Stability of Vicinal Surfaces: Beyond the Quasistatic Approximation

Scaling laws for step bunching on vicinal surfaces: Role of the dynamical and chemical effects

We study the evolution of step bunches on vicinal surfaces using a thermodynamically consistent step-flow model. By accounting for the dynamics of adatom diffusion on terraces and attachment-detachment at steps (referred to collectively as the dynamical effect), this model circumvents the quasistatic approximation that prevails in the literature. Furthermore, it generalizes the expression of the step chemical potential by incorporating the necessary coupling between the diffusion fields on adjacent terraces (referred to as the chemical effect). Having previously shown that these dynamical and chemical effects can explain the onset of step bunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier or other extraneous mechanisms, we are here interested in the evolution of step bunches beyond the linear-stability regime. In particular, the numerical resolution of the step-flow free boundary problem yields a robust power-law coarsening of the surface profile, with the bunch height growing in time as H∼t^{1/2} and the minimal interstep distance as a function of the number of steps in the bunch cell obeying ℓ_{min}∼N^{-2/3}. Although these exponents have previously been reported, the novelty of the present approach is that these scaling laws are obtained in the absence of an iES barrier or adatom electromigration. In order to validate our simulations, we take the continuum limit of the discrete step-flow system via Taylor expansions with respect to the terrace size, leading to a novel nonlinear evolution equation for the surface height. We investigate the existence of self-similar solutions of this equation and confirm the 1/2 coarsening exponent obtained numerically for H. We highlight the influence of the combined dynamical-chemical effect and show that it can be interpreted as an effective iES barrier in the setting of the standard Burton-Cabrera-Frank theory. Finally, we use a Padé approximant to derive an analytical expression for the velocity of steadily moving step bunches and compare it to numerical simulations.

In situ microbeam surface X-ray scattering reveals alternating step kinetics during crystal growth

The stacking sequence of hexagonal close-packed and related crystals typically results in steps on vicinal {0001} surfaces that have alternating A and B structures with different growth kinetics. However, because it is difficult to experimentally identify which step has the A or B structure, it has not been possible to determine which has faster adatom attachment kinetics. Here we show that in situ microbeam surface X-ray scattering can determine whether A or B steps have faster kinetics under specific growth conditions. We demonstrate this for organo-metallic vapor phase epitaxy of (0001) GaN. X-ray measurements performed during growth find that the average width of terraces above A steps increases with growth rate, indicating that attachment rate constants are higher for A steps, in contrast to most predictions. Our results have direct implications for understanding the atomic-scale mechanisms of GaN growth and can be applied to a wide variety of related crystals.