Step bunching

Another effect of elastic deformation is that it causes a long-range interaction between steps. From the continuum elasticity theory, two steps sepa-rated by a distance have a repulsive interaction proportional to l for homo- and to In i for hetero-epitaxial cases, respectively [84]. This interaction plays an important role, for example, in step fluctuations, terrace width distribution, step bunching, and so forth [7,85-88]. 

If spiral growth occurs due to the existence of screw dislocations, the results depend upon whether the diffusion length ijy is smaller or larger than the typical separation of the spiral arms i. In the first case the situation hardly changes from the purely kinetic situation without diffusion, but in the second case interaction between steps comes into effect [90] and phenomena such as step bunching [91] take place. We can estimate qualitatively the... 

S. R. Coriell, B. T. Murray, A. A. Chernov, G. B. McFadden. Step bunching on a vicinal face of a crystal growing in a flowing solution. J Cryst Growth 169 773, 1996. 

As a theoretical check of these ideas, Bartelteta/. (1994a) created in an SOS model a step bunch of 5 steps by initially confining them to half the lattice [in the k direction], then watching them evolve to nearly uniform spacing. There was no energetic interaction between... 

Fig. 2. Time evolution of the average position of steps in a step bunch relaxing back to their equilibrium distribution. The fluctuating lines are generated by Monte Carlo simulation, while the smooth curves come from the theory of Rettori and Villain (1988). From Bartelt et al. (1994a), with permission.

Multiple height step bunching, illustrated schematically in Figure 7... 

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. 

In this paper we review some of our recent work on the dynamics of step bunching and faceting on vicinal surfaces below the roughening temperature, concentrating on several cases where interesting two dimensional (2D) step patterns form as a result of kinetic processes. We show that they can be understood from a unified point of view based on an approximate but physically motivated extension to 2D of the kind of ID step models studied by a number of workers. For some early examples, see refs. [1-5]. We have tried to make the conceptual and physical foundations of our own approach clear, but have made no attempt to provide a comprehensive review of work in this active area. More general discussions from a similar perspective and a guide to the literature can be found in recent reviews by Williams and Williams and BartelF. 

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. 

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. 

Figure 1 Free energies for unreconstructed surface f, and reconstructed surface f, vs slope s. The critical slope, sc, and the slope of the surface at step bunches, St, are given by Sc = ,/ , and St = The thick curve in (a) represents the free energy of a hypothetical system... 

If thermal fluctuations were taken into account, the regular patterns selected by this kinetic mechanism would be expected to be less sharp. In particular, when wjwa, is not so small, the effects of mass conservation are spread out over many terraces and several terraces in front of the step bunch become larger than These would be particularly advantageous sites where thennal nucleation could occur, even before the induced width of the terrace as predicted by the deterministic models would exceed Wc. Thus nucleation sites and times are less precisely determined in this case, and we... 

Figure 3. The diffusion processes contributing to the fluctuations of a step edge in a step bunch.

These quantities are now independent for a uniform step train. One feature to note is that the 0(j ) term is missing for an infinite step train. It is oiily present in the steps at the beginning and end of a step bunch, or equivalently, if di. Thus the... 

When we observe the process of advancement of elemental spiral steps in situ, it is often noticed that two steps bunch together to form a step with the height of two layers as they advance. The advancing rate of the bunched layer is retarded... 

---