Showing papers in "Physical Review B in 1973"

Special Points in the Brillouin Zone

Abstract: We present sets of special points in the Brillouin zone from which the average over the Brillouin zone of a periodic function of wave vector (e.g., energy, charge density, dipole matrix elements, etc.) can be determined in a simple and accurate way once the values of the function at these points are specified. We discuss a method for generating the special-point sets and apply it to the case of crystals with cubic and hexagonal Bravais lattices.

Band Structures of Transition-Metal-Dichalcogenide Layer Compounds.

Abstract: The nonrelativistic augmented-plane-wave (APW) method is applied to calculate the electronic band structures of several transition-metal-dichalcogenide ($T{X}_{2}$) layer compounds, including materials with the $C6(1T\ensuremath{-}\mathrm{Hf}{\mathrm{S}}_{2},1T\ensuremath{-}\mathrm{Ta}{\mathrm{S}}_{2})$, $C27(2H\ensuremath{-}\mathrm{Ta}{\mathrm{S}}_{2},2H\ensuremath{-}\mathrm{Nb}{\mathrm{Se}}_{2})$, and $C7(2H\ensuremath{-}\mathrm{Mo}{\mathrm{S}}_{2})$ structure types These calculations involve crystal potentials that are derived from neutral-atom charge densities The results of these calculations confirm that the group-$\mathrm{IV}B$ ($1T\ensuremath{-}\mathrm{Hf}{\mathrm{S}}_{2}$) and group-$\mathrm{VI}B$ ($2H\ensuremath{-}\mathrm{Mo}{\mathrm{S}}_{2}$) compounds are semiconductors; the calculated indirect band gaps of 27 and 12 eV are in reasonable agreement with the observed values of 20 and 14 eV, respectively Metallic behavior is predicted for the intermediate group-$\mathrm{V}B$ compounds $1T\ensuremath{-}\mathrm{Ta}{\mathrm{S}}_{2}$, $2H\ensuremath{-}\mathrm{Ta}{\mathrm{S}}_{2}$, and $2H\ensuremath{-}\mathrm{Nb}{\mathrm{Se}}_{2}$ A novel feature of the metal $d$ bands in the $2H\ensuremath{-}T{X}_{2}$ compounds is the occurence of a 1-eV hybridization gap within the ${d}_{{z}^{2}}$ and ${d}_{xy}$, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ manifolds This splits off a pair of hybridized $d$ bands which are half-filled in $2H\ensuremath{-}\mathrm{Ta}{\mathrm{S}}_{2}$ and $2H\ensuremath{-}\mathrm{Nb}{\mathrm{Se}}_{2}$ and completely filled in $2H\ensuremath{-}\mathrm{Mo}{\mathrm{S}}_{2}$ As a result of this hybridization gap, the valence or conduction bandwidths in each of these $2H\ensuremath{-}T{X}_{2}$ compounds are reduced to about 1 eV

Stochastic Transport in a Disordered Solid. I. Theory

Abstract: A general theory of stochastic transport in disordered systems has been developed. The theory is based on a generalization of the Montroll-Weiss continuous-time random walk (CTRW) on a lattice. Starting from a general mobility formalism, specialized $\stackrel{\mathrm{\ifmmode\acute\else\textasciiacute\fi{}}}{\mathrm{t}}$o hopping conduction, an exact expression for the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ for the CTRW process is derived. The frequency dependence of $\ensuremath{\sigma}(\ensuremath{\omega})$ is determined by the Fourier transform of the zeroth and second spatial moments of the function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}},t)$, which is equal to the probability per unit time that the displacement and time between hops is $\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$, $t$. The conductivity corresponding to characteristically different types of hopping distributions is discussed, as well as the basic approximation in adopting a CTRW on a lattice to transport in disordered solids.

Theory of Metal Surfaces: Induced Surface Charge and Image Potential

Abstract: This paper contributes to the theory of the electron density distribution induced at a metal surface by a small static external charge distribution. As a first application, profiles of the charge induced by a uniform external electric field are obtained for metals of different bulk electron densities. A quantity of particular interest is the position of the center of mass, ${x}_{0}$, of these profiles, for which we present numerical values. (The $x$ axis is taken along the surface normal.) Next, the case of a small point charge $q$ with $x$ coordinate ${x}_{1}$ well outside the surface is treated. It is shown that the image potential experienced by such a charge has the form $\ensuremath{-}\frac{{q}^{2}}{[4({x}_{1}\ensuremath{-}{x}_{0})]}$, where ${x}_{0}$ is the above-mentioned quantity. We locate ${x}_{0}$, the effective position of the metal surface, relative to the last lattice plane of the crystal. We discuss the implications of these results for alkali adsorption on metal substrates, the capacitances of small-gap condensers, and field-emission experiments.

