Stochastic Study of Random-Ballistic Competitive Growth Model in 2 + 1 Dimension and Related Scaling Exponents

Abstract

In the fields of material science, surface and interface always play an extremely important role to explain any associated phenomena. With the advent of research on low-dimensional systems and thin-film growth, the importance has increased a million times. The properties of surfaces depend on how it gets evolved (both for physical and chemical growth). So far there are a number of proposed models that explain the physical growths of surfaces by deposition of particles/atoms. The present work reports variation of surface roughness, porosity, and height of the surfaces, evolves in 2 + 1 dimension through competitive growth between random and ballistic deposition (RD and BD) via a simulation study. The study has been done for different system sizes as well as for different competitive growth probabilities. From the simulation study, it has been found that growth has two different regions which in turn suggests two different critical times as well as two growth exponents ${\beta }_{1}$ and ${\beta }_{2}$. Of these two $\beta$ values one is found to be independent of both system size ($L$) and probability ($p$) whereas the other $\beta$ though is independent of $L$ but has a profound variation with the value of $p$. The variation of dynamic exponent ($z$) and critical times with both $L$ and $p$ has also been studied. It has been seen that either it is independent of the parameters or follows an exponential variation. The variation of fractional porosity of the surface is also not an exception. So far, though there are reports on the study of different growth models, however, most of them are in one dimension, or if they are in two dimensions the probabilistic approach (competitive) is missing. Thus it can be taken as the first effort to describe the surface evolved by competitive growth between RD and BD in the 2 + 1 dimension.