On numerical reconstructions of lithographic masks in DUV scatterometry

The solution of the inverse problem in scatterometry employing deep ultraviolet light (DUV) is discussed, i.e. we consider the determination of periodic surface structures from light diffraction patterns. With decreasing dimensions of the structures on photo lithography masks and wafers, increasing demands on the required metrology techniques arise. Scatterometry as a non-imaging indirect optical method is applied to periodic line structures in order to determine the sidewall angles, heights, and critical dimensions (CD), i.e., the top and bottom widths. The latter quantities are typically in the range of tens of nanometers. All these angles, heights, and CDs are the fundamental figures in order to evaluate the quality of the manufacturing process. To measure those quantities a DUV scatterometer is used, which typically operates at a wavelength of 193 nm. The diffraction of light by periodic 2D structures can be simulated using the finite element method for the Helmholtz equation. The corresponding inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. Fixing the class of gratings and the set of measurements, this inverse problem reduces to a finite dimensional nonlinear operator equation. Reformulating the problem as an optimization problem, a vast number of numerical schemes can be applied. Our tool is a sequential quadratic programing (SQP) variant of the Gauss-Newton iteration. In a first step, in which we use a simulated data set, we investigate how accurate the geometrical parameters of an EUV mask can be reconstructed, using light in the DUV range. We then determine the expected uncertainties of geometric parameters by reconstructing from simulated input data perturbed by noise representing the estimated uncertainties of input data. In the last step, we use the measurement data obtained from the new DUV scatterometer at PTB to determine the geometrical parameters of a typical EUV mask with our reconstruction algorithm. The results are compared to the outcome of investigations with two alternative methods namely EUV scatterometry and SEM measurements.