Kathie E. Newman

Scaling Field Representation of Wilson's Exact Renormalization Group Equation

Abstract A novel momentum-space renormalization-group (RG) technique, termed the scaling-field method, is proposed for the investigation of critical phenomena in three-dimensional systems. The approach provides a method for solving Wilsons exact functional differential equation for RG Hamiltonians H l [ σ ] by successive approximation, and allows the determination of both critical exponents and scaling functions. In other papers the method is used to calculate to high precision the critical exponents of the isotropic N -vector model in three dimensions and to investigate the Potts, percolation, cubic, and random Ising models in ranges of dimensions between the upper critical and three dimensions. The purpose of this article is to present the foundations of the method. The scaling-field representation of the Wilson functional RG equation is derived by using the expansion of the RG Hamiltonian H l [ σ ] in terms of the hierarchy of Gaussian operators of appropriate symmetries. The expansion coefficients are termed scaling fields. The algorithm for the computation of the scaling-field coupling coefficients is developed for the isotropic N -vector model. Technical details and examples are given in a series of Appendices. The invariance of the Wilson equation under physically redundant scale changes of the spin variable σ is carefully investigated for both the functional renormalization group and the scaling-field representation. The understanding of this invariance yields the operational definition of the spin-rescaling function that enters the Wilson equation as a parameter. Exact recursion relations for thermodynamic and correlation functions are derived by using the functional-integral representation of the Wilson equation. The recursion relations provide the starting point for the computation of scaling functions. Finally, as an example and a guide for the development of numerical techniques, the scaling-field equations for the isotropic N -vector model are solved by e expansion and in the spherical-model limit.

Effects of ordering on the band structure of III–V semiconductors

Abstract The band structures of five types of ordered compounds derived from parent zincblende alloys A1−xBxC and AC1−xDx have been determined. Included in this study are two novel x = 1 4 , 3 4 derived structures, luzonite and famatinite, and three x = 1 2 structures, chalcopyrite and two 1 × 1 superlattices oriented along the (0,0,1) and (1,1,1) directions. The theory combines an empirical tight-binding model for III-V compounds and a valence force-field model of strain. Strain-induced tetragonal and internal distortion as well as the spin-orbit interaction cause a splitting of the top of the valence band. Trends in this splitting and the band-gap variation are studied for the 18 combinations of III-V elements. The Hopfield quasicubic crystal-field model is found to accurately describe this splitting for all chalcopyrite compounds. But this model fails for several (0,0, 1)- and (1,1, l)-superlattice compounds containing large strain distortions. The extracted Hopfield crystal-field splitting parameter Δcf is found to scale linearly with tetragonal distortion for common-anion compounds ABC2, but follow curvilinearly internal distortion for common-cation compounds. Strain and natural lineup energy modify the band gap significantly from that found in the virtual-crystal approximation for the alloy. For the metastable alloy systems GaAs1−xSbx and GaP1−xSbx, the experimental bowing of the band gap passes quite close to the results for the band gaps of the seven ordered structures.

Real space electrostatics for multipoles. I. Development of methods

We have extended the original damped-shifted force (DSF) electrostatic kernel and have been able to derive three new electrostatic potentials for higher-order multipoles that are based on truncated Taylor expansions around the cutoff radius. These include a shifted potential (SP) that generalizes the Wolf method for point multipoles, and Taylor-shifted force (TSF) and gradient-shifted force (GSF) potentials that are both generalizations of DSF electrostatics for multipoles. We find that each of the distinct orientational contributions requires a separate radial function to ensure that pairwise energies, forces, and torques all vanish at the cutoff radius. In this paper, we present energy, force, and torque expressions for the new models, and compare these real-space interaction models to exact results for ordered arrays of multipoles. We find that the GSF and SP methods converge rapidly to the correct lattice energies for ordered dipolar and quadrupolar arrays, while the TSF is too severe an approximation to provide accurate convergence to lattice energies. Because real-space methods can be made to scale linearly with system size, SP and GSF are attractive options for large Monte Carlo and molecular dynamics simulations, respectively.

