Quantum Theory of Transport in Narrow Channels

Abstract

The electrical conductivity in a semiconductor surface channel or a thin film is written in terms of integrals over a retarded current-current correlation function and evaluated using a Green's-function formulation of perturbation theory. The perturbation theory exhibits four new features. (1) The boundary conditions at the surfaces of the channel are expressed in terms of a fluctuation potential rather than a Fuchs reflectivity parameter. (2) The quantization of the eigenvalues for motion normal to the channel is explicitly incorporated into the theory. (3) The averaging procedure used to obtain the diagrammatic definition of the propagators and correlation functions is extended to include the effects both of screening and of graded interface impurity doping by permitting summation of multiple-scattering effects within planes of impurities parallel to the surface prior to the consideration of interference between these planes. (4) The propagators and conductivity are evaluated at arbitrary temperatures, using the Matsubara formalism. The conductivity is calculated explicitly in the quantum limit that the energy spacings $\Delta E$ between the eigenvalues for motion normal to the surface satisfy $\Delta E\gg \mathrm{kT}$ for the occupied eigenstates. The approximations needed to reproduce the Boltzmann-equation analysis by Stern and Howard of the extreme quantum limit are delineated. The effects of dispersion and quantized-state mixing are examined for a $\delta$-function model of the fluctuation potential. They are found to be significant if a quantized eigenvalue is near the Fermi energy or if the doping in the channel is highly nonuniform.