Effects of next-nearest-neighbor interactions on the orientation dependence of step stiffness: Reconciling theory with experiment for Cu(001)

Within the solid-on-solid (SOS) approximation, we carry out a calculation of the orientational dependence of the step stiffness on a square lattice with nearest- and next-nearest-neighbor interactions. At low temperature our result reduces to a simple, transparent expression. The effect of the strongest trio (three-site, nonpairwise) interaction can easily be incorporated by modifying the interpretation of the two pairwise energies. The work is motivated by a calculation based on nearest neighbors that underestimates the stiffness by a factor of 4 in directions away from close-packed directions, and a subsequent estimate of the stiffness in the two high-symmetry directions alone that suggested that inclusion of next-nearest-neighbor attractions could fully explain the discrepancy. As in these earlier papers, the discussion focuses on Cu(001).

Figure 1

A finite-sized step edge whose projected length is $L$. The step has height $y_i$ at position $i$ $(0⩽i⩽L)$. The height difference $y_L-y_0$ is fixed; thus, the step edge makes an angle $\theta$ with the horizontal axis, and has an overall slope $m$ (shown as the top of the gray region). The energy of the step edge is found by counting the number of broken links required to form it. Here all NN and NNN broken links are shown.

Figure 2

The range of validity of Eq. (14) is examined by comparing it to exact numerical solutions of the SOS model at several temperatures. In the legend $T_c$ refers to the NN lattice-gas (Ising) model; for $\mid {ϵ}_1\mid =256\mathrm{meV}$, $T_c=1685\mathrm{K}$.

Figure 3

Equation (14) is plotted for a variety of different values of $D=-{ϵ}_2∕2k_{B}T$, where ${ϵ}_1$ and ${ϵ}_2$ are NN- and NNN-interaction energies, respectively, in a lattice-gas picture. The solid curve denoted “Ising NN” corresponds to ${ϵ}_2=0$. The dots labeled “Exp’t” are taken from Fig. 2 of Ref. 6 and were derived from the equilibrium shape of islands on Cu(001) at $302\mathrm{K}$, with the line segments serving as guides for the eye. To minimize clutter, we omit similar data derived from correlation functions of vicinal surfaces at various temperatures. Note that for ${ϵ}_2={ϵ}_1∕4$ a maximum has developed near $\tan \theta =1∕2$ that is not evident in the experimental data.