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

Abstract

coefficient and F is a force acting on the adatoms ( F is related to the electric current heating the crystal ). In the limit of fast surface diffusion and slow kinetics of atom attachment-detachment at the steps we formulate a model free of the quasi-static approximation in the calculation of the adatom concentration on the terraces. The linear stability analysis of a step train results in an instability condition in the formτ is the dimensionless life-time of an adatom before desorption, f and η are dimensionless electromigration force and the force of step repulsion whereas V and cr V are the velocity of steps in the train and the critical velocity respectively. As seen instability is expected when either the velocity V is larger than cr V ( this instability is related to the "kinetic memory effect" ), or steps. Numerical integration of the equations for the time evolution of the adatom concentrations and the equations of step motion reveals two different step bunching instabilities: 1) step density waves (small bunches which do not manifest any coarsening) induced by the kinetic memory effect and 2) step bunching with coarsening when the dynamics is dominated by the electromigration. The model developed in this paper also provides very instructive illustrations of the Popkov-Krug dynamical phase transition during sublimation and growth of a vicinal crystal surface. IntroductionIn 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. 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. We derived equations for the time evolution of the adatom concentration on the terraces and, also, equations for the step motion. In this way we were able to treat accurately the case when the time to reach steady state concentration of adatoms on the terraces is compatible with the time for nonnegligible change of the step configuration. In such a situation a new effect becomes important -the adatom concentration on a given terrace depends not only on the terrace size but on the "past of the terrace" as well. 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 -th...

Step bunching process induced by the flow of steps at the sublimated crystal surface

“…In most analytical models step movement factor is neglected 8,10-15 . However as it was discussed lately particle advection is in fact comparable with electromigration or Schwoebel barrier asymmetry 10,[34][35][36] . Results of our Monte Carlo (MC) simulations show that the step bunching happens at the ideal stepped surface with Schwoebel barrier assumed to be zero and no external particle driving present.…”

“…Surface evolution is an effect of individual particle jumps and as a result steps join together building bunches at the surface. It was shown that bunching can be caused by Schwoebel barriers 32 , diffusion at step edge 33 , impurities 28 or driven particle flow [8][9][10]34 . In most analytical models step movement factor is neglected 8,10-15 .…”

“…Depending on the assumed parameters step meandering, bunching or surface roughening is observed 8 . Results of many works devoted to analysis of both process show variety of possible mechanisms and scenario of surface pattern evolution [8][9][10][11][12][13][14][15][16][17][18] . We show that on annealing of quite simple (0001) vicinal face of the GaN crystal steps at the surface have a tendency of bunching rather than meandering.…”

Nonlinear Dynamics of Surface Steps

“…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]. In the presence of electromigration and sublimation, the non-quasistatic model reproduces the main features of the phase transition described above in Sect.1.3.4 [68].…”

Influence of electromigration field on the step bunching process on Si(111)

“…This observation is in agreement with the results of recent numerical modeling of the step bunching process which theoretically predicted the existence of a critical electromigration force as well as the formation of step density waves with relatively weak or zero electromigration. 16,17 These new models extend beyond the quasistatic approximation of earlier models 8,9 and take into account the movement of atomic steps as well as a different rate of adatom surface diffusion, step crossing, and attachment-detachment at the steps. 16,17 However, the formation of step density waves in these simulations 16 is observed at an electromigration force which is two orders of magnitude larger than the F cr = 2.7 ϫ 10 −18 N deduced in our experiment on the assumption that effective charge q eff of a Si adatom is equal to 1/3 of the elementary electric charge.…”