Mean-Value Point in the Brillouin Zone

Abstract: A new special point in the Brillouin zone is introduced. It is defined as the point such that the value which any given periodic function of wave vector assumes at this point is an excellent approximation to the average value of the same function throughout the Brillouin zone. This special point is termed the "mean-value point," and is dictated by the crystal symmetry. The coordinates of the mean-value point for cubic lattices are explicitly given.

Multiphonon Raman Spectrum of Silicon

Abstract: The energy and polarization characteristics of the one- and two-phonon Raman spectrum have been measured using a 180\ifmmode^\circ\else\textdegree\fi{} backscattering technique. The two-phonon spectrum was measured at 20, 80, and 305\ifmmode^\circ\else\textdegree\fi{}K. The one-phonon spectrum was measured at 17, 30, 80, and 305 \ifmmode^\circ\else\textdegree\fi{}K. The one-phonon line of symmetry ${\ensuremath{\Gamma}}_{25}$, was shown to be Lorentzian and to have a deconvoluted half-width at 17 \ifmmode^\circ\else\textdegree\fi{}K of 1.45 \ifmmode\pm\else\textpm\fi{} 0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. The two-phonon Raman spectrum was used to determine phonon energies at the four critical points $\ensuremath{\Gamma}$, $X$, $L$, and $W$.

Inhomogeneous Electron Gas

Abstract: This work is a generalization of the Hohenberg---Kohn---Sham theory of the inhomogeneous electron gas, with emphasis on spin effects. An argument based on quantum electrodynamics is used to express the ground-state energy of a system of interacting electrons as a functional of the current density. Expressions are derived for coefficients appearing in an expansion of the correlation functional in terms of the linear-response functions of the homogeneous system, for a gas of almost constant four-current density. The current density contains a spin-dependent term which leads, in the nonrelativistic limit, to a local potential which is also spin dependent. This potential is applied to the problems of spin splitting of energy bands in ferromagnets and spin-density-wave antiferromagnets. The relations between the present approach, that of Slater, and the collective electron theory of ferromagnetism of Stoner are described.

Infrared Lattice Vibrations and Free-Electron Dispersion in GaN

Abstract: Infrared reflectivity and absorption measurements have been made on single-crystal epitaxial GaN on (0001) $\ensuremath{\alpha}\ensuremath{-}$${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$ crystals. Analysis of the normal-incidence reflectance data on low-carrier-concentration layers using the Kramers-Kronig technique and dielectric oscillator fits yields the values ${\ensuremath{\omega}}_{\mathrm{TO}}^{\ensuremath{\perp}}=560$ ${\mathrm{cm}}^{\ensuremath{-}1}$ and ${\ensuremath{\omega}}_{\mathrm{LO}}^{\ensuremath{\perp}}=746$ ${\mathrm{cm}}^{\ensuremath{-}1}$ for the optical mode frequencies at 300 K. Adopting ${\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{\ensuremath{\perp}}=5.35$ from a fit to Ejder's refractive-index data the additional quantities ${\ensuremath{\epsilon}}_{o}^{\ensuremath{\perp}}=9.5$ for the static dielectric constant, ${e}_{B}^{*\ensuremath{\perp}}=2.65e$ for the Born effective charge, and ${\ensuremath{\alpha}}^{\ensuremath{\perp}}=0.44$ for the polaron coupling constant are derived. Reflectivity measurements at 50\ifmmode^\circ\else\textdegree\fi{} incidence with $s$ and $p$ polarizations show that the longitudinal lattice mode is nearly isotropic. Using the value ${\ensuremath{\omega}}_{\mathrm{TO}}^{\ensuremath{\parallel}}=533$ ${\mathrm{cm}}^{\ensuremath{-}1}$ from Raman data the values ${\ensuremath{\omega}}_{\mathrm{LO}}^{\ensuremath{\parallel}}=744$ ${\mathrm{cm}}^{\ensuremath{-}1}$, ${\ensuremath{\epsilon}}_{o}^{\ensuremath{\parallel}}=10.4$, ${e}_{B}^{*\ensuremath{\parallel}}=2.82e$, and ${\ensuremath{\alpha}}^{\ensuremath{\parallel}}=0.49$ are obtained from oscillator fits to the 50\ifmmode^\circ\else\textdegree\fi{} incidence data. Information on the free-carrier effects in GaN was obtained by studying the normal-incidence reflectance as a function of carrier concentration in the 2\ifmmode\times\else\texttimes\fi{}${10}^{17}$ to 1\ifmmode\times\else\texttimes\fi{}${10}^{20}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ range. By fitting the reflectance minima versus concentration data, a value of $\frac{{m}^{*}}{m}=(0.20\ifmmode\pm\else\textpm\fi{}0.02)$ for the optical effective mass is obtained. Measurements at 50\ifmmode^\circ\else\textdegree\fi{} incidence show that the plasma frequency is isotropic within experimental precision.

