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Analytical inversion of symmetric tridiagonal matrices

In this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices. The result is of relevance to the solution of a variety of problems in mathematics and physics. As an example, the formula is used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions.

A THEORY OF HIGH ELECTRIC FIELD TRANSPORT

The Generalized Quantum Langevin Equation (GLE) approach to the quantum transport of many electron systems is developed to study the high electric field transport. Two GLE equations, one for the center-of-mass momentum and the other for the center-of-mass energy, are obtained. The non-linear transport effects due to the presence of a high electric field are reflected directly in the memory functions of momentum and energy. By including velocity fluctuations, “collision broadening” and the “intra-collision field effect” appear in our transport equations quite naturally. In our theory, these quantum effects are illustrated by the phenomena of the level broadening, the velocity fluctuation relaxation and accelaration effects.

Dielectric response of a quasi-one-dimensional electron system

The dielectric response of a quasi-one-dimensional electron system is studied by including fluctuation effects (in the polarizability) and by using a recently derived analytic form for the electron-electron interactions. General forms for the polarizability matrices both for the intra- and inter-subband cases are presented. The generalized polarizability is analytic over the whole region of the wavevectors artd rigorously retains the number neutrality. Various differences between the intra- and inter-subband cases for the polarizability and the dielectric matrix function are studied. The theory is used to study impurity screening and plasmon excitations in the presence of multi-subbands. We show that the screened impurity potential of a quasi-one-dimensional electron system is a well defined quantity and, in contrast to its twc- and threedimensional counterparts, it is finite at the origin and has stronger Friedel oscillations. An explanation is given for the experimental results of Hansen et a1 conceming the relationship between the inter-subband plasmon frequencies and the electron densities.

Strong electric field effect on weak localization

The influence of an electric field on weak localization is studied by a recently proposed generalized quantum Langevin equation approach to the conductivity problem. A general formula for the memory function of the non-interacting electron gas, in the presence of high order impurity scattering and an arbitrary electric field, is derived. In the low field case, a scale and frequency dependent conductivity is obtained, which reduces to the well known scale dependent conductivity in the static limit. In the high field case, the conductivity is field dependent through the drift velocity. It is shown that the presence of strong electric fields tends to delocalize the one and two dimensional electron systems if one adopts the electron heating model. However, if the electron gas does not heat up, the conductivity will be field independent.

On the relationship between the quantum Langevin model and the Landauer formula

Abstract We show that, in the weak coupling limit, the main formalism of the phenomenological quantum Langevin model for single electron tunneling in small tunnel junctions is derivable within the framework of the rigorous generalized Landauer formula.

Inhomogeneous boundary effects in semiconductor quantum wires

Boundary effects play an essential role in determining the physical properties of semiconductor quantum wires. Additional features come into play when one considers inhomogeneous boundaries, i.e. wires whose widths are functions of distance along the length of the wire. The authors show that, in the adiabatic approximation (which assumes that the boundary potential fluctuation effects, or equivalently, the variations in the wire width, occur on a scale much larger than the inverse of Fermi wavelength), the boundary problem for a quantum wire is equivalent to a one-dimensional Schrodinger equation along the wire with: (i) an effective potential provided by the deviation from the homogeneous boundary and, (ii) a wave function coupled to the lateral direction. In the periodic boundary fluctuation case, the subbands of the system split into many mini-subbands, and become a useful system to test 1D band theory. When the boundary fluctuates randomly, there exists in each of the subbands, a mobility edge, below which the electron states are localized. The localization of the tail states of the top populated subbands makes the conductance drop smoothly whenever the Fermi energy passes the bottom of a new subband. The theory explains the recent experiments of Smith and Craighead (1990).

Electric field effect on weak localization in a semiconductor quantum wire

The influence of an electric field on weak localization in a semiconductor quantum wire is studied by a recently proposed generalized quantum Langevin equation approach to the conductivity problem. A new physical picture is presented. In our model the electronic motion is essentially one-dimensional, and the phase coherence length lφ is much larger than the elastic mean free path l2 of electrons. We find that when the electric field E exceeds a critical value Ec = KVFelφ, where VF is the Fermi velocity, it will introduce a new cut-off length Ls = (EcE)12lφ with implications for the experimental results on semiconductor quantum wires. Our theory is in good agreement with the experiments of Hiramoto and co-workers.

Weak localisation theory for lightly doped semiconductor quantum wires

A weak localisation theory for a semiconductor quantum wire, which has a width of the order of the Fermi wavelength, is presented. In our model the electronic motion is essentially one-dimensional and the localisation length L, is much larger than the mean free path I, so that, in contrast to conventional theories a non-localised quantum wire with a total length L L, the conductance of the quantum wire depends on L instead of L,, implying a temperature independent behaviour. Our theory explains recent experiments which found temperature independent transport behaviour at very low tem- perature for narrow AlGaAs/GaAs quantum wire. In studying the AC conductivity, our calculation predicts that, for the quantum wire with L > L,, there exists a critical value of the frequency above which the system is delocalised and the AC conductivity a(w) rises as w2.

Cyclotron resonance in GaAs/AlGaAs superlattices

Abstract The cyclotron resonance of the GaAs/AlGaAs superlattice is studied by the generalized quantum Langevin equation approach to electronic transport. Divergences are eliminated from the conventional forms for the memory functions by inclusion of the effects of the fluctuation of the center of mass. In our theory there exists two resonance peaks in the CR absorption spectrum: the conventional magnetoplasmon absorption peak at the frequency ω ⩾ ω c , and another peak at ω ⩽ ω c due to the fluctuation effect. The positions, amplitudes, and the relative spacing of these two peaks depend mainly on the magnetic field, the electron density, the mobility of the sample and the temperature. Our results for the CR spectrum of a single layer GaAs/AlGaAs heterojunction are consistent with the experiments of Schlesinger et al. and of Muro et al. In calculating the CR spectrum of an infinite layer GaAs/AlGaAs superlattice, we find that the peak shift becomes larger and the peak width is almost unchanged when the superlattice periodicity is reduced.

PHONON EFFECTS ON THE CYCLOTRON RESONANCE FOR A MANY-BODY SYSTEM: A GENERALIZED QUANTUM LANGEVIN EQUATION APPROACH

Abstract We have recently developed a new approach to the derivation of quantum transport results for an interacting many-body system of electrons, impurities and phonons. Now, using a similar philosophy, the cyclotron resonance lineshape (CRLS) of an electron-phonon many-body system is studied by the generalized quantum Langevin equation approach, in which the memory function is obtained directly by solving the Heisenberg equation of the motion for the fluctuating density of relative electrons. Frequency dependent line shift and line width results, for the case of interacting electrons in the RPA approximation, are derived in a very simple way. A systematic study of the temperature and magnetic field dependences of the CRLS of a two dimensional electron-phonon system is performed. Two resonance peaks, due to the acoustic phonon adsorption and emission process, are found in the vicinity of the cyclotron resonance frequency.