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PSTD Software for EUV Lithography
Fastlitho, the developer of the S-FDTD (subgrid and subcell FDTD) method [Ref. 1] for lithography simulation, has added fully implicit FDTD simulation capability to its software. With the fully implicit method, one can now run FDTD and S-FDTD simulations at many times the Courant stability limit!
Consider the numerical solution of the Maxwell equation,
The usual FDTD method of discretization is
Notice that the first term on the right hand side of the above equation contains only the known magnetic fields at the time step (n+1/2), while the second term contains both the unknown electric field at the time step (n+1) and the known electric field at the time step n. This is a semi-implicit method. As such, it is numerically stable only if the time step is less than the Courant stability limit:
Now consider a different discretization,
Notice that the first term on the right hand side now contains both the unknown magnetic fields at the time step (n+1) and the known magnetic fields at the time step n. This is a fully implicit method. As such, it is unconditionally stable. Any time step can now be used, provided that it gives the desired accuracy.
The drawback of the fully implicit method is that the unknown electric and magnetic fields in the computational domain are all coupled together through the spatial derivatives. Hence, they must be solved simultaneously by inverting a large matrix. However, our proprietary fully implicit software requires only about 3 times the computational effort of the much simpler, semi-implicit method. Substantial saving in computation time can therefore be obtained with the fully implicit method by choosing the time step large enough.
As the feature size in optical lithography is reduced, the grid size used in lithography simulation must also be reduced to resolve the smaller mask or wafer features. This is especially true in the simulation of wafer topography scattering, where a spatial grid resolution of 1nm on the wafer is desired in sub-45nm lithography. At this grid resolution, the Courant stability limit gives
which corresponds to more than 300 time samples per wave cycle for an exposure wavelength of 193nm. Clearly, this is oversampling in the time domain, since the waveform in the time domain is always smoothly varying and can therefore be accurately modeled by far fewer time samples per wave cycle. This is where the fully implicit FDTD method can be very useful.
As an example, consider the near-field simulation of topography scattering by a rectangular, 70nm quartz feature. The grid sizes used for the simulation are 1nm in X and Y and 5nm in Z. In the FDTD simulation, 292 time samples per wave cycle were used, which was close to the Courant limit of 276 time samples per wave cycle. However, it was found that only 12 time samples per wave cycle were needed to give good accuracy in the fully implicit FDTD simulation. This is 23 times the Courant limit!
Shown below are the plots of the magnitidue of the near-field instantaneous electric field obtained by the two methods, with the FDTD results on the left and the fully implicit FDTD results on the right. Notice how accurately the sharp discontinuities of the electric field at the feature edges in TM are resolved in both methods. This is because the two methods used the same spatial grid sizes.
TM:
TE:
In summary, Fastlitho is the only company that has incorporated all three advanced FDTD simulation capabilities in its software: (i) The subgrid method, (ii) the subcell method and (iii) the fully implicit method. As usual, our software runs on both the CPU and the GPU platforms. A brief description of one of our products is shown below.
Fastlitho also specializes in pseudo-spectral time-domain (PSTD) software, a highly accurate alternative to FDTD, and provides consultation service in lithography algorithm development. Please contact us by email (support@fastlitho.com) or by phone (408-287-0865) for inquiries about our business.
1. M. S. Yeung, "Fast and Accurate Subgrid and Subcell FDTD Methods for the Simulation of Mask Electromagnetic Effects in Sub-45nm Lithography", Proc. SPIE, vol. 7122, 71221T (2008)