grating reconstruction forward modeling part mark van kraaij casa phd-day tuesday 13 november 2007
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Grating reconstructionforward modeling part
Mark van KraaijCASA PhD-day
Tuesday 13 November 2007
Jamie and Adam explain Moore’s Law
Source: www.intel.com
Wafer: 300 mm
From wafer to grating
Intel Core2 Duo: 13.6 mm
1 m
Grating: 500 nm pitch, 70 nm linewidth
Source: www.intel.com
• 1. Project description
• 2. Forward modeling part– 2.1: Diffraction model
– 2.2: Error analysis
• 3. Improvements– 3.1: Finite differences
– 3.2: Adaptive spatial resolution
• 1. Project description
• 2. Forward modeling part– 2.1: Diffraction model
– 2.2: Error analysis
• 3. Improvements– 3.1: Finite differences
– 3.2: Adaptive spatial resolution
Outline
Microscope objective
Xenon lightsource
Wafer
,,
I
ASML Tool
Library of modeled data
Filter CCD,,
,, ,,
,,Library
search++
Reconstructed profile
1. Project description
A tool is needed that can measure profile information for CD and
Overlay metrology
Outline
• 1. Project description
• 2. Forward modeling part– 2.1: Diffraction model
– 2.2: Error analysis
• 3. Improvements– 3.1: Finite differences
– 3.2: Adaptive spatial resolution
• Assumptions (EM field):– Electromagnetic field quantities are time-harmonic
– Incident field is an arbitrary (linearly) polarized monochromatic
plane wave
• Assumptions (grating):– Media are isotropic, stationary → linear constitutive relations
– Grating is infinitely periodic and approximated with a layered structure
• Assumptions (grating):– Media are isotropic, stationary → linear constitutive relations
– Grating is infinitely periodic
2.1 Diffraction model: Assumptions
SEM: finite
periodic grating
Model: infinite
periodic grating
Rayleigh radiation condition
Rayleigh radiation condition
Pseudo-periodic boundary
condition
Continuity boundary condition
2.1 Diffraction model: Equations and bc’s
xz
• TM polarization:
• TE polarization:
• Expand electric field in top and bottom layer in eigenfunctions
(pseudo-periodic Fourier series)
• Expand electric field and permittivity function in intermediate
grating layers also in (pseudo-periodic) Fourier series
2.1 Diffraction model: Discretization and truncation
• Truncate series and solve 2nd order ODE in grating layers
2.1 Diffraction model: Final system of equations
• Use continuity boundary conditions at layer interfaces
• Solve system with stable condensation algorithm
Fundamental solutions
consist of– N growing components
– N decreasing components
Interface between
layer i
and i+1
Completely separated
boundary conditions
2.2 Error analysis: Number of harmonics
2.2 Error analysis: Number of harmonics
2.2 Error analysis: Number of harmonics
.
Outline
• 1. Project description
• 2. Forward modeling part– 2.1: Diffraction model
– 2.2: Error analysis
• 3. Improvements– 3.1: Finite differences
– 3.2: Adaptive spatial resolution
Bloch
3. Improvements
Separation of variables gives
with pseudo-periodic bc’s,
with continuity bc’s at layer interface,
where
Eigenvalues are related to the roots of
Eigenfunctions typically look like
Single domain approach
3.1 Improvements: Finite differences
• Discretize equation on most interior
points using central differences
• Discretize equation on some interior
points using modified central
differences
• Discretize equation on interior points
using central differences
• Discretize boundary condition on
boundary points using one-sided
differences
x0 x1 xM-1 xMxN-1 xN
Goal: Improve accuracy by replacing Fourier with finite
difference discretization (transitions modeled better)
Partitioned domain approach
x0 x1 xM-1 xM,a/b xM+1xN-1 xN
Single domain approach
Compute eigenvalues,
scheme overall O(h2):
Partitioned domain approach
Compute eigenvalues,
scheme overall O(h2):
Partitioned domain approach
Compute eigenvalues,
scheme overall O(h):
Eigenvalues computed using standard techniques for full matrices.
