Super-roughening versus intrinsic anomalous scaling of surfaces

TL;DR

This paper investigates the causes of anomalous scaling in kinetically rough surfaces, distinguishing super-roughening from intrinsically anomalous spectra, and provides methods to extract relevant exponents through analysis of correlation functions.

Contribution

It introduces a distinction between super-roughening and intrinsic anomalous spectra as causes of anomalous scaling, with new analysis techniques and exactly solvable examples.

Findings

Super-roughening occurs with a global roughness exponent ≥ 1.

Intrinsic anomalous spectra involve different scaling at small and large scales.

Both causes can be distinguished and analyzed through correlation functions.

Abstract

In this paper we study kinetically rough surfaces which display anomalous scaling in their local properties such as roughness, or height-height correlation function. By studying the power spectrum of the surface and its relation to the height-height correlation, we distinguish two independent causes for anomalous scaling. One is super-roughening (global roughness exponent larger than or equal to one), even if the spectrum behaves non anomalously. Another cause is what we term an intrinsically anomalous spectrum, in whose scaling an independent exponent exists, which induces different scaling properties for small and large length scales (that is, the surface is not self-affine). In this case, the surface does not need to be super-rough in order to display anomalous scaling. In both cases, we show how to extract the independent exponents and scaling relations from the correlation functions, and we illustrate our analysis with two exactly solvable examples. One is the simplest linear equation for molecular beam epitaxy , well known to display anomalous scaling due to super-roughening. The second example is a random diffusion equation, which features anomalous scaling independent of the value of the global roughness exponent below or above one.