Inverse problems for the scatterometric measuring of grating structures

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics PT B Inverse problems for the scatterometric measuring of grating structures Hermann Groß, Andreas Rathsfeld Mohrenstrasse Berlin Germany Tel IPMS, 23 May 2012

2 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 2 (42)

3 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 3 (42)

4 Lithography chip production like old-fashioned photography: photoresist layer illuminated by light scattered from mask, development: baking and etching procedures chip light source wafer photo resist mask standard test configurations: periodic line-space structure (lines formed by bridges with trapezoidal cross section) biperiodic array of trapezoidal blocks resp. holes Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 4 (42)

5 Scatterometry Spectroscopic Reflectrometer, Operating Range 13 nm, BESSY II, PTB Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 5 (42)

6 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 6 (42)

7 Periodic Grating incoming field reflected modes y θ i i (E,H ) photoresist z x chrome silicon oxide transmitted modes periodic gratings details of surface geometry in size of wavelength Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 7 (42)

8 Biperiodic (Doubly Periodic) Grating incoming plane wave θ reflected modes x 3 x 2 x 1 resist chrome silicon transmitted modes biperiodic surface structures (period d j in x j-direction, j = 1, 2) details of surface geometry in size of wavelength Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 8 (42)

9 Partial Differential Equation Plane-wave illumination. 3D time-harmonic Maxwell s equations reduce to: Curl-Curl equation for three-dimensional amplitude factor of time harmonic electric field (i.e. E(x 1, x 2, x 3, t) = E(x 1, x 2, x 3)e iωt ) E(x 1, x 2, x 3) k 2 E(x 1, x 2, x 3) = 0 Here: wave number k := ω µε scalar 2D Helmholtz equation if geometry is constant in x 3 direction and if direction of incoming plane wave is in x 1 x 2 plane v(x 1, x 2) + k 2 v(x 1, x 2) = 0, v = E 3, H 3 two coupled scalar 2D Helmholtz equations if geometry is constant in x 3 direction but direction of incoming plane wave is not in x 1 x 2 plane (conical diffraction) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 9 (42)

10 Partial Differential Equation Plane-wave illumination. 3D time-harmonic Maxwell s equations reduce to: Curl-Curl equation for three-dimensional amplitude factor of time harmonic electric field (i.e. E(x 1, x 2, x 3, t) = E(x 1, x 2, x 3)e iωt ) E(x 1, x 2, x 3) k 2 E(x 1, x 2, x 3) = 0 Here: wave number k := ω µε scalar 2D Helmholtz equation if geometry is constant in x 3 direction and if direction of incoming plane wave is in x 1 x 2 plane v(x 1, x 2) + k 2 v(x 1, x 2) = 0, v = E 3, H 3 two coupled scalar 2D Helmholtz equations if geometry is constant in x 3 direction but direction of incoming plane wave is not in x 1 x 2 plane (conical diffraction) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 9 (42)

11 Boundary Value Problem x2 x3 Domain of Computation: x 1 + Rectangular box over one Γ periodic cell Ω Non local boundary condition: Including radiation condition Mortaring with Fourier mode sol. (Coupling with boundary elements or Absorbing bound.cond. PML) Ω Transmission conditions: Continuous tangential traces Continuous tangential traces of curl period d 1 Γ Quasiperiodic boundary condition: v(x 1+d 1,x 2,x 3)=v(x 1,x,x ) exp(i αd 2 1) 3 v(x,x +d,x )=v(x,x,x ) exp(i β d ) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 10 (42)

12 Rayleigh Series Expansion Below/Above Computational Domain incident plane wave (inspecting w.): E inc (x 1, x 2, x 3) exp( iωt), ) E inc (x 1, x 2, x 3) := A inc exp (i k inc (x 1, x 2, x 3) E(x 1, x 2, x 3) = j,l= k inc = k + (sin θ cos φ, sin θ sin φ, cos θ) ( A j,l exp i k j,l (x 1, x 2, x 3) ), x 3 < x min, α j := k + sin θ cos φ + 2π d 1 j, k j,l := (α j, β l, γ j,l), β l := k + sin θ sin φ + 2π d 2 l, γ j,l := [k ] 2 [α j] 2 [β l ] 2 Rayleigh series: Rayleigh coefficients A j,l C3, A j,l k j,l, with k = k for x 3 < x min (similarly, coefficients A + j,l for expansion with x3 > xmax, where k = k+ ) Efficiency E ± j,l of (j, l) th mode: ratio of energy radiated into direction of mode E ± j,l := (γ j,l/γ ± 0,0) A + ± j,l 2 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 11 (42)

