Surface Science 424 ( 1999 ) L299–L308 Surface Science Letters Implications of random-matrix theory for terrace-width distributions on vicinal surfaces: improved approximations and exact results T.L. Einstein*, O. Pierre-Louis 1 Department of Physics, University of Maryland, College Park, MD 20742-4111, USA Received 20 October 1998; accepted for publication 18 December 1998 Abstract Quantitative measurement of the equilibrium terrace-width distribution of vicinal surfaces enables detailed investigation of step–step interactions. Using results from random-matrix theory, we point out simple analytical expressions that assist in this process, improving considerably over standard techniques and allowing assessment of weak repulsions, heretofore inaccessible except by indirect methods. This approach suggests new properties for experimentalists to measure and, by calibration with exact results, provides insights into controversies about assessing the interaction strength. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Equilibrium thermodynamics and statistical mechanics; Stepped single crystal surfaces; Surface structure, morphology, roughness, and topography; Surface energy; Vicinal single crystal surfaces On vicinal (stepped ) surfaces, terrace widths s kinetics and non-equilibrium responses of surfaces, can now be measured quantitatively using several such as step bunching [3,4]. Analyses typically rely different surface-sensitive real-space imaging tech- on the mapping of the associated set of configura- niques [1,2]. The resulting equilibrium terrace- tions in this two-dimensional problem onto the width distribution ( TWD) P (s) – where s is the ‘world lines’ (plots of the evolution) of non-cross- (dimensionless) ratio of s to its average value s – ing particles, thus analogous to fermions in 1+1 provides valuable information about the inter- dimensions: the along-step y ˆ direction is taken to action between steps. These interactions, in addi- be time-like. (Alternatively, the problem in 1+1D tion to the step stiffness, are crucial to determining can be recast in terms of hard bosons rather than the morphology of these surfaces [1,2]. They are fermions [5].) A crucial, viable assumption in this also vital to understanding phenomena involving mapping is that the interactions between the fermi- ons are instantaneous, i.e. only for the same value of y on the interacting steps. * Corresponding author. Fax: +1-301-314-9465; The non-crossing condition of the steps leads e-mail: einstein@physics.umd.edu. to a strongly temperature T-dependent entropic 1 Present address: Lab. Spectro. Phys., UJF (CNRS), Grenoble I, B.P. 87, 38042 Saint-Martin d’He `res, France. repulsion between them, since thermal fluctuations 0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0039-6028(99)00092-8