model adequately described general features of lattice dynamics of mixed alkali-halide crystals. This model uses two independent force constants f and f , which are obtained, as a rule, from the observed frequencies of LO phonons in pure substances, according to the expression f : mM/2(m ; M), where m and M(M) are masses of crystal-forming particles. The dependence of the force constant on concentration was described by equation F : f x 9 ( f 9 f )x by assuming a linear depend ence of f ( f ) on concentration x (see also Chang and Mitra, 1971). A more complex concentration dependence of the force constant was considered in detail in comprehensive reviews (Barker and Sievers, 1975; Elliott et al., 1974; Ipatova, 1988; Taylor, 1988), where the cluster model and isodisplacement model in lattice dynamics, based on the CPA (Taylor, 1967; Ehrenreich and Schwartz, 1976) or averaging of the T matrix (Taylor, 1975; Taylor, 1982), were also described (see also later).

2. First-Order Raman Spectra Elemental semiconductors (C, Si, Ge, -Sn) with diamondlike structures are ideal objects for the study of the isotopic effects by the Raman scattering method. High-quality isotopically enriched indicated crystals are also available. In this section we describe our understanding of the ﬁrst-order Raman spectra of the isotope-mixed elemental and compound semiconductors (CuCl, GaN, GaAs) with the zinc-blende structure. The materials with diamond structures discussed here have a set of threefold-degenerate phonons (frequency ) at the center (k : 0, point) of the Brillouin zone (BZ) (see also Chapter 3). These phonons are Raman active but infrared inactive (Lax and Burstein, 1955). Let us consider the case of Ge, with the ﬁve isotopes (Agekyan et al., 1989; Fuchs et al., 1991a). The uninitiated will ask whether one should see ﬁve phonons [or more if he or she knows that there are two atoms per primitive cell (PC)]

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Vladimir G. Plekhanov

Fig. 6. First-order Raman spectra of the L phonon of several Ge isotopes and natural Ge at 80 K. (After Fuchs et al., 1992.)

corresponding to the ﬁve different masses, or only one corresponding to the average mass. The reason why the Raman spectrum (see Fig. 6) of natural Ge does not show the local modes of the individual isotopes is that the scattering potentials for the phonons due to the mass defects (mass ﬂuctuations) are too small to induce bound states; that is, Anderson localization of the phonons (Anderson, 1955). Really, in a three-dimensional crystal ﬂuctuations in the parameters of the secular equation lead to localization if these ﬂuctuations [measured in units of frequency, i.e. (M/M) ] are larger than the bandwidth of the corresponding excitations. For optical phonons in Ge this bandwidth is :100 cm\ (see, e.g., Etchegoin et al., 1993) while (M/M) 0.4 ; 300 : 12 cm\. Hence no phonon localization (with lines corresponding to all pairs of masses) is expected, in agreement with the observation of only one line at 304 cm\ (:80 K) for natural Ge. Figure 6 shows superimposed Raman lines of natural and of isotopically enriched Ge single crystals recorded near the LNT (Fuchs et al., 1991a). The lines in these spectra are fully resolved instrumentally [the experimental resolution was better than 0.1 cm\ (Cardona et al., 1993)] and their width is caused by homogeneous broadening. The centroid of the Raman line shifts follow relation : M\. This behavior is expected within har monic approximation. Additional frequency shifts are observed (Zhang et al., 1998) for the natural and alloy samples, which arise from their isotope mass disorder. This additional shift is 0.34 <0.04 cm\ in natural Ge, and 1.06<0.04 cm\ (Fig. 7) in the 70/76Ge alloy sample, which has nearly the maximum isotopic disorder possible with natural isotopes. As is well known, natural diamond exhibits a single ﬁrst-order Raman peak at () : 1332.5 cm\. Figure 8 shows the ﬁrst-order Stokes *2Raman spectra for several samples with different isotope ratios (Hanzawa et al., 1996). The Raman energy is found to increase continuously, but nonlinearly, with decreasing x. The energy difference between the extreme

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95

Fig. 7. Raman frequency as a function of the average mass, measured at 10 K, for isotopically enriched and disordered Ge samples. The solid line is a calculation with : 2595.73/(M cm\. (After Zhang et al., 1998.)

Fig. 8. First-order Raman spectra of CC diamonds with different isotope composiV \V tions. Labels A, B, C, D, E, and F correspond to x : 0.989, 0.90, 0.60, 0.50, 0.30, and 0.01, respectively. The intensity is normalized at each peak. (After Hanzawa et al., 1996.)

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compositions is 52.3 cm\, which is consistent with the isotope mass ratio. Additional Raman data on isotopically mixed polycrystalline diamond ﬁlms (Chu et al., 1990) are not discussed here because of the numerous extrinsic factors (e.g., stress, impurities) (Knight and White, 1989) that can obscure the intrinsic composition dependence in such ﬁlms. For example, the linear Raman variation assumed by Chu et al. suggests that the x values determined in that work may be in error by as much as 0.1 (for details see Hass et al., 1992). The ﬁrst-order Raman spectra of -Sn, as with the other semiconductors with diamond—like structures, consists of a lone LO line (D. T. Wang et al., 1997). The dependence of the Raman frequency on the isotope concen tration in -Sn is depicted in Figure 9. The Raman frequencies of the samples with low mass variance follow an :M\ dependence with small deviations of less than a tenth of a wave number. These deviations are, however, consistent in several series of measurements by Wang et al., and can be explained qualitatively by the mass dependence of the anharmonic contribution (self-energy) to the Raman frequency. The Raman frequencies of the samples with large mass variance are considerably higher than expected from the :M\ dependence of the average mass. The deviation

Fig. 9. Raman frequency as a function of the average mass, measured at :10 K using laser wavelength of 676 and 647 nm for ﬁve thin -Sn ﬁlms on InSb 001 with different Sn isotope compositions: pure Sn, Sn, Sn, and Sn and Sn (50/50) mixtures. Measurements carried out on a SPEX double monochromator are marked MPV; those carried out on the DILOR, marked MP111. (After D. T. Wang et al., 1997.)

4 Raman Spectra of Isotopically Mixed Crystals

97

is about 0.7 cm\ for the 112/116Sn sample and 1.8 for the 112/124Sn sample. These shifts (see later) can be attributed to isotopic disorder. In addition to the elemental semiconductors in Gobel et al. (1997) and Zhang and co-workers (1997b), the Raman spectra of compound semiconductors have been studied. In contrast to elemental semiconductors, the changes in phonon frequencies and atomic displacements in compounds strongly depend on the phonon branch and wave vector q when atoms of the compound constituents are isotopically substituted. Since the acoustic and optic branches of the phonon dispersion are affected differently by the substitution of heavy or light atoms, their energies can be tuned almost independently (see also Plekhanov and Altukhov, 1983), in particular at the X point. This enables one to probe the efﬁciency of the anharmonic decay of the point optical phonons into lower-lying acoustic bands. One of the prominent features of CuCl is its anomalous Raman spectrum even at low temperature (2 K). Besides being a unanimously accepted LO phonon (:209 cm\) it exhibits a broad structure with at least two peaks (145—175 cm\) in the TO region (Garro et al., 1996a; Gobel et al., 1996a, 1996b, 1997; Park and Chadi, 1996) instead of the single TO phonon predicted by group theory for zinc-blende compounds (Figs. 10—12). Two interpretations of this anomaly have been offered: on the one hand, Krautzman and co-workers (1974) modeled the anomalous TO spectrum of CuCl by a Fermi resonance in which the optical phonon is repelled on a two-phonon contribution band. Subsequent detailed lattice dynamical calculations by Kanelis and co-workers (1986) showed that third-order anharmonic coupling is sufﬁcient to account for both peaks of the TO phonon structure. On the other hand, a model in which a substantial fraction of the Cu atoms are located an nonideal sites, giving rise to additional vibrational modes, has also been proposed. In this model, the large anharmonicity is held for secondary minima in the lattice potential, which gives rise to Cu displacement from the standard zinc-blende sites (for details see Park and Chadi, 1996, and references therein). Figure 10 shows the LO() Raman spectra of isotopically modiﬁed CuCl. The reduced mass of the samples increases from the bottom to the top spectrum. Accordingly, the LO() phonon frequency (:209 cm\) decreases from the bottom to the top. The peak intensities are normalized to 1. The LO() frequency as a function of the reduced mass is shown in Fig. 11. The error bars indicate the mean square deviation as determined from several measurements on the respective sample. The dashed line corresponds to the reduced mass behavior with respect to the experimental frequency of the natural compound. The agreement with the prediction for an optical point phonon is excellent. Gobel and co-workers determined by ﬁtting the spectra to a Voigt *-A proﬁle, implicitly assuming a Lorentzian line shape for LO( ) (see also Menendez and Cardona, 1984).

98

Vladimir G. Plekhanov

Fig. 10. Raman spectra of the LO phonon of isotopically modiﬁed CuCl at 2 K. The reduced mass increases from the bottom to the top; LO() shifts according to :\, as expected for a zone center phonon. (After Gobel et al., 1997.)

The TO structure has two main features: a very narrow line at :173 cm\, usually denoted as TO(), and a broad maximum around :155.5 cm\, labeled TO() (Fig. 12). The spectra of Fig. 12 are normalized with respect to the TO() and were taken during the same scan as LO() the spectra in Fig. 10. Note that the spectra in Fig. 12 are displayed in a different order as compared to Fig. 10. The vertical bars indicate the position of TO() as determined by averaging the results of several measurements. The Raman shifts of TO() versus copper mass of the respective compound are shown in Fig. 13. TO() shifts by 2.3 < 0.2 cm\ when Cu is replaced by Cu in the samples containing natural chlorine; that is, a much greater shift than 1.0 cm\ is expected from the corresponding change in the reduced mass. Samples of the same copper mass have similar TO() frequencies and the shifts observed for chlorine substitution are much

4 Raman Spectra of Isotopically Mixed Crystals

99

Fig. 11. Raman frequencies of the LO() phonon at 2 K. The data correspond to different samples with isotope mixtures. The two data points for natural CuCl stem from samples of independent origin. The solid line represents a reduced mass behavior :\. (After Gobel et al., 1997.)

smaller than expected from their respective reduced masses (for details see Gobel et al., 1997). The TO() peak also shifts as a result of isotope substitution. The detection of small changes in its Raman shift, however, is obscured by its considerable width and small intensity for some of the compounds (see Fig. 14). As was shown by Gobel et al. the Raman spectra of CuCl with different isotopic composition at 2 K is in good agreement with the Fermi resonance model. In this scheme, anharmonic interactions, in which a zone center optic phonon decays into two acoustic phonons of opposite wave vector, renormalize the mode frequencies and alter the Raman line shape (Figs. 10 and 12). In CuCl, the unrenormalized TO zone center phonon interacts with a two-phonon combination band that is strongly peaked. Consequently, the TO phonon is partly pushed out of the two-phonon continuum, which results in the TO() peak. Its position is closely tied to the edge of the two-phonon states. For samples having the natural copper abundance, the

100

Vladimir G. Plekhanov

Fig. 12. Raman spectra of the anomalous TO structure for several isotopically modiﬁed CuCl samples. The spectra of samples with the same copper mass are grouped together. Within these groups the reduced mass decreases from the bottom to the top. The TO() line does not shift according to changes in , but with changes of the copper mass. The broad TO() line shifts according to changes in . (After Gobel et al., 1997).

linewidth of the TO() peak increases with the chlorine mass. The changes in the TO() frequency are reduced-mass-like, a fact that is also expected within the Fermi resonance model. For unaltered copper composition, the ratio of the peak intensities of TO() to TO() decreases with increasing chlorine mass. The unrenormalized LO phonon interacts with an almost ﬂat two-phonon density of states. As a result, it shifts according to the expected reduced mass behavior of a zone center phonon. The linewidth of LO() comprises anharmonic as well as isotope-disorder-induced broadening. In contrast, off-center models do not account for the Raman data. The quantitative description of an earlier model by Livescu and Brafman (1986) implies two reduced-mass-like oscillators, which does not explain the

4 Raman Spectra of Isotopically Mixed Crystals

101

Fig. 13. Raman frequencies of the TO() peak as a function of the copper mass (T : 2 K, dashed line M\ scaled to the frequency in Cu Cl): O, measured values, X, calculated ! values, as discussed in text. For clarity, the theoretical values have been shifted by 90.2 amu. The error bars represent the mean square deviation of several measurements. (After Gobel et al., 1997.)

different mass dependences of TO() and TO() (for details see Gobel et al., 1997). Similar to CdS and ZnO, hexagonal GaN has a wurtzite structure, possessing two formula units per primitive unit cell. Group theory predicts the following point optical phonon modes for the wurtzite structure (point group C , Birman, 1974): A ; E ; 2E ; 2B , which are all Raman T active except for the silent B modes (Zhang et al., 1997). Figures 15a to 15c show Raman spectra of GaN, GaN N , and GaN, respectively, measured at 10 K in the frequency region between 460 and 780 cm\. All Raman-active phonons for the two modes in the alloy sample, which has a high carrier concentration resulting in strong phonon—plasmon coupling, are observed in the scattering geometries given by the selection rules. Several E and A modes from sapphire substrate are also observed. These lines are E E quite narrow and were used for calibration purposes. All Raman peaks were

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Fig. 14. Raman frequencies of the TO() peak versus the reduced mass (T : 2 K, dashed line :\). (After Gobel et al., 1997.)

Fig. 15. Raman spectra of GaN (a), GaN N (b), and GaN (c), respectively. The spectra were taken at 10 K with 514.5-nm excitation using the scattering geometries indicated in the ﬁgure. Peaks marked by asterisks are phonon from sapphire substrates. (After Zhang et al., 1997a.)

4 Raman Spectra of Isotopically Mixed Crystals

103

ﬁtted with Lorentzian lineshapes. As was shown by Zhang et al. (1997b), the Raman frequencies (except the low-frequency E mode) had nonlinear dependence on isotope concentration in GaN. The alloy sample with large N-mass ﬂuctuation has Raman frequencies that are lower than those expected from its average mass. These additional frequency shifts are 93.4 < 0.7 cm\, 93.1%< 0.7 cm\, and 92.5 < 0.7 cm\ for A (TO), E (TO), and the high-frequency E mode, respectively. The already cited authors attribute these shifts to a mass disorder effect (for details see Zhang et al., 1997b). Thus, Figs. 6—15 show experimental results that testify to the nonlinear dependence Raman frequency shift on the isotope concentration.

3. Second-Order Raman Spectra The second-order Raman spectra for natural and isotope-mixed crystals of diamond were investigated by Chrenko (1988) and Hass and co-workers (1992). Second-order Raman spectra for the synthetic diamonds are shown in Fig. 16. The second-order spectra were measured by Hass and co-workers with slightly lower resolution (:4 cm\ ) than the ﬁrst-order spectra because of the much lower count rate. The results of Hass et al. for 1.1 at. % C agree well with previous measurements for natural diamond (Solin and Ramdas, 1970). The spectra for 0.07 and 99 at. % C also look similar, if one ignores the shifts that occur as a result of differences in M. More signiﬁcant differences are observed for the more heavily mixed crystals: the 34.4 and 65.7 at. % C results are noticeably broader and do not appear to exhibit the sharp peak near the high-frequency cutoff. As already shown, this peak at the top of the second-order spectrum (2667 cm\ for 1.1 at. % C) has been the subject of intense controversy. Chrenko (1988) also examined the second-order spectra of his samples and claims that he was able to see this peak at all compositions except 68 at. % C. His measurements may have been of somewhat higher resolution than those of Hass et al., but it is clear that even in his 89 at. % C spectrum (which is the only raw data presented), some broadening of this peak has occurred. The second-order phonon Raman spectrum (Fuchs et al., 1991a) and IR transmission spectra (Fuchs et al., 1992) of natural Ge and isotopically enriched Ge were taken at 100 K. In Fig. 17 a comparison is made between the bond—charge model overtone density of states of Ge and natural Ge, the latter obtained with (CPA) corrections. We can see that the dominant effects of isotopic shifts and broadening take place in the highfrequency optical phonons. Energies and broadenings for phonons away from the zone center (q"0) were obtained from acoustic and optical two-phonon spectra (Figs 18 and 19). As opposed to the phonons at the point in the ﬁrst-order spectra, the 2TO(W ) and 2TO(L ) phonon peaks are broader in natural Ge than in Ge.

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Fig. 16. Second-order Raman spectra for synthetic diamond with indicated compositions at room temperature. (After Hass et al., 1992.)

4. Two-Mode Behavior of the LO Phonon: The Case of LiH D V \V The IR absorption in mixed crystals can change in two ways, depending on the concentration: one-mode and two-mode behavior (see, for example, Elliott et al., 1974). In the case of one-mode behavior, the spectrum always exhibits a single band whose maximum gradually shifts from one extreme

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105

Fig. 17. Calculated overtone phonon density of states of Ge compared with natural Ge. The natural Ge curve takes into account multiple-scattering corrections within the CPA. The density of states has been smoothed to eliminate some numerical noise, however the plot gives accurate shifts and broadenings. (After Fuchs et al., 1991a.)

Fig. 18. Second-order acoustic overtone and combination Raman spectrum of isotopically enriched Ge (upper curve) compared to natural Ge (lower curve). The observed peak positions are listed in Table II (p. 114). Data were taken at 80 K with a laser excitation energy of E : 2.183 eV. (After Fuchs et al., 1991a.) *

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Fig. 19. Second-order optical Raman spectrum of isotopically enriched Ge (upper curve) and natural Ge (lower curve). The observed peak positions are listed in Table II and the arrows indicate the linewidth discussed in the text. Data were taken at 80 K with laser excitation energy of E : 2.183 eV. (After Fuchs et al., 1991a.) *

position to another. The two-mode behavior corresponds to the presence of two bands in the spectrum, which are characteristic for each of the components of a mixed crystal. As the concentration of components changes, these bands shift, and their intensities undergo a strong redistribution. In principle, the same system can exhibit different types of behavior at the opposite ends (Barker and Sievers, 1975). This classiﬁcation is only a qualitative one, and it is seldom realized in its pure form (Chang and Mitra, 1971). The appearance of the localized mode in the limit of the isolated defect is considered the most important necessary condition for the twomode behavior of phonons (and also electrons; Ipatova, 1988). In Elliott and co-workers (1974), a simple quantitative criterion was suggested for determining the type of behavior of the IR absorption in a crystal of the NaCI type (see also Ipatova, 1988). Because the square of the frequency of the TO() phonon is inversely proportional to the reduced mass of the unit cell M, the shift caused by the defect is equal to /M ). : (1 9 M (5) 2This shift is compared in Elliott et al., (1974) with the width of the phonon optical zone. This width in the parabolic dispersion approximation, neglecting the acoustic branches, is 9 . W : 2- ;

(6)

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107

The localized or gap mode appears, provided W /2. However, as was noted by Elliott et al. (1974), in order for the two peaks to be retained up to a concentration of about 0.5, the stricter condition of W should be satisﬁed. The substitution of numerical values into Eqs. (5) and (6) shows that the relation

W

(7)

for LiH (LiD) is always valid, because : 0.44 and W : 0.58 . This 22means that the localized mode should be observed at low concentrations. This conclusion agrees with the experimental data described earlier (Fig. 1). As for the second theoretical relation W, as noted, for LiH (LiD) crystals, the reverse relation W is always valid (Plekhanov, 1993). We consider this question in more detail after a discussion of the Raman spectra of mixed LiH D crystals at high isotope concentrations. Figure 20 V \V shows the second-order Raman spectra of mixed LiH D crystals at V \V room temperature (Plekhanov, 1995b). Along with the properties of Raman spectra at high concentrations discussed in Plekhanov (1997a), note also that as the hydrogen concentration further increases (x 0.15), the intensity of the 2LO() phonon peak in a LiD crystal decreases, while the intensity of the highest frequency peak in mixed LiH D crystals increases. The V \V latter peak is related to the renormalized LO() modes in a mixed crystal (Plekhanov, 1993). Thus, comparison of Raman spectra l and 2 in Fig. 20 shows that in the concentration range of 0.1 x 0.45, the Raman spectrum exhibits LO() phonon peaks of a pure LiD and mixed LiH D V \V crystal. A further increase in x 0.45 is accompanied by two effects observed in the Raman spectra of mixed crystals. The ﬁrst effect is manifested in a substantial rearrangement of the acoustooptical part of the spectrum (spectra 1—3 in Fig. 20), and the second one consists in a further blue shift of the highest frequency LO() phonon peak. This peak shifts up to the position of peak 12 in the spectrum of a pure LiH crystal (Plekhanov, 1997a). This is most clearly seen from comparison of spectra 2 and 4 in Fig. 20. Note that the resonance increase in the intensity of the highest frequency peak is observed in all mixed crystals for x 0.15 (Plekhanov, 1988). The dependence of the frequency of the highest-frequency peak in the Raman spectra of pure and mixed crystals on the concentration of isotopes is presented in Fig. 21 (curve 1). Figure 21 also shows the concentration dependence of the frequency of the TO() phonon measured from IR absorption spectra of thin LiH D ﬁlms (Montgomery and Hardy, 1965) V \V (curve 2). One can see that both these dependences are nonlinear. Curve 2 was already discussed in the analysis of the concentration dependence of the frequency and intensity of the localized mode. Note here that the concentration dependence in a broad range has not yet been self-consistently described (Tripathi and Behera, 1974). Note once more that the theory is in

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Vladimir G. Plekhanov

Fig. 20. Second-order Raman spectra of mixed LiH D crystals excited at ! : 448.0 nm at V \V room temperature: spectra (1), (2), (3) and (4) for x : 0.0, 0.42, 0.76, and 1, respectively. The arrows show the bands corresponding to LO() phonons. (After Plekhanov, 1995b.)

good agreement with experimental data in the region of low concentrations of isotopes, which is not the case for high concentrations. One can see from Fig. 21 that for concentrations of x 0.45, the LO() phonons exhibit two-mode behavior. This contradicts the prediction of the CPA (Elliott et al., 1974; Elliott and Leath, 1975), according to which the width W of the optical vibrations band should be smaller than the frequency shift () of the transverse optical phonon. However, calculations according to Eqs. (6) and (7) show that in LiH (LiD) crystals, the reverse inequality takes place; that is, W . According to Plekhanov (1995b), this discrepancy between experimental results and the theory based on the CPA (Elliott et al., 1974) is mainly explained by the strong potential of scattering of phonons, caused by a large change in the mass on substitution of deuterium for hydrogen. This is also conﬁrmed by the observation of the localized mode in these

4 Raman Spectra of Isotopically Mixed Crystals

109

Fig. 21. Dependence of (a) the half width and (b) the frequency of optical phonons on the isotope concentration in the second-order Raman spectra of mixed LiH D crystals. Curves V \V (1) and (2) in (a) ! 253.7 and 488.0 nm, respectively. In (b) curves (1) (Plekhanov, 1995a) and (2) data on IR absorption spectra for TO( ) phonons in thin LiH D ﬁlms V \V (Montgomery and Hardy, 1965).

systems. The latter fact is consistent with the results obtained in Fuchs et al. (1991a) and Etchegoin et al. (1993), where it was stated that the observation of the localized mode directly indicates a strong scattering potential of phonons on isotopic substitution. For this reason, experimental results obtained for mixed LiH D crystals cannot be described in the CPA in V \V the case of weak scattering of phonons.

III. Disorder Effects in Raman Spectra of Mixed Crystals 1. Coherent-Potential-Approximation Formalism In this section, two methods that have been employed to evaluate phonon energy shifts and broadening due to isotopic disorder are described. The ﬁrst is the coherent potential approximation and the second is lowest-order

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Vladimir G. Plekhanov

perturbation theory. In tetrahedrally coordinated semiconductors, the second-order vibrational Raman spectrum for parallel incident and scattered photon polarization closely follows the phonon overtone density of states. Therefore the ﬁrst approach is to evaluate the effects of isotopic disorder on the overtone density of states by means of the coherent potential approximation (Taylor, 1967; Ehrenreich and Schwartz, 1976) and to compare the results of these calculations directly with the second-order Raman spectra. By using the CPA, the usual approximations inherent in this method are made, mainly such that only the effects of differences in isotopic masses are taken into account (not the force constant changes), and the isotopes are randomly distributed on the lattice (not clustered). Note that Fuchs and co-workers (1991b) found that self-energy corrections due to isotopic disorder for certain phonons may be quite large, thus a multiple scattering formalism such as the CPA is expected to provide a more accurate estimation of such corrections than lowest-order perturbation theory. Hass et al., (1992) and Fuchs and co-workers (1991b) treat scattering in the CPA with respect to the virtual crystal approximation (VCA), deﬁned to be a perfect diamond-structure crystal with each site occupied by an atom of atomic mass equal to the concentration-weighted average isotopic mass. In the VCA, the phonon frequencies are given by , where q is the wave OH vector and j is the branch index. Scattering is treated by considering an effective medium characterized by the dimensionless ‘‘self-energy’’ () with respect to the VCA. In this condition, the VCA DOS is equal (normalized to unity) g () : (6N)\ ( 9 ), 4! OH OH

(8)

where N is the number of unit cells in a normalized volume. As mentioned, this medium is characterized by a dimensionless self-energy (), which represents a complex ‘‘mass defect’’ relative to M (in the case of diamond : 12 ; x) at frequency . Self-consistency is imposed by the condition M that the average scattering from a single site in the effective medium vanishes. It is convenient to express this condition as (Ehrenreich and Schwartz, 1976) ) () : M

x(1 9 x)(M)F() , 1 ; [(1 9 2x)M ; M ) ()]F()

(9)

where M is the 1-amu mass difference (between C and C) and F() is the site of Green’s function (see Economou, 1983):

g (*) d* 1 4! F() : (6NM )\ G (q, ) : . H [1 9 ) ()] 9 * M OH

(10)

4 Raman Spectra of Isotopically Mixed Crystals

111

The second term of Eq. (10) follows from the deﬁnition of the q space of Green’s function 1 G (q, ) : , H [1 9 ) ()] 9 ( ) OH

(11)

whose negative imaginary part deﬁnes the CPA spectral function. As mentioned, with matrix element effects neglected, this spectral function is proportional to the Raman- and neutron-scattering cross section (Elliott et al., 1974). In the absence of disorder, -ImG (q, ) reduces to a function H at . To a good approximation (Fuchs et al., 1991b), the real and OH imaginary parts of ) () describe the shift of the spectral peak to a frequency of ) : [1 9 Re ) ( )]\ and broaden it to a full width at half O H O H O H maximum (FWHM) of 9) Im ) () ). The shift and broadening are related O H O H by the Kramers—Kronig relation: Re ) () :

2

* Im[)(*)] d* . * 9

(12)

The normalized one-phonon CPA DOS is given by (Taylor, 1967; Fuchs et al., 1991b) g() : 9

2

g

4!

(*) d* Im

* . [1 9 ) ()] 9 *

(13)

The CPA self-energy ) () is formally independent of q. An implicit q dependence does arise, however, through Eq. (11). The lack of an explicit q dependence is a consequence of the single-site nature of the CPA. In the 1) and/or dilute concentrations the CPA limits of weak scattering (M/M self-energy becomes exact and the q independence is rigorous (Ehrenreich and Schwartz, 1976). In other regimes, the CPA is a highly successful interpolation scheme and q-dependent corrections should only appear at relatively high order in the scattering strength. Equation (9) in the weak-scattering reduces to ) () : x(1 9 x)(M)F M

() , 4!

(14)

where F () is the VCA site of Green’s function, obtained by setting 4! ) () : 0 in Eq. (10). Using the fact that 9Im F

4!

() : g ()/(2M ), 4!

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we obtain the perturbative ‘‘golden-rule’’ expression for the FWHM broadening: 9 Im ) () :

M g (). x(1 9 x) 4! M 2

(15)

The full CPA Eqs. (9) and (10) are easily solved numerically by an iterative process beginning with VCA DOS (for details see Hass et al., 1992). 2. Disorder Shift and Broadening of the Raman Spectra As described in the preceding sections, the isotopically enriched samples of Ge show a frequency inversely proportional to the square root of the average mass (see Fig. 7), and a linewidth inversely proportional to the mass (Fig. 22), in agreement with the harmonic approximation and Fermi’s golden rule, respectively. However, in natural Ge and Ge alloy isotopes, we have an additional disordered Gaussian shift of the phonon frequency and a broadening of the linewidth. This additional shift is 0.34 < 0.04 cm\ in natural Ge and 1.06 < 0.4 cm\ in the Ge alloy sample, which has nearly the maximum isotopic disorder possible with natural isotopes. Single-site CPA calculations (Fuchs et al., 1991b; Cardona et al., 1993)

Fig. 22. Intrinsic phonon linewidth of isotopic Ge versus average mass, measured at 10 K, with an excitation of 6471 Å. The ab initio linewidth reported by Debernardi and coworkers (1996) for isotopically pure Ge with M : 72.6 is indicated by a circle. (After Zhang et al., 1998.)

4 Raman Spectra of Isotopically Mixed Crystals

113

based on the density of states obtained within the harmonic approximation predict that the disorder-induced frequency shift of the point optical phonon amounts to 0.4 < 0.1 cm\ for natural Ge and 1.2 < 0.1 cm\ for the alloy sample, depending on the lattice-dynamical model used to obtain the phonon density of states. A simple estimate from second-order perturbation theory can also provide a resonable prediction of these disorderinduced frequency shifts (Menendez et al., 1984). The disorder-induced shift is given by

1 : g N ( ) d , 12 G 9 B G G

(16)

where the phonon density of states (Nelin and Nilsson, 1972) are normalized through $ N ( ) d : 6. The mass-ﬂuctuation parameter g is given by B G G

M g : x 19 G , G M G

(17)

where x is the fraction of isotope i, M is its mass, and M is the average G G mass. This equation yields g : 5.87;10\ for natural Ge and g : 1.53;10\ for the alloy, respectively. Using Eq. (16), Zhang et al. (1998) obtain a disorder-induced frequency shift of 0.41 cm\ for the natural sample, and 1.07 cm\ for the Ge alloy sample, respectively. An additional broadening (0.03 < 0.03 cm\ for natural Ge and 0.06 < 0.03 cm\ for the alloy sample) is clearly seen (see Fig. 22) in the two isotopically mixed samples. Their linewidths lie signiﬁcantly above those expected from the inverse-average-mass rule. This can be attributed to a combination of disorder-induced scattering and anharmonic decay (see also Fuchs et al., 1991a). Including the anharmonic broadening in the phonon density of states N () yields a nonvanishing scattering probability on the B mass defects. One can thus estimate the mass disorder broadening of the optical phonon with the expression (Cardona et al., 1993): :g N (). 12 B

(18)

This calculation gives an additional broadening of 0.017 cm\ for natural Ge, and 0.046 cm\ for the Ge Ge alloy, in agreement with the measured data. Turning to the second-order spectra, Table II summarizes the observed shifts of peak positions. Agreement with the calculated shifts within the framework of CPA is good. From the preceding results, it can be concluded that the maximal shift possess of LO() phonons, but broadening-acoustic and optical phonons with q"0.

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Vladimir G. Plekhanov TABLE II

Observed Peaks in the Second-Order Raman Spectra of Isotopically Enriched Ge and Natural Ge with Frequency Shifts Compared to CPA Calculations (After Fuchs et al., 1991) Phonon energies (cm\) in Identiﬁcation 2TA(X) 2TA(W-K, L -W ) O() TO(X);TA(X) 2LA(K-L ) 2TO(X) 2TO(W ) 2TO(L ) 2O()

Relative shift of observed peaks in natural Ge

Ge Expt.

Natural Ge Expt.

CPA

Expt.

165.1 235.9 309.4 361.1 389.8 554.8 562.8 586.7 618.8

162.5 231.2 304.5 354.6 382.3 546.0 553.6 577.5 609.0

161.8 231.5 304.8 354.0 382.1 542.5 553.9 575.2 609.6

1.6 2.0 1.6 1.7 1.9 1.6 1.7 1.6 1.6

Figure 23 compares the composition of the Raman frequency in the VCA and CPA according to Hass et al. (1992) and Spitzer et al. (1993). The present Raman data in Fig. 23a are in excellent agreement with those of Chrenko (1988) and Hanzawa et al. (1996). Both sets of data exhibit a pronounced bowing (nonlinearity) relative to the VCA that is described very well by CPA. Hass et al. concluded that the bowing is a direct consequence of the scattering due to isotopic disorder. Similar nonlinearities are observed in many other properties of alloy systems; for example, the bandgaps of semiconductor alloys (Efros and Raikh, 1988) and isotope-mixed crystals (Plekhanov, 1988). The deviation from linearity is approximately 5 cm\ near the middle of the composition range. This is much larger than the experimental uncertainties (about the size of the data points) and should certainly be considered if the Raman frequency is to be used as a measure of isotopic composition (Hass et al., 1991). The measured Raman linewidths (Fig. 23b) are larger near the center of the composition range than near the end points. The variation is not symmetric in x and (1—x) and (as in LiH, see Fig. 21) the maximum width occurs at approximately 70 at. % C. The CPA curves represent intrinsic contributions to the Raman linewidth due to the disorder-induced broadening of the zone-center optic mode. The observed widths, according to Hass and co-workers (1992) contain additional contributions due to instrumental resolution (:1.8 cm\) and anharmonic decay (Fuchs et al., 1991b; Wang et al., 1990). The anharmonic broadening of the Raman line was calculated for diamond by Wang et al. (1990) to be on the order of 1 cm\ at 300 K. Contributions other than disorder thus account well for the observed widths

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115

Fig. 23. Isotopic composition dependence of (a) the diamond Raman frequency and (b) the linewidth (FWHM). CPA results for pure and adjusted Tubino et al. models and VCA results in (a) shown as dashed, solid, and dotted lines, respectively. Experimental data as indicated. Chrenko’s data are taken from Chrenko (1988). (After Hass et al., 1992.)

near x : 0 and 1. Assuming that such contributions are reasonably constant across the entire composition range, we see that both CPA calculations account very well for the qualitative trend in the data, including the peak near x : 0.7. The pure valence force ﬁeld model (Tubino et al., 1972) underestimates the magnitude of the variation, however, by a factor of about 2 (for details see Hass et al., 1992; Spitzer et al., 1993). Detailed calculations of the self-energy [Eqs. (12) to (15)] and the ﬁrstorder Raman lineshape were performed by Spitzer and co-workers (1993). They obtained a qualitative agreement with experimental results. Comparing the Raman lineshape of Ge and C, note that the presence of a large isotopic broadening for diamond, contrary to the small broadening observed for Ge (compare Figs. 23 and 7; 22). The reason lies in the fact that k : 0 is not the highest point of the phonon dispersion relation in the case

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Fig. 24. Vertically expanded ﬁrst-order Raman line of Ge with different amount of isotopic disorder, exhibiting the disorder-induced scattering. (After Fuchs et al., 1993b.)

of diamond (Schwoerer-Bohning et al., 1998). This maximum lies somewhat off k : 0, resulting in a nonvanishing density of states at , considerably larger than that found from relation N : Re[ 9 ; i( /2)] (CarB dona, 1994). This density of states is strongly asymmetric about , a fact that yields an asymmetric phonon lineshape (Spitzer et al., 1993). This asymmetry also results in a lopsided dependence of the linewidth versus concentration (Figs. 21 and 23), which disagrees with the symmetric dependence expected from the proportionality to g [Eq. (17)]. As already mentioned, because of the isotopic disorder, not only k : 0 phonons should contribute to the ﬁrst-order Raman spectrum, but also all others, with maximum contributions for in the regions where the density of states has a maximum, especially for the TO phonon branches. The experimental results for Ge are shown in Fig. 24, where the vertical scale has been normalized to be 1 at the peak of the Raman peak. The disorderinduced contribution is rather weak (less than 0.4% of the peak) but clearly identiﬁable, since it increases from natural Ge to Ge Ge and is absent for Ge (Fuchs et al., 1993a). The two structures observed (at 275 and 290 cm\) correspond to maxima in the density of TO phonons. The disorder-induced contribution of Fig. 24 can be theoretically calculated using CPA techniques. Fuchs and co-workers found two different contributions: a ‘‘coherent’’ one, due to the structure in the imaginary part of the self-energy of the k : 0 phonon, plus an ‘‘incoherent’’ part due to noncon-

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117

Fig. 25. Coherent and incoherent components of the disorder-induced Raman scattering of natural Ge calculated with the CPA. The sum of these two components (open dots) is compared with the experimental results (ﬁlled dots). The agreement is excellent, both for the lineshape and the absolute intensity. (After Cardona, 1994.)

servation of k (see also Etchegoin et al., 1993). The two contributions, together with their sum, are compared with the experimental results in Fig. 25. The agreement is excellent (see, however, Gobel et al., 1998). This effect has also been observed for diamond (Spitzer et al., 1993), -Sn (Wang et al., 1997), as well as for LiH D (Plekhanov and Altukhov, 1985; see also V \V Fig. 26). Effects of the isotopic disorder observed in the Raman spectra of LiH D crystals are similar, but have some important differences (see V \V also Plekhanov, 1995b). In contrast to germanium and diamond, whose ﬁrst-order Raman spectra exhibit the one-mode behavior of LO phonons over the entire range of concentrations (see Figs. 7 and 23), the Raman spectra of LiH D crystals show the one- and two-mode behavior of V \V LO() phonons and the localized mode at low x (see Fig. 21). Figure 26 shows the dependence of the half width of the 2LO() phonon line in the Raman spectrum on the isotope concentration. As the excitation wavelength increases, the LO phonon line slightly broadens both in pure (Tyutyunnik and Tyutyunnik, 1990; Plekhanov and Altukhov, 1985) and mixed crystals (Plekhanov, 1990b), which was earlier observed by Klochikhin et al. (1973).