Spherical Model of Shallow Acceptor States in Semiconductors

Abstract: The effective-mass approximation for shallow acceptor states in cubic semiconductors with degenerate valence bands is reformulated. The Hamiltonian is written as the sum of a spherical term and a cubic corection, thus pointing out the relevance of the spherical symmetry in the acceptor problem and the strong similarity to the case of atoms with the spin-orbit interaction. Without the introduction of any explicit representation of the Hamiltonian, the present formulation yields a meaningful classification of the acceptor states and reduces the eigenvalue problem to simple radial Hamiltonians. These radial Hamiltonians are explicitly given for the most improtant acceptor states and are shown to apply also to the description of the exciton problem. The variational method is used in the numerical calculation. The resulting eigenvalues, eigenfunctions, and related quantities are given as functions of the relevant parameters. The theoretical ionization energies are compared with available experimental data.

SPIN POLARIZATION OF ELECTRONS TUNNELING FROM FILMS OF Fe, Co, Ni, AND Gd.

Abstract: The spin polarization of electrons tunneling from films of Fe, Co, Ni, and Gd to superconducting Al films is determined from conductance measurements. The phenomenological theory of superconducting-normal-metal tunneling is modified to describe superconducting-ferro-magnetic tunneling in a magnetic field. The experimental technique and the method of analysis of the conductance curves to obtain the electron polarization are both described. The observed polarization is positive (majority spin direction predominating) for all the metals; the values obtained were Fe, +44%; Co, +34%; Ni, +11%; and Gd, +4.3%.

Stochastic Transport in a Disordered Solid. II. Impurity Conduction

Abstract: In a previous paper, the authors have developed a general theory of stochastic transport in disordered systems. In the present paper, the theory is applied, in detail, to a prototype of transport in a disordered system - impurity conduction in semiconductors. The complete frequency dependence of the real and imaginary part of the conductivity is calculated. In particular, the calculation details the transition from an ${\ensuremath{\omega}}^{s}$ dependence to essentially dc behavior (at a finite frequency), where $s\ensuremath{\sim}0.6\ensuremath{-}0.8$, depending on temperature and concentration. The theoretical results for frequency, temperature, and concentration dependence of the conductivity are shown to be in good agreement with the measurements of Pollak and Geballe (PG). In addition, the ac conductivity data of PG interpreted with the present theory yield experimental evidence for the existence of two-channel hopping in $n$-type Si.

Theory of Isothermal Currents and the Direct Determination of Trap Parameters in Semiconductors and Insulators Containing Arbitrary Trap Distributions

Abstract: Equations are presented that permit the calculation of the isothermal current- (I) vs-time ($t$) characteristics for defect insulators and semiconductors in which the field is sufficiently high and the active region sufficiently thin so that recombination of free carriers is negligible. More important, however, it is shown that by plotting $\mathrm{It}$ vs ${log}_{10}t$ the resulting characteristic is either (a) in the case of distributed traps, a direct image of the trap distribution, or (b) in the case of discrete traps, exhibits a number of sharp maxima at various times, depending on the trap parameters. The technique permits the direct determination of the trap distribution without having to have an a priori knowledge of the trap parameters and without the necessity of laborious analyses.

