  Inhomogeneous dynamical mean-field theory is employed to calculate the
vertex-corrected electronic charge transport for multilayered devices composed
of semi-infinite metallic lead layers coupled through a strongly correlated
material barrier region. The barrier region can be tuned from a metal to a Mott
insulator through adjusting the interaction strength and the particle filling.
We use the Falicov-Kimball model to describe the barrier region because an
exact expression for the vertex corrections is known, allowing us to determine
their effect on transport. The dc conductivity is calculated and we find the
effects of the vertex corrections are relatively small, manifesting themselves
in a small reduction in the resistance-area product. This reduction saturates
in absolute magnitude as the barrier layer becomes thick, as expected due to
the vanishing nature of the vertex corrections in bulk. The vertex corrections
have a larger relative effect on the resistance-area product for more metallic
and thinner devices.
