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

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ASML Tool

Library of modeled data

Filter CCD,,

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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