New type of step bunching instability at vicinal surfaces in crystal evaporation affected by electromigration

Abstract

A continuum model of crystal evaporation in the case of “transparent” steps (kinetics without local adatom conservation) and electromigration of the adatoms is developed. It manifests step bunching at an electromigration in step-up direction and displays a size scaling law l b∼ N − α , where l b is the average interstep distance in the bunch and N is the number of steps in it. The size scaling exponent α is shown to be the key to the problem of determining the distance dependence of the step–step repulsion from step bunching experiments. Thus the value α=1/2 corresponds to n=2 in the step–step repulsion of the type U( l)= A/ l n whereas α=2/3 corresponds to n=1 [in general α=2/( n+2)]. These values of the scaling exponent are obtained for near-to-equilibrium evaporation kinetics. In the opposite case of far-from-equilibrium evaporation one obtains α=3/5 at n=2 and α=3/4 at n=1. Besides the size scaling relation for the steady state interstep distance l b, the time evolution of the step bunching instability is studied and the average number of steps in the bunch is shown to increase with the evaporation time as N av ∼ t (the derivation of this scaling law is correct only for the later stages of the bunching process, when the terraces separating the bunches of steps have a width larger than the average diffusion distance λ s). The bunch relaxation, driven by a downhill electromigration, is treated in the framework of the proposed model and the slope S b in the core of the bunch is found to decrease with the evaporation time. The initial exponential decrease is shown to turn to S b∼ N 2/ t with the increase of the bunch width during the relaxation process.