Showing papers in "Physical Review B in 1976"

Special points for brillouin-zone integrations

Abstract: A method is given for generating sets of special points in the Brillouin zone which provides an efficient means of integrating periodic functions of the wave vector. The integration can be over the entire Brillouin zone or over specified portions thereof. This method also has applications in spectral and density-of-state calculations. The relationships to the Chadi-Cohen and Gilat-Raubenheimer methods are indicated.

Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism

Abstract: The aim of this paper is to advocate the usefulness of the spin-density-functional (SDF) formalism. The generalization of the Hohenberg-Kohn-Sham scheme to and SDF formalism is presented in its thermodynamic version. The ground-state formalism is extended to more general Hamiltonians and to the lowest excited state of each symmetry. A relation between the exchange-correlation functional and the pair correlation function is derived. It is used for the interpretation of approximate versions of the theory, in particular the local-spin-density (LSD) approximation, which is formally valid only in the limit of slow and weak spatial variation in the density. It is shown, however, to give good account for the exchange-correlation energy also in rather inhomogeneous situations, because only the spherical average of the exchange-correlation hole influences this energy, and because it fulfills the sum rule stating that this hole should contain only one charge unit. A further advantage of the LSD approximation is that it can be systematically improved. Calculations on the homogeneous spin-polarized electron liquid are reported on. These calculations provide data in the form of interpolation formulas for the exchange-correlation energy and potentials, to be used in the LSD approximation. The ground-state properties are obtained from the Galitskii-Migdal formula, which relates the total energy to the one-electron spectrum, obtained with a dynamical self-energy. The self-energy is calculated in an electron-plasmon model where the electron is assumed to couple to one single mode. The potential for excited states is obtained by identifying the quasiparticle peak in the spectrum. Correlation is found to significantly weaken the spin dependence of the potentials, compared with the result in the Hartree-Fock approximation. Charge and spin response functions are calculated in the long-wavelength limit. Correlation is found to be very important for properties which involve a change in the spinpolarization. For atoms, molecules, and solids the usefulness of the SDF formalism is discussed. In order to explore the range of applicability, a few applications of the LSD approximation are made on systems for which accurate solutions exist. The calculated ionization potentials, affinities, and excitation energies for atoms propose that the valence electrons are fairly well described, a typical error in the ionization energy being 1/2 eV. The exchange-correlation holes of two-electron ions are discussed. An application to the hydrogen molecule, using a minimum basis set, shows that the LSD approximation gives good results for the energy curve for all separations studied, in contrast to the spin-independent local approximation. In particular, the error in the binding energy is only 0.1 eV, and bond breaking is properly described. For solids, the SDF formalism provides a framework for band models of magnetism. An estimate of the splitting between spin-up and spin-down energy bands of a ferromagnetic transition metal shows that the LSD approximation gives a correction of the correct sign and order of magnitude to published $X\ensuremath{\alpha}$ results. To stimulate further use of the SDF formalism in the LSD approximation, the paper is self-contained and describes the necessary formulas and input data for the potentials.

Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields

Abstract: An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

Quantum critical phenomena

Abstract: This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space Since the time variable lies in the finite interval [$0,\ensuremath{-}i\ensuremath{\beta}$], where $\ensuremath{\beta}$ is the inverse temperature, the resulting description of a $d$-dimensional system shares some features with that of a ($d+1$)-dimensional classical system which has finite extent in one dimension However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional in the same way The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply Of these, the itinerant ferromagnet is studied in greater detail The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but $4\ensuremath{-}z$ dimensions, where $z$ depends on the way the frequency enters the generalized free-energy functional When it does so in the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; $z=1$ It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds $z=2, 3, \mathrm{and} 4$, respectively (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for $dl4$) very close to the critical value of the coupling ${U}_{c}$, when $\frac{(U\ensuremath{-}{U}_{c})}{{U}_{c}}\ensuremath{\ll}{(\frac{T}{{U}_{c}})}^{\frac{2}{z}}$ $\frac{z}{2}$ is therefore the quantum-to-classical crossover exponent

Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm

Abstract: A new approach for the determination of the critical percolation concentration, percolation probabilities, and cluster size distributions is presented for the site percolation problem. The novel "cluster multiple labeling technique" is described for both two- and three-dimensional crystal structures. Its distinctive feature is the assignment of alternate labels to sites belonging to the same cluster. These sites are members of a simulated finite random lattice. An algorithm useful for the determination of the critical percolation concentration of a finite lattice is also presented. This algorithm is especially useful when applied in conjunction with the cluster multiple labeling technique. The basic features of this technique are illustrated by applying it to a small planar square lattice. Numerical results are given for a triangular subcrystal containing up to 9 000 000 sites. These results compare favorably with the exact value of the infinite lattice critical percolation concentration.

Nonlocal pseudopotential calculations for the electronic structure of eleven diamond and zinc-blende semiconductors

Abstract: An empirical nonlocal pseudopotential scheme is employed to calculate the electronic structure of eleven semiconductors: Si, Ge, $\ensuremath{\alpha}\ensuremath{-}\mathrm{Sn}$, GaP, GaAs, GaSb, InP, InAs, InSb, ZnSe, and CdTe. Band structures, reflectivity spectra, electronic densities of states, and valence charge densities are presented and compared to experimental results. Improved optical gaps, optical critical-point topologies, valence-band widths, and valence charge distributions are obtained as compared to previous local pseudopotential results.

Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition

Abstract: The lowest-energy state of the incommensurate charge density wave (CDW) near the lock-in transition is found to be a distorted plane wave. An exact solution is found in the weak-coupling limit and the lock-in phase transition is continuous. A new defect is found in the commensurate CDW, a discommensuration, in which the phase of the CDW slips by $\frac{2\ensuremath{\pi}}{3}$ relative to the perfectly locked in CDW. The lock-in phase transition is interpreted as a defect melting transition with a finite density of discommensurations in the incommensurate state.

Quasiparticle and phonon lifetimes in superconductors

Abstract: Low-energy quasiparticle scattering, recombination, and branch-mixing lifetimes and phonon pair-breaking and scattering lifetimes are calculated for superconductors. The quasiparticle calculations relate these lifetimes to the low-frequency behavior of ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})$. Results are obtained using the low-frequency approximate form ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})=b{\ensuremath{\Omega}}^{2}$, with $b$ determined from electron tunneling measurements. For the strong-coupling superconductors Pb and Hg, the full tunneling form for ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})$ is used. The phonon lifetimes are shown to depend on ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})$. Results are compared with experiment.

Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Abstract: We compute exactly the spin-spin correlation functions 〈σ0,0σM,N〉 for the two-dimensional Ising model on a square lattice in zero magnetic field for T>Tc and T

Thermopower in the correlated hopping regime

Abstract: The high-temperature limit for the thermopower of a system of interacting localized carriers is governed entirely by the entropy change per added carrier. The calculation of this quantity reduces to a simple combinatorial problem dependent only on the density of carriers and the interactions stronger than the thermal energy. We have thus been able to generalize the Heikes formula to include several cases of interacting Fermi systems with spin.

Van der Waals interaction between an atom and a solid surface

Abstract: : This paper contributes to the theory of the long-ranged attractive polarization force between a neutral atom and a crystalline solid surface in the non-relativistic limit. The first two terms in the asymptotic expansion of the polarization energy are used to define an atom-solid potential of the form V(pol) = -C(Z - Zo) to the (-3) power. The constant C appearing in this expression is known from the earlier work of E. M. Lifshitz. The present paper gives a theory of the position of the 'reference plane,' Z(o), which is important in applications to physisorption. An explicit expression for Z(o) is first derived for atoms interacting with a jellium metal and with an insulating crystal consisting of atoms which interact via dipole-dipole forces. These model calculations are then incorporated into a computation of the polarization energies of rare gas atoms physisorbed on noble metal surfaces. The computed energies are found to be consistent with observed adsorption energies.