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Fig. 26. Lineshapes of the excited light (1), 2LO() line scattering at 4.2 K in LiH (2), and mixed LiH D (3) crystals excited at ! : 253.7 nm. (After Plekhanov, 1990a.) V \V

However, the half width of this line (Fig. 26) strongly depends on the isotope concentration. Because the width of the optical bands of pure LiH and LiD crystals are comparable (Plekhanov, 1990b), it is reasonable to assume as was done by Plekhanov (1995b) that the broadening of LO() phonon lines in the Raman spectra of LiH D crystals is partially caused by the V \V isotopic disorder of a crystal lattice. This assumption is consistent with a nonlinear dependence of the half width of LO() phonon lines in the second-order Raman spectra of LiH D crystals (Fig. 21) as well as with V \V additive structure in the resonant Raman spectra of these crystals (Plekhanov and Altukhov, 1985). It is not improbable that the shift and the broadening of LO() phonon lines in the Raman spectra of LiH D V \V crystals can also be described within the framework of CPA in the anharmonic approximation (see also Plekhanov, 2000). Returning to the Raman spectra of GaN crystals, note the following. The calculation of CPA is more complicated in polyatomic crystals since mass ﬂuctuations can exist at different atomic sites, where the atoms have different eigenvectors. In addition, the phonon density of states of GaN with wurtzite structure, whose accuracy should be as high as possible when it is used in a CPA calculation, is not available at present. Zhang et al. performed a rough estimate of the disorder-induced frequency shifts from second-order perturbation theory (for details see Zhang et al., 1997b).

SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 5

Excitons in LiH Crystals

I. The Comparative Study of the Band-Edge Absorption in LiH, Li O, LiOH, and Li CO . . . . . . . . . . . . . . . . . . . II. Exciton Reflection Spectra of LiH Crystals . . . . . . . . III. Band Structure of LiH . . . . . . . . . . . . . . . . . . . . IV. Dielectric—Metal Transition under External Pressure . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

119 122 126 132

I. The Comparative Study of the Band-Edge Absorption in LiH, Li2O, LiOH, and Li2CO3 This section brieﬂy analyzes ﬁndings concerning the history of band-edge absorption investigations and what approaches the essence of the question. Although the ﬁrst measurements of the thin ﬁlm absorption spectra of LiH were carried out in the 1930s (Bach and Bonhoeffer, 1933; Kapustinsky et al., 1937; Rauch, 1939), some questions concerning band-edge absorption remain unanswered (Gavrilov, 1959; Pretzel et al., 1960; Gavrilov et al., 1971, 1974). Our ﬁrst step is connected with other works, such as Kapustinsky et al. (1937), Bach and Bonhoeffer (1933), and Rauch (1939), who interpreted the observations of Bach and Bonhoeffer and Kapustinsky et al., that at room temperature, the absorption band in the vicinity of 4.9 eV is the F center absorption. The main reason for the discrepancy between the results of different authors is the hygroscopic nature of LiH and the fact that it is highly reactive. However, the maintenance of the precaution measure when fabricating LiH thin ﬁlms enabled Pretzel et al., (1960) to not only reproduce the results of Bach and Bonhoeffer (1933) and Kapustinsky et al. (1937) but also to determine the order of the coefﬁcient value of the longwavelength absorption band [actually the exciton band (see Fig. 1) with a maximum at 4.905 eV at room temperature (Plekhanov et al., 1977)]. The value of the absorption coefﬁcient is 10 cm\. The last argument enables us to connect the origin of the long-wavelength band in fundamental absorption spectra to exciton generation in LiH. Despite the circumstances of this paper, for more than 15 years (see, e.g., Gavrilov et al., 1974) the 119

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Vladimir G. Plekhanov

Fig. 1. Discrete and continuous (hatched area) of Wannier—Mott exciton spectrum taking into account its kinetic energy. Dashed line connects to the dispersion of light in medium.

fundamental absorption edge, in accordance with the Rauch results, was considered to lie in the vicinity of the vacuum ultraviolet spectra. At the time, this assumption resulted in an absence of information concerning the fundamental absorption-edge-related compounds (Li O, LiOH, etc.) that are formed on the corroding surface of the LiH crystals (for details see Holcombe and Powell, 1973). Later investigations (Plekhanov et al., 1976; Klochikhin and Plekhanov, 1980; Harbach and Fisher, 1975; Pilipenko et al., 1986; Barker et al., 1991) signiﬁcantly altered the situation. Now the numerous measurements of the fundamental absorption edge carried out on LiH crystals from different branches show that the absorption edge is located in the spectral vicinity of 5 eV (see Fig. 2). On comparison of the results depicted in Fig. 2, we can see that among indicated compounds, LiH possesses a fundamental absorption edge at a longer wavelength. The fundamental absorption edge of Li O was shown by Uchida et al., (1980) to be the closest to that of LiH (see also Barker et al., 1991). According to the results of Harbach and Fischer (1975) and Betenekova (1977) LiOH and Li CO crystals possess shorter wavelength absorption. All the enumerated compounds except LiOH (Betenekova, 1977) have direct electron transitions, which form the longwavelength edge of the fundamental absorption (see also Fig. 3). The absorption edge of LiOH crystals possesses a lingering tail that may indicate not only indirect electron transition, but also the poor quality of these crystals. The absence of detailed low-temperature investigations of absorption (reﬂection) spectra for these substances makes a complete interpretation difﬁcult at this time. It should be added here that the more detailed

5 Excitons in LiH Crystals

121

Fig. 2. Absorption spectra of crystals: curve 1, LiH (Plekhanov et al., 1988); curve 2, Li O (Uchida et al., 1980); curve 3, LiOH (Plekhanov et al., 1988); and curve 4, Li CO , (Betenekova, 1977) at room temperature.

investigations carried out by Kink and co-workers (1987) show that the changes in the reﬂection spectrum (in the vicinity of 6—15 eV) of LiH crystals are due to the photolysis but not oxidation or hydratation of the LiH surface. The results presented in this section show that the comparative investigations of the fundamental absorption edge of LiH, Li O, LiOH, and Li CO compounds beneﬁt from the fact that LiH possesses a minimum value of E between these substances. E

Fig. 3. The reﬂection spectra of LiH (1) and LiD (2) according to Betenekova (1977) and absorption spectra of thin ﬁlms of LiH (1) and LiD (2) according to Kapustinsky et al., (1937). All spectra measured at room temperature.

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II. Exciton Reﬂection Spectra of LiH Crystals At room temperature and ambient pressure, lithium hydride forms a NaCl-type cubic lattice; that is it consists of two centered cubic lattices shifted relative to each other by half the lattice constant (see, e.g. Calder et al., 1962). Compounds with such crystalline structure belong to the O space F group. According to the type of chemical bond, LiH is generally assigned to I—VII ionic compounds [an analog of alkali-halide crystals (AHC)]. However, more detailed investigation shows that its ionic character lies in the range 0.8—1 (see, e.g., Calder et al., 1962). A comparison of ionic radii r > : 0.68 Å and r \ : 1.5 Å with lattice constant l : 2.042 Å shows that * & the lithium and hydrogen atoms in LiH are strongly overlapped (see also Pauling, 1960). More detailed information on LiH and its physiochemical parameters are presented in Shpil’rain et al. (1983). Table I shows the main structure of LiH (and its isotope analogs) in comparison with a typical representative of AHC—LiF as well as semiconducting crystals of Ge and C. Samples of LiH were grown from the melt using modiﬁed Bridgman— Stockburger techniques, which have been described many times elsewhere (see, e.g., Tyutyunnik et al., 1984). To improve the hydrogen (deuterium) stoichiometric composition, the bulk crystals were additionally annealed in a hydrogen or deuterium atmosphere of 2—5 atm at a temperature of 500—550°C. The method of growing Ge crystals by isotope substitution is described in many papers (see, e.g., Agekyan et al., 1989; Itoh et al., 1993).

TABLE I The Main Parameters of LiH, LiD, LiT, LiF, Ge, and C Crystals Parameter Lattice constant (Å) Density (g/cm) Binding energy (kcal/mol) Melting temperature (K) Refraction index T (K) " E (eV) E Stable isotope

LiH

LiD

4.084

4.068

0.775 9217.8

0.802 9218.8

961 1.9847 3.61 12.9<0.5 1080 4.92 (2 K) 5

LiT 4.0633

964 1.9856 3.63 14.0<0.5 1032 5.095 (2 K) 5

LiF

Ge

C

4.017

5.658

3.567

2.640 9243.6

5.327 2.650

3.51

1115

1210

1.3916 1.96 9.01

5

4

4.0055 16.5 406 (300 K) 0.898 (2 K) 5

Data from Plekhanov (1997b), Vavilov et al. (1985), and Blistanov et al. (1982).

2.417 5.7 1860 5.48 2

5 Excitons in LiH Crystals

123

The growth of isotopically controlled diamond strongly differs from the standard semiconductor crystal growth techniques and requires some special comments. Banholzer and Anthony have performed most of the recent isotopically controlled diamond crystal growth using a two-step process (for details see Banholzer and Anthony, 1992; Haller, 1995; Handbook of Industrial . . . , 1998). Taking into account the high reactivity of the freshly cleaved surface of lithium hydride crystals in air, it was necessary to develop a method of crystal cleavage that would enable us not only to obtain a pure surface but also to retain it for several hours (the time required to perform the experiment). The well-known technique of crystal cleavage directly in a helium bath of an optical cryostat in liquid or superﬂuid helium proved to be suitable for this purpose (Plekhanov et al., 1984). This technique enables one to obtain samples with a pure surface. We found no changes in the shape of reﬂection, luminescence (Plekhanov, 1990a) and resonant Raman scattering (RRS) spectra (Plekhanov and Altukhov, 1985) of samples cleaved in such a way over 10—16 h of the experiment. Using the cleavage facility with three degrees of freedom and a degree of rotation of up to 90° enabled us to carry out our experiments. As a rule, the investigated samples of LiH crystals were cleaved from bulk quality crystals. In addition to chemical and mass spectrometric determination (Betenekova, 1977) of the isotope concentration in a solid LiH D solution, the crystal composition V \V was also checked by the position of the maximum of the zero phonon line in the free exciton emission (Plekhanov and Altukhov, 1983). X-ray diffraction studies show (Zimmerman, 1972) that mixed LiH D crystals form V \V a continuous series of solid solution and behave like virtual crystals with the crystal lattice constantly changing according to Vegard’s law. The experimental setup for measuring low-temperature (2—300 K) reﬂection and luminescence (RRS) spectra was described elsewhere (see, e.g., Plekhanov et al., 1984; Plekhanov and Altukhov, 1985; Plekhanov, 1993). The results described in this part of the book were obtained for pure crystals surface cleaved directly in liquid helium, as mentioned. Samples characterized by a weak topographic dependence of exciton reﬂection and luminescence spectra were selected for study (see also Plekhanov, 1990b). Figure 4 shows the specular reﬂection spectrum of a LiH crystal with a pure surface at 4.2 K. One can see that the spectrum exhibits ﬁne structure. The long-wavelength reﬂection maximum, which shows the singularity from the short-wavelength side at 4.960 eV (Plekhanov and Altukhov, 1985), corresponds to the energy E : 4.950 eV. The analogous structure of this Q maximum has been observed numerously in the reﬂection spectra of crystals with pure surfaces (Kink et al., 1987). In the short-wavelength region after the dispersion dip, a second maximum at E : 4.982 eV is observed. The Q maximum of this peak shifts to the red with increasing temperature (Fig. 5), its intensity decreases and cannot be detected above 115 K (Fig. 6). This

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Vladimir G. Plekhanov

Fig. 4. Mirror reﬂection spectrum of LiH crystal cleaved in liquid helium; angle of incidence is 45°.

suggests that a decrease in the E peak intensity results from ionization of Q the states related to this peak. As in Segall and Marple (1967), the ionization energy of this state can be roughly estimated assuming that E : k T Q (where k is Boltzmann’s constant). This estimate gives E : 10 meV. Q Note that the red shift of this state with increasing temperature differs from the shift of the long-wavelength reﬂection band (see later). High values of the reﬂection coefﬁcient (0.7—0.9) for the ﬁrst long-wavelength peak and its comparatively small half width (E25 meV) along with a sharp temperature dependence (see later) suggest that the E and E peaks are related to Q Q the ﬁrst two terms of the hydrogen-like series of the Wannier—Mott exciton. According to Plekhanov et al. (1976), the electronic transition is direct and occurs at the point X at the Brillouin zone boundary (Kunz and Mickish, 1975; Baroni et al., 1985) where the distance between points X and X is a minimum (Fig. 7). The assumption that the exciton spectrum of LiH crystals is hydrogen-like (especially in the parabolic region of the exciton

Fig. 5. Temperature dependence of the energy of maximum of peak of n : 2s state of exciton in reﬂection spectrum of LiH crystal.

5 Excitons in LiH Crystals

125

Fig. 6. Reﬂection spectrum of LiH crystal measured at: 2 K, curve 1; 71 K, curve 2; 138 K, curve 3; 220 K, curve 4; and 288 K, curve 5.

band for energies 40 meV (Plekhanov and Altukhov, 1981) seems natural for the following reasons. First, the ratio of intensities observed for the ﬁrst two peaks in the reﬂection spectrum is close to the n\ dependence, according to the theory (Elliott, 1957) for direct allowed transitions, where n is the principal quantum number. Second, the distance between these peaks is substantially smaller than the LO phonon energy ( : 140 meV *(Plekhanov and O’Konnel-Bronin, 1978a). For LiH, this ratio /

: 0.3 (see, e.g., Plekhanov, 1997b). For comparison, note that for *excitons in CdS, where the large-radius exciton model is applied, this ratio is 0.74 (Segall and Mahan, 1968). Third, the applicability of the Wannier— Mott model to excitons in LiH is supported by the fact that a similar structure consisting, as a rule, of two lines was observed in reﬂection spectra

Fig. 7. Band structure of LiH crystal as calculated in (1) Kunz and Mickish (1975), (2) Perrot (1976), and (3) Baroni et al., (1985).

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of many semiconducting crystals (Agekyan, 1977; Permogorov, 1986; Segall and Mahan, 1968; Sturge and Rashba, 1982). According to the preceding, and taking into account the energies of the long-wavelength peaks E and Q E , the following values for the exciton Rydberg constant in LiH are Q obtained: E : (4/3) : 42 meV (where : E 9 E and E : 11 @ Q Q Q Q 1 meV). Here, E is the binding energy of the ﬁrst excited exciton state. The Q E value found from the hydrogen-like expression for the exciton-binding Q energy correlates well with the preceding estimates of the thermal ionization energy for the n : 2s exciton state. The exciton binding energy in these crystals was also determined from the temperature quenching of luminescence of free excitons. By assuming that it results from ionization of the exciton ground state, the binding energy in LiH was obtained to be 40 < 3 meV (see, e.g., Plekhanov, 1997b). A small energy gap between the exciton n : 1s and n : 2s levels suggests a strong screening of the Coulomb potential coupling of an electron and hole. Strong screening of the Coulomb potential is directly indicative of a comparatively large radius of the exciton state and, hence, of slow relative motion of the bound carriers (Plekhanov et al., 1976). According to Haken (1976), the Coulomb potential should include the total permittivity value. A comparatively strong exciton scattering by longitudinal optical (LO) phonons (for details see later) would result in a noticeable deviation of the exciton n : 1s level from the Rydberg series (see also Fedoseev, 1973). However, for LiH this is not the case, since the deviation of the exciton n : 1s level from the hydrogen-like position does not exceed 0.1E (Plek@ hanov, 1996c). The maximum of the n : 3s line from this series should be located at 4.987 eV. This state is not observed in reﬂection spectra because of the extremely low oscillator strength (cf. Thomas and Hopﬁeld, 1959). The interband transition energy E : E ; E : 4.992 eV (T : 2—4.2 K) E Q @ is determined from the exciton binding energy. For comparison, note that according to band calculations, this value (the direct X —X transition) ranges from 6.61 (Kunz and Mickish, 1975) to 5.24 (Baroni et al., 1985), which is indicative of the lack of good agreement between theory and experiment (see later).

III. Band Structure of LiH It is impossible to comprehend the optical properties of solids without a detailed knowledge of their electron energy structure. The advent of fast and powerful computers in the 1960s stimulated a huge number of calculations of the band structures of various substances, including ionic crystals (see, e.g., Harrison, 1970, 1980). Many different methods of calculation have been developed (see, e.g., Jones, 1962; Callaway, 1964). A critical analysis of the

5 Excitons in LiH Crystals

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results of calculations for broadgap ionic crystals using the example of A B was performed (Poole et al., 1975a, 1975b), together with a consistent comparison between theoretical calculations and experimental data (see also Shirley, 1998). This comparison threw light on a number of conceptual problems that indicate that the calculation of the band structure of ionic crystals is not just a ‘‘technical feat.’’ Moreover, some recognized experts (see, for example, Harrison, 1970) tend to conclude that the very concept of band structure does not apply to ionic crystals. The spectrum of one-electron states of crystal is determined by the solution of the Schro¨dinger equation for the ‘‘extra’’ particle (hole or electron) moving in the averaged ﬁeld created by all the remaining electrons and nuclei:

9

; V (r ) k (r) : E (k)k (r), L 2m

(1)

where the notation is conventional, and V (r) is the periodic potential. The existing calculation techniques differ in the method of constructing the electron potential V (r), the approximation of the wave function k (r), the ways of ensuring self-consistency, the reliance on empirical parameters, and so on. In particular, two factors are especially important in the case of ionic crystals: (1) the inclusion of exchange interaction and (2) the inclusion of polarization of the electron and ion subsystems of crystal by the extra particle (Mott and Herni, 1948). The one-electron potential of any many-electron system is nonlocal because of the exchange interaction between the electrons. It is very difﬁcult to take this interaction into account. Because of this, the exact potential in the band theory is often replaced by the local potential of the form V (r) . [ (r)], where (r) is the charge density function, and the constant is selected in the range from 1 (Slater potential) to 2/3 (Cohn— Sham potential) (Slater, 1975). Two effects are associated with the local exchange. First, the results of calculation depend strongly on the numerical value of , and second, this approximation always underestimates the values of E and E (the width of E T the valence band). It is the low values of E obtained by many authors in T the approximation of local exchange that are responsible for the wrong conclusion concerning the inapplicability of band theory to ionic dielectrics. These problems reﬂect the fundamental drawback of the one-electron approximation, which does not take into account the reciprocal effect of the selected electron (hole) on the rest of the system. This effect consists of the polarization of the crystal by the particle, and is generally made up of two parts: the electron polarization (inertialess) and the lattice polarization (inertial). In the common optical phenomena related to the absorption or

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Vladimir G. Plekhanov

scattering of photons, the lattice polarization is not important, because the frequency of optical transitions is much higher than the average frequencies of phonons. The electron polarization is different. The extra particle (electron, hole) is regarded by this theory as the slowest particle in the system — in other words, all the remaining electrons adiabatically follow it. Hence it follows that the inertialess polarization deﬁnitely must be included in the calculation of the energy spectrum. An important feature of ionic crystals is that the polarization energy E is of the same order of magnitude N as the bandwidth. Such a correction obviously cannot be regarded as small. In the extreme case of a particle at rest, the polarization energy can be calculated by methods of classical electrostatics (the Mott—Littleton method; Mott and Herni, 1948), or by the newer and more accurate technique proposed by Fowler (1966). The value of E for AHC found by N this method is 2—3 eV for each of the quasi-particles (E 0 for electrons N and E 0 for holes). This implies that the inclusion of electron polarization N will reduce the magnitude of E by 4—5 eV (Poole et al., 1975a). By E assumption, the electron bands are displaced rigidly, without changing the dispersion law E(k). The simple electron structure of lithium hydride (combined with the negligibly small spin—orbit interaction) is very helpful for calculating the band structure: all electron shells can easily be taken into account in the construction of the electron potential. The ﬁrst calculations of band structure of lithium hydride were carried out as early as 1936 by Ewing and Seitz (1936) using the Wigner—Seitz cell method. This method consists essentially of the following. The straight lattice is divided into polyhedra in such a way that the latter ﬁll the entire space; inside each polyhedron is an atom forming the basis of the lattice (Wigner—Seitz cells). The potential inside each cell is assumed to be spherically symmetrical and coinciding with the potential of free ions. This approximation works well for ions with closed shells. The radial Schro¨dinger equation in the coordinate function R (r) is J solved within each selected cell, the energy being regarded as a parameter. Then the Bloch function is constructed in the form of expansion: J k (r) : C (k)Y ( , +)R (r, E ), JK JK J J K\J

(2)

where r, , and + are the spherical coordinates (with respect to the center of the cell); and Y are spherical functions. The coefﬁcients C (k) and the JK JK energy E(k) are found from conditions of periodicity and continuity on the boundaries of the cell. If r and r are the coordinates of two points on the surface of the Wigner—Seitz cell, linked by the translation vector R , then the J boundary conditions are (Ewing and Seitz, 1936) k (r ) : exp(ikR )k (r ), J

(3)

5 Excitons in LiH Crystals

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and k (r ) : exp(9ikR ) k (r ), L J L where is the gradient normal to the surface of the cell. We see that the L method of cells differs from the problem of free atoms only in the boundary conditions. Owing to the complex shape of the cell, however, the construction of boundary conditions is a very complicated task, and this method is rarely used nowadays. The plane-associated waves (PAW) method for calculating the band structure and the equation of state for LiH was used in Perrot (1976). According to this method, the crystal potential is assumed to be spherically symmetrical within a sphere of radius r described around each atom, and constant between Q the spheres [the so-called cellular mufﬁn-tin (MT) potential]. Inside each sphere, as in the Wigner—Seitz method, the solutions of the Schro¨dinger equation have the form of spherical harmonics; outside the spheres they become plane waves. Accordingly, the basis functions have the form k (r) : exp(ikr) (r 9 r ) ; a Y ( , +)R (E, r) (r 9 r), Q JK JK J Q

(4)

where (x) : 1 at x ( 0, and (x) : 0 at x 0. The coefﬁcients a can be JK easily found from the condition of sewing on the boundary of the sphere. This is an important advantage of the PAW method over the cells method. The calculations of Perrot (1976) are self-consistent, and the local potential is used in the Cohn—Sham form. The correlation corrections were neglected. The Corringi—Cohn—Rostocker (CCR) method, or the Green’s functions method, was used to calculate the band structure of LiH in Zavt et al. (1976) (only concerned with the valence band) and in Kulikov (1978). Calculation of the band structure of LiH in Grosso and Paravicini (1979) was based on the wave function used in the orthogonalized plane waves (OPW) method of the form k (r) : exp(ikr) 9 exp (ikr) X X (r), A A A

(5)

where X is the atomic function of state of the skeleton, and exp(ikr) X A A is the integral of the overlapping of the plane wave with the skeleton function (see also Baroni et al., 1985). The method of the linear combination of local basis functions was applied to the calculation of the band structure of LiH in Kunz and Mickish (1975). This method is based on constructing the local orbitals for the occupied atom states, based on certain invariant properties of the Fock operator. The main feature of local orbitals is that they are much less extensive than the atomic orbitals. Importantly, the

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correlation correction is taken into account in Kunz and Mickish (1975). Due to the high polarizability of hydrogen molecules, the correlation effect in lithium hydride is exceptionally strong. Yet another calculation of the band structure of LiH was carried out in Zavt et al. (1976a) using the so-called extended elementary cell method (Evarestov et al., 1983). This approach is based on the semiempirical techniques of the theory of molecules, and is quite similar to the cluster calculations. Note also that the cluster is selected in such a way that the quasi-molecular wave function transforms in accordance with the group symmetry of certain wave vectors in the Brillouin zone. This method yields the energy values only at the points of high symmetry. In addition, Hama and Kawakami (1988, 1989), in connection with the study of high-pressure effects on the transition of NaCl—CsCI in lithium hydride, analyzed in detail the band structure and the equation of state of the latter. The calculated band structures of LiH are compared in Fig. 7. We see that the overall picture given by various methods is generally the same, despite the vast spread of the transition energy values (see Table II). Looking at the structure of the valence band we see that it is very similar to the s-band in the method of strong bond (see also Poole et al., 1975b; Shirley, 1998). This is surprising, given the strong overlap of the anion s-functions in lithium hydride. The wave functions in this band are almost entirely composed of the Is states of hydrogen ion. Different authors place the ceiling of the band either at point X or at point W of the Brillouin zone. Although in all cases the energies of the states X and W differ little (0.3 eV), the question of the actual location of the top of the valence band may be important for the dynamics of the hole. Different calculations also disagree on the width of the valence band. For example, the width of the valence band in LiH without correlation is, according to Kunz and Mickish (1975), E : 14.5 eV, and the T value of E is reduced to one half of this when correlation is taken into T account. This shows how much the polarization of crystal by the hole affects the width of the valence band E . According to Perrot (1976), the width of T the valence band in LiH is 5.6 eV. The density of electron states in the valence band of LiH was measured in Betenekova et al. (1978) and Ichikawa et al. (1981). In Betenekova et al. (1978), the measurements were carried out with a magnetic spectrometer having the resolution of 1.5 eV, whereas the resolution of the hemispherical analyzer used in Ichikawa et al. (1981) was 1.1 eV. From experimental data, the width of the valence band is 6 eV according to Betenekova et al., and 6.3 eV according to Ichikawa et al. Observe the good agreement with the calculated value of E in this theory. Note also that the measured distribuT tion of the electron density of states in the valence band of LiH exhibits asymmetry typical of s-bands (for more details see Betenekova et al., 1978; Ichikawa et al., 1981). The lower part of the conduction band is formed wholly by p-states and displays an absolute minimum at point X, which

5 Excitons in LiH Crystals

131

TABLE II Calculated Energy Values of Some Direct Optical Transitions in LiH Reduced to the Experimental Value of E : 4.992 eV (Plekhanov, 1990c) E Source Transition K —K W —W L —L W —W X —X K —K L —L —

1

2

3

4

6.9 8.0 9.2 12.6 12.9 14.7 19.7 24.5

7.5 7.9 9.6 14.9 13.8 16.1 20.9 25.3

6.5 7.3 9.0 12.2 13.6 15.0 20.7 33.3

6.4 7.4 9.1 — — — — —

1. Kunz, A. B., and D. J. Mickish, Phys. Rev. B11, 1700 (1975). 2. Perrot, F., Phys. Stat. Solidi B77, 517 (1976). 3. Grosso, G., G. P. Paravicini, Phys. Rev. B20, 2366 (1979); Baroni, S., G. Pastori Paravicini, and G. Pezzica, Phys. Rev. B32, 4077 (1985). 4. Kulikov, N. I., Fiz. Tverd. Tela 20, 2027 (1978).

corresponds to the singlet symmetry state X . The inversion of order of s and p-states in the spirit of the LCAO method can be understood as the result of the s-nature of the valence band. Mixing of s-states of the two bands leads to their hybridization and spreading, which changes the sequence of levels. If we compare the structure of the conduction band with the p-band of the method of strong bond (Pantelides, 1975), we see that the general structure and the sequence of levels are the same except for some minor details [the location of the L level, and the behavior of E(k) in the neighborhood of ]. In other words, the lower part of the conduction band in lithium hydride is very close to the valence p-band of AHC. The direct optical gap in LiH according to all calculations is located at the X point and corresponds to the allowed transition X —X . The indirect transition X —X ought to have a similar energy. According to the above calculations, the energies of these transitions differ by 0.03—0.3 eV. The different values of E for LiH obtained by different E authors are apparently due to the various methods used for taking into account the exchange and correlation corrections (see earlier). As follows from Table II, the transitions at critical points in the low-energy region form two groups at 7—9 and 13—15 eV. Measurements of reﬂection spectra in the 4—25-eV range at 5 K (Kink et al., 1987) and 4—40 eV at 300 K throw new light on the results of calculations (see also Plekhanov, 1997a). The

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Vladimir G. Plekhanov

singularities occurring at 7.9 and 12.7 eV in the reﬂection spectra are associated in the cited papers with the interband transitions W —W and X —X , respectively. From the standpoint of dynamics of quasi-particles, an important consequence of such band structure is the high anisotropy of the tensor of effective mass of electrons and (especially) holes. The estimated mass of electrons in the neighborhood of X is, according to Kunz and Mickish (1975), (m ) < 0.3 m in the direction X—, and (m ) : (m ) < 0.8 m in the CV CW CX direction X—W. Similarly, the mass of the hole in the neighborhood of X is X— in the direction (m ) < 0.55 m and about the same in the neighborCV hood of W . It is assumed that the transverse components of m are greater F by several orders of magnitude (Zavt et al., 1976a). Note also that, according to Baroni et al. (1985), the estimated masses of carriers are: m : 0.121, CJ m : 0.938, m : 0.150, and m : 4.304 m , where the subscripts l and t CR FJ FR C denote, respectively, the longitudinal (in the direction —X) and the transverse (in the direction X—W ) components. This high anisotropy of masses of electrons and holes ought to have resulted in the high anisotropy of the reduced (1/ : I/m ; 1/m ) and the translation (M : m ; m ) C F C F masses of exciton. This, however, is not the case. Moreover, the study of Plekhanov and Altukhov (1981) reveals that with a good degree of conﬁdence one can assume that in the energy range E 40 meV the exciton band is isotropic and exhibits parabolic dispersion. As will be shown later, the studies of the exciton—phonon luminescence of free excitons and RRLS in LiH crystals (Plekhanov and Altukhov, 1985; Plekhanov, 1990c) reveal that the kinetic energy of excitons in these crystals is greater than E by an order @ of magnitude exactly because of the very small masses of exciton and hole.

IV. Dielectric--Metal Transition under External Pressure Let us now consider the problem of the dielectric—metal transition in LiH under external pressure. As the lattice constant decreases, the forbidden gap in dielectrics narrows and eventually vanishes — the transition from dielectric to metal takes place. For typical dielectrics, the critical pressure exceeds several Mbar. The interest in the ‘‘metallization’’ of lithium hydride is primarily due to the fact that its properties in the metallic phase may, to a certain extent, simulate the properties of metallic hydrogen. Indeed, owing to the presence of the broad hydrogen s-band and the strong expulsion of the s-band of Li in LiH, the Fermi level in the metallic phase ought to fall into the band created primarily by the electrons of hydrogen. Because of the high frequencies of optical oscillations, one might expect the metallic lithium hydride to go over into a superconducting state at a high enough temperature (Ginsburg, 1997). The usefulness of hydrides of alkaline metals at

5 Excitons in LiH Crystals

133

superhigh pressures as systems simulating metallic hydrogen is discussed in detail in Maximov and Pankratov (1975). Intuitively, the idea of the metallization of lithium hydride at relatively low pressures seems quite plausible. As a matter of fact, the outer electron of H\ is rather strongly delocalized even at normal pressure. Since the overlap of wave functions increases exponentially as the lattice constant decreases, one can expect that a moderate amount of compression will result in complete delocalization of this electron. The pressure of the dielectric-metal transition in LiH was calculated in the following papers: Behringer (1959), Trubitzin and Ulinich (1963), Perrot (1976), Vaisnys and Zmuidzinas (1978), Kulikov (1978), and Hama and Kawakami (1988). Such a calculation is usually carried out in two steps: ﬁrst, the width of the forbidden gap (or the entire band structure) is calculated as a function of the lattice parameter, and then the equation of state that links the lattice constant and the pressure is expressed in some way. Strictly speaking, these two problems ought to be solved concurrently; so far, however, this has been done only in Ewing and Seitz (1936) and Hama and Kawakami (1989), and then only for the electron contribution to the pressure. Early calculations (Behringer, 1959; Trubitzin and Ulinich, 1963) gave discouragingly high values of the critical pressure, 23—25 Mbar, and accordingly, very small values of a in the metallic phase, from 1.3 to 2.2 Å. Later, however, these results were found to be incorrect because of a number of unreasonable assumptions. In Behringer (1959), for example, it was assumed that the gap in LiH is indirect and corresponds to the W —L interval (see Fig. 7). Then, the overlap of electron distributions in LiH was neglected, although the calculated value of the lattice constant (1.3 Å) implies that the distance between lithium ions is much less than the sum of their ionic radii. A more detailed criticism of these studies can be found in various papers (Perrot, 1976; Kulikov, 1978; Vaisnys and Zmuidzinas, 1978; Hama and Kawakami, 1988). Of much greater interest are the results obtained thereafter. In Perrot (1976) it is demonstrated that at l : 2.9 Å, which corresponds to a pressure of 2 Mbar; the gap X —X closes. The bands, however, just come into contact with each other without overlapping, the latter being forbidden at point X from considerations of symmetry. Accordingly, the Fermi surface is a point, and the system corresponds to a semimetal. As the lattice constant decreases further, the structure of the bands changes dramatically; in particular, the minimum of the valence band goes over from point to point L . At l : 1.96 Å the bands overlap, and the system becomes a metal. The critical pressure turns out to be 30 Mbar. The results of Kulikov (1978) are similar to those obtained by Perrot; the attention is drawn, however, to the following interesting circumstance. If one calculates the band spectrum under certain assumptions concerning the hypothetical structure of the CsCl-type structure, the gap (indirect) of lithium hydride in the CsCl

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structure turns out to be close to zero. The critical pressure for this structure is estimated at 0.5 to 1 Mbar. In other words, the structural phase transition NaCl ; CsCl occurs earlier than the gap in the NaCl structure closes, and is accompanied by metallization of the crystal. This very interesting conclusion was developed further in a paper by Hama and Kawakami (1989). In this study, a method similar to that employed by Perrot (1976) was used to express the equation of state for LiH in the shock wave. According by Hama and Kawakami, the bands also overlap at point X with subsequent lowering of point X (the bottom of the p-type conduction band) with respect to point X (the top of the s-valence band). The overlap starts at P : 2.26 Mbar, which is 13% higher than reported by Perrot. The relatively low pressure of the dielectric—metal transition in LiH (:0.5 Mbar) is also predicted by Ruoff et al. (1996), whereas according to Kondo and Asaumi (1988) the critical pressure is not less than 4 Mbar (see also Chandehari et al. (1995). The calculation by Ruoff et al. indicates that structural phase transition B1—B2 (NaCl ; CsCl) ought to occur at the pressure 1.3—1.4 Mbar and the temperature 200 K. The study by Ruoff et al. (1996) was concerned with the behavior of E and the E refraction index of CsH crystals (E : 4.4 eV at P : 0 Mbar) as a function E of the external pressure. No structural phase transition in CsH crystals could be observed in the broad range of 0—2.51 Mbar. According to estimates made in this paper, the conduction and valence bands may start to overlap at about 10 Mbar. We see that studies of these aspects of LiH are quite far from conclusive.

SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 6

Exciton—Phonon Interaction

I. Interaction between Excitons and Nonpolar Optical Phonons . . II. Polarization Interaction of Free Excitons with Phonons . . . . III. Effects of Temperature and Pressure on Exciton States . . . . . 1. T heoretical Background . . . . . . . . . . . . . . . . . . . . 2. Experimental Results . . . . . . . . . . . . . . . . . . . . . IV. Isotopic Effect on Electron Excitations . . . . . . . . . . . . 1. Renormalization of Energy of Band-to-Band Transitions in the Case of Isotopic Substitution in L iH Crystals . . . . . . . . . . . 2. T he Dependence of the Energy Gaps of A B and A B Semiconducting Crystals on Isotope Masses . . . . . . . . . . . . . . . . . . 3. Renormalization of Binding Energy of Wannier—Mott Excitons by the Isotopic Effect . . . . . . . . . . . . . . . . . . . . . . . . 4. L uminescence of Free Excitons in L iH Crystals . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

135 136 139 139 146 156

. . .

156

. . .

158

. . . . . .

168 174

I. Interaction between Excitons and Nonpolar Optical Phonons Foundations of the theory of exciton—phonon interaction were laid in the 1950s. The interaction between an electron and a nonpolar optical phonon in a crystal can be described simply in terms of a deformation potential (see, e.g., Bir and Picus, 1972). The net effect of the lattice displacement on the electron is assumed to be a small shift in the electronic energy band of the crystal. The constant of proportionality between this energy shift and the lattice displacement is deﬁned as the deformation potential. The exciton— phonon (deformational potential) Hamiltonian can therefore be expressed as (Ansel’m and Firsov, 1956) H : #.

(W 9 W ) C F a >> a [b ; b> ], \O I >O I O 2N a

where W and W denote the deformation potentials of the electron and hole, C Fand a are the creation and annihilation operators of an respectively; a> k k exciton with wave vector k ; b>q and bq are the creation and annihilation \ 135

136

Vladimir G. Plekhanov

operators of an optical phonon with momentum q ; is the reduced mass of the atoms in the unit cell; N is the number of unit cells in the crystal; a is the lattice constant of the crystal; and is the energy of the optical phonon. II. Polarization Interaction of Free Excitons with Phonons Apart from pioneering the study of Ansel’m and Firsov (1956), the interaction of excitons with longitudinal optical phonons was considered by many authors (Knox, 1963; Haken, 1976; Thomas, 1967; Firsov, 1975; Permogorov, 1982). In ionic crystals, there are two main mechanisms of interaction of excitons with lattice vibrations. One — the mechanism of the short-range deformation interaction — is caused by modulation of the wave function of excitons by longitudinal vibrations. The magnitude of this interaction is characterized by the deformation potential (see earlier). The deformation interaction strongly affects the energy spectrum and dynamics of excitons of relatively small radius (e.g., the ground state of excitons in AHC and crystals of inert gases; Knox, 1963). As the radius of the exciton increases, this interaction becomes less important, since the wave vector of actual phonons is q . r\ (Toyozawa, 1958), where r is the exciton radius, and the number of such phonons is proportional to q. The second mechanism — the polarization or Fro¨hlich interaction (Fro¨hlich, 1954) — is caused by the Coulomb interaction of the charge carriers forming the exciton with macroscopic ﬁeld created by longitudinal optical oscillations (see, e.g., Pekar, 1951; Firsov, 1975). If the exciton radius is much greater than the lattice constant, then the exciton—phonon interaction can be regarded as the sum of independent interactions of electrons and holes with phonons (see Klochikhin, 1980). The interaction operator of charge and mass m (m or m ), neglecting the dispersion of the latter, is C F (Fro¨hlich, 1954) > ; bq), H : Wq exp(iqr)(b 9q q where Wq :

4q

CF *,

q

V

(1) (2)

V is the volume of the system, and r are the coordinates of the particles. In this expression, we introduce the main parameters that determine the interaction of the electron (hole) with optical vibrations: the polaron ‘‘radius’’:

r* : m , CF CF *2

(3)

137

6 Exciton—Phonon Interaction

and the dimensionless Fro¨hlich constant of interaction,

e 1 1 1 g : 9 . CF r *- CF

(4)

The ﬁrst of these quantities characterizes the size of the polarization region of the lattice by the extra charge, and the second describes the strength of the electron—phonon interaction (see Knox, 1963; Haken, 1976). As follows from Eq. (1), the interaction operator Wq does not depend on the mass of the quasiparticle, and is the same for electrons and holes. Accordingly, the interaction Hamiltonian of Wannier—Mott excitons with optical phonons has the same form as Eq. (1), the only difference is that exp(iqr) is replaced with exp(iq r) 9 exp(iq r), where r are the coordinates CF C F of the electron (hole). In the center-of-mass system, the interaction operator becomes (Ansel’m and Firsov, 1956) H : Wq [exp(iq r) 9 exp(iq r)](b > 9q ; bq). #6* C F q

(5)

Replacing m by the reduced mass by analogy with Eq. (1), we can deﬁne CF the characteristic size of the polarization region r : [(r* ) ; (r* )] and C F the interaction constant g . Making use of the characteristics of the Wannier—Mott exciton (r E ), one can express the latter as @

E r* @ 9 1 : 9 1 , g :

r *

(6)

where r : a (m ), and a : /m e : 0.53 Å is the Bohr radius of the I hydrogen atom. The scattering of excitons by LO phonons is determined by the magnitude and the wave vector dependence of the matrix element HHH : H , #6* H #6* H

(7)

where and are the wavefunctions of the initial and ﬁnal states of the H H exciton with the wave vectors k and k : k ; q. The properties of the matrix element of exciton—phonon scattering, as ﬁrst noted in Bulyanitza (1970), depend crucially on the parties of the initial ! and ﬁnal ! states. If the parity is the same (scattering occurs within the same band, as 1s—1s or 2s—2s, etc., or in the case of interband scattering 1s—2s, 1s—3s, etc.), this mechanism of exciton—phonon scattering is forbidden, because HHH ; 0 #6* when q ; 0. When excitons are scattered in a ground state band (1s—1s), as

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Vladimir G. Plekhanov

shown in the paper of Ansel’m and Firsov, the matrix element has the form HHH . #6*

g *- q

r

1;

q r \ q r \ C 9 1 ; F . 2 2

(8)

When q is small (it is the small values that are of special importance; see later), the matrix element is proportional to

m 9m C qr . HHH . F #6* m ;m F C

(9)

As follows from Eq. (9), the Fro¨hlich mechanism of intraband scattering is absolutely forbidden when the effective masses are equal. This is because the centers of the distribution of masses and charges coincide at m : m , so C F the polarization interactions of electrons and holes cancel out completely. In the general case, the matrix element, Eq. (8), arrives at maximum near qr : 1, and then falls off rapidly (Fig. 1). Similar behavior is displayed by the matrix elements of the scattering processes between the same symmetry (is 9 js, ip 9 jp, etc.). In such cases, the matrix element attains its maximum at the value of the inverse radius (r ) of the corresponding exciton state, that L is, 1 q : , r : nr ,

nr L L

(10)

According to the results of the paper by Bulyanitza, when scattering occurs between the bands of different symmetry (is 9 kp) when q ; 0, the matrix element tends to be a constant (nonzero) value, and such processes are therefore allowed. As in our ﬁrst case, however, the matrix element HHH #6*

Fig. 1. Dependence of matrix element of intraband scattering on qr for the mass ratio CV m /m : 3.5. (After Plekhanov, 1997b.) F A

6 Exciton—Phonon Interaction

139

falls off rapidly as q q r\ increases. The dependence of matrix elements on q for some cases of allowed and forbidden scattering for CdS is discussed in Permogorov (1982). The behavior of matrix elements of Eq. (8) is deﬁnitive for the structure and properties of luminescence spectra of free excitons and Raman scattering in the resonance region. The existing theory of exciton—phonon interaction describes the case of isotropic band with parabolic dispersion of exciton states. Its successful application to LiH (with the high anisotropy of the valence band, see Fig. 7 in Chapter 5), once again testiﬁes to the fact that the dispersion of the exciton band is indeed parabolic (Plekhanov and Altukhov, 1981) in the range of low kinetic energies of exciton. Note also that, according to Permogorov (1982), the exciton band exhibits parabolic dispersion even for such anisotropic crystal as CdS, which is supported by the successful application of the theory of exciton—phonon interaction to the analysis of luminescence spectra and RRLS of free excitons in CdS crystals.

III. Effects of Temperature and Pressure on Exciton States 1. Theoretical Background Even after the ﬁrst works on the spectroscopy of large-radius excitons (see Gross, 1976) it became clear that the location of the edge of fundamental absorption (and hence the exciton structure) in a solid depends on the temperature. Further studies also revealed (see also Knox, 1963; Cardona, 1969) that the temperature dependence of the absorption edge may be caused by two factors: the expansion of the lattice and the lattice vibrations. Vibrations of the lattice will cause not only displacement but also broadening of the energy levels of electron excitations. As ﬁrst shown by Fan (1951), the change in the energy of band-to-band transitions E in most substances E is caused primarily by the displacement of the energy levels rather than by their broadening, as had been assumed before. It was demonstrated that in the mechanism of deformation potential of electron—phonon interaction the quantity E is directly proportional not E only to the square of deformation potentials of the valence band and the conduction band, but also to the sum of q and q , where q is the

maximum value of the wave vector of phonon. If the displacement of electron bands (or exciton levels) associated with the Fro¨hlich mechanism of electron—phonon interaction is taken into account, the matrix element is inversely proportional to the wave vector of phonons [see Eqs. (3) and (8)]. There is an important distinction between displacement and broadening of energy levels. The point is that in the case of displacement the theory considers virtual transitions, whereas in the case of broadening the

140

Vladimir G. Plekhanov

transitions are real. Real transitions require conservation of quasimomentum and energy, whereas only quasi-momentum must be conserved in the case of virtual transitions. In the mechanism of deformation potential this circumstance is especially important for acoustic vibrations, being associated with the directly proportional dependence of the matrix element of electron transition in the magnitude of the wave vector of phonons. And since virtual transitions may involve phonons with not necessarily small values of q, it becomes clear why in the mechanism of deformation potential of electron—phonon interaction the displacement of energy levels greatly exceeds their broadening (for more details see Fan, 1967). The contemporary microscopic treatment of this problem (see, for example, Heine and van Vechten, 1976) singles out four contributions to the temperature dependence of E (see also Zollner et al., 1992; Plekhanov, E 1993): 1. The Debye—Waller factor in the Fourier expansion of the periodical part of crystal potential; 2. The term related to intraband transitions, now commonly referred to as the Fan term (Cohen and Chadi, 1980); 3. The contribution from band-to-band transitions, which currently is hard to evaluate; 4. The contribution from thermal expansion of the crystal lattice.

As a rule, the majority of experimental studies concerned with this problem (see, for example, Cohen and Chadi, 1980, and the references therein) consider two contributions: the contribution from electron—phonon interaction, expressed as either the Debye—Waller or the Fan term, and the contribution from thermal expansion of the lattice. A consistent study of the effects of temperature on the energy of band-to-band transitions for a large number of semiconductor compounds (Zollner et al., 1992; Logothetidis et al., 1985, 1991; Cardona and Gopalan, 1989) indicates that theory can be brought into good agreement with experiment when at least the ﬁrst three factors are taken into account. Note also that, apart from the microscopic approach to the temperature dependence E (T ), the attempts to explain the E empirical formula of this dependence have continued for over three decades. First, we should note the well-known Varshni formula, widely used in experimental works (Varshni, 1967), and the currently no less popular Manoogian—Leclerk relation (see Manoogian, 1982; Quintero et al., 1991, and references therein). The dependence of the electronic energy gap E on the isotopic mass M I (where k is the atomic species) at a constant temperature can be separated into two contributions, similar to those responsible for the temperature

6 Exciton—Phonon Interaction

141

dependence of the gaps (see, e.g., Pankov, 1971):

E E E : ; . M M M I 2 I #. I 2#

(11)

The ﬁrst term is the contribution of the electron—phonon interaction (EP) at constant volume. The second one is due to the change of the lattice constant or, equivalently, of the crystal volume with the isotopic mass. Anharmonic corrections to the crystal volume at low temperature depend on the atomic masses through the ‘‘zero-point’’ vibrational amplitudes. The origin of this term is equivalent to that of the thermal expansion (TE) at low temperature; by analogy we call it the zero-point thermal expansion term. As mentioned earlier, the renormalization of the unperturbed band energy n with wave vector kn (of the state k k and band index n) by the electron—phonon interaction (see Fig. 2) can be written as , 1# Ekn (T ) : kn ; "5 kn ; kn ; ikn

(12)

where k"5 n is the shift of the band energy induced by the Debye-Waller (DW) term. The complex self-energy (SE) term has a real part 1# kn , which gives rise to an energy shift of the band states and an imaginary part kn , which causes a lifetime broadening of these states.

Fig. 2. (a) Feynman diagrams for the self-energy of electrons due to interaction with phonons, (b) Feynman diagrams for two-phonon Raman scattering, and (c) renormalization vertex corresponding to the sum of Debye-Waller (DW) and self-energy (SE) or Fan term. (After Cardona and Gopalan, 1989.)

142

Vladimir G. Plekhanov TABLE I

Parameters Needed for the Evaluation of the Thermal Expansion Contribution to the Temperature Shifts of Bandgaps

Material

Gap

(E/p) 2 (meV/GPa)

Diamond Diamond Diamond LiH Ge

E E G E X —X —X

7.0? 6.4@ 10A 90.11B 914D

B (GPa)

(300 K) (10\ K\)

442

1.0

31.2C 75

3 5.90

? M. Cardona and N. E. Christensen, Solid State Commun. 58, 421 (1986). @ P. E. Van Camp, V. E. Van Doren, and J. T. Devreese, Phys. Rev. B34, 1314 (1986). A O. Madelung, in Numerical Data and Functional Relationships in Science and Technology, O. Madelung, ed. (Springer, Berlin, 1987), Vols. 17a and 22a. B Y. Kondo and K. Asaumi, J. Phys. Soc. Japan 57, 367 (1988). C M. W. Guinan and C. F. Cline, J. Nonmetals 1, 11 (1972). D K. J. Chang, S. Froyen, and M. L. Cohen, Solid State Commun. 50, 105 (1984).

In the following description of the different terms, Eqs. (11) and (12), we are very close to Zollner et al. (1992). By the thermal expansion (TE) the lattice constant increases and thus the band gaps shrink, if they have a positive pressure coefﬁcient E/p. The shift for the gaps are found to be (Cohen and Chadi, 1980)

E T

E p

, (13) 2# 2 where (T ) is the temperature-dependent thermal expansion coefﬁcient and B is the bulk modulus. This term can be evaluated very easily using the values listed in Table I and is small compared to those due to the electron—phonon interaction as described later (see also Plekhanov, 1993). The Debye—Waller term arises from the simultaneous interaction of an electron (with wave vector k in band n) with two phonons of the wave vector and mode j (electron— two phonon interaction). In the rigid-ion approxiQ mations Zollner et al. assumed that the potential V of the atom of type ? moves rigidly with the atom in a phonon vibration. Then the DW contri for frozen-in lattice displacement bution to the shifts of an electronic state nk l is given by Cohen Ul of one atom of type located at the atomic site R and Chadi (1980) (E nk)

"5

: n,k l l

: 93(T )B

(n, (n, B k, l , , n, k ) l B k , l , , n, k ) R, : Ul l U l 9(E nk 9 Enk )

(14)

6 Exciton—Phonon Interaction

143

where the sum runs over all intermediate electron states n, k , all lattice vectors l , and the basis () of the lattice. The angular brackets with superscript t denote the thermal or temporal average. The energies necessary to distort the lattice (phonon energies) have been assumed to be much smaller than the usual electronic bandgaps and thus are neglected in the denominator of Eq. (14). Thus (n, V ( l ) nk B k, l , , n, k ) : nk r 9R ?

(15)

is the matrix element of the gradient of the potential V of one atom of type ? l . The notation used located at the atomic site R here is somewhat symbolic but very intuitive, to emphasize the inﬂuence of the isotopic mass l . For explicit expressions applicable entering the phonon displacements U for a lattice with basis, see Gopalan et al. (1987). Zollner et al. used an empirical pseudopotential band structure (Cohen and Chadi, 1980) and assumed that the matrix element of the atomic potential with the true wave functions is the same as that of the pseudopotential evaluated with the pseudowave functions (Sham, 1961) (rigid pseudoion method). To the second order in phonon displacement, Eq. (14) for the Debye—Waller term is equivalent to simply multiplying the structure factors S(G ) used in the band-structure calculations by Debye—Waller factors exp(92 W) with /12. This term is the dominant contribution to the temperaW : u G ture shifts of bandgaps. It has been evaluated for several materials (Gamassel and Auvergne, 1975), but usually overestimates the shifts when considered without the other electron—phonon term discussed next. The third contribution to the temperature shifts of electronic states is the real part of the self-energy term, which arises from the interaction of an electron with one phonon taken to the second order in perturbation theory (see the second term in Fig 2a). This term is usually somewhat smaller than the Debye—Waller term, but opposite in sign. Therefore it should be taken into account in a realistic calculation of temperature shifts, although it requires a Brillouin-zone integration to be evaluated. According to Zollner et al., it can be expressed as (E nk) : 1#

n,k l l

(n, (n, B k, l , , n, k ) l B k , l , , n, k) l l U l R, : U (E nk 9 Enk )

(16) The imaginary part of the self-energy causes lifetime broadenings of critical points and is responsible for intervalley scattering processes (see also Zollner et al., 1991). For numerical reasons, Zollner et al. (1992) Fourier-transformed Eqs. (14) and (16) to the phonon representation (thus replacing the sum over lattice

144

Vladimir G. Plekhanov

sites by an integration over all phonons in the ﬁrst Brillouin zone) and and branch labeled the contribution of a single phonon with wave vector Q j (with the occupation number NQ j : 1) as (Enk /NQ j) , where K stands ) for DW or SE. From these electron—phonon couplings Zollner et al. deﬁned the dimensionless electron—phonon spectral functions:

Enk ( 9 Q j), gF (n, k, ) : I NQ j ) Q j

(17)

in such a way that the temperature shifts due to the DW or SE terms are given by (E nk) : )

1 dgF (n, k, ) N ; . I 2

(18)

, the cited authors For the integration over all phonon wave vectors Q used the tetrahedron method with 89 points in the irreducible wedge of the Brillouin zone. For the numerical evaluation of the terms in Eqs. (14) and (16) Zollner et al. assumed that only the phonon amplitudes and not their energies depend on the temperature and isotope concentration. The calculated energy of the E gap in natural diamond C as a function of temperature (assuming E : 7.3 eV at 0 K), including all three contribu tions, is shown by the solid line in Fig. 3. The contribution of thermal expansion (dashed-dotted line) and self-energy (dashed-double-dotted line) are displayed separately. It can be seen that the shifts due to thermal

Fig. 3. Shifts of the E gap in natural diamond C (solid line) including thermal expansion (dashed-dotted line), self-energy (dashed-double-dotted line), and Debye—Waller term (not shown separately). (After Zollner et al., 1992.)

6 Exciton—Phonon Interaction

145

expansion are small, especially because of the very small pressure dependence of the E gap commonly found in semiconductors (Madelung, 1987). Furthermore, the TE contributions are rather small for all gaps in diamond because of the large bulk modulus (442 GPa; see Van Camp et al., 1986). The DW contribution is dominant up to about 400 K, but at 700 K the SE accounts for 40% of the shifts. The only experimental data for the shifts of the E gap in diamond known to us are the reﬂection measurements of Clark et al. (1964) reporting a shift of the direct gap of 100 meV between 133 and 295 K. This result is much larger than calculated shifts (only 15 meV), but it is not clear whether the assigment of the experimental peak to E transition is correct. By the way, we should note that in silicon, the E and E critical points are almost degenarate (Lautenschlager et al., 1987). The temperature dependence of the indirect gap E (0 K) : 5.41 eV in G diamond C is shown in Fig. 4. It can be seen that the DW term dominates the shifts, in contrast to the results for the E critical point (see Fig. 3). Figure 4 also shows the experimental data of Clark et al. (1964) (dashed line), which are in good agreement with the calculated results. We can see that at given temperature, isotope effects enter in Eqs. (14) and (16) through the mean-squared phonon amplitude

U Q jR :

[1 ; 2NQ j (T )], 2M NQ j ?

(19)

Fig. 4. Shifts of the E (0 K) : 5.41 eV gap in natural diamond C (solid line) including G thermal expansion (dashed-dotted line), self-energy (dashed-double-dotted line), and Debye— Waller term (not shown separately) The dashed line shows the experimental data of Clark et al., (1964). (After Zollner et al., 1992.)

146

Vladimir G. Plekhanov

where M is the mass of one atom of type and NQ j (T ) the occupation ? , branch j, and energy . number of the phonon with wave vector Q Qj Neglecting the small isotope dependence of the matrix elements of Eq. (15), the Debye—Waller and self-energy terms cause not only temperature shifts but also an isotope-dependent renormalization of the electron energies at zero temperature. At 0 K (when N : 0), we have

U Q jR . M\, ?

(20)

since the phonon frequency is proportional to M\. ? 2. Experimental Results The temperature shift of the E in diamond was investigated by Collins et G al. (1990). The results of Collins et al. are depicted in Fig. 5. According to Collins et al., the temperature dependence of the indirect gap in diamond has the following form: E (T ) : E ; EV

df () n(, T ) ;

1 9 a(c ; 2c )V (T )/3V. (21) 2

Here n(, T ) is the Bose—Einstein occupation number, and f () d is the difference in the electron—phonon coupling for the conduction-band minima and the valence-band maximum for those modes in the frequency range to ; d, the energy gap at 0 K being E ; $ df (). The ﬁnal term in Eq. (21) allows for the temperature-dependent lattice expansion, where V (T )/V is the fractional volume expansion to temperature T, c and c are elastic constants, and a is the change in energy per unit compressional hydrostatic stress, measured as a : 5 < 1 meV GPa\. The volume expansion accounts for only about 6% of the temperature dependence of E EV (Fig. 5). Most of the measured temperature dependence arises from the term $ df ()n(, T ). The functional form of f () is not known. However, Collins et al. found that a precise ﬁt can be made to the temperature dependence of E using f () : cg(), where g() is the density of phonon EV states (Dolling and Cowley, 1966) for C diamond, and the constant c of proportionality is given by the ﬁt to E (T ). At 0 K, changing the isotopes EV gives a contribution to E —E of EV EV 1 : 2

12 91 13

f () d : 13.5 < 2 meV.

(22)

As was shown by Collins et al., although the best ﬁt to the data in Fig. 5

147

6 Exciton—Phonon Interaction

Fig. 5. Squares: experimental data of Clark et al., (1964) for the temperature dependence of the indirect energy gap of diamond C. The calculated shift (thick line) is the sum of contributions from the lattice expansion (curve a) and the electron coupling (curve b), using f () : cg(). (After Collins et al., 1990.)

is obtained when f () : cg(), adequate ﬁts are obtained for f () : c g() to f () : c g() producing most of the uncertainty shown in Eq. (22). The 20% uncertainty contributes a further 0.1 meV to the uncertainty in Eq. (22). A second contribution to the isotopic dependence of E comes (as was EV mentioned) from the volume change produced by changing the isotope. The difference in molar volume of C and C is V—V :

2(c

3 G ; 2c ) G

12 91 , 13

(23)

where the frequencies are for C, is the Gruneisen parameter of the ith G

148

Vladimir G. Plekhanov

Fig. 6. Temperature dependence of the lowest indirect energy gap of Ge. Points show the trend of experimental data from McClean (1960), the ﬁne curve ‘‘a’’ shows the effect of volume expansion and the full curve shows the total ﬁt of Eq. (21) with f () : cNg() (p : 0.4). (After Davies et al., 1992.)

mode, and the density of phonon states is known (Dolling and Cowley, 1966). Taking into account : 1.15—1.6, — < 50 cm\, is ** equal to 3 < 1.3 meV (Collins et al., 1990). Thus it was shown that the indirect energy gap of diamond changes by 13.6 < 0.2 meV when the isotope composition changes from C to C. Most of this shift is caused by the isotopic dependence of the electron— phonon coupling, giving a shift estimated from the temperature dependence of energy gap as : 13.5 < 2 meV. An additional smaller contribution, estimated to be : 3 < 1.3 meV, comes from the difference in molar volume of C and C diamond. The temperature dependence of the lowest indirect energy gap of Ge is shown in Fig. 6 (Davies et al., 1992). Here also the two main contributions to the isotope dependence of the indirect energy gap are from electron— phonon coupling and from the dependence of the atomic spacing on the isotope. The volume change on changing the isotope from the natural isotopic content (effective mass number 72.6) to nominal Ge (with an average mass of A : 73.9) has been measured at low temperature (Buschert

6 Exciton—Phonon Interaction

149

V— V : (14.9 < 0.3) ; 10\. V

(24)

et al., 1988).

The change in energy with volume of the indirect gap is also known (Schmid and Christensen, 1990): : V dE/dV : 93800 meV.

(25)

Consequently the change in exciton energy with mass number A from the volume change is only about 12% of the total shift (dE/dA)

: 0.044 meV,

(26)

and most of the shift arises from the isotope dependence of the electron— phonon coupling. A simple estimate of this contribution (as in the case of diamond) to the energy shift can be made from measurements of the temperature dependence of the indirect energy gap. Evaluation of Eqs. (22) and (23) yields, for the range p : 0—0.8 (Davies et al., 1992) an electron— phonon contribution (dE/dA)

C\NF

: 0.22—0.30 meV.

(27)

The total predicted shift is the sum of the contributions in Eqs. (26) and (27) and is (dE/dA)

: 0.26—0.34 meV.

(28)

The observed shift in the indirect energy gap at the isotope effect in Ge of : 0.36 meV/amu (see also Agekyan et al., 1989; Cardona, 1994). G The electron—phonon interaction renormalization of a bandgap Enk in CdS can be expressed as (Zhang et al., 1998)

a 1 b 1 E(M, T ) : 9 n( , T ) ; 9 n( , T ) ; , (29) M ! 1 2 M 2 ! ! 1 1 where represents the unperturbed gap energy and the additional two terms on the right-hand side correspond to the contributions from the acoustic phonons (Cd vibrations) and optic phonons (S vibrations), respectively. Using an average acoustic-phonon frequency of < 60 cm\ and ! an average optic-phonon frequency of < 270 cm\, a least-squares ﬁt to 1 experimental data of the temperature dependence of the A and B excitons

150

Vladimir G. Plekhanov

Fig. 7. Temperature dependence of the A and B excitons in CdS. The ﬁlled circles and open squares display wavelength-modulated reﬂectivity data (Anedda and Fortin, 1976), while the diamonds and triangles represent the measured gap energy of the A bandgap reduced by the exciton binding energy of 27 meV (Benoit a la Guillaume et al., 1969; Seiler et al., 1982). The solid lines are least-squares ﬁts to the data performed with Eq. (29), the average phonon frequencies < 60 cm\ and < 270 cm\. The dashed and the dotted lines represent the ! 1 individual contributions of acoustic phonons (Cd vibrations with average frequency ) and ! optic phonons (S vibrations with average frequency ) to the shift of the B exciton, 1 respectively. (After Zhang et al., 1998.)

in CdS (Benoit a la Guillaume et al., 1969; Anedda and Fortin, 1976; Seiler et al., 1982) as shown in Fig. 7, yields a : 0.0134 and b : 0.1310 eV amu for the A exciton and a : 0.0159 and b : 0.0999 eV amu for the B exciton, respectively. The zero-temperature isotopic-mass-dependent renormalization of the electron energies can also be predicted using Eq. (29). Using the approximate expression . 1/(M one can ﬁnd: !1 !1 E a : ; M 4 M ! ! !

E b : . M 4 M 1 1 1

(30)

151

6 Exciton—Phonon Interaction TABLE II Isotope Shifts of the Excitonic Energies in CdS for Cd Substitution Measured by Zhang et al. (1998) Compared with the Results for S Substitution Extracted from Reflectivity Spectra in Kreingol’d et al. (1984) at T < 6 K. The Slopes Are Given in Units of eV/ amu PL

PR

Exciton

*(A)

(A)

*(A)

*(B)

E/M ! E/M 1

68 < 13

68 < 20

61 < 20 740 < 100

40 < 22 740 < 100

In this manner, Zhang et al. obtained E/M : 36 eV/amu (42 eV/ ! amu) and E/M : 950 eV/amu (724 eV/amu) for the A(B) exciton, 1 respectively. Given the simpliﬁed treatment of the CdS lattice dynamics by means of only two averages, the results are in surprisingly good agreement with the results for isotope substitution of the Cd and S atoms listed in Table II (Kreingol’d et al., 1984; Zhang et al., 1998). In all the preceding cases, the deformation potential interaction via acoustic phonons was considered. Since the calculations include only short-range (deformation-potential) electron—phonon interaction, it may be concluded that the long-range Fro¨hlich interaction (especially actualized for polar materials CdS and GaAs) is not important for the phenomenon treated in this section. The similar long-wavelength structure of spectra of mirror reﬂection of pure (LiH, LiD) and isotopically mixed crystals (see later and also Fig. 9 on p. 157) enables us to attribute it to the excitation of the ﬁrst and the second exciton states (Klochikhin and Plekhanov, 1980). As the temperature increases, the exciton reﬂection spectra of pure (see Fig. 2) and mixed crystals shift toward the longer wavelengths as a whole. The exciton structure of reﬂection spectra broadens. In the temperature range from 2 to 200 K, the line of the ground state of the exciton broadens approximately ﬁvefold. At T ( 130—140 K the peak caused by excitation of the exciton in the state n : 2s becomes indiscernible in the spectrum (Fig. 8). The reﬂection spectra of LiH D crystals, including LiD, behave in a similar V \V way. The ﬁgure shows the temperature dependence of the energy of long-wavelength maximum in the reﬂection spectrum for three crystals. For all three crystals, the dependence E : f (T ) in the temperature range Q T ( 140 K is well approximated by a linear function. The temperature coefﬁcients of linear shift found in this way are given in Table III. We see that the temperature coefﬁcient dE/dT is larger for the heavier isotope. Assuming that the temperature shift is caused by the interaction of excitons

152

Vladimir G. Plekhanov

Fig. 8. Temperature shift of location of maximum of 1s exciton state in the reﬂection spectrum of crystals LiH, curve 1; LiH D , curve 2; and LiD, curve 3. Experimental values shown by points, calculated values by solid lines. (After Plekhanov, 1997b.)

with optical vibrations, we have dE 4 dE E (0) —E (0) : ; , dT 3 dT (T )

(31)

where dE /dT :dE (LiD)/dT, dE /dT :dE (LiH)/dT ; LQ LQ T : (LiD)/k ; *and E (0) and E (0) are the values of the maximum of exciton band of LiD and LiH, respectively, at T : 0 K. Substituting the values of the quantities involved [ : 104 meV (Plekhanov, 1997b)] we ﬁnd that the magnitude *of dE /dT for LiD is 0.34 meV/K. Comparing this with the experimental LQ value (0.28 meV/K), we see that there is a rather large discrepancy, which is discussed a little later. In accordance with the arguments developed earlier, the temperature dependence of E (as well as that of E ) is shaped by two contributions: LQ E the lattice expansion, and the exciton—(electron)—phonon interaction, which is a sum of three terms. The ﬁrst of these can be found by evaluating the baric shift of E . It is well known that the effects of pressure on the LQ spectra of fundamental absorption is generally associated with changes in the width of the forbidden band, in the probabilities of transitions, and in

153

6 Exciton—Phonon Interaction

the effective masses of carriers (see, e.g., Bir and Pikus, 1972). For nondegenerate states, usually only the change in E is of practical importance. E Then one can expect that the exciton series shifts as a whole, without any signiﬁcant changes in the binding energy and the intensities of individual lines (see also Gross, 1976). When the pressure P is relatively small (when the pressure-induced change in the energy of a given state is much less than the distance to the adjacent levels), the band shift may be assumed to be a linear function of the pressure. In this approximation, the pressure dependence of E (and hence E ) can be expressed as E LQ E (P) : E (0) 9 (EA 9 ET )k P, 2 E E

(32)

where

1 V k :9 2 V P

2

E E ; E :V , V V

2

:

1 E :E . B k P 2

(33)

Here k is the isothermal compressibility and the hydrostatic deformation 2 potential (E ); and superscripts c and v stand for the conduction band and B the valence band. The deformation potential E in LiH we estimate from the B results of the paper by Kondo and Asaumi (1988). This study was concerned with measuring the energy of the maximum of the long-wavelength peak of mirror reﬂection of LiH crystals as a function of external hydrostatic pressure (P330 kbar). The absence of exciton structure (except for the n : Is state) in the reﬂection spectrum (see also Fig. 3 in Chapter 5) at room temperature leads to the assumption that the binding energy of the exciton does not depend on the pressure, and therefore the function E : f (P) LQ uniquely deﬁnes E : f (P), and vice versa. In the paper of Kondo and E Asaumi it was shown that in the pressure range 40 P 330 kbar this shift is linear with the baric coefﬁcients E/P : 91.1 meV/kbar. According to Guinan and Cline (1972), k : 3 ; 10\ bar\. Then E : 0.36 eV for LiH. 2 B Apart from this low value of E , one should also note the nonmonotonic B dependence E : f (P): at P 40 kbar, according to the calculations of LQ the band structure (Perrot, 1976; Kulikov, 1978), the quantity E (and LQ hence E ) increases and, after P ( 40 kbar, decreases linearly.* This behavE ior of the exciton (n : Is) maximum in the reﬂection spectrum may be attributed to two factors: on the electron transition 1. Removal of the ban with respect to K W —X , which is estimated theoretically to lie 0.03 eV below the direct X —X transition in LiH (for more details see Plekhanov et al., 1976).

