Terrace-width distributions on vicinal surfaces: generalized Wigner surmise and extraction of step–step repulsions

From quantitative measurement of the equilibrium terrace-width (ℓ) distribution (TWD) of vicinal surfaces, one can assess the strength A of elastic step–step repulsions A/ℓ2. Generally the TWD depends only on Ã=A×(step stiffness)/(kBT)2. From ideas of fluctuation phenomena, TWDs should be describable by the “generalized Wigner distribution” (GWD), essentially a power-law in ℓ/〈ℓ〉 times a “Gaussian decay” in ℓ/〈ℓ〉. The power-law exponent is related simply to Ã. Alternatively, the GWD gives the exact solution for a mean-field approximation. The GWD provides at least as good a description of TWDs as the standard fit to a Gaussian (centered at 〈ℓ〉). It works well for weak elastic repulsion strengths à (where Gaussians fail), as illustrated explicitly for vicinal Pt(1 1 0). Application to vicinal copper surfaces confirms the viability of the GWD analysis. The GWD can be treated as a two-parameter fit by scaling ℓ using an adjustable characteristic width. With Monte Carlo and transfer-matrix calculations, we show that for physical values of Ã, the GWD provides a better overall estimate than the Gaussian models. We quantify how a GWD approaches a Gaussian for large à and present a convenient, accurate expression relating the variance of the TWD to Ã. We describe how discreteness of terrace widths impacts the standard continuum analysis.