Electronic properties and superlattice formation in the semimetal TiSe 2

Abstract: Neutron-diffraction studies of ${\mathrm{TiSe}}_{2}$ show that a second-order structural phase transition occurs at ${T}_{0}=202$ K involving transverse atomic displacements with wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=(\frac{1}{2},0,\frac{1}{2})$. The electronic transport properties of the most nearly stoichiometric crystals show that ${\mathrm{TiSe}}_{2}$ is a semimetal with ${n}_{e}={n}_{h}={10}^{20}$/${\mathrm{cm}}^{3}$. Small impurity concentrations or deviations from stoichiometry reduce ${n}_{h}$, increase ${n}_{e}$, and suppress the phase transition. This reduction together with the observed displacement pattern lead us to speculate that the transition is driven by an electron-hole coupling.

Spontaneous breakdown of continuous symmetries near two dimensions

Abstract: The long-distance properties of classical Heisenberg ferromagnets below the transition point are related to a continuous-field theory, the nonlinear $\ensuremath{\sigma}$ model. The renormalizability of this model in two dimensions and its ultraviolet asymptotic freedom are used to derive renormalization-group equations valid above $d=2$. It is argued that this model is renormalizable up to four dimensions. The scaling properties which incorporate critical and Goldstone singularities follow. Explicit calculations of exponents and of correlation functions in powers of $d\ensuremath{-}2$ are given. A technique is proposed to make calculations in the symmetric phase applicable even in two dimensions.

Photoemission of spin-polarized electrons from GaAs

Abstract: The spin polarization of electrons photoemitted from (110) GaAs by irradiating with circularly polarized light of energy 1.5 & heu & 3.6 eU was measured by Mott scattering. The GaAs surface was treated with cesium and oxygen to obtain a negative electron af6nity (NEA), The spectrum of spin polarization P(hem) exhibits a peak (P = 40%) at threshold arising from transitions at I, and positive (P = 8%) and negative (P = —8%) peaks at 3.0 and 3.2 eV, respectively, arising from transitions at L (A). Anomalous behavior, consisting of a depolarization at threshold and an increase and shift in the peak polarization to 54% at 1.7 eV, is attributed to a small positive electron a%nity (PEA) characteristic of some samples. Restriction of the photoelectron emission angle by the PEA leads directly to the anomalously high P. Results of calculations show that P cannot be increased above 50% for emission arising from transitions at I in NEA GaAs. Our detailed interpretation of the spectra indicates how spin-polarized photoemission can be used to study the spin-

Stress-induced ferroelectricity and soft phonon modes in SrTi O 3

Abstract: By means of dielectric measurements and a Raman-scattering experiment, the uniaxial stress dependence of the ferroelectric and structural phonon modes in SrTi${\mathrm{O}}_{3}$ crystal has been studied at liquid-helium temperature. The ferroelectric phase transitions were induced by a stress normal to the (100) or (110) face. The inverse dielectric susceptibilities were found to change linearly with applied stress, and the phonon frequencies of corresponding ferroelectric modes were found to vary following the Lyddane-Sachs-Teller relation. These characteristics were analyzed by using the phenomenological free energy which contains as interaction terms ${Q}_{\ensuremath{\lambda}\ensuremath{\mu}}{X}_{\ensuremath{\lambda}}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}})}_{\ensuremath{\mu}}+{R}_{\ensuremath{\lambda}\ensuremath{\mu}}{X}_{\ensuremath{\lambda}}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\Phi}}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\Phi}}})}_{\ensuremath{\mu}}+{t}_{\ensuremath{\lambda}\ensuremath{\mu}}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}})}_{\ensuremath{\lambda}}{(\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\Phi}}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\Phi}}})}_{\ensuremath{\mu}}$. From the behavior of the dielectric constants below the critical stress and that of the soft-phonon-mode frequencies above the critical stress, the coupling coefficients ${Q}_{\ensuremath{\lambda}\ensuremath{\mu}}$, ${R}_{\ensuremath{\lambda}\ensuremath{\mu}}$, ${t}_{\ensuremath{\lambda}\ensuremath{\mu}}$ and other parameters in the free energy have been determined consistently. Anticrossing between the ferroelectric and structural modes was observed for an oblique-wave-vector phonon. Anomalous increase of the damping of the total symmetric ferroelectric mode near the transition stress has been found and discussed.

