Terrace Width Distribution and First Passage Probabilities for Interacting Steps

Abstract

Stochastic behavior of steps and inter-step distance is studied using Monte Carlo simulation. Terrace-step-kink model is used to represent vicinal surfaces. These vicinal surfaces consist of steps and the space between the steps called terraces.

In the first part, the distribution of the width of the terraces and its relation with the strength of step-step interaction is studied. Step positions on vicinal surfaces can be mapped into the world line of fermionic particles in one dimension. The distribution of the inter-particle distance in one dimension is in turn related to the distribution of energy levels one obtains from Random Matrix theory. The energy level distribution in Random Matrix theory is nicely approximated by Wigner distribution for three symmetries described by three parameters. These parameters correspond to the step-step interaction strength in vicinal surfaces. However, when we consider vicinal surfaces the three values of step-step interaction strength are not special. Therefore, they are generalized to include all interaction strengths and it is called the generalized Wigner distribution. The Monte Carlo simulation results show that the generalized Wigner distribution is a very accurate description for the terrace width distribution.

Analytical and simulation results of study of the evolution of the variance of the terrace width distribution for different physically interesting and experimentally testable situations are also presented. The analytical result is based on Fokker-Planck formalism obtained from the mapping of the vicinal surfaces into one-dimensional spinless fermionic particles.

In the second part, we present the study of the effect of step-step interaction on several scaling laws one obtains from the Langevin formalism of step fluctuations. Based on the limiting processes responsible for fluctuations of isolated step, the mechanisms are divided into three universality classes: attachment-detachment, step-edge diffusion and terrace diffusion. Using Monte Carlo simulation of an attachment-detachment type process, we show that the scaling laws for width of fluctuation, correlation time and survival probabilities are affected by interaction of steps. In contradiction to what one expects from the analytical results obtained using the Gruber-Mullins picture, We also show that the correlation time increases with interaction strength.