Resistivity scaling due to electron surface scattering in thin metal layers

The effect of electron surface scattering on the thickness-dependent electrical resistivity \ensuremath{\rho} of thin metal layers is investigated using nonequilibrium Green's function density functional transport simulations. Cu(001) thin films with thickness $d=1\ensuremath{-}2$ nm are used as a model system, employing a random one-monolayer-high surface roughness and frozen phonons to cause surface and bulk scattering, respectively. The zero-temperature resistivity increases from $9.7\ifmmode\pm\else\textpm\fi{}1.0\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$ at $d=1.99$ nm to $18.7\ifmmode\pm\else\textpm\fi{}2.6\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$ at $d=0.90$ nm, contradicting the asymptotic $T=0$ prediction from the classical Fuchs-Sondheimer model. At $T=900$ K, $\ensuremath{\rho}=5.8\ifmmode\pm\else\textpm\fi{}0.1\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$ for bulk Cu and $\ensuremath{\rho}=13.4\ifmmode\pm\else\textpm\fi{}1.1$ and $22.5\ifmmode\pm\else\textpm\fi{}2.4\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$ for layers with $d=1.99$ and 0.90 nm, respectively, indicating an approximately additive phonon contribution which, however, is smaller than for bulk Cu or atomically smooth layers. The overall data indicate that the resistivity contribution from surface scattering is temperature-independent and proportional to $1/d$, suggesting that it can be described using a surface-scattering mean-free path ${\ensuremath{\lambda}}_{s}$ for 2D transport which is channel-independent and proportional to $d$. Data fitting indicates ${\ensuremath{\lambda}}_{s}=4\ifmmode\times\else\texttimes\fi{}d$ for the particular simulated Cu(001) surfaces with a one-monolayer-high surface roughness. The $1/d$ dependence deviates considerably from previous $1/{d}^{2}$ predictions from quantum models, indicating that the small-roughness approximation in these models is not applicable to very thin ($<2$ nm) layers, where the surface roughness is a considerable fraction of $d$.