Inward and Outward Integral Equations and the KKR Method for Photons

In the case of electromagnetic waves it is necessary to distinguish between inward and outward on-shell integral equations. Both kinds of equation are derived. A correct implementation of the photonic KKR method then requires the inward equations and it follows directly from them. A derivation of the KKR method from a variational principle is also outlined. Rather surprisingly, the variational KKR method cannot be entirely written in terms of surface integrals unless permeabilities are piecewise constant. Both kinds of photonic KKR method use the standard structure constants of the electronic KKR method and hence allow for a direct numerical application. As a by-product, matching rules are obtained for derivatives of fields on different sides of the discontinuity of permeabilities. Key words: The Maxwell equations, photonic band gap calculationsComment: (to appear in J. Phys. : Cond. Matter), Latex 17 pp, PRA-HEP 93/10 (exclusively English and unimportant misprints corrected

Korringa ratio of ferromagnetically correlated impure metals

The Korringa ratio, $\cal K$, obtained by taking an appropriate combination of the Knight shift and nuclear spin-lattice relaxation time, is calculated at finite temperature, $T$, in the three-dimensional electron gas model, including the electron-electron interaction, $U$, and non-magnetic impurity scatterings. $\cal K$ varies in a simple way with respect to $U$ and $T$; it decreases as $U$ is increased but increases as $T$ is raised. However, $\cal K$ varies in a slightly more complicated way with respect to the impurity scatterings; as the scattering rate is increased, $\cal K$ increases for small $U$ and low $T$, but decreases for large $U$ or high $T$ regime. This calls for a more careful analysis when one attempts to estimate the Stoner factor from $\cal K$.Comment: 7 pages including 3 figures. To be published in Phys. Rev. B, Dec.

Energy-resolved inelastic electron scattering off a magnetic impurity

We study inelastic scattering of energetic electrons off a Kondo impurity. If the energy E of the incoming electron (measured from the Fermi level) exceeds significantly the Kondo temperature T_K, then the differential inelastic cross-section \sigma (E,w), i.e., the cross-section characterizing scattering of an electron with a given energy transfer w, is well-defined. We show that \sigma (E,w) factorizes into two parts. The E-dependence of \sigma (E,w) is logarithmically weak and is due to the Kondo renormalization of the effective coupling. We are able to relate the w-dependence to the spin-spin correlation function of the magnetic impurity. Using this relation, we demonstrate that in the absence of magnetic field the dynamics of the impurity spin causes the electron scattering to be inelastic at any temperature. Quenching of the spin dynamics by an applied magnetic field results in a finite elastic component of the electron scattering cross-section. The differential scattering cross-section may be extracted from the measurements of relaxation of hot electrons injected in conductors containing localized spins.Comment: 15 pages, 9 figures; final version as published, minor changes, reference adde

Coulomb "blockade" of Nuclear Spin Relaxation in Quantum Dots

We study the mechanism of nuclear spin relaxation in quantum dots due to the electron exchange with 2D gas. We show that the nuclear spin relaxation rate is dramatically affected by the Coulomb blockade and can be controlled by gate voltage. In the case of strong spin-orbit coupling the relaxation rate is maximal in the Coulomb blockade valleys whereas for the weak spin-orbit coupling the maximum of the nuclear spin relaxation rate is near the Coulomb blockade peaks.Comment: 4 pages, 3 figure

Three disks in a row: A two-dimensional scattering analog of the double-well problem

We investigate the scattering off three nonoverlapping disks equidistantly spaced along a line in the two-dimensional plane with the radii of the outer disks equal and the radius of the inner disk varied. This system is a two-dimensional scattering analog to the double-well-potential (bound state) problem in one dimension. In both systems the symmetry splittings between symmetric and antisymmetric states or resonances, respectively, have to be traced back to tunneling effects, as semiclassically the geometrical periodic orbits have no contact with the vertical symmetry axis. We construct the leading semiclassical ``creeping'' orbits that are responsible for the symmetry splitting of the resonances in this system. The collinear three-disk-system is not only one of the simplest but also one of the most effective systems for detecting creeping phenomena. While in symmetrically placed n-disk systems creeping corrections affect the subleading resonances, they here alone determine the symmetry splitting of the 3-disk resonances in the semiclassical calculation. It should therefore be considered as a paradigm for the study of creeping effects. PACS numbers: 03.65.Sq, 03.20.+i, 05.45.+bComment: replaced with published version (minor misprints corrected and references updated); 23 pages, LaTeX plus 8 Postscript figures, uses epsfig.sty, espf.sty, and epsf.te

