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 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

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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

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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

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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 . . . .

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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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211 214 215 216 216 221

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

225

References

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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

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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

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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 : \ 2-

(3)

90

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

4 Raman Spectra of Isotopically Mixed Crystals

91

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

92

Vladimir G. Plekhanov

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.

4 Raman Spectra of Isotopically Mixed Crystals

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)]

94

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 : 0.68 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 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 HHH‚ :  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 HHH‚ ; 0 #6* when q ; 0. When excitons are scattered in a ground state band (1s—1s), as

138

Vladimir G. Plekhanov

shown in the paper of Ansel’m and Firsov, the matrix element has the form HHH‚ . #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 . HHH‚ . 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 HHH‚ #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.)

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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

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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

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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

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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

154

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

156

Vladimir G. Plekhanov

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

157

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

158

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 ) 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

7 The Emission Spectrum of Polaritons

187

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

189

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

190

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

7 The Emission Spectrum of Polaritons

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.

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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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Vladimir G. Plekhanov

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

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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

8 Isotopic Disordering of Crystal Lattices

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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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Vladimir G. Plekhanov

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

8 Isotopic Disordering of Crystal Lattices

203

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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Vladimir G. Plekhanov

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

8 Isotopic Disordering of Crystal Lattices

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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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Vladimir G. Plekhanov

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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Vladimir G. Plekhanov

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

209

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

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

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

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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

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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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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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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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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

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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

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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

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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

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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