Indirect Interaction between Adatoms on a Tight-Binding Solid

Abstract: The indirect interaction between adatom pairs on the (100) surface of a simple-cubic tight-binding solid is investigated within a molecular-orbital approach. A general scheme for calculating the surface-density-of-states change and the interaction energy of one and two single-level adatoms is presented, and contact (and a correction) is made with Grimley's formulation. The method permits binding above surface atoms, at bridge sites, or at centered positions, and yields interaction energy as a function of band filling, adatom energy level, and a general hopping potential $V$ between an adatom and the nearest surface atom(s). Calculations have been carried out for $\frac{V}{{W}_{b}}$ in the range 1/12-1/2, the upper limit giving split-off states (${W}_{b}\ensuremath{\equiv}\mathrm{bandwidth}$). The single-atom interaction shows little dependence on binding type, in all three cases being most attractive when the Fermi energy equals the noninteracting adatom level, with a strongly $V$-dependent strength. For the pair interaction, one finds a strength at nearest-neighbor separation of about an order of magnitude smaller than the absorption energy of a single adatom. This interaction has an exponentiallike dropoff and sign alternations as one moves along the $〈10〉$ direction. Under reasonable conditions, the nearest-neighbor interaction is often repulsive while the next nearest, third nearest, or fourth nearest is attractive, suggesting the patterns $c(2\ifmmode\times\else\texttimes\fi{}2)$, (2 \ifmmode\times\else\texttimes\fi{} 2), and $c(4\ifmmode\times\else\texttimes\fi{}2)$, respectively, which are frequently observed in the adsorption of simple gases on the (100) surfaces of transition metals. On the basis of two-dimensional Ising-model calculations including second-neighbor interactions, one can estimate the strength of $V$ from the observed disordering temperature of the adatom lattice; the result is similar to that obtained from estimates based on the heat of adsorption.

Lattice Modes in Ferroelectric Perovskites: PbTi O 3

Abstract: A complete study of the lattice-dynamical behavior of the ferroelectric tetragonal perovskite PbTi${\mathrm{O}}_{3}$ has been carried out using Raman spectroscopy. The temperature dependence of all the long-wavelength mode frequencies are determined. We show that a damped-harmonic-oscillator model with a frequency-independent damping coefficient is sufficient to explain the observed shape of the soft $E(1\mathrm{TO})$ mode in the ferroelectric phase at all temperatures below ${T}_{c}=493$ \ifmmode^\circ\else\textdegree\fi{}C. Moreover, by combining our experimental mode-frequency data with pyroelectric measurements of the change in spontaneous polarization ${P}_{s}$ with temperature, we obtain values of ${P}_{s}(25 ^{\ensuremath{\circ}}\mathrm{C})=81$ \ensuremath{\mu}C/${\mathrm{cm}}^{2}$ and ${P}_{s}({T}_{c})=42$ \ensuremath{\mu}C/${\mathrm{cm}}^{2}$. This is in excellent agreement with a recent direct experimental measurement. The temperature dependence of all the mode strengths has also been determined to ${T}_{c}$. These results are used to extract the temperature dependence of the clamped dielectric constants. In PbTi${\mathrm{O}}_{3}$ the dielectric constant along the ferroelectric axis ${\ensuremath{\epsilon}}_{c}$ is determined primarily by the lowest frequency ${A}_{1}(1\mathrm{TO})$ mode at all temperatures to ${T}_{c}$, in contrast to BaTi${\mathrm{O}}_{3}$ where in the ferroelectric phase the lowest mode determines only $\ensuremath{\approx}25%$ of ${\ensuremath{\epsilon}}_{c}$. In PbTi${\mathrm{O}}_{3}$ the dielectric constant perpendicular to the $c$ axis ${\ensuremath{\epsilon}}_{a}$ is also determined by the lowest $E(1\mathrm{TO})$ mode at all temperatures. The possibility of observing critical effects near ${T}_{c}$ in the soft-mode data of PbTi${\mathrm{O}}_{3}$ has been examined. These effects are not observed. Also, it is shown that very careful fitting of the soft-mode temperature dependence to a functional form containing a minimum number of separately determined parameters is required before critical effects can be invoked in first-order phase transitions of the displacive type.