13 Rayleigh Series Expansion Below/Above Computational Domain incident plane wave (inspecting w.): E inc (x 1, x 2, x 3) exp( iωt), ) E inc (x 1, x 2, x 3) := A inc exp (i k inc (x 1, x 2, x 3) E(x 1, x 2, x 3) = j,l= k inc = k + (sin θ cos φ, sin θ sin φ, cos θ) ( A j,l exp i k j,l (x 1, x 2, x 3) ), x 3 < x min, α j := k + sin θ cos φ + 2π d 1 j, k j,l := (α j, β l, γ j,l), β l := k + sin θ sin φ + 2π d 2 l, γ j,l := [k ] 2 [α j] 2 [β l ] 2 Rayleigh series: Rayleigh coefficients A j,l C3, A j,l k j,l, with k = k for x 3 < x min (similarly, coefficients A + j,l for expansion with x3 > xmax, where k = k+ ) Efficiency E ± j,l of (j, l) th mode: ratio of energy radiated into direction of mode E ± j,l := (γ j,l/γ ± 0,0) A + ± j,l 2 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 11 (42)

14 Variational Equation 1 Ω with E F = Ω k 2 E F E + ν F + Γ + Γ + ν (E E + ) F + E ν F + ν (E E ) F + finite rank terms Γ Γ E inc ν F + ν E inc F + + finite rank term, F, F ± Γ + Γ ( ) a (E, E +, E ), (F, F +, F ) ( ) = F (F, F +, F ), F H(curl, Ω), F ± Rayleigh expansions solution (E, E +, E ): E H(curl, Ω) and E ± Rayleigh expansions in H loc (curl) Ω domain of computation, Γ ± upper/lower boundary face, ν normal at Γ ± H(curl, Ω) := {F [L 2 (Ω)] 3 : F quasi-per., piecewise F [L 2 (Ω)] 3, ν F continuous over interfaces} Theorem (Hu, R. 12) For profile gratings with( perfectly conducting ) substrate: operator corresponding to sesqui-linear A 0 0 form a takes the form + T, where A 0, A 1 strongly elliptic and T compact. 0 A 1 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 12 (42)

15 Variational Equation 1 Ω with E F = Ω k 2 E F E + ν F + Γ + Γ + ν (E E + ) F + E ν F + ν (E E ) F + finite rank terms Γ Γ E inc ν F + ν E inc F + + finite rank term, F, F ± Γ + Γ ( ) a (E, E +, E ), (F, F +, F ) ( ) = F (F, F +, F ), F H(curl, Ω), F ± Rayleigh expansions solution (E, E +, E ): E H(curl, Ω) and E ± Rayleigh expansions in H loc (curl) Ω domain of computation, Γ ± upper/lower boundary face, ν normal at Γ ± H(curl, Ω) := {F [L 2 (Ω)] 3 : F quasi-per., piecewise F [L 2 (Ω)] 3, ν F continuous over interfaces} Theorem (Hu, R. 12) For profile gratings with( perfectly conducting ) substrate: operator corresponding to sesqui-linear A 0 0 form a takes the form + T, where A 0, A 1 strongly elliptic and T compact. 0 A 1 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 12 (42)

16 Variational Equation 2 finite rank term to cope with the case of plane-wave modes in Rayleigh expansion propagating in the direction of the x 1 x 2 plane (i.e., for k ± := ω µε ±, there exist indices j, l with γ ± j,l = 0) finite rank term := η ν (E E ± ) (ν U j,l ) Γ ± (j,l) Υ ± Γ ± ν (E E ± ) (ν U j,l ) where: Υ ± is any fixed finite index set containing all (j, l) with γ ± j,l = 0 η is a fixed constant U j,l is special mode in Rayleigh expansion defined by 1 U j,l (x 1, x 2, x 3) := ( β l, α j, 0) αj 2 + β2 l exp (i k ±j,l (x 1, x 2, x 3) ) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 13 (42)

