Ž . Thin Solid Films 323 1998 1–5 Letter Analysis of X-ray reflectivity curves of non-Gaussian surfaces G. Vignaud a, ) , A. Gibaud a , F. Paris b , D. Ausserre b , G. Grubel c ´ ¨ a UniÕersite du Maine, URA 807 CNRS, Faculte des Sciences Le Mans Cedex 72017, France ´ ´ b UniÕersite du Maine, URA 509 CNRS, Faculte des Sciences, Le Mans Cedex 72017, France ´ ´ c ESRF, Experimental diÕision, AÕenue des Martyrs, BP 220, Grenoble Cedex F 38043, France Received 30 September 1997; accepted 30 October 1997 Abstract Surfaces of symmetric diblock copolymers thin films exhibiting non-Gaussian distribution of height are studied by X-ray reflectivity Ž . and atomic force microscopy AFM . When deposited on a silicon substrate, the surface is essentially flat and its roughness may be described by a Gaussian distribution of height. Upon annealing, films operate a two-dimensional phase transformation and form islands at the free surface having height and size that evolve as a function of annealing time. The height probability function cannot be represented by a Gaussian distribution anymore, and the question that arises is how to take into account the morphology of such surfaces in the reflectivity calculations. In a first approach, we show that the height distribution function derived from AFM measurements is directly transferable to analyze X-ray reflectivity curves according to a formalism that we present. In a second part, we determine the height distribution function from a fit to the observed reflectivity. q 1998 Elsevier Science S.A. All rights reserved. Keywords: X-ray reflectometry; Thin film structure and morphology; Polymers; Elastomers and plastics 1. Introduction X-ray and neutron reflectivity have recently been known as a veritable explosion of interest in the scientific commu- nity working on the structural characterization of thin wx films. Dating back to the work of Parratt 1 who initiated the recursive technique, reflectivity curves are now mainly analyzed via the matrix technique. More recently, the Born w x approximation and the distorted Born approximation 2–6 were used to model the diffuse scattering that is inevitably observed in off-specular directions as soon as the surface presents some kind of roughness with correlations between height fluctuations. A wide variety of surfaces and inter- faces occurring in nature are well represented by a kind of roughness associated with self-affine fractal scaling, de- wx fined by Mandelbrodt 7 in terms of fractional Brownian motion. An isotropic rough surface can be described by the mean-square height difference given by: 2 ² : GR s hr y hO , 1 Ž . Ž . Ž . Ž. Ž. where hr stands for the height of the surface at the ²: in-plane position r and the symbol denotes an ensem- ) Corresponding author. Ž . ble average. For any physical surface, GR will saturate to a mean-square roughness s at sufficiently large hori- zontal lengths, i.e., when R is larger than the roughness w x correlation length j 8,9 . For surfaces in which the correlation length of the fluctuations of height is smaller than the coherence length of the beam, the reflectivity measurements are frequently analyzed by means of two Ž. quantities such as: 1 the mean-square roughness s of the surface that produces a deviation of the intensity decay from the Fresnel reflectivity in the specular direction; and Ž. 2 the height–height correlation function that is the rele- vant quantity intervening in the analysis of the diffuse scattering. This simple description is only possible if the height distribution of the surface is Gaussian, or if the roughness is sufficiently weak, i.e., q s - 1. In such a description, z ²Ž Ž . only the second moment of the distribution s s hr y Ž .:. 2 : 1r2 - hr , i.e., the roughness of the surface, is suffi- cient to model the decay of the specular reflectivity curve. Although this assumption is frequently acceptable, it hap- pens that some surfaces can in no way be described by such a distribution. In such cases, the specular reflectivity Ž. can only be calculated by means of the probability pz of finding some points of height z at the surface. Up to now, little work on X-ray reflectivity has been presented on the 0040-6090r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved.