Backside Feature Transfer during
Electrostatic Chuckingof Masks*
Sanjay Govindjee, PhD, PEGerd Brandstetter, MSc
University of California, Berkeley
*Research sponsored by ©Intel Corporation
2
IntroductionPattern Placement Error Sources:• Reticle deformation occurs during e-chucking in the
exposure tool− In-plane distortion (IPD)
I
−Out-of-plane distortion (OPD)
O
• IPD and OPD cause pattern placement error (PPE)
I
−Slope changes ( ) are known to be negligible
PPE
IPD
PPE
OPD
IncidentLight
Reticle
PPE
3
IntroductionEWOC Strategy *
• Even for extremely flat masks (~50 nm) overlay-error due to PPE is greater than budget
• Idea: Compensate for pattern placement during e-beam write step by knowledge of IPD, OPD at e-chucking in exposure tool
• Here: Development of an analytical model to predict for IPD, OPD during e-chucking of masks
−Provides important knowledge of underlying physics−Fast computational alternative to expensive finite
element simulations of two body chuck-reticle contact problem
−Basis for possibly new standards
* Chandhok, M.; Goyal, S.; Carson, S., et al., “Compensation of overlay errors due to mask bending and nonflatness for EUV masks”, Proc. SPIE, 72710G (2009)
,
4
U
Reticle Frontside Non-flatness After E-Chucking at Voltage U
Chuck
Reticle
IntroductionAnalytical Model for IPD and OPD Prediction
• Reticle deformation at frontside during e-chucking is a combination of backside feature transfer and chuck feature transfer
• Here: Focus on backside feature transfer; i.e. assume flat chuck and reticle frontside before e-chucking
• Concentrate on OPD (IPD follows analogously)
C
Reticle BacksideNon-flatnessBefore E-Chucking
0 V
Chuck
Reticle
Electrodes
Dielectric
ConductingBack-side
0 V 0 V U
4
5
• Reticle thickness h, infinite x-domain with a cosine patterned backside with n waves over span L
• The backside can be ideally flattened with a harmonic load with pressure amplitude A
• Backside amplitude ubs before e-chucking
• Frontside amplitude ufs after e-chucking
Ideal Flattening in 2DThe Mechanical Boundary Value Problem
n = 6
6
Ideal Flattening in 2DExact Solution (Linear Elasticity)
E
• Usage of Airy-stress function
• Satisfies bi-harmonic equation
• Incorporate boundary conditions, linear elastic constitutive relation
• Resulting vertical displacement field is of the form:
7
Here: L = 152 mmA = 15 kPanu = 0.2E = 70 GPa
Ideal Flattening in 2DResults
• Relation for backside amplitude which can be ideally flattened by an applied pressure A
−ubs increases with higher pressure A−ubs decreases with increasing wavenumber n−ubs decreases with increasing reticle thickness h
d
8
d
Ideal Flattening in 2DResults
• Transmission Coefficient: Fraction of backside feature amplitude which transfers to the frontside
− Increase wavenumber n → Increase damping for backside feature transfer to the frontside
−More damping for thicker reticles
9
Electrostatic Chucking in 2DIdeal, Complete E-Chucking
• Infinite x-domain
• E-chuck provides a uniform pressure with the following voltage dependency
• Hertzian contact force ansatz
• Total backside pressure is cosine and thus has direct relation to the ideal flattening pressure
10
Electrostatic Chucking in 2DMaximum Allowable Backside Amplitude
• Estimate of the maximum amplitude of a harmonic backside displacement that can be flattened at e-chucking with voltage U
U
• For higher wavenumbers n, extremely high voltages are expected to be necessaryHere: L = 152 mmh = 6.35 mmnu = 0.2E = 70 Gpaeps = 8delta_d = 150 mum
11
Electrostatic Chucking in 2DIdeal, Non-complete E-chucking
• If actual exceeds
→ Only the fraction can be transferred to the frontside
Reticle Reticle
Chuck Chuck
12
k = 0
k = 1
k = 2
k = 3
k = 4
Electrostatic Chucking in 2DDiscrete Cosine Transform
• One Dimensional DCT (N data points xi)
O
• Our harmonic displacement
• Connection by setting
13
Electrostatic Chucking in 2DAnalytical Prediction of Arbitrary Feature Transfer• “Algorithm”
• Example: Analytical prediction of feature transfer compared to a finite element computation (see Slide 19)
Data ybsDCT
Data YbsFilter iDCT
Yfs yfs
cT
* yfeap: result from finite element verification (see Slide 14)
*
*
DCT
iDCT
14
Electrostatic Chucking in 2DFinite Element Analysis in FEAP• Simulation properties
−Rigid chuck mesh−Reticle (spline surface)
R
−Contacts, gap dependant pressure−Finite x-domain
• FEA verifies the analytical prediction of high frequency cut-off
• Relative deviation between the FEA and analytical computation to the peak-to-valley range is ~10%
Chuck
Reticle
15
Three Dimensional CaseApproximation via Energy Minimization
• Analog to 2D ideal flattening
• Usage of potential function
• Require
• Get
• Infinite x-, z-domain
• Stress boundary condition
16
Three Dimensional CaseArbitrary Surface Shape
• Same techniques as in two dimensions
• 2D discrete cosine transform
• Filter in frequency domain by usage of maximum allowable backside amplitude and transmission coefficient
17
Three Dimensional CaseAnalytical Prediction of Front-side Shape at 2000 V
• After applying filter, back transformation using inverse discrete cosine transform -> get frontside shape
• Wavenumbers n > 5 (corresponding ki > 10) are not transmitted
• That is, reticle backside features of wavelength < 30 nm cannot be observed at the frontside after chucking
18
Three Dimensional CaseFinite Element Analysis in FEAP
• Two body contact problem w/ gap dependant pressure
−Finite x-, z-domain• Matches the analytical prediction
of high frequency cut-off
• CPU- time ~ 15 min
19
2D Example for OPD/IPD PredictionAnalytical Model vs. Finite Element Calculation
• Top: Reticle backsurface shape ybs before e-chucking.
• Middle: Frontside OPD prediction.
• Bottom: Frontside IPD prediction according to the analytical treatment of ideal e-chucking at voltage U = 2000 V and result from finite element simulation (FEAP out) of real e-chucking.
20
ConclusionSummary• Ideal flattening model
− Theoretical derivation of transmission coefficient (2D/3D)− Prediction of high frequency damping for transmitted
waves from backside to frontside• Ideal e-chucking
− Theoretical prediction of maximum backside amplitude (2D/3D) which can removed by ideal e-chucking
− Accounts for chuck dielectric properties and applied voltage
− Numerical example shows high frequency cut-off, resp. feature wavelengths < 30 nm not being transmitted
0.76
0.99
0.99
Transmissioncoefficient
5
2
1
n
115 nm51 nm76 mm
5.4 nm
1600 nm
Maximum allowableamplitude (3000 V)
Backside feature
wavelength
Maximum allowableamplitude (2000 V)
152 mm 710 nm
30 mm 2.4 nm
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ConclusionSummary
• Real e-chucking−Finite element calculation of two body chuck-reticle
contact problem−Comparison to theoretical prediction of an arbitrary
feature transfer shows good agreement; analytic results good for estimation and understanding but FEA needed for precision (~10% deviation)
−Finite element computation about 15 min vs. analytical computation << 1 min