A Model for Conductive Percolation in Ordered Nanowire Arrays

TL;DR

This paper models the clumping behavior of nanowires during alumina etching using percolation theory, revealing critical exponents and a percolation probability scaling law.

Contribution

It introduces a novel elastic rod model for nanowire clumping and applies percolation theory to analyze the process.

Findings

Percolation probability scales as (t - t_c)^x with x=2.8.

Critical exponents for the system are identified, e.g., β=2.1, γ=0.57.

Clumping behavior is characterized by a percolation transition.

Abstract

The combined processes of anodization and electrodeposition lead to highly ordered arrays of cylindrical nanowires. This template-based self-assembly fabrication method yields nanowires embedded in alumina. Commonly, chemical etching is used to remove the alumina and free the nanowires. However, it has been experimentally observed during the etching process that the nanowires tend to form clumps. In this work, the nanowires are modeled as elastic rods subject to surface interaction forces. The dynamics of the model give rise to the aforementioned clumping behavior which is studied via percolation theory. This work finds that percolation takes place with probability $P \sim (t-t_c)^x$, where the exponent $x = 2.8$ and $t_c$ is the time at which percolation takes place. The critical exponents which entirely determine the system are found to be for (dimension) $d = 2$, $\beta = 2.1, \gamma = 0.57, \Delta = 2.7, \alpha = -2.8, \nu = 2.4,$ and $\delta = 1.3$.