Height and roughness distributions in thin films with Kardar–Parisi–Zhang scaling

Abstract

We study height and roughness distributions of films grown with discrete Kardar–Parisi–Zhang (KPZ) models in a small time regime which is expected to parallel the typical experimental conditions. Those distributions are measured with square windows of sizes 8 ⩽ r ⩽ 128 gliding through a much larger surface. Results for models with weak finite-size corrections indicate that the absolute value of the skewness and the value of the kurtosis of height distributions converge to 0.2 ⩽ ∣ S∣ ⩽ 0.3 and 0 ⩽ Q ⩽ 0.5, respectively. Despite the low accuracy of these results, they give additional support to a recent claim of KPZ scaling in oligomer films. However, there are significant finite-size effects in the scaled height distributions of models with large local slopes, such as ballistic deposition, which suggests that comparison of height distributions must not be used to rule out KPZ scaling. On the other hand, roughness distributions of the same models show good data collapse, with negligible dependence on time and window size. The estimates of skewness and kurtosis for roughness distributions are 1.7 ⩽ S ⩽ 2 and 3 ⩽ Q ⩽ 7. A stretched exponential tail was found, which seems to be a particular feature of KPZ systems in 2 + 1 dimensions. Moreover, the KPZ roughness distributions cannot be fitted by those of 1/ f α noise. This study suggests that the roughness distribution is the best option to test KPZ scaling in the growth regime, and provides quantitative data for future comparison with other models or experiments.