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E-BLOW: E-B

• Beam

eam Lit Lithogr hography aphy Ov Over erlapping lapping aw awar are e Stencil encil Planning lanning for

• r MCC System

em

Bei Yu, Kun Yuan*, Jhih-Rong Gao, and David Z. Pan ECE Dept. University of Texas at Austin, TX *Cadence Inc., CA Supported in part by NSF and NSFC

Int ntroduct

• duction–

ion– E-B

• Beam

eam

t E-Beam lithography:

› Several decades, for mask manufacturing › Candidate for next generation lithography, with MPL/EUV/DSA

t Conventional E-Beam system:

› variable shaped beams (VSB): shaping aperture + second aperture › Character Projection (CP): a pattern (character) is pre-designed on the stencil, then it can be printed in one electronic shot;

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Electrical Guns Wafer 2nd Apenture Shaping Apentures Electrical Gun Wafer Stencil Shaping Apenture

1 2 3

Int ntroduct

• duction–

ion– MCC system em

t Multi-Column Cell (MCC) system

› Several independent character projections (CP) to speed-up › Each CP is applied on one section of wafer. › Share one stencil design

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Electrical Guns Shaping Apentures w1 w2 w3 w4 4 Regions on Wafer Stencils

MCC system with 4 CPs 4 stencils share 1 design

Introduction– MCC system Shot#

t MCC system with:

› P CPs, wafer is divided into P regions › n character candidates (patterns) {c1, …, cn} › For ci, its VSB shot# is ni; repeat on region › ai: indicate whether ci is selected on stencil

t Total shot# for region : t MCC system writing time:

4

tic

wc

wc

With stencil, CP shot# Without stencil, VSB Shot#

Problem

• blem For

Formulat mulation ion

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Overlapping aware Stencil Planning (OSP) Problem:

t Input: set of characters; MCC system info t Output: selected characters, pack them on stencil t Objective: minimize MCC system writing time t 1D-OSP and 2D-OSP problems:

A B C D E F A B C D E F

Problem Formulation-- Complexity

t Lemma 1: 1D-OSP is NP-hard

› Reduced from Multiple-Knapsack problem

t Lemma 2: 2D-OSP is NP-hard

› Reduced from Strip Packing problem

t New challenges for MCC system:

① New total shot# functions ② More character number (more than 4000)

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A B C D E F A B C D E F

E-BLOW

E-BLOW for 1D-OSP

t ILP formulation

› NP-hard to solve ILP, runtime penalty. › LP relaxation cannot be applied here. Why? (aik = ajk = 0.5)

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E-BLOW for 1D-OSP (cont.)

t Simplified ILP formulation t Theorem: The LP Rounding solution of (3) can be a 0.5/α

− approximation to program (3’), where (3’) is a similar multiple knapsack problem.

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E-B

• BLOW

LOW for

• r 1D

1D-OS

• OSP (cont

cont.) .)

t Novel iterative solving framework to near-optimal solution t LP relaxation with lower bound theoretically t Successive rounding t Dynamic programming based refinement

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Apply S-Blank Assumption Successive Rounding Simplified LP Formulation Refinement Output 1D-Stencil Solve New LP Finish? Update LP No Yes

Regions Info Characters Info

E-BLOW for 2D-OSP

t Simulated annealing based framework. t Sequence Pair as topology representation. t Pre-filter process to remove bad characters. t Clustering is applied to achieve speedup.

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KD-Tree based Clustering Simulated Annealing based Packing Output 2D-Stencil Pre-Filter

Regions Info Characters Info

E-BLOW for 2D-OSP (cont.)

t KD-Tree based Clustering

› Speed-up the process of finding available pair; › From O(n) to O(logn); › For c2 to find another candidate with the similar space, only scan c1 − c5.

11 c5 c2 c4 c3 c1 c7 c6 c8 c9 Horizontal Space Vertical Space

c5 c2 c3 c4 c1 c7 c6 c8 c9

1D-OSP Writing Time Comparison

t For 1D cases, greedy algorithm introduces 47% more wafer writing

time, and [TCAD’12] introduces 19% more wafer writing time.

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900,000 1,000,000 1 D − 1 1 D − 2 1 D − 3 1 D − 4 1 M − 1 1 M − 2 1 M − 3 1 M − 4 1 M − 5 1 M − 6 1 M − 7 1 M − 8 Shot Number for 1D cases

Greedy in [TCAD’12] [TCAD’12] E−BLOW

100,000 200,000 300,000 400,000 500,000 600,000 700,000 800,000

2D-OSP Writing Time Comparison

t For 2D cases, greedy introduces 30% more wafer writing time, while

[TCAD’12] introduces 14% more wafer writing time.

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2D−2 2D−3 2D−4 2M−1 2M−2 2M−3 2M−4 2M−5 2M−6 2M−7 2M−8 Shot Number for 2D cases

Greedy in [TCAD’12] [TCAD’12] E−BLOW

200,000 400,000 600,000 800,000 1,000,000 1,200,000 1,400,000 2D−1

CPU Runtime Comparison

t Compared with [TCAD’12], E-BLOW can reduce 34.3% of runtime

for 1D cases, while 2.8× speedup for 2D cases.

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2M−2 2M−3 2M−4 2M−5 2M−6 2M−7 2M−8 Runtime (s)

[TCAD’12] E−BLOW

1 10 100 1,000 10,000 1D−1 1D−2 1D−3 1D−4 1M−1 1M−2 1M−3 1M−4 1M−5 1M−6 1M−7 1M−8 2D−1 2D−2 2D−3 2D−4 2M−1

Conclusion

t E-BLOW, a tool to solve OSP problem in MCC system. t E-BLOW can achieve better performance in terms of wafer

writing time and CPU runtime, for both MCC system and traditional E-Beam system.

t E-Beam is under heavy R&D, including massive parallel

writing.

› More research to improve the throughput of E-Beam › More research on the E-Beam-aware design

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t Thank You

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