We propose an integrated modeling approach to the fundamental problem of vicinal crystal surfaces destabilized by step-down (SD) and step-up (SU) currents with focus on both the initial and the intermediate stages of the process. We reproduce and analyze quantitatively the step bunching (SB) instability, caused by the two opposite drift directions in the two situations of step motion mediating sublimation and growth. For this reason we develop further our atomistic scale model (vicCA) of vicinal crystal growth (Gr) destabilized by SD drift of the adatoms in order to account for also the vicinal crystal sublimation (Sbl) and the SU drift of the adatoms as an alternative mode of destabilization. For each of the four possible casesGr + SD, Gr + SU, Sbl + SD, Sbl + SU, we find a self-similar solutionthe time-scaling of the number of steps in the bunch N, = N T 2 /3, where T is the time, rescaled with a combination of model parameters. In order to study systematically the emergence of the instability, we use N further as a measure and probe the model's stability against SB on a dense grid of points in the parameter space. Stability diagrams are obtained, based on simulations running to fixed moderate rescaled times and with small-size systems. We confirm the value of the numerical prefactor in the time scaling of N, 2/ 3 by results obtained from systems of ordinary differential equations for the step velocity that contain, in contrast to vicCA, step−step repulsions. This last part of our study provides also the possibility to distinguish between diffusion-limited and kinetic-limited versions of the step bunching phenomenon.

Abstract. We devise a new 1D atomistic scale model of vicinal growth based on Cellular Automaton. In it the step motion is realized by executing the automaton rule prescribing how adatoms incorporate into the vicinal crystal. Time increases after each rule execution and then n DS diffusional updates of the adatoms are performed. The increase of n DS switches between the diffusion-limited (DL, n DS =1) and kinetics-limited (KL, n DS >> 1) regimes of growth. We study the unstable step motion by employing two alternative sources of instability -biased diffusion and infinite inverse Ehrlich-Schwoebel barrier (iiSE). The resulting step bunches consist of steps but also of macrosteps since there is no step-step repulsion incorporated explicitly into the model. This complex pattern formation is quantified by studying the time evolution of the bunch size N and macrostep size N m in order to find the proper parameter combinations that rescale the time and thus to obtain the full time-scaling relations including the pre-factors. For the case of biased diffusion the time-scaling exponent β of N is 1/2 while for the case of iiSE it is 1/3. In both cases the time-scaling exponent β m of N m is ~3β/4 in the DL regime and 3β/5 in the KL one.

The bunching of steps at the surface of growing and sublimating crystals can be induced by both directions of the electromigration force acting on the adatoms, step-up and stepdown. The processes happen in different intervals of the adatom concentration and differ in character. In this modeling study, we show that the overall picture of the bunching process changes entirely when steps cannot overlap, thus forming macrosteps. The step−step exclusion rule that forbids the steps to coalesce is introduced in our atomistic scale model of vicinal crystal growth, and we show how it distinguishes between the step-up and step-down directions of the destabilizing diffusional bias; the model surface destabilized by the step-up driving force smoothens, while the surface destabilized by the step-down driving force changes its character from vertical (01)-faceted macrosteps toward regular (11)-faceted bunches. We also clearly demonstrate how the step− step exclusion switches on another second length scale next to the bunch height, that is the bunch width, instead of the macrostep height. The bunch profiles, stability diagrams, and time-scaling dependences of the bunch properties are heavily affected by the step−step exclusion, and we quantify this effect by showing how it changes the overall scaling behavior of the phenomenon. Making step−step exclusion conditional by introducing a probability for exclusion brings additional features, and a new characteristic time scale emerges as an effect of the interplay between (01)-faceted macrosteps and (11)-faceted bunches.

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We report for the first time the observation of bunching of monoatomic steps on vicinal W(110) surfaces induced by step up or step down currents across the steps. Measurements reveal that the size scaling exponent γ, connecting the maximal slope of a bunch with its height, differs depending on the current direction. We provide a numerical perspective by using an atomistic scale model with a conserved surface flux to mimic experimental conditions, and also for the first time show that there is an interval of parameters in which the vicinal surface is unstable against step bunching for both directions of the adatom drift.