# Temporal Evolution of Step-Edge Fluctuations Under Electromigration Conditions

## TL;DR

This paper investigates how electromigration influences step-edge fluctuations over time, revealing deviations from classical scaling laws and providing a model to estimate electromigration forces from fluctuation data.

## Contribution

It introduces a continuum Langevin model for step-edge fluctuations under electromigration and derives a critical time scale where behavior deviates from classical models.

## Key findings

- Deviations from $t^{1/4}$ scaling occur at a critical time $ au$.
- A superposition of $t^{1/4}$ and $t^{3/4}$ laws describes early-time behavior.
- Monte-Carlo simulations confirm the theoretical predictions.

## Abstract

The temporal evolution of step-edge fluctuations under electromigration conditions is analysed using a continuum Langevin model. If the electromigration driving force acts in the step up/down direction, and step-edge diffusion is the dominant mass-transport mechanism, we find that significant deviations from the usual $t^{1/4}$ scaling of the terrace-width correlation function occurs for a critical time $\tau$ which is dependent upon the three energy scales in the problem: $k_{B}T$, the step stiffness, $\gamma$, and the bias associated with adatom hopping under the influence of an electromigration force, $\pm \Delta U$. For ($t < \tau$), the correlation function evolves as a superposition of $t^{1/4}$ and $t^{3/4}$ power laws. For $t \ge \tau$ a closed form expression can be derived. This behavior is confirmed by a Monte-Carlo simulation using a discrete model of the step dynamics. It is proposed that the magnitude of the electromigration force acting upon an atom at a step-edge can by estimated by a careful analysis of the statistical properties of step-edge fluctuations on the appropriate time-scale.