Abstract
The determination of concentration profiles of impurities in silicon from angle scans of emitted x-ray fluorescence intensities using the maximum-entropy method is studied. Existence and convergence properties of the maxium- entropy method are discussed. The application of the maximum-entropy method to Grazing emission X-Ray Fluorescence Spectromety is compared with an analytical method. It is found that, provided noise levels are sufficiently low, concentration profiles can be reconstructed without using a priori knowledge.
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R.S. Becker, J.A. Golovchenko and J.R. Patel, X-ray evanescent-wave absorption and emission. Phys. Rev. Lett. 50 (1983) 153–156.
Y.C. Sasaki and K. Hirokawa, Depth profile measurement by using a refracted x-ray fluorescence method. Appl. Phys. Lett. 58 (1991) 1384–1386.
S. Hasegawa, Y. Ino, and H. Daimon, Chemical analysis of surfaces by total-reflection-angle x-ray spectroscopy in RHEED experiments, Jpn. J. Appl. Phys. 24 (1985) L387–L398.
H. Schwenke, J. Knoth, L. Fabry, S. Pahlke, R. Scholz and L. Frey,Measurement of shallow arsenic impurity profiles in semiconductor silicon using time-of-flight secondary ion mass spectrometry and total reflection x-ray fluorescence spectrometry. J. Electrochem. Soc. 144 (1997) 3979–3983.
P.K. de Bokx and H.P. Urbach, Laboratory grazing-emission x-ray fluorescence spectrometer. Rev. Sci. Instrum. 66 (1995) 15–19.
H.P. Urbach and P.K. de Bokx, Calculation of intensities in grazing-emission x-ray fluorescence, Phys. Rev. B 53 (1996) 3752–3763.
H.P. Urbach and P.K. de Bokx, Grazing Emission x-ray fluorescence from multilayers. Phys. Rev. B 63 (2001) 1–17.
P.R. Chernoff, Spectral representation of the Laplace transform. Am. Math. Monthly 101 (1994) 366–367.
D.S. Gilliam, J.R. Schulenberger, and J.L. Lund, Spectral representation of the Laplace and Stieltjes transform. Mat. Applic. 7 (1998) 101–107.
A.O. Istratov and O.F. Vyvenko, Exponential analysis in physical phenomena. Rev. Sci. Instr. 70 (1999) 1233–1257.
E.T. Jaynes, Information theory and statistical mechanics. Phys. Rev. 106 (1957) 620–630.
E.T. Jaynes, Information theory and statistical mechanics II. Phys. Rev. 108 (1957) 171–190.
E.T. Jaynes, On the rationale of maximum-entropy methods. Proc. IEEE 70 (1982) 939–952.
R.T. Rockafellar, Integrals which are convex functionals, II. Pacific J. Math. 39 (1971) 439–469.
J.M. Borwein and A.S. Lewis, Convergence of best entropy estimates. SIAM J. Optim. 1 (1991) 191–205.
P.P.B. Eggermont, Maximum entropy regularization for Fredholm integral equations of the first kind. SIAM J. Math. Anal. 24 (1995) 1557–1576.
I. Ekeland and R. Teman, Convex Analysis and Variational Problems. Amsterdam: North-Holland (1976) 401 pp.
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. Washington: Winston and Sons (1977) 258 pp.
G. Wahba, Spline Models for Observational Data. Philadelphia: SIAM (1990) 169 pp.
C. Kok and H.P. Urbach, On the regularization of the inverse Laplace transform in grazing-emission x-ray fluorescence spectroscopy. Inv. Probl. Eng. 7 (1999) 433–470.
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Smolders, S., Urbach, H. On the determination of dopant-concentration profiles by grazing emission X-ray fluorescence spectroscopy using the maximum-entropy method. Journal of Engineering Mathematics 43, 115–134 (2002). https://doi.org/10.1023/A:1020324522086
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DOI: https://doi.org/10.1023/A:1020324522086