Low-temperature orientation dependence of step stiffness on {111} surfaces

Abstract

For hexagonal nets, descriptive of ͕111͖ 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 ͑NN͒ interactions. Our formula agrees well with experimental data for both Ag and Cu͕111͖ surfaces, indicating that NN interactions alone can account for the data in these cases ͑in contrast to results for Cu͕001͖͒. 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 six-fold symmetry, in contrast to the three 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 three-fold symmetry in a lattice gas model, we invoke an orientation-dependent trio interaction and examine its consequences.

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

“…where V 0 (m) ≡ V (m, ζ, ζ), and γ + ∂ 2 θ γ is known as the "step stiffness" [1,44] and expresses the inertia of the step in the presence of driving forces; see (5.2).…”

Biased Surface Fluctuations due to Current Stress

“…(1) and (4), given the step stiffness , which is calculated [40,41] using an effective kink energy of " 0:117 eV [42,43]. The four measurements of the electromigration time constant yield average values of the force per step-edge atom of ÿ2:7 10 ÿ5 eV=nm for J nom 4 10 5 A=cm 2 (325-350 K) and ÿ9:7 10 ÿ6 eV=nm for J nom 1 10 5 A=cm 2 (370 K).…”

Spontaneous magnetization of the square 2D Ising lattice with nearest- and weak next-nearest-neighbour interactions

“…In the literature, several routes to determine the boundary tension of a number of 2D and 3D lattices without crossing bonds have been put forward [7][8][9][10]. Recently, we derived an expression for the boundary tension (or boundary free energy) along the high symmetry (10) direction, F (10) , of a square 2D Ising model with crossing bonds [11].…”