Weak Coupling Case Research Articles

Magnitude and size scaling of intervalley coupling in semiconductor alloys and superlattices

Coupling between different \ensuremath{\Gamma}, $X,$ and $L$ band-structure valleys is responsible for (a) level anticrossing in superlattices as a function of period, pressure, and electric field and for (b) ``optical bowing'' of band gaps in random alloys. We investigate the symmetry, magnitude, and size scaling of intervalley coupling in semiconductor superlattices and alloys by direct supercell calculations, performed with screened pseudopotentials and a plane-wave basis, considering up to ${10}^{6}$ atoms/supercell. Projecting the calculated electronic wave functions ${\ensuremath{\psi}}_{i}$ of alloys or superlattices onto the bulk states of the constituent zinc-blende materials shows that ${\ensuremath{\psi}}_{i}$ contain a ``majority representation'' from one or more zinc-blende states \ensuremath{\gamma}. The intervalley coupling $E(i,j)$ between the alloy states ${\ensuremath{\psi}}_{i}$ and ${\ensuremath{\psi}}_{j}$ then includes a term $2F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ due to the ``majority representations'' \ensuremath{\gamma} and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ of ${\ensuremath{\psi}}_{i}$ and ${\ensuremath{\psi}}_{j},$ respectively, plus residual terms due to the minority representations. We find the following: (i) In alloys, the orbital overlap function $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ is large, since the wave functions are extended. The intervalley coupling element $V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ exhibits simple selection rules: being zero for $({\ensuremath{\Gamma}}_{1c}{,X}_{1c}),$ $({\ensuremath{\Gamma}}_{1c}{,L}_{3c}),$ ${(X}_{1c}^{x}{,X}_{1c}^{y}),$ etc. (``weak coupling''), and nonzero for $({\ensuremath{\Gamma}}_{1c}{,X}_{3c}),$ $({\ensuremath{\Gamma}}_{1c}{,L}_{1c}),$ ${(L}_{3c}{,X}_{1c}),$ etc. (``strong coupling''). This explains why the $\overline{\ensuremath{\Gamma}}$-like conduction band of mixed-cation alloys contains zinc-blende ${\ensuremath{\Gamma}}_{1c}$ and ${L}_{1c}$ character, but not ${X}_{1c}.$ In the case of strong coupling, $E(i,j)$ scales as $1/\sqrt{\ensuremath{\Omega}},$ where \ensuremath{\Omega} is the volume, while in the weak-coupling case the entire coupling originates from the ``minority representation,'' and is 20--100 times smaller. The minority representation, however, contributes to the bowing of the band gap vs composition. (ii) In superlattices, although the above selection rule for $V(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ still exists, the magnitude of the intervalley coupling is governed by the overlap function $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}}).$ For simple superlattices, $F(\ensuremath{\gamma},{\ensuremath{\gamma}}^{\ensuremath{'}})$ is small, since the wave functions are localized in particular segments (``weak coupling''). Consequently, the ``majority representation'' contributes 5--100 times less than in the analogous case of alloys. Furthermore, $E(i,j)$ scales as ${1/n}^{3},$ where $n$ is the superlattice period.

Kink’s internal modes in the Frenkel-Kontorova model

We consider a generalized Frenkel-Kontorova model, describing the dynamics of a chain of particles in a periodic substrate potential, and analyze the effect of discreteness on the existence and properties of internal (or shape) modes of kinks, topological excitations of the chain. In particular, we show that kink's internal modes can appear not only below but also above the phonon spectrum band and, in the latter case, the localized mode describes out-of-phase oscillations of the kink's shape. For the sinusoidal on-site potential, when the model is described by the discrete sine-Gordon equation, we reveal, in sharp contrast with the continuum limit, the existence of the kink's internal mode in a narrow region of the discreteness parameter. We apply two different analytical techniques to describe the cases of weak and strong coupling between particles, explaining qualitatively and even quantitatively the main features of the kink oscillations observed in numerical simulations. We also discuss the effect of nonlinearity on the existence and properties of kink's internal modes and show, in particular, that a nonlinearity-induced frequency shift of the lattice vibrations can lead to the creation of the nonlinear kink's internal modes, which, however, slowly decay due to a generation of radiation through higher-order harmonics.

