Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both $d=1+1$ and $2+1$, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the “KPZ ansatz” to nMBE models, we estimate the cumulants of the $1+1$ HDs. Spatial covariance for the flat subclass is hallmarked by a minimum, which is not present in the ES one. Temporal correlations are shown to decay following well-known conjectures.

Figure 1

(a) Convergence of effective growth exponents ${\beta }_{\text{eff}}\equiv \frac{1}{2}\frac{d\left(ln{〈h^2〉}_c\right)}{d\left(lnt\right)}$. (b) Estimates of the cumulants of $\chi$ from ${〈h^n〉}_c/{\left(\mathrm{\Gamma }t\right)}^{n\beta }$ for the $1+1$ CRSOS1 model. (c) Extrapolation of skewness (higher) and kurtosis (lower values) for $d=1+1$ (top) and $2+1$ (bottom panel). (d) Estimates of $g_2$ from ${〈h^2〉}_c/t^{2\beta }$. All data in (b)–(e) are for flat ICs.

Figure 2

Rescaled HDs for models in (a) $d=1+1$ and (b) $2+1$. Distributions for flat (open: inner) and ES (solid symbols: outer curves, with $v_d=2$ in $2+1$) ICs are shown. The insets show the same data in a linear scale.

Figure 3

Rescaled spatial covariances for models with flat (open: bottom) and ES (solid symbols: top) ICs, in (a) $d=1+1$ and (b) $2+1$ (with $v_d=2$ in ES ICs).

Figure 4

Rescaled temporal covariances for flat (open: bottom) and ES (solid symbols: top) ICs, in (a) $1+1$ and (b) $2+1$ (with $v_d=2$ for ES ICs). Dashed lines have the indicated slopes. The inset shows nonextrapolated curves for the ES case, in $d=2+1$, for different $t_0\in [125,1000]$.