Bond-Charge Calculation of Nonlinear Optical Susceptibilities for Various Crystal Structures

Abstract: A simple localized-bond-charge model for the calculation of nonlinear optical susceptibilities is presented. We find that there are three important contributions to the nonlinearity, namely, the bond ionicity, the difference in atomic radii of the bonded atoms, and $d$-electron contributions. By including these effects we are able with one simple theory to accurately treat a wide variety of different types of compounds including ${A}^{\mathrm{III}}{B}^{\mathrm{V}}$ (e.g., GaAs, GaP, InSb), ${A}^{\mathrm{II}}{B}^{\mathrm{VI}}$ (e.g., ZnS, ZnO, BeO), ${A}^{\mathrm{I}}{B}^{\mathrm{VII}}$ (e.g., CuCl, CuBr, CuI), ${A}^{\mathrm{IV}}{B}_{2}^{\mathrm{VI}}$ (e.g., Si${\mathrm{O}}_{2}$), multibond crystals [e.g., ${A}^{\mathrm{I}}{B}^{\mathrm{III}}{C}_{2}^{\mathrm{VI}}$ (LiGa${\mathrm{O}}_{2}$, AgGa${\mathrm{S}}_{2}$, CuIn${\mathrm{S}}_{2}$, CuGa${\mathrm{Se}}_{2}$), ${A}^{\mathrm{II}}{B}^{\mathrm{IV}}{C}_{2}^{\mathrm{V}}$ (CdGe${\mathrm{P}}_{2}$, CdGe${\mathrm{As}}_{2}$, ZnGe${\mathrm{P}}_{2}$), ${A}^{\mathrm{III}}{B}^{\mathrm{V}}{C}_{4}^{\mathrm{VI}}$ (AIP${\mathrm{O}}_{4}$), also K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$], highly anisotropic crystals (e.g., HgS, Se, Te), as well as ferroelectrics (e.g., LiNb${\mathrm{O}}_{3}$, ${\mathrm{Ba}}_{2}$Na${\mathrm{Nb}}_{5}$${\mathrm{O}}_{15}$, LiTa${\mathrm{O}}_{3}$).

Quantum Dielectric Theory of Electronegativity in Covalent Systems. III. Pressure-Temperature Phase Diagrams, Heats of Mixing, and Distribution Coefficients

Abstract: Electronegativity difference was redefined in Paper I of this series as a scaling parameter which combines the concepts of valence and size differences. A procedure has been developed for its evaluation in terms of a two-band model. In Paper II of this series it was shown that this model describes and predicts the ionization potentials and electronic interband gaps of binary ${A}^{N}{B}^{8\ensuremath{-}N}$ compounds and their alloys. Here the energy of this model semiconducting-insulating solid is evaluated relative to a free-electron gas, i.e., an idealized metal, as a function of composition, pressure, and temperature. Using this highly simplified scaling approach, we obtain suprisingly accurate predictions for the heat of fusion, melting point, and pressure-temperature phase diagrams of these materials. A revised method of calculating the excess heat of mixing of a substitutional alloy is presented. This calculation is extended to the case of an arbitrary dilute impurity in an arbitrary semiconducting host; the distribution coefficient at the melting point of the host is obtained.

Model for an Exciton Mechanism of Superconductivity

Abstract: The exciton mechanism of superconductivity is discussed with respect to a particular model, a thin metal layer on a semiconductor surface. In this model, the metal electrons at the Fermi surface tunnel into the semiconductor gap where they interact with virtual excitons, producing a net attractive interaction among the electrons in direct analogy with the phonon mechanism of superconductivity. The physical requirements for successful realization of the exciton mechanism in a metal-semiconductor system are explored in detail, and the relevant parameters are described. Estimates are made for electron tunneling and band-bending effects, and an electron-exciton coupling constant is defined and estimated. Finally, an appropriately modified integral equation for the superconducting energy gap is solved numerically to yield transition temperatures both for a pure-exciton mechanism and for the exciton and phonon mechanisms acting simultaneously.

Metal-Insulator Transitions in Pure and Doped V 2 O 3

Abstract: The addition of ${\mathrm{Ti}}^{3+}$ and ${\mathrm{Mg}}^{2+}$ to ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$ leads to the suppression of the antiferromagnetic insulating phase; whereas the addition of ${\mathrm{Ti}}^{4+}$, ${\mathrm{Zr}}^{4+}$, and ${\mathrm{Fe}}^{3+}$ results in a first-order transition from a metallic to an insulating state. The effect of impurity ions is discussed in terms of the changes they cause in the bandwidth in analogy with the effect of pressure. The Hall coefficient of metallic ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$ at 4.2 \ifmmode^\circ\else\textdegree\fi{}K and 20 kbar is ${R}_{H}=+(3.5\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{3}$/C which is close to the value measured at 150 \ifmmode^\circ\else\textdegree\fi{}K and 1 atm. The residual resistivity of metallic ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$ is strongly impurity dependent (140 \ensuremath{\mu}\ensuremath{\Omega} cm/at.% Cr and 35 \ensuremath{\mu}\ensuremath{\Omega} cm/at.% Ti). These results are not completely consistent with current theories for the metal-insulator transition in ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$ but the best available model still seems to involve a localized-to-nonlocalized transition within the $d$ band primarily involving orbitals in the basal plane.

Low-Temperature Specific Heat and Thermal Conductivity of Noncrystalline Dielectric Solids

Abstract: The specific heat and thermal conductivity of amorphous ${\mathrm{As}}_{2}$${\mathrm{S}}_{3}$, ${\mathrm{B}}_{2}$${\mathrm{O}}_{3}$, 3Si${\mathrm{O}}_{2}$ \ifmmode\cdot\else\textperiodcentered\fi{} ${\mathrm{Na}}_{2}$O, CaK${(\mathrm{N}{\mathrm{O}}_{3})}_{3}$, Ge${\mathrm{O}}_{2}$, and GE No. 7031 varnish has been measured between 0.05 and 2 K. Their properties were found to be very similar to those of the previously measured glasses: Si${\mathrm{O}}_{2}$, Corning code 7740, Be${\mathrm{F}}_{2}$, Se, polymethylmethacrylate (PMMA), polystyrene (PS), Lexan, and glycerol. They all have a specific heat ${C}_{v}={c}_{1}T+{c}_{3}{T}^{3}$, where ${c}_{1}$ varies from 7 to 50 erg/g K, and ${c}_{3}$ from 1.2 to $3{c}_{\mathrm{Deb}}$ depending on the material (${c}_{\mathrm{Deb}}$ is the coefficient calculated with the Debye model). They also all have a conductivity, for $T0.5$ K, of $\ensuremath{\kappa}=\ensuremath{\beta}{(\frac{T}{\ensuremath{\alpha}})}^{\ensuremath{\delta}}$ where $\ensuremath{\beta}$ varies from 1.6 to 16 W/cm K, the exponent $\ensuremath{\delta}$ spans the range 1.9 \ifmmode\pm\else\textpm\fi{} 0.1, and $\ensuremath{\alpha}=1$ K. This uniformity of thermal properties among the diverse group of glasses measured is as difficult to explain as their temperature dependence, and is so far not understood.

Electron-Hole Liquids in Semiconductors

Abstract: In this paper the energetics of the formation of electron-hole metallic liquids in semiconductors is examined. The ground-state energies of electron-hole metals are calculated using Hubbard's approximate treatment of the electron gas for the following cases: (a) germanium, (b) germanium with a large (111) strain, (c) silicon, and (d) GaAs. The simple case of a single isotropic maximum for the valence band and a single minimum for the conduction band is also treated. It is shown that for both Si and Ge, the binding energy of the metallic state relative to free excitons is 5.7 and 1.7 meV, respectively. These values and the values of the equilibrium density are in good agreement with experiment. In the isotropic model the metallic state is not bound while for GaAs and strained Ge the metallic-state energy per electron is essentially equal to that for a gas of free excitons. The low-density limit of the isotropic band model is examined and the ground state for this system is predicted to be a dilute gas of molecules. It is argued that the forces between molecules are repulsive and will cause this state to break up at relatively low densities. If the density is increased, the system will undergo a first-order transition to the metallic state. The relevance of these calculations to the metal-insulator transition problem is discussed. It is pointed out that the fact that anisotropic and many-valleyed bands favor the metallic state means that the metal-insulator transition must ultimately be first order.

