Linear stability of circular micro- and nanowires with facets

Abstract

Micro- and nanowires are commonly used in biological sciences, micro- and nanoelectronics, and optoelectronics, and their morphological stability needs to be understood and controlled. We study the linear stability of equilibrium circular wires with length to diameter ratio of 1, 2, 3.5, 6, and 11, assuming that the wire surface can deform by capillarity-driven surface diffusion. The facetted equilibrium wire shape is modeled by the Dirac delta function and is perturbed by an infinitesimal axisymmetric disturbance, leading to an eigenvalue problem for the growth rate, which is solved by a finite-difference method. Numerical accuracy is checked by grid refinement. All converged eigenvalues are negative, indicating that the wires are linearly stable. The first six eigenvalues are listed for all the wires which show that, for the same eigenmode, the eigenvalue decreases in magnitude as the wire length increases. The eigenfunctions for the longest wire studied are plotted and reveal how a non-equilibrium wire finally approaches the equilibrium state. The linear-stability formulation is then extended to an infinitely-long circular wire. The wire is stable for all wavelengths if its surface coincides with a facet plane. Hence, Rayleigh’s instability is completely suppressed in faceted circular wires.