Scaling of local roughness distributions

TL;DR

This paper investigates local roughness distributions in surface growth models across different universality classes, revealing universal behaviors and limitations in experimental data interpretation due to finite-size effects and measurement accuracy.

Contribution

It provides a quantitative analysis of the universal properties of local roughness distributions in KPZ and VLDS growth models, highlighting the importance of scaling methods and extrapolation techniques.

Findings

Plateaus of C and S observed for r <~ 0.3 xi with small time dependence

Extrapolation with power-law corrections confirms universality in 1+1 dimensions

Scaling by the average improves accuracy over variance-based scaling

Abstract

Local roughness distributions (LRDs) are studied in the growth regimes of lattice models in the Kardar-Parisi-Zhang (KPZ) class in 1+1 and 2+1 dimensions and in a model of the Villain-Lai-Das Sarma (VLDS) growth class in 2+1 dimensions. The squared local roughness w_2 is defined as the variance of the height inside a box of lateral size r and the LRD P_r(w_2) is sampled as this box glides along a surface with size L >> r. The variation coefficient C and the skewness S of the distributions are functions of the scaled box size r / xi, where xi(t) is a correlation length. For r <~ 0.3 xi, plateaus of C and S are observed, but with a small time dependence. For a quantitative characterization of the universal LRD, extrapolation of these values with power-law corrections in time are performed. The reliability of this procedure is confirmed in 1+1 dimensions by comparison of results of the restricted solid-on-solid model and theoretically predicted values of Edwards-Wilkinson interfaces. For r >> xi, C and S vanish because the LRD converges to a Dirac delta function. This confirms the inadequacy of extrapolations of amplitude ratios to r -> infinity, as proposed in recent works. On the other hand, it highlights the advantage of scaling LRDs by the average instead of scaling by the variance due to the usually higher accuracy of C compared to S. The scaled LRD of the VLDS model is very close to the KPZ one due to the small difference between their variation coefficients and the plateaus of C and S are very narrow due to the slow time increase of xi. These results suggest that experimental LRDs obtained in short growth times and with limited resolution may be inconclusive to determine their universality classes if data accuracy is low and/or data extrapolation to the long time limit is not feasible.