Current-Induced Stabilization of Surface Morphology in Stressed Solids

We examine the surface morphological evolution of a conducting crystalline solid under the simultaneous action of an electric field and mechanical stress based on a fully nonlinear model and combining linear stability theory with self-consistent dynamical simulations. We demonstrate that electric current, through surface electromigration, can stabilize the surface morphology of the stressed solid against cracklike surface instabilities. The results also have more general implications for the morphological response of solid surfaces under the simultaneous action of multiple external forces.

Figure 1

Dispersion relations for various $\Xi$ values giving the dependence of the characteristic dimensionless frequency, $\omega$, for the growth ($\omega >0$) or decay ($\omega <0$) rate of a shape perturbation from a planar surface morphology on the dimensionless wave number, $\tilde{k}$, of the perturbation for surface diffusional anisotropy parameters $A=10$, $m=3$, and $\varphi =-15°$.

Figure 2

Surface morphological evolution, $\tilde{h}(\tilde{x},\tilde{t})$, starting with a perturbation $\tilde{h}(\tilde{x},0)={\tilde{\Delta }}_0\sin (\tilde{k}\tilde{x})$, at (a) $\Xi =0.0$ for $\tilde{k}=0.8$ and ${\tilde{\Delta }}_0/\tilde{\lambda }=0.05$, where $\tilde{\lambda }\equiv \lambda /l$; (b) $\Xi =0.60$ for $\tilde{k}=0.50$ and ${\tilde{\Delta }}_0/\tilde{\lambda }=0.02$; (c) same shape perturbation and $\Xi$ value as in (b) but the electric current is turned off from $t=0.38$ to $t=0.81\tau$ and then it is turned back on; and (d) $\Xi =0.16$ for $\tilde{k}=0.35$ and ${\tilde{\Delta }}_0/\tilde{\lambda }=0.005$. The anisotropy parameters are the same with those that yielded the dispersion curves of Fig. 1. The evolution sequences are from the bottom to the top, the electric field is applied from right to left, and the stress is tensile and applied uniaxially along $x$. The snapshots shown correspond to (a) $t=0$, 1.00, 1.54, 1.76, 1.93, 2.06, 2.17, 2.34, and $2.40\times {10}^{-2}$ $\tau$, (b) $t=0$, 0.05, 0.16, 0.47, 0.77, 1.05, 1.51, 1.94, 2.36, 2.78, 3.23, 3.70, 4.17, 4.67, 5.20, 5.76, 6.37, 7.00, 7.64, 8.27, 8.90, 9.53, 10.1, 11.2, 12.2, 14.3, and $18.5\times {10}^{-1}$ $\tau$, (c) $t=0$, 0.06, 0.10, 0.15, 0.21, 0.27, 0.38, 0.66, 0.76, 0.81, 0.85, 0.89, 0.97, 1.08, 1.28, 1.56, and 1.90 $\tau$; and (d) $t=0$, 0.79, 1.63, 2.14, 3.16, 4.43, 5.89, 10.8, 17.7, 21.7, 23.5, and 25.3 $\tau$. (e) Evolution of the perturbation amplitude, $\tilde{\Delta }$, for cases (b),(c), and an additional case with $\Xi =1.0$ for $\tilde{k}=0.65$ and ${\tilde{\Delta }}_0/\tilde{\lambda }=0.05$. (f) Evolution of $\tilde{\Delta }$ for case (d). In (e) and (f), solid lines represent the simulation results according to the fully nonlinear model, while the data points [open circles in (e) and star symbols in (e) and (f)] correspond to the LST results for the same parameter sets and initial surface morphologies.