Relation between conductivity and transmission matrix

Abstract

The dc conductance $\ensuremath{\Gamma}$ of a finite system with static disorder is related to its transmission matrix $t$ by the simple relation $\ensuremath{\Gamma}=(\frac{{e}^{2}}{2\ensuremath{\pi}\ensuremath{\hbar}}) \mathrm{Tr}({t}^{\ifmmode\dagger\else\textdagger\fi{}}t)$. This relation is derived from the Kubo formula and is valid for any number of scattering channels with or without time-reversal symmetry. Differences between various definitions of the conductance of a finite system are discussed.