Real space electrostatics for multipoles. II. Comparisons with the Ewald sum

We report on tests of the shifted potential (SP), gradient shifted force (GSF), and Taylor shifted force (TSF) real-space methods for multipole interactions developed in Paper I of this series, using the multipolar Ewald sum as a reference method. The tests were carried out in a variety of condensed-phase environments designed to test up to quadrupole-quadrupole interactions. Comparisons of the energy differences between configurations, molecular forces, and torques were used to analyze how well the real-space models perform relative to the more computationally expensive Ewald treatment. We have also investigated the energy conservation, structural, and dynamical properties of the new methods in molecular dynamics simulations. The SP method shows excellent agreement with configurational energy differences, forces, and torques, and would be suitable for use in Monte Carlo calculations. Of the two new shifted-force methods, the GSF approach shows the best agreement with Ewald-derived energies, forces, and torques and also exhibits energy conservation properties that make it an excellent choice for efficient computation of electrostatic interactions in molecular dynamics simulations. Both SP and GSF are able to reproduce structural and dynamical properties in the liquid models with excellent fidelity.

Elastic wave scattering from irregular voids

Results are presented for elastic wave scattering from irregular voids embedded in Ti alloy by the diffusion bonding process. The defects examined are: two overlapping spherical voids of unequal radii, two overlapping voids consisting of a sphere and a prolate spheroid, and a spherical void with an encircling crack. Representative plots are given for the raw waveforms, magnitude of the deconvolved Fourier transform, and in some cases the time impulse response function. The data are compared to and analyzed in terms of two current theoretical approaches. While good quantitative agreement was observed over certain ranges, the comparisons point to definite (in some cases not unexpected) limitations in either the pertaining theory or experiment or both. The results are discussed with an eye toward applications to nondestructive evaluation.

Effects of strain on structural properties in ordered semiconductors

Abstract Semiconductor alloys of the types A1−xBxC and AC1−xDx can be grown in ordered states for the composition values x = 1 4 , 1 2 , or , 3 4 . The structural properties of five forms of ordered III–V compounds have been investigated using a phenomenological model of strain due to Harrison. Among properties predicted are bond lengths and strain energies in the new compounds. Types of order include layered 1×1 superlattices oriented along the (0,0,1) or (1,1,1) directions, and chalcopyrite, famatinite, and luzonite structures. The strain results imply that if chemical effects favor ordering, the chalcopyrite and famatinite structures should be the most stable forms. The ordered-structure results are also compared directly with EXAFS results for first- and second-neighbor distances in ternary alloys. The alloy Ga1−xInxAs is well described by the family of compounds with an ordering wave vector k = (1,0, 1 2 ) .

Effects of an order–disorder transition on surface deep levels in metastable (GaAs)1−x Ge2x

The major chemical trends in the energy levels of deep point defects at the (100) surfaces of (GaAs)1−x Ge2x alloys are elucidated. These levels are predicted to exhibit bifurcation as functions of alloy composition x (at xc ≂0.3) because of an order–disorder phase transition from a zincblende phase for x xc. Levels associated with distinct anion and cation sites of the zincblende structure coalesce into a single level of the diamond structure.

The distorted wave Born approximation: Application to elastodynamics

An approximate theory for scattering of elastic waves by general shaped defects has been developed. A defect of arbitrary shape can be represented by a sphere S and a remainder volume R. Using the exact solution for a sphere and treating R as a perturbation, the solution corresponding to the distorted wave Born approximation is obtained. This solution contains nontrivial frequency dependence and phase information. Preliminary comparisons with experiments are presented.

EXAFS Studies of Interfaces in ZnTe/CdSe Superlattices

Polarization-dependent grazing incidence EXAFS, employing the electron yield technique, has been used to study the local atomic structure of two different short-period ZnTe/CdSe (001) superlattices grown by MBE. This data indicate Zn-Se and Cd-Te coordination numbers greater than those expected for sharp interfaces in the superlattice. Although strain would appear to suppress interdiffusion, we show that the results are consistent with an interchange of Zn and Cd atoms across the Zn-Se interface, and Se and Te atoms across the Cd-Te interface. We discuss accommodation of this increased strain through bond bending and bond stretching. We also discuss other experiments in progress to more fully understand these interfaces.

Metastable (III–V)1−xIV2x alloys

Abstract Zincblende-to-diamond-lattice structural phase transitions should occur in metastable (III–V)1−xIV2x alloys at a transition composition xc that is controllable by growth conditions. The effect of this transition should be visible both in the electronic as well as the vibrational properties of these alloys. For example, in the prototypical (III–V)1−xIV2x alloy, (GaAs)1−xGe2x, the observed anomalous V-shaped bowing of the direct gap is explained in terms of the phase transition, which occurs at the minimum of the “V.” Predictions are made for the band gaps of new metastable alloys, such as (GaSb)1−xSn2x. Consequences of this transition for (III–V)-IV superlattices are also discussed.