Average T-matrix Approximation Research Articles

Optical properties of high aspect ratio plasma etched silicon nanowires: fabrication-induced variability dramatically reduces reflectance

In this work we investigate both experimentally and theoretically the optical properties of aligned, perpendicular to the substrate, high aspect ratio (AR), plasma etched Si nanowires (SiNWs) with controlled variability. We focus on the role of imperfections in fabrication, which manifest themselves as dimensional variability of SiNW, lattice defects or positional randomization. SiNW arrays are fabricated by e-beam lithography (perfectly ordered array) or colloidal particle self-assembly (quasi-ordered array) followed by cryogenic Si plasma etching, which offers fast etch rate (up to 3 μm min−1) combined with clean, smooth, and controllable sidewall profile, but induces some dimensional variability on the diameters of the SiNWs. Sub-200 nm diameter SiNWs having AR as high as 37:1 are demonstrated. The total reflectance of SiNWs is below 2% in a wide range of the optical spectrum. We experimentally demonstrate improved light absorption when moving from a perfectly ordered (after e-beam lithography) to a defective and quasi-ordered (after colloidal self-assembly) SiNW array. In addition our measured reflectivity (for both ordered and quasi-ordered SiNWs) is much lower compared to the one predicted theoretically for a perfect SiNWs array, using full-electrodynamic calculations with the layer-multiple-scattering method. To explain such low reflectivity, we model the influence of disorder using the average T-matrix approximation and show that even small dimensional variability (10–20%) leads to dramatic reduction of the reflectance (matching the experimental results) and increased light trapping inside the SiNW justifying their possible application in photovoltaic devices.

Interacting quasi-band model for electronic states in alloy semiconductors: Relation to average t-matrix approximation and band anticrossing model

A new variational theory is proposed for electronic states in alloy semiconductors with arbitrary diagonal and off-diagonal randomness and any concentration. In AcABcB substitutional alloys, the theory derives the mutual interaction between two quasi-A and -B electronic bands whose effective band widths are proportional to cA and cB, respectively, i.e., an interacting quasi-band model. The model provides satisfactory results near-band-edge states, especially for band bowing. For diagonal randomness, the theory corresponds to the average t-matrix approximation, and in the dilute limit, a formula similar to the band anticrossing model, which has been frequently applied to GaN-related alloys, is obtained.

Dynamics of impurity modes and electron–phonon interaction in Heavy Fermion (HF) systems

A theoretical explanation is provided to understand the effect of small concentration of impurities characterized by change in mass and nearest neighbor force constants on the phonon spectrum as well as on the electron–phonon interaction in some Heavy Fermion (HF) systems in the normal state within theoretical framework of the Periodic Anderson Model (PAM). Three different mechanisms of the electron–phonon interactions, namely, the usual interaction between the phonons with the electrons in the f-bands, electrons arising from that of hybridization term of PAM and the local electron–phonon coupling at the impurity sites are considered. Coherent Potential Approximation (CPA) is used to evaluate the configuration averaged self–energy and the total Green function. For simplicity of calculation the CPA self–energy is evaluated in Average t -matrix Approximation (ATA). The analytical analysis is carried out for finite T in the long wavelength limit. The influence of impurity mass parameter λ and other system parameters such as d, the position of f-level, the effective coupling strength g on the calculated re-normalized phonon frequency and the excitation spectrum through the spectral function is studied. The numerical analysis of the results does show the influence of impurities as evident from different plots in this paper.

Effective medium approach to thermal conductivity: applying to core-shell nanocomposites

The effective medium method is employed to study thermal conductivity of nanostructured phases consisting of embedded particles in a host. Beyond the average T-matrix single-particle approximation, the effect of higher-order inter-particle terms on heat transport is estimated to be small, which can be validated by using a re-formulated differential effective medium (DEM) approach. However, the effect of higher-order terms on heat transport is found to become significant when phonon scattering from embedded particles is considered. Applying the DEM approach to two-phase nanocomposites, particularly core-shell materials that mimic a three-dimensional superlattice, it results in a large reduction of thermal conductivity, and potentially high ZT.

