Electronic transport through a planar defect in the bulk

The extra resistivity ρₑₓₜᵣₐ due to a planar perturbation in an otherwise homogeneous bulk medium is considered. The current density incident on the barrier and the density distribution in the bulk are calculated self-consistently as a solution of a classical diffusion problem. Numerical calculations show fluctuations of the density within a few bulk mean free paths in the vicinity of the barrier. They depend sensitively on the transmission and reflection coefficients. Using the Einstein equivalence, we obtain ρₑₓₜᵣₐ from the long-range density drop across the barrier. The occurrence of the density fluctuations generally prevents the derivation of a simple generalization of the one-dimensional Landauer formula. Instead, a more involved expression for ρₑₓₜᵣₐ is found in a simplified model of discrete angles of carrier motion.