At the moment not able to exploit sparse structure of matrix…
Partitioned domain approach Single domain approach
3.1 Improvements: Finite differences
By a change of variable in each layer the spatial
resolution is increased around the discontinuities
Properties:
– The electric field and permittivity in each layer i are expanded in a layer
specific basis which depends on the locations of the
transition points
– The basis functions in each layer are projected on the plane wave basis
when connecting layers
3.2 Improvements: Adaptive spatial resolution
3.2 Improvements: Adaptive spatial resolution
Question1: Should all eigenvalues be used?
500 nm silicon block, non-normal incidence,0==1mu, (TE, N=10)
Exact eigenvalues (Bloch)
-1.349076641333984e+001
-1.135076524908338e+001
-7.892853673610714e+000
-3.454082776318814e+000
1.276327724136246e-001
7.878383953399616e-001
4.951776353671870e+000
6.064364267444717e+000
1.286665389409586e+001
1.347902372017457e+001
2.288512237078319e+001
…
prop
agat
ing
evan
esce
nt
500 nm silicon block, non-normal incidence,0==1mu, (TM, N=10)
Exact eigenvalues (Bloch)
-1.323802014939081e+001
-1.032763462047682e+001
-5.576329792050528e+000
-1.266781396948507e+000
1.154532768840351e-001
1.029193134524742e+000
3.848084904617279e+000
7.334522367194479e+000
1.139286665954855e+001
1.467711974044035e+001
2.201015238581615e+001
…
3.2 Improvements: Adaptive spatial resolution
Question2: How to compute projection matrix ?
• Numerical quadrature difficult due to high frequencies
• FFT might be possible since integral can be seen as computing
a Fourier coefficient
• Rewrite into standard Bessel related integrals:
Summary
• Stability RCWA understood and linked to standard techniques
• Gaining insights in error estimates
• Finite differences alternative for Fourier but not yet competitive
• ASR another alternative for standard Fourier but still work to be done on– choosing optimal stretching parameter
– implementing special functions
2.2 Error analysis: Number of harmonics
2.2 Error analysis: Number of harmonics
2.2 Error analysis: Number of harmonics
.
Special functions
Anger function:
Properties:– When integer: regular Bessel function
– When non-integer and• small z : Power series expansion with Lommel functions
• large z : Asymptotic expansions with second kind Lommel functions and regular cylindrical Bessel functions
– Recurrence relation for
3.2 Improvements: Adaptive spatial resolution
Special functions
Weber function:
Properties:– When integer: series expansion with Gamma and Struve functions
– When non-integer and• small z : Power series expansion with Lommel functions
• large z : Asymptotic expansions with second kind Lommel functions and irregular cylindrical Bessel functions
– Recurrence relation for .
3.2 Improvements: Adaptive spatial resolution
Old results: ASR improves convergence of diffraction efficiencies in TE polarization, but TM fails…
Diffraction efficiency 0th order
N Old New
5 0.1164746 0.1406958
10 0.1011264 0.1317428
15 0.1208512 0.1317091
20 0.1269484 0.1317092
25 0.1291739 0.1317092
30 0.1302258 0.1317092
35 0.1307600 0.1317092
40 0.1310702 0.1317092
200 0.1317039
Diffraction efficiency 1st order
N Old New
5 0.8528182 0.7286252
10 0.7622694 0.7342224
15 0.7440938 0.7342789
20 0.7385729 0.7342789
25 0.7365554 0.7342788
30 0.7356116 0.7342788
35 0.7351296 0.7342788
40 0.7348521 0.7342788
200 0.7342836
300R0
R1
=m
=m
n = 0.22-6.71i
d = 1m
3.2 Improvements: Adaptive spatial resolution
Jamie and Adam explain Moore’s Law
Source: www.intel.com
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