Reconstruction of secular polynomials for Hubbard model from energy perturbation series for weak and strong coupling cases

An interpolation technique which is based on the strongly and weakly correlated cases of the one-dimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of N=6 sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for N=6 agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.

Disorder effects in superconductors with anisotropic pairing: From Cooper pairs to compact bosons

In weak coupling BCS-theory normal impurities do not influence superconducting T c in case of s-wave pairing. In case of d-wave pairing these are strongly pair-breaking. This fact is in rather strong contradiction with many experiments on disordered high- T c superconductors. With the growth of electron attraction within the Cooper pair the system smoothly crosses over from BCS-pairs to compact Boson picture of superconductivity. Significant deviations from universal Abrikosov-Gorkov dependence of T c on disorder are expected in this crossover region with superconducting state becoming more stable than in the weak coupling case. As high- T c superconductors are actually in this intermediate region, we can understand the relative stability of their d-wave pairing state under disordering.

Bolometric response of optically thick superconducting films

A one-dimensional heat conduction model is established to interpret the chopping frequency dependence of the bolometric response of optically thick superconducting films. The key feature of this model is that it takes into account the thermal boundary conductance of both the film/substrate and the substrate/heat-sink interfaces. The model is found to be suitable for both the strong and weak thermal coupling cases. By comparing the results with a simplified bolometric model, explicit expressions of the thermal conductance, thermal capacitance, radiation absorption efficiency and response time constant of the bolometric system have been deduced, where the influences of the film, substrate and the heat conduction boundaries are all considered.

Reduced density matrix approach for gas-solid scattering

A short time propagation algorithm for the reduced density matrix is derived which describes the interaction between a particle hitting a surface and the thermal vibrations of the lattice. The algorithm, including the dissipative terms, is derived directly from the full particle-bath density matrix and Hamiltonian. The model is used to examine the scattering of He, Ne and Ar from a model Cu surface over a wide range of energy and temperature. The method provides a good description of particle behavior for weak, intermediate and strong coupling cases.

Superconductivity in the Strong Coupling Limit: Van Hove Singularity and Extended Saddle Point Singularity Scenario

The effect of orthorhombic distortion and second nearest neighbor hopping on the transition temperature (T c ) and the isotope-shift exponent (α) have been studied for a two dimensional system, within the framework of Eliashberg formalism in the van Hove singularity scenario. The behavior of T c and α have been discussed for the case of extended saddle point singularity. A comparison of the strong coupling theory results is made with the results of weak coupling theory. From our study it reveals that the effect of singularity in the density of states (DOS) in the strong coupling limit is not so significant as that in weak coupling case.

Coupling of ηi and trapped electron modes in plasmas with negative magnetic shear

In toroidal collisionless plasmas, the ion temperature gradient (ITG or ηi) and the trapped electron (TE) modes are shown to be weakly (strongly) coupled when both the temperature gradients and the driving mechanism of the trapped electrons are moderate to strong (weak but finite). In the regime of strong coupling, there is a single hybrid mode, unstable for all ηi in plasmas with positive magnetic shear. For the weak coupling case, two independent unstable modes, one in the ion and the other in the electron diamagnetic direction, are found to coexist. In either situation, the negative magnetic shear exerts a strong stabilizing influence; the stabilizing effect is considerably enhanced by the presence of trapped particles. It is predicted that for a given set of plasma parameters, it will be much harder to simultaneously excite the two modes in a plasma with negative shear. The results of this study are significant for tokamak experiments.

Disordering effects in superconductors with anisotropic pairing: From Cooper pairs to compact bosons

In the weak coupling BCS-approximation normal impurities do not influence superconducting T_{c} in significant manner in case of isotropic s-wave pairing. However, in case of d-wave pairing these are strongly pair-breaking. This fact is in rather strong contradiction with many experiments on disordered high-T_{c} superconductors assuming the d-wave nature of pairing in these systems. With the growth of electron attraction within the Cooper pair the system smoothly crosses over from BCS-pairs to compact Boson picture of superconductivity. As pairing strength grows and pairs become compact significant deviations from universal Abrikosov-Gorkov dependence of T_{c} on disorder appear in case of d-wave pairing with superconducting state becoming more stable than in the weak coupling case. As high-T_{c} superconductors are actually in the intermediate region with Cooper pairs size of the order of few interatomic lengths, these results can explain the relative stability of d-wave pairing under rather strong disordering.