*Observe also that, according to the results of Ghandehari et al. (1995) the baric coefﬁcient takes on two values: 0 P 300 Kbar E/P : 94.6 < 2 meV/kbar, and E/P : 90.237 < 0.001 meV/kbar in the range 300 P 2510 kbar. Note that, in contrast to LiH, the value of E in CsH starts to decrease immediately after application of external pressure. E

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Vladimir G. Plekhanov

2. Excitation, as the pressure increases, of the higher p-states in the conduction band, as noted earlier in Plekhanov et al. (1984) and Hama and Kawakami (1988). These two factors may also work simultaneously, which, on the other hand, may be responsible for the low value of E characteristic of the B electron transitions occurring at points other than the point of the Brillouin zone (cf. Wolford, 1987). Note also that for most semiconductor compounds with the structure of diamond (or zinc blende) the rate of change (E /P) of the quantity E at points , X, and L is, on the average, E E 12, 91.5, and 5 meV/kbar, respectively (Pankov, 1971; Moss et al., 1973), and thus differs not only in magnitude but also in sign (for more details, see Bir and Pikus, 1972). The value of E found enables one to estimate the contribution of the B lattice expansion into the change in E . It constitutes 12% at low LQ temperatures and 20% at room temperature of the entire shift for LiH crystals. We see that the main change in E (E ) comes from the LQ E Debye—Waller term and the term that accounts for the internal energy (the Fan term). As demonstrated in papers on semiconductors (see earlier), the inclusion of the latter two terms (which, incidentally, have opposite signs) along with the lattice expansion brings the theory into good agreement with the experimentally observed dependence E : f (T ). E The microscopic calculation of the temperature shift of E indicates B (Harrison, 1970) that this shift can only be related to those terms in the expansion of the potential energy that are quadratic with respect to displacement (see later and Zollner et al., 1992; Cohen and Chadi, 1980). In the one-oscillator Einstein model, the dependence of E on T can be E represented as E(T ) : E(0) 9 A coth

, 2k T

(34)

where E(0) is the value of E (E ) at T : 0, and is the effective E LQ frequency of the phonon in the model under consideration. The temperature dependence of the location of the maximum of the exciton peak in the reﬂection spectra of pure and mixed crystals, calculated according to Eq. (34), is shown in Fig. 8 by a solid line. As follows from Fig. 8, there is good enough agreement between theoretical curves and experimental data. The calculated energy of actual phonons , as can be seen from Table III, falls into the range of acoustic vibrations. The latter seems to point to the domination of the mechanism of deformation potential of electron—(exciton)—phonon interaction. This is a reasonable conclusion if we recall that the energy of the longitudinal optical phonon is 104 meV

155

6 Exciton—Phonon Interaction TABLE III

Values of Parameters Calculated by Formula (34) and Values of Temperature Coefficient of Line Shift Crystals LiH LiH D LiD

E (0) (meV) K

A (meV)

(meV)

dE /dT (meV/K) LQ

4961 5018 5061

12 < 1 15 < 1 17 < 1

11 < 1 12 < 1 13 < 1

0.19 < 0.01 0.22 < 0.01 0.25 < 0.01

for LiD, and 140 meV for LiH. This apparently also explains the discrepancy between theoretical and experimental values of the linear temperature coefﬁcient dE/dT (see earlier). Linear extrapolation (harmonic approximation) of the location of maximum of the long-wavelength exciton peak E to T : 0 K yields E(0) : LQ 4962 meV for LiH and 5066 meV for LiD. These results agree well with the calculated values (see second column in Table III). The difference between these two extrapolations is 104 meV, which practically coincides with the experimental value at T : 2 K; E : E (LiD) 9 E (LiH) : 103 meV (see, E E E e.g., Plekhanov, 1996a). Assuming that this value is wholly determined by the interaction of electrons with zero oscillations of the lattice, we can evaluate it using the Fan approximation (Fan, 1951):

M 91 , E (exp) : E E E M

(35)

where M and M are the masses of light and heavy isotopes, E is the E narrowing of the forbidden gap because of the preceding interaction, F (exp) is the experimentally observed narrowing of E equal to 103 meV. E E Obviously, in the case of LiH and LiD, the masses M and M must be replaced with the reduced masses of the elementary cell: : 7/8 and * & : 14/9. Substituting these values into Eq. (35), we ﬁnd that E : * " E 412 meV. Then the ‘‘actual’’ (the crystal lattice not perturbed by zero oscillations) width of the forbidden gap in LiH is E : 4992 ; 412 : 5404 E meV, which is greater than E at 2 K for LiD (5095 meV, see Table III) by E more than 300 meV (more than twice the energy for LiH). This *mismatch between experiment and theory (harmonic approximation) is too large. Observe that for isotopic substitution in ZnO (Kreingol’d, 1978), Ge (Agekyan et al., 1989), and diamond (Collins et al., 1990) a similar theoretical evaluation of E by Eq. (35) is in good agreement with the E experiment. Today, the reason behind such striking disagreement in the case of LiH is not quite clear. There are, however, at least two features that fundamentally distinguish LiH from ZnO, Ge, and C — to wit, isotopic

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substitution greatly changes the scattering potential (Plekhanov, 1995d), and zero oscillations give a substantial contribution to anharmonizm. Indeed, the energy of zero oscillations in the Debye approximation for a two-atom crystal is E (9/8k ,) where , is the Debye temperature. Given that , : 1190 < 80 K for LiH (Yates et al., 1974), we ﬁnd that E 115 meV, which is close to the energy of the LO phonon and much greater than E — in other words, it is not at all small. To conclude this @ section, note that the different temperature dependence of exciton peaks of n : Is and 2s states leads to the temperature dependence of the binding energies of Wannier—Mott excitons — this problem has not received adequate treatment. More speciﬁcally, the energy E in LiH crystals @ (Klochikhin and Plekhanov, 1980) decreases with increasing temperature, whereas E increases for excitons of the green and yellow series in Cu O @ crystals (Itoh and Narita, 1975).

IV. Isotopic Effect on Electron Excitations 1. Renormalization of Energy of Band-to-Band Transitions in the Case of Isotopic Substitution in LiH Crystals Isotopic substitution affects only the wavefunction of phonons; therefore, the energy values of electron levels in the Schro¨dinger equation ought to have remained the same. This, however, is not so, since isotopic substitution modiﬁes not only the phonon spectrum, but also the constant of electron— phonon interaction (see earlier). Therefore, the energy values of purely electron transitions in hydride and deuteride molecules are found to be different (Herzberg, 1945). This effect is even more prominent when we are dealing with a solid (Kapustinsky et al., 1937). Intercomparison of absorption spectra for thin ﬁlms of LiH and LiD at room temperature revealed that the long-wavelength maximum (as we now know, the exciton peak; Plekhanov et al., 1976) moves 64.5 meV toward the shorter wavelengths when H is replaced with D. For obvious reasons, this fundamental result could not then receive consistent and comprehensive interpretation, which does not belittle its importance today. As we see later, this effect becomes even more pronounced at low temperatures. The mirror reﬂection spectra of mixed and pure LiD crystals cleaved in liquid helium are presented in Fig. 9. For comparison, in the same diagram we also plotted the reﬂection spectrum of LiH crystals with clean surfaces. All spectra have been measured with the same apparatus under the same conditions. As the deuterium concentration increases, the long-wavelength maximum broadens and shifts toward the shorter wavelengths. As can clearly be seen in Fig. 9, all spectra exhibit a similar long-wavelength

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6 Exciton—Phonon Interaction

Fig. 9. Mirror reﬂection spectra of crystals: LiH, curve 1; LiH D , curve 2; and LiD, curve V \V 3 at 4.2 K. Light source without crystal, curve 4. Spectral resolution of instrument indicated in the diagram. (After Plekhanov, 1997b.)

structure. This circumstance enables us to attribute this structure to the excitation of the ground (Is) and the ﬁrst excited (2s) exciton states. The energy values of exciton maxima for pure and mixed crystals at 2 K are presented in Table IV. The binding energies of excitons E , calculated by the @ hydrogenlike formula, and the energies of interband transitions E are also E given in Table IV. The ionization energy, found from the temperature quenching of the peak of reﬂection spectrum of the 2s state in LiD is 12 meV. This value agrees fairly well with the value of E calculated by the hydrogenlike Q formula. Moreover, E : 52 meV for LiD agrees well with the energy of @ activation for thermal quenching of free-exciton luminescence in these crystals (Plekhanov, 1990b). Going back to Fig. 9, it is hard to miss the growth of (Plekhanov, 1996a), which in the hydrogenlike model causes an increase of the exciton Rydberg with the replacement of isotopes (Fig. 10). When hydrogen is completely replaced with deuterium, the exciton Rydberg (in the Wannier— Mott model) increases by 20% from 40 to 50 meV, whereas E exhibits a E TABLE IV Values of the Energy of Maxima in Exciton Reflection Spectra of Pure and Mixed Crystals at 2 K and Energies of Exciton-Binding E , Band-to-Band Transitions E @ E Energy (meV)

LiH

LiH D

LiH D

LiD

LiH (78 K)

E Q E Q E @ E E

4950 4982 42 4992

4967 5001 45 5012

5003 5039 48 5051

5043 5082 52 5095

4939 4970 41 4980

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Vladimir G. Plekhanov

Fig. 10. Binding energy of Wannier—Mott excitons as function of reduced mass of ions based on values of reduced mass of ions for LiH, LiH, LiD, LiD, and LiT. (After Plekhanov, 1996b.)

2% increase, and at 2—4.2 K is E : 103 meV. This quantity depends on E the temperature, and at room temperature is 73 meV, which agrees well enough with E : 64.5 meV, as found in the paper of Kapustinsky et al. E The continuous change of the exciton Rydberg was earlier observed in the crystals of solid solutions A B (Nelson, 1982; Nelson et al., 1976; Monemar et al., 1976) and A B (Radautsan et al., 1971; Brodin et al., 1984). Isotopic substitution of the light isotope (S) by the heavy one (S) in CdS crystals (Kreingol’d et al., 1984) reduces the exciton Rydberg, which was attributed to the tentative contribution from the adjacent electron bands (see also Bobrysheva et al., 1982), which, however, are not present in LiH (for more details see Kunz and Mickish, 1975; Baroni et al., 1985). The single-mode nature of the exciton reﬂection spectra of mixed crystals LiH D agrees V \V qualitatively with the results obtained with the virtual crystal model (see, e.g., Elliott et al., 1974; Onodera and Toyozawa, 1968), being at the same time its extreme realization, since the difference between ionization potentials (-) for this compound is zero. According to the virtual crystal model, - : 0 implies that E : 0, which is in contradiction with the experimental E crystals. By now the change in E caused by isotopic results for LiH D V \V E substitution has been observed for many broadgap and narrow-gap semiconductor compounds (see Section III.2). 2. The Dependence of the Energy Gaps of A B and A B Semiconducting Crystals on Isotope Masses In this section we brieﬂy discuss the variation of the electronic gap (E ) E of semiconducting crystals with its isotopic composition. In the last section the whole row of semiconducting crystals was grown. These crystals are

6 Exciton—Phonon Interaction

159

diamond (Collins et al., 1990; Collins, 1998), copper halides (Garro et al., 1996a; Go¨bel et al., 1997), germanium (Haller, 1995), and GaAs (Garro et al., 1996b). All enumerated crystals show the dependence of the electronic gap on the isotope masses. Before we complete the analysis of these results, we should note that before these investigations, studies were carried out on the isotopic effect on exciton states for a whole range of crystals by Kreingol’d and co-workers (Kreingol’d, 1978, 1985; Kreingol’d et al., 1976, 1977; Kreiongol’d and Kulinkin, 1986). We don’t know why these papers are unknown in Western scientiﬁc literature. First, the following are the classic crystals Cu O (Kreingol’d et al., 1976, 1977, 1984) with the substitution O ; O and Cu ; Cu. Moreover, there have been some detailed investigations of the isotopic effect on ZnO crystals, where E was seen to increase by 55 cm\ E (O ; O) and 12 cm\ (at Zn ; Zn) (see Fig. 11) (Kreingol’d, 1978; Kreingol’d and Kulinkin, 1986). In Kreingol’d et al. (1984) it was shown that the substitution of a heavy S isotope for a light S isotope in CdS crystals resulted in a decrease in the exciton Rydberg constant (E ), which was @ explained tentatively (Bobrysheva et al., 1982) by the contribution from the nearest electron energy bands, which, however, are absent in LiH crystals (see also Fig. 7 in Chapter 5). More detailed investigations of the exciton reﬂectance spectrum in CdS crystals were done by Zhang et al. (1998). The photoreﬂectance spectra of CdS and CdS are depicted in Fig. 12. Besides the exciton ground state of A and B excitons we clearly see the excited states of the A and B excitons.

Fig. 11. The reﬂection spectrum of ZnO crystal (solid line) and ZnO O crystal dashed line). (After Kreingol’d and Kulinkin, 1986.)

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Vladimir G. Plekhanov

Fig. 12. Photoreﬂectance of (a) CdS and (b) CdS at 6 K. The assignment of the spectral features to various components of the series of A and B excitons is indicated. (After Zhang et al., 1998.)

Polarization measurements indicate the features at about 2.574 and 2.590 eV arise from the A and B exciton transitions to the n : 2 excited states (2s), respectively. This enabled Zhang et al. to directly determine the binding energy (E ) and the corresponding bandgaps (E ) from a hydrogenlike @ E model (Knox, 1963): E : E 9 E /n. L E @

(36)

In this manner, it was possible to obtain the binding energies of A excitons of 26.4 < 0.02 meV and 26.8 < 0.02 meV in CdS and CdS, respectively. The corresponding bandgaps E ( — ) are, as denoted in E A T Fig. 13 by n : -, 2.5806 (2) eV and 2.5809 (2) eV, respectively. In the case of B excitons, these values are E : 27.1 < 0.2 meV (27.1 < 0.2 meV) and @ E ( — ) : 2.5964 (2) [2.5963 (2) eV] for the CdS ( CdS) sample. E A T Unfortunately, the n : 2 excited states of the A and B excitons could not be observed in other isotopic CdS samples. Better samples are required for such measurements.

6 Exciton—Phonon Interaction

161

For GaAs or ZnSe, isotope substituents of either type should lead to shifts of the E gap, which have been calculated to be 430 (420) and 310 (300) eV/amu for cation (anion) mass replacement, respectively (Garro et al., 1996). These values are in reasonable agreement with data measured for GaAs [E /M : 390 (60 eV/amu] (Garro et al., 1996a, b) and prelimi % nary results for isotopic ZnSe obtained by Zhang et al. (1998) based on photoluminescence measurements of the bound exciton (neutral acceptor I ) [E/M : 140 < 40 eV/amu and E/M : 240 < 40 eV/amu]. 1 8 Such behavior, however, is not found in wurtzite CdS. A previous reﬂectivity and photoluminescence study of CdS and CdS shows (Kreingol’d et al., 1984) that for anion isotope substitution the ground state (n : 1) energies of both A and B excitons have a positive energy shift with the large rate of E/M : 740 < 100 eV/amu. This value is more than one 1 order of magnitude larger than E/M obtained by Zhang et al. (see also ! Fig. 13). The electronic band structures of semiconductors with a diamond or zinc-blende crystal lattice have a degenerate valence-band maximum for : 0, point) as light and heavy holes at the center of the Brillouin zone (k well as a split-off valence band with its maximum at the same location in k space. Three conduction-band minima are observed at the high-symmetry points , L , and X. The lowest energy conduction-band minimum occurs in Ge at the L point and in Si near the X point, forming indirect bandgaps in these two elemental semiconductors. The width and the character (direct or indirect) of this lowest energy gap is of paramount importance for a large number of semiconductor properties and, in turn, for all semiconductor

Fig. 13. Dependence of the ground-state energy of the *(A) (a) and *(B) excitons (b) on the isotopic mass of Cd obtained from photoreﬂectance measurements at 6 K. The solid lines represent the best ﬁt with a straight line. (After Zhang et al., 1998.)

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Vladimir G. Plekhanov

devices. Because of this great signiﬁcance there exists a strong interest in all effects that inﬂuence the band structure. The isotopic composition affects the bandgaps through the electron— phonon coupling and through the change of volume with isotopic mass. Several groups have conducted low-temperature studies of the direct and indirect bandgaps of natural and isotopically controlled Ge single crystals. For the ﬁrst time, Agekyan et al. (1989) used photoluminescence, infrared absorption, and Raman spectroscopy with Ge crystals of natural composition and crystals with 85% Ge and 15% Ge. They found an indirect (see also Fig. 14) bandgap change E : 0.9 meV and a direct bandgap change E E : 1.25 meV with an error of <0.05 meV. Etchegoin et al. (1992) and E Davies et al. (1992) reported photoluminescence studies of natural and several highly enriched, high-quality single crystals of Ge. Measurement of the energies of impurity-bound excitons by Davies et al. enabled the direct determination of bandgap shifts with the crystal isotope mass because the radiative recombination does not require phonon participation. Figure 15 shows the no-phonon energies of excitons E bound to P and Cu in several ,. isotopically controlled crystals. As expected from the very large Bohr orbit of the excitons (see Davies et al., 1992 and the references therein), their binding energy depends only on the average isotope mass and not on the isotopic disorder. The rate of bandgap energy change with isotope mass as determined by Davies et al. is dE /dA : dE /dA : 0.35 < 0.02 meV/amu. '% ,.

(37)

Etchegoin et al. (1992) obtained a very similar value. The contribution to the bandgap shift originating in the volume change can be estimated using the results of Buschert et al. (1988) for the lattice-constant change with isotope mass and the published dependence of

Fig. 14. Transmission spectra of Ge (1) and Ge (2) in the vicinity of the direct excitons transitions at 1.7 K. (After Agekyan et al., 1989.)

6 Exciton—Phonon Interaction

163

Fig. 15. Energies of the no-phonon lines of excitons bound to Cu acceptors (squares) and P donors (circles). (After Davies et al., 1992.)

E on volume (Schmid and Christensen, 1990). They found '% (dE /dA) : 0.132 meV/amu. '%

(38)

This is the smaller contribution to the experimentally determined energy— gap change with isotope mass. It is in reasonable agreement with the earlier estimates of Agekyan et al. The main contribution to dE/dA can be directly related to the change of the energy gap with temperature (see also earlier). This change is described by structure factors that contain electron—phonon interaction terms (Debye—Waller factors) and self-energy terms. For practical calculations these terms are expanded in a power series of the atomic displacements. The leading terms are proportional to the mean-square displacements u of each atom. Describing the lattice atoms in terms of harmonic oscillators, one ﬁnds

u :

1 ;n 2

M.

(39)

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Vladimir G. Plekhanov

With increasing temperature, both n and u increase, leading to the observed reduction of the energy gap. At low temperatures, n : 0 and we deal only with the zero-point oscillation. Combining the dependence of on M with the preceding equation, one ﬁnds

u . 1/(M.

(40)

Zollner et al. (1992) have performed a numerical calculation (see also earlier) of the electronic bands using an empirical pseudopotential method including the necessary lattice dynamics. They found for Ge (dE /dA) : '% C\N 0.41 meV. The total calculated shift of the indirect bandgap energy with isotope mass adds up to (dE /dA) : 0.48 meV. This result compares '% favorably with the experimental values stated in the preceding by Davies et al. and by Etchegoin et al., who reported (dE /dA) : 0.37 < '% 0.01 meV/amu (see also Fig 16). : 0) in the Measurements of the direct bandgap at the point (k Brillouin zone have also been performed. Although the direct bandgap is technologically less important than the minimum indirect bandgap, determining the dependence of this gap on isotope mass is of the same fundamental signiﬁcance as the indirect bandgap studies. Davies et al. (1993) used

Fig. 16. Atomic mass dependence of the indirect gap E of Ge at T : 6 K. (After Parks et al., E 1994.)

6 Exciton—Phonon Interaction

165

Fig. 17. Isotopic mass (in amu) dependence of the (a) E and (b) E ; direct energy gaps obtained from photomodulated reﬂectivity measurements at T : 6 K. The curves are the best to relation E : E ; C/(M, where M is the atomic mass and E is the energy gap at M : -. The ﬁtting yields E : 959 meV and C : 9606 meV/amu. (After Parks et al., 1994.)

low-temperature optical-absorption measurements of very thin samples of Ge single crystals with natural composition and three different, highly enriched isotopes. They found dE/dA : 0.49 < 0.03 meV/amu

(41)

for the temperature extrapolated to zero. Parks et al. (1994) used piezo- and photomodulated reﬂectivity spectra of one natural and four monoisotopic Ge crystals. These techniques do not require the extreme sample thinning, which is necessary for optical-absorption measurements, and the derivative nature of the spectra emphasizes the small changes. The excellent signal-tonoise ratio and the superb spectral resolution enabled a very accurate determination of the dependence of E on isotopic mass (Fig. 17). At very "% low temperatures an inverse square-root dependence accurately describes the bandgap dependence: E

C : E ; . "% (M "%

A ﬁt through ﬁve data points yields E : 959 meV "%

and

C : 9606 meV/amu.

(42)

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Vladimir G. Plekhanov

Written as a linear dependence for the small range of isotopic masses, Parks et al. found dE /dA : 0.49 meV/amu, in perfect agreement with the results "% of Davies et al. (1993). Parks et al. also determined the isotope mass dependence of the sum of the direct gap and the split-off valence band ( ) and found d(E ; )/dA : 0.74 meV/amu. The experimental results can "% be compared to the Zollner et al. (1992) calculations, which are found to be of the correct order of magnitude. The theoretical estimates for the contributions of the linear isotope shifts of the minimum, indirect gaps, which are caused by electron—phonon interaction, are too large by a factor of :1.7 and for the smallest direct gap, they are too large by a factor of :3.2. Substitution of Ga on the Ga increases the bandgap in GaAs (Garro et al., 1996a) on 10.5 cm\ (see also Table V). The interesting results were communicated in the papers of Cardona and co-workers (Garro et al., 1996a; Go¨bel et al., 1997), where the dependence of E on the isotope effect E in CuCl crystals was studied. When the Cu on the Cu is substituted, the value of E in CuCl crystals decreased by 1.24 cm\, for example, the E isotope effect on the electronic excitation has an opposite sign. Considering the series of Ge, GaAs, ZnSe, CuBr, for example, the 3d states of the ﬁrst constituent play an increasing role in determining the band structure. In Ge, these states can be considered as localized core states (atomic energy level 930 eV). In GaAs, however, they have already moved up in energy by 10 eV, and their hybridization with the top of the valence band affects the gap (see, e.g., Harrison, 1970). Proceeding further in the series, this effect becomes more important, and the CuBr and Cu 3d states even overlap in energy with halogen p-states, with which they strongly hybridize. Therefore, we cannot exclude that the main reason for the opposite sign of the isotopic effect in these compounds may be connected to the different character of the d-electron—phonon interactions in these semiconductors (Plekhanov, 1998). The change of the indirect gap of diamond between pure C and C was determined by Collins et al. (1990), using for this purpose the luminescence spectra of diamond. The luminescence spectra of the natural (C) and synthetic (C) diamond were investigated by Collins et al. (1990), Ruf et al. (1998), and Collins (1998). Figure 18 compares the edge luminescence for a natural diamond with that for a synthetic diamond. The peaks labeled A, B, and C are due, respectively, to the recombination of a free exciton with the emission of transverse-acoustic, transverse-optic, and longitudinal-optic phonons having wave vector
et al. (1964):

: 87 < 2, : 141 < 2, : 163 < 1 meV. 2 2*Features B and B are further free-exciton processes involving the preced ing TO phonon with one- and two-zone center optic phonons, respectively.

6 Exciton—Phonon Interaction

167

Fig. 18. Spectra measured at 77 K of the phonon-assisted free exciton cathodeluminescence feature (A, B, and C) and the phonon-assisted bound-exciton features (D) from a natural semiconducting C diamond and a C synthetic diamond. (After Collins et al., 1990.)

Boron forms an effective-mass-like acceptor in diamond, and both specimens used in Fig. 18 are slightly semiconducting with uncompensated boron concentrations around 5 ; 10 cm\ in the natural diamond and 3 ; 10 cm\ in the synthetic diamond. Peaks labeled D are associated with the decay of excitons bound to the boron acceptors (for details see Collins et al., 1990). Comparison of the data from the two diamonds shows that the zero-phonon lines D and D are 14 < 0.7 meV higher for C than for C diamond, and that the LO and TO phonon energies are lower by a factor of 0.96 (cf. Subsection 2 of Section IV), equal within experimental error to the factor (12/13) expected to be ﬁrst order when the lattice is changed from C to C. The low-energy thresholds of the free-exciton peaks A, B, and C are given by Collins et al. (1990): E (A) : E 9 , E (B) : E 9 , EV 2 EV 2-

and E (C) : E 9 . EV *-

As was shown by Collins et al., the predicted thresholds are entirely consistent with the experimental data. From the results of Collins et al., it was concluded that the dominant contribution arises from electron—phonon

168

Vladimir G. Plekhanov

Fig. 19. Cathode-luminescence spectra of isotopically modiﬁed diamond at 36 K. Intrinsic phonon-assisted recombination peaks are labelled in the top spectrum, those from boronbound excitons in that at the bottom. The spectra are normalized to the intensity of the B peak and vertically offset for clarity. (After Ruf et al., 1998.)

coupling, and that there is a smaller contribution due to a change in volume of the unit cell produced by changing the isotope. These two terms were calculated as 13.5 < 2.0 and 3.0 < 1.3 meV, respectively. The more detailed and quantitative investigations of E : f (x), where x is the isotope concenE tration, were done by Ruf et al. (1998), who studied ﬁve samples of diamond with different concentrations x (Fig. 19). From these data Ruf et al. determined the linear variation of E : f (x) for diamond (Fig. 20). Linear E ﬁts of the experimental data of Ruf et al. (solid line in Fig. 20) yield a slope of 14.6 < 0.5 meV/amu, close to the theoretical predictions. All of these results are documented in Table V, where the variation of E , E E , and are shown at the isotope effect. We should highlight here that @ *the most prominent isotope effect is observed in LiH crystals, where the dependence of E : f (C ) is also observed and investigated. To end this @ & section, we note that E decreases by 97 cm\ when Li is replaced with Li E (see Table V). 3. Renormalization of Binding Energy of Wannier—Mott Excitons by the Isotopic Effect In the original work of Plekhanov et al. (1976) the exciton binding energy E was found to depend on the isotopic composition and this change in E @ @ was attributed to the exciton—phonon interaction (originally with LO

6 Exciton—Phonon Interaction

169

Fig. 20. Energy of the D multiplet in isotopic diamond at 36 K. The ﬁlled symbols are for the main component, the open ones for weaker side peaks. The solid lines are linear ﬁts to the data. (After Ruf et al., 1998.)

phonons). The preferential interaction of excitons with LO phonons in LiH (LiD) crystals was later repeatedly demonstrated in the luminescence spectra (Plekhanov, 1990a) and resonant Raman light scattering (Plekhanov and O’Konnel-Bronin, 1978a; Plekhanov and Altukhov, 1985), which consist of a phononless line (in the former case) and its LO repetitions. The effects of the Fro¨hlich mechanism of exciton—phonon interaction on the energy spectrum of Wannier—Mott exciton has been considered over and over again (Peierls, 1932; Haken, 1976; Segall and Marple, 1967; Segall and Mahan, 1968; Fedoseev, 1973; Ansel’m and Firsov, 1956; Thomas, 1967; Firsov, 1975; Toyozawa, 1958; Fro¨hlich, 1954; Klochikhin, 1980; Bulyanitza, 1970; Fan, 1967; Singh, 1984). Today we know that the main consequences of the electron and hole interaction in excitons with polarization vibrations are the static screening of the lattice charges (introducing ) and the change in the effective masses of the particles. Both these effects of electron—(hole)— phonon interaction can easily be taken into account, and lead to a change in the exciton Rydberg E . These corrections do not destroy the hydrogen@ like structure of the exciton spectrum. At the same time, the non-Coulombic corrections to the electron—hole Hamiltonian modify the hydrogenlike structure — removing, for example, degeneration of levels with respect to orbital and magnetic quantum numbers (see, e.g., Fedoseev, 1973). The very fact, however, that the problem of renormalization of energy spectra of Wannier—Mott excitons does not result in an exact solution even in the limiting cases, often gives rise to a situation in which there is no agreement between the results obtained by different authors. Starting with the classical works of Haken (1976), all papers can be divided into two broad classes depending on how they deal with the Coulomb interaction: between ‘‘bare’’

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electrons and holes, or between electrons and holes in the polaron state. In other words, ﬁrst the interaction of band electrons and holes with LO optical phonons is taken into account, and then the Coulomb interaction between electrons and holes clad in the ‘‘polarization coats’’ is considered. As shown in the following, the study of exciton—phonon interaction in crystals with isotopic effect not only provides entirely new information, but also enables us to reconstruct experimentally the values of Fo¨hlich and Coulomb interaction constants. From Fig. 10 we see that when hydrogen is completely replaced with deuterium, the binding energy of the exciton exhibits a 20% increase from 42 to 52 meV (Plekhanov et al., 1976). It is easy to see that in the model of virtual crystals the binding energy of the exciton in LiT crystals (Plekhanov, 1996c) must be equal to 57 meV (see Fig. 10). Hence it follows that in the linear approximation the isotopic dependence of binding energy of Wannier—Mott excitons may be expressed as E : E (0) (1 ; ). @ @

(43)

where E (0) is the purely Coulombic binding energy of the exciton (in the @ frozen lattice), which in our case is equal to 31.5 meV, and the angular coefﬁcient is : 12.18 meV/M, where M is the reduced mass of ions of lithium and hydrogen (deuterium, tritium) ions; : M/E (0) (see also @ Plekhanov, 1996c). From the standard equation for the Coulomb binding energy of the exciton, e E : , @ e

(44)

we get the dimensionless constant of Coulomb interaction: E (0) * : @ : 0.47.

*Comparing the value of * : 0.47 and the constant of Fro¨hlich exciton— phonon interaction g : 0.33 (Plekhanov and Altukhov, 1981) we see that they are close enough. This implies that both the Fro¨hlich and the Coulomb interactions between electrons (holes) and LO phonons in the exciton must be treated with equal attention, as has already been emphasized in Klochikhin (1980). This paper deals from the start with ‘‘bare’’ electrons and holes, and all renormalizations are calculated in the two-particle conﬁguration. Such an approach enables us to avoid the considerable difﬁculty that arises when polarons (Sak, 1972) are used as start-up particles. This difﬁculty is primarily associated with the fact that the momentum of each particle is

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6 Exciton—Phonon Interaction

conserved when the particles are treated separately, whereas it is the center-of-mass momentum that is conserved when a pair moves as a whole. As demonstrated in Klochikhin (1980), this approach also makes it possible to calculate the higher-order corrections to the exciton—phonon interaction. It was also shown that the use of the pole parts of polaron Green functions in place of complete expressions in Sak (1972) and Mahanti and Varma (1972) leads to a situation where the corrections of the order of *g and g to the potential energy are lost because the corrections to the vertex parts and Green functions cancel out. The quantity lost is of the same order (g) as the correction to the residue but has the opposite sign (for more details see Sak, 1972; Mahanti and Varma, 1972). The approach developed in Klochikhin (1980) enabled the calculation of corrections of the order of *g and g, the latter is comprised of the correction to the Fro¨hlich vertex and the correction to the Green functions in the exciton—phonon loop. It is important that the latter have opposite signs and cancel out exactly in the limit E . As a result, because of the potential nature @ *of the start-up Coulomb interaction, the correction to the Coulomb vertex of the order of *g does not vanish. As a result, the following expression was obtained in Klochikhin’s paper for the binding energy E of Wannier— @ Mott excitons when E (the spectrum of exciton remains hydrogen@ *like): E : @ *-

* 9 g ; *g(c ; v) 2

(45)

where c, v : (m /) and m and m are the electron and hole masses. AT A T Now E depends explicitly on g (the Fro¨hlich constant of exciton—phonon @ interaction), and hence depends on the isotopic composition of the lattice, whereas the standard expression for the binding energy E : (* 9 g) : e/2 , @ * which describes the exciton spectrum of many semiconductors accurately enough, exhibits no dependence on the isotopic effect. In the case of Eq. (45) the exciton spectrum remains hydrogenlike. When the higher-order corrections are taken into account, Eq. (45) becomes

e m ;m 19 T ; g - ; (m ; m ) . (46) E : 1 ; g A @ 2 A T The order-of-magnitude evaluation of the coefﬁcients - , - gives - 0.15 and - 0.02; when g(m ; m ) 3.3, the correction of the order A T of *g is much less than the term of the order of *g (Klochikhin, 1980).

172

Vladimir G. Plekhanov TABLE V The Change of the Value of the Exciton Binding Energy (E ), Band-to-Band @ Transitions (E ) at the 100% Isotopic Substitution E

Compound Ge ; Ge GaAs (Ga ; Ga) Cu O (O ; O) Cu O (O ; O) CdS (S ; S) CdS (Cd ; Cd) CuCl (Cu ; Cu) ZnO (O ; O) ZnO (Zn ; Zn) C ; C LiH (H ; D) LiH (Li ; Li) All 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

E E Indirect (L point) 2.1 meV : 16.94 cm\ [1,2,3] Direct ( point) 0.85 meV : 10.5 cm\ [4] Direct (yellow) 2.2 meV : 18 cm\ [5] Direct (green) 2.2 meV : 18 cm\ [5] — Direct 0.003 meV ( point) 90.155 meV : 91.24 cm\ [9] ( point) 6.82 meV : 55 cm\ [10,11] ( point) 1.5 meV : 12 cm\ [10,11] Indirect ( point) 13.6 meV : 109.7 cm\ [12] Direct 103 meV : 831 cm\ [15,16,17] Direct 5 meV : 41 cm\ [16]

E

@

— — — — 91.6 meV [7] 0.4 meV [8] — — — — 10 meV [16] 3 meV [16]

values are in meV or cm\. V. F. Agekyan et al., Fizika Tverdogo Tela 31, 101 (1989). C. Parks et al., Phys. Rev. B49, 14244 (1994). J. J. Haller, J. Appl. Phys. 77, 2857 (1995). N. Garro et al., Phys. Rev. B54, 4732 (1996). F. I. Kreingol’d et al., Pis’ma v ZET PH 23, 679 (1976). F. I. Kreingol’d, Fizika Tverdogo Tela 27, 2839 (1985). F. I. Kreingol’d et al., Fizika Tverdogo Tela 26, 3490 (1984). J. M. Zhang et al., Phys. Rev. B57, 9716 (1998). N. Garro et al., Solid State Commun. 98, 27 (1996). F. I. Kreingol’d Fizika Tverdogo Tela 20, 3138 (1978). F. I. Kreingol’d and B. S. Kulinkin, Fizika Tverdogo Tela 28, 3164 (1986). T. Collins et al., Phys. Rev. L ett. 65, 891 (1990). R. M. Chrenko, J. Appl. Phys. 63, 5873 (1988). K. S. Hass et al. Phys. Rev. B54, 7171 (1992). A. A. Klochikhin and V. G. Plekhanov, Fizika Tverdogo Tela 22, 585 (1980). V. G. Plekhanov, Uspekhi Fiz. Nauk 167, 577 (1997b). A. F. Kapustinsky et al., Physicochimica USSR 7, 799 (1937).

So from Eq. (46) we see that the correction to the purely Coulombic binding energy, Eq. (44), is important. This is primarily because the values of * and g are close to each other. Setting m /m : 3.5 and g/* : I 9 / , and T A ( / ) :( / : 1/3.5, in Klochikhin and Plekhanov (1980) it was 2- *found that E : 48 and 42 meV for LiD and LiH, respectively. Com@ paring these results with the experimental values (see Table V) we observe good agreement between theory and experiment. Hence it follows a natural conclusion that the isotopic dependence of the exciton binding energy is due primarily to the Fro¨hlich interaction mechanism between excitons and phonons.

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6 Exciton—Phonon Interaction

Fig. 21. Temperature dependence of polaron contribution to the change in E for isotopic E substitution in LiH crystal. (After Plekhanov, 1990a.)

