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

TL;DR

This paper develops a theoretical model for step position fluctuations on crystal surfaces, showing how the variance depends on measurement length and establishing a scale where theoretical and observed distributions agree.

Contribution

It introduces the Pairwise Einstein Model for step fluctuations, predicting a Gaussian form for the Step Position Distribution and identifying a key length scale for experimental agreement.

Findings

The SPD is well approximated by a Gaussian with finite variance.

Measured variance of SPD diverges as measurement length increases.

The length scale L_W is approximately 14.2 times the correlation length, independent of step-step repulsion strength.

Abstract

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 $\Delta y$, the length of step over which it is calculated, with the measured variance diverging in the limit $\Delta y \to \infty$. As a result, a length scale $L_{\rm W}$ can be defined as the value of $\Delta y$ at which the measured and theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model indicate that $L_{\rm W} \approx 14.2 \xi_Q$, where $\xi_Q$ is the correlation length in the direction parallel to the steps, independent of the strength of the step-step repulsion. $L_{\rm W}$ 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.