Adiabatically switched-on electrical bias in continuous systems, and the Landauer-Buttiker formula

TL;DR

This paper rigorously derives the Landauer-Buttiker formula for electrical conductance in a three-dimensional lead-sample system under adiabatic bias, using finite volume regularization to define current and analyze linear response.

Contribution

It provides a rigorous mathematical proof that the Landauer-Buttiker formula applies to continuous 3D systems with adiabatic bias, extending prior theoretical results.

Findings

Conductivity tensor given by Landauer-Buttiker formula in most physical situations

Finite volume regularization effectively defines current as charge derivative

Adiabatic switching of bias analyzed in a 3D multi-lead system

Abstract

Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-B{\"u}ttiker type formula.