In the preceding section we saw that isotopic substitution affects not only E , but also E . For the LiH—LiD system at low temperatures @ E the difference is E : 103 meV. Apart from the zero oscillations conE sidered in the previous section (see also Kreingol’d, 1978; Allen, 1994), this change is also contributed to by the polaron shift, which explicitly depends on the temperature. Figure 21 shows the temperature dependence of E : E (LiD) 9 E (LiH), which is generally similar to E : f (T ). LQ LQ E From this diagram, we see that as the temperature rises from 2 to 300 K, the quantity E decreases from 103 to 73.5 meV. The last result agrees well with the value of 65 meV obtained by Kapustinsky et al. In light of the Fro¨hlich mechanism of exciton—phonon interaction considered in the preceding, the magnitude of the polaron shift may be estimated by the following expression:

91 E : 9E @ *-

(1 ; m ; A

1;

1 . m T

This estimate gives about 20 meV (Plekhanov, 1997b), which is about 1/4E; on the other hand, it agrees well enough with the magnitude of the polaron shift, as derived from the temperature dependence of the shift (see Fig. 21). Although the isotopic change in E has been observed in a large number E of compounds (dielectrics and semiconductors), the number of studies of isotopic effect on the exciton binding energy is limited to four cases. As demonstrated earlier, the replacement of a light isotope with a heavy one in LiH crystals (Plekhanov et al., 1976) leads to an increase in E ; the binding @ energy decreases in CdS crystals (Kreingol’d et al., 1984) and remains the same in the crystals of germanium (Parks et al., 1994) and diamond (Collins

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Vladimir G. Plekhanov

et al., 1990). Quantitative measurements of the isotopic effect on the levels of large-radius exciton allow for experimental reconstruction of the Fro¨hlich and Coulomb constants. 4. Luminescence of Free Excitons in LiH Crystals Because of the low intensity of scattered light, and thanks to the high resolution of modern spectroscopic instruments, the development of highly sensitive techniques for the detection of weak optical signals (photon counting mode, optical multichannel analyzers, optical linear arrays, and other specialized systems (see, e.g., Chang and Long (1982), the luminescence method has become one of the most common techniques for studying excitons in dielectrics and semiconductors. While the structure of spectra of fundamental reﬂection (absorption) depends on the internal degrees of freedom of the Wannier—Mott exciton, the structure and shape of the luminescence spectrum are determined primarily by its external degrees of freedom. The latter are associated with the translation motion of largeradius excitons as a whole, with the translation mass M : m ; m (Knox, A T 1963). The results on the luminescence (RRLS) of LiH D crystals V \V presented in the following were obtained from the clean surfaces of crystals cleaved directly under liquid superﬂuid helium in the cell of optical cryostats (Plekhanov et al., 1984). The effects of surface habitus on optical spectra (including the luminescence spectra) of excitons in hygroscopic LiH and LiD crystals were brieﬂy described earlier (Plekhanov et al., 1984; Pilipenko and Gavrilov, 1985; Plekhanov, 1987). The luminescence of LiH crystals was ﬁrst observed in 1959 by Gavrilov. The broadband luminescence in the red part of the spectrum was attributed to the defects related to the nonstoichiometric composition of crystals (see also Plekhanov et al., 1988). Later the broadband luminescence of pure and activated (mostly with mercurylike ions with s outer electron conﬁgurations LiH crystals were studied in a large number of works (see the review of Plekhanov et al., 1988, and the references therein). The ﬁrst results on photoluminescence (Pustovarov, 1976) of LiH crystals near the edge of fundamental absorption, like the ﬁrst results on cathode-ray luminescence (Zavt et al., 1976a) were rather qualitative. The results on cathode-ray luminescence spectra of LiH crystals at 6 K are described in greater detail in Zav’yalov et al. (1985); they are analyzed and compared with the photoluminescence spectra in Plekhanov et al. (1988). LO repetitions were discovered in the spectra of x-ray luminescence of LiH crystals cleaved in the inert gas environment (Plekhanov et al., 1977). The discovery of the LO structure of luminescence spectra and luminescence excitation spectra, and later that of RRLS (Plekhanov and O’Konnel-Bronn, 1978b) in LiH (LiD) crystals, has presented the opportunity for spectroscopic

6 Exciton—Phonon Interaction

175

Fig. 22. Emission spectra of free excitons at 2 K in LiH crystals cleaved in liquid helium. Spectral resolution of instrument indicated on diagram. (After Plekhanov, 1995a.)

studies of energy relaxation processes in the course of interaction with phonons (Plekhanov, 1981). As demonstrated earlier, most low-energy electron excitations in LiH crystals are the large-radius excitons. Exciton luminescence is observed when LiH crystals are excited in the midst of fundamental absorption. The spectrum of exciton photoluminescence of crystals of lithium hydride cleaved in liquid helium consists of a narrow (in the best crystals its half-width is E 10 meV; Plekhanov et al., 1984; Plekhanov and Altukhov, 1985) phononless emission line and its broader phonon repetitions, which arise due to radiative annihilation of excitons with the production of 1—5 longitudinal (LO) phonons (Fig. 22). The phononless emission line coincides in an almost resonant way with the reﬂection line of the exciton ground state (Plekhanov, 1990b), which is another indication of direct electron transition. The lines of phonon replicas form an equidistant series biased toward the lower energies from the resonance emission line of excitons. The energy difference between these lines, as in Plekhanov (1981) is about 140 meV, which is close to the calculated energy of LO phonons in the middle of the Brillouin zone (Verble et al., 1968) and measured by Plekhanov and O’Konnel-Bronin (1978a). The most important distinctions between the exciton luminescence spectrum shown in Fig. 22 and those measured earlier (Plekhanov et al., 1977) are the following: (a) the presence of a second series of LO repetitions counted from the level of the n : 2s exciton state, (b) comparable intensities of the phononless emission line and its 2LO replica, and (c) noticeable narrowing of the observed lines (see also Plekhanov, 1987). Here note also an overall increase in the intensity over the entire luminescence spectrum. Evidently, the intensity of phononless emission lines of free excitons (proof of the existence of quasimomentum is given in the next subsection) increases because the rate of emissionless recombination on the pure surface decreases. This seems natural for the

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surface of a specimen cleaved in liquid helium because the surface states (as a rule, of extrinsic origin; Fischer and Stolz, 1982; Schultheis and Balslev, 1983) and their electric ﬁelds (recall that the value of the exciton Rydberg is relatively low, E : 40 meV) lead not only to broadening of the lumines@ cence lines, but also to quenching of their intensity, and ﬁrst of all, quenching of the intensity of the zero-phonon line.

a. Proof of Existence of Quasi-Momentum of Excitons in LiH (LiD) Crystals It is well known that one of the main properties of the exciton is its ability to move freely over the crystal lattice (Knox, 1963; Gross, 1976). In the effective-mass approximation (Dresselhaus, 1956), an exciton is regarded as a quasi-particle associated with a certain value of the wave vector (quasimomentum) k, the change whereof characterizes the motion of the exciton. The proof of motion of the exciton, especially in semiconductor crystals, has been the subject of many papers (see, e.g., Knox, 1963; Gross, 1976; Agekyan, 1977; Segall and Marple, 1967; Segall and Mahan, 1968; Thomas, 1967; Gross et al., 1971; Permogorov, 1975). It was ﬁrst demonstrated by Gross and co-workers (1971) that the exciton in a crystal may possess a considerable amount of kinetic energy. When the exciton distribution is in equilibrium, the kinetic energy is known to be determined by the temperature of the crystal lattice. The studies of the lineshapes and the temperature dependence of their relative intensity in the spectra of exciton—phonon luminescence have supplied the most comprehensive information concerning the exciton motion (and have proved this motion to be free), which enabled the establishment of the exciton distribution law within the band (Plekhanov, 1997b). Indeed, according to the law of conservation of quasimomentum, the phononless emission is possible only for those excitons whose wave vector is of the same order of magnitude as the wave vector of 0), and whose kinetic energy is practically zero. the exciting photons (k Then, the phononless emission line is narrow. The emission processes, however, which are associated with excitation of one or several optical phonons, may involve excitons with arbitrary kinetic energy and wave vector values. In this case, the excess thermal motion wave vector of the exciton is passed on to the phonons. Since the energy of optical phonons, as a rule, does not depend on the wave vector, the emission spectra will contain lines whose long-wavelength edges are displaced (see Fig. 22) by the energy of a whole number of optical phonons with respect to the energy of the phonon ground state — the bottom of the exciton band (see also Gross et al., 1971). The shape of the emission lines, as indicated earlier, reﬂects the distribution of excitons with respect to their kinetic energy E . Such LO structure has actually been observed in the spectra of intrinsic luminescence

6 Exciton—Phonon Interaction

177

of a large number of semiconductor crystals. The shape of LO phonon repetitions is described by the Maxwell distribution of excitons with respect to kinetic energy:

dW F : W E exp 9 , dE k T

(47)

where W is the probability of exciton—phonon interaction, which depends on k in the case of one-phonon scattering, and does not depend on k in the case of two-phonon scattering (Gross et al., 1971; Klochikhin et al., 1976). Figure 23 shows the comparison of experimental ﬁrst- and second-order lines of exciton luminescence in LiH crystals with the shape described by Eq. (47). In general, the agreement between theory and experiment is reasonably good. At the same time, we notice that the half width of the line is somewhat larger, and its shape on the long-wavelength side differs from that described by Eq. (47). As shown in the following (see also Plekhanov et al., 1988; Plekhanov, 1994a), this deviation is mainly determined by the magnitude of the longitudinal-transverse splitting — that is, by the strength of exciton—photon interaction (Knox, 1963; Benoit a la Guillaume et al., 1970). Reasonable accuracy in the description of the shape of ILO and 2LO repetitions can be achieved by assuming that the temperature of the exciton gas is 200 K (Plekhanov, 1997b). This value is more than three times as high as the temperature of crystal in the cryostat, which suggests that there is no thermodynamic equilibrium between the excitons and the lattice. This is one of the reasons why the emission lines of LO replicas in crystals cleaved in liquid helium exhibit broadening even at the lowest temperature. This circumstance has already been noted in Plekhanov et al. (1984). A detailed study of the temperature evolution of emission spectra of free excitons in LiD crystals (Plekhanov et al., 1988) revealed a pronounced short-wavelength shift of the lines of the ILO and 2LO replicas with the increasing temperature (see also Fig. 26), which is obviously associated with

Fig. 23. Lineshapes of 1LO and 2LO replicas in exciton luminescence spectrum of LiH crystal at 62 K, and Maxwell approximation of distribution of excitons with respect to kinetic energy (dashed line at 200 K). (After Plekhanov et al., 1984.)

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Vladimir G. Plekhanov

the shift of the Maxwell distribution. This is a clear indication that excitons in these crystals do possess kinetic energy. This conclusion ﬁts in well with the results on the temperature dependence of the relative intensity of ILO and 2LO repetitions. In the temperature range where the exciton distribution is in equilibrium with respect to kinetic energy, this dependence is well described by a linear function (Toyozawa, 1958). Thus we can conclude that the preceding results and their consistent interpretation clearly indicate that the observed emission is the emission of free excitons possessing a considerable amount of kinetic energy. Accordingly, the motion of excitons is coherent and complies with the law of conservation of quasi-momentum k.

b. Band Relaxation of Free Excitons Detailed information regarding the kinetic energy of free excitons can be extracted from the luminescence excitation spectra of free excitons (Gross, 1976; Gross et al., 1970). This function directly reﬂects the process of exciton relaxation over the band. The important circumstance is that while the kinetic energy of excitons in the case of exciton—phonon luminescence is usually of the order of k T, it can considerably exceed the binding energy of the exciton in the case of luminescence excitation spectra, as ﬁrst demonstrated in Gross et al. (1970). As an example, let us consider the exciton luminescence excitation spectrum (Fig. 24) of LiH crystals cleaved in liquid helium. The most striking difference between the results displayed in Fig. 24

Fig. 24. Luminescence excitation spectrum of the 2LO replica line in a LiH crystal at 4.2 K and original record in the region of the exciton ground and ﬁrst excited state. (After Plekhanov, 1987.)

6 Exciton—Phonon Interaction

179

Fig. 25. Luminescence excitation spectrum of the 1LO replica line in a LiH crystal at 4.2 K (After Plekhanov et al., 1988.)

and the excitation spectra of crystals cleaved in hot air (Plekhanov, 1981) show the presence of ﬁne structure exactly in the exciton region. The observed structure of the excitation spectrum is related primarily to the manifestation of the n : 2s exciton state. In addition, the long-wavelength wing (see arrow I in Fig. 24) features a kind of ‘‘singularity’’ at a distance of 8 meV. This singularity may be related to the intervalley splitting (Plekhanov and Altukhov, 1981), given the existence of three equivalent X valleys in the band structure of these crystals (Baroni et al., 1985) where direct and allowed transition takes place (Plekhanov et al., 1976). As the exciton kinetic energy increases, the observed structure smears out (for more details see Plekhanov, 1987). This may be caused both by the reduced probability of indirect transitions in the course of absorption at large wave vector values (Ansel’m and Firsov, 1956; Yu, 1979), and by the reduced probability of the generation of excitons from recombination of electron—hole pairs (Lipnik, 1964, Zinov’ev et al., 1983; Abakumov et al., 1980). The results presented in Fig. 24 lead to the conclusion that the excitons in these crystals (as well as in LiD; Plekhanov, 1981) may have very large kinetic energy (up to 0.5 eV). This is more than 10 times their binding energy. Such a high value of the kinetic energy of excitons once again points to their high temperature, which ﬁts in well with the large half width of the phonon repetitions of exciton luminescence. The high velocity of these hot excitons ensures large travel paths within the crystal, thus facilitating their capture by the defects (impurities) in the LiH lattice. This apparently also explains the low quantum yield of free exciton luminescence in these crystals (* : 3—0.1%; Plekhanov, 1997b). A similar structure consisting of LO replicas is also displayed by the excitation spectrum of emission lines of ILO repetition (Fig. 25) in LiH crystals. As in the previous case (see Fig. 24), the highest intensity in the spectrum corresponds to the line removed from the emission line by the energy of the 2LO phonons. The low intensity of the luminescence line of ILO phonons did not enable the measurement of its spectrum over a broad range of energies. It is clear, however, that the intensity of the spectrum decreases and the lines broaden as the kinetic energy of excitons

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Vladimir G. Plekhanov

Fig. 26. Temperature evolution of lineshapes of the 1LO and 2LO replica lines in the luminescence spectra of free excitons in LiD crystals. (After Plekhanov et al., 1988.)

increases. In this way, accurate measurements of the spectra of multiphonon luminescence and its excitation in crystals with clean surfaces enable us not only to demonstrate the existence of the quasimomentum of such excitons, but also to trace the band relaxation of excitons (see also Fig. 26).

SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 7

Isotopic Effect in the Emission Spectrum of Polaritons

I. Theory of Polaritons . . . . . . . . . II. Experimental Results . . . . . . . . . III. Resonance Light Scattering Mediated by LiH (LiD) Crystals . . . . . . . . . .

I.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitonic Polaritons in . . . . . . . . . . . . . . .

181 186 190

Theory of Polaritons

As is well known, transverse excitons interact with light, because they carry polarization via their dipole moment p (see, e.g., Knox, 1963). This interaction couples the exciton to the photon and must be taken into account. It leads to the true electromagnetic mode of the system, the exciton—polariton (Pekar, 1983). To obtain the polariton dispersion relation, it is not necessary to quantize the light ﬁeld and introduce photons: a classical treatment, based on Maxwell’s equations, is sufﬁcient and leads to the following implicit equation for (q), the wave-vector dependent frequency of the exciton—polariton (see also Agranovich and Ginsburg, 1979): ,( q, ) : cq,

(1)

where ,( q, ) is the transverse dielectric function of the crystal. The interaction with a transverse external ﬁeld is given by: (R )E,(R ), H : 9v P U G G G

(2)

(R ) is the polarization operator of the interband transition and where P G E,(R ) the transverse electric ﬁeld. Since we are dealing with an allowed G (R ) is deﬁned as transition, P G

p (R ) : (c> v ; v ; c ), P G G G v G G U

(3)

with p, the dipole moment of the transition, given by E (0>) 9 E (0>) : ,

p /v , of course, p . E,. U 181

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Vladimir G. Plekhanov

After Fourier transforming H into q-space, and using the exciton representation (see, e.g., Knox, 1963), we have v (9q )E ,( q H :9 UP ), N q

(4)

p (N [+ nq (0)S nq> ; (+ n\q (0)*)S n\q]. v U L

(5)

with (9p ) : P

and hence S nq are the operators Since the electric ﬁeld is transverse, q Q P for the transverse excitons, as deﬁned in Eq. (6)

S nq :

1 (N

+L(k, q )Sk ( q ).

(6)

The occurrence of the prefactor + nq (0) indicates that for allowed transitions, only s-type excitons are coupled to the light. We now have to determine the equation of motion of S nq in the presence ,: of the driving ﬁeld E i S/ nq : [S nq , H ; H ] : EL ( q )S nq ; [S nq , H ] . , \ \

(7)

Neglecting the analytical part of w( q ), w(0) : p /v and w,(0) : 0, U according to the following relation

M

p 1 L , cos( p, q ) ; V ; w( q : 0>) : HHHH v k "0 k U L L

(8)

where k are reciprocal lattice vectors and M matrix elements of the dipole L L operator. The sum over k contains the so-called ‘‘local-ﬁeld’’ effects and is L analytic at q : 0. If it is neglected, one ﬁnds the classical Lorentz correction 1 p , V : HHHH 3v U

(9)

for the dipole self-interaction. Absorbing V into w( q : 0) has the HHHH advantage that this quantity acts on longitudinal states only (see also Knox, 1963). Within this approximation, EL (0) equals the exact energies EL(0), as , deﬁned in the following equation: [S nq , H ; H ] : EL( q )S nq . S

(10)

7 The Emission Spectrum of Polaritons

183

Inserting Eq. (5) into Eq. (4) and using [S nq , S n> q ] : , LLY \

(11)

p [S nq , H ] : 9 + nq (0)E,( q ). \ (N

(12)

we obtain

We can now take the expectation value of Eq. (6), or simply regard S nq as a classical observable transform and obtain:

S nq() : 9

+ nq (0) (q E , ). (N 9 EL,(q ) , p

(13)

The same can be done for S nq>; inserting these results into Eq. (6) yields for the polarization

+ n9q (0)

p

+ nq (0) ,( q ; E , ) v EL (9q ) ; EL ( q ) 9 , , U L : (,(q, ) 9 )E ,( q, ).

(q

P , ) :

(14)

In the second line we have deﬁned the transverse dielectric function ,( q, ). Its dependence on q is called spatial dispersion (Hopﬁeld, 1958; Pekar, 1983). In addition, ,( q, ) depends also on the density of free carriers produced by the light: they contribute to the screening of the Coulomb interaction, which in turn leads to density dependent energies EL( q ) and oscillator strengths +(0) . In this way, nonlinear effects are contained in the theory (see also Ho¨nerlage et al., 1985). Several model calculations of the excitonic dielectric function exist. Bendow (1979) used a truncated hydrogenic model consisting of the 1s and 2s states and a continuum ranging from E to E . E E Stahl (1981) has discussed the polariton resulting from the continuum states only. Egri (1985) calculated the dispersion relation of the contact exciton. As energies below E , q() is not a single valued function, owing to spatial E dispersion. This is the origin of the additional boundary condition (ABC) problem (Pekar, 1983), because the conventional electrodynanic boundary conditions specify only the total polarization, but not the individual contributions from each of the solutions. An additional complication arises from the ﬁnite size of the exciton. Naively speaking, it cannot get closer to the crystal surface than its Bohr radius, leading to an excitonic dead layer at the surface (see, e.g., Stahl, 1981). We now discuss the solutions of the dispersion relation Eq. (2) for q : 0. Obviously, : 0 is a solution; it corresponds to the ‘‘photon-like’’ (see

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Vladimir G. Plekhanov

Fig. 1) branch of the polariton. Other solutions are given by ,(0, ) : 0. J

(15)

Inserting Eq. (14) we obtain 1:9

+L (0)

p

+L (0) ; . v EL(0) 9 EL(0) 9 U L J J

(16)

Comparing this condition with the relation

+ nq(0)

+ n9q (0) 1 : 9w( q ) ; EL( q ) 9 E( q ) EL(9q) ; E( q ) L

(17)

evaluated at q : 0 we see that E(0) : . This means that the zeros of the J transverse dielectric function are indeed the longitudinal excitons and the plasmon. Alternatively, one can say that the microscopic deﬁnition of longitudinal elementary excitations, namely, Eq. (17) coincides with the macroscopic deﬁnition of longitudinal excitations as given by the zeros of the dielectric function Eq. (16) (see also Knox, 1963). We conclude this section by discussing how the dilemma of additional boundary conditions can be resolved, if the center of mass and the internal motion of the exciton are taken into account correctly. As has been shown by Stahl (1981) this is possible in the real space representation. To this end, one considers the Fourier transform of S k ( q ), given by 1 ) : v>c : % exp#9ik %S ( q R S(R , R exp#i(k ; q )R ). F C N F C C F k k, q

(18)

This is the operator for an electron—hole pair, destroyed at lattice sites R C and R , respectively. It is related to the polarization operator by F

p (R ): ) ; S>(R ,R )). (S(R , R P G G G G G v U

(19)

The equation of motion for S reads i

) : [S(R ,R ), H] ; [S(R ,R ), H ]. S(R , R F C F C F C t

(20)

As was shown by Egri (1985), H ; H can be absorbed into H , if the T U (R , t) is replaced by (R the total ﬁeld (R , t) : E , t) ; external ﬁeld E G G G

7 The Emission Spectrum of Polaritons

185

(R , t). Then the last relation can be written as G i

), H ; H ] ; [S(R ,R ), H ], S(R , R ) : [S(R , R F C F C L F C t

(21)

with (R ) (R , t). H0 : 9v P G G U G

(22)

The second commutator yields simply ), H0 ] : 9p (R , t) . [S(R , R C CF F C

(23)

The ﬁrst commutator is usually evaluated in the Wannier model and yields:

2

2 e ), H ; H ] : E 9 r 9 R 9 S(R , r ) [S(R , R E 2 F C L 2M 4r

(24)

with r :R

F

9R

C

: and R

1 ;m R ). (m R C C M F F

(25)

The continuum limit on Eq. (22) is performed by introducing M( r ) : lim p , ? CF

(26)

a sharply peaked function of the dimensions of the unit cell. As a ﬁnal step, the average of Eq. (22) is taken, and one obtains

i

2 e 9 E ; 2r ; R ; Y (R , r ) : 9M( r ) (R , t), (27) E 2 t 2M 4r

where Y (R , r ) : S(R , r ) is the electron—hole pair amplitude. This is a Schro¨dinger-like differential equation for Y with an inhomogeneity representing the source term. On the other hand, Maxwell’s equation yields for the transverse light ﬁeld (see, e.g., Knox, 1963) 1 (R ) 9 cR (R ) : 9

D (R ) t t 2 :9 dr Re#Y (R , r )%M( r ), (28) v t U

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Vladimir G. Plekhanov

where we have used Eqs. (19) and (26). These two equations represent the ( polariton in real space formulation. Note the polarization P r ) is replaced by the exciton amplitude Y (R , r ), for which the usual quantum mechanical boundary condition applies. For example, if we model the crystal surface of an inﬁnite barrier at z : 0, the boundary condition for Y reads Y (R , r ) : 0, if z : 0 or z : 0, C F

(29)

and R and are related to where z and z are the z components of R r and C F C F via Eq. R (25). ) is Roughly speaking, the center-of-mass motion (represented by R responsible for spatial dispersion, whereas the internal structure (represented by r ) induces the dead layer.

II. Experimental Results Since polaritons are derived from an exact solution of an interaction Hamiltonian, they provide the most physically satisfying basis for the description of a variety of optical effects, among them absorption (Hopﬁeld, 1958; Pekar, 1983; Suzura et al., 1996), luminescence (Tait and Weiher, 1969; Koteles, 1982), nonlinear processes (Ovander, 1965), and light scattering (Weisbuch and Ulbrich, 1982; Haug and Koch, 1993, and for details see the following). The energy range of polariton resonance is determined primarily by the strength of the oscillator, and is characterized by two quantities. The ﬁrst of these is the energy E corresponding to the bottom of the band of transverse 2 mechanical excitons (Agranovich and Ginsburg, 1979). The second is the energy E (see Fig. 1) corresponding to the bottom of the band of * longitudinal excitons, which cannot be excited directly by transverse electromagnetic waves (Knox, 1963). This approach enables us to say that there are two polariton branches above the energy E in the energy spectrum of * electron excitations: the lower polariton branch (LPB) and the upper polariton branch (UPB). The expedience of the polariton model for the qualitative description of the experimentally observed features of resonance exciton emission was ﬁrst demonstrated in Benoit a la Guillaume et al. (1970) and Gross et al. (1972). Indeed, notwithstanding the successful theoretical description (Segall and Marple, 1967; Segall and Mahan, 1968) of the emission line shapes in LO repetitions and their relative intensities, the experimentally measured (Permogorov, 1975; Klochikhin et al., 1976; Plekhanov, 1994b) half width of the zero-phonon emission line of free excitons and the relative intensity of phonon repetitions with respect to the phononless line do not ﬁt in with the theoretical results for a model with an

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Fig. 1. Dispersion curves for excitonic polaritons (i.e., the exciton—photon coupled mode excitations) in the vicinity of a single exciton resonance E . Note that the asymptotic 2 photon-like polariton states above E are renormalized by all excitations of the crystal above * E (photon ‘‘dressed’’ according to the background dielectric constants ) and below E by the * @ 2 same plus the excitonic level (total dielectric constant ; 4). Vacuum photon and @ longitudinal exciton (uncoupled to light) dispersion curves are also shown.

equilibrium energy distribution of excitons. A characteristic feature of the observed manifestation of polariton effects in most nonmetallic crystals is the doublet shape of the resonance luminescence lines of free excitons at low temperatures (Brodin et al., 1984; Plekhanov, 1997b). Today there is a huge body of experimental material on free exciton luminescence and its dependence on various factors, such as surface quality (Lester et al., 1988), the

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Vladimir G. Plekhanov

presence of defects created, for example, with gamma radiation (Zhilyaev et al., 1988), scattering by neutral and charged impurities (Koteles et al., 1985), as well as reabsorption (Lester et al., 1988; Sermage et al., 1979) and the damping constant (Akhmediev, 1980; Strashnikova and Bessonov, 1978; Pentke and Broser, 1993; Pevtzov et al., 1985; Mariette et al., 1987). Moreover, while the theoretical interpretation of experimental results in Pevtzov et al. (1985) involved only the kinetic effects on the energy distribution function, the description of which can be found in Tavnikov and Krivolapchuk (1983) where the authors took into account both the energy and the space distributions of polaritons. As a matter of fact, it is when not only the energy distribution but also the polariton space distribution are taken into account, that good agreement is achieved between theory and experiment. Thorough investigations of the shape of zero-phonon emission lines in mixed A B crystals over a wide range of temperatures have brought Mariette et al. (1987) to the conclusion that they were able to observe experimentally the polariton effects in these crystals (see also Pekar, 1983; Agranovich and Ginsburg, 1979; Neu et al., 1984). Figure 2 shows the luminescence spectra of LiH and LiD crystals, cleaved and measured at 2 K. A more or less pronounced short-wavelength singularity of the phononless line is displayed by practically all crystals with clean surface (Plekhanov, 1995a). The luminescence spectrum of LiD crystals shown in Fig. 2 is much more similar to the spectrum of intrinsic luminescence of LiH crystals (see Fig. 22 of Chapter 6). There are, however, some distinctions: one is related to the unequal intensities of the phononless line (or, more precisely, of its long-wavelength component) and its 2LO repetition, whereas these intensities in LiH are practically the same. Another distinction is the clearly visible doublet nature of the phononless emission

Fig. 2. Luminescence spectra of (1) LiH and (2) LiD crystals cleaved and measured at 2 K. The inserts shows the zero-phonon emission line of free excitons in a mixed crystal at 78 K. (After Plekhanov, 1995d.)

7 The Emission Spectrum of Polaritons

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Fig. 3. Luminescence of polaritons in LiH crystals cleaved in liquid helium at (1) 4.2 and (2) 105 K. (After Plekhanov, 1995d.)

line of free excitons in LiD crystals even at a low temperature. On top of that, the magnitude of longitudinal—transverse splitting for LiD is greater than for LiH, and constitutes 25 < 2 meV. A pronounced doublet structure of the phononless emission line with magnitude of : 18 < 2 meV for *2 LiH crystal is observed at an elevated temperature (Fig. 3). This value of is somewhat larger than the magnitude of splitting observed by *2 Plekhanov and O’Konnel-Bronin (1978b) with LiH crystals cleaved in a jet of hot air. This behavior of also agrees with the results of Lester et al. *2 (1988), where the surface quality was demonstrated to dramatically affect not only the structure but also the intensity of polariton luminescence. The measured value of : 18 meV for LiH crystals ﬁt in well with the *2 magnitude of splitting observed in the reﬂection (Plekhanov and Altukhov, 1985; Kink et al., 1987) and RRLS (Plekhanov and Altukhov, 1985) spectra of crystals with clean surfaces (see also later). The doublet structure of the phononless emission line under consideration can be interpreted with the aid of the polariton dispersion curve. It is known (see, e.g., Brodin et al., 1984) that this curve represents a gradual transition to the dispersion curve from the dispersion of photons in the crystal E : ck of excitons E : E ; k/M (see also Fig. 1), where k and M are the wave vector and the translation mass of the excitons. As ﬁrst shown in Toyozawa (1959), it is this region, aptly referred to as a ‘‘bottleneck,’’ that is responsible for the processes of light absorption and emission by the crystal (see also Pekar, 1983). According to this approach, the doublet structure of the phononless emission line is caused by the radiative decay of the states on

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Vladimir G. Plekhanov

the upper and lower polariton branches (the approximate locations of these are indicated in Figs. 2 and 3 by arrows). As the temperature increases, the maximum of the polariton energy distribution in the crystal moves toward higher energies, and the population of UPB increases. According to the preceding, and as follows from Fig. 3, at T : 105 K the intensity of the short-wavelength component of the phononless emission line increases relative to the long-wavelength component. This result agrees well with the numerous data reported in the literature (Pekar, 1983; Brodin et al., 1984; Weisbuch and Ulbrich, 1982). For a mixed crystal (x : 0.55), the emission spectrum also displays a similar doublet structure of the phononless emission line with a somewhat larger half width than that in the original binary crystals (for more details see Plekhanov, 1995d). Going back to the results displayed in Fig. 2 and looking at the magnitude of , we see that it increases from 18 meV for LiH to 25 meV *2 for LiD crystals. Since in the transition from LiH to LiD only the energy of LO phonons is changed, one can assume that the main cause of renormalization of is the change in the polariton—phonon interaction. Account*2 ing for the different values of E (see Table IV of Chapter 6) and the @ dependence of on the exciton transition oscillator strength (Knox, *2 1963), we can write : E/E (see also Nelson et al., 1988). Substituting @ E *2 the values of E and E for LiH and LiD into this expression, we ﬁnd that @ E / : 0.65 for LiH crystals (Plekhanov, 1995a). This is somewhat less *2 *2 than the experimental value of : 25 meV. This discrepancy (we estimate *2 it at about 25%) may be caused by the polariton—phonon renormalization. Currently, the mechanism of this renormalization is not clear, since the branches of acoustic phonons are practically not affected by the isotopic substitution. On the other hand, since the energy of LO phonons depends nonlinearly on the isotope concentration (Plekhanov, 1994b), one can expect that the dependence of on the isotope concentration will also be *2 nonlinear. We see that the experimental material presented in this section gives convincing evidence that the magnitude of longitudinal—transverse splitting of polaritons in LiH crystals increases as hydrogen is replaced with deuterium.

III. Resonance Light Scattering Mediated by Excitonic Polaritons in LiH (LiD) Crystals The importance of polaritons for light scattering phenomena was recognized very early, mainly in the pioneering work of Ovander (1965). As discussed in greater detail by Bendow (1979), two main approaches to describe light scattering by excitonic polaritons have been pursued. The ﬁrst

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191

(called approach A in the following) is to consider a scattering event as the succession of three steps: (1) transmission of an incoming photon at the interface as a polariton inside the crystal, (2) scattering from one polariton state to another inside the crystal, and (3) subsequent propagation and transmission of the scattered polariton outside the crystal as a photon. In such a sequence, the overall scattering probability P (or efﬁciency) can be factored as P

: P T ( )T ( ), G Q

(30)

where T ( ) and T ( ) are the transmission coefﬁcients of ingoing and G Q outgoing polaritons at the crystal interface and P is the scattering probability of polaritons inside the crystal. When the outside probability is expressed per unit angle, care must be exerted to relate outside angles d to the corresponding inside angle d . A very common approximation for near-normal angles is d : nd , where n is the relative index of refraction (see also Lax and Nelson, 1976). The second approach (B) considers quantum states extending over the whole space, asymptotically behaving as photon states or polariton states far away on one side or the other of the crystal—vacuum interface (Zeyher et al., 1974). A scattering event then consists of the transition from an incident photon state to another (scattered) photon state. In approach B, the whole resonance behavior is taken into account in the branching coefﬁcients (such as transmission, exciton—phonon coupling) but has a resonant behavior through polariton densities of states and velocities. Whereas approach B is more rigorous in principle, it has been little studied, and we therefore use the ﬁrst approach, which has the advantage of pointing out more clearly the basic physical assumptions and consequences of the polariton description of light scattering. It also enables a rather direct comparison of experimental results with theory (see Weisbuch and Ulbrich, 1982). Working out both theories down to their ultimate consequences should, in principle, yield equivalent results. The second approach accounts in a natural way for interference effects between multibranch polaritons. In the polariton formalism used here, scattering events are ascribed to transitions between polariton states within the crystal. The kinematic properties of polaritons (i.e., direct consequences from the dispersion curve) will therefore play a major role. From the examination of dispersion curves such as shown in Fig. 4, a number of distinctive properties of polaritonmediated scattering can be expected (see also Weisbuch and Ulbrich, 1982). The spectra of resonant Raman scattering (RRS) in pure LiH (LiD) crystals was investigated by Plekhanov and Altukhov (1985) and in mixed LiH D by Plekhanov (1988, 1995d). For a displacement of the excitation V \V line frequency toward long wavelengths by an amount compared to the exciton resonance, for example, E R E (where E : 4.950 eV at 2 K LQ LQ G

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Vladimir G. Plekhanov

Fig. 4. Excitonic polariton dispersion curve of the transverse polaritons in the crystals. Possible one- and two-phonon scattering channels with participation of optical and acoustic phonons are illustrated. (After Plekhanov and Altukhov, 1985.)

is the energy of the exciton ground state, intense light scattering is observed (Fig. 5). As in the luminescence (see earlier), the process of energy relaxation take place, mainly with emission of LO phonons. This is shown by the character of the structure in the scattering spectrum. Indeed, the energy difference between the peaks in the scattered spectrum equals the energy of the LO phonon in the -point of the Brillouin zone (Verble et al., 1968). The relatively large half width of the scattered peaks should be noted. Additional investigations have shown that their half widths are always larger than that of the excitation line (Plekhanov, 1990a). Thus the half width of the 2LO line is 18 meV, whereas the half width of the 1LO scattered line is approximately 12 meV and is mostly determined by the characteris-

Fig. 5. Resonant Raman scattering spectrum of a LiD crystal at 4.2 K (E : 4.992 eV). The G arrow at 1s indicates the energy position of the ground exciton state.

