Novel method for error limit determination in x-ray reflectivity analysis
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چکیده
A novel error limit determination method for x-ray reflectivity (XRR) analysis is developed and applied to data measured from atomic-layer-deposited aluminium oxide on silicon. The analysis here is based on Parratt’s formalism and on a fitness defined as a mean-squared error between a measurement and a fit in logarithmic scales. The mathematically derived upper bound for an error uses a trick which divides the fitness into two parts. The divided original fitness equals the fitness between a measurement and a numerically optimal but unknown fit (the first part) plus the fitness between the unknown optimal fit and the known original fit (the second part). In practical error determination, the fitness in the first part is the fitness of noise and it is approximated using a separate simulation and in the second part, the unknown optimal fit is considered as a variable to be optimized. An efficient implementation is presented for the error determination and the determined parameters were 42.4 ± 0.12 nm (0.3%), 3.15 ± 0.11 g cm−3 (3.5%) and 0.80 ± 0.06 nm (7.5%) for the thickness, the mass density and the surface roughness, respectively. Although the formalized error may need some fine tuning as future work since it gives an asymptotic estimate, it still gave reasonable results in the case of systematic error caused by nonideal fit.
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تاریخ انتشار 2008