Terrace-Width Distributions and Step-Step Repulsions on Vicinal Surfaces: Symmetries, Scaling, Simplifications, Subtleties, and Schr\"odinger

Abstract

For more than three decades, measurement of terrace width distributions (TWDs) of vicinal crystal surfaces have been recognized as arguably the best way to determine the dimensionless strength $\tilde{A}$ of the elastic repulsion between steps. For sufficiently strong repulsions, the TWD is expected to be Gaussian, with $\tilde{A}$ varying inversely with the squared variance. However, there has been a controversy over the proportionality constant. From another perspective the TWD can be described as a continuous generalized Wigner distribution (CGWD) essentially no more complicated than a Gaussian but a much better approximation at the few calibration points where exact solutions exist. This paper combines concisely the experimentally most useful results from several earlier papers on this subject and describes some advancements that are in progress regarding numerical tests and in using Schr\"odinger-equation formalism to give greater understanding of the origin of the CGWD and to give hope of extensions to more general interaction potentials between steps. There are many implications for future experiments.