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

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)

7 pages, 2 figures; based on talk presented at TH-2002, UNESCO, Paris, July 2002; to be published in Ann. Henri Poincare'

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