Crossover effects in the Wolf-Villain model of epitaxial growth in 1+1 and 2+1 dimensions

TL;DR

This study uses extensive simulations to reveal complex crossover behaviors and anomalous scaling in the Wolf-Villain epitaxial growth model across 1+1 and 2+1 dimensions, challenging existing continuum theories.

Contribution

It uncovers unexpected crossover phenomena and anomalous scaling in the Wolf-Villain model, providing new insights into epitaxial growth dynamics in multiple dimensions.

Findings

Crossover from β_eff ≈ 0.37 to 0.33 in 1+1 D

Additional crossovers resembling Edwards--Wilkinson behavior

Anomalous scaling due to power-law growth of step height

Abstract

A simple model of epitaxial growth proposed by Wolf and Villain is investigated using extensive computer simulations. We find an unexpectedly complex crossover behavior of the original model in both 1+1 and 2+1 dimensions. A crossover from the effective growth exponent $\beta_{\rm eff}\!\approx\!0.37$ to $\beta_{\rm eff}\!\approx\!0.33$ is observed in 1+1 dimensions, whereas additional crossovers, which we believe are to the scaling behavior of an Edwards--Wilkinson type, are observed in both 1+1 and 2+1 dimensions. Anomalous scaling due to power--law growth of the average step height is found in 1+1 D, and also at short time and length scales in 2+1~D. The roughness exponents $\zeta_{\rm eff}^{\rm c}$ obtained from the height--height correlation functions in 1+1~D ($\approx\!3/4$) and 2+1~D ($\approx\!2/3$) cannot be simultaneously explained by any of the continuum equations proposed so far to describe epitaxial growth.