The rigorous coupled wave analysis (RCWA) is a widely used method for simulating diffraction from periodic structures. Since its recognized formulation by Moharam [J. Opt. Soc. Am. A12, 1068 and 1077 (1995)], there still has been a discussion about convergence problems. Those problems are more or less solved for the diffraction from line gratings, but there remain different concurrent proposals about the convergence improvement for crossed gratings. We propose to combine Popov and Nevière's formulation of the differential method [Light Propagation in Periodic Media (Dekker, 2003) and J. Opt. Soc. Am. A18, 2886 (2001)] with the classical RCWA. With a suitable choice of a normal vector field we obtain a better convergence than for the formulations that are known from the literature.
Early formulations of the RCWA yield, implicated by the erroneous application of factorization rules to discrete Fourier transformations, poor convergence in certain cases. An explanation for this finding and an approach to overcome the problem for crossed gratings was first given by Li [J. Opt. Soc. Am. A 13, 1870 (1996) and 14, 2758 (1997)]. A further improvement was achieved by Schuster et al. [J. Opt. Soc. Am. A 24, 2880 (2007)], using a structure dependent normal vector (NV) field. While it is trivial to create those NV fields for simple geometrical shapes, to our knowledge an appropriate algorithm for arbitrary shapes does not exist, yet. In this work we present such an algorithm.
In Fourier modal methods like the RCWA and the Differential Method the Li-rules for products in truncated Fourier space have to be obeyed in order to achieve good convergence of the results with respect to the mode number. The Lirules have to be applied differently for parts of the field that are tangential and orthogonal to material boundaries. This is achieved in the Differential Method by including a field of vectors in the calculation that are normal to the material boundaries. The same can be done laterally in each layer of an RCWA calculation of a 2-D periodic structure. It turns out that discontinuities in the normal vector field can disturb the computation especially when metallic materials are dominant in the structure which would make the usefulness of the normal vector method questionable. So it is of great importance to investigate how normal vector fields can be established with as few discontinuities as possible. We present various methods for the 2-D RCWA and the 1-D and 2-D Differential Method and compare the respective convergence behaviors. Especially we emphasize methods that are automatic and require as few user input as possible.
Scatterometry or optical CD metrology (OCD) has become one of the most common techniques in quantitative wafer metrology within the recent years. Different tool configurations are either available in commercial inspection tools or subject of recent and present research activities. Among these are normal incidence reflectometry, 2-θ scatterometry, spectroscopic ellipsometry and angle resolved Fourier scatterometry. The two latter techniques appear to be promising for future use in semiconductor fabs. Spectroscopic ellipsometry is well established, and Fourier scatterometry has become of increasing interest within the recent time. Line edge roughness, i.e. an edge position variation of printed lines in lithography, has been of less importance up to now, as its amplitude could largely be neglected with respect to the feature dimensions. This will, however, not be the case for future nodes, as on the one hand CDs are getting smaller and smaller, and on the other hand, even the absolute amplitude is expected to increase due to the higher complexity of lithography and etch processes. In this paper a comparison of scatterometric reconstructions in both spectroscopic and angle resolved techniques considering LER afflicted samples is presented. The validity and benefit of a simple effective medium model is investigated.
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