Unified model for the study of diffusion localization and dissipation

A model that generalizes the study of quantum Brownian motion (BM) is constructed. We consider disordered environment that may be either static (quenched), noisy or dynamical. The Zwanzig-Caldeira-Leggett BM model formally constitutes a special case where the disorder autocorrelation length is taken to be infinite. Alternatively, a localization problem is obtained if the noise autocorrelation time is taken to be infinite. Also the general case of weak nonlinear coupling to a thermal, possibly chaotic bath is handled by the same formalism. A general, Feynman-Vernon type path-integral expression for the propagator is introduced. A Wigner transformed version of this expression is utilized in order to facilitate comparison with the classical limit. It is demonstrated that nonstochastic genuine quantal manifestations are associated with the model. It is clarified that such effects are absent in the standard BM model, either the disorder or the chaotic nature of the bath are essential. Quantal correction to the classical diffusive behavior is found even in the limit of high temperatures. The suppression of interference due to dephasing is discussed, leading to the observation that due to the disorder the decay of coherence is exponential in time, and no longer depends on geometrical considerations. Fascinating non-Markovian effects due to time-correlated (colored) noise are explored. For this, a strategy is developed in order to handle the integration over paths. This strategy is extended in order to demonstrate how localization comes out from the path-integral expression.

Exact time evolution and master equations for the damped harmonic oscillator

Using the exact path integral solution for the damped harmonic oscillator it is shown that in general there does not exist an exact dissipative Liouville operator describing the dynamics of the oscillator for arbitrary initial bath preparations. Exact non-stationary Liouville operators can be found only for particular preparations. Three physically meaningful examples are examined. An exact new master equation is derived for thermal initial conditions. Second, the Liouville operator governing the time-evolution of equilibrium correlations is obtained. Third, factorizing initial conditions are studied. Additionally, one can show that there are approximate Liouville operators independent of the initial preparation describing the long time dynamics under appropriate conditions. The general form of these approximate master equations is derived and the coefficients are determined for special cases of the bath spectral density including the Ohmic, Drude and weak coupling cases. The connection with earlier work is discussed.

Localized global ICRF eigenmodes and conversion zones in a two-ion species tokamak

The location of the slow - fast wave conversion zones is analysed for a tokamak in ICRF. The obtained dispersion relation which determines the surfaces of wave conversion describes the cases of weak and strong mode coupling as well as the intermediate one. Using the same approach, the localized global eigenmodes are studied. With respect to their nature, they are separated into two types. The first is the TLE - TAE-like eigenmode. Like the TAE it is the result of mode coupling. The second one is the single-mode eigenmodes. This type is represented by two eigenmodes. One is the extension of the GAE (global Alfvén eigenmode) to the two-ion species plasma. The other one is OHE (original hybrid eigenmode) which exists in the case of strong mode coupling. The field structure and dispersion of these waves are discussed.

Oscillatory dephasing in the presence of inhomogeneous broadening.

Oscillatory dephasing is a characteristic signature of non-Markovian relaxation phenomena. From its analysis one can obtain valuable information about the dynamics of the surroundings of a given system. In this paper we calculate the transient four-wave mixing signal in the short pulse limit from two-level systems coupled to a bath of harmonic oscillators for arbitrary coupling strengths and in the presence of inhomogeneous broadening. The strong and weak coupling cases having already been analyzed, we focus on the study of the intermediate case where oscillatory dephasing is observed. We specifically show that even in the presence of inhomogeneous broadening, information about the surroundings can still be extracted in the intermediate coupling regime. The formation and the evolution of the photon echo in such experiments are also analyzed. \textcopyright{} 1996 The American Physical Society.

STATISTICAL ENERGY ANALYSIS OF MULTIPLE, NON-CONSERVATIVELY COUPLED SYSTEMS

In what follows, the analysis of Keane [1] for the energy flows around an arbitrary configuration of coupled multi-modal subsystems is extended to the case of non-conservative coupling. Here, the joints between any two subsystems are modelled by a spring anda damper, thus allowing for dissipation of energy to occur at the joint. The aim of this study is to give greater insight into the problem of energy flows through non-conservative couplings which has not been extensively discussed in the literature. Interest is focused on the effect of damping in the joints on the magnitudes of energy flows between, and energy levels in, each subsystem. The model derived is used to demonstrate the well known fact that selecting the correct level of damping in the joints surrounding a driven subsystem may cause a large percentage of the power input to the subsystem to be dissipated in these joints. This minimizes the overall power dissipated within the subsystems and thus the system total energy level. A Statistical Energy Analysis (SEA) model for non-conservatively coupled systems is then suggested, in which coupling damping loss factors are introduced into the main SEA energy balance equations to account for the energy dissipation in the joints. This model is shown to be exactly correct for the limiting case of weak coupling. Numerical examples which illustrate these various ideas are presented, with the use of parameters adopted in previous studies.

