T PB Direct Problem: Diffraction by Grating Structures. 1 W - - PowerPoint PPT Presentation

T

PB

W eierstraß-InstitutfürAngewandteAnalysisundStochastik

• H. Groß, A. Rathsfeld

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 1 (44) Outline

Outline 1

Direct Problem: Diffraction by Grating Structures.

2

Reconstruction of Periodic Surface Structures

3

Sensitivity Analysis Tasks of Sensitivity Analysis Uncertainty Estimates for Derived Values

4

Stochastic Geometries

5

Conclusions

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 2 (44) Direct Problem: Diffraction by Grating Structures.

.

θ z x y

reflected modes transmitted modes

chrome photoresist

incoming field (E ,H )

i i

silicon oxide

periodic gratings (similarly, biperiodic surface structures) details of surface geometry in size of wavelength

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 4 (44) Direct Problem: Diffraction by Grating Structures.

Lithography chip production like old-fashioned photography: photoresist layer illuminated by light scattered from mask, development: baking and etching procedures − → chip

mask photo resist wafer light source

standard test configurations: – periodic line-space structure (lines formed by bridges with trapezoidal cross section) – biperiodic array of trapezoidal blocks resp. holes

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 5 (44)

Direct Problem: Diffraction by Grating Structures.

.

Scatterometry

• ✁

2θ θ Sample Lightsource Detector rot.Analyser Polarizer Variable Retarder

• M. Wurm, B. Bodermann, and W. Mirandé

"Eval. of scatterom. tools for critical dim. metrology" DGaO Proc. 2005, www.dgao-proceedings.de, P 25

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 6 (44) Direct Problem: Diffraction by Grating Structures.

Spectroscopic reflectrometer operating in the EUV range around 13 nm BESSY II, PTB

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 7 (44) Direct Problem: Diffraction by Grating Structures.

.

Plane wave illumination. 3D time-harmonic Maxwell’s equns. reduce to: – Curl-Curl equation for three-dimensional amplitude factor of time harmonic electric field (i.e. E(x1, x2, x3, t) = E(x1, x2, x3)e−iωt) ∇ × ∇ × E(x1, x2, x3) − k2E(x1, x2, x3) = 0 – scalar 2D Helmholtz equation if geometry is constant in x3 direct- ion and if direction of incoming plane wave is in x1 − x2 plane ∆v(x1, x2) + k2v(x1, x2) = 0, v = E3, H3 – two coupled scalar 2D Helmholtz equations if geometry is constant in x3 direction but direction of incoming plane wave is not in x1 − x2 plane (conical diffraction)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 8 (44) Direct Problem: Diffraction by Grating Structures.

x1 x

2

x

3

Domain of Computation: Coupling with boundary elements Including radiation condition (or Absorbing bound. cond. PML

• r mortaring with Fourier mode sol.)

periodic cell Ω Non−local boundary condition: Transmission conditions: Quasiperiodic boundary condition: Continuous tangential traces

1 2 1 2 3 3

period d1

1 2 1 1 2 2 3 3

α β

1

v(x ,x +d ,x )=v(x ,x ,x ) exp(i d )

2

v(x +d ,x ,x )=v(x ,x ,x ) exp(i d ) Continuous tangential traces of curl Rectangular box over one

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 9 (44)

Direct Problem: Diffraction by Grating Structures.

.

E(x1, x2, x3) =

∞

• j,l=−∞

A−

j,l exp

• i k−

j,l · (x1, x2, x3)⊤

, x3 < xmin, αj := k+ sin(θ) cos(φ) + 2π d1 j, k−

j,l := (αj, βl, γ− j,l)⊤, βl := k+ sin(θ) sin(φ) + 2π

d2 l, γ−

j,l := −

• [k−]2 − [αj]2 − [βl]2

Rayleigh series: Rayleigh coefficients A−

j,l ∈ C3, A− j,l⊥k− j,l

(coefficients A+

j,l for expansion with x2 > xmax)

Efficiency E±

j,l of (j, l)−th mode: ratio of energy radiated into direction

• f mode

E±

j,l

:= (γ±

j,l/γ+ 0,0)|A± j,l|2

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 10 (44) Direct Problem: Diffraction by Grating Structures.