7 The Emission Spectrum of Polaritons

193

Fig. 6. The shape of the 2LO line in the resonant secondary emission of a LiH crystal for excitation below (solid line) and above (broken line) the longitudinal exciton energy. (After Plekhanov and Altukhov 1985).

tics of measured equipment. The half width of the 3LO and 4LO lines and their intensities are similar, being 38 and 43 meV, respectively. In addition to the LO lines the RRS spectrum contains two more bands, the maxima of which are displaced by twice the energy of the TO() and LO(X) phonons from the exciting line (for LiH : 76 meV and : 117 meV 2- *-6 (Verble et al., 1968; Plekhanov and Altukhov, 1985). The ratio of the intensities of the ﬁrst- to the second-order lines as a function of the excitation frequency is described by the relation I /I : k. For excita*- *tion below the excitonic resonance by an amount approximately equal to the Rydberg exciton E , the intensity of the 2LO scattering line is more than @ 10 times that of one of the LO replicas (see also Fig. 5). However, when the energy of the exciting polariton approaches resonance, the intensity of the 1LO scattering line rises more quickly than that of the 2LO line (for details see Plekhanov, 1997a). The upper polariton branch (Fig. 4) begins to appear when the energy of the light quanta is slightly greater than the longitudinal exciton energy, E . * This situation is illustrated in Fig. 6, which shows the change in the line shape on scanning the excitation frequency within the LO—TO exciton gap (see earlier). The contribution to the secondary emission line of the upper polariton branch states can be clearly seen (dotted line in Fig. 6). For excitation at shorter wavelengths, the contribution of the low polariton branch to the RRS cross section decreases (for details see Plekhanov and Altukhov, 1985). Thus investigation of the ratio of one to two LO phonon scattering efﬁciencies permits one to determine experimentally a portion of the low polariton branch as well as the translational mass of the exciton.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 8

Isotopic Disordering of Crystal Lattices

I. Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . II. Effects of a Disordered Lattice on the Energy of Interband Transitions in LiH D Crystals . . . . . . . . . . . . . . . . . . . . . . . . V \V III. Broadening of Exciton Ground State Lines in the Mirror Reflection Spectrum of LiH D Crystals . . . . . . . . . . . . . . . . . . . V \V IV. Nonlinear Dependence of Binding Energy on Isotope Concentration . . V. Effects of Disordering on Free Exciton Luminescence Linewidths . . .

195 199 201 203 204

According to the classical deﬁnition of Lifshitz (Lifshitz, 1987b), there are two types of disordering in crystal lattices: site disordering and structure disordering. In the ﬁrst case, we deal with the random distribution of atoms (ions) with different scattering properties in the lattice sites. Structural disordering is the distortion of the statistical distribution of the lattice sites. Obviously, the isotopic disordering ought to be classiﬁed as site disordering of the crystal lattice. Even though a special volume has been published in a well-known series on solid state physics devoted to the optical properties of mixed crystals with different types of disordering (Elliott and Ipatova, 1988) (and where, incidentally, the isotopic model is used, see, e.g., Ipatova, 1988), there are simply no experimental results regarding the physical characteristics of isotopically mixed crystals. In this respect the present chapter is an important addendum to Elliott and Ipatova (1988), since here we are mainly concerned with the description of experimental results and their consistent comparison with existing theoretical models. Let us add that the fact that these effects are common for a large number of crystals (C, LiH, ZnO, Cu O, GaAs, GaN, CuCl, CdS, Si, Ge, and -Sn) with different binding and different physical characteristics enables us to ﬁnd the limits of model applicability over a broad range of microscopic parameters.

I. Models of Disorder In this section the main theoretical models describing the energy spectra of solid solutions are brieﬂy discussed (see Ziman, 1979; Efros and Raikh, 195

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Vladimir G. Plekhanov

1988). Most of the work in this ﬁeld is concerned with ternary or quaternary solutions of A B compounds. However, there has been increasing activity in the studies of A B solutions with isoelectronic substitution either in anionic or cationic sublattices (Permogorov and Reznitsky, 1992; Klochikhin et al., 1997). In this chapter, we follow the results of Ipatova (1988) and Permogorov and Reznitsky (1992). In a solid solution formed by isoelectronic substitution the disorder is usually weak, which enables an easy comparison of the solution’s optical spectra with the spectra of its parent compounds. As a rule, the atoms in such solutions keep on average the same positions so that the long-range order exists and perfect samples can be grown. Moreover, isoelectronic alloying does not change the nature of the chemical bonding so that the main features of the band structure remain, unlike the case of hydrogenated amorphous Si or Ge. As is well known (Elliott and Ipatova, 1988), the formation of the tails in the density of states at the band edges essentially depends on the effective masses and energy ﬂuctuations in the particular bands forming the bandgap. In A B compounds, the effective mass of holes usually by far exceeds that of electrons. Moreover, in accordance with the common ion rule (Harrison, 1980) it can be supposed that the energy ﬂuctuations mainly correspond to the band originating from the wave functions of the ion that is substituted. Thus, it can appear that the tailing of the density of states and carrier localization will take place mostly for the carriers of one sign, which also simpliﬁes the analysis. The simplest theoretical model for the description of solid solutions is the so-called virtual crystal approximation. It supposes that the real atoms in solution can be replaced by some hypothetical atoms with the atomic potentials smoothly varying between that of the substitutional species. In this case, all the electronic properties such as the ionic radii, band positions, and effective masses should vary smoothly at the change of composition. In fact, this approximation does not take into account the disorder at all. However, it gives a satisfactory picture of the compositional dependence of the band energy positions and lattice constant in the solid solution. For the study of the broadening of the electronic states, models that explicitly consider the disorder should be used (for details see also Schwabe and Elliott, 1996). In what follows, we brieﬂy discuss some models of this kind, giving special attention to exciton states that are responsible for the detailed shape of the fundamental absorption edge in most direct-gap semiconductors. In some cases, the discussion will start with the limit of very diluted solutions where the substitution atoms of small abundance can be considered as impurities. The dependence of the solid solution energy spectrum on composition critically depends on the position of the electronic level produced by the isolated atom of the substituting impurity with respect to the energy bands of the solvent crystal.

8 Isotopic Disordering of Crystal Lattices

197

In the case where the impurity level falls inside the forbidden gap, the isolated potential wells are formed. If these wells are deep enough, the localized states for carriers of corresponding sign can appear. The condition for the appearance of such localized states is E q /m*a, U

(1)

where E is the well depth, m* is the carrier mass, and a is the spatial size U of the well, which has the order of the interatomic distance in the case of short-range potential of the isoelectronic impurity. The localized carrier, in turn, binds a carrier of the opposite sign with Coulomb forces, forming an exciton localized by the isoelectronic impurity. Localized exciton states of this kind have been observed for many A B compounds (see Dean, 1983). The localization energy of a carrier from an isoelectronic impurity usually far exceeds the exciton binding energy and amounts to 300—400 meV for the typical cases of A B compounds. At small concentrations of isoelectronic impurity a narrow local level is shifted by the energy E to lower energies with respect to the band edge E . However, the optical spectra of excitons E bound to isoelectronic impurities usually show very broad emission and absorption bands with mirror symmetry. The width of these bands is mainly due to the electron—phonon interaction and does not reﬂect the width of exciton levels. In emission, the excitons localized by the isoelectronic impurities constitute the dominating recombination channel, which makes such systems promising materials for light sources. With increasing impurity concentration, the localized levels corresponding to excitons bound to pairs, triads, and other multiatomic impurity clusters appear in the density of states spectrum (for details see Agekyan et al., 1987). Simultaneously the interimpurity interaction causes a broadening of the cluster levels and blurring of the absorption edge. In the limiting case of very high impurity concentration the cluster levels should completely merge into a new absorption edge at the position of the band edge of the second parent compound. Such behavior corresponds to the ‘‘persistenttype’’ two-mode evolution of the energy spectra of solid solutions, as discussed by Onodera and Toyozawa (1968). The opposite case, when the level of isolated impurity falls within the band of allowed states, has been discussed quite intensively (Efros and Raikh, 1988; Klochikhin et al., 1997). In this case, the interaction of resonant impurity levels with the band states in a regular shift of the band edge corresponds to ‘‘amalgamation-type’’ behavior of the energy spectra (Onodera and Toyozawa, 1968). In a virtual crystal approximation, this shift reﬂects the variation of the average potential acting on carriers. In most cases, the compositional shift of the bandgap is nonlinear and monotonous. However, if the nonlinearity is high enough, the minimum value of the bandgap can correspond to some intermediate concentration of the solid

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solution, as in the case for SnSe Te (Permogorov and Reznitsky, 1992). \V V Another composition-dependent effect in amalgamation-type solid solution is a broadening absorption edge with the formation of localized states owing to compositional ﬂuctuations. The statistical distribution of substitutional atoms over the crystal sites produces, besides the change of the average potential, a spatially variable potential relief. However, a rather extended volume is necessary for the formation of a potential well, which can produce the localized state, an essential deviation of the local concentration from the mean values. In other words, only the so-called long-range ﬂuctuations will be responsible for the creation of the localized states. Localization of an exciton as a hole within the potential well was considered by Efros and Raikh (1988). It has been shown that the localized states form the tail at the absorption edge with the density asymptotically decreasing as (see, e.g., Efros and Raikh, 1988) (E) exp[9E/E ],

(2)

1 Mx(1 9 x) E : . 178

N

(3)

where

Here x is the mean solution concentration, : dE /dx is the rate of band E change with concentration, E is the localization energy measured from the mobility edge, M is the exciton translation mass, and N is the number of elementary cells in the unit volume. For characteristic parameters of A B compounds, E has values from several fractions up to several units of meV. It should be noted that the position of the mobility edge does not correspond to any singular point in the density-of-states spectrum and should be determined from some experiment (see also Elliott and Ipatova, 1988). The shape of the exciton absorption line for this model has been calculated by Klochikhin et al. (1997). It has been found that the absorption spectrum has the same low-energy asymptote, Eq. (2) in the density of states. The half width of the absorption line is approximately 14E . The low-energy wing of the absorption line is formed by transitions into the localized states and the spectrum is inhomogeneously broadened in this region. The broadening on the high-energy side of the line comes from the exciton scattering by potential ﬂuctuations and has a homogeneous nature. Strictly speaking, the model of exciton localization as a whole is applicable only for localization energies smaller than the exciton binding energy. A similar approach can be used to describe some other models of exciton localization by long-range ﬂuctuations (Efros and Raikh, 1988). In the case when

8 Isotopic Disordering of Crystal Lattices

199

potential ﬂuctuations correspond mainly to valence band it is possible to assume that only the holes are localized, whereas the electrons bound to the holes by Coulomb forces are moving outside the potential wells (Elliott and Ipatova, 1988). As noted in the paper of Efros and Raikh the asymptotic f of the absorption spectrum for this case should also keep the form of Eq. (2) with and M replaced by the corresponding values for the valence band.

II. Effects of a Disordered Lattice on the Energy of Interband Transitions in LiHx D19 9x Crystals Numerous studies of reﬂection (absorption) spectra of mixed compounds A B , A B , and A B , carried out over the past four decades, point to the existence of large-radius excitons in these materials. The widths of exciton bands were shown to depend on the composition of the solid solution. The random relief of the potential is caused by disordering of the crystal lattice (see earlier). According to modern views, if the kinetic energy of an exciton exceeds the potential energy of localization, the exciton will not ‘‘sense’’ the random relief of the potential. If the kinetic energy of the exciton is small, then the exciton will be localized by the potential created by ﬂuctuations of the composition of the crystal lattice. Since the localization energy is different at different points of the sample, the exciton transition energy will also be different, which leads to broadening of the exciton line. The localized and delocalized states are separated by the mobility threshold (Belitz and Kirkpatrick, 1994). From the preceding it becomes clear that exciton states in solid solutions will be observed when and only when the exciton is not ionized by the ﬁeld of such ﬂuctuation, and broadening of the exciton lines is much less than the binding energy E of the exciton in such states. In the @ localized exciton states not only the relative motions of excitons and the hole is localized, but also the translation motion of the exciton as a whole. Note also that quasi-momentum is no longer a good quantum number for excitons with low kinetic energy, and therefore excitons with different energies may be generated through the absorption of light. The density of states in the exciton band in the low-energy range is much different from the density of states of free excitons (Fig. 1). In the paper by Efros and Raikh (1988) the method of optimal ﬂuctuation (for more details see Lifshitz, 1987b) was used to ﬁnd expressions of the absorption coefﬁcient k(E) in the region of the long-wavelength wing of the exciton line, as well as the density of the localized states (E) [see formula (2)]. Both these functions fall off toward the depth of the forbidden zone as

K(E) . (E) . exp

E . E

(4)

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Fig. 1. Densities of states (E) in the exciton band for noncoherent and coherent motion of exciton. Dashed line shows the density of states for free excitons.

In chemically mixed semiconductor crystals, especially those of the A B group (Klochikhin et al., 1997), smearing of the conduction and valence band edges occurs because of ﬂuctuations in the electron potential; the band is smeared more the heavier the carrier. Numerous comparisons of the results on the broadening of exciton lines by large-scale compositional ﬂuctuations (whose geometrical size is commensurate with the wavelength of the exciton) with the proposed theoretical model point to good qualitative and quantitative agreement, especially for A B compounds (see earlier). As follows from Fig. 9 of Chapter 6, excitons in LiH D crystals display V \V a unimodal character, which facilitates the interpretation of their concentration dependence. Figure 2 shows the concentration dependence of the energy of interband transitions E . Each value of E was found by adding E E together the energy of the long-wavelength band in the reﬂection spectrum and the binding energy of the exciton. The latter was found from the hydrogenlike formula using the experimental values of the energies levels of 1s and 2s exciton states. We see that the 100% replacement of hydrogen with deuterium changes E by E : 103 meV at T : 2 K (Plekhanov, 1996c). E E This constitutes 2% of the energy of the electron transition, which is two orders of magnitude greater than the value corresponding to the isotopic replacement of atomic hydrogen with deuterium reported earlier (Plekhanov, 1996b). The nonlinear concentration dependence of E can be sufﬁciently well E approximated with a second order polynomial E (x) : E ; (E 9 E 9 b)x 9 bx, (5) E @ ? @ where E and E are the values of E for LiD and LiH, respectively, and b ? @ E is the curvature parameter equal to 0.046 eV. This result generally agrees

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201

Fig. 2. Energy of band-to-band transitions E as function of isotope concentration in mixed E LiH D crystals at 2 K: 1, linear dependence of E on x in virtual crystal model; 2, V \V E calculation according to Eq. (5), points derived from reﬂection spectra indicated by crosses, from luminescence spectra by triangles. (After Plekhanov, 1990b.)

with the published data (see also Elliott and Ipatova, 1988, and the references therein). For comparison, note that in the case of isotopic substitution in germanium the energy E depends linearly on the isotopic E concentration for both direct (E , E ; , E ; ) and indirect electron transitions (Parks et al., 1994). Unfortunately, today there is no information on the form of the function E . f (x) for isotopic substitution in C, ZnO, E CdS, CuCl, Cu O, GaAs, GaN, Si, Ge, and other such crystals, although, as noted earlier, the values of E have been measured for isotopically pure E crystals (see Section III.2 in Chapter 6).

III. Broadening of Exciton Ground State Lines in the Mirror Reﬂection Spectrum of LiHx D19 9x Crystals As follows from Fig. 9 of Chapter 6, the addition of deuterium leads not only to the short-wavelength shift of the entire exciton structure (with different rates for 1s and 2s states (Klochikhin and Plekhanov, 1980), but also to a signiﬁcant broadening of the long-wavelength exciton reﬂection line. This line is broadened 1.5- to 3-fold upon transition from pure LiH to pure LiD. The measure of broadening was the half width of the line measured in the standard way (see, e.g., Permogorov and Reznitsky, 1992) as the distance between the maximum and the minimum in the dispersion gap of the reﬂection spectrum, taken at half height. The concentration dependence of the half width (E0) of the long-wavelength band in the exciton reﬂection spectrum at 2 K is shown in Fig. 3. Despite the large spread and the very limited number of concentrations used, one immediately

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Fig. 3. Concentration dependence of half width of the line of ground state of exciton in mirror reﬂection spectrum at 2 K: 1, approximation of virtual crystal model; 2, calculation according to Eq. (6), experimental points indicated by crosses. (After Plekhanov, 1996a.)

recognizes the nonlinear growth of E0 with decreasing x. A similar concentration dependence of E0 in the low-temperature reﬂection spectra of solid solutions of semiconductor compounds A B and A B has been reported more than once (see, e.g., the review of Elliott and Ipatova, 1988, and the references therein). The observed broadening of exciton lines is caused by the interaction of excitons with the potential of large-scale ﬂuctuations in the composition of the solid solution. Efros and colleagues (see, e.g., Efros and Raikh, 1988) used the method of optimal ﬂuctuation (Lifshitz, 1987b) to express the formula for the concentration dependence of the broadening of exciton reﬂection lines: E0 : 0.5

x(1 9 x) . Nr

(6)

where : dE /dx; r is the exciton radius, which varies from 47 to 42 Å on E transition from LiH to LiD (Plekhanov, 1997b). The value of coefﬁcient was found by differentiating Eq. (5) with respect to x — that is, dE /dx : E : E 9 E 9 b : 2bx. The results of calculation according to Eq. (6) are ? @ shown in Fig. 3 by a full curve. The experimental results lie much closer to this curve than to the straight line plotted from the virtual crystal model. At the same time it is clear that there is only qualitative agreement between theory and experiment at x 0.5. Nevertheless, even this qualitative analysis clearly points to the nonlinear dependence of broadening on the concentration of isotopes, and hence to the isotopic disordering. Since isotopic substitution only affects the energy of optical phonons, and, as a consequence, the constant of exciton— phonon interaction (in the ﬁrst place, the Fro¨hlich interaction g), the nonlinearity of functions E . f (x), E0 . f (x) is mainly related to the E nonlinear behavior of g . f (x). In this way, the experimental study of the concentration dependence of the exciton—phonon interaction constant may

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throw light on the nature and mechanism of the large-scale ﬂuctuations of electron potential in isotopically disordered crystals.

IV. Nonlinear Dependence of Binding Energy on Isotope Concentration A principal matter for further theoretical development is the question concerning the effect of crystal lattice disordering on the binding energy E @ of Wannier—Mott excitons. This problem has been treated theoretically in the papers of Elliott and co-workers (Kanehisa and Elliott, 1987; Schwabe and Elliott, 1996), where they study the effect of weak disordering on E (the @ disordering energy is comparable with E ). The binding energy indicated in @ the papers was calculated under the coherent potential approximation by solving the Bethe—Salpeter equation as applied to the problem of Wannier— Mott excitons in a disordered medium. One of the principal results of this paper (Kanehisa and Elliott, 1987) is the nonlinear dependence of E on the @ concentration. As a consequence, the binding energy E at half-and-half @ concentrations is less than the value derived from the virtual crystal model. The exciton binding energy is reduced because the energy E is less, owing E to the ﬂuctuation smearing of the edges of the conduction and valence band. This conclusion is in qualitative agreement, although not in quantitative agreement, the discrepancy being about an order of magnitude (see also Kanehisa and Elliott, 1987) with the experimental results for the mixed crystal GaAs P with x : 0.37, where the reﬂection spectra exhibited two \V V exciton maxima (see also Fig. 9 of Chapter 6) used for ﬁnding the value of E (see Nelson et al., 1976 and the references therein). Let us add that the @ pivotal feature of the model of Elliott and co-workers is the short-range nature of the Coulomb potential (for more details see Plekhanov, 1996b). The data from Table IV of Chapter 6 and other published sources (Plekhanov, 1981, 1997b) were used for plotting the energy E as a function @ of isotopic concentration x in Fig. 4. The values of binding energy E were @ calculated using the hydrogenlike formula (see later) with the energies of exciton levels of 1s and 2s states being found from the reﬂection spectra (see Fig. 9 of Chapter 6). Theoretical description of the binding energy of Wannier—Mott excitons as a function of x was based on the polynomial derived by Elliott and co-workers (Kanehisa and Elliott, 1987):

19W 9E , 2U E : x(1 9 x) A ? , W

E : E 9 E @ @

W 9 W, E : U ; @ 2U

(7) (8) (9)

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Fig. 4. Concentration dependence of binding energy of Wannier—Mott exciton: 1, approximation of virtual crystal model; 2, calculation according to Eq. (9), experimental points indicated by triangles. (After Plekhanov, 1996a.)

where W : W ; W , and W and W are the widths of the conduction band A T A T and the valence band, which are equal to 21 eV (Hama and Kawakami, 1989) and 6 eV (Betenekova et al., 1978), respectively. Here E is the curvature parameter found from the function E . f (x); and are the E A T magnitudes of the ﬂuctuation smearing of the valence band and the conduction band edges, : 0.103 eV and : 90.331 eV. As follows from A T Fig. 4, these values of the parameters give a good enough description of the nonlinear dependence of the binding energy of Wannier—Mott excitons in a disordered medium. This agreement between theory and experiment once again proves the inherent consistency of the model proposed by Kanehisa and Elliott, since the isotopic substitution affects the short-range part of the interaction potential. In this way, the nonlinear dependence of the binding energy of Wannier—Mott excitons is caused by isotopic disordering of the crystal lattice.

V. Effects of Disordering on Free Exciton Luminescence Linewidths When light is excited by photons in a region of fundamental absorption in mixed LiH D crystals at low temperature, line luminescence is V \V observed (Fig. 5), as in the pure LiH and LiD crystals. As before (Plekhanov and Altukhov, 1983), the luminescence spectrum of crystals cleaved in liquid helium consists of the relatively narrow zero-phonon line and its wide LO replicas. For the sake of convenience, and without sacriﬁcing generality, Fig. 5 shows the lines of two replicas. Usually up to ﬁve LO repetitions are observed in the luminescence (excitation) spectrum, as described in detail in

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Fig. 5. Photoluminescence spectra of free excitons in (1) LiH, (2) LiH D , and (3) LiD V \V crystals at 4.2 K. Spectrometer resolution is shown. (After Plekhanov et al., 1988.)

Plekhanov et al. (1988). In Fig. 5 we see immediately that the structure of all three spectra is the same. The difference is in the distance between the observed lines, as well as in the energy at which the luminescence spectrum begins, and in the half widths of the line. The ﬁrst feature is explained by the gradual and smooth decrease of the energy of LO phonons upon transition from LiH to LiD. This is related to the unimodal nature of LO phonons in mixed LiH D crystals at high concentrations of isotopes V \V (0.4 x 1.0; Plekhanov, 1994b). This conclusion was made in the cited paper from the RRLS spectra. The second feature is attributed to the phonon renormalization (mainly through interaction with LO phonons (Plekhanov, 1981) for the energy of band-to-band transitions. The change in E is also smooth and continuous though nonlinear (see Fig. 2). It is hard E to miss the broadening of exciton emission lines of RRLSs in mixed crystals as compared with pure crystals. This is especially clear for the lines of LO repetitions as described earlier for mixed crystals of the A B group in the cited review by Elliott and Ipatova (1988). In our case, the line broadening for some crystals is as large as three or four widths of the lines of pure crystals (see Fig. 22 of Chapter 6 and also Fig. 2 of Chapter 7). A more detailed quantitative study of the lineshapes in repetitions has been carried out using the example of a line in the 2LO repetition. The results of this study are presented in Fig. 6. Here curve 1 is the Maxwellian distribution of excitons with respect to kinetic energy [see Eq. (47) of Chapter 6]. Curve 2 is the result of convolution of the Lorentz and Gauss curves of the form (see, e.g., Bodart and Feldman, 1985): I (E) : 2

\

I (E) exp 94 ln 2 $

E 9 E dE,

(10)

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Fig. 6. Photoluminescence spectrum of LiH D crystal at 4.2 K, cleaved in liquid helium: 1, V \V Maxwell distribution calculated for virtual crystal model; 2, convolution of Lorentzian and Gaussian. (After Plekhanov, 1997b.)

where I (E) is the Maxwell distribution of excitons, and is the Gaussian $ broadening of emission line due to disordering of anions (isotopes) in the crystal lattice (see Plekhanov, 1997b). The comparison of calculated results with the experimental data points clearly (see Fig. 6) to better agreement with curve 1. It is important that, as before (Plekhanov, 1990a), for reconciliation between theory and experiment one assumes that the temperature of excitons is 100 times as high as the temperature of the lattice.* Such large broadening cannot be explained solely by the third-order anharmonism, given the close values of the widths of the branches of optical phonons in LiH and LID (Plekhanov, 1990). There follows a reasonable assumption that some of the experimentally observed broadening is deﬁnitely caused by disordered isotopic substitution in the anion sublattice. An additional argument in favor of this assumption is the experimentally measured dependence of the half width of the line of 2LO repetition on the concentration x shown in Fig. 7. Here curve 1 also represents the concentration dependence of the half width of the line of 2LO repetition in the virtual-crystal model — that is, : x ; (I 9 x) , where and are the half widths of the lines of 2LO repetitions for pure LiH and LiD crystals. Although, as noted earlier, the change of the lattice constant of LiH D crystals with x is directly described by the virtualV \V crystal model, from Fig. 7 we see that the linear approximation alone is not sufﬁcient to describe the experimental dependence . f (x). Curve 2 is ** The existence of kinetic energy of heated excitons in mixed CdS Se crystals was recently V \V reported in Permogorov and Reznitsky (1992). Note that while the conclusion concerning the existence of quasi-momentum of free excitons made in Permogorov and Reznitsky’s paper holds only for small concentrations, in the case of LiH D the same is true for arbitrary V \V concentrations of isotopes.

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Fig. 7. Concentration dependence of half width of the luminescence line of 2LO replica in LiH D crystals. The virtual crystal modes (1) and (2) calculated according to Eqs. (11) and V \V (3) experimental data. (After Plekhanov, 1997b.)

much closer to the experimental results, according to the model of Singh and Bajaj (1984). In this model the dependence of on x is found with the *aid of perturbation theory in the virtual-crystal approximation. The perturbation is represented by ﬂuctuations in the crystal potential, which in Singh and Bajaj (1984) were treated according to the Lifshitz method of optimal ﬂuctuations (Lifshitz, 1987b)

x(1 9 x)1.4r A , (11) r dE r : xr ; (1 9 x)r , : E , (12) dx where r is the size of cluster (ﬂuctuation) (in the calculations the value of A r was assumed to be equal to the lattice constant of mixed LiH D A V \V crystals (Zimmerman, 1972), r is the radius of exciton in mixed crystals found from Eq. (6) of Chapter 6, r and r are the exciton radii in pure crystals. The values of r found from the hydrogenlike formula ﬁt in with these values (see Section 2 of Chapter 6). The dependence of the half width of the free-exciton emission lines in a mixed crystal on the composition of its lattice as calculated from Eq. (11) is also plotted in Fig. 7. Triangles on the same diagram depict the experimental points derived from the spectra of intrinsic luminescence of mixed LiH D crystals. Generally, there is V \V good enough agreement between theory and experiment. This agreement relates essentially to the nonlinear behavior of . f (x). At the same *time we see that the maximum values of in theory and in experiment are not the same. The largest broadening of the line of 2LO repetition in LiH D crystals is experimentally observed at x 40%. This, on the V \V other hand, is in qualitative agreement with the results of Neu et al. (1984), where the asymmetry of the function . f (x) is attributed to the fact *that r and r are not the same. The common feature of the preceding results is the maximum amount of broadening: three- or fourfold according to both :2

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Fig. 8. (a) Luminescence spectra of ZnO (1) and ZnO (2) at 77 K (after Kreingol’d, 1978); (b) LA and TO phonon replicas of electron-hole liquid line emission of Ge (1) and Ge (2) at 1.7 K. (After Agekyan et al., 1989.)

calculations and the experiment, respectively. Notwithstanding the agreement between theoretical and experimental results, we must admit that it has not been possible to achieve quantitative agreement between the experiment and the model of Singh and Bajaj (1984). A more common and precise model was developed by Langer et al. (1992). In this paper it was shown that the half width is : 2.35E G

x(1 9 x) , 8a

(13)

8 Isotopic Disordering of Crystal Lattices

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where is the volume of the primitive cell, x represents the defects (isotope) concentration, a is the Bohr radius, and E is the defect energy level. G However, the use of this formula requires precise values of m a , and E , CF G but m has not yet been measured for LiH crystals. The crude estimation CF of the value in Eq. (13) yields a value that is closer to experimental values than that obtained with the Singh and Bajaj Eq. (11). Note, however, that the results of quantitative study of the shape and half width of emission lines of free excitons in the mixed crystals described by Plekhanov (1997b) are, to our knowledge, presented for the ﬁrst time. Further studies are certainly required to obtain the ﬁnal form of the function . f (x), which should take due account of the concentration dependence of anharmonism (Maradudin and Califano, 1993; Plekhanov, 1995b). Even these ﬁrst results, however, obtained with a limited number of mixed crystals, are universal enough. Indeed, the existence of the kinetic energy of excitons in mixed LiH D crystals (and hence the existence of quasiV \V momentum k ) is another independent proof of the importance of ﬁne ﬂuctuations in the potential for the localization of free excitons. The mismatch between the experimental results on the broadening of the emission lines of free excitons and the linear approximation of the virtualcrystal model is a clear indication that it is necessary to take into account the lattice disordering even for isotopically mixed crystals (Plekhanov, 1996b). We should add that isotopic substitution in diamond (Collins et al., 1990), ZnO (Kreingol’d, 1978), and Ge (Agekyan et al., 1989; Davies et al., 1992, 1993; Etchegoin et al., 1992; Cardona et al., 1992) also resulted in a shift of the free exciton luminescence spectra (Fig. 8). Moreover, isotopic substitution in Ge leads not only to the shift of the luminescence spectrum, but also to the nonlinear concentration dependence of the emission line half width, which, as in the case of lithium hydride, was attributed to the isotopic disordering of the crystal lattice (see also Davies et al., 1993).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 9

Future Developments and Applications

I. II. III. IV. V. VI.

Isotopic Confinement of Light . . . . . . . . . . . . Isotopic Information Storage . . . . . . . . . . . . Neutron Transmutations . . . . . . . . . . . . . . Isotopic Structuring for Fundamental Studies . . . . Isotope Diffusion in Semiconductors . . . . . . . . . Other Unexplored Applications of Isotopic Engineering

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211 214 215 216 216 221

The fact that most chemical elements have more than one stable isotope is, of course, commonly known. Yet, until now there have been surprisingly few attempts to explore the potential application of structures made with ordered distributions of different isotopes of the same chemical element(s). Berezin and Haller identiﬁed some emerging possibilities of purposeful isotopic structuring for various microelectronic applications (Berezin, 1989; Haller, 1995).

I. Isotopic Conﬁnement of Light As a ﬁrst example of possible major applications of isotopic engineering (Berezin, 1988) we consider isotopic ﬁber-optics and isotopic optoelectronics at large. It is known that for typical solids the lattice constant variations of isotopically different samples are usually within the limits (see Chapter 3) d : 10\—10\. d

(1)

Let us deﬁne an isotopic ﬁber as a structure in which core and cladding have the same chemical content but different isotopic composition. The boundary between different isotopic regions form an isotopic interface. The difference in the refractive index on both sides of the isotopic interface could lead to the possibility of total internal reﬂection of light and, consequently, could provide an alternative route to the conﬁnement of light. For a 211

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Vladimir G. Plekhanov

Fig. 1. Isotopic ﬁber in which the core and cladding are both pure SiO but with a different isotopic composition. (After Berezin, 1989.)

quantitative estimate, Berezin considered a boundary between SiO (the main component of silica) where body sides are identical chemically and structurally but have a different isotopic content — for example, SiO and SiO , respectively (Fig. 1). In the ﬁrst approximation the refractive index n is proportional to the number of light scatterers in the unit volume. From the Clausius—Mosoti relation for the refractive index one can deduce the following proportion (at n n) n d < 3c , n d

(2)

where c is a dimensionless adjustment factor of the order of unity. Substituting Eq. (1) for Eq. (2) we can obtain n : 3 ; 10\—10\. n

(3)

Using the Snell law of light refraction we obtain the following expression for the ray bending angle ' when the light travels through the refractive boundary:

n sin , (4) n where is the angle between the falling ray and the direction normal to the interface. For a sliding ray ( < 90°), which is the control case for light ' < arcsin

9 Future Developments and Applications

213

conﬁnement in ﬁbers, combining Eqs. (3) and (4) leads to an estimate ' : 1.5—4.5°. Thus, in the isotopic ﬁbers in which core and cladding are made of different isotopes the half-angle of the acceptance core could be up to several degrees. The resulting lattice mismatch and strains at the isotopic boundaries are correspondingly, according to Berezin, one part per few thousand and, therefore, could be tolerated. Further advancement of this ‘‘isotopic option’’ could open the way for an essentially monolithic optical chip with built-in isotopic channels inside the fully integrated and chemically uniform structure. Another application of isotopic engineering is the isotopic superlattices, which is supposed to be by Haller in 1990 (Haller, 1995). The ﬁrst experimental results on the Raman spectra of a series of isotopic (Ge) (Ge) L L superlattices with 2 n 32 was by Spitzer and co-workers (1994). The motivation for vibrational studies of isotopic superlattices lies in their one-dimensional character, which should make localization possible, regardless of the magnitude of the mass difference. A plane of Ge embedded in bulk Ge, for instance, should always lead to a localized vibrational mode above the Raman frequency of Ge. Since it is difﬁcult to see experimentally just one plane, Spitzer et al. considered the possibility of a periodic superlattice with the repeat unit consisting of several planes of Ge followed by several planes of Ge. Thus, such structures have been grown along the

001 direction, with periods of Ge Ge ranging from n : 2 to n : 32. L L Figure 2 shows the Raman spectra obtained for a series of these superlattices ranging from 2 to 32. Next to it, model calculations of these spectra, based on a planar force constants description of the lattice dynamics, are displayed. The model included partial mixture of the two monolayers forming the interface. A number of peaks are observed in these Raman spectra, which correspond to the various so-called ‘‘conﬁned’’ modes in which the vibrations occur predominantly in either Ge or Ge. From this result it follows that these samples represented an excellent model system for the investigation of conﬁnement of optical phonons in superlattices. A number of other possible applications can be envisaged for isotopically tailor-made semiconductor crystals. In the case of Ge one should bear in mind the copious use made of this material as a radiation detector, embracing the range from the IR to -rays (Haller, 1995). Noise (so-called spikes) and background signal often arises from the capture of cosmic-rayinduced particles (protons, neutrons) by the nuclei and subsequent decay. It is clear that the corresponding response will depend on the type of nuclei, that is, on the isotopic composition (Barthelmy et al., 1993; Gehrels, 1990). In the case of germanium these processes are strongest for the Ge (Gehrels, 1990). As an example we show in Fig. 3 the response of a natural

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Fig. 2. (a) Measured and (b) calculated Raman spectra of a series of isotopic GeGe L L superlattices showing various conﬁned modes of Ge and Ge layers. The measurements were performed with the 514.5-nm line of an Ar> ion laser at a temperature of 10 K. (After Spitzer et al., 1994.)

germanium detector and a Ge detector to background radiation, that is, what one can call the ‘‘dark signal,’’ in the 10—250 K region. It is clear that this deleterious background is considerably weaker in the enriched Ge detector than in that made out of natural germanium.

