Temperature-dependent surface relaxations of Ag(111)

The temperature-dependent surface relaxation of Ag(111) is calculated by density-functional theory. At a given temperature, the equilibrium geometry is determined by minimizing the Helmholtz free energy within the quasiharmonic approximation. To this end, phonon dispersions all over the Brillouin zone are determined from density-functional perturbation theory. We find that the top-layer relaxation of Ag(111) changes from an inward contraction $(-0.8\%)$ to an outward expansion $\mathrm{}(+6.3\%)\mathrm{}$ as the temperature increases from $T=0\mathrm{}\mathrm{K}$ to 1150 K, in agreement with experimental findings. Also, the calculated surface phonon dispersion curves at room temperature are in good agreement with helium-scattering measurements. The mechanism driving this surface expansion is analyzed, and the physical picture developed by Narasimhan and Scheffler is essentially confirmed.