D. M. Wood

The structure and implementation of a new general iterative method for diagonalising large matrices (the 'residual minimisation/direct inversion in the iterative subspace' method of Bendt and Zunger) are described and contrasted with other more commonly used iterative techniques. The method requires the direct diagonalisation of only a small submatrix, does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure. Numerical results for two rather different matrices (a real 50*50 non-diagonally dominant matrix and a complex Hermitian 181*181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set) are used to compare the new method with the competing methods. the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements; it is quite insensitive to the properties of the matrices used. This technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required.

Received 24 November 1980DOI:https://doi.org/10.1103/PhysRevLett.46.749©1981 American Physical Society

Existing effective-medium-type theories for the propagation of long-wavelength electromagnetic radiation in heterogeneous media are examined, and structural effects, neglected by such theories, are introduced by a multiple-scattering approach that yields an effective propagation wave vector. Results are presented for propagation through an infinite periodic array of small spheres immersed in a host of different permittivity (or permeability). The procedure is generalized to aperiodic systems to include the lowest-order corrections for small-sphere volume fill fraction $\ensuremath{\eta}$ (for arbitrary scattering strength) and for weak scattering (for arbitrary $\ensuremath{\eta}$). In all cases significant effects due to structure-induced multipole fields are seen to occur. A simple parametrization of deviations from the lowest-order result, the Maxwell-Garnett expression, is proposed in order to extract information on structural multipoles or clustering effects from experimental data. We present the results of calculations for mixtures of real dielectrics and for small metal spheres embedded quasirandomly in a dielectric host, and describe generalizations to include the effects of particle coating and size distributions on optical properties.

The electric-dipole susceptibility of small metal particles of characteristic dimension $L$ is calculated within the random-phase approximation on the assumption that ${k}_{F}^{\ensuremath{-}1}<<L<<\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the wavelength of the electromagnetic field and ${k}_{F}$ the Fermi wave vector for bulk metal. Electron scattering is introduced in a number-conserving relaxation time approximation, and the optical conductivity of a single particle and the absorption coefficient for a suspension of such particles are determined. The matrix elements for cubical particles are sufficiently tractable that the evolution of the optical properties with particle size may be followed down to a metal-insulator transition demonstrated to occur for particle dimensions ($\frac{\ensuremath{\sim}1}{{k}_{F}}$) consistent with the Ioffe-Regel localization criterion. The far-infrared absorption coefficient is found to diverge as the critical particle size is approached. The surface plasmon is monotonically red-shifted and considerably broadened by Landau damping. Criteria for observing the discrete optical structure in small metallic particles are presented.

We examine theoretically structural and electronic properties of thin ${\mathrm{Si}}_{\mathrm{n}}$${\mathrm{Ge}}_{\mathrm{n}}$ superlattices for n=1, 2, 4, and 6, grown on (001)-oriented substrates. The increased repeat distance along the growth direction leads to folding of conduction-band states to the \ensuremath{\Gamma} point of the superlattice Brillouin zone, resulting in a significant reduction in the minimum direct band gap. Transitions to these folded-in states have nonzero dipole matrix elements because of (i) atomic relaxation, leading to the accommodation of distinct Si-Si and Ge-Ge bond lengths and (ii) the superlattice ordering potential. Our calculations show that superlattices grown pseudomorphically on a Si substrate remain indirect-band-gap structures, with a minimum gap from \ensuremath{\Gamma} to \ensuremath{\Delta} (near the X point) of the fcc Brillouin zone. We find, however, that increasing the lattice parameter ${a}_{s}$ of the substrate will further reduce the direct band gap. For ${a}_{s}$\ensuremath{\gtrsim}\ifmmode \bar{a}\else \={a}\fi{}, where \ifmmode \bar{a}\else \={a}\fi{} is the average of the lattice constants for Si and Ge, we predict a nearly direct band gap: For ${\mathrm{Si}}_{6}$${\mathrm{Ge}}_{6}$ the indirect band gap for ${a}_{s}$=\ifmmode \bar{a}\else \={a}\fi{} is only \ensuremath{\sim}0.01 eV smaller than the direct band gap. The lowest conduction-band states in this case are localized on the Si sublattice.