17 Variational Equation 3 E F k 2 E F E + ν F + ν (E E + ) F + Ω Ω Γ + Γ + +η ν (E E + ) ν (F F + ) Γ + E ν F + ν (E E ) F Γ Γ +η ν (E E ) ν (F F ) Γ = E inc ν F + ν E inc F + + η ν E inc ν (F F + ), F, F ± Γ + Γ Γ + ( ) a (E, E +, E ), (F, F +, F ) ( ) = F (F, F +, F ), (F, F +, F ) H b (curl) theorem now true for H b (curl): E H(curl, Ω), E ± Rayleigh expansions in H loc (curl), and ν (E E ± ) Γ ± L 2 t (Γ ± ) 3 modification of mortar method by Huber/Schoeberl/Sinwel/Zaglmayr (analysis of FEM open) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 14 (42)

18 Finite Element Methods and Alternatives 2D theory: H. Urbach, G. Bao, D.C. Dobson, J.A. Cox, J. Elschner, R. Hinder, and G. Schmidt generalized finite elements (trial space of piecewise Helmholtz solutions) for faster convergence and for highly oscillating solutions 3D theory: G. Schmidt for the approximation of the divergence free magnetic field Edge elements of Nédélec (cf. P. Monk, S. Burger, J. Pomplun, A. Schädle, L. Zschiedrich, F. Schmidt, M. Huber, J. Schöberl, A. Sinwel, S. Zagelmayr) alternatives: rigorous coupled wave analysis (RCWA) and integral equation methods (BEM, e.g., B. Bugert and G. Schmidt) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 15 (42)

19 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 16 (42)

20 Inverse Problem Given: the diffraction properties of grating (e.g. efficiencies E ± j,l ) Seek: the grating, i.e., the geometry of the domains filled with different materials and the refractive indices of these materials severely ill-posed problem: small errors in data lead to large errors for the solution contributions to 2D case by F. Hettlich, A. Kirsch, G. Bruckner, J. Elschner, G. Schmidt, D.C. Dobson, G. Bao, A. Friedman, M. Yamamoto, J. Cheng, K. Ito, F. Reitich T. Arens, and G. Hu contributions to 3D case by H. Ammari, G. Bao, Z. Zhou, H. Zhang, and J. Zou, Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 17 (42)

21 Uniqueness Theorem 3D perfectly conducting profile grating (PCPG): two materials, perfectly conducting substrate, dielectric cover material, interface graph {(x, x 3) R 3 : x 3 = f(x )} of a function f PCPG with small f with finite number of Fourier modes, Rayleigh coefficients measured for propagating plane-wave modes: uniqueness by Ammari general grating, Rayleigh coefficients measured for propagating reflected plane-wave modes (even for all angles of incidence): no reconstruction, 2D counter example by Kirsch PCPG with small and smooth function, electric field measured over plane above interface: certain uniqueness by Bao/Zhou PCPG with polyhedral function, single incoming plane wave, electric field measured over plane above interface: interface not identifiable if in one of 7 classes of Bao/Zhang/Zou Theorem (Bao, Zhang, Zou 11) Two polyhedral PCPGs. One interface has two faces forming an angle different from π/4, π/3, π/2, and 2π/3. For single incoming wave, corresponding reflected fields coincide over plane above interfaces. Then: PCPGs coincide. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 18 (42)

22 Uniqueness Theorem 3D perfectly conducting profile grating (PCPG): two materials, perfectly conducting substrate, dielectric cover material, interface graph {(x, x 3) R 3 : x 3 = f(x )} of a function f PCPG with small f with finite number of Fourier modes, Rayleigh coefficients measured for propagating plane-wave modes: uniqueness by Ammari general grating, Rayleigh coefficients measured for propagating reflected plane-wave modes (even for all angles of incidence): no reconstruction, 2D counter example by Kirsch PCPG with small and smooth function, electric field measured over plane above interface: certain uniqueness by Bao/Zhou PCPG with polyhedral function, single incoming plane wave, electric field measured over plane above interface: interface not identifiable if in one of 7 classes of Bao/Zhang/Zou Theorem (Bao, Zhang, Zou 11) Two polyhedral PCPGs. One interface has two faces forming an angle different from π/4, π/3, π/2, and 2π/3. For single incoming wave, corresponding reflected fields coincide over plane above interfaces. Then: PCPGs coincide. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 18 (42)

23 Class of Gratings avoid full inverse problems by using more a priori information on the grating: seek grating in class defined by a few parameters h = (h j) J j=1 (parameter identification) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 19 (42)

