Numerical study of discrete models in the class of the nonlinear molecular beam epitaxy equation

TL;DR

This study numerically investigates discrete growth models within the nonlinear molecular beam epitaxy (VLDS) class, providing refined estimates of scaling exponents and exploring universality of height distribution ratios across different models and dimensions.

Contribution

It offers the most accurate estimates of scaling exponents for CRSOS models in 1D, confirms higher order corrections in VLDS theory, and demonstrates universality of amplitude ratios in the VLDS class.

Findings

CRSOS model in 1D yields alpha=0.94±0.02, z=2.88±0.04.

In 2D, roughness exponent alpha approximates 2/3, matching one-loop predictions.

Amplitude ratios S and Q are universal across models in the VLDS class.

Abstract

We study numerically some discrete growth models belonging to the class of the nonlinear molecular beam epitaxy equation, or Villain-Lai-Das Sarma (VLDS) equation. The conserved restricted solid-on-solid model (CRSOS) with maximum heights differences H_m=1 and H_m=2 was analyzed in substrate dimensions d=1 and d=2. The Das Sarma and Tamborenea (DT) model and a competitive model involving random deposition and CRSOS deposition were studied in d=1. For the CRSOS model with H_m=1 we obtain the more accurate estimates of scaling exponents in d=1: roughness exponent alpha = 0.94 +- 0.02 and dynamical exponent z = 2.88 +- 0.04. These estimates are significantly below the values of one-loop renormalization for the VLDS theory, which confirms Janssen's proposal of the existence of higher order corrections. The roughness exponent in d=2 is very near the one-loop result alpha=2/3, in agreement with previous works. The moments W_n of orders n=2,3,4 of the heights distribution were calculated for all models and the skewness S = W_3/{W_2}^{3/2} and the kurtosis Q = W_4/{W_2}^{2}-3 were estimated. At the steady states, the CRSOS models and the competitive model have nearly the same values of S and Q in d=1, which suggests that these amplitude ratios are universal in the VLDS class. The estimates for the DT model are different, possibly due to their typically long crossover to asymptotic values. Results for the CRSOS models in d=2 also suggest that those quantities are universal.