Mask Design for Resolution Enhancement: An Inverse Problem in
Optical Microlithography
June 25, 2007Conference on Applied Inverse Problems (AIP)
Amyn Poonawala and Peyman MilanfarUniversity of California, Santa Cruz
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Outline
Introduction to Optical Lithography and RET
Inverse Lithography Framework
Regularization Framework
Results
Double Exposure Lithography and Conclusions
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Optical Microlithography
Figure from F. Schellenbeg, “A Little Light Magic”, IEEE Spectrum, 40 (2003)
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Distortions
DiffractionEffects
Input(Mask Pattern)
Projection Optics
Resist Action
Output(Wafer Pattern)
Non-linear
Band limited higher frequencies lost.
Corner rounding, line-end shortening, poor fidelity.
Pattern collapse
Mask
Mask
Wafer
Wafer
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Resolution Limit and RET
λ Wavelength.NA Numerical Aperture.k1 Process Constant (min=0.25).wmin Minimum line-width.
Decrease the wavelengthDecrease the wavelength.Options
Increase NAIncrease NA.
Decrease k1Decrease k1 Resolution Enhancement Techniques (RET)
Optical Proximity Correction (OPC)Phase Shift Masks (PSM)
Assist BarsDouble Exposure
Mask Design Problem
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Optical Proximity Correction (OPC)Pre-distort the pattern such that it cancels out for the process losses to come.
Figure from F. Schellenbeg, “A Little Light Magic”, IEEE Spectrum, 40 (2003)
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180 degree phase-shift creates destructive interference giving a dark line and good contrast.
Phase-Shift Masks (PSM)
Amplitude(Single Aperture)
Amplitude(Single Aperture)
Glass
Glass
Chrome
Chrome
Phase Shifter
Intensity
Intensity
Con
vent
iona
lPS
M
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Inverse Lithography Mask Design (ILT)
Output intensity function Input intensity function
Known Forward Model
ImagingSystem: T{.}
(target)T-1{.}
??
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Imaging System
Non-linear (Heaviside) operation.
Coherent, incoherent, or partially coherent.
DiffractionEffects
Thresholding
Binary Input Mask
Projection Optics
Aerial Image (Gray)
Resist Action
Binary Output (Wafer) Pattern
T{.}
Past Work: Integer Optimization, Genetic Algorithms, etc.Our Goal: Continuous function formulation.
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Approximated Forward Model (Coherent Imaging)
Sampling and lexicographic ordering.
Close to binary
Sigmoid
ThresholdingIntensity
| . | 2Convolution (H)
Cutoff = NA/λ
Aerial Image Resist
J1(.) Bessel function of the first kind
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Sigmoid Function and Process Model
Continuous function which approximates the hard-threshold operation.
a steepness of the function.tr resist threshold.tr
for i = 1,…,MN
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Optimization Problem
subject to the following constraints:
-1 or 0 or +1Strong PSM (With chrome)-1 or +1Strong PSM (100% transmission)
-0.4243 or +118% attenuated PSM-0.2449 or +16% attenuated PSM
0 or +1OPC
Allowable transmission values (constraints on mj for j=1,…,MN)
RET
Integer Optimization Problem!
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Continuous Function Formulation
Parametric transformation
is the unconstrained parameter vector
Relax the constraints
Bound constraints (continuous)
Consider OPC mask design for coherent systems
mj = 0 or 1
Analytic gradient
convolution element-by-element multiplication
convolution
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Regularization Framework
Inverse Lithography is an ill-posed problem.
Estimated masks still have continuous transmission values.
Pixel-based masks are complex and difficult to manufacture.
We prefer certain solutions more than others. This information can be incorporated as a prior knowledge about the solution.
Regularization framework can be employed to promote certain desirable properties in the solution.
Contour Fidelity term
Weighing parameters
Regularization function (penalty term)
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Regularization Framework
Discretization Penalty (Rdis(m)) Penalize and curb transmission values not equal to 0 or 1 using quadratic or quartic penalty terms.
0 0.5 10
0.2
0.4
0.6
0.8
1
mj
Pena
lty C
ost
Penalty for binary masks
Complexity Penalty (RTV(m))An L1 norm based penalty term to curb the complexity of the masks,
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Error = 140
1: OPC for Incoherent System
NA=1.2, λ=193nm, σ=1, tr=0.5, γfid=1, 5nm sampling
60nm features k1 = 0.37
BinaryMasks
Binary Output
The main feature is broken!
Error = 1150
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1: OPC for Incoherent System (Reg)
γfid = 1, γd = 0.005, γc=0.2
BinaryMasks
Binary Output
60nm features k1 = 0.37
Error = 1150 Error = 177
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2: Compare with Integer Optimization
S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process., vol. 4, no.9, pp. 1252–1257, Sep. 1995.
Fidelity was the same (equal to zero).Run-time 5 seconds (compared to 20-25 minutes reported in 1995).
Square aperture, incoherent illumination.
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3: Assist bars for 100nm contact hole (Coherent System)
NA = 0.85, λ=193nm, σ=0, tr=0.3, k1=0.44, 10nm pixels.
BinaryMasks
BinaryOutput
γc = 0.001 γc = 0(No Complexity Penalty)
100nm
Aerial Image Slices
• Assist features are small features below the resolution limit of the imaging system.
• Help improve contrast of the main feature.
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4: 50nm feature using PSM (NA=0.85, λ=193nm, σ=0, tr=0.3)
Estimated mask Aerial Image Binary Output
Black = -0.2449 White = 1
6% EPSM
Gray = -1Black = 0White = 1
Strong PSM
Black = -1White = 1
100% transmission PSM
UnintuitiveMasks!!
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4: PSM Aerial Image Slices
The aerial image slices show a good contrast especially for strong PSM.
Side-lobes are below the resist threshold.
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5: Intel Fab Results (0.93NA/60nm feature/k1=0.289)
sigma = 0.2 (partial coherence)
CPL: Black=-1, White=1
Courtesy Intel Corporation
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Impact and Conclusions
Mask design problem was formulated as continuous function unconstrained optimization problem.
First ones to calculate and report analytic gradient. Computationally efficient extendible to full-chip level.
Regularization framework introduced to incorporate user-constraints.
Analytic gradient-based ILT solutions are now widely adopted by the lithography industry: Luminescent Tech (SPIE 2005/Patent 2007), Mentor Graphics (SPIE 2005/06), Invarium Tech (SPIE 2007), Magma (2007), Synopsys (SPIE 2007).
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Double Exposure
Aerial image contrast poor for very low k1 values Sensitive to process variations.
Single exp has phase conflict problem if odd number cycles present.
DEL-ILT
??
Desired
??
Exposure # 1
Exposure # 2 Input Mask # 2
Develop
Output (Wafer) Pattern
T{.}
Input Mask # 1
Why Double Exp?
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DEL-ILT Result
Single Exp
Combined Aerial ImageAerial Image 1 Aerial Image 2