⊲ 2D theory by 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

⊲ shape derivatives of diffracted fields w.r.t. parameters describing

grating geometry computed by finite elements, almost no additional computing time

⊲ 3D theory by G. Schmidt ⊲ 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, conical diffraction by G. Schmidt)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 11 (44) Direct Problem: Diffraction by Grating Structures.

.

Component of electric field in groove direction

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 12 (44) Direct Problem: Diffraction by Grating Structures.

3D example FEM grid and real part of x1 component of electric field

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 13 (44)

Reconstruction of Periodic Surface Structures

.

Full 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 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, and T. Arens

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 15 (44) Reconstruction of Periodic Surface Structures

Uniqueness theorem 2D (Elschner/Yamamoto) Theorem Suppose: 1) gratings G1 and G2 profile gratings (two materials: cover material and substrate, separated in cross section by profile curve) 2) substrate perfectly conducting 3) polarization state TE or TM 4) general polygonal (piecewise linear) profile curves of G1 and G2 5) transversal components of scattered fields for G1 and G2 coincide

• ver upper boundary line Γ+ for two resp. four plane waves

incident from different directions = ⇒ two gratings G1 and G2 coincide!

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 16 (44) Reconstruction of Periodic Surface Structures

.

period d CrO Cr SiO −substrate x y p p

2

p p p p p p

5 6 7 8 4 3 2 1

hi := pi, i = 1, 4, hi :=

pi period, i = 2, 5, 7,

hi :=

pi pi−1 , i = 3, 6, 8

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 17 (44) Reconstruction of Periodic Surface Structures

Optimization Problem: measured data, efficiencies or phase shifts: Emeas = (Emeas

m

)m∈M comp.data corresponding to parameters h: E(h) = (Em(h))m∈M minimize objective functional f

• Em(h)
• −

→ inf f

• Em(h)
• :=
• m∈M

ωm |Em(h) − Emeas

m

|2 , ωm := 1 σ(Emeas

m

)2 box constraints hmin

j

≤ hj ≤ hmax

j

, use: conjugate gradients method, interior point method, Levenberg-Marquardt algorithm

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 18 (44)

Reconstruction of Periodic Surface Structures

.

Solution of Operator Equation: E(hsol) = Emeas Gauß-Newton method with modification for box constraints (SQP type method): compute hk+1 = hk + ∆h with ∆h the optimal solution of convex quadratic optimization problem with box constraints: min

∆h: hmin

j

≤[hk

j +∆h]j≤hmax j

• E(hk) + ∂E

∂h (hk)∆h − Emeas

• 2

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 19 (44) Reconstruction of Periodic Surface Structures

period d CrO Cr SiO −substrate x y p p

2

p p p p p p

5 6 7 8 4 3 2 1

hi := pi, i = 1, 4, hi :=

pi period, i = 2, 5, 7,

hi :=

pi pi−1 , i = 3, 6, 8

wavelength of inspecting light: 632.8 nm exact solution: period 1.12µm, top CD 580nm, side wall angles 73◦, bridge height 50nm, layer width 23nm

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 20 (44) Reconstruction of Periodic Surface Structures

.

lev. h1 h3 h4 h5 h6 h8 3 0.04719 0.26449 0.02184 0.73551 0.32085 0.27829 4 0.04939 0.27645 0.02276 0.73134 0.29526 0.26069 5 0.04988 0.27897 0.02295 0.73035 0.29177 0.25755 6 0.04997 0.27954 0.02299 0.73016 0.29099 0.25674 ex. 0.05000 0.27967 0.02300 0.73007 0.29068 0.25638 Reconstructed parameters depending on the level of FEM discretization (halved meshsize for transition to next higher level)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 21 (44) Reconstruction of Periodic Surface Structures

1 2 3 4 5 6 8 −6 −4 −2 2 4 6 8 10 x 10

−3

deviation from the expected value index of optimised parameter 1 2 3 4 5 6 8 −3 −2 −1 1 2 3 4 x 10