24 Class of Gratings y Reasonable initial values for optimizations bottomcd: ~ 140 nm m 1 Vac. m 3 TaO m 2 TaN p7 p6 htao: 12 nm SWA: 82,6 o htan: 54.9 nm SWA: 90 o m 4 Gla m 4 Gla Si p2 hgla(buff): 8 nm SWA: 90 o hgla(capping): nm hsi(capping): nm x 50x [ MoSi-Mo -MoSi Si] 50x h [ nm ] mask for EUV lithography: wavelength of inspecting light 13 nm period d Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 20 (42)

25 Equivalent Optimization Problem measured data, efficiencies or phase shifts: E meas = (Em meas ) m M computed data corresponding to parameters h: E(h) = (E m(h)) m M minimize objective functional ( ) F E m(h) inf ( F(E):= F E(h) ):= m M ω m E m(h) E meas m 2, ω m := 1 σ(e meas m ) 2 standard deviation of measurement: σ(em meas ) box constraints h min j h j h max j use: conjugate gradients method, interior point method, Levenberg-Marquardt algorithm or SQP Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 21 (42)

26 Convergence of Gauß-Newton iteration (SQP) similar to statistics: least squares estimator proposed by Drège, Al-Assad, Byrne for grating reconstruction without box constraints convergence properties: limit satisfies KKT condition necessary for minimum if measurement data is exact: h k+1 h sol c h k h sol 2 small error in measurement data: h k h sol cq k, 0 < q < 1 large error ε in measurement data: h k h sol cε if k k 0 advantages in comparison to gradient based method for optimization of objective functional: fast: small number of iterations, no line search no trouble with computation of small gradients at iterates close to optimum more robust w.r.t. nonoptimal scaling Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 22 (42)

27 Convergence of Gauß-Newton iteration (SQP) similar to statistics: least squares estimator proposed by Drège, Al-Assad, Byrne for grating reconstruction without box constraints convergence properties: limit satisfies KKT condition necessary for minimum if measurement data is exact: h k+1 h sol c h k h sol 2 small error in measurement data: h k h sol cq k, 0 < q < 1 large error ε in measurement data: h k h sol cε if k k 0 advantages in comparison to gradient based method for optimization of objective functional: fast: small number of iterations, no line search no trouble with computation of small gradients at iterates close to optimum more robust w.r.t. nonoptimal scaling Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 22 (42)

28 Convergence of Gauß-Newton iteration (SQP) similar to statistics: least squares estimator proposed by Drège, Al-Assad, Byrne for grating reconstruction without box constraints convergence properties: limit satisfies KKT condition necessary for minimum if measurement data is exact: h k+1 h sol c h k h sol 2 small error in measurement data: h k h sol cq k, 0 < q < 1 large error ε in measurement data: h k h sol cε if k k 0 advantages in comparison to gradient based method for optimization of objective functional: fast: small number of iterations, no line search no trouble with computation of small gradients at iterates close to optimum more robust w.r.t. nonoptimal scaling Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 22 (42)

29 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 23 (42)

30 Material Derivative vector field on Ω vanishing on boundary Ω: χ( x) generates family of automorphisms of domain Ω: Φ t( x) := x + tχ( x), 0 t < 1 generates family of solutions over transformed domain: E t solution of time-harmonic Maxwell equation with coefficient k replaced by k t := k Φ 1 t material derivative of F(E) w.r.t. χ: t[f(e t)] t=0 := lim t 0 F(E t) F(E 0) t If grating geometry is polyhedral, if the only non-vanishing component of χ is χ xi, if χ xi is piecewise linear and linear over all faces of the grating geometry, if χ xi (P ) = 1 for the vertex P, and if χ xi (Q) = 0 for all other vertices Q, then t[f(e t)] t=0 is the shape derivative of F(E) w.r.t. the x i-component of the vertex P Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 24 (42)

31 Derivation of Formula classical techniques of shape derivatives (cf. e.g. Sokolowski/Zolesio) unfortunately, energy space H(curl, Ω) not invariant w.r.t. transform Φ t fortunately, magnetic permeability is constant and magnetic field is in H 1 (Ω) over each subdomain filled with one material derive shape derivative for the magnetic solution by classical method and, in the final formula, replace magnetic field by electric field Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 25 (42)