Multiphonon Relaxation of Rare-Earth Ions in Yttrium Orthoaluminate

Abstract: Nonradiative relaxation by multiple-phonon emission was investigated for excited electronic states of rare-earth ions in YAl${\mathrm{O}}_{3}$. Ions studied included ${\mathrm{Nd}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Ho}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Tm}}^{3+}$. Rates of multiphonon emission were determined from the difference of measured excited-state lifetimes and calculated radiative lifetimes. Electric dipole transition probabilities were computed using Judd-Ofelt intensity parameters for rare earths in YAl${\mathrm{O}}_{3}$. Multiphonon decay rates, measured for seventeen different levels with energies to the next-lower level ranging from 1400 to 4700 ${\mathrm{cm}}^{\ensuremath{-}1}$, exhibited an approximately exponential dependence on energy gap $\ensuremath{\Delta}E$ given by $W(0){e}^{\ensuremath{-}\ensuremath{\alpha}\ensuremath{\Delta}E}$, where $W(0)=5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ and $\ensuremath{\alpha}=4.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ cm. Exceptions to the exponential law occur only when selection rules severely restrict the number of terms in the ion-lattice interaction active in inducing transitions. The phonon frequency distribution and ion-phonon coupling in YAl${\mathrm{O}}_{3}$ were examined from infrared, Raman, and vibronic spectra. Although phonon energies range up to 750 ${\mathrm{cm}}^{\ensuremath{-}1}$, measurements of the temperature dependence of multiphonon emission indicate that phonons of energies $\ensuremath{\sim}550\ensuremath{-}600$ ${\mathrm{cm}}^{\ensuremath{-}1}$ make the dominant contribution to the relaxation at temperatures 77-700 \ifmmode^\circ\else\textdegree\fi{}K.

Pressure and Temperature Dependences of the Raman-Active Phonons in Sn O 2

Abstract: The temperature (10-500 \ifmmode^\circ\else\textdegree\fi{}K) and pressure (0-4 kbar) dependences of the four Raman-active phonons ${B}_{1g}$, ${E}_{g}$, ${A}_{1g}$, and ${B}_{2g}$ as well as thermal expansion (93-700 \ifmmode^\circ\else\textdegree\fi{}K) and isothermal compressibility (0-3 kbar) in Sn${\mathrm{O}}_{2}$ were measured. These measurements allowed us to determine the mode Gr\"uneisen parameters for the Raman-active phonons and to separate the isobaric temperature dependence of each frequency into pure-volume and pure-temperature contributions. By this procedure the cubic and quartic anharmonicities responsible for the pure-temperature contributions to the mode frequencies were determined. The ${B}_{1g}$ mode in Sn${\mathrm{O}}_{2}$ exhibited anomalous temperature and pressure dependences in that $\ensuremath{\omega}({B}_{1g})$ increased with temperature and decreased with pressure. The remaining modes exhibited decreases in frequency with increasing temperature and increases in frequency with increasing pressure, characteristic of ionic crystals. The results are compared with the recent results on the isomorphic compound tetragonal Ti${\mathrm{O}}_{2}$.

Lattice Dynamics of Gold

Abstract: The complete phonon dispersion relations for gold in the high-symmetry directions have been measured at room temperature by the coherent inelastic scattering of neutrons. It is found that the forces in gold are not homologous with the other noble metals, the frequencies of gold lying appreciably higher than those "scaled" from copper and silver. An analysis of the data in terms of different force-constant models reveals that a general tensor force is required for the first-neighbor interaction, whereas for neighbors beyond the first either general tensor or axially symmetric forces give an excellent fit to the data. The axially symmetric model alone does not adequately describe the data even when forces extending to ninth-nearest neighbors are included in the fit. In addition, simple screened-pseudopoential models were fit to the data and these results also indicate the need for the first-neighbor interaction to be general. Frequency distribution functions and related thermodynamic quantities were calculated from the various force-constant models. The Debye temperature ${\ensuremath{\Theta}}_{C}$ versus temperature curves obtained show an anomaly at low temperatures consistent with the ${\ensuremath{\Theta}}_{C}(T)$ obtained from specific-heat measurements. The relation between this anomaly and the character of the dispersion curves is given.

Optical Intensities of Rare-Earth Ions in Yttrium Orthoaluminate

Abstract: Optical absorption and emission intensities are investigated for trivalent rare earths in YAl${\mathrm{O}}_{3}$. Ions examined included Pr, Nd, Eu, Tb, Ho, Er, Tm, and Yb. Oscillation strengths of $f\ensuremath{\rightarrow}f$ transitions between $J$ manifolds were measured at room temperature and compared with calculated electric and magnetic dipole oscillator strengths. The Judd-Ofelt approach and phenomenological parameters for each ion were used to derive the electric dipole intensities. The intensity parameters, which were obtained for a least-square-fitting procedure, exhibited a general decrease with increasing number of $4f$ electrons throughout most of the series. The intensities for Pr and Tb were not satisfactorily accounted for in the present theory; some reasons for this are presented.

