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