Flicker (1f) noise: Equilibrium temperature and resistance fluctuations

Abstract: We have measured the $\frac{1}{f}$ voltage noise in continuous metal films. At room temperature, samples of pure metals and bismuth (with a carrier density smaller by ${10}^{5}$) of similar volume had comparable noise. The power spectrum ${S}_{V}(f)$ was proportional to $\frac{{\overline{V}}^{2}}{\ensuremath{\Omega}{f}^{\ensuremath{\gamma}}}$, where $\overline{V}$ is the mean voltage across the sample, $\ensuremath{\Omega}$ is the sample volume, and $1.0\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\lesssim}1.4$. $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}$ was reduced as the temperature was lowered. Manganin, with a temperature coefficient of resistance ($\ensuremath{\beta}$) close to zero, had no discernible noise. These results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance to give ${S}_{V}(f)\ensuremath{\propto}\frac{{\overline{V}}^{2}{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}$, where ${C}_{V}$ is the total heat capacity of the sample. The noise was spatially correlated over a length $\ensuremath{\lambda}(f)\ensuremath{\approx}{(\frac{D}{f})}^{\frac{1}{2}}$, where $D$ is the thermal diffusivity, implying that the fluctuations obey a diffusion equation. The usual theoretical treatment of spatially uncorrelated temperature fluctuations gives a spectrum that flattens at low frequencies in contradiction to the observed spectrum. However, the empirical inclusion of an explicit $\frac{1}{f}$ region and appropriate normalization lead to $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}\ensuremath{\propto}\frac{{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}[3+2\mathrm{ln}(\frac{l}{w})]f$, where $l$ is the length and $w$ is the width of the film, in excellent agreement with the measured noise. If the fluctuations are assumed to be spatially correlated, the diffusion equation can yield an extended $\frac{1}{f}$ region in the power spectrum. We show that the temperature response of a sample to $\ensuremath{\delta}$- and step-function power inputs has the same shape as the autocorrelation function for uncorrelated and correlated temperature fluctuations, respectively. The spectrum obtained from the cosine transform of the measured step-function response is in excellent agreement with the measured $\frac{1}{f}$ voltage noise spectrum. Spatially correlated equilibrium temperature fluctuations are not the dominant source of $\frac{1}{f}$ noise in semiconductors and discontinuous metal films. However, the agreement between the low-frequency spectrum of fluctuations in the mean-square Johnson-noise voltage and the resistance fluctuation spectrum measured in the presence of a current demonstrates that in these systems the $\frac{1}{f}$ noise is also due to equilibrium resistance fluctuations.