Optical-Absorption Edge and Raman Scattering in Ge x Se 1 − x Glasses

Abstract: Optical-absorption-edge measurements and Raman scattering experiments on ${\mathrm{Ge}}_{x}{\mathrm{Se}}_{1\ensuremath{-}x}$ glasses are reported for the range $0\ensuremath{\le}x\ensuremath{\le}0.4$. The change in the magnitude of the main peaks of Raman spectra (localized about 195, 215, and 250 ${\mathrm{cm}}^{\ensuremath{-}1}$) and the variation of the optical-absorption edge as a function of $x$ lead to a model of local structure for $x\ensuremath{\le}\frac{1}{3}$. In this range of concentrations, germanium atoms are coordinated with four selenium atoms. There is no Ge-Ge bond, and furthermore, Ge-Se-Ge sequences remain scarce as long as the germanium concentration of the mixture makes it possible.

Electronic Structure of AgIn Se 2 and CuIn Se 2

Abstract: We report electroreflectance and photoluminescence studies of the chalcopyrite compounds AgIn${\mathrm{Se}}_{2}$ and CuIn${\mathrm{Se}}_{2}$. Observation of photoluminescence at low temperatures at the same energy as the direct energy gaps located by electroreflectance measurements confirms that both compounds have direct band gaps. At 300 \ifmmode^\circ\else\textdegree\fi{}K, the values for the energy gaps are 1.24 and 0.96 eV, respectively. The spin-orbit splittings of the uppermost valence bands as observed in electroreflectance measurements are considerably less than expected for $p$ levels, a result which we attribute to \ensuremath{\sim} 17% hybridization of Ag $4d$ levels, and \ensuremath{\sim} 34% hybridization of Cu $3d$ levels, with the otherwise $p$-like valence bands. An ultraviolet electroreflectance structure observed in CuIn${\mathrm{Se}}_{2}$ may result from transitions from the $d$ levels themselves to the lowest conduction-band minimum. The crystal-field and spin-orbit parameters for the uppermost valence bands of CuIn${\mathrm{Se}}_{2}$ disagree with values found in a recent energy-band calculation ignoring $d$ bands, a calculation which also predicted that CuIn${\mathrm{Se}}_{2}$ has an indirect energy gap. We also observe an anomalous temperature dependence of the energy gap in AgIn${\mathrm{Se}}_{2}$. Whereas the energy gap in CdSe (the binary analog of AgIn${\mathrm{Se}}_{2}$) decreases by approximately 80 meV as the temperature increases from 77 to 300 \ifmmode^\circ\else\textdegree\fi{}K, the energy gap of AgIn${\mathrm{Se}}_{2}$ is independent of temperature over this range within experimental error (\ifmmode\pm\else\textpm\fi{} 5 meV).

Thermal Conductivity of Complex Dielectric Crystals

Abstract: The theory of the thermal resistivity of dielectric crystals at ordinary and high temperatures in terms of anharmonic three-phonon interactions is reformulated. The resistivity is similar in form to that obtained by Leibfried and Schl\"omann, but larger by a factor of 6.8. The theory is then extended to crystals where the unit cell contains many atoms. To include all three-phonon interactions one sums over all harmonics of the reciprocal-lattice vectors in an extended-zone representation. This sum increases the scattering rate. However, the matrix elements of the three-phonon processes are reduced in the case of large unit cells, because coherence is lost in the Fourier transform of the different bonds in each cell. A simplified model is chosen, and in this case the latter effect cancels the former, so that the anharmonic relaxation rate is substantially independent of the number of atoms per unit cell. However, the zone boundaries affect the phonon dispersion curves and reduce the group velocity of most modes. Using a model proposed by Slack, in which only the acoustic phonons of the fundamental zone contribute to the conductivity, and invoking the independence of the relaxation time with cell size here derived, the conductivity varies as the inverse cube root of the number of atoms per cell. The conductivity varies inversely with temperature, even if the phonon mean free path is shorter than the cell dimensions, because the major contribution to the anharmonic interaction comes from the highest harmonics of the fundamental reciprocal-lattice vectors.