Birth and Morphological Evolution of Step Bunches under Electromigration

We analyze the dynamics of electromigration-induced step bunching in the absence of desorption. We show that, even when the instability occurs at long wavelength, hinting to a smooth morphology, the surface suddenly splits into bunches escorted with wide terraces, in agreement with several observations. As the size of the bunches increases, a nonstandard regime is exhibited, namely, the bunches do not match tangentially to the facet, as would the classical Pokrosvky-Talapov shape dictate. This Letter presents a complete scenario of evolution of bunches from their birth up to their ultimate stage.

Figure 1

Upper panel: Sketch of a vicinal surface. Middle panel: Numerical solution of the dynamics for a train of 200 steps with periodic boundary conditions. $x$ is the step position, and $t$ is the time. Lower panel: Solution of the highly nonlinear equation. Step density $\rho$ as a function of $y$ at different times.

Figure 2

Upper panel: Potential $\mathcal{V}$. The continuous steady state are oscillations between 0 and $u^{*}$. Solutions I and II explore the linear part at $u>u^{*}$, and regime III corresponds to an oscillation from 0 to $u^{*}$. Lower panel: Dashed arrows indicate the trajectory of the surface into the $N,L_0$ plane. The dotted line indicates the continuous steady state, and solid line indicates the discontinuous steady state.

Figure 3

Analysis of one bunch in regime II. The parameters are $d=0.1$, $\xi =500$, and $A=0.001$. The lines indicates the analytical prediction of regime II. The squares, triangles, and diamonds, represent numerical results for $L_{\min }$, $W/N$, and $L_e$, respectively. These results are found for a periodic box size of $L_{\mathrm{box}}=5000$, but do not depend on $L_{\mathrm{box}}$ as long as $L_{\mathrm{box}}\gg \xi$.