II. Isotopic Information Storage Isotopic information storage may consist of assigning the information ‘‘zero’’ or ‘‘one’’ to monoisotopic microislands (or even to a single atom)

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Fig. 3. Background spectra for normal and enriched germanium detectors in the low-energy -ray regime. (After Barthelmy et al., 1993.)

within a bulk crystalline (or thin ﬁlm) structure. This technique could lead to a very high density of ROM-type information storage, probably up to 10 bits/cm. The details are discussed by Berezin et al. (1988). Note here only that the use of tri-isotopic systems (e.g., Si, Si, Si) rather than di-isotopic (e.g., C, C) could naturally lead to direct three-dimensional color imaging without the need for complicated redigitizing (it is known that any visible color can be simulated by a properly weighted combination of three primary colors, but not of two).

III. Neutron Transmutations The method of neutron transmutation doping (NTD) is presently used for doping of the different semiconductors (Haller, 1995; Magerle et al., 1995; Kurijama and Sakai, 1996). In all cases, it is based on a selective transmu-

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tation of a particular isotope under neutron ﬂux irradiation. For example, out of three stable isotopes of Si only one (Si) transforms into phosphorus according to the two-step nuclear reaction (Schnoller, 1974): Si ; n ; Si ; @\ ; P. 2

(5)

Phosphorus is an n-type dopant. It is easy to see that a uniform neutron irradiation of isotopically nonuniform material leads to a correspondingly nonuniform distribution of P-atoms. This distribution of induced phosphorus atoms acts as a developed ‘‘hidden image’’ quantitatively reﬂecting the originally built-in isotopic nonuniformity (for details see the review of Haller, 1995). The method of neutron transmutation is currently the easiest way to achieve semiconductor crystals with the uniform distribution of the neutral impurity of the crystal volume (Magerle et al., 1995; Kuriyama and Sakai, 1996). Besides that, isotope substitution is opening new possibilities in the investigation of carrier scattering by neutral impurity (Erginsoy, 1950), the problem of which is actually more than a half century old (Ansel’m, 1978). Note also that the mechanism causing the appearance of the double structure in the polariton emission spectra (e.g., the crystal with isotope effect, Plekhanov, 1998) is the polariton scattering on the neutral impurity (e.g., donors, Koteles et al., 1985). IV. Isotopic Structuring for Fundamental Studies It is widely known that the melting and boiling points of ordinary water and heavy water (D O) differ by a few degrees centigrade. For elements heavier than hydrogen, the isotopic differences in melting points (T ) of elemental and complex solids are generally smaller but also detectable. It is quite surprising, however, that there are almost no reports of direct measurement of these differences in the literature (Berezin, 1988). Another noticeable fact is that sometimes the isotope effect shows a drastic ‘‘self-ampliﬁcation,’’ for example, isotopic replacements of Ba and Ti in BaTiO (both are heavy elements) can shift the phase transition tempera tures by as much as 200 K (Hidaka and Oka, 1987). The reason(s) for such selective anomalies are not yet clearly established.

V. Isotope Diffusion in Semiconductors One of the fundamental processes occurring in all matter is the random motion of its atomic constituents. In semiconductors, much has been learned in recent years about the motion of host and impurity atoms as well as

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native defects such as vacancies and interstices. A number of excellent review papers have been written (see, e.g., Frank et al., 1984; Fuchs et al., 1995; Haller, 1995; and references therein). When a concentration gradient dN/dx is introduced, random motion leads to a net ﬂux of matter J that is proportional to the gradient (Fick’s ﬁrst law; see, e.g., Tan et al., 1991): J : 9D dN/dx.

(6)

The diffusion coefﬁcient D can in many (though not all) cases be described by a thermally activated constant: D : D exp(9E/k T ).

(7)

Impurity diffusion in semiconductors plays a key role in the fabrication of electronic devices. For example, diffusion can be utilized as a desirable process enabling the introduction of dopant impurities into areas deﬁned by a mask on a semiconductor wafer. Diffusion can also act as a determinant process, broadening narrow impurity implantation proﬁles or rapidly admitting diffusion of undesirable impurities. Very extensive literature exists on diffusion studies for most semiconductors. The ﬁeld is extremely active, especially for semiconductors, which have recently become important for high-temperature electronics (e.g., C, SiC), light-emitting devices working in the green and blue ranges of the visible spectrum (e.g., ZnSe, GaN), and IR records (Ge). As simple as diffusion may appear to be, at least conceptually, many basic unanswered questions still exist. Results from supposedly identical experiments conducted by different groups often scatter by signiﬁcant factors. This clearly indicates that there are still hidden factors that need to be determined. Even for the most thoroughly studied crystalline solid, Si (see, e.g., Gusev et al., 1995; Baumwol et al., 1999; and references therein), we still do not know with certainty the relative contributions of vacancies and interstices to self- and impurity-diffusion as a function of temperature, the position of the Fermi level, and external effects, such as surface oxidation and nitridation (see the previously cited references for more details). Fuchs et al. (1995) presented results of a very accurate method to measure the self-diffusion coefﬁcient of Ge that circumvents many of the experimental problems encountered in the conventional methods. Fuchs et al. used Ge isotopic heterostructures (stable isotope), grown by molecular-beam epitaxy (MBE) (see, e.g., Haller, 1990). In general, isotope heterostructures consist of layers of pure (e.g., Ge, Ge) or deliberately mixed isotopes of a chemical element. Fuchs et al. used the isotope heterostructure growing on the Ge substrate. At the interface only the atomic mass is changing, while (to ﬁrst order) all the other physical properties stay the same. In the as-grown samples, this interface is atomically ﬂat with layer thickness

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ﬂuctuations of about two atomic ML (Itoh et al., 1993). Upon annealing, the isotopes diffuse into each other (self-diffusion) with a rate that depends strongly on temperature. The concentration proﬁles are measured with SIMS (secondary-ion-mass spectroscopy), after pieces of the same samples have been separately annealed at different temperature. This allows an accurate determination of the self-diffusion enthalpy as well as the corresponding entropy. The isotopic heterostructures are unique for the selfdiffusion studies in several respects. 1. The interdiffusion of Ge isotopes takes place at the isotopic interface inside the crystal, unaffected by possible surface effects (e.g., oxidation, strains, and impurities) encountered in the conventional technique. 2. One sample annealed at one temperature provides ﬁve more or less independent measurements: Germanium consists of ﬁve stable isotopes. Their initial respective concentrations vary for the different layers of the as-grown isotope heterostructure. After annealing, the concentration proﬁle of each of the ﬁve isotopes can be analyzed separately to obtain ﬁve data points for each annealing temperature. Tan et al. (1992) were the ﬁrst to make use of GaAs isotope superlattices (GaAs; GaAs) on a Si-doped substrate to study Ga self-diffusion. Unfortunately, their analysis was only partially successful because native defects and silicon outdiffusion from the doped substrate into the superlattice obscured their results (see also Fuchs et al., 1995). Contrary to the short-period superlattices required for Raman experiments, Fuchs et al. (1995) used sufﬁciently thick layers to access (Dt products of one to several micrometers. This ﬁrst study focused on Ge self-diffusion in undoped material. They used bilayers of Ge and Ge (each 1000 or 2000 Å thick) that were grown by MBE on a natural substrate. Disregarding for the moment the small differences in diffusity caused by different isotope masses (see also Campbell, 1975), they expected the Ge isotopes to diffuse symmetrically into each other following complementary error functions. There was no net ﬂow of Ge atoms and the atomic concentrations added up to unity at every point: [Ge] ; [Ge] : 1. The individual proﬁles are described by [Ge] : 0.5[Ge] #1 9 erf(x/2(Dt)% V [Ge] : 0.5[Ge] #1 9 erf(x/2(Dt)% V

(8) (9)

The interface is located at x : 0 and [Ge] : [Ge] 4.4 ; 10 cm\. For the experiments, Fuchs et al. chose ﬁve diffusion temperatures and adjusted the times so as to obtain similar (Dt products. Figure 4 shows SIMS results for the as-grown sample and for the sample diffused at 586°C for the 55.55 h. Because the isotopes are only enriched into the high 90%range, they obtained SIMS data from some of the residual minor Ge isotopes. This

9 Future Developments and Applications

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Fig. 4. Experimental depth of the atomic fraction of Ge, Ge, Ge, Ge, and Ge of a diffusion-annealed sample. The solid line is a theoretical ﬁt of Ge. (After Fuchs et al., 1995.)

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Vladimir G. Plekhanov

redundancy in data is very useful in the deconvolution of the SIMS instrument function and in improving the accuracy of the data. The results obtained by Fuchs et al. are in excellent agreement with previously published values. Ga self-diffusion in GaAs (Wang et al., 1996) and GaP (L. Wang et al., 1997) was measured directly in isotopically controlled GaAs and GaP heterostructures. In the case of GaP, for the experiment, GaP and GaP epitaxial layers 200 nm thick were grown by solid source MBE at 700°C on undoped GaP substrates. The natural Ga isotope composition in the GaP substrates is 60.2% Ga and 39.8% Ga. The compositions in the isotopically controlled epilayers, on the other hand, were 99.6% Ga (Ga) and 0.4% Ga Ga). In the SIMS measurement, the primary ion beam was formed with 5.5 keV Cs> ions. GaCs> molecules were detected as secondary species as the sputtering proceeded. As before, assuming Fick’s equations describe the self-diffusion process and the diffusion coefﬁcient D is constant (Crank, 1993), the concentrations of the Ga isotopes can be expressed as C ;C C 9C 9 erf(x/R), C(x) : 2 2

(10)

where x : 0 at the epitaxial interface, C and C are the initial Ga isotope concentrations at the left- and right-hand sides of the interface, respectively, and erf(y) is the error function. The characteristic diffusion length R was deﬁned as R : 2(Dt,

(11)

where D is the Ga self-diffusion coefﬁcient and t is the annealing time. The SIMS data can then be compared with calculated values of C(x). Adjusting the diffusion length R, a ﬁt of C(x) to the SIMS proﬁle can be made. Figure 5 shows the SIMS proﬁles (solid lines) and the calculated C(x) of Ga (circles) and Ga (continuous line) in a sample annealed at T : 111°C for 3 h and 51 min. Excellent agreement was obtained between the measured and the calculated proﬁles over two and a half orders of magnitude in concentration. From these results for GaP, L. Wang et al. (1997) obtained the values of the activation enthalpy H1" and self-diffusion entropy S1" equal to 4.5 eV and 4k , respectively. For comparison, L. Wang et al. (1997) obtained the activation enthalpy and entropy for GaAs as 4.24 eV and 7.5k , respectively. The signiﬁcant difference in values of S1", according to Wang et al., indicates profound variation in the way that the mediating native defects are formed or migrate in GaP as compared to GaAs. The small value S1" in GaP may be connected to the stronger Ga—P bond compared to the Ga—As bond (see also Beernik et al., 1995). Recently, Bracht et al. (1999) studied (see also Fig. 6) the Ga self-diffusion and Al—Ga interdiffusion with isotope heterostructures of AlGaAs/GaAs.

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Fig. 5. SIMS depth proﬁles of Ga and Ga isotope epilayers annealed at 1111°C for 231 min. The ﬁlled circles represent the calculated Ga concentration proﬁle. (After L. Wang et al., 1997).

Ga diffusion in Al Ga As with x : 0.41, 0.62, 0.68, and 1.0 was found to V \V decrease with increasing Al concentration. The intermixing observed at AlGaAs/GaAs interfaces was accurately described if a concentrationdependent interdiffusion coefﬁcient was assumed. The higher Al diffusity in GaAs as compared to Ga self-diffusion was attributed to the higher jump in frequency of Al as compared to Ga, caused by the difference in their masses. The lower Ga diffusity in AlAs compared to GaAs was proposed to be due to lower thermal equilibrium concentrations of vacancies (C) in T ALAs as compared to GaAs. The different values C in these materials were T explained by the differences in the electronic properties between AlAs and GaAs. We should add here that the value of activation enthalpy Q of studied heterostructures lies in the range 3.6 < 0.1, which is consistent with the results of Wee et al. (1997). Readers can ﬁnd more details in the forthcoming volume in this series devoted to the isotope diffusion in semiconductors.

VI. Other Unexplored Applications of Isotopic Engineering Here we shall brieﬂy list a few additional possibilities of isotopic structuring. 1. L aser application. The discovery of the linear luminescence of free excitons observed in the wide temperature range has put the lithium

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Vladimir G. Plekhanov

Fig. 6. SIMS depth proﬁles of Al (*), Ga ()) and Ga (;) in the as-grown AlGaAs/ AlGaAs/GaAs heterostructure (b) [see (a)] and after annealing at 1050°C for 1800 s [see (b)]. (After Bracht et al, 1999.)

2.

3. 4.

5.

hydride, as well as diamond crystals (Takiyama et al., 1996) in the row of the prospective sources of the coherent radiation in the UV spectral range. For LiH, the isotope tunning of the exciton emission has also been shown (Plekhanov and Altukhov, 1983). Another direction of isotope engineering could be based on exploiting the differences in thermal conductivity (see earlier) between isotopically pure and isotopically mixed solids for purposes such as phonon focusing, precise thermometry based on isotopically gradient structures, and so on. The use of isotopically structured Ni ﬁlms for neutron interference ﬁlters has been reported by Antonov et al. (1986). Isotopically structured light devices. This could slightly shift the spectral characteristics and lead to some changes in the kinetics of energy transfer, modify the lifetimes, recombination rate, and so on. Since the speed of sound is proportional to (M, variations in isotopically structured acoustoelectronic devices (transducers, surface

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acoustic wave devices, etc.) could be signiﬁcant, especially in achieving phase differences over the relatively short isotopically distinguished paths. We have outlined several, mostly untested possibilities arising from exploiting differences in various stable isotopes and purposeful isotopic structuring. These examples of the potential capabilities of isotopic engineering by no means form an exhaustive list.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

CHAPTER 10

Conclusions

In this review, we have presented the results of experimental and theoretical studies of the effects of the isotopic substitution of a crystal lattice on elastic, thermal, and vibrational properties of semiconducting and insulating crystals. A detailed study of the thermal conductivity of isotope pure and isotope mixed semiconducting crystals is needed to reﬁne the existing theoretical model. Further interesting effects of isotopic substitution should result from investigations of low-dimensional structures: isotopic superlattices, quantum wells, and wires or quantum dots. Considering what has been achieved with isotopically controlled diamond and Ge (as well as the other crystals), we can be conﬁdent that a very large number of interesting experiments can be performed if there is enough commitment to obtaining the necessary materials. There can be little doubt that we will see experiments with isotopically controlled Si-built crystals (as in the case of thermal conductivity) and perhaps a SiSi superlattice. The studies performed with Ge should deﬁnitely be repeated with Si to verify and solidify the understanding gained with Ge. The limitation of the conventional method for determining the energy spectrum of the disordered systems, based on the separation of the ordered part of the potential and a small disordered perturbation, becomes evident, because this method, which is valid in the case of weak scattering of phonons in isotopically mixed germanium and diamond crystals, fails to describe media with a strong scattering potential (for example, LiH D V \V crystals), resulting in localization of LO() phonons in such systems. Therefore, a new approach should be developed that would yield a selfconsistent model of the lattice dynamics capable of describing not only localized modes (at low concentrations), but also crystal vibrations of mixed crystals over the entire range of concentrations of components. Also, a more consistent consideration of the anharmonicity is required, probably begin225

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ning from the model of isotopic defect, because neither elastic nor vibrational properties of isotopically mixed crystals can be described without taking anharmonicity into account. In my opinion, this enables one to develop a model of lattice dynamics to describe not only weak but also strong scattering of phonons caused by the isotopic disorder. In conclusion, note also the possible applications of isotopically mixed crystals. The wide possibilities of isotopic engineering brieﬂy mentioned in this review (Chapter 9) hold the greatest promise for application in solid-state and quantum electronics, optoelectronics, and many other modern and new technologies that are even now difﬁcult to imagine. The universality of a broad class of phenomena related to the isotopic effect on large-radius excitons in crystals with different structure and type of chemical bond (C, LiH, CsH, ZnO, ZnSe, CuCl, CdS, Cu O, GaAs, Si, and Ge) enables us to speak of the emerging new direction of research: the spectroscopy of Wannier—Mott excitons in crystals with different isotopic composition. Numerous practical applications of such research provide a strong and independent stimulus for its advancement. The existence of a large number of stable (or long-lived) isotopes, together with well-developed separation techniques, facilitate the development of isotopic engineering, which requires a deep understanding of the fundamental processes that take place in such compounds. The experimental results presented in this review indicate that isotopic substitution for a light isotope with a heavy one leads to an increase in the band-to-band transitions with a nonlinear dependence on the concentration of isotopes, although the lattice constant of most isotopically mixed crystals (with the exception of diamond) complies with Vegard’s law. A comparative study of the temperature and isotopic shift of the edge of fundamental absorption for a large number of different crystals indicates that the main (but not the only) contribution to this shift comes from the zero oscillations whose magnitude may be quite considerable and comparable with the energy of longitudinal optical phonon. The replacement of the light isotope with a heavier one causes a nonlinear increase in the binding energy of Wannier—Mott excitons, and to an increase in the energy of longitudinal-transverse splitting — the latter is especially hard to interpret theoretically. Theoretical description of the experimentally observed dependence of the binding energy of excitons on nuclear mass requires simultaneously taking into account the exchange of LO phonons between electron and hole in the exciton, and the separate interaction of carriers with LO phonons. Apart from the isotopic shift, all lines of exciton spectra and luminescence (RRLS) exhibit additional broadening. The nonlinear concentration dependence of the energy E . f (x) and the half width E of the long-wavelength exciton peak in the reﬂection spectrum E0 . f (x) is caused primarily by isotopic disordering of the crystal lattice in mixed LiH D crystals. The nonlinear dependence of the binding energy V \V E . f (x) of Wannier—Mott excitons on the concentration x is due to the @

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227

ﬂuctuation smearing of the edges of the conduction band and the valence band. The experimental dependence E . f (x) for LiH D crystals ﬁts in @ V \V well enough with the calculations according to the model of large-radius exciton in a disordered medium; hence it follows that the ﬂuctuation smearing of the band edges is caused by isotopic disordering of the crystal lattice. The identical structure of spectra of intrinsic luminescence of pure crystals and mixed LiH D crystals, which consists of the narrow V \V zero-phonon line and its broader LO replicas, and the peculiar temperature dependence of the intensity of these lines are deﬁnitely associated with the emission by free excitons. Free excitons may carry kinetic energy that is as high as ten times their binding energy. The experimentally observed considerable broadening of the lines of LO repetitions in mixed crystals is also caused by isotopic disordering of the crystal lattice. A further argument in favor of the preceding is provided by the nonlinear dependence of the half width of LO replicas in the emission spectrum of free excitons on the isotope concentration x. The observed dependence ﬁts in qualitatively with the theory of the emission of excitons in a disordered medium. It is also assumed that anharmonism of the third order depends nonlinearly on the isotope concentration. Finally, note the universality of phenomena associated with isotopic effect and the effect of disordering in the crystals of diamond and lithium hydride as well as silicon and germanium.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 68

Index

Copper halides, dependence of energy gaps, 158—168 Corringi—Cohn—Rostocker (CCR) method, 129 Coulomb force matrix, 59 CuCl, ﬁrst-order Raman spectra, 97—101

A Ab initio calculations, 69, 70 Additional boundary conditions (ABC), 183, 184—186 Anharmonicity of vibrations, 6—7, 10, 15, 29 Anti-Stokes, 79

D B Debye formula, interpolation, 26, 32 Debye temperature, 4, 6, 8, 23—26 Debye—Waller factor, 140—146, 154, 163 Deformation dipole model, (DDM), 61—62 Deformation potential, 135, 136, 139, 151 Density of states (DOS), 58 Raman spectra and, 73—83 Diamond dependence of energy gaps, 158—168 dependence of thermal conductivity on isotopic composition, 28—46 experimental results, 37—42 ﬁrst-order Raman spectra, 94—96 lattice constants and, 51—53 Raman spectra and density of phonon states, 81—83 second-order Raman spectra, 103—104 Diamond, elastic properties, 9—17 temperature dependence, 17—22 Dielectric-metal transition, 132—134 Diffusion in semiconductors, 216—221 Dipole models (DMs) deformation, (DDM), 61—62 shell model (SM), 60—61 Dispersion law, 63 Dulong and Petit law, 24, 25, 26, 31

Band-edge absorption, comparative study of, 119—121 Band-to-band transitions, 140 renormalization of energy, 156—158 Bethe—Salpeter equation, 203 Bloch function, 128 Boltzmann’s constant, 4, 24, 34 Born—Mayer potentials, 59 Bose—Einstein factor, 54, 79, 146 Boundary scattering, 42, 43 Brillouin scattering of light, 2, 6, 7 diamonds and, 9—17 Bulk modulus deﬁned, 12 zone center optical mode and, 12—17

C Callaway’s model, 33—34, 43—44 Cauchy’s relation, 4 Christoffel equation, 1 Clausius—Mosoti relation, 212 Coherent potential approximation (CPA), 109—112 Cohn—Sham form, 129

243

244

Index E

Einstein approximation, 8, 24, 25 Einstein free energy function, 46 Elastic properties background information on constant measurements, 1—2 bulk modulus, 12 diamond, 9—17 experimental results, 3—17 LiH crystals, 3—9 propagation modes, 1 temperature dependence in diamond, 17—22 zone center optical mode, 12—17 Energy conservation law, 30 Exciton—phonon interaction See also Lithium hydride crystals, excitons in dependence of energy gaps, 158—168 effects of temperature and pressure, 139—156 interaction between excitons and nonpolar optical phonons, 135—136 isotopic effects, 156—180 luminescence of free excitons in LiH and LiD crystals, 174—180 polarization interaction of free excitons with phonons, 136—139 renormalization of binding energy and Wannier—Mott excitons, 168—174 renormalization of energy, 156—158 Extended elementary cell method, 130 Extended shell model (ESM), 61 F Fan term, 140, 154 First-order Raman spectra, 93—103 Force constants, 57—59 Fourier law, 28 Fourier transform, 59—60 Frequency distribution function, 63, 73—74 Fro¨hlich interaction, 136—139, 151, 169—174 G GaAs dependence of energy gaps, 158—168 lattice constants and, 54—55

GaN, ﬁrst-order Raman spectra, 101—103 Gap modes, 85—92 dependence of energy gaps, 158—168 Germanium crystals dependence of energy gaps, 158—168 dependence of thermal conductivity on isotopic composition, 28—46 experimental results, 42—46 ﬁrst-order Raman spectra, 93—94 lattice constants and, 50—51 second-order Raman spectra, 103—104 Green’s functions, 86, 88, 110—111, 129, 171 Gruneisen constant, 6, 41, 46

H Holland’s model, 34—36, 45 Hooke’s law, 23

I Information storage, 214—215 Isotopic composition dependence of lattice constant on, 46—55 dependence of thermal conductivity of diamond, Ge, and Si crystals on, 28—46 effects on speciﬁc heat, 27—28 Isotopic disordering of crystal lattices broadening of ground state, 201—203 effects on interband transitions, 199—201 effects on luminescence linewidths, 204— 209 models of disorder, 195—199 nonlinear dependence of binding energy, 203—204 site versus structure disordering, 195 Isotopic effects, electron excitations and dependence of energy gaps, 158—168 luminescence of free excitons in LiH and LiD crystals, 174—180 renormalization of binding energy and Wannier-Mott excitons, 168—174 renormalization of energy, 156—158 Isotopic effects, emission spectrum of polaritons and experimental results, 186—190 resonance light scattering in LiH and LiD crystals, 190—193

245

Index theory of polaritons, 181—186 Isotopic engineering applications, 221—223 conﬁnement of light, 211—214 diffusion in semiconductors, 216—221 information storage, 214—215 neutron transmutations, 215—216 structuring for fundamental studies, 216

K Klemens—Callaway model, 39 Kramers—Kronig relation, 111

L Lattice constant, dependence on isotopic composition, 46—55 Lattice dynamics, models of dipole models (DMs), 60—62 formal force constants, 57—59 rigid-ion model (RIM), 59—60 valence force ﬁeld model (VFFM), 62—63 Lattice spectral function, 55 Light, isotopic conﬁnement of, 211—214 Lithium hydride and lithium deuteride crystals elastic properties, 3—9 isotopic composition effects of, on speciﬁc heat, 27—28 lattice constants and, 48—50 localized, resonant, and gap modes, 85—92 luminescence of free excitons in, 174—180 Raman spectra and density of phonon states, 73—83 resonance light scattering in, 190—193 two-mode behavior of LO phonon, 104—109 Lithium hydride crystals, excitons in band structure, 126—132 comparative study of band-edge absorption, 119—121 dielectric-metal transition, 132—134 exciton reﬂection spectra, 122—126 Local-density-approximation (LDA), 81

Local-ﬁeld effects, 182 Localized modes, 85—92 Lorentz correction, 182 Lowest- and second-order perturbation theory, 109—110, 112—118 Luminescence of free excitons in LiH and LiD crystals, 174—180 effects of disordering, 204—209

M Madelung constants, 59 Maxwell’s equations, 181, 185—186 Mixed crystals. See Raman spectra, mixed crystals and Motion, vector equation of, 58 Mufﬁn tin (MT) potential, 129 Mungham equation of state, 15

N Neutron scattering method, 63—72 Neutron transmutations, 215—216 Normal three-phonon scattering, 42, 44—45

O Optimal ﬂuctuation method, 199, 202, 207 Orthogonalized plane waves (OPW) method, 129

P Phonon density of states (DOS), 58 Phonon dispersion, neutron scattering method and, 63—72 Phonon interaction. See Exciton—phonon interaction Phonon states, Raman spectra and density of, 73—83 Planck’s constant, 34 Planck’s equation, 4, 24

246

Index

Plane-associated waves (PAW) method, 129 Polaritons, isotopic effects and experimental results, 186—190 resonance light scattering in LiH and LiD crystals, 190—193 theory of, 181—186 Polarization interaction of free excitons with phonons, 136—139 Pressure effects on exciton states, 139—156

Stokes, 79 Szigetti charge, 61

T

Rayleigh term, 41 Raman spectra, density of phonon states and, 73—83 Raman spectra, mixed crystals and coherent potential approximation, 109— 112 disorder effects, 109—118 ﬁrst-order, 93—103 localized, resonant, and gap modes, 85—92 lowest- and second-order perturbation theory, 109—110, 112—118 second-order, 103—104 two-mode behavior of LO phonon, 104—109 Resonance light scattering in LiH and LiD crystals, 190—193 Resonant modes, 85—92 Rigid-ion model (RIM), 59—60 Rydberg exciton, 157—158, 176

Temperature effects on exciton states, 139—156 Thermal conductivity Callaway’s model, 33—34, 43—44 compound semiconductors and lattice constants, 54—55 dependence of lattice constant on isotopic composition, 46—55 dependence of thermal conductivity on isotopic composition, 28—46 diamond, experimental results, 37—42 diamond and lattice constants, 51—53 ﬁnite value of the heat of conductivity, 29 germanium, experimental results, 42—46 germanium and lattice constants, 50—51 historical background, 28—32 Holland’s model, 34—36, 45 isotopic composition effects, 27—28 lithium hydride and lattice constants, 48—50 silicon, experimental results, 42—46 speciﬁc heat and Debye temperature, 23—26 Thermal expansion of crystal lattice, 140 Transfer, process of, 30 Two-mode behavior of LO phonon, 104— 109

S

U

R

Schro¨dinger equation, 127, 128, 129, 156, 185, 186 Second-order Raman spectra, 103—104 Self-energy term, 141, 143—144 Shell model (SM), 60—61 Silicon crystals, dependence of thermal conductivity on isotopic composition, 28—46 experimental results, 42—46 Silicon crystals, Raman spectra and density of phonon states, 81—83 Sn, ﬁrst-order Raman spectra, 96—97 Snell law of light refraction, 212

Ultrasonic methods, 6, 7 diamonds and, 9—10 U (Umklapp) process, 30, 34, 36, 39, 42, 43—46

V Valence force ﬁeld model (VFFM), 10, 62—63 Vibrational properties dipole models (DMs), 60—62 formal force constants, 57—59

247

Index neutron scattering method and phonon dispersion, 63—72 Raman spectra and density of phonon states, 73—83 rigid-ion model (RIM), 59—60 valence force ﬁled model (VFFM), 62—63 Virtual crystal approximation (VCA), 110, 111, 196, 197

nonlinear dependence of binding energy, 203—204 renormalization of binding energy, 168— 174 Wave vector conservation law, 30 Wigner—Seitz cell method, 128

Z W Wannier—Mott exciton, 124, 125—126, 137, 156, 226—227

Zero-point displacement, 46 ZnSe and lattice constants, 54—55 Zone center optical mode, 12—17

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Contents of Volumes in This Series

Volume 1 Physics of III‒V Compounds C. Hilsum, Some Key Features of III—V Compounds F. Bassani, Methods of Band Calculations Applicable to III—V Compounds E. O. Kane, The k-p Method V. L . Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure D. Long, Energy Band Structures of Mixed Crystals of III—V Compounds L. M. Roth and P. N. Argyres, Magnetic Quantum Effects S. M. Puri and T. H. Geballe, Thermomagnetic Effects in the Quantum Region W. M. Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H. Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H. Weiss, Magnetoresistance B. Ancker-Johnson, Plasma in Semiconductors and Semimetals

Volume 2 Physics of III—V Compounds M. G. Holland, Thermal Conductivity S. I. Novkova, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R. Drabble, Elastic Properties A. U. Mac Rae and G. W. Gobeli, Low Energy Electron Diffraction Studies R. Lee Mieher, Nuclear Magnetic Resonance B. Goldstein, Electron Paramagnetic Resonance T. S. Moss, Photoconduction in III—V Compounds E. Antoncik and J. Tauc, Quantum Efﬁciency of the Internal Photoelectric Effect in InSb G. W. Gobeli and I. G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in III—V Compounds M. Gershenzon, Radiative Recombination in the III—V Compounds F. Stern, Stimulated Emission in Semiconductors

249

250

Contents of Volumes in This Series

Volume 3 Optical of Properties III—V Compounds M. Hass, Lattice Reﬂection W. G. Spitzer, Multiphonon Lattice Absorption D. L. Stierwalt and R. F. Potter, Emittance Studies H. R. Philipp and H. Ehrenveich, Ultraviolet Optical Properties M. Cardona, Optical Absorption above the Fundamental Edge E. J. Johnson, Absorption near the Fundamental Edge J. O. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lax and J. G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Carries on Optical Properties E. D. Palik and G. B. Wright, Free-Carrier Magnetooptical Effects R. H. Bube, Photoelectronic Analysis B. O. Seraphin and H. E. Bennett, Optical Constants

Volume 4 Physics of III—V Compounds N. A. Goryunova, A. S. Borschevskii, and D. N. Tretiakov, Hardness N. N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A'''B4 D. L. Kendall, Diffusion A. G. Chynoweth, Charge Multiplication Phenomena R. W. Keyes, The Effects of Hydrostatic Pressure on the Properties of III—V Semiconductors L. W. Aukerman, Radiation Effects N. A. Goryunova, F. P. Kesamanly, and D. N. Nasledov, Phenomena in Solid Solutions R. T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors H. Levinstein, Characterization of Infrared Detectors P. W. Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors I. Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides D. Long and J. L. Schmidt, Mercury-Cadmium Telluride and Closely Related Alloys E. H. Putley, The Pyroelectric Detector N. B. Stevens, Radiation Thermopiles R. J. Keyes and T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F. R. Arams, E. W. Sard, B. J. Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Macrowave-Based Photoconductive Detector R. Sehr and R. Zuleeg, Imaging and Display

Volume 6 Injection Phenomena M. A. Lampert and R. B. Schilling, Current Injection in Solids: The Regional Approximation Method R. Williams, Injection by Internal Photoemission A. M. Barnett, Current Filament Formation

Contents of Volumes in This Series

251

R. Baron and J. W. Mayer, Double Injection in Semiconductors W. Ruppel, The Photoconductor-Metal Contact

Volume 7 Application and Devices Part A J. A. Copeland and S. Knight, Applications Utilizing Bulk Negative Resistance F. A. Padovani, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower, W. W. Hooper, B. R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs Field-Effect Transistor M. H. White, MOS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heterojunction Properties

Part B T. Misawa, IMPATT Diodes H. C. Okean, Tunnel Diodes R. B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices R. E. Enstrom, H. Kressel, and L. Krassner, High-Temperature Power Rectiﬁers of GaAs P \V V

Volume 8 Transport and Optical Phenomena R. J. Stirn, Band Structure and Galvanomagnetic Effects in III—V Compounds with Indirect Band Gaps R. W. Ure, Jr., Thermoelectric Effects in III—V Compounds H. Piller, Faraday Rotation H. Barry Bebb and E. W. Williams, Photoluminescence I: Theory E. W. Williams and H. Barry Bebb, Photoluminescence II: Gallium Arsenide

Volume 9 Modulation Techniques B. O. Seraphin, Electroreﬂectance R. L. Aggarwal, Modulated Interband Magnetooptics D. F. Blossey and Paul Handler, Electroabsorption B. Batz, Thermal and Wavelength Modulation Spectroscopy I. Balslev, Piezopptical Effects D. E. Aspnes and N. Bottka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

Volume 10 Transport Phenomena R. L. Rhode, Low-Field Electron Transport J. D. Wiley, Mobility of Holes in III—V Compounds C. M. Wolfe and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals R. L. Petersen, The Magnetophonon Effect

252

Contents of Volumes in This Series

Volume 11 Solar Cells H. J. Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efﬁciency; Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12 Infrared Detectors (II) W. L. Eiseman, J. D. Merriam, and R. F. Potter, Operational Characteristics of Infrared Photodetectors P. R. Bratt, Impurity Germanium and Silicon Infrared Detectors E. H. Putley, InSb Submillimeter Photoconductive Detectors G. E. Stillman, C. M. Wolfe, and J. O. Dimmock, Far-Infrared Photoconductivity in High Purity GaAs G. E. Stillman and C. M. Wolfe, Avalanche Photodiodes P. L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Putley, The Pyroelectric Detector — An Update

Volume 13 Cadmium Telluride K. Zanio, Materials Preparations; Physics; Defects; Applications

Volume 14 Lasers, Junctions, Transport N. Holonyak, Jr. and M. H. Lee, Photopumped III—V Semiconductor Lasers H. Kressel and J. K. Butler, Heterojunction Laser Diodes A Van der Ziel, Space-Charge-Limited Solid-State Diodes P. J. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters B. L. Sharma, Ohmic Contacts to III—V Compounds Semiconductors A. Nussbaum, The Theory of Semiconducting Junctions J. S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te H. Kressel, The Effect of Crystal Defects on Optoelectronic Devices C. R. Whitsett, J. G. Broerman, and C. J. Summers, Crystal Growth and Properties of Hg Cd Se alloys \V V M. H. Weiler, Magnetooptical Properties of Hg Cd Te Alloys \V V P. W. Kruse and J. G. Ready, Nonlinear Optical Effects in Hg Cd Te \V V

Volume 17 CW Processing of Silicon and Other Semiconductors J. F. Gibbons, Beam Processing of Silicon A. Lietoila, R. B. Gold, J. F. Gibbons, and L. A. Christel, Temperature Distributions and Solid Phase Reaction Rates Produced by Scanning CW Beams

