On Nanowire Morphological Instability and Pinch-Off by Surface Electromigration

TL;DR

This work analyzes electromigration-driven morphological instability in axisymmetric nanowires by developing two weakly nonlinear analyses beyond the linear Rayleigh-Plateau framework. It derives a Sivashinsky-type amplitude equation near the longwave threshold, predicting finite-time blow-up (pinch-off) with self-similar spike scaling that depends on the electric field $E$; it also performs a weakly nonlinear multi-scale analysis to identify the fastest-growing instability modes and their time- and field-dependent amplitudes, highlighting dominant Fourier components $\cos ky$, $\cos 2ky$, and $\sin 2ky$. Together, these results quantify how surface diffusion and surface electromigration interact to cause pinch-off and segment formation, including estimates of pinch-off times and spike dimensions for realistic metal wires. The findings provide theoretical predictions for the spacing and breakup dynamics of nanowires under current, informing design and interpretation of experiments in nanoscale contacts and molecular electronics. Overall, the paper advances understanding of electromigration-induced instabilities and offers a framework for predicting pinch-off behavior in metallic nanowires.

Abstract

Surface diffusion and surface electromigration may lead to a morphological instability of thin solid films and nanowires. In this paper two nonlinear analyses of a morphological instability are developed for a single-crystal cylindrical nanowire that is subjected to an axial current. These treatments extend the conventional linear stability analyses without surface electromigration, that manifest a Rayleigh-Plateau instability. A weakly nonlinear analysis is done slightly above the Rayleigh-Plateau (longwave) instability threshold. It results in a one-dimensional Sivashinsky amplitude equation that describes a blow-up of a surface perturbation amplitude in a finite time. This is a signature of a pinching singularity of a cylinder radius, which leads to a wire separation into a disjoint segments. The time- and electric field-dependent dimensions of the focusing self-similar amplitude profile approaching a blow-up are characterized via the scaling analysis. Also, a weakly nonlinear multi-scale analysis is done at the arbitrary distance above a longwave or a shortwave instability threshold. The time- and electric field-dependent Fourier amplitudes of the major instability modes are derived and characterized.

Figures (10)

• Figure 1: Two situations for a free-standing unperturbed cylindrical nanowire (the base state). Left: a nanowire suspended in 3D space, say, by hanging it on a thread attached to a support. Right: a nanowire grown vertically on a horizontal substrate. $j_e$ is the surface electric current that drives the surface electromigration of adatoms via the "electron wind" HuntingtonHoKwokREW.
• Figure 2: Unperturbed cylindrical nanowire grown laterally on a substrate with a 90$^\circ$ contact angle.
• Figure 3: The neutral stability curve $E(k)$.
• Figure 4: (a,b) Sketch of a cross-section of a bump on a wire surface by a plane that is parallel to a wire axis. A projection of a constant electric field $E_0$ onto a surface, $E_{loc}$, drives the electromigration adatom flux $j$ in the opposite direction, i.e. in the direction of the electron flow $j_e$. This contributes to either (a) a bump smoothing (stabilization), or to (b) a bump growth (destabilization).
• Figure 5: (a) Spike formation via computation of Eq. (\ref{['mKS3']}) with $c=1/4$. The initial condition is a small Gaussian-shaped perturbation of the base state $\phi=1$ on the interval $0\le X\le 2\sqrt{2}\pi$, where $2\sqrt{2}\pi$ is the most dangerous instability wavelength (the one at which the growth rate $\omega$ is the maximum value). Boundary conditions are periodic. The last shown profile corresponds to $\tau_s=43.35$. Computation was done on a fixed grid in $X$ using 3000 grid points, a fourth-order finite difference discretization formulas in $X$ and a backward difference formulas of variable order in $\tau$. (b) Evolution of the spike dimensions (Eqs. (\ref{['spd']})-(\ref{['spdisp']})) at $\Delta E=0.01, a=2, b=1, t_s=1.3\times 10^7$. Proportionality constants $c_{1,2,3}=1$ are chosen to facilitate plotting three functions on the same plot.