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