Low-Temperature Orientation Dependence of Step Stiffness on {111} Surfaces

TL;DR

This paper derives a low-temperature formula for the orientation dependence of step stiffness on {111} surfaces, showing good agreement with experiments and revealing insights into symmetry and interactions.

Contribution

It introduces a simple, low-temperature combinatoric formula for step stiffness on {111} surfaces based on the Ising model, including novel insights into symmetry and interactions.

Findings

Formula agrees with experimental data for Ag and Cu{111} surfaces.

Step line tension cannot be directly extracted from stiffness.

Low-temperature stiffness exhibits 6-fold symmetry, contrasting with crystal shape symmetry.

Abstract

For hexagonal nets, descriptive of {111} fcc surfaces, we derive from combinatoric arguments a simple, low-temperature formula for the orientation dependence of the surface step line tension and stiffness, as well as the leading correction, based on the Ising model with nearest-neighbor (NN) interactions. Our formula agrees well with experimental data for both Ag and Cu{111} surfaces, indicating that NN-interactions alone can account for the data in these cases (in contrast to results for Cu{001}). Experimentally significant corollaries of the low-temperature derivation show that the step line tension cannot be extracted from the stiffness and that with plausible assumptions the low-temperature stiffness should have 6-fold symmetry, in contrast to the 3-fold symmetry of the crystal shape. We examine Zia's exact implicit solution in detail, using numerical methods for general orientations and deriving many analytic results including explicit solutions in the two high-symmetry directions. From these exact results we rederive our simple result and explore subtle behavior near close-packed directions. To account for the 3-fold symmetry in a lattice gas model, we invoke a novel orientation-dependent trio interaction and examine its consequences.