Measurement of layer thicknesses with an improved optimization method for depolarizing Mueller matrices

Measurement of layer thicknesses with an improved optimization method for depolarizing Mueller matrices Tobias Grunewald http://orcid.org/0000-0003-4909-1129 Matthias Wurm Sven Teichert Bernd Bodermann Johanna Reck Uwe Richter Abstract

There are some commonly-used optimization techniques for the analysis of measured data in spectroscopic Mueller matrix ellipsometry (MME) used, for example, to calculate the layer thicknesses of samples under test. Concentrating on the metrological aspects of MME, we identified a non-optimal treatment of depolarization in all these techniques. We therefore recently developed an improved optimization method to adequately take depolarization in MME into account. In a further step, we also included statistical measurement noise and derived a likelihood function, which enabled us to apply both the maximum likelihood method and Bayesian statistics as well as the Bayesian information criterion for data evaluation. In this paper we concentrate on the application of this new method to measurements of SiO 2 -layer thicknesses on silicon. With a state-of-the-art SENTECH SENresearch 4.0 Mueller ellipsometer, we measured standard samples of different SiO 2 -layer thicknesses, whose calibrated thicknesses were between about 6 nm and 1000 nm. The MME results were compared to the calibration data. For all samples, an SiO 2 -SiO double-layer model turned out to be optimal. The measured total oxide layer thicknesses matched excellently with the calibration values, within the estimated range of uncertainties. All the results are presented here. This is the first comparison with traceable reference measurements demonstrating the validity of our novel MME analysis method.
08 27 2020 11 01 2020 115010 http://dx.doi.org/10.1088/crossmark-policy iopscience.iop.org Measurement of layer thicknesses with an improved optimization method for depolarizing Mueller matrices Measurement Science and Technology paper © 2020 The Author(s). Published by IOP Publishing Ltd 2019-12-18 2020-05-22 2020-08-27 http://creativecommons.org/licenses/by/4.0 https://iopscience.iop.org/info/page/text-and-data-mining 10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf https://iopscience.iop.org/article/10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da https://iopscience.iop.org/article/10.1088/1361-6501/ab95da/pdf Opt. Express Wurm 10.1364/OE.381244 28 8108 2020 Some aspects on the uncertainty calculation in Mueller ellipsometry Grunewald 120 2019 Proc. SPIE Hasche 10.1117/12.512216 5190 165 2003 Metrological characterization of nanometer film thickness standards for XRR and ellipsometry applications Metrologia Seah 10.1088/0026-1394/45/1A/08013 45 08013 2008 CCQM-K32 key comparison and P84 pilot study: amount of silicon oxide as a thickness of SiO2 on Si The BIPM key comparison database Proc. SPIE Cloude 10.1117/12.962889 1166 177 1989 Conditions for the physical realisability of matrix operators in polarimetry Opt. Lett. Ossikovski 10.1364/OL.40.000954 40 954 2015 Integral decomposition and polarization properties of depolarizing Mueller matrices Opt. Lett. Hingerl 10.1364/OL.41.000219 41 219 2016 General approach for modeling partial coherence in spectroscopic Mueller matrix polarimetry Opt. Lett. Ossikovski 10.1364/OL.41.004044 41 4044 2016 General formalism for partial spatial coherence in reflection Mueller matrix polarimetry Thin Solid Films Rossow 10.1016/S0040-6090(97)00777-3 313–4 97 1998 Depolarization/mixed polarization corrections of ellipsometry spectra J. Opt. Necas 10.1088/2040-8978/13/8/085705 13 2011 Variable-angle spectroscopic ellipsometry of considerably non-uniform thin films J. Appl. Phys. Herzinger 10.1063/1.367101 83 3323 1998 Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation Palik 1998 J. Opt. Soc. Am. Ibrahim 10.1364/JOSA.61.001622 61 1622 1971 Parameter-correlation and computational considerations in multiple-angle ellipsometry J. Opt. Soc. Am. Bu-Abbud 10.1364/AO.20.003020 20 3020 1981 Parameter correlation and precision in multiple-angle elipsometry Statist. Sci. Sherlock 10.1214/10-STS327 25 172 2010 The random walk metropolis: linking theory and practice through a case study Comput. Stat. Haario 10.1007/s001800050022 14 375 1999 Adaptive proposal distribution for random walk Metropolis algorithm Metrologia Heidenreich 10.1088/1681-7575/aae41c 55 201 2018 Bayesian approach to determine critical dimensions from scatterometric measurements J. Stat. Softw. van den Meersche 10.18637/jss.v030.c01 30 1 2009 xsample(): an R function for sampling linear inverse problems Joint Committee for Guides in Metrology, Technical Report 2008 Evaluation of measurement data—guide to the expression of uncertainty in measurement Busch Private comm. on calibration certificates 2015 J. Opt. Soc. Am. A Gil 10.1364/JOSAA.17.000328 17 328 2000 Characteristic properties of Mueller matrices J. Opt. Soc. Am. A Ossikovski 10.1364/JOSAA.27.000808 27 808 2010 Alternative depolarization criteria for Mueller matrices Opt. Acta Gil 10.1080/713821924 33 185 1986 Depolarization and polarization indices of an optical system Optik Cloude 75 26 1986 Group theory and polarisation algebra Appl. Phys. Lett. Jellison 10.1063/1.107871 61 512 1992 Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry Brosseau 1998 Brosseau 1998 Hecht 7 2018