Extraction of Step-Repulsion Strengths from Terrace Width Distributions: Statistical and Analytic Considerations

TL;DR

This paper refines the statistical analysis of terrace width distributions on vicinal surfaces, extending the generalized Wigner distribution to better extract step-repulsion strengths, especially under weak interactions, using analytical and numerical methods.

Contribution

It introduces a two-parameter extension of the generalized Wigner distribution for more accurate step-repulsion analysis from terrace width data.

Findings

The extended distribution improves fit for weak repulsions.

Correlations significantly reduce independent measurements in STM images.

Discreteness effects can bias interaction estimates at high misorientation.

Abstract

Recently it has been recognized that the so-called generalized Wigner distribution may provide at least as good a description of terrace width distributions (TWDs) on vicinal surfaces as the standard Gaussian fit and is particularly applicable for weak repulsions between steps, where the latter fails. Subsequent applications to vicinal copper surfaces at various temperatures confirmed the serviceability of the new analysis procedure but raised some theoretical questions. Here we address these issues using analytical, numerical, and statistical methods. We propose an extension of the generalized Wigner distribution to a two-parameter fit that allows the terrace widths to be scaled by an optimal effective mean width. We discuss quantitatively the approach of a Wigner distribution to a Gaussian form for strong repulsions, how errors in normalization or mean affect the deduced interaction, and how optimally to extract the interaction from the variance and mean of the TWD. We show that correlations reduce by two orders of magnitude the number of {\em independent} measurements in a typical STM image. We also discuss the effect of the discreteness ("quantization") of terrace widths, finding that for high misorientation (small mean width) the standard continuum analysis gives faulty estimates of step interactions.