Random-matrix theory of quantum transport

Abstract

This is a review of the statistical properties of the scattering matrix of a mesoscopic system. Two geometries are contrasted: A quantum dot and a disordered wire. The quantum dot is a confined region with a chaotic classical dynamics, which is coupled to two electron reservoirs via point contacts. The disordered wire also connects to two reservoirs, either directly, or via a point contact or tunnel barrier. One of the two reservoirs may be in the superconducting state, in which case conduction involves Andreev reflection at the interface with the superconductor. In the case of the quantum dot, the distribution of the scattering matrix is Dyson's circular ensemble for ballistic point contacts, or the Poisson kernel for point contacts containing a tunnel barrier. In the case of the disordered wire, the distribution of the scattering matrix is obtained from the Dorokhov-Mello-Pereyra-Kumar equation, which is a one-dimensional scaling equation. The equivalence is discussed with the non-linear σ model, which is a supersymmetric field theory of localization. The distribution of scattering matrices is applied to a variety of physical phenomena, including universal conductance fluctuations, weak localization, Coulomb blockade, sub-Poissonian shot noise, reflectionless tunneling into a superconductor, and giant conductance oscillations in a Josephson junction. To be published in Rev. Mod. Phys. (1997).

Semiclassical prediction for shot noise in chaotic cavities

“…One lead is assumed to support N 1 ingoing channels and the second one N 2 outgoing channels. In contrast to the random-matrix treatment of [5,7] and work on quantum graphs in [8], our results cover all orders in the inverse number of channels, N = N 1 + N 2 , and thus apply to the experimentally relevant case of few channels [9]. Previously unknown and surprisingly simple expressions for the shot noise arise, both with and without time reversal invariance (see Eq.…”

“…denotes an average over a small energy interval. Previous work had involved averages over ensembles of matrices t and obtained [5,7] (1) here β = 1 refers to the so-called "orthogonal case" of time-reversal invariant dynamics; if a magnetic field is applied to break time-reversal invariance ("unitary case", β = 2), the second ("weak localization") term disappears. Higher orders in 1 N are as yet unknown.…”

“…denotes an average over a small energy interval. Previous work had involved averages over ensembles of matrices t and obtained [5,7] …”

Anderson Localization of Thermal Phonons Leads to a Thermal Conductivity Maximum

“…!" = 6.2) and anisotropy stemming from the superlattice structure are potential causes for the failure of the Thouless relationship at higher frequencies [30]. Furthermore, since GaAs reservoirs are coupled to the superlattice, some phonon modes in the ordered system will still be scattered due to contact resistance, i.e.…”

Universal scheme to generate metal–insulator transition in disordered systems

“…Our results are averaged over an ensemble of disorder configurations to reach convergence. Finally, the zero-temperature conductance is [3,37]. In the site representation, the Schrödinger equation H|Ψ = E|Ψ can be expressed as…”