General laws of X-ray reflection from rough surfaces: II. Conformal roughness
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RACTION AND SCATTERING OF IONIZING RADIATIONS
General Laws of XRay Reflection from Rough Surfaces: II. Conformal Roughness I. V. Kozhevnikov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 119333 Russia email: [email protected] Received February 14, 2011
Abstract—Is shown that, if the expansions of the Debye–Waller formulas for the reflection and total scatter ing coefficients in the roughness height σ are limited to terms of order σ2, these expressions are valid for any layered inhomogeneous medium with conformal (depthperiodic) roughness and for any distribution func tion of the roughness heights if the roughness correlation length along the surface is sufficiently large. The advantages of measuring the total reflection coefficient, which characterizes the total intensity of radiation (both specularly reflected and diffusively scattered) directed by a rough surface back into vacuum, for solving the inverse problem of Xray reflectometry (i.e., the reconstruction of the permittivity profile from a mea sured reflection curve) are discussed. DOI: 10.1134/S1063774512030133
INTRODUCTION Currently, measurements of angular and spectral dependences of hard and soft Xray reflection coeffi cients are widely used to study the surface layers of var ious materials and make it possible to determine the distributions of permittivity [1–3], the atomic con centrations of chemical elements [4], and the chemi calbond structure in materials [4, 5] with a subna nometer depth resolution. Obviously, no real surface can be ideally smooth for X rays. Roughness gives rise to scattered radiation and, therefore, changes the shape of the reflection curve. As a result, the reconstructed permittivity profile ε0(z) can be significantly deformed: the ignored effect of roughness leads to a number of artifacts, for example, smoothed distribution ε0(z) at interfaces of different materials [3]. At the same time, an analysis of the fine structure of interfaces is of particular interest for microelectronics and multilayer Xray optics technol ogies. The most successive approach to solving this prob lem is to perform a reconstruction of the permittivity profile explicitly taking into account the effect of roughness on the reflection curve based on a simulta neous analysis of the latter and a set of angular scatter ing distributions measured at different angles of inci dence of the probe beam [6]. However, this selfcon sistent approach is fairly difficult from both experimental and computational points of view. Therefore, an analysis of general features of Xray reflection and scattering from an arbitrary layered inhomogeneous rough structure is of great interest, because its results can be used to take into account the roughness effect when reconstructing the permittivity
profile and to determine the experimental conditions under which the effect of roughness on the reflection curve is minimal. The simplest expression for the reflection coeffi cient of a rough surface is well known
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