Domain Decomposition Method for Electromagnetic Scattering Problems: Application to EUV Lithography

Transcription

1 Domain Decomposition Method for Electromagnetic Scattering Problems: Application to EUV Lithography Lin Zschiedrich, Sven Burger, Achim Schädle, Frank Schmidt Zuse Institute Berlin, JCMwave GmbH NUSOD, 21 September 2005 DFG Research Center Matheon

2 Photolithography Imprint the integrated circuits to a semi-conductor chip Transfer mask pattern by optical projection Light Source Mask Wafer EUV Light Source EUV Mask Wafer Mirror Left: State-of-the-art tools, wavelengths of 250 and 190 nm. Minimum feature sizes down to 65nm. Right: Extreme Ultraviolet (EUV) radiation system, 13nm wavelengths. Dramatically reduced resist feature sizes.

3 EUV Mask z Illuminating Plane Wave (kx, ky, kz) y x Air Line Multi Layer Stack Substrate Pitch Multilayer coating serves as a mirror. Leads to huge computational domains.

4 Ein Domain Decomposition Computational Domain y=h Solve scattering problem in upper domain with incoming waves E in and E n Material 1 Outgoing Wave Reflected Wave Eout Material 1 y=0 En y=0 Compute E out Solve scattering problem in lower domain with incoming wave E out n n+1 Multi Layer Stack Substrate Compute reflected wave E n+1 See also: Despres ( 97), Toselli ( 98), Colino & Joly ( 00), Gander ( 02)

5 DD: Building Blocks Scattering Problems: Finite elements with transparent boundary condition in structure blocks and for perturbed layer blocks. Fourier-mode expansion in non-perturbed layer block (semi-analytic solution) Coupling Problems: FEM FEM Coupling (with transparent boundary conditions!) FEM Fourier Mode Coupling Fourier Mode FEM Coupling

6 DD: Building Blocks FEM FEM FOURIER

7 DD: Maxwell s Equations Periodic and z invariant Media: ε(x + a, y) = ε(x, y), µ(x + a, y) = µ(x, y) Bloch periodic incoming field: E inc (x + a, y, z) = E inc (x, y)e ik xa e ik zz Maxwell s equation: with curl kz = curl, z ik z curl kz µ 1 curl kz E ω 2 ɛe = 0 Bloch Periodicity: E(x + a, y) = E(x, y)e ik xa

8 Fourier Modes Expansion Most simple situation: µ and ε are constant E sc E inc E out Fourier expansion of E sc : E sc (x, y) = n e i(n2π/a+k x)x ( e n,+ e ik y,ny + e n, e ik y,ny ) with k y,n = k 2 0 (n2π/a + k x) 2 k 2 z Radiation boundary condition: e n, = 0

9 Anomalous Modes Real(k y ) > (Radiating mode, Sommerfeld like boundary condition) Imag(k y ) > (Evanescent mode) k y = 0 (Anomalous mode)

10 Fourier Coupling Fourier expansion of incoming field and reflected field: E inc (x, y) = n E out (x, y) = n e i(n2π/a+k x)x e n,inc e ik y,ny e i(n2π/a+k x)x e n,out e ik y,ny k y,n threshold : Dirichlet Coupling e n,inc n ( e n,out n, ik y,n e n,out n) No evaluation of curl E inc n. k y,n < threshold : Mixed Dirichlet Neumann Coupling Splitting into two polarization directions. Use weak evaluation of curl E inc n.

11 Finite Element Problem E inc E out y=0 Ω + = [0, a] [0, ) Ω = [0, a] [ h, 0] Ω = [0, a] (, h] E inc E out y= h Split field: with E = ( E E z ) E H B ( curl ) := { u H ( curl ) u(x + a, y) = e ik xa u(x, y) } E z H 1 B := { u H 1 u(x + a, y) = e ik xa u(x, y) } See also: Elschner et al. (1999) Theory & (E z, H z ) formulation

12 Bloch Periodic Finite Elements Phase jump exp(ik xa) pitch=a Discretization of H with Nedelec elements, H z with Lagrange elements.

13 Coupled Exterior Interior System Ω + = [0, a] [0, ) Interior Problem curl kz µ 1 curl kz E ω 2 εe = 0 Exterior Problem curl kz µ 1 + curl k z E sc ω 2 ε + E sc = 0 Matching condition at y = 0 E n = (E sc + E inc ) n µ 1 curl E = ( µ 1 + E sc + µ 1 + E ) inc n

14 Perfectly Matched Layer Idea: Complex continuation (y (1 + iσ)y) of E sc (x, y) E PML (x, y) = n e n,+ e ik y,n(1+iσ)y e i(n2π/a+k x)x Exponential decaying field: E PML (x, y) e κy with κ = min n {Rk y,n, Ik y,n }. Truncate exterior domain: Ω ρ = [0, a] [0, ρ] κ = 0 in the presence of anomalous modes!

15 PML for anomalous modes The (near) anomalous modes are slowly varying in y. Idea: Use very, very large PML layer but very few discretization points. Interior Domain Exterior Domain h 1 h j Heuristical choice of h j+1 : Determine k y,max such that e ik y,maxρ j threshold, ρ j = h j j j Waves with λ min <= 2π/k y,max are sufficiently damped h j = max{h 1, λ min /N}

16 Total reflection E inc E sc E out k z k x Totalreflection Rel.Err Inclination Error at wood anomaly: 4.78e-3, 1.1e-3, 2.9e-4, 7.2e-5 (uniform interior grid refinement)

17 Real world example 10 1 Reflection off Dense Line, 40 MoSi layers, α = 4deg, θ = 30deg, P Pol. Regular Refinement, zero order Regular Refinement, first order Regular Refinement, seventh order Regular Refinement, eighth order Reflection off Dense Line, 40 MoSi layers, α = 4deg, θ = 30deg, P Pol., Domain Decomp. Zero order First order Seventh order Eighth order Relative Error 10 3 Relative Error sec 94sec 587sec N 32sec 78sec 316sec 1330sec N

18 Outlook 3D Photolithography mask. All building blocks for domain decomposition available in JCMharmony (by JCMwave) Isolated Structures, e.g. contact holes. Finite Element solver available. Coupling of domains requires exterior field evaluation (pole condition). Talk by Frank Schmidt et al.: Benchmark of FEM, waveguide, and FDTD algorithms for rigorous mask simulation, Photomask Technology 2005, Monterey, 4 October 2005