  Landauer's formula relates the conductance of a quantum wire or interface to
transmission probabilities. Total transmission probabilities are frequently
calculated using Green function techniques and an expression first derived by
Caroli. Alternatively, partial transmission probabilities can be calculated
from the scattering wave functions that are obtained by matching the wave
functions in the scattering region to the Bloch modes of ideal bulk leads. An
elegant technique for doing this, formulated originally by Ando, is here
generalized to any Hamiltonian that can be represented in tight-binding form. A
more compact expression for the transmission matrix elements is derived and it
is shown how all the Green function results can be derived from the mode
matching technique. We illustrate this for a simple model which can be studied
analytically, and for an Fe|vacuum|Fe tunnel junction which we study using
first-principles calculations.
