Electron transport through a circular constriction

TL;DR

This paper derives an exact formula for the electrical conductance of a circular constriction, smoothly transitioning between diffusive and ballistic regimes, providing a precise interpolation between two classical conductance models.

Contribution

It presents an exact calculation of conductance for a circular constriction, bridging diffusive and ballistic transport regimes with explicit Green's function formulation.

Findings

Conductance formula deviates less than 11% from naive interpolation.

Provides explicit Green's function for the linearized Boltzmann operator.

Interpolates smoothly between Maxwell and Sharvin conductance.

Abstract

We calculate the conductance of a circular constriction of radius $a$ in an insulating diaphragm which separates two conducting half-spaces characterized by the mean free path $\ell$. Our exact result interpolates between the Maxwell conductance in diffusive ($\ell \ll a$) and the Sharvin conductance in ballistic ($\ell \gg a$) transport regime. Following the earlier approach of Wexler we find the explicit form of the Green's function for the linearized Boltzmann operator. The formula for the conductance deviates by less than 11% from the naive interpolation formula obtained by adding resistances in the diffusive and the ballistic regime.