−3

deviation from the expected value index of optimised parameter

simulated noise of levels 1.1 · 10−4 and 0.55 · 10−4 superimposed onto measurement data (several samples of noisy data sets stochastically generated)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 22 (44)

Reconstruction of Periodic Surface Structures

. y

50x [ MoSi - Mo - MoSi – Si ] hSi(capping): 12.536 nm

Si

hTaO: 12 nm SWA: 82,6o hTaN: 54.9 nm SWA: 90o hGla(buff): 8 nm SWA: 90o

hGla(capping): 1.246 nm m4 Gla

m4 Gla

m1 Vac. 50x h [ 0.5–2.259– 1.263– 3.077 nm ] Reasonable initial values for optimizations

bottomCD: ~ 140 nm

m3 TaO m2 TaN

p2 p6 p7

mask for EUV lithography: wavelength of inspecting light 13 nm

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 23 (44) Reconstruction of Periodic Surface Structures

reconstruction from all initial vectors with p2 ∈ {0.57333, 0.59333} p6 ∈ {0.0450, 0.0570} p7 ∈ {0.57333, 0.59333} example of a reconstruction: Discr.Level(Nmb.of Iterations) p2 p6 p7 initial solution 0.580000 0.050000 0.583330 3(5) 0.583486 0.052037 0.584732 3(11) 0.583284 0.054903 0.583340 exact solution 0.583333 0.054900 0.583333

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 24 (44) Reconstruction of Periodic Surface Structures

.

f(p6,p7) for optimal 12 efficiencies, weights chosen by N2s-12

f(p6,p7) for SimH4,N2s−12,GFEM(2,29,4)−Lev.3,BND−MESH−SIZE=0.4

p6 [h(TaN) um] p7 [~topCD(TaN)]

0.045 0.05 0.055 0.06 0.065 0.574 0.576 0.578 0.58 0.582 0.584 0.586 0.588 0.59 0.592 0.2 0.4 0.6 0.8 1 1.2 1.4 0.045 0.05 0.055 0.06 0.065 0.575 0.58 0.585 0.59 10

−4

10

−3

10

−2

10

−1

10 10

1

p7 [~topCD(TaN)]

f(p6,p7) for SimH4,N2s−12,GFEM(2,29,4)−Lev.3,BND−MESH−SIZE=0.4

p6 [h(TaN) um]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 25 (44) Reconstruction of Periodic Surface Structures

f(p6,p7) from an optimal subset of 12 efficiencies, weights chosen by N2m-12

f(p6,p7) for H4,N2m−12,GFEM(2,29,4)−Lev.3,BND−MESH−SIZE=0.4

p6 [h(TaN) um] p7 [~topCD(TaN)]

0.045 0.05 0.055 0.06 0.065 0.574 0.576 0.578 0.58 0.582 0.584 0.586 0.588 0.59 0.592 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0.045 0.05 0.055 0.06 0.065 0.575 0.58 0.585 0.59 10

p7 [~topCD(TaN)]

f(p6,p7) for H4,N2m−12,GFEM(2,29,4)−Lev.3,BND−MESH−SIZE=0.4

p6 [h(TaN) um]

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 26 (44)

Reconstruction of Periodic Surface Structures

.

Typical result of optimizations for EUV_H4 of DS1 (N2m-69, minim.(1)):

−15 −10 −5 5 10 15 10−2 10−1 100 101 102

• rders

efficiency [%] Comparison of meas. & sim. efficiencies for the optimal solution; EUV−H4 at 13.389 nm simulated efficiencies measured efficiencies sum(((E−C)./C).2)/N = 0.0477 −15 −10 −5 5 10 15 10−3 10−2 10−1 100 101 102

• rders

efficiencies [%]

Comparison meas. & sim. efficiencies for the optimal solution; EUV−H4 (DS1) at 13.655 nm simulated efficiencies measured efficiencies sum(((E−C)./C).2)/N = 0.1541

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 27 (44) Reconstruction of Periodic Surface Structures

Comparison Scatterometry+DIPOG and SEM results for the fields of DS2

D4 H4 F6 D8 H8 520 530 540 550 560 570 580 590 600 610 620

field CD /nm

comparison SEM and DIPOG results for the different fields of data set DS2 SEM DIPOG (minimum 1) DIPOG (minimum 2) pitch = 720 nm CD(SEM) = pitch−space

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 28 (44) Sensitivity Analysis Tasks

Tasks of Sensitivity Analysis

⊲ Theoretically: Huge amount of direct measurement data possible.

