1

 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 UrbanaChampaign

2

 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)
 CGFFT needs special modification for X Architecture lithography
 Capabilities
 Volumesurface 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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 Singlelayer Cr (e_{r}=0.77j*1.51)
photo mask: 8x8x0.1l^{3
}slab, l=157nm
 Observation plane: 4.5x4.5l^{2}
plane, 0.05l below the
mask
 Evaluating current on the volume mesh (70,282 tetrahedron bases) by a
fivelevel 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.)

4

 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 (e_{r}=0.77j*1.51)
 Observation plane: 4.5x4.5l^{2}
plane, 0.05l above the
patterns
 Evaluating current on the mesh (11,053 surface bases, 18,756 tetrahedron
bases) by a fourlevel 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.)

6

 Hybridize FAFFA with MLFMA.
 Up to 40 % speedup for 1 million unknowns.

7

 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 surfacewire 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 surfacewire 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 MaxwellWagner 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

15

 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
