Improved x-ray reflectivity calculations for rough surfaces and interfaces

Abstract

We have investigated the fact that the calculated x-ray reflectivity based on the Parratt formalism, coupled with the use of the Nevot-Croce representation of roughness, show a strange phenomenon where the amplitude of the oscillation due to interference effects increases for a rougher surface. Here, we propose that the strange result has its origin in a currently used equation due to a serious mistake in which the Fresnel transmission coefficient in the reflectivity equation is increased at a rough interface, and the increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. The mistake in Nevot and Croce's treatment originates in the fact that the modified Fresnel coefficients were calculated based on the theory which contains the x-ray energy conservation rule at surface and interface. In their discussion, the transmission coefficients were replaced approximately by the reflection coefficients by the ignoring diffuse scattering term at the rough interface, and according to the principle of conservation energy at the rough interface also. The errors of transmittance without the modification cannot be ignored. It is meaningless to try to precisely match the numerical result based on a wrong calculating formula even to details of the reflectivity profile of the experimental result. Thus, because Nevot and Croce's treatment of the Parratt formalism contains a fundamental mistake regardless of the size of roughness, this approach needs to be corrected. In the present study, we present a new accurate formalism that corrects this mistake, and thereby derive an accurate analysis of the x-ray reflectivity from a multilayer surface, taking into account the effect of roughness-induced diffuse scattering. The calculated reflectivity obtained by the use of this accurate reflectivity equation gives a physically reasonable result, and should enable the structure of buried interfaces to be analyzed more accurately.