Contents of Volumes in This Series

253

A. Leitoila and J. F. Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N. M. Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K. F. Lee, T. J. Stultz, and J. F. Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications, and Techniques T. Shibata, A. Wakita, T. W. Sigmon, and J. F. Gibbons, Metal-Silicon Reactions and Silicide Y. I. Nissim and J. F. Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride P. W. Kruse, The Emergence of (Hg Cd )Te as a Modern Infrared Sensitive Material \V V H. E. Hirsch, S. C. Liang, and A. G. White, Preparation of High-Purity Cadmium, Mercury, and Tellurium W. F. H. Micklethwaite, The Crystal Growth of Cadmium Mercury Telluride P. E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy and V. J. Mazurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A. K. Soad, and T. J. Tredwell, Photovoltaic Infrared Detectors M. A. Kinch, Metal-Insulator-Semiconductor Infrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neumark and K. Kosai, Deep Levels in Wide Band-Gap III—V Semiconductors D. C. Look, The Electrical and Photoelectronic Properties of Semi-Insulating GaAs R. F. Brebrick, Ching-Hua Su, and Pok-Kai Liao, Associated Solution Model for Ga-In-Sb and Hg-Cd-Te Y. Ya. Gurevich and Y. V. Pleskon, Photoelectrochemistry of Semiconductors

Volume 20 Semi-Insulating GaAs R. N. Thomas, H. M. Hobgood, G. W. Eldridge, D. L. Barrett, T. T. Braggins, L. B. Ta, and S. K. Wang, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits C. A. Stolte, Ion Implantation and Materials for GaAs Integrated Circuits C. G. Kirkpatrick, R. T. Chen, D. E. Holmes, P. M. Asbeck, K. R. Elliott, R. D. Fairman, and J. R. Oliver, LEC GaAs for Integrated Circuit Applications J. S. Blakemore and S. Rahimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21 Hydrogenated Amorphous Silicon Part A J. I. Pankove, Introduction M. Hirose, Glow Discharge; Chemical Vapor Deposition Y. Uchida, di Glow Discharge T. D. Moustakas, Sputtering I. Yamada, Ionized-Cluster Beam Deposition B. A. Scott, Homogeneous Chemical Vapor Deposition

254

Contents of Volumes in This Series

F. J. Kampas, Chemical Reactions in Plasma Deposition P. A. Longeway, Plasma Kinetics H. A. Weakliem, Diagnostics of Silane Glow Discharges Using Probes and Mass Spectroscopy L. Gluttman, Relation between the Atomic and the Electronic Structures A. Chenevas-Paule, Experiment Determination of Structure S. Minomura, Pressure Effects on the Local Atomic Structure D. Adler, Defects and Density of Localized States

Part B J. I. Pankove, Introduction G. D. Cody, The Optical Absorption Edge of a-Si: H N. M. Amer and W. B. Jackson, Optical Properties of Defect States in a-Si: H P. J. Zanzucchi, The Vibrational Spectra of a-Si: H Y. Hamakawa, Electroreﬂectance and Electroabsorption J. S. Lannin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R. A. Street, Luminescence in a-Si: H R. S. Crandall, Photoconductivity J. Tauc, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. E. Vanier, IR-Induced Quenching and Enhancement of Photoconductivity and Photo luminescence H. Schade, Irradiation-Induced Metastable Effects L. Ley, Photoelectron Emission Studies

Part C J. I. Pankove, Introduction J. D. Cohen, Density of States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Taylor, Magnetic Resonance Measurements in a-Si: H K. Morigaki, Optically Detected Magnetic Resonance J. Dresner, Carrier Mobility in a-Si: H T. Tiedje, Information about band-Tail States from Time-of-Flight Experiments A. R. Moore, Diffusion Length in Undoped a-Si: H W. Beyer and J. Overhof, Doping Effects in a-Si: H H. Fritzche, Electronic Properties of Surfaces in a-Si: H C. R. Wronski, The Staebler-Wronski Effect R. J. Nemanich, Schottky Barriers on a-Si: H B. Abeles and T. Tiedje, Amorphous Semiconductor Superlattices

Part D J. I. Pankove, Introduction D. E. Carlson, Solar Cells G. A. Swartz, Closed-Form Solution of I—V Characteristic for a a-Si: H Solar Cells I. Shimizu, Electrophotography S. Ishioka, Image Pickup Tubes

Contents of Volumes in This Series

255

P. G. LeComber and W. E. Spear, The Development of the a-Si: H Field-Effect Transistor and Its Possible Applications D. G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-State Image Sensor M. Matsumura, Charge-Coupled Devices M. A. Bosch, Optical Recording A. D’Amico and G. Fortunato, Ambient Sensors H. Kukimoto, Amorphous Light-Emitting Devices R. J. Phelan, Jr., Fast Detectors and Modulators J. I. Pankove, Hybrid Structures P. G. LeComber, A. E. Owen, W. E. Spear, J. Hajto, and W. K. Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22 Lightwave Communications Technology Part A K. Nakajima, The Liquid-Phase Epitaxial Growth of InGaAsP W. T. Tsang, Molecular Beam Epitaxy for III—V Compound Semiconductors G. B. Stringfellow, Organometallic Vapor-Phase Epitaxial Growth of III—V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs M. Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of Ga In AsP Alloys V \V \W P. M. Petroff, Defects in III—V Compound Semiconductors

Part B J. P. van der Ziel, Mode Locking of Semiconductor Lasers K. Y. Lau and A. Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers C. H. Henry, Special Properties of Semiconductor Lasers Y. Suematsu, K. Kishino, S. Arai, and F. Koyama, Dynamic Single-Mode Semiconductor Lasers with a Distributed Reﬂector W. T. Tsang, The Cleaved-Coupled-Cavity (C) Laser

Part C R. J. Nelson and N. K. Dutta, Review of InGaAsP InP Laser Structures and Comparison of Their Performance N. Chinone and M. Nakamura, Mode-Stabilized Semiconductor Lasers for 0.7—0.8- and 1.1—1.6-m Regions Y. Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 m B. A. Dean and M. Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R. H. Saul, T. P. Lee, and C. A. Burus, Light-Emitting Device Design C. L. Zipfel, Light-Emitting Diode-Reliability T. P. Lee and T. Li, LED-Based Multimode Lightwave Systems K. Ogawa, Semiconductor Noise-Mode Partition Noise

256

Contents of Volumes in This Series

Part D F. Capasso, The Physics of Avalanche Photodiodes T. P. Pearsall and M. A. Pollack, Compound Semiconductor Photodiodes T. Kaneda, Silicon and Germanium Avalanche Photodiodes S. R. Forrest, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Campbell, Phototransistors for Lightwave Communications

Part E S. Wang, Principles and Characteristics of Integrable Active and Passive Optical Devices S. Margalit and A. Yariv, Integrated Electronic and Photonic Devices T. Mukai, Y. Yamamoto, and T. Kimura, Optical Ampliﬁcation by Semiconductor Lasers

Volume 23 Pulsed Laser Processing of Semiconductors R. F. Wood, C. W. White, and R. T. Young, Laser Processing of Semiconductors: An Overview C. W. White, Segregation, Solute Trapping, and Supersaturated Alloys G. E. Jellison, Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon R. F. Wood and G. E. Jellison, Jr., Melting Model of Pulsed Laser Processing R. F. Wood and F. W. Young, Jr., Nonequilibrium Solidiﬁcation Following Pulsed Laser Melting D. H. Lowndes and G. E. Jellison, Jr., Time-Resolved Measurement During Pulsed Laser Irradiation of Silicon D. M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors D. H. Lowndes, Pulsed Beam Processing of Gallium Arsenide R. B. James, Pulsed CO Laser Annealing of Semiconductors R. T. Young and R. F. Wood, Applications of Pulsed Laser Processing

Volume 24 Applications of Multiquantum Wells, Selective Doping, and Superlattices C. Weisbuch, Fundamental Properties of III—V Semiconductor Two-Dimensional Quantized Structures: The Basis for Optical and Electronic Device Applications H. Morkoc and H. Unlu, Factors Affecting the Performance of (Al, Ga)As/GaAs and (Al, Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N. T. Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Abe et al., Ultra-High-Speed HEMT Integrated Circuits D. S. Chemla, D. A. B. Miller, and P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing F. Capasso, Graded-Gap and Superlattice Devices by Band-Gap Engineering W. T. Tsang, Quantum Conﬁnement Heterostructure Semiconductor Lasers G. C. Osbourn et al., Principles and Applications of Semiconductor Strained-Layer Superlattices

Contents of Volumes in This Series

257

Volume 25 Diluted Magnetic Semiconductors W. Giriat and J. K. Furdyna, Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap A'' Mn B Alloys at V '4 '\V Zero Magnetic Field S. Oseroff and P. H. Keesom, Magnetic Properties: Macroscopic Studies T. Giebultowicz and T. M. Holden, Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors J. Kossut, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. A. Gaj, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Mycielski, Shallow Acceptors in Diluted Magnetic Semiconductors: Splitting, Boil-off, Giant Negative Magnetoresistance A. K. Ramadas and R. Rodriquez, Raman Scattering in Diluted Magnetic Semiconductors P. A. Wolff, Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors

Volume 26 III—V Compound Semiconductors and Semiconductor Properties of Superionic Materials Z. Yuanxi, III—V Compounds H. V. Winston, A. T. Hunter, H. Kimura, and R. E. Lee, InAs-Alloyed GaAs Substrates for Direct Implantation P. K. Bhattacharya and S. Dhar, Deep Levels in III—V Compound Semiconductors Grown by MBE Y. Ya. Gurevich and A. K. Ivanov-Shits, Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-Dimensional Organic Crystals E. M. Conwell, Introduction to Highly Conducting Quasi-One-Dimensional Organic Crystals I. A. Howard, A Reference Guide to the Conducting Quasi-One-Dimensional Organic Molecular Crystals J. P. Pouquet, Structural Instabilities E. M. Conwell, Transport Properties C. S. Jacobsen, Optical Properties J. C. Scott, Magnetic Properties L. Zuppiroli, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28 Measurement of High-Speed Signals in Solid State Devices J. Frey and D. Ioannou, Materials and Devices for High-Speed and Optoelectronic Applications H. Schumacher and E. Strid, Electronic Wafer Probing Techniques D. H. Auston, Picosecond Photoconductivity: High-Speed Measurements of Devices and Materials J. A. Valdmanis, Electro-Optic Measurement Techniques for Picosecond Materials, Devices, and Integrated Circuits. J. M. Wiesenfeld and R. K. Jain, Direct Optical Probing of Integrated Circuits and High-Speed Devices G. Plows, Electron-Beam Probing A. M. Weiner and R. B. Marcus, Photoemissive Probing

258

Contents of Volumes in This Series

Volume 29 Very High Speed Integrated Circuits: Gallium Arsenide LSI M. Kuzuhara and T. Nazaki, Active Layer Formation by Ion Implantation H. Hasimoto, Focused Ion Beam Implantation Technology T. Nozaki and A. Higashisaka, Device Fabrication Process Technology M. Ino and T. Takada, GaAs LSI Circuit Design M. Hirayama, M. Ohmori, and K. Yamasaki, GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Watanabe, T. Mizutani, and A. Usui, Fundamentals of Epitaxial Growth and Atomic Layer Epitaxy S. Hiyamizu, Characteristics of Two-Dimensional Electron Gas in III—V Compound Heterostructures Grown by MBE T. Nakanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers T. Nimura, High Electron Mobility Transistor and LSI Applications T. Sugeta and T. Ishibashi, Hetero-Bipolar Transistor and LSI Application H. Matsueda, T. Tanaka, and M. Nakamura, Optoelectronic Integrated Circuits

Volume 31 Indium Phosphide: Crystal Growth and Characterization J. P. Farges, Growth of Discoloration-free InP M. J. McCollum and G. E. Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. Inada and T. Fukuda, Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method O. Oda, K. Katagiri, K. Shinohara, S. Katsura, Y. Takahashi, K. Kainosho, K. Kohiro, and R. Hirano, InP Crystal Growth, Substrate Preparation and Evaluation K. Tada, M. Tatsumi, M. Morioka, T. Araki, and T. Kawase, InP Substrates: Production and Quality Control M. Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T. A. Kennedy and P. J. Lin-Chung, Stoichiometric Defects in InP

Volme 32 Strained-Layer Superlattices: Physics T. P. Pearsall, Strained-Layer Superlattices F. H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y. Marzin, J. M. Gera´ rd, P. Voisin, and J. A. Brum, Optical Studies of Strained III—V Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Conﬁnement M. Jaros, Microscopic Phenomena in Ordered Superlattices

Volume 33 Strained-Layer Superlattices: Materials Science and Technology R. Hull and J. C. Bean, Principles and Concepts of Strained-Layer Epitaxy W. J. Schaff, P. J. Tasker, M. C. Foisy, and L. F. Eastman, Device Applications of Strained-Layer Epitaxy

Contents of Volumes in This Series

259

S. T. Picraux, B. L. Doyle, and J. Y. Tsao, Structure and Characterization of Strained-Layer Superlattices E. Kasper and F. Schaffer, Group IV Compounds D. L. Martin, Molecular Beam Epitaxy of IV—VI Compounds Heterojunction R. L. Gunshor, L. A. Kolodziejski, A. V. Nurmikko, and N. Otsuka, Molecular Beam Epitaxy of II—VI Semiconductor Microstructures

Volume 34 Hydrogen in Semiconductors J. I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. Seager, Hydrogenation Methods J. I. Pankove, Hydrogenation of Defects in Crystalline Silicon J. W. Corbett, P. Deák, U. V. Desnica, and S. J. Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J. Pearton, Neutralization of Deep Levels in Silicon J. I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M. Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S. J. Pearton, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A. D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N. M. Johnson, Hydrogen Migration and Solubility in Silicon E. E. Haller, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J. Chevalier, B. Clerjaud, and B. Pajot, Neutralization of Defects and Dopants in III—V Semiconductors G. G. DeLeo and W. B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R. F. Kieﬂ and T. L. Estle, Muonium in Semiconductors C. G. Van de Walle, Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

Volume 35 Nanostructured Systems M. Reed, Introduction H. van Houten, C. W. J. Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Büttiker, The Quantum Hall Effects in Open Conductors W. Hansen, J. P. Kotthaus, and U. Merkt, Electrons in Laterally Periodic Nanostructures

Volume 36 The Spectroscopy of Semiconductors D. Heiman, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields A. V. Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques A. K. Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors O. J. Glembocki and B. V. Shanabrook, Photoreﬂectance Spectroscopy of Microstructures D. G. Seiler, C. L. Littler, and M. H. Wiler, One- and Two-Photon Magneto-Optical Spectroscopy of InSb and Hg Cd Te \V V

260

Contents of Volumes in This Series

Volume 37 The Mechanical Properties of Semiconductors A.-B. Chen, A. Sher and W. T. Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys D. R. Clarke, Fracture of Silicon and Other Semiconductors H. Siethoff, The Plasticity of Elemental and Compound Semiconductors S. Guruswamy, K. T. Faber and J. P. Hirth, Mechanical Behavior of Compound Semiconductors S. Mahajan, Deformation Behavior of Compound Semiconductors J. P. Hirth, Injection of Dislocations into Strained Multilayer Structures D. Kendall, C. B. Fleddermann, and K. J. Malloy, Critical Technologies for the Micromachining of Silicon I. Matsuba and K. Mokuya, Processing and Semiconductor Thermoelastic Behavior

Volume 38 Imperfections in III/V Materials U. Scherz and M. Schefﬂer, Density-Functional Theory of sp-Bonded Defects in III/V Semiconductors M. Kaminska and E. R. Weber, El2 Defect in GaAs D. C. L ook, Defects Relevant for Compensation in Semi-Insulating GaAs R. C. Newman, Local Vibrational Mode Spectroscopy of Defects in III/V Compounds A. M. Hennel, Transition Metals in III/V Compounds K. J. Malloy and K. Khachaturyan, DX and Related Defects in Semiconductors V. Swaminathan and A. S. Jordan, Dislocations in III/V Compounds K. W. Nauka, Deep Level Defects in the Epitaxial III/V Materials

Volume 39 Minority Carriers in III—V Semiconductors: Physics and Applications N. K. Dutta, Radiative Transitions in GaAs and Other III—V Compounds R. K. Ahrenkiel, Minority-Carrier Lifetime in III—V Semiconductors T. Furuta, High Field Minority Electron Transport in p-GaAs M. S. Lundstrom, Minority-Carrier Transport in III—V Semiconductors R. A. Abram, Effects of Heavy Doping and High Excitation on the Band Structure of GaAs D. Yevick and W. Bardyszewski, An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in III—V Semiconductors

Volume 40 Epitaxial Microstructures E. F. Schubert, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A. Gossard, M. Sundaram, and P. Hopkins, Wide Graded Potential Wells P. Petroff, Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Temkin, D. Gershoni, and M. Panish, Optical Properties of Ga In As/InP Quantum \V V Wells

Contents of Volumes in This Series

261

Volume 41 High Speed Heterostructure Devices F. Capasso, F. Beltram, S. Sen, A. Pahlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frank, S. L. Wright, and F. Canora, GaAs-Gate Semiconductor—InsulatorSemiconductor FET M. H. Hashemi and U. K. Mishra, Unipolar InP-Based Transistors R. Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. G. Sollner, High-Frequency-Tunneling Devices H. Ohnishi, T. More, M. Takatsu, K. Imamura, and N. Yokoyama, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42 Oxygen in Silicon F. Shimura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T. J. Schaffner and D. K. Schroder, Characterization Techniques for Oxygen in Silicon W. M. Bullis, Oxygen Concentration Measurement S. M. Hu, Intrinsic Point Defects in Silicon B. Pajot, Some Atomic Conﬁgurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R. C. Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W. J. Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrems, Simulation of Oxygen Precipitation K. Simino and I. Yonenaga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tsuya, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications R. B. James and T. E. Schlesinger, Introduction and Overview L. S. Darken and C. E. Cox, High-Purity Germanium Detectors A. Burger, D. Nason, L. Van den Berg, and M. Schieber, Growth of Mercuric Iodide X. J. Bao, T. E. Schlesinger, and R. B. James, Electrical Properties of Mercuric Iodide X. J. Bao, R. B. James, and T. E. Schlesinger, Optical Properties of Red Mercuric Iodide M. Hage-Ali and P. Siffert, Growth Methods of CdTe Nuclear Detector Materials M. Hage-Ali and P Siffert, Characterization of CdTe Nuclear Detector Materials M. Hage-Ali and P. Siffert, CdTe Nuclear Detectors and Applications R. B. James, T. E. Schlesinger, J. Lund, and M. Schieber, Cd Zn Te Spectrometers for \V V Gamma and X-Ray Applications D. S. McGregor, J. E. Kammeraad, Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund, F. Olschner, and A. Burger, Lead Iodide M. R. Squillante, and K. S. Shah, Other Materials: Status and Prospects V. M. Gerrish, Characterization and Quantiﬁcation of Detector Performance J. S. Iwanczyk and B. E. Patt, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R. B. James, and T. E. Schlesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

262

Contents of Volumes in This Series

Volume 44 II‒IV Blue/Green Light Emitters: Device Physics and Epitaxial Growth J. Han and R. L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSe-based II—VI Semiconductors S. Fujita and S. Fujita, Growth and Characterization of ZnSe-based II—VI Semiconductors by MOVPE E. Ho and L. A. Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap II—VI Semiconductors C. G. Van de Walle, Doping of Wide-Band-Gap II—VI Compounds — Theory R. Cingolani, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A. Ishibashi and A. V. Nurmikko, II—VI Diode Lasers: A Current View of Device Performance and Issues S. Guha and J. Petruzello, Defects and Degradation in Wide-Gap II—VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization H. Ryssel, Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wang and Teng-Cai Ma, Electronic Stopping Power for Energetic Ions in Solids S. T. Nakagawa, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Müller, S. Kalbitzer and G. N. Greaves, Ion Beams in Amorphous Semiconductor Research J. Boussey-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M. L. Polignano and G. Queirolo, Studies of the Stripping Hall Effect in Ion-Implanted Silicon J. Stoemenos, Transmission Electron Microscopy Analyses R. Nipoti and M. Servidori, Rutherford Backscattering Studies of Ion Implanted Semiconductors P. Zaumseil, X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M. Fried, T. Lohner and J. Gyulai, Ellipsometric Analysis A. Seas and C. Christoﬁdes, Transmission and Reﬂection Spectroscopy on Ion Implanted Semiconductors A. Othonos and C. Christoﬁdes, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Inﬂuence of Annealing C. Christoﬁdes, Photomodulated Thermoreﬂectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U. Zammit, Photothermal Deﬂection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films A. Mandelis, A. Budiman and M. Vargas, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. Kalish and S. Charbonneau, Ion Implantation into Quantum-Well Structures A. M. Myasnikov and N. N. Gerasimenko, Ion Implantation and Thermal Annealing of III-V Compound Semiconducting Systems: Some Problems of III-V Narrow Gap Semiconductors

Contents of Volumes in This Series

263

Volume 47 Uncooled Infrared Imaging Arrays and Systems R. G. Buser and M. P. Tompsett, Historical Overview P. W. Kruse, Principles of Uncooled Infrared Focal Plane Arrays R. A. Wood, Monolithic Silicon Microbolometer Arrays C. M. Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D. L . Polla and J. R. Choi, Monolithic Pyroelectric Bolometer Arrays N. Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M. F. Tompsett, Pyroelectric Vidicon T. W. Kenny, Tunneling Infrared Sensors J. R. V ig, R. L . Filler and Y. Kim, Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays P. W. Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes G. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M. G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F. M. Steranka, AlGaAs Red Light Emitting Diodes C. H. Chen, S. A. Stockman, M. J. Peanasky, and C. P. Kuo, OMVPE Growth of AlGaInP for High Efﬁciency Visible Light-Emitting Diodes F. A. Kish and R. M. Fletcher, AlGaInP Light-Emitting Diodes M. W. Hodapp, Applications for High Brightness Light-Emitting Diodes I. Akasaki and H. Amano, Organometallic Vapor Epitaxy of GaN for High Brightness Blue Light Emitting Diodes S. Nakamura, Group III-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices D. J. Lockwood, Light Emission in Silicon G. Abstreiter, Band Gaps and Light Emission in Si/SiGe Atomic Layer Structures T. G. Brown and D. G. Hall, Radiative Isoelectronic Impurities in Silicon and Silicon-Germanium Alloys and Superlattices J. Michel, L. V. C. Assali, M. T. Morse, and L. C. Kimerling, Erbium in Silicon Y. Kanemitsu, Silicon and Germanium Nanoparticles P. M. Fauchet, Porous Silicon: Photoluminescence and Electroluminescent Devices C. Delerue, G. Allan, and M. Lannoo, Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites L. Brus, Silicon Polymers and Nanocrystals

Volume 50 Gallium Nitride (GaN) J. I. Pankove and T. D. Moustakas, Introduction S. P. DenBaars and S. Keller, Metalorganic Chemical Vapor Deposition (MOCVD) of Group III Nitrides W. A. Bryden and T. J. Kistenmacher, Growth of Group III—A Nitrides by Reactive Sputtering N. Newman, Thermochemistry of III—N Semiconductors S. J. Pearton and R. J. Shul, Etching of III Nitrides

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Contents of Volumes in This Series

S. M. Bedair, Indium-based Nitride Compounds A. Trampert, O. Brandt, and K. H. Ploog, Crystal Structure of Group III Nitrides H. Morkoc, F. Hamdani, and A. Salvador, Electronic and Optical Properties of III—V Nitride based Quantum Wells and Superlattices K. Doverspike and J. I. Pankove, Doping in the III-Nitrides T. Suski and P. Perlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemar, Optical Properties of GaN W. R. L. Lambrecht, Band Structure of the Group III Nitrides N. E. Christensen and P. Perlin, Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and H. Amano, Lasers J. A. Cooper, Jr., Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 51A Identiﬁcation of Defects in Semiconductors G. D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaeth, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy and E. R. Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K. H. Chow, B. Hitti, and R. F. Kieﬂ, SR on Muonium in Semiconductors and Its Relation to Hydrogen K. Saarinen, P. Hautojärvi, and C. Corbel, Positron Annihilation Spectroscopy of Defects in Semiconductors R. Jones and P. R. Briddon, The Ab Initio Cluster Method and the Dynamics of Defects in Semiconductors

Volume 51B Identiﬁcation of Defects in Semiconductors G. Davies, Optical Measurements of Point Defects P. M. Mooney, Defect Identiﬁcation Using Capacitance Spectroscopy M. Stavola, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander, W. D. Rau, C. Kisielowski, M. Gribelyuk, and A. Ourmazd, Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy N. D. Jager and E. R. Weber, Scanning Tunneling Microscopy of Defects in Semiconductors

Volume 52 SiC Materials and Devices K. Järrendahl and R. F. Davis, Materials Properties and Characterization of SiC V. A. Dmitriev and M. G. Spencer, SiC Fabrication Technology: Growth and Doping V. Saxena and A. J. Steckl, Building Blocks for SiC Devices: Ohmic Contacts, Schottky Contacts, and p-n Junctions M. S. Shur, SiC Transistors C. D. Brandt, R. C. Clarke, R. R. Siergiej, J. B. Casady, A. W. Morse, S. Sriram, and A. K. Agarwal, SiC for Applications in High-Power Electronics R. J. Trew, SiC Microwave Devices

Contents of Volumes in This Series

265

J. Edmond, H. Kong, G. Negley, M. Leonard, K. Doverspike, W. Weeks, A. Suvorov, D. Waltz, and C. Carter, Jr., SiC-Based UV Photodiodes and Light-Emitting Diodes H. Morkoç, Beyond Silicon Carbide! III--V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Subject and Author Index Including Tables of Contents for Volume 1—50

Volume 54 High Pressure in Semiconductor Physics I W. Paul, High Pressure in Semiconductor Physics: A Historical Overview N. E. Christensen, Electronic Structure Calculations for Semiconductors under Pressure R. J. Neimes and M. I. McMahon, Structural Transitions in the Group IV, III-V and II-VI Semiconductors Under Pressure A. R. Goni and K. Syassen, Optical Properties of Semiconductors Under Pressure P. Trautman, M. Baj, and J. M. Baranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs M. Li and P. Y. Yu, High-Pressure Study of DX Centers Using Capacitance Techniques T. Suski, Spatial Correlations of Impurity Charges in Doped Semiconductors N. Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

Volume 55 High Pressure in Semiconductor Physics II D. K. Maude and J. C. Portal, Parallel Transport in Low-Dimensional Semiconductor Structures P. C. Klipstein, Tunneling Under Pressure: High-Pressure Studies of Vertical Transport in Semiconductor Heterostructures E. Anastassakis and M. Cardona, Phonons, Strains, and Pressure in Semiconductors F. H. Pollak, Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures A. R. Adams, M. Silver, and J. Allam, Semiconductor Optoelectronic Devices S. Porowski and I. Grzegory, The Application of High Nitrogen Pressure in the Physics and Technology of III-N Compounds M. Yousuf, Diamond Anvil Cells in High Pressure Studies of Semiconductors

Volume 56 Germanium Silicon: Physics and Materials J. C. Bean, Growth Techniques and Procedures D. E. Savage, F. Liu, V. Zielasek, and M. G. Lagally, Fundamental Crystal Growth Mechanisms R. Hull, Misﬁt Strain Accommodation in SiGe Heterostructures M. J. Shaw and M. Jaros, Fundamental Physics of Strained Layer GeSi: Quo Vadis? F. Cerdeira, Optical Properties S. A. Ringel and P. N. Grillot, Electronic Properties and Deep Levels in Germanium-Silicon J. C. Campbell, Optoelectronics in Silicon and Germanium Silicon K. Eberl, K. Brunner, and O. G. Schmidt, Si C and Si Ge C Alloy Layers \W W \V\W V W

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Contents of Volumes in This Series

Volume 57 Gallium Nitride (GaN) II R. J. Molnar, Hydride Vapor Phase Epitaxial Growth of III-V Nitrides T. D. Moustakas, Growth of III-V Nitrides by Molecular Beam Epitaxy Z. Liliental-Weber, Defects in Bulk GaN and Homoepitaxial Layers C. G. Van de Walle and N. M. Johnson, Hydrogen in III-V Nitrides W. Götz and N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride B. Gil, Stress Effects on Optical Properties C. Kisielowski, Strain in GaN Thin Films and Heterostructures J. A. Miragliotta and D. K. Wickenden, Nonlinear Optical Properties of Gallium Nitride B. K. Meyer, Magnetic Resonance Investigations on Group III-Nitrides M. S. Shur and M. Asif Khan, GaN and AlGaN Ultraviolet Detectors C. H. Qiu, J. I. Pankove, and C. Rossington, III-V Nitride-Based X-ray Detectors

Volume 58 Nonlinear Optics in Semiconductors I A. Kost, Resonant Optical Nonlinearities in Semiconductors E. Garmire, Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport D. S. Chemla, Ultrafast Transient Nonlinear Optical Processes in Semiconductors M. Sheik-Bahae and E. W. Van Stryland, Optical Nonlinearities in the Transparency Region of Bulk Semiconductors J. E. Millerd, M. Ziari, and A. Partovi, Photorefractivity in Semiconductors

Volume 59 Nonlinear Optics in Semiconductors II J. B. Khurgin, Second Order Nonlinearities and Optical Rectiﬁcation K. L. Hall, E. R. Thoen, and E. P. Ippen, Nonlinearities in Active Media E. Hanamura, Optical Responses of Quantum Wires/Dots and Microcavities U. Keller, Semiconductor Nonlinearities for Solid-State Laser Modelocking and Q-Switching A. Miller, Transient Grating Studies of Carrier Diffusion and Mobility in Semiconductors

Volume 60 Self-Assembled InGaAs/GaAs Quantum Dots Mitsuru Sugawara, Theoretical Bases of the Optical Properties of Semiconductor Quantum Nano-Structures Yoshiaki Nakata, Yoshihiro Sugiyama, and Mitsuru Sugawara, Molecular Beam Epitaxial Growth of Self-Assembled InAs/GaAs Quantum Dots Kohki Mukai, Mitsuru Sugawara, Mitsuru Egawa, and Nobuyuki Ohtsuka, Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAs/GaAs Quantum Dots Emitting at 1.3 m Kohki Mukai and Mitsuru Sugawara, Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara, The Photon Bottleneck Effect in Quantum Dots Hajime Shoji, Self-Assembled Quantum Dot Lasers Hiroshi Ishikawa, Applications of Quantum Dot to Optical Devices Mitsuru Sugawara, Kohki Mukai, Hiroshi Ishikawa, Koji Otsubo, and Yoshiaki Nakata, The Latest News

Contents of Volumes in This Series

267

Volume 61 Hydrogen in Semiconductors II Norbert H. Nickel, Introduction to Hydrogen in Semiconductors II Noble M. Johnson and Chris G. Van de Walle, Isolated Monatomic Hydrogen in Silicon Yurij V. Gorelkinskii, Electron Paramagnetic Resonance Studies of Hydrogen and HydrogenRelated Defects in Crystalline Silicon Norbert H. Nickel, Hydrogen in Polycrystalline Silicon Wolfhard Beyer, Hydrogen Phenomena in Hydrogenated Amorphous Silicon Chris G. Van de Walle, Hydrogen Interactions with Polycrystalline and Amorphous Silicon — Theory Karen M. McNamara Rutledge, Hydrogen in Polycrystalline CVD Diamond Roger L. Lichti, Dynamics of Muonium Diffusion, Site Changes and Charge-State Transitions Matthew D. McCluskey and Eugene E. Haller, Hydrogen in III-V and II-VI Semiconductors S. J. Pearton and J. W. Lee, The Properties of Hydrogen in GaN and Related Alloys Jörg Neugebauer and Chris G. Van de Walle, Theory of Hydrogen in GaN

Volume 62 Intersubband Transitions in Quantum Wells: Physics and Device Applications I Manfred Helm, The Basic Physics of Intersubband Transitions Jerome Faist, Carlo Sirtori, Federico Capasso, Loren N. Pfeiffer, Ken W. West, Deborah L. Sivco, and Alfred Y. Cho, Quantum Interference Effects in Intersubband Transitions H. C. Liu, Quantum Well Infrared Photodetector Physics and Novel Devices S. D. Gunapala and S. V. Bandara, Quantum Well Infrared Photodetector (QWIP) Focal Plane Arrays

Volume 63 Chemical Mechanical Polishing in Si Processing Frank B. Kaufman, Introduction Thomas Bibby and Karey Holland, Equipment John P. Bare, Facilitization Duane S. Boning and Okumu Ouma, Modeling and Simulation Shin Hwa Li, Bruce Tredinnick, and Mel Hoffman, Consumables I: Slurry Lee M. Cook, CMP Consumables II: Pad François Tardif, Post-CMP Clean Shin Hwa Li, Tara Chhatpar, and Frederic Robert, CMP Metrology Shin Hwa Li, Visun Bucha, and Kyle Wooldridge, Applications and CMP-Related Process Problems

Volume 64 Electroluminescence I M. G. Craford, S. A. Stockman, M. J. Peanasky, and F. A. Kish, Visible Light-Emitting Diodes H. Chui, N. F. Gardner, P. N. Grillot, J. W. Huang, M. R. Krames, and S. A. Maranowski, High-Efﬁciency AlGaInP Light-Emitting Diodes R. S. Kern, W. Götz, C. H. Chen, H. Liu, R. M. Fletcher, and C. P. Kuo, High-Brightness Nitride-Based Visible-Light-Emitting Diodes Yoshiharu Sato, Organic LED System Considerations V. Bulovic´, P. E. Burrows, and S. R. Forrest, Molecular Organic Light-Emitting Devices

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Contents of Volumes in This Series

Volume 65 Electroluminescence II V. Bulovic´ and S. R. Forrest, Polymeric and Molecular Organic Light Emitting Devices: A Comparison Regina Mueller-Mach and Gerd O. Mueller, Thin Film Electroluminescence Markku Leskelä, Wei-Min Li, and Mikko Ritala, Materials in Thin Film Electroluminescent Devices Kristiaan Neyts, Microcavities for Electroluminescent Devices

Volume 66 Intersubband Transitions in Quantum Wells: Physics and Device Applications II Jerome Faist, Federico Capasso, Carlo Sirtori, Deborah L. Sivco, and Alfred Y. Cho, Quantum Cascade Lasers Federico Capasso, Carlo Sirtori, D. L. Sivco, and A. Y. Cho, Nonlinear Optics in Coupled-Quantum-Well Quasi-Molecules Karl Unterrainer, Photon-Assisted Tunneling in Semiconductor Quantum Structures P. Haring Bolivar, T. Dekorsy, and H. Kurz, Optically Excited Bloch Oscillations — Fundamentals and Application Perspectives

Volume 67 Ultrafast Physical Processes in Semiconductors Alfred Leitenstorfer and Alfred Laubereau, Ultrafast Electron—Phonon Interactions in Semiconductors: Quantum Kinetic Memory Effects Christoph Lienau and Thomas Elsaesser, Spatially and Temporally Resolved Near-Field Scanning Optical Microscopy Studies of Semiconductor Quantum Wires K. T. Tsen, Ultrafast Dynamics in Wide Bandgap Wurtzite GaN J. Paul Callan, Albert M.-T. Kim, Christopher A. D. Roeser, and Eriz Mazur, Ultrafast Dynamics and Phase Changes in Highly Excited GaAs Hartmut Haug, Quantum Kinetics for Femtosecond Spectroscopy in Semiconductors T. Meier and S. W. Koch, Coulomb Correlation Signatures in the Excitonic Optical Nonlinearities of Semiconductors Roland E. Allen, Traian Dumitricaˇ, and Ben Torralva, Electronic and Structural Response of Materials to Fast, Intense Laser Pulses E. Gornik and R. Kersting, Coherent THz Emission in Semiconductors

Isotope Effects in Solid State Physics, Volume 68 (Semiconductors and Semimetals)