Analysis of terrace-width distributions using the generalized Wigner surmise: Calibration using Monte Carlo and transfer-matrix calculations

Abstract

Measurement of terrace-width distributions (TWD's) of vicinal surfaces is used routinely to find the dimensionless strength $\~A$ of the elastic repulsion between steps. For sufficiently strong repulsions, the TWD can be described by a Gaussian about the mean step spacing, but controversy has arisen on the correct prefactor in the relation of the TWD variance to $\~A.$ Instead of the various Gaussian approximations, we have advocated for several years that the TWD be fit with the generalized Wigner distribution, essentially a gamma distribution in the normalized squared TWs. The basis for this idea stems from a mapping of the step model to the Sutherland model of fermions in one dimension. While several applications to experiment have been successful, definitive comparison of the various approximations requires high-quality numerical data. We report transfer matrix and extensive Monte Carlo simulations of terrace-step-kink models to support our contentions. Our work includes investigation of finite-size effects and of the breakdown of the continuum picture for values of $\~A$ larger than in typical experiments.