Resonance-Induced Effects in Photonic Crystals

For the case of a simple face-centered-cubic photonic crystal of homogeneous dielectric spheres, we examine to what extent single-sphere Mie resonance frequencies are related to band gaps and whether the width of a gap can be enlarged due to nearby resonances. Contrary to some suggestions, no spectacular effects may be expected. When the dielectric constant of the spheres $\epsilon_s$ is greater than the dielectric constant $\epsilon_b$ of the background medium, then for any filling fraction $f$ there exists a critical $\epsilon_c$ above which the lowest lying Mie resonance frequency falls inside the lowest stop gap in the (111) crystal direction, close to its midgap frequency. If $\epsilon_s <\epsilon_b$, the correspondence between Mie resonances and both the (111) stop gap and a full gap does not follow such a regular pattern. If the Mie resonance frequency is close to a gap edge, one can observe a resonance-induced widening of a relative gap width by $\approx 5$.Comment: 14 pages, 3 figs., RevTex. For more info look at http://www.amolf.nl/external/wwwlab/atoms/theory/index.htm

A simple formula for the L-gap width of a face-centered-cubic photonic crystal

The width $\triangle_L$ of the first Bragg's scattering peak in the (111) direction of a face-centered-cubic lattice of air spheres can be well approximated by a simple formula which only involves the volume averaged $\epsilon$ and $\epsilon^2$ over the lattice unit cell, $\epsilon$ being the (position dependent) dielectric constant of the medium, and the effective dielectric constant $\epsilon_{eff}$ in the long-wavelength limit approximated by Maxwell-Garnett's formula. Apparently, our formula describes the asymptotic behaviour of the absolute gap width $\triangle_L$ for high dielectric contrast $\delta$ exactly. The standard deviation $\sigma$ steadily decreases well below 1% as $\delta$ increases. For example $\sigma< 0.1$ for the sphere filling fraction $f=0.2$ and $\delta\geq 20$. On the interval $\delta\in(1,100)$, our formula still approximates the absolute gap width $\triangle_L$ (the relative gap width $\triangle_L^r$) with a reasonable precision, namely with a standard deviation 3% (4.2%) for low filling fractions up to 6.5% (8%) for the close-packed case. Differences between the case of air spheres in a dielectric and dielectric spheres in air are briefly discussed.Comment: 13 pages, 4 figs., RevTex, two references added. For more info see http://www.amolf.nl/external/wwwlab/atoms/theory/index.htm

First-principles Calculation of the Formation Energy in MgO-CaO Solid Solutions

The electronic structure and total energy were calculated for ordered and disordered MgO-CaO solid solutions within the multiple scattering theory in real space and the local density approximation. Based on the dependence of the total energy on the unit cell volume the equilibrium lattice parameter and formation energy were determined for different solution compositions. The formation energy of the solid solutions is found to be positive that is in agreement with the experimental phase diagram, which shows a miscibility gap.Comment: 11 pages, 3 figure

Photonic Band Gaps of Three-Dimensional Face-Centered Cubic Lattices

We show that the photonic analogue of the Korringa-Kohn-Rostocker method is a viable alternative to the plane-wave method to analyze the spectrum of electromagnetic waves in a three-dimensional periodic dielectric lattice. Firstly, in the case of an fcc lattice of homogeneous dielectric spheres, we reproduce the main features of the spectrum obtained by the plane wave method, namely that for a sufficiently high dielectric contrast a full gap opens in the spectrum between the eights and ninth bands if the dielectric constant $\epsilon_s$ of spheres is lower than the dielectric constant $\epsilon_b$ of the background medium. If $\epsilon_s> \epsilon_b$, no gap is found in the spectrum. The maximal value of the relative band-gap width approaches 14% in the close-packed case and decreases monotonically as the filling fraction decreases. The lowest dielectric contrast $\epsilon_b/\epsilon_s$ for which a full gap opens in the spectrum is determined to be 8.13. Eventually, in the case of an fcc lattice of coated spheres, we demonstrate that a suitable coating can enhance gap widths by as much as 50%.Comment: 19 pages, 6 figs., plain latex - a section on coated spheres, two figures, and a few references adde