Surface energy and stress release by layer relaxation

Abstract

The surface energy $(\ensuremath{\gamma})$ and surface stress $(\ensuremath{\tau})$ for semi-infinite close-packed surfaces of $4d$ transition metals have been calculated using ab initio total-energy methods. The moderate agreement between the present and former theoretical data for $\ensuremath{\tau}$ indicates the high level of numerical difficulty associated with such calculations. For the most close-packed surfaces, the present unrelaxed $\ensuremath{\tau}$ values follow the typical trend characteristic for the cohesive energy in nonmagnetic transition-metal series, whereas the relaxed $\ensuremath{\tau}$ values group around $\ensuremath{\sim}1\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕{\mathrm{m}}^{2}$, obtained for Y, Zr, and Ag, and $\ensuremath{\sim}3\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}∕{\mathrm{m}}^{2}$, calculated for Nb, Mo, Tc, Ru, Rh, and Pd. We have found that the average surface energy reduction upon layer relaxation is around 4%. At the same time, a large part of the surface stress is released during the surface relaxation process. To explain the observed behaviors, we have established a simple relationship, which connects the variations of $\ensuremath{\gamma}$ and $\ensuremath{\tau}$ to the layer relaxation. This relation reveals the principal factors determining the difference between the surface energy and stress release rates at $4d$ transition-metal surfaces.