Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates

Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for extreme ultraviolet lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. The inverse problem can be formulated as a nonlinear operator equation in Euclidean space. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gauss-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period in the range of 420 nm-840 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.