On the regularization of the inverse laplace transform in grazing-emission X-ray fluorescence spectroscopy

The determination of shallow concentration profiles of dopants in silicon wafers using grazing-emission X-ray fluorescence spectroscopy is studied. In grazing-emission X-ray fluorescence spectrometry, fluorescence intensities are measured at grazing angles to the sample surface. The concentration of the dopant is determined as a function of depth from an angle scan of the emitted fluorescence radiation by inversion of a truncated Laplace transform. Tikhonov's regularization method is applied to this ill-posed inversion. Two methods to determine the regularization parameter, namely Morozov's discrepancy method and generalized cross-validation, are compared. The inversion method is applied to some typical profiles of arsenic in silicon. The numerical experiments suggest that provided the X-ray sourer that is used to induce the fluorescence is strong enough, the reconstruction of dopant concentration profiles is possible.