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 organometallic vapor phase epitaxy give values of the kinetic coefficients for this system. In Appendixes, 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.
Highlights
The atomic-scale mechanisms of crystal growth are often described within the framework of Burton-Cabrera-Frank (BCF) theory [1,2,3,4,5], in which deposited adatoms diffuse on top of the exposed atomic layers of the crystal surface, until they either attach to existing steps at terrace edges, join together to nucleate a new terrace, or evaporate
The lowest-energy steps are often normal to 0110 -type directions, and the two resulting step structures are conventionally labeled A and B [28,29]. (Face-centered-cubic materials have A- and B-type steps on close-packed {111} surfaces, but they do not alternate between successive terraces for a given step orientation [28].) The kinetics of adatom attachment at A and B steps have been predicted to differ [26,29,30,31,32,33,34,35,36,37], which can explain the alternating terrace widths and step morphologies often observed in hcp-type systems [30,38,39,40,41,42,43,44,45,46]
We develop quasi-steady-state solutions for the adatom density distributions and the dynamics of the α and β terrace fractions, and investigate how the full-steady-state terrace fraction depends upon growth rate and kinetic parameters
Summary
The atomic-scale mechanisms of crystal growth are often described within the framework of Burton-Cabrera-Frank (BCF) theory [1,2,3,4,5], in which deposited adatoms diffuse on top of the exposed atomic layers (terraces) of the crystal surface, until they either attach to existing steps at terrace edges, join together to nucleate a new terrace, or evaporate. Most implementations of one-dimensional BCF theory presume that all steps have identical kinetic properties This is based on the assumption that steps have full-unit-cell heights, and identical structures owing to the crystal lattice periodicity. When steps have fractional-unit-cell heights, the kinetic properties can differ from step to step This generally occurs for crystal symmetries which contain screw axes or glide planes, and can lead to fundamentally different growth behavior [27]. On a vicinal surface the orientation of the atomic arrangements alternates between each α and β layer, so that the structure and properties of the steps alternate. For such hcp-type systems, the adatom diffusivity is isotropic and equal. Terrace Top Layer Height 5c/2 2c 3c/2 c c/2 0 x/a A step B step A step B step A step
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