Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth

Abstract

We argue that symmetries and conservation laws greatly restrict the form of the terms entering the long wavelength description of growth models exhibiting anomalous roughening. This is exploited to show by dynamic renormalization group arguments that intrinsic anomalous roughening cannot occur in local growth models. However some conserved dynamics may display super-roughening if a given type of terms are present. PACS numbers: 81.15.Aa,64.60.Ht,05.70.Ln Recent theoretical and experimental studies of self-affine kinetic roughening have uncovered a rich variety of novel features [1]. In particular, the existence of anomalous roughening has received much attention. Anomalous roughening refers to the observation that local and global surface fluctuations may have distinctly different scaling exponents. This leads to the existence of an independent local roughness exponent α loc that characterizes the local interface fluctuations and differs from the global roughness exponent α. More precisely, global fluctuations are measured by the global interface width, which for a system of total lateral size L scales according to the Family-Vicsek ansatz [2] aswhere the scaling function f (u) behaves asThe roughness exponent α and the dynamic exponent z characterize the universality class of the model under study. The ratio β = α/z is the time exponent. In contrast, local surface fluctuations are given by either the height-height correlation function,, where the average is calculated over all x (overline) and noise (brackets), or the local width,where · · · l denotes an average over x in windows of size l. For growth processes in which an anomalous roughening takes place these functions scale as w(l, t) ∼ G(l, t) = t β f A (l/t 1/z ), with an anomalous scaling function [3,4] given byinstead of Eq.(2). The standard self-affine Family-Vicsek scaling [2] is then recovered when α = α loc . This singular phenomenon was first noticed in numerical simulations of both continuous and discrete models of ideal molecular beam epitaxial growth [3,4,5,6,7,8,9,10,11]. Anomalous roughening has later on been reported to occur in growth models in the presence of disorder [12,13] Nowadays it has become clear [3,23] that anomalous kinetic roughening is related to a non-trivial dynamics of the average surface gradient (local slope), so that (∇h) 2 ∼ t 2κ . Anomalous scaling occurs whenever κ > 0, which leads to the existence of a local roughness scaling with exponent α loc = α − zκ [23]. Also, it has recently been shown that the existence of power-law scaling of the correlation functions (i.e. scale invariance) does not determine a unique dynamic scaling form of the correlation functions [24]. On the one hand, there are super-rough processes, α > 1, for which α loc = 1 always. On the other hand, there are intrinsically anomalous roughened surfaces, for which the local roughness α loc < 1 is actually an independent exponent and α may take values larger or smaller than one depending on the universality class (see [4,24] and references therein). T...

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