Burton-Cabrera-Frank theory for surfaces with alternating step types

TL;DR

This paper extends Burton-Cabrera-Frank theory to surfaces with alternating step types, providing analytical solutions and fitting experimental data to determine kinetic coefficients for crystal growth.

Contribution

It develops a BCF model for surfaces with alternating step properties, including step transparency and step-step repulsion, with analytical solutions and experimental validation.

Findings

Derived explicit expressions for terrace widths and fractions.

Fitted theoretical models to GaN surface data to extract kinetic coefficients.

Connected kink diffusion models to terrace step kinetics.

Abstract

Burton-Cabrera-Frank (BCF) theory has proven to be a versatile framework to relate surface morphology and dynamics during crystal growth to the underlying mechanisms of adatom diffusion and attachment at steps. For an important class of crystal surfaces, including the basal planes of hexagonal close-packed and related systems, the steps in a sequence on a vicinal surface can exhibit properties that alternate from step to step. Here we develop BCF theory for such surfaces, relating observables such as alternating terrace widths as a function of growth conditions to the kinetic coefficients for adatom attachment at steps. We include the effects of step transparency and step-step repulsion. A general solution is obtained for the dynamics of the terrace widths, assuming quasi-steady-state adatom distributions on the terraces. An explicit simplified analytical solution is obtained under widely applicable approximations. From this we obtain expressions for the full-steady-state terrace fraction as a function of growth rate. Fits of the theoretical predictions to recent experimental determinations of the steady-state and dynamics of terrace fractions on GaN (0001) surfaces during organo-metallic vapor phase epitaxy give values of the kinetic coefficients for this system. In Appendices, we also connect a model for diffusion between kinks on steps to the model for diffusion between steps on terraces, which quantitatively relates step transparency to the kinetics of atom attachment at kinks, and consider limiting cases of diffusion-limited, attachment-limited, and mixed kinetics.