Characterization of a layer stack by wavelet analysis on X-ray reflectivity data

X-ray reflectometry allows the determination of the thickness, density, absorption and rms roughness of a stack of thin layers on a substrate from several nanometres to some hundred nanometres. Inversion of the experimental reflectivity data is usually realized by a trial-and-error method based on the theoretical computation of the reflectivity curve after having extracted initial values of the layer thicknesses from the result of a classical FFT of the reflectivity data. However, the order information of the layers is lost during classical FFT. The order of the layers then has to be known a priori. Besides, the classical Fourier transform does not reveal anything about the stack parameters (density, absorption and the rms roughness). As this trial-and-error method is efficient provided that one has a good idea of the stack parameter values, it is important to extract some valuable information directly from the experimental reflectivity. In this paper, it will be shown that the order of the layers can be obtained by the so-called joint time-frequency representations. Furthermore, the continuous wavelet transform allows qualitative determination of the stack parameters and helps in the determination of an appropriate starting model for the trial-and-error method. The points of interest of this method are illustrated by experimental examples.