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- H. Y. Chao, C. Lin, and W. C. Chew
- Center for Computational Electromagnetics
- and Electromagnetics Laboratory
- Department of Electrical and Computer Engineering
- University of Illinois at Urbana-Champaign
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- Motivation
- Apply MLFMA for photolithography simulation
- Fast calculation of aerial images for critical features (gates,
contacts, vias, etc.)
- Avoid grid dispersion errors in FEM and FDTD. No ABC.
- Expedite the traditional method of moments (MOM)
- CG-FFT needs special modification for X Architecture lithography
- Capabilities
- Volume-surface integral equation (VSIE) for modeling inhomogeneous
dielectric and PEC scatterers (photomasks, multilayered reflective
coating)
- Impedance boundary condition (IBC) for modeling thin dielectric coating
and lossy metal surfaces (material surfaces beneath PR)
- Resistive surface formulation for modeling thin dielectric sheets (TDS)
(add phase shifts to incident field)
- The code is originally designed for wire antenna simulations and
contains multiple types of basis functions: surface, wire, junction,
and volume.
- Examples
- Formulation
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- Single-layer Cr (er=-0.77-j*1.51)
photo mask: 8x8x0.1l3
slab, l=157nm
- Observation plane: 4.5x4.5l2
plane, 0.05l below the
mask
- Evaluating current on the volume mesh (70,282 tetrahedron bases) by a
five-level MLFMA: 3 hrs., 510 MB of memory, 10 iterations for error=10-3
- Evaluating field on 90x90 observation points: 12 mins., 200 MB of memory
(direct computation: 740 mins.)
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- With a circularly polarized source, only the optical proximity between
two horizontal bars needs to be corrected.
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- Scattered field from a PEC mask with Cr patterns (er=-0.77-j*1.51)
- Observation plane: 4.5x4.5l2
plane, 0.05l above the
patterns
- Evaluating current on the mesh (11,053 surface bases, 18,756 tetrahedron
bases) by a four-level MLFMA: 8 hrs, 680 MB of memory, 3130 iterations
for error=10-3
- Evaluating field on 90x90 observation points: 22 mins., 160 MB of memory
(direct computation: 337 mins.)
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- Hybridize FAFFA with MLFMA.
- Up to 40 % speedup for 1 million unknowns.
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- MLFMA has been applied for calculating currents on photomasks and aerial
images on object planes
- The computational complexities for memory requirement and CPU time are
O(N) and O(N log N), respectively, where N is the number of unknowns
- Fields in dielectric materials are modeled by a volume integral equation
- Number of unknowns and simulation time can be significantly reduced if
SIE (PMCHW formulation) is used instead of VSIE
- VSIE is more efficient for simulating structures with thin dielectric
layers
- Fast field calculation by MLFMA applies to both VSIE and SIE
formulations
- Promising technologies for fast aerial image calculation for the next
generation photolithography simulation
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- For objects with imperfectly conducting surfaces , PEC wires , PEC surface-wire junctions , and inhomogeneous dielectric
regions , the VSIE is
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- The EFIE and MFIE operators are
- IBC for lossy conducting surfaces or conducting surfaces with thin
dielectric coatings
- For resistive sheets (RS) and thin dielectric sheets (TDS), the VSIE is
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- Represent current distribution on surfaces and wires by surface, wire,
and surface-wire junction basis functions
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- The current in dielectric region is discretized by volume basis
functions
- Limitation
- Wires cannot connect to impedance surfaces
- Basis functions cannot intersect each other
- Using the method of moments, one obtains a matrix equation
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- The components of impedance matrix for b=S,W,J are
- The vector and scalar potentials for b=S,W,J are
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- For volume basis functions
- The last term in is
associated with the Maxwell-Wagner charges at an interface of dissimilar
media.
- The excitation vector is
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- Multipole expansion is performed for EFIE and MFIE operators separately
- The radiation and receiving patterns are
- Multilevel implementation: grouping subscatterers recursively and
performing interpolation and anterpolation to calculate radiation and
receiving patterns for groups in each level
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- Using the symmetry of radiation
and receiving patterns, the memory requirement can be reduced by 58%.
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- Multipole expansion is performed for EFIE and MFIE operators separately
- The radiation and receiving patterns are
- Multilevel implementation: grouping subscatterers recursively and
performing interpolation and anterpolation to calculate radiation and
receiving patterns for groups in each level
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