Step Position Distributions and the Pairwise Einstein Model for Steps on Crystal Surfaces

The Pairwise Einstein Model (PEM) of steps not only justifies the use of the Generalized Wigner Distribution (GWD) for Terrace Width Distributions (TWDs), it also predicts a specific form for the Step Position Distribution (SPD), i.e., the probability density function for the fluctuations of a step about its average position. The predicted form of the SPD is well approximated by a Gaussian with a finite variance. However, the variance of the SPD measured from either real surfaces or Monte Carlo simulations depends on $\mathrm{\Delta }y$, the length of step over which it is calculated, with the measured variance diverging in the limit $\mathrm{\Delta }y\to \mathrm{\infty }$. As a result, a length scale $L_{\mathrm{W}}$ can be defined as the value of $\mathrm{\Delta }y$ at which the measured and theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model indicate that $L_{\mathrm{W}}\approx 14.2{\xi }_Q$, where ${\xi }_Q$ is the correlation length in the direction parallel to the steps, independent of the strength of the step-step repulsion. $L_{\mathrm{W}}$ can also be understood as the length over which a {\em single} terrace must be sampled for the TWD to bear a “reasonable” resemblence to the GWD.