Solid on Solid Model for Surface Growth in 2+1 Dimensions

TL;DR

This study investigates the surface growth behavior of the Restricted-Solid-On-Solid (RSOS) model in 2+1 dimensions, revealing a crossover from Random-Deposition to KPZ universality class and analyzing the roughness exponent through structure functions.

Contribution

The paper introduces an alternative Markovian approach to determine the roughness exponent for the RSOS model in 2+1 dimensions and characterizes the crossover behavior between universality classes.

Findings

Crossover time scales with S as a power law with exponent ~2.

The roughness exponent remains constant across different S values.

The crossover from RD to KPZ universality class occurs at S ≥ 8.

Abstract

We analyze in detail the Solid-On-Solid model (SOS) for growth processes on a square substrate in 2+1 dimensions. By using the Markovian surface properties, we introduce an alternative approach for determining the roughness exponent of a special type of SOS model-the Restricted-Solid-On-Solid model (RSOS)- in 2+1 dimensions. This model is the SOS model with the additional restriction that the height difference must be S=1. Our numerical results show that the behaviour of the SOS model in 2+1 dimensions for approximately $S\geq S_{\times}\sim 8$ belongs to the two different universality classes: during the initial time stage, $t< t_{\times}$ it belongs to the Random-Deposition (RD) class, while for $t_{\times}<t\ll t_{sat}$ it belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The crossover time ($t_{\times}$) is related to S via a power law with exponent, $\eta=1.99\pm0.02$ at $1\sigma$ confidence level which is the same as that for 1+1 dimensions reported in Ref. \cite{e1}. Using the structure function, we compute the roughness exponent. In contrast to the growth exponent, the roughness exponent does not show crossover for different values of S. The scaling exponents of the structure function for fixed values of separation distance versus S in one and two space dimensions are $\xi=0.92\pm0.05$ and $\xi=0.86\pm0.05$ at $1\sigma$ confidence level, respectively.