Step repulsions

The relaxation of isolated, pairs of and ensembles of steps on crystal surfaces towards equilibrium is reviewed, for systems both above and below the roughening transition temperature. Results of Monte Carlo simulations are discussed, together with analytic theories and experimental findings. Elementary dynaniical processes are, below roughening, step fluctuations, step-step repulsion and annihilation of steps. Evaporation kinetics arid surface diffusion are considered. 

A series or bunch of m initially straight and parallel steps, between heights 0 and m, may be expected to relax with the same asymptotics as a pair of steps. Modifications may occur, already for a pair of steps, when step-step interactions are present in addition to the entropic step repulsion. Here, we merely refer to recent reviews on experiments and theoretical analyses " on the much studied phenomenon of step bunching for vicinal surfaces, which is accompanied by interesting phase transistions. 

Additional insight may be gained by considering the time scales introduced by the meandering of isolated top steps, by the step-step repulsion stemming from the steps below and by the top step annihilation. Indeed, in the evaporation case, the top step annihilation seems to determine the scaling behavior, with the step-step repulsion acting on the same time scale. 

To proceed, we must describe the effective driving force and the effective interactions between steps on this mesoscopic scale. We focus here on two cases of recent experimental and theoretical interest current-induced step bunching on Si( 111) surfaces - " and reconstruction-induced faceting as seen a number of systems including the O/Ag(110) and Si(lll) surfaces" In both cases interesting 2D step patterns can arise from the competition between a driving force that promotes step bunching, and the effects of step repulsions, which tend to keep steps uniformly spaced. 

TREATMENT OE STEP REPULSIONS AND ELUCTUATIONS 2.1 Physical origin... 

In the ID limit, Eqs. (7) and (8) and related equations have been used to analyze the relaxation of non-equilibrium step profile - and in a variety of other application We will not review this work here, but instead turn directly to two cases where characteristic 2D step patterns and step bunching are found as a result of the competition between the step repulsions and a driving force favoring step bunching. Perhaps the simplest application arises as a result of surface reconstmction. 

We adopt the following simple picture. Initially, we assume that steps are far enough apart that the effects of step repulsions can be ignored. The relevant physics for evaporation involves the detachment of adatoms from step edges, their surface diffusion on the adjacent terraces, and their eventual evaporation. This is quite well described by a generalization of the classical BCF model "- which considers solutions to the adatom diffusion equation with boundary conditions at the step edges. 

Direct step-step interaction terms in the step energy ( direct interactions are entropic repulsion, strain terms, electronic structure effects etc.) do influence the step fluctuations, and they also drive the spreading of step trains, wires and bumps. Nevertheless, it is instructive to first ignore these direcf step-step repulsion, as is done in... 

Deposition of Au onto this surface leads to the nucleation of Au islands at the intersection of clean Cu stripes thus leading to a square island lattice with a period of 50 A [83,86-88]. The N-covered Cu(100) surface has also been used for the growth of so far less well-ordered lattices of Fe and Cu [89], Co [90-92], Ag [93,94], and Ni [95], We note that square lattices can in principle also be created on Au(f4,f5,f5) since this miscut leads to 70 A step distance, which is equal to the reconstruction period. However, the steps are already far apart reaching the limit of the elastic step repulsions which may render global order difficult. Finally we note that another interesting alternative square template, although with smaller lattice constant, is presented by the (3 /3 x 5)-phase of V-oxide on Rh(lll) [96]. 

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