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A Comprehensive Model for Line-Edge Roughness
Chris Mackwww.lithoguru.com
© 2010 by Chris Mack EUV Lithography Workshop 2010
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Outline:There’s a lot going on in LER
• Modeling Approaches• Photon and acid shot noise• Reaction-diffusion kinetics• Development and dynamical scaling• Overall model for LER• What’s missing – future work• Conclusions
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Modeling LER
• Describe every event probabilistically• Law of Large Numbers – stochastic models become
continuum models when the number of events becomes very large (mean field theory)
• First Stochastic Method: Monte Carlo– Can be very rigorous and very useful, but very slow– Can be difficult to optimize materials and processes (slow, hard to
gain intuition)• Second Stochastic Method: Approximate analytic solution
– True analytic solution is not possible (how good is the approximation?)
– Goal: Predict standard deviation, correlation length, and roughness exponent (i.e., predict the full PSD)
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Stochastic View of Chemical Concentration
• Model atom/molecule as a point located at its center of mass• Consider a volume V – is the molecule in the volume or not?
– This is a binary proposition, governed by the binomial distribution: P(n) = probability of finding n molecules in V
– The binomial probability distribution becomes a Poisson distribution with average concentration C
( ) CVn
en
CVnP −=!
)(
CVnnn 11
==σ
CVn = CVn =2σ
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Stochastic View of Exposure Reaction
• Including photon shot noise, acid uncertainty is
• The pure photon shot noise contribution is usually much, much smaller than acid shot noise. It can be ignored for 193nm lithography
• For EUV, an extra noise term must be added (of unknown form)
( ) ( )[ ]photonPAGPAG
h nnhh
nh
−−
−−+=
0
2
0
2 1ln1σ
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Stochastic View of Reaction-Diffusion
• Reaction is catalyzed by the diffusing species:
CH2-CH
M
H+
von Smoluchowski Trap:
Reaction can occur once acid approaches the blocking group within its capture radius, a.
aRate∝
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Stochastic View of Reaction-Diffusion
• Deriving the statistics of reaction-diffusion is hard! For the details, please see: Chris A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication, John Wiley & Sons, (London: 2007).
RDPSFhheff ⊗=
hD
ha
effσ
σσ ⎟
⎟⎠
⎞⎜⎜⎝
⎛≈
2Derivation of this is approximate – more work is needed
∫=PEBt
PEBdtDPSF
tRDPSF
0
1
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Stochastic View of Exposure + Reaction-Diffusion
• Final expression for the uncertainty in deblocked polymer concentration over some volume V:
( ) ( ) ( )[ ]⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −−+⎟⎟
⎠
⎞⎜⎜⎝
⎛+=⎟
⎟⎠
⎞⎜⎜⎝
⎛
−−− photonPAGPAGDPEBamp
blocked
m
nnhh
nhatK
mnm 0
2
0
22
0
21ln121
σσ
Photon shot noise
PAG concentration,
exposure
Reaction-diffusion
Deblocked concentration
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Dose Dependence of σm
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0 20 40 60 80 100
σm
Exposure Dose (mJ/cm 2)
σD/a = 5
σD/a = 10
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Impact of Gradient on LER
5/
333 +=dxdmLERσBlue line:
T. B. Michaelson, et al., “The Effects of Chemical Gradients and Photoresist Composition on Lithographically Generated Line Edge Roughness”, Advances in Resist Technology and Processing XXII, SPIE Vol. 5753 (2005) pp. 368-379.
LER
, nm
Protected Polymer Gradient, 1/um
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Line-Edge Roughness(Tying it all Together)
• Consider a small deviation deblocking level. Assuming threshold development, the resulting change in resist edge position will be approximately
• Data suggests an Overall Model for LER:
mdmdxx Δ=Δ
0/σσσ +=
dxdmm
LER
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Line-Edge Roughness(Tying it all Together)
• How to improve LER:– Increase latent image gradient– Decrease σm
– Decrease σo
• These terms sometimes work against each other– Polymer size determines the volume over which the
chemistry is averaged, but also σo
– Increasing polymer size will decrease σm but increase σo
– There will be an optimum polymer size for minimum LER
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Line-Edge Roughness and Acid Diffusion
0
1
2
3
4
5
0 5 10 15 20 25 30Acid Diffusion Length (nm)
LER
(Arb
. Uni
ts)
σmdx/dm
σLER
dxdmm
LER /σ
σ ∝
Trap Radius a = 1 nm
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Line-Edge Roughness and Acid Diffusion
0
1
2
3
4
5
0 5 10 15 20 25 30Acid Diffusion Length (nm)
LER
(Arb
. Uni
ts)
a = 0.5 nm
a = 1.5 nm
Optical/Thermal Dose Trade-off
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Average deblocking level held constantDiffusion length held constant
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Future Work (What’s Missing)
• EUV resist exposure mechanism• Base quencher has been ignored (by me) to date
– Quencher is at lower concentrations than acid – more uncertainty– Quencher improves the latent image gradient (there has to be an
optimum quencher concentration)• Development rate uncertainty – there’s more to do
– Examine impact of correlations of development rate noise– How does a development rate gradient affect things?– What happens as the dissolution rate becomes very slow – will we
move into the directed percolation depinning (DPD) universality class?• Other things
– Speckle – correlated photons– PAG and/or quencher aggregation?– Metrology vs. reality, calibrating the model– Device impact – how good must the model be?
Conclusions
• LER is the ultimate limiter to resolution in optical lithography (for both EUV and 193i)
• A good LER model is needed to optimize resist process and material properties and to find the minimum possible LER– Progress is being made, but a (good enough) predictive
LER model does not yet exist– How low can LER go? What is the ultimate resolution
limit? Will we understand LER before it is too late?
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