Scaling properties of step bunches induced by sublimation and related mechanisms

This work provides a ground for a quantitative interpretation of experiments on step bunching during sublimation of crystals with a pronounced Ehrlich-Schwoebel (ES) barrier in the regime of weak desorption. A strong step bunching instability takes place when the kinetic length $d_{+}=D_s∕K_{+}$ is larger than the average distance $l$ between the steps on the vicinal surface; here $D_s$ is the surface diffusion coefficient and $K_{+}$ is the step kinetic coefficient. In the opposite limit $d_{+}⪡l$ the instability is weak and step bunching can occur only when the magnitude of step-step repulsion is small. The central result are power law relations of the form $L\sim H^{\alpha }$, $l_{\min }\sim H^{-\gamma }$ between the width $L$, the height $H$, and the minimum interstep distance $l_{\min }$ of a bunch. These relations are obtained from a continuum evolution equation for the surface profile, which is derived from the discrete step dynamical equations for the case $d_{+}⪢l$. The analysis of the continuum equation reveals the existence of two types of stationary bunch profiles with different scaling properties. Through comparison with numerical simulations of the discrete step equations, we establish the value $\gamma =2∕(n+1)$ for the scaling exponent of $l_{\min }$ in terms of the exponent $n$ of the repulsive step-step interaction, and provide an exact expression for the prefactor in terms of the energetic and kinetic parameters of the system. For the bunch width $L$ we observe significant deviations from the expected scaling with exponent $\gamma =1-1∕\alpha$, which are attributed to the pronounced asymmetry between the leading and the trailing edges of the bunch, and the fact that bunches move. Through a mathematical equivalence on the level of the discrete step equations as well as on the continuum level, our results carry over to the problems of step bunching induced by growth with a strong inverse ES effect, and by electromigration in the attachment∕detachment limited regime. Thus our work provides support for the existence of universality classes of step bunching instabilities [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002)], but some aspects of the universality scenario need to be revised.

Figure 1

Schematic of a bunched vicinal surface, illustrating the definition of the bunch width $L$, the bunch height $H$, and the bunch spacing $\xi$.

Figure 2

Sketch of the potential $V\left(u\right)$ appearing in the mechanical analog. The dashed arrows illustrate trajectories of type I and type II.

Figure 3

Stationary bunch shapes computed by numerical integration of Eq. (58) with $S=0.14$ and $\beta =0.01$. The figure shows the dimensionless slope $(l∕h_0)m\left(x\right)$ as a function of the rescaled coordinate $x∕l$ for different values of the maximum slope. The horizontal dotted line shows the value $m=h_0∕l$ corresponding to the mean slope of the unperturbed surface.

Figure 4

Profile of a crystal surface after some time of sublimation, with dimensionless parameters $l_0∕l=0.24$, $\beta =0.01$, $d_{+}∕l=10$, and ${\lambda }_s∕l=100$. The inset shows the enlargement of an individual bunch.

Figure 5

Numerical data for the minimum interstep distance, measured in units of $l$, as a function of bunch size. Open symbols show results obtained in the natural bunching geometry with 500 steps, while asterisks show data obtained from computations with a single bunch. The interaction strength was $l_0∕l=0.24$ in all cases. Open squares and diamonds show data for $d_{+}∕l=10$ and ${\lambda }_s∕l=100$, triangles show data for $d_{+}∕l=150$ and ${\lambda }_s∕l=200$, and asterisks show data for $d_{+}∕l={\lambda }_s∕l=100$; other parameters are given in the figure. Bold lines show the theoretical prediction (55) for $l_{\min }$.

Figure 6

Numerical results for the bunch width $L$ as a function of bunch size. Data are shown for two of the parameter sets displayed in Fig. 5: $\beta =0.1$, ${\lambda }_s∕l=200$, $d_{+}∕l=150$ (diamonds), and $\beta =0.01$, ${\lambda }_s∕l=100$, $d_{+}∕l=10$ (crosses). In both cases $l_0∕l=0.24$ and the system contained 500 steps. Dashed and full lines show the predictions for type I and type II solutions, respectively. A power law fit to the data yields $L\sim N^{0.44}$.

Figure 7

Scaling of the first and last terrace size in the bunch with bunch size for the same parameter sets as in Fig. 6. The full lines show the theoretical prediction for type II solutions.

Figure 8

Time dependence of the mean bunch size for the parameter sets in Figs. 6, 7, and an additional set with $\beta =0.1$, ${\lambda }_s∕l=100$, $d_{+}∕l=100$, $l_0∕l=0.24$, and step interaction exponent $n=3$. The bold lines illustrate the predictions of the scaling theory for $\varrho =-1$ $(\alpha ∕z=1∕2)$ and for $\varrho =0$, $n=2$ $(\alpha ∕z=3∕4)$.