GaAs lower conduction-band minima: Ordering and properties

Abstract: Synchrotron-radiation Schottky-barrier electroreflectance spectra from the $\mathrm{Ga} 3{d}^{V}$ core levels to the lower $s{p}^{3}$ conduction band have shown that the ${L}_{6}^{C}$ lower conduction-band minima are located 170 \ifmmode\pm\else\textpm\fi{} 30 meV in energy below the ${X}_{6}^{C}$ minima in GaAs. Here, we investigate the implications of this ordering, which is opposite to that commonly accepted as correct. We find that, without exception, the results of previous experiments that apparently supported the opposite ordering can be reinterpreted within the ${\ensuremath{\Gamma}}_{6}^{C}\ensuremath{-}{L}_{6}^{C}\ensuremath{-}{X}_{6}^{C}$ model. By performing a line-shape analysis, we resolve an apparent discrepancy between intraconduction band absorption measurements of the ${X}_{6}^{C}\ensuremath{-}{\ensuremath{\Gamma}}_{6}^{C}$ energy separation. By comparing these optical results with other modulation spectroscopic ($s{p}^{3}$ valence-conduction-band electroreflectance, high-precision reflectance) data, combining these with the results of photoemission, transport (high pressure and high temperature), semiconductor alloy, and luminescence measurements, nonlocal pseudopotential calculations $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ theory, the rigid-valence-band hypothesis, and using the systematics of other tetrahedrally bonded semiconductors with temperature and pressure, we obtain a set of consistent parameters describing the ${\ensuremath{\Gamma}}_{6}^{C}$, ${L}_{6}^{C}$, and ${X}_{6}^{C}$ lower conduction-band minima of GaAs. This model resolves the former contradictions in the apparent indirect threshold energy as determined previously by photoemission, transport, and optical measurements. Previous photoemission data for cesiated GaAs show clearly after structure reassignment that hot electrons thermalize in the ${L}_{6}^{C}$ minima. This implies that Gunn oscillator operation in GaAs involves the ${L}_{6}^{C}$, and not ${X}_{6}^{C}$, conduction-band minima. We obtain the variation of these minima with composition, $x$, in the $\mathrm{Ga}{\mathrm{As}}_{1\ensuremath{-}x}{\mathrm{P}}_{x}$ alloy series, and show that the increase in binding energy of the N isoelectronic trap with increasing As fraction in this series is in qualitative agreement with the prediction of a two-level model wherein a Koster-Slater isoelectronic trap potential interacts with the densities of states of both ${L}_{6}^{C}$ and ${X}_{6}^{C}$. These results have clear implications for the theory of operation of light-emitting diodes of GaAs and its alloys.

Application of the renormalization group to phase transitions in disordered systems

Abstract: The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $mg{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $ml{m}_{c}$, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{ u}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{ e}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{ u}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

Acoustic phonon instabilities and structural phase transitions

Abstract: Structural phase transitions are considered in which the order parameter is a homogeneous deformation of the crystal. The fluctuations at these transitions are the acoustic modes, and it is shown that an effective Hamiltonian may be constructed describing the homogeneous deformations and their fluctuations. There are three cases which result, those in which there are no fluctuations with wavelengths less than the crystal dimensions, those in which the acoustic modes have strongly temperature-dependent velocities for wave vectors on particular lines of reciprocal space, and those for which the velocities are temperature dependent for wave vectors within planes in reciprocal space. In many cases, the transitions are expected to be first order because of the presence of cubic invariants in the effective Hamiltonian. In those which may be continuous, the behavior is shown by use of renormalization-group theory to be that of classical Landau theory, with the possibility of logarithmic corrections in one particular instance. Unfortunately, we are unaware of any examples of this case, but in the other cases, the results are in accord with experimental results.

Electronic structure of a metal-semiconductor interface

Abstract: The electronic structure of a jellium-Si interface is calculated using a jellium density corresponding to Al and self-consistent Si pseudopotentials. Local densities of states and charge densities are used to study states near the interface. At the Si surface, a high density of extra states is found in the energy range of the Si fundamental gap. These states are bulklike in jellium and decay into Si with a high concentration of charge in the dangling-bond free-surface-like Si state. Truly localized interface states are also found but at lower energies. The calculated barrier height is in excellent agreement with recent experimental results.

Optical properties of hexagonal boron nitride

Abstract: Optical absorption, reflectivity, and photoconductivity in the near-uv range (1950-3200 \AA{}) of a thin film of hexagonal boron nitride were measured. The main absorption peak was observed at 6.2 eV. A sharp fall at about 5.8 eV was attributed to the direct band gap. The temperature dependence of the band gap was found to be less than 4 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ eV/\ifmmode^\circ\else\textdegree\fi{}K. Self-consistent tight-binding band-structure calculations were performed on a two-dimensional hexagonal crystal model, using Hamiltonian matrix elements calculated by semiempirical LCAO (linear combination of atomic orbitals) methods. The calculated value for the band gap of hexagonal BN was in reasonably good agreement with the experimental value obtained in the present work, as well as with values reported earlier from electron-energy-loss and photoelectron-emission measurements. The calculations also predicted a very small change in the band gap with temperature, in agreement with the experimental observations.