32 Shape-Derivative Formula a 1(E, F ) := + k 2 E Ω ( k 2 E F χ E F χ Ω Ω ) x1 F, x2 F, x3 F + k 2( ) χ x1 F, χ x2 F, χ x3 F F Ω ] + [ E] [[ F ] x1 χ x1 + [ F ] x2 χ x2 + [ F ] x3 χ x3 Ω ] + [[ E] x1 χ x1 + [ E] x2 χ x2 + [ E] x3 χ x3 [ F ] Ω differentiated sesqui-linear form ( ) a (E, E +, E ), (F ad, F ad, + Fad) = [ν E] ψ, E, E +, E Γ + ψ := [ ] c mω m Em E m E meas m M m adjoint solution (F ad, F + ad, F ad ) (here cm certain constants) final formula: t[f(e t)] t=0 = Re a 1(E 0, F ad ) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 26 (42)

33 Shape-Derivative Formula a 1(E, F ) := + k 2 E Ω ( k 2 E F χ E F χ Ω Ω ) x1 F, x2 F, x3 F + k 2( ) χ x1 F, χ x2 F, χ x3 F F Ω ] + [ E] [[ F ] x1 χ x1 + [ F ] x2 χ x2 + [ F ] x3 χ x3 Ω ] + [[ E] x1 χ x1 + [ E] x2 χ x2 + [ E] x3 χ x3 [ F ] Ω differentiated sesqui-linear form ( ) a (E, E +, E ), (F ad, F ad, + Fad) = [ν E] ψ, E, E +, E Γ + ψ := [ ] c mω m Em E m E meas m M m adjoint solution (F ad, F + ad, F ad ) (here cm certain constants) final formula: t[f(e t)] t=0 = Re a 1(E 0, F ad ) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 26 (42)

34 Shape-Derivative Formula a 1(E, F ) := + k 2 E Ω ( k 2 E F χ E F χ Ω Ω ) x1 F, x2 F, x3 F + k 2( ) χ x1 F, χ x2 F, χ x3 F F Ω ] + [ E] [[ F ] x1 χ x1 + [ F ] x2 χ x2 + [ F ] x3 χ x3 Ω ] + [[ E] x1 χ x1 + [ E] x2 χ x2 + [ E] x3 χ x3 [ F ] Ω differentiated sesqui-linear form ( ) a (E, E +, E ), (F ad, F ad, + Fad) = [ν E] ψ, E, E +, E Γ + ψ := [ ] c mω m Em E m E meas m M m adjoint solution (F ad, F + ad, F ad ) (here cm certain constants) final formula: t[f(e t)] t=0 = Re a 1(E 0, F ad ) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 26 (42)

35 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 27 (42)

36 Parameters to be Reconstructed reconstruct the three parameters a, b, and c for array of contact holes, contact hole symmetric frustum of pyramid in middle of periodic cell Ω a b a b g 3.15 nm a 3.85 nm, 4.55 nm b 5.25 nm, 3.85 nm c 4.55 nm period in x- and y- direction: 16.8 nm and 14 nm inspecting wave length: 13.7 nm side material SiO 2, substrate layer Ta, lower multilayer system Mo and Si normal incidence, five measured efficiencies E m of order (-1,0), (0,-1), (0,0), (0,1), (1,0) corresponding weights ω m: 1000, 2000, 4, 2000, 1000 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 28 (42)

37 Objective Functional F and Numerical Gradients Obj.funct.(col.isols.)+Grads.(arrs.) param.b param.a Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 29 (42)

38 Convergence of Reconstruction Algorithm mesh size nmb.of its. a in nm b in nm g in nm initial vals exact Number of iterations and reconstruction of parameters for contact hole. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 30 (42)

39 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 31 (42)

40 Analysis Done for 2D Case Sensitivity analysis: Optimize choice of measurement data and weights using the Jacobians Given distribution and variance of measurement values: estimate uncertainties (standard deviation) of reconstructed values covariance matrix estimates by inverse Jacobians Monte Carlo methods Estimate the impact of model errors on the reconstruction, Maximum-likelihood methods Stochastic deviations of the widths in the multilayer system Corner roundings and footage Linewidth roughness (LWR) and lineedge roughness (LER) of line-space structures Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 32 (42)

