Structural refinement of superlattices from x-ray diffraction

We present a general procedure for quantitative structural refinement of superlattice structures. To analyze a wide range of superlattices, we derived a general kinematical diffraction formula that includes random, continuous, and discrete fluctuations from the average structure. We show that only the structure factor of one single layer of each material has to be averaged over the random variables and prove that this relation is equivalent to earlier, less general models. Implementing a nonlinear-fitting algorithm to fit the entire x-ray-diffraction profile, refined parameters that describe the average superlattice structure and deviations from this average are obtained. We compare the results of structural refinement to results obtained independently from other measurements. The roughness introduced artificially during growth in Mo/Ni and Nb/Cu superlattices is accurately reproduced by the refinement. The lattice parameters of Ag/Mn obtained from this refinement procedure are in very good agreement with the values obtained from independent extended x-ray-absorption fine-structure and x-ray photoelectron diffraction studies. The relative thicknesses of the layers can be accurately determined, as proved for Cu/Ni in comparison with chemical analysis, for W/Ni compared to the calibrated sputtering rate, and for Mo/Ni compared to the low-angle profile.