Applications of Ideas from Random Matrix Theory to Step Distributions on "Misoriented" Surfaces

TL;DR

This paper applies random matrix theory to analyze the distribution of step spacings on tilted surfaces, revealing a generalized Wigner surmise that better describes experimental data than traditional methods.

Contribution

It introduces a novel application of RMT to surface step distributions, deriving a generalized Wigner surmise that fits experimental data more accurately than previous models.

Findings

Distribution of step spacings follows a generalized Wigner surmise.

The method outperforms Gaussian and mean-field approaches.

Application to experimental data on metals and semiconductors.

Abstract

Arising as a fluctuation phenomenon, the equilibrium distribution of meandering steps with mean separation $<\ell>$ on a "tilted" surface can be fruitfully analyzed using results from RMT. The set of step configurations in 2D can be mapped onto the world lines of spinless fermions in 1+1D using the Calogero-Sutherland model. The strength of the ("instantaneous", inverse-square) elastic repulsion between steps, in dimensionless form, is $\beta(\beta-2)/4$. The distribution of spacings $s< \ell>$ between neighboring steps (analogous to the normalized spacings of energy levels) is well described by a {\it "generalized" Wigner surmise}: $p_{\beta}(0,s) \approx a s^{\beta}\exp(-b s^2)$. The value of $\beta$ is taken to best fit the data; typically $2 \le \beta \le 10$. The procedure is superior to conventional Gaussian and mean-field approaches, and progress is being made on formal justification. Furthermore, the theoretically simpler step-step distribution function can be measured and analyzed based on exact results. Formal results and applications to experiments on metals and semiconductors are summarized, along with open questions. (conference abstract)