Evaporation and growth of crystals: Propagation of step-density compression waves at vicinal surfaces

Abstract

We studied the step dynamics during crystal sublimation and growth in the limit of fast surface diffusion and slow kinetics of atom attachmentdetachment at the steps. For this limit we formulate a model free of the quasi-static approximation in the calculation of the adatom concentration on the terraces at the crystal surface. Such a model provides a relatively simple way to study the linear stability of a step train in a presence of step-step repulsion and an absence of destabilizing factors (as Schwoebel effect, surface electromigration etc.). The central result is that a critical velocity of the steps in the train exists which separates the stability and instability regimes. Instability occurs when the step velocity exceeds its critical valuewhere K is the step kinetic coefficient, Ω is the area of one atomic site at the surface, and the energy of step-step repulsion iswhere l is the interstep distance. Integrating numerically the equations for the time evolution of the adatom concentrations on the terraces and the equations of step motion we obtained the step trajectories. When the step velocity exceeds its critical value the plot of these trajectories manifests clear space and time periodicity (step density compression waves propagate on the vicinal surface). This ordered motion of the steps is preceded by a relatively short transition period of disordered step dynamics.

Instabilities at vicinal crystal surfaces: Competition between electromigration of adatoms and kinetic memory effect

“…[1] to account for the electromigration of the adatoms. The model is based on the following assumption -adatoms are subject of electromigration force (see Fig.1), so that they jump more frequently in the direction of this force than in the opposite direction [6,12].…”

“…In a recent paper [1] we advanced a new model for the step dynamics during sublimation and growth of a vicinal crystal surface. This model contains the same physics as the classical Burton, Cabrera, Frank (BCF) theory [2,3,4] but the mathematical treatment deviates from the BCF procedure.…”

“…We call this a "kinetic memory effect". This effect provides a ground for a new type of instability of the regular step distribution.Step density compression waves appear at the vicinal surface as shown by both linear stability analysis and numerical integration of the equations for step motion [1]. Here our aim is to see how the "kinetic memory effect" competes with another (well known) destabilizing factor -the electromigration of the adatoms [5][6][7][8][9][10][11].…”

“…This model contains the same physics as the classical Burton, Cabrera, Frank (BCF) theory [2,3,4] but the mathematical treatment deviates from the BCF procedure. In contrast to their assumption that the adatom concentrations i n on the terraces reach instantly their steady state for a given step configuration we analyzed the non-steady state problem [1]. This is relatively easy in the limiting case of fast surface diffusion and slow kinetics of atom attachment and detachment at the steps, since the fast diffusion provides for a constant value of the adatom concentration all over a given terrace.…”

Nonlinear Dynamics of Surface Steps

“…Remarkably, in this setting the equidistant step train may undergo an instability into a new dynamic phase characterized by step compression waves [67], even if it would be completely stable in the quasistatic limit. The instability is caused solely by the time delay that is introduced into the interaction between steps by the finite time scale of the adatom dynamics, similar to the instabilities induced in follow-the-leader models of highway traffic by the finite reaction time of drivers [52,53].…”