Dimensional crossover in surface growth on rectangular substrates

TL;DR

This study investigates dimensional crossovers in surface growth models across various universality classes on rectangular substrates, revealing how roughness and height distributions transition between 2D and 1D behaviors as aspect ratios change.

Contribution

It extends previous work on KPZ to other classes, demonstrating universal crossover behaviors in roughness scaling and height distributions through extensive simulations.

Findings

Roughness scales as t^{β_{2D}} for t << t_c and t^{β_{1D}} for t >> t_c.

Height distribution approaches 2D form at short times and crosses over to 1D form.

Steady state regimes interpolate between 2D and 1D in roughness and height distributions.

Abstract

In a recent work [Phys. Rev. E 109, L042102 (2024)], interesting dimensional crossovers [from two- to one-dimensional (2D to 1D) scaling] were found in the growth of Kardar-Parisi-Zhang (KPZ) interfaces on rectangular substrates, with lateral sizes $L_y > L_x$. Here, we extend this study to other universality classes for interface growth -- specifically, the Edwards-Wilkinson (EW), the Mullins-Herring (MH), and the Villain-Lai Das Sarma (VLDS) classes. From extensive simulations, we demonstrate that, in all systems with sufficiently large aspect ratio $\mathcal{R}=L_y/L_x$, the roughness $W$ scales with time $t$ in the growth regime as $W \sim t^{\beta_{\text{2D}}}$ for $t \ll t_c$ and $W \sim t^{\beta_{\text{1D}}}$ for $t \gg t_c$, where $t_c \sim L_x^{z_{2\text{D}}}$ in most cases. For the VLDS class, this crossover is also observed in the height distribution (HD), which approaches its characteristic probability density function for the 2D case at short times ($t \ll t_c$) and then crosses over to the asymptotic 1D HD. Dimensional crossovers are also found in the steady state regime, both in the roughness scaling as well as in the VLDS HD, which interpolate between the 2D and 1D ones as $\mathcal{R}$ increases. The particular case $L_x = L_y^{\delta}$, with $0 < \delta < 1$, is also discussed in detail and reveals interesting features of the investigated systems. For instance, there exist a `special' exponent $\delta^* = z_{1\text{D}}/z_{2\text{D}}$ such that the temporal crossover cannot be observed for $\delta > \delta^*$. Moreover, this leads the saturation roughness to display a nonuniversal scaling: $W_s \sim L_y^{\Lambda}$, with $\Lambda = (1-\delta) \alpha_{1\text{D}} + \delta \alpha_{2\text{D}}$.