Segregation and interlayer relaxation at the NiPd(111) surface

The composition and structure of the NiPd(111) alloy surface are measured using low energy electron diffraction (LEED). The diffracted intensities are analyzed using tensor-LEED methods with the average t-matrix approximation. The major composition results of the study are that the surface is enriched in palladium (≈80% Pd) while the second layer is enriched in nickel (≈28% Pd). No significant structural changes were found, with all measured interlayer displacements being smaller than the corresponding measurement uncertainties. Comparisons of these results are made with literature for other NiPd low-index surfaces, for the well-studied NiPt(111) surface, and with the growing body of theoretical literature for NiPd alloy surfaces.

Local structural and compositional determination via electron scattering: Heterogeneous Cu(001)-Pd surface alloy

We have measured the structure and chemical composition of ultrathin Pd films on Cu(001) using low-energy electron microscopy. We determine their local stoichiometry and structure, with 8.5 nm lateral spatial resolution, by quantitatively analyzing the scattered electron intensity and comparing it to dynamical scattering calculations, as in a conventional low-energy electron diffraction (LEED)-IV analysis. The average t-matrix approximation is used to calculate the total atomic scattering matrices for this random substitutional alloy. As in the traditional LEED analysis, the structural and compositional parameters are determined by comparing the computed diffraction intensity of a trial structure to that measured in experiment. Monte Carlo simulations show how the spatial and compositional inhomogeneity can be used to understand the energetics of Cu-Pd bonding.

Theory of Vibrational Excitations in a Solid Solution with the Inclusion of Their Scattering by Multi-Impurity Clusters

A theory of vibrational excitations in a solid solution is proposed. This theory allows for scattering of excitations by multi-impurity clusters and offers an adequate description of the complex spectra of the solid solution. The main feature of the theory is that the averaging and the Fourier transformation with respect to the coordinate of the impurity cluster as a whole are performed at the first stage, whereas the averaging over the distances between impurities in the cluster is carried out at the final stage. The calculations are performed in the approximation similar to the average t-matrix approximation and also the coherent-potential approximation. The theory is illustrated by calculating the spectra and dielectric functions of disordered linear chains with one and two atoms per unit cell. The calculated vibrational spectra are in good agreement with the computer calculations performed by Dean.

New approach to the theory of lattice vibrations of random alloys

The theory of vibrational excitations of random alloys is presented, which takes into account the scattering of excitations on n-impurity complexes (clusters) and is capable to describe complex structure of their spectra. The essential feature of the theory consists in two-stage disorder averaging procedure. First, theory equations are averaged and Fourier transformed over the coordinate characterizing the cluster position as a whole. Then they are averaged over inter-impurity distances in the cluster. The derivation of the Green function equations is carried out both in the version similar to average-t-matrix approximation (ATA) and in that similar to coherent potential approximation (CPA). (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Initial Growth of Co on Cu{001} Studied with LEED I(V)

The initial growth of Co deposited at room temperature on Cu{001} was studied with low energy electron diffraction (LEED) and temperature-programmed desorption (TPD). Measured I(V) spectra were compared with calculated spectra from several model structures, including substitutionally disordered alloys. The averaged T-matrix approximation (ATA) was used to model the random alloy layers. According to the I(V) analysis, alloying occurs in the first stages of the growth. TPD of CO indicates that both large areas of Co and areas of a surface alloy are already present at the lowest coverage. Both methods show that a transition to layer-by-layer growth occurs as the coverage increases.

A tensor LEED determination of the structure and compositional profile of a Cu{ [formula omitted]}-c(2×2)-Pt surface alloy

The geometric structure and compositional profile of a Cu{1 0 0}-c(2×2)-Pt surface alloy formed by thermal activation of a monolayer Pt film has been determined by tensor low energy electron diffraction (LEED). A wide range of models have been tested. The favoured model consists of an ordered c(2×2) CuPt underlayer below a Cu terminated surface. Models involving a mixed ordered CuPt layer outermost may be definitively ruled out. The average T-matrix approximation (ATA) has been applied allowing variable Pt concentrations to be introduced into both the outermost layer and deeper into the selvedge (layers 3 and 4) in the form of a random substitutionally disordered Cu x Pt 1− x alloy. The favoured concentration profile corresponds to an almost pure outermost Cu monolayer ( Θ Pt=10±10 at.%) with Pt concentrations of 20±20 and 30±30 at.% in layers 3 and 4 respectively. Introduction of Pt into the surface layers induces a significant expansion of the selvedge yielding modification of the outermost three interlayer spacings to 1.84±0.02 Å ( Δdz 12=+1.9±1.1%), 1.91±0.03 Å ( Δdz 23=+5.8±1.7%) and 1.89±0.03 Å ( Δdz 34=+4.7±1.7%). The rippling in the first mixed CuPt monolayer is small and of amplitude 0.03±0.04 Å with Pt rippled outwards towards the solid–vacuum interface.

