Grazing incidence x-ray fluorescence based characterization of nanostructures for element sensitive profile reconstruction

Abstract

For the reliable fabrication of the current and next generation of nanostructures it is essential to be able to determine their material composition and dimensional parameters. Using the grazing incidence X-ray fluoresence technique, which is taking advantage of the X-ray standing wave field effect, nanostructures can be investigated with a high sensitivity with respect to the structural and elemental composition. This is demonstrated using lamellar gratings made of Si 3 N 4 . Rigorous field simulations obtained from a Maxwell solver based on the finite element method allow to determine the spatial distribution of elemental species and the geometrical shape with sub-nm resolution. The increasing complexity of nanostructures and demanded sensitivity for small changes quickly turn the curse of dimensionality for numerical simulation into a problem which can no longer be solved rationally even with massive parallelisation. New optimization schemes, e.g. machine learning, are required to satisfy the metrological requirements. We present reconstruction results obtained with a Bayesian optimization approach to reduce the computational effort.

Recent advances in Bayesian optimization with applications to parameter reconstruction in optical nano-metrology

“…We have also shown that it outperforms the conventional BO, which has been successfully employed in the reconstruction of parameters in optical metrology. 29,30 We are therefore preparing a publication in which we are investigating the applicability of the BTVO to a comparable problem from optical metrology.…”

Shape- and Element-Sensitive Reconstruction of Periodic Nanostructures with Grazing Incidence X-ray Fluorescence Analysis and Machine Learning

“…The is high where we have not evaluated the function or found values close to a minimum of the function. In a previous work [ 27 , 41 ], it was shown that the BO performs much better than other metaheuristic optimization approaches with respect to the computing time needed to find the global minimum. Since BO considers all previous function evaluations, it can be more efficient than other metaheuristic global optimization strategies [ 27 ] and local optimization strategies [ 42 ].…”

“…This is a crucial benefit here, as one model calculation takes several minutes on a standard desktop computer. For a detailed benchmark study of the different optimizer methods in comparison to BO, see [ 27 , 41 ]. For our problem, the BO gives good results in a reasonable amount of time [ 41 ].…”

“…For a detailed benchmark study of the different optimizer methods in comparison to BO, see [ 27 , 41 ]. For our problem, the BO gives good results in a reasonable amount of time [ 41 ]. Here, we use an implementation of BO that is part of the JCMsuite software package [ 43 ].…”

Shape- and element-sensitive reconstruction of periodic nanostructures with grazing incidence X-ray fluorescence analysis and machine learning

“…BO uses a stochastic model, a Gaussian process, of an unknown objective function to be minimized in order to determine promising parameter values [24,37]. In a previous work [25,38], it was shown that the BO performs much better than other meta-heuristic optimization approaches with respect to the computing time needed to find the global minimum. Since BO considers all previous function evaluations, it can be more efficient than other meta-heuristic global optimization strategies [25] and local optimization strategies [39].…”