Generalization of the Landauer conductance formula

We study the electrical current transport in conductor-insulator-conductor structures, where the charge carriers are assumed to traverse the insulating layer by tunneling.
The current flow in the conductor contacts is treated by solving self-consistently the Poisson equation and the relaxation time form of the Boltzmann equation for an inhomogeneous electron system. The tunneling is taken into account by the appropriate boundary conditions for the electronic distribution functions in the two contacts. It is shown that the tunneling current density consists of two contributions, the first of which is a direct generalization of the celebrated Landauer conductance formula into the nonlinear voltage regime.
The second contribution or the correction term originates from the screening of the electrical potential across the insulating layer and from the matching of the distribution functions on the opposite sides of the barrier. In the linear voltage regime for well-conducting contacts the tunneling current density is given by the ordinary Landauer result, but for semiconducting contacts the second contribution may become comparable to the first one.
Moreover, it is shown that in the correction term the matching of the distribution functions is always negligible when compared to the screening effect. Finally, the limits of validity of our results are discussed.