Surface alloying and magnetism of ultrathin Fe films on Pd(001)

The atomic structure and the chemical composition of ultrathin $(<3$ monolayer) Fe films grown on a clean Pd(001) at room temperature are studied by low energy electron diffraction (LEED) $I/V$ (spot intensity versus beam energy) analysis with average t-matrix approximation. We find sizable amount of alloying between Fe film and Pd substrate near the interface, and the alloyed Fe films show large tetragonal distortion. Fully relativistic ab initio band structure calculation is performed to elucidate the stability and the correlation between the surface alloying and the magnetic properties of the alloyed Fe film. We find that the sublayer Fe film below 1 ML Pd capping layer is stable compared with the Fe overlayer, supporting efficient inter-diffusion between Fe and Pd, and that lattice spacing in the alloyed Fe films contract as observed by the LEED $I/V$ analysis. The magnetic moments of Fe are found strongly enhanced, and the induced magnetic moments of Pd are also substantial.

Photoelectron diffraction from random surface alloys: critique of calculational methods

For random single crystal alloys, the lack of periodicity parallel to the surface, which is consequence of the substitutional disorder, complicates the calculation of photoelectron diffraction (PD) patterns. One way to calculate PD spectra from random surface alloys is to evaluate the spectra for each member of the complete ensemble of configurations and then to sum appropriately to generate the PD spectra from the alloy. The number of different configurations is very large, which makes this approach very time consuming computationally. A computationally efficient approximation, which has had success in applications to low energy electron diffraction from random alloys, is the average t-matrix approximation (ATA), where the scattering properties are described by an effective t-matrix. In this work we compare the ATA with the average over configuration within the context of the cluster-based PD theory and compare the results of both methods with experimental data from a random surface alloy, namely Pd on Cu(1 1 1). The results of this study indicate that the ATA is efficient in extracting structural and concentration information. Cluster details can be obtained from the configuration average method. Furthermore, the ATA is useful in determining the best parameters (energy, angles, etc.) to be used in an experiment.

Electronic structure of MnSi 0.7Al 1.3 and related transition metal alloys with the TiSi 2 structure

The densities of states of some ternary alloys (NbSi 1.4Al 0.6, MoAl 1.3Si 0.7, MoGa 1.4Ge 0.6, MnAl 1.3Si 0.7) with the TiSi 2 structure have been obtained with the tight binding-linear muffin tin orbital (TB-LMTO)-average T matrix approximation (ATA) method and have been found to be similar to those of the chimney-ladder structures, previously computed with the same method. This is in agreement with the similar behaviour of their valence electron concentration. MnAl 1.3Si 0.7 has been found found to be magnetic and a more elaborate KKR-CPA calculation predicts a magnetic moment of 0, 4 μ B for Mn in this alloy.

Effective Dielectric Response of Composites Containing Uniaxial Inclusions

An approach is proposed for the calculation of an effective dielectric function for a composite medium containing randomly oriented particles with uniaxial polarisability (the medium is isotropic on average). A scalar renormalised average polarisability of the particles is introduced, which takes complete account of the tensor nature of the dipole-dipole interaction between them. Fluctuations originating from their random positions are treated in the spirit of the average T-matrix approximation. The renormalised polarisability is related to the effective dielectric function via the Maxwell-Garnett equation. The theory is illustrated by calculating the reflectivity and absorption of a colloidal solution of hexagonal CdS particles.

