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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