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

We study numerically some discrete growth models belonging to the class of the nonlinear molecular beam epitaxy equation, or the Villain–Lai–Das Sarma (VLDS) equation. The conserved restricted solid-on-solid model (CRSOS) with maximum height differences ΔH_max=1 and ΔH_max=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 ΔH_max=1, we obtain the more accurate estimates of scaling exponents in d=1: roughness exponent α=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 α=2/3, in agreement with previous works. The moments W_n of orders n=2, 3, 4 of the height distribution were calculated for all models, and the skewness S≡W₃∕W₂^3∕2 and the kurtosis Q≡W₄∕W₂²−3 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.