Finite resolution effects in the analysis of the scaling behavior of rough surfaces Javier Buceta, Juanma Pastor, Miguel A. Rubio, and F. Javier de la Rubia Departamento de Fı ´sica Fundamental, Universidad Nacional de Educacio ´n a Distancia, Apartado Correos 60.141, 28040 Madrid, Spain Received 12 April 1999 We investigate the influence of finite spatial resolution in the analysis of the scaling behavior of rough surfaces. We analyze such an effect for two usual measurement methods: the local width and the height-height correlation function. We show that while the correlation function is insensitive to finite resolution effects for practical purposes, the local width presents correction terms to the scaling law, leading to an effective value of the local roughness exponent smaller than the theoretically expected. We also show that a functional scaling relation can only be properly formulated in terms of the height-height correlation function. PACS numbers: 68.35.Ct, 05.10.-a, 68.35.Fx, 81.15.Aa To characterize the roughness of a surface is an important issue in science and technology. Mechanical problems con- cerning friction, wear, or adhesion show a crucial depen- dence on the smoothness of the surfaces that get into contact. It is also known that surface roughness affects dramatically the electrical and optical properties of thin films, which makes the development of better controlled surface growth techniques an important line of research. Many of these tech- niques show growth regimes with common spatiotemporal features, as for instance the appearance of scale invariant rough surfaces. Natural processes such as the infiltration of water in porous rocks or the growth of bacterial colonies also show scale invariant behavior 1. Scale invariance is revealed by scaling exponents and functions that may be measured to classify the growth pro- cesses into universality classes. To characterize experimental results or numerical simulations, it is necessary to study the scaling properties of some functions related to the surface profile. One of the most widely used is the so-called local interface width, W l 2 ( t ), defined as the rms fluctuations of the interface height h ( x, t ), i.e., W l 2 ( t ) =  h ( x, t ) -h l ( x, t )  2  x, l , where l is the size of a measurement win- dow, h l ( x, t ) is the mean height in the window, and  •  x, l indicates averages within a window and over the windows of the same size there is also an implicit average over realiza- tions when this is needed. After a saturation time that scales as t sat l z ( z is the dynamic exponent, the local width satu- rates, and a power law can be defined for small l such that W l 2 ( t t sat ) l 2  l , where  l is the local roughness exponent. Other relevant quantities, both in experiments and simula- tions, are the height-height correlation function HHCF, G l 2 ( t ) =  h ( x+l ) -h ( x)  2  x , which scales in the same way as the local width assuming that any possible overall slope in the interface has been removed, and the power spectrum of the interface, S ( q , t ) 1. A common characteristic in experiments and numerical simulations is the finite spatial resolution in the data. In this paper we analyze the limitations introduced in the analysis of scale invariant growth regimes by that unavoidable finite resolution when measuring the local roughness exponent. By considering various scaling behaviors, we show that the local width depends on the spatial resolution in a relevant way, introducing corrections to W l 2 ( t ) that may lead to wrong val- ues for the local roughness and preclude the possibility of formulating a functional scaling relation in terms of the local width. Moreover, we also show that the corrections to the HHCF can be neglected for practical purposes, and therefore we claim that in experiments and numerical simulations this technique is more adequate to study the scaling properties of rough surfaces. To put our work in a proper context, we briefly review some of the concepts used in the analysis of growth pro- cesses. The usual way to theoretically formulate the scaling properties of rough interfaces is in terms of the global width, W L 2 ( t ), where L is the system size. The functional behavior of the global width is summarized in the Family-Vicsek FV scaling relation 2, W L 2 ( t ) L 2  g f ( L / t 1/z ), where f ( u ) const for u 1, and f ( u ) u -2  g for u 1. The dynamic exponent z reflects the lateral correlation length dependence on time, and  g is the global roughness exponent. In an experimental situation, the system size is usually a fixed parameter, whereas in numerical simulations changing the system size is not the optimal way to compute the rough- ness exponent. Therefore, other ways of measurement, not showing these limitations, have to be considered. One possi- bility is to compute the power spectrum, S ( q , t ), so that when FV scaling is satisfied it behaves as S ( q , t ) =q -(2  g +d ) g ( q -1 t -1/z ), where g ( u ) const for u 1 and g ( u ) u -(2  g +d ) for u 1, and where d denotes the spatial dimension of the interface. However, there are models in the literature for which S ( q , t ) does not show this scaling behav- ior. It is known that if the noise is not renormalized NRN, g ( u ) behaves as u -z for u 1 3, whereas in some cases, named intrinsic anomalous IA, g ( u ) u -2(  g - l ) for u 1 4,5. However, the power spectrum technique is not free of problems. In experiments where the primary data are the topography of the interfaces, the power spectrum is obtained through a Fourier transformation which is usually rather noisy. On the other hand, experiments that yield directly the power spectrum must be cross-checked with other comple- mentary methods giving the topography of the interfaces, because a spurious q decay behavior, which would yield an incorrect  g value, might be introduced in several ways 6. Therefore, it is common to focus on the local width function W l 2 ( t ) or on the height-height correlation function G l 2 ( t ), that, as already said, scale for t t sat as l 2  l . For the global PHYSICAL REVIEW E MAY 2000 VOLUME 61, NUMBER 5 PRE 61 1063-651X/2000/615/60154/$15.00 6015 ©2000 The American Physical Society