Which part of this data is really needed for an accurate and fast reconstruction of the entities to be “measured” indirectly? = ⇒ minimize the condition numbers of the mapping: directly measured data → derived data (greedy algorithm)

⊲ Knowing the uncertainties of the direct measurement data,

estimate the measurement uncertainties of the indirect measurement values! = ⇒ Monte Carlo method covariance matrix estimates

⊲ Estimate the uncertainties of the direct measurement data!

= ⇒ maximum likelihood estimator

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 30 (44) Sensitivity Analysis Uncertainty Estimates, Derived Vals.

Uncertainty Estimates for Derived Values

Monte Carlo approach Measurement uncertainty simulated: – Normal distribution Emeas

m

∼ N(Em(hexact), σm) – Noise level (measurement uncertainty, standard deviation) um := σm =

• (fEm)2 + bgn2, with background

noise bgn and relative noise fEm (f constant). Reconstruct solution from noisy measurement data: check distribution of reconstructed parameters

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 31 (44)

Sensitivity Analysis Uncertainty Estimates, Derived Vals.

.

Typical uncertainties of optimizations with perturbed 12 efficiencies (optimal subset of meas.) for a grating similar to EUV-H4 illuminated with λ in the range around 13 nm:

1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 x 10

−3

perturbed efficiency [%] index of efficiency value u(e)=sqrt((f*e)2+bgn2) f: 0.01 bgn: 0.001 sample size: 32 6 7 16 −6 −4 −2 2 4 6 x 10

−3

deviation from the expected value index of optimized parameter Noise test (f: 0.01 bgn: 0.001); data sample size:39 σ6: 1.10e−03 µm σ7: 1.03e−03 µm σ16: 8.02e−04 µm

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 32 (44) Sensitivity Analysis Uncertainty Estimates, Derived Vals.

Typical uncertainties of optimizations with perturbed 415 efficiencies (complete dataset) for a grating similar to EUV-H4:

201 212 2 4 6 8 10 12 x 10

−3

perturbed efficiency [%] index of efficiency value u(e)=sqrt((f*e)2+bgn2) f: 0.01 bgn: 0.001 sample size: 32

. . . . . . . . . . . . . . . . (a part of the complete perturbed data set) 6 7 16 −6 −4 −2 2 4 6 x 10

−3

deviation from the expected value index of optimised parameter Noise test (f: 0.01 bgn: 0.001); data sample size: 25 σ6: 1.22e−05 µm σ7: 5.09e−05 µm σ16: 1.61e−04 µm (calculated with complete data set: N2s−415)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 33 (44) Sensitivity Analysis Uncertainty Estimates, Derived Vals.

.

Two examples for distributions of multilayer system (MLS) models: heights of capping layers (indices 1,2), heights of 4 layers of MoSi group in multilayer system (indices 3,4,5,6), statistics of 128 samples, – left: 1% perturbation of all six parameters – right: 1% and 0.1% noise for capping and MLS components, resp.

1 2 3 4 5 6 −3 −2 −1 1 2 3 x 10

−4

perturbed width [µ m] index of ML width Normaly distrib. ML widths (cap:1%; group:1%) 1 2 3 4 5 6 −3 −2 −1 1 2 3 x 10

−4

perturbed width [µ m] index of ML width Normaly distrib. ML widths (cap:1%; group:0.1%)

result: cf. poster by H. Groß

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 34 (44) Sensitivity Analysis Uncertainty Estimates, Derived Vals.

