Phase-Field Modeling of Step Dynamics on Growing Crystal Surface: Direct Integration of Growth Units to Step Front

We propose a new formulation for numerically simulating step dynamics on growing crystal surfaces in the framework of a phase-field technique. The step advancement rate is proportional to a supersaturation at the crystal surface when the growth units in the ambient phase are integrated to the step front directly (direct integration hypothesis). We conduct numerical simulations of some standard step dynamics problems: the advancement of a straight step, the growth or dissolution of a two-dimensional island, and the vertical growth of the crystal surface due to single or multiple screw dislocations. During evaluations, our phase-field model accurately calculated the rate of advancement of a straight step for various supersaturations. The calculated time variation of the radius of the two-dimensional island showed good agreement with the exact solution. The vertical growth rate due to screw dislocations qualitatively agreed with the predictions of the classical theory of Burton, Cabrera, and Frank. Our simple formulation requires only a single parabolic partial differential equation to be solved numerically. Thus, our phase-field model provides a simple numerical tool for a quantitative step-by-step trajectory calculation, when the advancing velocity of each step follows the direct integration hypothesis.