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FDTD and SFDTD Software for DUV Lithography
Why is a new EMF simulator needed for EUV lithography?
The FDTD method, which is widely used in DUV lithography simulation, turns out to be completely unsuitable for EUV lithography simulation, because of the following limitations:
1. All FDTD cells must have the same cell size
2. FDTD has large numerical dispersion error which varies with the direction of propagation
3. FDTD has difficulty in modeling buried defects in multilayered mirrors
These limitations are all overcome by the PSTD method, as explained below.
Consider an FDTD grid used to model a single bilayer of a multilayered mirror:
In the above example, there are 4 FDTD cells in the molybdenum region and 6 in the silicon region. Since FDTD employs the finite-difference approximation for the spatial derivatives,
in order to achieve second-order accuracy, all the FDTD cells must have the same cell size in the Z direction, namely, 0.7nm in the above example. This means that the thicknesses of the molybdenum and silicon layers are forced to be in the ratio of small integers, namely, 4:6 in the above example. This is a major limitation of FDTD, since in reality the thicknesses of the molybdenum and silicon layers can be quite different, for example, 2.78nm and 4.17nm, respectively. For a 40-bilayer mirror with 2.78nm/4.17nm Mo/Si thicknesses, the exact TE reflectivity is 0.8367 at normal incidence and 0.4222 at 17 degrees incidence, whereas with 2.8nm/4.2nm Mo/Si thicknesses, the exact TE reflectivity is 0.8014 at normal incidence and 0.2912 at 17 degrees incidence. Thus, by forcing the Mo/Si thicknesses to be 2.8nm/4.2nm, FDTD immediately incurs an error of 4.4% at normal incidence and a whopping 45% at 17 degrees incidence! This error is in additon to other errors of the FDTD method, in particular, numerical dispersion error.
Next, consider a PSTD grid used to model the bilayer:
Notice that the actual Mo/Si thicknesses (2.78nm/4.17nm in the above example) are used in the PSTD grid and that the PSTD nodes in the Mo and Si layers have different spacings depending on the layer thicknesses. Furthermore, the PSTD nodes in each layer are not spaced evenly, but, rather, are positioned at the Gauss-Lobatto quadrature points for that layer. This is to ensure maximum accuracy in the polynomial interpolation of the fields within each layer.
In the PSTD and other spectral methods, the polynomials used for field interpolation are the Jacobi polynomials, examples of which are the Chebyshev and Legendre polynomials. The nature of these polynomials is such that the convergence is exponentially fast, that is, faster than any algebraic power of the polynomial order. Once the field within each layer is approximated by an expansion in these polynomials, the spatial derivative of the field is calculated by direct differentiation of the polynomials themselves, without using any finite-difference approximation, as shown below for the Mo layer. This fact alone accounts for the very small numerical dispersion error of the PSTD method.
To see how accurate the PSTD method is, consider the reflectivity of a 40-bilayer flat mirror. First, the results of such a reflectivity calculation using the FDTD method (taken from the literature) are shown below, as a function of the angle of incidence:
Notice how inaccurate the FDTD results are, especially at large angles of incidence. This is because FDTD has large numerical dispersion error which varies with the direction of propagation. It is possible to employ dispersion compensation technique, such as by tweaking the wavelength or the material parameters, to compensate for the numerical dispersion error for a single direction of propagation of the EM wave, but not for all the directions of propagation of the diffracted waves present in a typicial mask diffraction simulation.
A corresponding calculation using the PSTD method would yield results which are almost indistinguishable from the analytic curve on the scale of the above plot. To be precise, consider the reflectivity at 17 degrees angle of incidence. The PSTD results are shown below:
Here, it can be seen that even with low order polynomials, the PSTD results are within 0.5% in amplitude and 6 degrees in phase of the exact results. Such accuracy is usually adequate for the purpose of lithography simulation. However, for metrolgy simulation purposes, this may not be enough. Fortunately, it is easy to increase the accuracy of PSTD by simply increasing the polynomial order, keeping the cell size fixed, because of the exponential convergence nature of the PSTD method. This way, one easily obtains an accuracy of 0.1% in amplitude and 0.3 degree in phase for the computed reflectivity, as shown above.
However, such an approximation is known to be very inaccurate unless the cell size is extremely small, which would render the FDTD simulation very slow.
In the PSTD method, the buried defect is modeled simply by deforming the PSTD cells into a nonplanar shape to fit nicely the nonplanar layer interfaces, as shown below for a vertical slice through the PSTD computation domain. This way, the boundary conditions on the nonplanar layer interfaces are satisfied accurately, resulting in very accurate simulation of the buried defect.
