Orientation-dependent surface and step energies of Pb from first principles

Abstract

The orientation-dependent surface energies of 35 low-index and vicinal Pb surface orientations, located in the [001], $[\overline{1}10]$, and $[01\overline{1}]$ zones, have been calculated by density-functional theory within the local-density approximation. The highest surface energy anisotropies in these zones are at the (210), (110), and (311) directions. Surface relaxation decreases the surface energy anisotropy significantly. For misorientations smaller than 12\ifmmode^\circ\else\textdegree\fi{} the (projected) surface energy in a given zone increases linearly with step density, while curvature is found at higher misorientations, indicative of repulsive step-step interactions. These results are fully consistent with the orientation-dependent surface energy predicted by the statistical mechanics of the terrace-step-kink model of vicinal surfaces. The step formation energies and surface and step relaxation energies are derived and analyzed. There is good agreement with available experimental data. The calculated surface energies in eV/atom correlate linearly with the number of broken surface bonds. Deviations from perfect linearity are found to be essential for a proper description of the equilibrium crystal shape of Pb.