Nordheim dependence in the surface resistivity of disordered overlayers

A multiple scattering calculation is used to compute the surface resistivity of Cu(100), Al(100), and Al(111) surfaces as a function of coverage by a disordered overlayer of Cu or Al adatoms. The Bloch states of the semi-infinite bulk are described by a layer-Korringa–Kohn–Rostoker calculation combined with the coherent potential approximation to represent the carrier scattering by the disordered overlayer. The diffuse scattering of carriers by the surface disorder results in a Nordheim, or quasiparabolic, dependence of the induced surface resistivity upon the coverage. This result confirms the qualitative behavior observed in a prior calculation using the average t-matrix approximation where the surface was modeled as a random distribution of s-wave scatterers in front of a hard-wall potential representing the surface barrier [D. L. Lessie and E. R. Crosson, J. Appl. Phys. 59, 504 (1986)].

Theory of incompletely isovalent -doped semiconductors

In order to calculate the dependence of the electronic structure of single -doped layers on the area concentration c of isovalent impurities, the averaged t-matrix approximation (ATA) is applied to the case of a few impurities within a layer, and generalized to the case of a few missing impurities in an otherwise completely filled layer. In this way, exact results for the limits and are obtained. An interpolation between these limits results in Green's functions which allow the calculation of c-dependent electronic properties. In the case of GaAs:In, the top of the heavy-hole subband is calculated for a short-range (-like) difference between the host atom (Ga) and the impurity (In) potentials on the basis of a parabolic host band structure as well as by applying a three-band Hamiltonian. The comparison with the results from a virtual-crystal approximation proves the latter to describe the subbands well, provided that single impurities do not bind charge carriers. The subband dispersion for heavy holes, calculated for AlAs:Ga on the basis of a realistic ten-band Hamiltonian, shows a camel-back structure caused by the warping of the host heavy-hole band. This structure is proved to result in a pronounced high-energy peak of the subband density of states, in contrast to its step-like behaviour in the case of parabolic bands.

Multiple-scattering theory for mean responses in a plate with sprung masses

Diagrammatic multiple-scattering theory is applied to the case of an infinite homogeneous plate in flexure attached to a random distribution of sprung masses. This system is a prototypical example of a wave-bearing master structure with a locally reacting “fuzzy” substructure. Results for mean fields are obtained from the first-order smoothing approximation, the Foldy average t-matrix approximation, and Soven’s coherent-potential approximation. The study of mean-square responses is reserved to a later paper. It is found that the attenuation as calculated by Pierce et al. [J. Vib. Acoustics 117, 339–348 (1995)] differs from that of the multiple-scattering theory by a fractional amount which is small if the individual sprung masses are weak.

The initial growth mode of Ni deposited on a Ag(001) surface

The initial growth mode of Ni deposited on a Ag(001) surface at room temperature was studied by analyzing LEED (low-energy electron diffraction) I- V (intensity versus beam voltage) characteristics in an average t-matrix approximation (ATA). Substantial alloying of Ni in the form of clusters with the hosting Ag atoms was inferred. To gain a detailed idea of the alloying behavior of Ni, Monte Carlo simulation was performed with the Hamiltonian based on the Ising model. The formation of the Ni clusters of various sizes near the surface is also predicted.

Kinetics of diffusion-influenced reversible reaction A+B ⇌ C in solutions

Reversible diffusion-influenced pseudo first order reaction A+B ⇌C with static particles in excess is rigorously studied. Under most general assumptions, the problem of the reversible reaction kinetics is reduced to the consideration of the effective irreversible reaction studied by conventional methods. In the framework of the average t-matrix approximation (ATA) we reproduce some results derived earlier and establish their applicability limits. Rigorous investigation of the kinetics behavior at long times shows that the t−3/2 law predicted earlier and reproduced by ATA has a different concentration-dependent amplitude. On the basis of diagrammatic summation, providing correct long-time asymptotics, a modified theory has been developed. The range of validity of the modified theory is much wider than that of ATA and similar theories.

A relation between multiple-scattering theory and micromechanical models of effective thermoelastic properties

A relation between multiple-scattering theory and micromechanical models of effective elastic material properties of heterogeneous materials has been established within a unified theoretical framework, and exemplified on three important approximations, the average t-matrix (or Mori-Tanaka), symmetric self-consistency (coherent potential), and asymmetric self-consistency approximations.