J. Heinrichs

Tight-binding surface states in finite crystals

The electronic states of a finite crystal are studied using Goodwins model of a tight-binding linear chain of N one-level atoms with nearest-neighbour overlap. Using a transfer matrix approach we obtain the explicit form of the secular equation which correctly yields N eigenvalues in the interval (0,π) of wavenumber q, unlike Goodwins equation which involves spurious solutions at q = 0 and q = π. We present a new general analysis of bulk- and surface-state eigenvalues as a function of the parameter e0/γ describing the difference (e0) of Coulomb integrals for surface and bulk atoms relative to the overlap integral γ. We identify four distinct domains of values of |e0/γ| in three of which one or two surface states of different origins exist, which we determine explicitly. Our discussion is valid for both signs of e0/γ and differs considerably in detail from Goodwins analysis. In particular, it does not require distinct analyses for chains with even and odd numbers of sites.

Force between metallic films at small separation

An exact expression for the force between two dissimilar metal films in the limit of vanishing separation is derived in terms of bulk properties of the metals. This force is compared with the results of an approximate numerical treatment of Ferrante and Smith for finite separations between the films.

Self-consistent theory of localization in weakly disordered systems

An analytic treatment of localization in a weakly disordered system is presented for the case where the real lattice is approximated by a Cayley tree. Contrary to a recent assertion we find that the mobility edge moves inwards into the band as disorder increases from zero.

Conductance and localization in disordered wires: The role of evanescent states

This paper extends an earlier analytical scattering matrix treatment of conductance and localization in coupled two and three Anderson chain systems for weak disorder when evanescent states are present at the Fermi level.Such states exist typically when the interchain coupling exceeds the width of propagating energy bands associated with the various transverse eigenvalues of the coupled tight-binding systems. We calculate reflection and transmission coefficients in cases where, besides propagating states, one or two evanescent states are available at the Fermi level for elastic scattering of electrons by the disordered systems. We observe important qualitative changes in these coefficients and in the related localization lengths due to ineffectiveness of the evanescent modes for transmission and reflection in the various scattering channels. In particular, the localization lengths are generally significantly larger than the values obtained when evanescent modes are absent. Effects associated with disorder mediated coupling between propagating and evanescent modes are shown to be suppressed by quantum interference effects, in lowest order for weak disorder.

Localization from conductance in few-channel disordered wires

We study localization in two- and three-channel quasi-one-dimensional (1D) systems using multichain tight-binding Anderson models with nearest-neighbor interchain hopping. In the three-chain case we discuss the cases of both free and periodic boundary conditions between the chains. The finite disordered wires are connected to ideal leads and the localization length is defined from the Landauer conductance in terms of the transmission coefficients matrix. The transmission and reflection amplitudes in properly defined quantum channels are obtained from S matrices constructed from transfer matrices in Bloch wave bases for the various quasi-1D systems. Our exact analytic expressions for localization lengths for weak disorder reduce to the Thouless expression for 1D systems in the limit of vanishing interchain hopping. For weak interchain hopping the localization length decreases with respect to the 1D value in all three cases. In the three-channel cases it increases with interchain hopping over restricted domains of large hopping.

Stochastic treatment of AC hopping conduction in disordered systems

As was shown recently, the continuous-time random-walk model of hopping conduction introduced by Scher and Lax (1973) should give no frequency dependence of the conductivity. This stochastic model is generalized, taking into account the relatively slow polarization response of the charge carriers to the electric field produced by instantaneous hops of a test carrier. This modified stochastic model leads to AC conduction as a result of the polarization response, which is treated as a relaxation process. An exact expression is derived for the AC conductivity in terms of a distribution of waiting times between successive hops of a carrier. In the frequency range of interest, an omega v dependence (0<v<1) is found for the real and imaginary parts of the conductivity which satisfy the expected form of the Kramers-Kronig relation. The treatment suggests that v increases towards unity with decreasing temperature, in agreement with experimental findings.

Transmission, reflection and localization in a random medium with absorption or gain

We study reflection and transmission of waves in a random tight-binding system with absorption or gain for weak disorder, using a scattering matrix formalism. Our aim is to discuss analytically the effects of absorption or gain on the statistics of wave transport. Treating the effects of absorption or gain exactly in the limit of no disorder allows us to identify short- and long-length regimes relative to absorption or gain lengths, where the effects of absorption/gain on statistical properties are essentially different. In the long-length regime, we find that a weak absorption or a weak gain induce identical statistical corrections in the inverse localization length, but lead to different corrections in the mean reflection coefficient. In contrast, a strong absorption or a strong gain strongly suppress the effect of disorder in identical ways (to leading order), both in the localization length and in the mean reflection coefficient.

Anomalous scaling of conductance cumulants in one-dimensional Anderson localization

The mean and the variance of the logarithm of the conductance (lng) in the localized regime in the one-dimensional Anderson model are calculated analytically for weak disorder, starting from the recursion relations for the complex reflection and transmission amplitudes. The exact recursion relation for the reflection amplitudes is approximated by improved Born approximation forms which ensure that averaged reflection coefficients tend asymptotically to unity in the localized regime, for chain lengths . In contrast, the familiar Born approximation of perturbation theory would not be adapted for the localized regime since it constrains the reflection coefficient to be less than one. The proper behaviour of the reflection coefficient (and of other related reflection parameters) is responsible for various anomalies in the cumulants of lng, in particular for the well-known band centre anomaly of the localization length. While a simple improved Born approximation is sufficient for studying cumulants at a generic band energy, we find that a generalized improved Born approximation is necessary to account satisfactorily for numerical results for the band centre anomaly in the mean of lng. For the variance of lng at the band centre, we reveal the existence of a weak anomalous quadratic term proportional to L2, besides the previously found anomaly in the linear term. At a generic band energy the variance of lng is found to be linear in L and is given by twice the mean, up to higher order corrections which are calculated. We also exhibit the offset terms independent of L in the variance, which strongly depend on reflection anomalies.

Relation between Anderson and invariant imbedding models for transport in finite chains and study of phase and delay time distributions for strong disorder

The invariant-imbedding evolution equations for the amplitude reflection and transmission coefficients of a disordered one-dimensional chain are shown to follow from the continuum limit, for weak disorder, of recursion relations between reflection (transmission) coefficients of Anderson chains of N and

Absence of localization in a disordered one-dimensional ring threaded by an Aharonov-Bohm flux.

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