LANDAUER CONDUCTANCE IN THE DIFFUSIVE REGIME

We present a derivation of the de Landauer conductance formula in thre e dimensions starting from a mean-field approximation of the Kubo form ula using a Green function for electron propagation that contains a se lf-energy that may be complex. Unlike some previous derivations of the Landauer formula from the Kubo formula, we find a nonzero contributio n to the current from the volume integral over the leads, even without the mean-field approximation. Because we use a complex self-energy to describe the sample, the sum of the reflection and the transmission p robabilities is in general less than one. In the four-probe case the c onductance of a sample with length L, electron mean free path I, trans mission amplitude t(k(parallel to)), and reflection amplitude r(k(para llel to)) is given by (e(2)/h)2N Sigma(k parallel to)[1-\r\(2)-(1-\r\( 2)-\t\(2))C]/Sigma(k parallel to)(1+\r\(2)-\t\(2)) for each spin, wher e N is the number of channels and C=1/(1-e(-L/lcos theta))-lcos theta/ L with cos theta=\k(parallel to)\/k(F). In the diffusive limit L much greater than l we obtain the three-dimensional Boltzmann solution. In the ballistic limit L much less than l we obtain one-dimensional Boltz mann solutions for N independent channels. If one applies the multipro be one-dimensional Buttiker formula to a system where R+T less than or equal to 1, one reproduces our result for a single channel.