Long-wave model for strongly anisotropic growth of a crystal step

A continuum model for the dynamics of a single step with the strongly anisotropic line energy is formulated and analyzed. The step grows by attachment of adatoms from the lower terrace, onto which atoms adsorb from a vapor phase or from a molecular beam, and the desorption is nonnegligible (the “one-sided” model). Via a multiscale expansion, we derived a long-wave, strongly nonlinear, and strongly anisotropic evolution PDE for the step profile. Written in terms of the step slope, the PDE can be represented in a form similar to a convective Cahn-Hilliard equation. We performed the linear stability analysis and computed the nonlinear dynamics. Linear stability depends on whether the stiffness is minimum or maximum in the direction of the step growth. It also depends nontrivially on the combination of the anisotropy strength parameter and the atomic flux from the terrace to the step. Computations show formation and coarsening of a hill-and-valley structure superimposed onto a long-wavelength profile, which independently coarsens. Coarsening laws for the hill-and-valley structure are computed for two principal orientations of a maximum step stiffness, the increasing anisotropy strength, and the varying atomic flux.

Figure 1

Faceted step bunches (dark lines) and individual steps (light lines) on the Nb(011) surface. (Reprinted with permission from Ref. [1]. Copyright 2002, American Vacuum Society.)

Figure 2

Late-time step morphology. $\alpha =0.1$, $F=2$. Dashed line: Long-wavelength modulation of the mean step position. This line connects the grid nodes whose numbers are the average values of all grid nodes on an uphill or a downhill; thus the locations of such grid nodes do not coincide with a midpoint on a slope, as can be seen in the bottom inset (but they are close). Bottom inset: Zoom into the middle section of the main figure. Top inset: Zoom into the graph of the step slope, $H_x$. Note: The axes labels in the figures are capitalized for visibility.

Figure 3

Coarsening of the long-wavelength modulation of the mean step position. $\alpha =1.6$, $F=2$. For better view, the profiles corresponding to different times have been bunched together.

Figure 4

Evolution of the morphology from the small random perturbation to the final computed shape. $F=2$. Top panel: $\alpha =0.1$, bottom panel: $\alpha =1.6$.

Figure 5

Coarsening of the hill-and-valley structure for different anisotropy strengths. Coarsening laws are shown for the case $\alpha =0.1$. Inset shows the time to reach the steady state (the horizontal part of the curve in the main figure) vs $\alpha$. (The curve in the inset is only a guide for the eye.) $F=2$.

Figure 6

Step velocity vs the time for different anisotropy strengths. Inset shows the velocity in the steady state (the horizontal part of the curve in the main figure) vs $\alpha$. The slope of this line is $1.7\times {10}^{-4}$. (The curve in the inset is only a guide for the eye.) $F=2$.

Figure 7

Coarsening of the hill-and-valley structure for different fluxes $F$. Inset shows the time to reach the steady state vs $F$. (The curve in the inset is only a guide for the eye.) $\alpha =0.4$.

Figure 8

Step velocity vs the time for different fluxes $F$. The curve that corresponds to $F=150$ is not shown in order to not scale down the view of the other four curves. Inset shows the velocity in the steady state. (The solid curve in the inset is only a guide for the eye.) The dashed curve is the five-point fit $1.16\times {10}^{-7}{F}^2+4.26\times {10}^{-6}F+5.35\times {10}^{-5}$. $\alpha =0.4$.

Figure 9

Coarsening of the hill-and-valley structure for different anisotropy strengths. Coarsening laws are shown for the case $\alpha =-0.1$.