Lattice-Gas Modeling of Stiffness on fcc(111) Surfaces: General Results

In a nearest-neighbor (NN) lattice-gas model of the island atoms on a (111) fcc surface,\footnote{T. J. Stasevich, Hailu Gebremariam, T. L. Einstein, M. Giesen, C. Steimer, and H. Ibach, submitted to PRB [cond-mat/0412002].} the entropy of the leading term in the low-temperature expansion of the orientation-dependent free energy—relevant for understanding experiments on noble metals--comes exclusively from geometric considerations.\footnote{ C. Rottman and M. Wortis, PRB {\bf 24}, 6274 (1981).} There are several remarkable consequences:$^2$ 1) The lowest-order stiffness (LOS) has no contribution from the energy. 2) Therefore, the step line tension cannot be extracted from the LOS. 3) The LOS has 6-fold symmetry, even though the line tension has only 3-fold symmetry. 4) The reduced LOS has the strikingly simple form $2\surd 3/\sin(3\theta)$. Near close-packed orientations, the LOS is not adequate at these temperatures; by explicating the exact implicit solution for the hexagonal lattice gas,\footnote{R.K.P. Zia, J. Stat. Phys. {\bf 45}, 801 (1986).} we provide quantitative markers for where this breakdown occurs. Lastly, we show how to account for the energy difference between A and B steps by invoking a novel orientation-dependent trio interaction between atoms forming an equilateral triangle with NN legs.