Cluster size and boundary distribution near percolation threshold

Abstract: It is shown that the shape of the large, random clusters, near the critical percolation concentration ${c}_{0}$, is such that their mean boundary $〈b〉$ is proportional to their mean bulk $〈n〉$ and this is illustrated by an argument which shows that the dimension of the boundary is the same as that of the bulk. The resulting ratio $\frac{〈b〉}{〈n〉}$ is simply related to the critical concentration ${c}_{0}$. The detailed results of a Monte Carlo calculation, previously reported, are given for $cl{c}_{0}$ on a simple square lattice; they yield an empirical formula for the probability distribution $\mathcal{P}(n,b)$, for finding a cluster of size $n$ and boundary $b$, that is proportional to a Gaussian in $\frac{b}{n}$, which is independent of concentration and which narrows to a $\ensuremath{\delta}$ function at $\frac{b}{n}={\ensuremath{\alpha}}_{0}$, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. The asymptotic behavior of the Gaussian form gives the critical exponents $\ensuremath{\beta}=0.19\ifmmode\pm\else\textpm\fi{}0.16$, and $\ensuremath{\gamma}=2.34\ifmmode\pm\else\textpm\fi{}0.3$, and ${\ensuremath{\alpha}}_{0}$, gives the critical concentration ${c}_{0}=0.587\ifmmode\pm\else\textpm\fi{}0.14$, in agreement with previous determinations.

Defects and the central peak near structural phase transitions

Abstract: The effect of defect or impurities on the static and dynamic response, near a displacive structural phase transition, depends on the symmetry and dynamics of the defect cell. It is shown that a small concentration of defect cells, in which the order parameter relaxes on a slow time scale between different equivalent orientations, may account for the narrow "central peak" as well as for the temperature dependence of the "soft-mode" frequency in the perovskites near structural phase transitions. The case of a frozen defect cell is also discussed. Calculations are performed for the pure and impure systems using mean-field theory, and corrections are discussed using the universality hypotheses and renormalization-group calculations for dynamic critical behavior.

Finite-size behavior of the Ising square lattice

Abstract: An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of $N$ and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.

Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals

Abstract: Using the wedge technique we have directly compared the second-order nonlinear susceptibilities of infrared and visible nonlinear crystals. The measured nonlinear coefficient ratios at 2.12 \ensuremath{\mu}m relative to ${d}_{31}(\mathrm{LiI}{\mathrm{O}}_{3})$ are: for $\mathrm{LiNb}{\mathrm{O}}_{3}({d}_{33})$, 4.53 \ifmmode\pm\else\textpm\fi{} 4.3; $\mathrm{GaP} ({d}_{36})$, 12.1 \ifmmode\pm\else\textpm\fi{} 1.7; $\mathrm{GaAs} ({d}_{36})$, 26.9 \ifmmode\pm\else\textpm\fi{} 2.1; $\mathrm{AgGa}{\mathrm{Se}}_{2} ({d}_{36})$, 10.5 \ifmmode\pm\else\textpm\fi{} 1.2; $\mathrm{CdSe} ({d}_{33})$, 10.2 \ifmmode\pm\else\textpm\fi{} 1.2. The measured ratios at 1.318 \ensuremath{\mu}m relative to ${d}_{31}(\mathrm{LiI}{\mathrm{O}}_{3})$ are: for $\mathrm{LiI}{\mathrm{O}}_{3} ({d}_{33})$, 0.990 \ifmmode\pm\else\textpm\fi{} 0.05; $\mathrm{LiNb}{\mathrm{O}}_{3} ({d}_{31})$, 0.870 \ifmmode\pm\else\textpm\fi{} 0.07; $\mathrm{LiNb}{\mathrm{O}}_{3} ({d}_{33})$, 4.66 \ifmmode\pm\else\textpm\fi{} 0.56; $\mathrm{K}{\mathrm{H}}_{2}{\mathrm{PO}}_{4} ({d}_{36})$, 0.088 \ifmmode\pm\else\textpm\fi{} 0.01; $\mathrm{GaP} ({d}_{36})$, 12.0 \ifmmode\pm\else\textpm\fi{} 1.2. We have used the parametric fluorescence method to accurately measure the absolute second-order susceptibility of $\mathrm{LiI}{\mathrm{O}}_{3} ({d}_{31})$ and $\mathrm{LiNb}{\mathrm{O}}_{3} ({d}_{31})$ at 4880 and 5145 \AA{}. Our recommended values for ${d}_{31}(\mathrm{LiIO}{\mathrm{O}}_{3})=(7.31\ifmmode\pm\else\textpm\fi{}0.62)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ and ${d}_{31}(\mathrm{LiNb}{\mathrm{O}}_{3})=(5.82\ifmmode\pm\else\textpm\fi{}0.70)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ m/V agree very well with previous independent absolute measurements. By scaling the nonlinear susceptibilities through the relatively dispersionless Miller's $\ensuremath{\Delta}$ and using the wedge ratio results, we have, for the first time, established a uniform scale of nonlinear susceptibility values relative to ${d}_{31}(\mathrm{LiI}{\mathrm{O}}_{3})$ that extends from 0.488 to 10.6 \ensuremath{\mu}m in the infrared.

Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride

Abstract: We have measured the heat capacity and thermal conductivity of the layered material hexagonal pyrolytic boron nitride from 2 to 10 K by a pulse-heating technique. The thermal conductivity has also been determined by a steady-state technique from 1.5 to 350 K. At low temperatures, the heat capacity has a ${T}^{3}$ dependence and the thermal conductivity a ${T}^{2.4}$ dependence. Crystallite size was found from transmission electron micrographs. The influence of dislocations on the thermal conductivity is discussed.

Magnetic susceptibility of mixed-valence rare-earth compounds

Abstract: Many rare-earth compounds (e.g., Sm chalcogenides, ${\mathrm{YbAl}}_{3}$) exhibit temperature-independent magnetic susceptibility at low temperatures in their mixed valence phase despite the fact that the ionic configuration in at least one of the valences is such as to lead to a Curie-Weiss behavior. An essential feature of these compounds is that the Fermi level is pinned to the $f$ levels. We first examine the effect of this feature in the Anderson model for an isolated impurity and find through a strong-coupling variational wave function as well as through a simple Green's-function treatment that the susceptibility is finite at $T\ensuremath{\rightarrow}0$\ifmmode^\circ\else\textdegree\fi{}K and of order $\frac{{\ensuremath{\mu}}^{2}}{\ensuremath{\Gamma}}$ where $\ensuremath{\Gamma}$ is the virtual width of the $f$ level corrected for correlation effects. For the compounds, a two-band ("$f$" and "$d$" with orbital degeneracies neglected) Hubbard-like model leads in the same treatment to a finite susceptibility at $T=0$, where now $\ensuremath{\Gamma}$ is essentially the $f\ensuremath{-}d$ hybridization energy. Order-of-magnitude agreement with experiments is obtained with a reasonable value of the $f\ensuremath{-}d$ mixing interaction. The physics of the finite susceptibility at $T=0$ is the renormalization of the local moments by the conduction electrons which is strongest when the $f$ levels are at the Fermi level.

Simple model of multiple charge states of transition-metal impurities in semiconductors

Abstract: The Anderson model for magnetic impurities in metals is extended to semiconductors. It is shown how self-consistent Hartree-Fock solutions can exist in the gap for many different charge states of the impurity, providing the matrix elements coupling the impurity and substrate are large enough.