41 Analysis Done for 2D Case Sensitivity analysis: Optimize choice of measurement data and weights using the Jacobians Given distribution and variance of measurement values: estimate uncertainties (standard deviation) of reconstructed values covariance matrix estimates by inverse Jacobians Monte Carlo methods Estimate the impact of model errors on the reconstruction, Maximum-likelihood methods Stochastic deviations of the widths in the multilayer system Corner roundings and footage Linewidth roughness (LWR) and lineedge roughness (LER) of line-space structures Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 32 (42)

42 Analysis Done for 2D Case Sensitivity analysis: Optimize choice of measurement data and weights using the Jacobians Given distribution and variance of measurement values: estimate uncertainties (standard deviation) of reconstructed values covariance matrix estimates by inverse Jacobians Monte Carlo methods Estimate the impact of model errors on the reconstruction, Maximum-likelihood methods Stochastic deviations of the widths in the multilayer system Corner roundings and footage Linewidth roughness (LWR) and lineedge roughness (LER) of line-space structures Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 32 (42)

43 Lineedge Roughness and Linewidth Roughness stochastic perturbation in line space structure Lineedge roughness (LER) Linewidth roughness (LWR) Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 33 (42)

44 Simple 2D Simulation of LWR/LER simplifying assumption: assume perturbation of lines is slow in the direction of the lines locally: like geometry of perfect parallel lines with perturbations in line widths and distances between lines approximated by periodic grating: super cell with large period containing many (e.g. 48) lines with variable line widths and distances cross section of super cell cross section of single line Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 34 (42)

45 Simple 2D Simulation of LWR/LER simplifying assumption: assume perturbation of lines is slow in the direction of the lines locally: like geometry of perfect parallel lines with perturbations in line widths and distances between lines approximated by periodic grating: super cell with large period containing many (e.g. 48) lines with variable line widths and distances cross section of super cell cross section of single line Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 34 (42)

46 Monte-Carlo Results for LER: Systematic Error normalized deviations order n j (a) normalized standard deviation FEM results: σ xi = 5.6 nm [1 exp( σ r 2 kj 2 /3)]/sqrt(2) σ r = 5.55 nm; k j = 2 π n j /d d = 280 nm (period) order n j (b) (a) Normalized deviations from the efficiencies of the unperturbed reference line structure, depicted as circles; diamonds represent the mean over the deviations of all 100 samples; dashed lines indicate the mean efficiency ± standard deviation; random perturbations of the center positions (LER): σ xi = 5.6 nm. (b) Standard deviations relative to the mean perturbed efficiencies shown as circles for the given example; diamonds depict best approximation by an exponential function. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 35 (42)

47 Monte-Carlo Results for LWR: Systematic Error normalized deviations normalized standard deviation FEM results: σ CDi = 5.6 nm [1 exp( σ r 2 kj 2 /3)]/sqrt(2) σ r = 2.74 nm; k j = 2π n j /d d = 280 nm (period) order n j (a) order n j (b) (a) Normalized deviations from the efficiencies of the unperturbed reference line structure, depicted as circles; diamonds represent the mean over the deviations of all 100 samples; dashed lines indicate the mean efficiency ± standard deviation; random perturbations of the line widths (LWR): σ CDi = 5.6 nm. (b) Standard deviations relative to the mean perturbed efficiencies shown as circles for the given example; diamonds depict an approximation by an exponential function. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 36 (42)

48 Correction Factor for LER/LWR Perturbations A. Kato and F. Scholze: formula for impact of LER/LWR if Fraunhofer approximation applies E + j,pert exp ( [ ] 2 2πj σ) E + j,no d pert, where: d period, j order of mode, σ is the standard deviation of the stochastic perturbation of the line edges. similar exponential function for variances by our experiments: formulas confirmed for rigorous Maxwell s equations and slow perturbation along the lines knowing the LER/LWR perturbation: incorporate formula into numerical reconstruction algorithm Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 37 (42)

49 Correction Factor for LER/LWR Perturbations A. Kato and F. Scholze: formula for impact of LER/LWR if Fraunhofer approximation applies E + j,pert exp ( [ ] 2 2πj σ) E + j,no d pert, where: d period, j order of mode, σ is the standard deviation of the stochastic perturbation of the line edges. similar exponential function for variances by our experiments: formulas confirmed for rigorous Maxwell s equations and slow perturbation along the lines knowing the LER/LWR perturbation: incorporate formula into numerical reconstruction algorithm Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 37 (42)

