Evaluation of uncertainty in measuring thin crystal thickness and extinction distance by Kossel-Möllenstedt pattern analysis

Abstract

In this paper, the local thickness of single crystal Si film sample and the extinction distance <inline-formula><tex-math id="M11">\begin{document}$ {\xi }_{400} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M11.png"/></alternatives></inline-formula> of the (400) plane of Si crystal are obtained by analyzing the double-beam converging beam diffraction (CBED) pattern of single crystal Si film sample under the 200 kV of accelerated voltage. The factors affecting the measurement uncertainty are analyzed, and the influence coefficients of each factor on the measurement uncertainty are discussed by using the concept of first-order partial derivative. The measurement uncertainty of thin crystal thickness and extinction distance are evaluated and expressed according to national standards GB/T 27418-2017. The conclusions are as follows. The local thickness of the measured Si crystal is estimated at 239 nm, the combined standard uncertainty is 5 nm, and the relative standard uncertainty is 2.2%. With the inclusion probability being 0.95, the coverage factor is 2.07 and the expanded uncertainty is 11 nm. With the accelerated voltage being 200 kV, the extinction distance of Si crystal (400) plane is estimated at 194 nm, the combined standard uncertainty of the extinction distance is 20 nm, and the relative standard uncertainty of the extinction distance is 10%. With the inclusion probability being 0.85, the coverage factor is 1.49 and the expanded uncertainty is 30 nm. The main factors that can affect the combined standard uncertainty of sample thickness <i>t</i>₀ are camera constant, accelerating voltage and sample thickness, while the factors that influence the combined standard uncertainty of extinction distance are camera constant, accelerating voltage and extinction distance. The influence of the uncertainties of the measurement data of the Kossel-Möllenstedt pattern on the uncertainty of the extinction distance is <inline-formula><tex-math id="M12">\begin{document}${n}_{i}{\left( {\xi }/{t}\right)}^{3}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M12.png"/></alternatives></inline-formula> times that on the sample thickness, and their influence on the slope of the fitting line is about <inline-formula><tex-math id="M13">\begin{document}$ {n}_{i} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M13.png"/></alternatives></inline-formula> times that on the intercept of the line, where <inline-formula><tex-math id="M14">\begin{document}$ {n}_{i} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M14.png"/></alternatives></inline-formula> is a positive integer and greater than or equal to 1. If the sample is not too thin, that is, <inline-formula><tex-math id="M15">\begin{document}$ {n}_{i} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212271_M15.png"/></alternatives></inline-formula> is greater than 1, then the uncertainty of crystal thickness will be smaller than the uncertainty of extinction distance.