Universality and crossover behavior of single-step growth models in $1+1$ and $2+1$ dimensions

We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter $p$ in $1+1$ and $2+1$ dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any $p <\frac{1}{2}$. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function $p$. The effective nonuniversal parameters are continuously decreasing with $p$, but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for $p \neq \frac{1}{2}$ belongs to the KPZ universality class in $2+1$ dimensions.