Forward diffraction modelling : analysis and application to grating reconstruction

Abstract

The semiconductor industry uses lithography machines for manufacturing complex integrated circuits (also called ICs) onto wafers. Because an IC is built up layer by layer and feature sizes get smaller and smaller, tight control of the lithography process is required to guarantee a fast production of working ICs. Typically a lot of information on the lithography process can be obtained by measuring test structures or gratings which are scattered over the wafer. These gratings are tiny periodic structures much smaller than ICs. First these gratings are illuminated and its response (a scattered intensity) is measured. For certain applications like overlay metrology the asymmetry in this measured signal (due to an offset between two gratings) can be used to align the lithographic process. For other applications like critical dimension (CD) metrology one is interested in the shape of the grating lines that produced the measured signal. Since this information is not directly available but encrypted in the measurement, a reconstruction algorithm is used to extract it. The reconstructed values like height, width and sidewall angle can then be related to machine settings like dose and focus which control the lithographic process. In particular the CD metrology application requires rigorous mathematical models that solve optical diffraction problems for periodic gratings in combination with advanced reconstruction algorithms. This thesis focuses on the optical diffraction problem for 1D periodic gratings. Starting from Maxwell's equations a reduced model is derived by simplifying both the grating and the incident electromagnetic field. The former is approximated with an infinitely periodic layered structure with isotropic non-magnetic materials. The latter is approximated with a time-harmonic incident plane wave. The reduced model is discretised using two different mode expansion methods, Bloch and the Rigorous Coupled-Wave Analysis (RCWA). Bloch expands the electromagnetic field in each layer in terms of the exact eigenfunctions whereas RCWA only uses approximate eigenfunctions. After truncation of the involved series a transmission problem is derived by matching the fields at the layer interfaces. Having solved the resulting linear system, the scattered field can be computed easily. Both mode expansion methods solve a similar linear system containing a large but sparse block-structured coefficient matrix. However, special care needs to be taken when solving this system stably and efficiently. Therefore a stable condensation algorithm is derived based on Riccati transformations that decouples the exponentially growing and decaying terms that are present in the solution. This separation or decoupling is the key feature explaining the stability which is not always clear in alternative condensation algorithms. Furthermore the algorithm is optimised for speed by using a two-stage approach. Finally it is shown that the resulting stable recursions are identical to those used in the 'enhanced transmittance matrix (a frequently used condensation algorithm), thereby confirming its stability as well. This thesis also examines and extends both mode expansions methods. The Bloch method is generalised to deal with multiple material transitions inside a grating layer covering a wider range of applications. However, lossy or fully asymmetric gratings are still hard to solve. On the other hand the Fourier discretisation used in RCWA is much more exible but only approximates the more exact discretisation of Bloch. Therefore two RCWA modifications have been investigated to improve the accuracy while keeping its exibility and relatively straightforward implementation. Adaptive Spatial Resolution applies an additional layer specific coordinate transformation before Fourier discretising the problem again. A good transformation not only refines near a material interface but also does this in a smooth way. A significant improvement in accuracy is observed that approaches and sometimes outperforms the results obtained with the Bloch method. The second modification removes the Fourier discretisation completely and uses a finite difference approximation in the periodic direction. Although this approach allows for a better discretisation near a material interface, the sparsity of the resulting matrices could not be exploited to make a competitive implementation within the standard RCWA framework. Finally the integration of the forward diffraction model in the CD reconstruction application is discussed. Either a library based or real-time regressions approach can be used for this reconstruction. Both approaches rely heavily on having an accurate and fast forward model. By exploiting additional symmetries and smart reuse of information, acceptable library fill times and real-time reconstructions are now feasible.