Approximate covariance matrix proposed for the grating reconstruction by Drège, Al-Assad, Byrne Weighted Jacobian: Jσ

M

:= ∂Em(hsol) ∂hj σ

• Em(hsol)

−1

• m∈M, j=1, ... ,J

covariance matrix (E is expectation): Cov(h) :=

• E
• (hj − E(hj))(hj′ − E(hj′))
• j,j′=1, ... ,J

∼

• Jσ ⊤

M Jσ M

−1 standard deviation σ(hj) of hj is square root of jth main diagonal entry

• f covariance matrix

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 35 (44)

Sensitivity Analysis Uncertainty Estimates, Derived Vals.

.

N2s-12 N2s-415 N2m-12 N2m-69 σ6 [nm] 0.81 0.012 0.33 0.13 σ7 [nm] 0.86 0.054 0.13 0.034 σ16 [nm] 0.91 0.15 0.35 0.17 σSWA(TaN) [◦] 1.3 0.14 0.47 0.21

Table: Standard deviations of the reconstructed parameters for different subsets of simulated resp. measured efficiencies. The standard deviations are computed by the covariance matrix method. The standard deviations of the measurement input has been set to σ(Emeas

m

) =

• (0.01Em)2 + 0.0012.

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 36 (44) Sensitivity Analysis Uncertainty Estimates, Derived Vals.

Assume normal distribution of measurement values with uncertainty defined by an unknown multiplicative parameter f: Emeas

m

∼ N(Em(hexact), σm) σm =

• [fEm(hexact)]2 + bgn2

Maximum Likelihood Estimator: minimizer (hopt, fopt) of functional f(h, f) :=

• m
• log
• [fEm(h)]2 + bgn2

+ (Emeas

m

− Em(h))2 [fEm(h)]2 + bgn2

• Provides reconstructed parameter set + factor of uncertainty estimate.

Conclusion: Factor f = 0.01 is too optimistic: 0.05 < f < 0.1

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 37 (44) Stochastic Geometries

.

smaller details: production of masks with ideal interfaces is too difficult or too expensive = ⇒ stochastic models for rough surfaces D.G. Stearns et al.: modified Fresnel formulas for reflectivity r and transmittivity t (refl./transm.efficiency), e.g. Debye-Waller and Nevot-Croce models rideal = nup cos(θ) − nlow cos(θ′) nup cos(θ) + nlow cos(θ′) , θ′ : nup sin(θ) = nlow sin(θ′) , rrough = rideal w 4π cos(θ) λ

• ,

w(x3) := d dx3

• ε(x1, x2, x3)dx1dx2

[εup − εlow]

• dx1dx2

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 39 (44) Stochastic Geometries

Cross section of multilayer stack with “stochastic” interfaces Comp.reflectivities depending on number of “stochastic” interfaces

5 10 15 20 25 30 35 40 45 50 4 6 8 10 12 14 16 Number of multilayers with random interfaces Efficiency of reflection

IMD DiPoG: 50 random geometries with 200 Corner Points DiPoG: without random interfaces Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 40 (44)

Stochastic Geometries

.

Stochastic perturbations of line-space structures top view

Lineedge roughness (LER) Linewidth roughness (LWR)

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 41 (44) Conclusions

Conclusions

⊲ Finite element package DiPoG developed at WIAS ⊲ Reconstruction of geometric parameters possible

(beyond diffraction limit)

⊲ Optimization of measurement data helpful ⊲ Uncertainties of reconstructed parameters can be estimated even

by fast covariance-matrix method

⊲ To get results closer to those obtained by alternative measurement

methods: better model e.g. for multi-layer stack is needed For more details: cf. posters

• H. Groß: Tue, 21 July, 2009, 17:15-18:15, Foyer

M.-A. Henn: Tue, 21 July, 2009, 17:15-18:15, Foyer

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 43 (44) Conclusions

.

Acknowledgment The research has been supported by the BMBF (Federal Ministry of Education and Research): – Project ABBILD (Imaging Methods for Nano-Electronic Devices) – 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, F

. Scholze (PTB), and T. Arnold (WIAS)

Thank you for your attention!

Numerical Aspects of the Scatterometric Measurement of Periodic Surface Structures M37, July 21 44 (44)