Forward diffraction modelling : analysis and application to grating reconstruction

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 approach" (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

Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures

Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products

Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures

Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products

Forward diffraction modelling : analysis and application to grating reconstruction

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 approach" (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

A more efficient rigorous coupled-wave analysis algorithm

We present a modification of a well-known mathematical model based on the Rigorous Coupled-Wave Analysis (RCWA) that can be used to solve optical diffraction problems on periodic structures (both 1-D and 2-D gratings with approximated layer-structure). The algorithm calculates the reflected and transmitted field which in turn determine the diffraction efficiencies for all reflected and transmitted orders. Results created with a Matlab implementation of the modified RCWA algorithm (MSolver) show excellent overlap with other published and measured data

2D TM scattering problem for finite dielectric objects in a dielectric stratified medium employing Gabor frames in a domain integral equation

We present a method to simulate two-dimensional scattering by dielectric objects embedded in a dielectric layered medium with transverse magnetic polarization through a domain integral equation formulation. A mixed spatial-spectral discretization is employed with both a spatial and a spectral representation along the direction of the layer interfaces. In the spectral domain, a discretization on a path through the complex plane is used on which the Green function is well behaved. To calculate the field-material interaction in the spatial domain, an auxiliary field is employed similar to the Li factorization rules. Numerical results show that this auxiliary-field formulation significantly improves accuracy, compared to a formulation that directly employs the electric field

Methods and apparatus for determining electromagnetic scattering properties and structural parameters of periodic structures

Numerical calculation of electromagnetic scattering properties and structural parameters of periodic structures is disclosed. A reflection coefficient has a representation as a bilinear or sesquilinear form. Computations of reflection coefficients and their derivatives for a single outgoing direction can benefit from an adjoint-state variable. Because the linear operator is identical for all angles of incidence that contribute to the same outgoing wave direction, there exists a single adjoint-state variable that generates all reflection coefficients from all incident waves that contribute to the outgoing wave. This adjoint-state variable can be obtained by numerically solving a single linear system, whereas one otherwise would need to solve a number of linear systems equal to the number of angles of incidence

A pseudo-spectral longitudinal expansion in a spectral domain integral equation for scattering by periodic dielectric structures

We explore several options to introduce a pseudo-spectral expansion along the longitudinal direction in a spectral-domain integral equation for scattering by periodic dielectric structures. To this end we first simplify the integral equation to the formulation for a one-dimensional dielectric slab and consider the computational efficiency, convergence, and conditioning of several schemes. One scheme has been implemented in the domain integral equation and it exhibits exponential convergence with respect to the number of expansion functions in the longitudinal direction