The dissipative dynamics of a pair of paraelectric tunnelling impurities

Paraelectric impurities contribute significantly to the low-temperature properties of alkali halide crystals. Even at very low density the dipolar interaction of adjacent defect ions may lead to deviations from the behaviour expected for isolated impurities; in a certain range of concentration it is sufficient to consider pairs of coupled defects. Applying a projection operator method, previous work on this pair model is extended to the case of finite asymmetry and weak coupling to acoustic phonons. After performing the ensemble averaging, the specific heat and the zero-frequency susceptibility are calculated and compared with experimental data on KCl:Li and KCl:CN. The isotope effect on the Rabi frequency and relaxation rate is discussed.

Theory of the Seebeck coefficient in LaCrO3 and related perovskite systems.

We consider the Seebeck coefficient in ${\mathrm{LaCrO}}_{3}$ and related transition-metal-oxide perovskites using a model for electronic conduction based on the electronic structure of the 3d orbitals of the B-site transition-metal cations. Relations for the Seebeck coefficient are presented for those perovskite systems in which electronic conduction is through the ${\mathit{t}}_{2\mathit{g}}$ states of the B-site transition-metal cations. High- and low-temperature limits for the Seebeck coefficient are identified for the cases of both strong and weak magnetic coupling between electron spins. In these high- and low-temperature limits, the Seebeck coefficient is determined as a function of carrier concentration. Results are applied to an analysis of experimental data for the (La,Sr)${\mathrm{CrO}}_{3}$ series. \textcopyright{} 1996 The American Physical Society.

ON THE PHONON-INDUCED SUPERCONDUCTIVITY OF DISORDERED ALLOYS

A model of alloy is considered which includes both quenched disorder in the electron subsystem (“alloy” subsystem) and electron-phonon interaction. For given approximate solution for the alloy part of the problem, which is assumed to be conserving in Baym’s sense, we construct the generating functional and derive the Eliashberg-type equations which are valid to the lowest order in the adiabatic parameter. The renormalization of bare electron–phonon interaction vertices by disorder is taken into account consistently with the approximation for the alloy self-energy. For the case of exact configurational averaging the same set of equations is established within the usual T-matrix approach. We demonstrate that for any conserving approximation for the alloy part of the self-energy the Anderson’s theorem holds in the case of isotropic singlet pairing provided disorder renormalizations of the electron-phonon interaction vertices are neglected. Taking account of the disorder renormalization of the electron-phonon interaction we analyze general equations qualitatively and present the expressions for Tc for the case of weak and intermediate electron-phonon coupling. Disorder renormalizations of the logarithmic corrections to the effective coupling, which arise when the effective interaction kernel for the Cooper channel has the second energy scale, as well as the renormalization of the dilute paramagnetic impurity suppression are discussed.

Quantum Mechanical Theory of Electron Transfer Reactions at Metal Electrodes

Abstract The quantum theory of electron transfer reactions at metal electrodes was studied. The obtained potential dependence of the electron transfer rate in the weak coupling case resembles the Butler–Volmer equation of classical electrochemistry. The volcano-shaped dependence of the hydrogen exchange current on the adsorption energy of hydrogen on various metals was explained microscopically, based on this theory. The mechanism of hydrogen evolution was also explained microscopically based on this theory. The exchange current density for hydrogen evolution at Pt electrode calculated quantitatively agreed well with the experimental value.

Band spectra of rectangular graph superlattices.

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-S" type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling.

Theoretical wave packet study on pump-probe stimulated emission signals from electron transfer systems in condensed phases

The pump-probe stimulated emission signals from the electron transfer system in a condensed phase are studied theoretically. To show how the system dynamics affect the stimulated emission signal, both weak and strong electronic coupling cases are investigated. The effects of temperature on the dynamics of the system are reported.