Scaling and Dynamic Stability of Model Vicinal Surfaces

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 = 2 root T/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/root 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.