50 Correction Factor for LER/LWR Perturbations A. Kato and F. Scholze: formula for impact of LER/LWR if Fraunhofer approximation applies E + j,pert exp ( [ ] 2 2πj σ) E + j,no d pert, where: d period, j order of mode, σ is the standard deviation of the stochastic perturbation of the line edges. similar exponential function for variances by our experiments: formulas confirmed for rigorous Maxwell s equations and slow perturbation along the lines knowing the LER/LWR perturbation: incorporate formula into numerical reconstruction algorithm Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 37 (42)

51 LEWR 1% perturbed & bias corrected datasets x 10 2 x 10 Improved 1 σ 2 : 1.62 Numerical nm σ 6 : Reconstruction 0.04 nm 2 0 x LEWR 1% perturbed & bias corrected σ 2 7 : 0.48 datasets nm x x σ : 1.62 nm 2 σ 6 : 0.04 nm σ 6 : 0.20 nm σ : 0.35 nm σ : 0.48 nm σ : 0.73 nm σ : 0.20 nm 3 6 σ : 0.35 nm 2 8 p p σ 2 : 0.73 nm reconstruction 0 p2 parameter (b) 3 (a) p7 (b) 50 8 p reconstruction 0 p2 2 6 parameter (a) 94 (b) (b) CDm : nm mean SWA σ : o CDm : 0.72 nm 92 mean CDm : nm mean σ : 1.82 o 91 σ SWA CDm : 0.72 nm SWA mean : o SWA : o 91 σ SWA : 0.82 o 0.09 mean SWA mean : o σ : 1.82 o 90 1 σ (d) SWA SWA : 0.82 o 1% 0.09 standard deviation (c) of edges with LER/LWR, left reconstruction (d) 89 without correction factor, 1 1 right 92with correction factor, (c) deviation around nominal value (d) CD at middle height / µm deviation from reference /µm deviation from deviation reference from / /µm reference / µm CD at middle height / µm side wall angle side wall / o angle / o 1 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 38 (42) (d) deviation from deviation reference from / µm reference / µm side wall side wall angle / o angle / o

52 Outline 1 Scatterometry. 2 Direct Problem: Diffraction by Grating Structures. 3 Reconstruction of Periodic Surface Structures. 4 Shape Derivatives. 5 Numerical Example. 6 Statistics and Further Issues. 7 Conclusions. Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 39 (42)

53 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

54 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

55 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

56 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

57 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

58 Conclusions Simulation of scattered field by biperiodic grating possible with FEM and mortar method (unfortunately: slow) Shape derivatives can be computed by FEM Reconstruction of geometric parameters possible (beyond diffraction limit) Uncertainties of reconstructed parameters can be estimated even by fast covariance-matrix method To get results closer to those obtained by alternative measurement methods: model errors need to be analyzed, e.g., LER, LWR For more details: S. Heidenreich: today s talk, M10, Salon B, 16:50 17:15 Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 40 (42)

59 Some References [1] M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, On numerical reconstructions of lithographic masks in DUV scatterometry, Proc. SPIE, 7390 (2009), 73900Q. [2] H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, Profile reconstruction in extreme ultraviolet scatterometry: Modeling and uncertainty estimates, Measurement Science and Technology, 20 (2009), pp [3] H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, On stochastic modeling of line roughness for the diffraction intensities and its impact on the reconstructed critical dimensions in scatterometry, (2012), to appear. [4] A. Rathsfeld, Shape derivatives for the diffraction intensities scattering by biperiodic structures, WIAS Preprint, Berlin, 1640 (2011). [5] G. Hu and A. Rathsfeld, Scattering of time-harmonic electromagnetic planar waves by perfectly conducting diffraction gratings, WIAS Preprint, Berlin, 1694 (2012). Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 41 (42)

60 Acknowledgment The research has been supported by the BMBF (Federal Ministry of Education and Research): BMBF-Project CDuR 32: Schlüsseltechnologien zur Erschliessung der Critical Dimension (CD) und Registration (REG) Prozess- und Prozesskontrolltechnologien für die 32 nm-maskenlithographie We appreciate the cooperation/consultation with: M. Bär, B. Bodermann, M.-A. Henn, S. Heidenreich, A. Kato, F. Scholze (PTB), and T. Arnold, J. Elschner, G. Hu, (WIAS) Thank you for your attention! Inverse problems for scatterometric measuring of gratings IPMS, 23 May 2012 Page 42 (42)