Kinetic Pattern Formation at Solid Surfaces

Introduction to Step Dynamics and Step Instabilities

“…One finds, in particular, that the minimal terrace size l min inside the step bunch (the inverse of the maximal slope) scales with the number of steps N in the bunch as l min ∼ N −2/3 , which is close to the behavior observed in experiments [87,88]. At present it is not clear to what extent continuum equations like (5.5) are capable of also describing the time-dependent behavior of step bunches [62]. Equation (5.5) is a representative of a larger class of evolution equations which have been proposed within a general classification scheme for step bunching phenomena [92].…”

“…Putting together the destabilizing and stabilizing effects, the resulting surface evolution equation takes the general form where the coefficients A and B are positive, and the constant on the right hand side is the deposition flux. The same evolution equation has been obtained for step bunching induced by a normal ES effect during sublimation [62] and for electromigration-induced step bunching in the limit of slow attachment/detachment kinetics (l ± ≫ l) [83]. It applies whenever the destabilizing part of the step dynamical equations, which is responsible for the step bunching instability, becomes linear in the step spacings [as in (5.1)].…”

“…Additional complications, not discussed here, arise when the current direction is varied continuously relative to the orientation of the steps [62,80,81].…”

“…Some of the bunch properties can be derived analytically by passing from the discrete step dynamics to a continuum description of the one-dimensional height profile h(y, t) perpendicular to the steps [62]. The continuum limit is rather straightforward for the case of step flow growth, because the creation and annihilation of steps by island nucleation and coalescence does not have to be taken into account.…”

“…Clearly y j (t) can be replaced by a continuous function y(h, t), where h = −aj (compare to Fig.5), by simply expanding the difference on the right hand side of (5.1). The right hand side of this equation is naturally interpreted as a conservation law with a growth-induced current [3,4,55,62,63], which is inversely proportional to the surface slope and directed downhill (recall that ∂h/∂y < 0). The step-step interaction terms arising from (3.9) can be handled in a similar way [90].…”