Terrace Width Distribution and First Passage Probabilities for Interacting Steps

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Date
2005-12-05Author
Bantu, Hailu Gebremariam
Advisor
Einstein, Theodore L
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Show full item recordAbstract
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.