  A realistic tight-binding model is developed and employed to elucidate the
resistivity size effect due to steps on Ru thin films. The resistivity of two
different film orientations, $(0001)$ and $(1 \bar{1}00)$, is computed for
transport along a $[1 1 \bar{2} 0]$ direction both for smooth surfaces and for
surfaces with monolayer-high steps. In the case of smooth films, the systems
are also studied using solutions to the Boltzmann transport equation (BTE).
Interestingly, the resistivity of $(1 \bar{1}00)$ surfaces exhibits a
significant size effect even in the absence of surface steps. When
monolayer-high steps are spaced $\sim 10$ nm apart, the resistivity is shown to
increase due to scattering from the steps. However, only a small increase was
found which cannot explain the large effect seen in recent experiments with Ru
thin films. This highlights the need for further elucidation of the resistivity
size effect. Theoretical analysis suggest that films made from materials with a
relatively large ballistic conductance per area like Ru should exhibit a
reduced resistivity size effect. This result points to Ru as a promising
interconnect material. Finally, because a very efficient algorithm for
computing resistivity based on the kernel polynomial method (KPM) is used, the
approach fulfills a need for realistic models that can span length scales
directly relevant to experimental results. The calculations described here
include films approaching $5$ nm in thickness, with in-plane distances up to
$\sim 160$ nm and $3.8\times10^{5}$ atomic sites.
