E-BLOW: E-Beam Lithography Overlapping aware Stencil Planning for MCC System

Transcription

1 E-BLOW: E-Beam Lithography Overlapping aware Stencil Planning for MCC System 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

2 Introduction E-Beam 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; Electrical Guns Electrical Gun Shaping Apentures Shaping Apenture nd Apenture Stencil Wafer 2 Wafer

3 Introduction MCC system 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 Electrical Guns MCC system with 4 CPs Shaping Apentures 4 stencils share 1 design Stencils w1 w2 w3 w4 4 Regions on Wafer 3

4 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 w c t Total shot# for region : t ic w c With stencil, CP shot# Without stencil, VSB Shot# t MCC system writing time: 4

5 Problem Formulation 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: D E F D E F A B C A B C 5

6 Problem Formulation-- Complexity t Lemma 1: 1D-OSP is NP-hard Reduced from Multiple-Knapsack problem D E F A B C t Lemma 2: 2D-OSP is NP-hard Reduced from Strip Packing problem D E F A B C t New challenges for MCC system: New total shot# functions More character number (more than 4000) E-BLOW 6

7 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) 7

8 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. 8

9 E-BLOW for 1D-OSP (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 Regions Info Characters Info Apply S-Blank Assumption Simplified LP Formulation Update LP Solve New LP Successive Rounding Refinement No Finish? Yes Output 1D-Stencil 9

10 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. Regions Info Characters Info Pre-Filter KD-Tree based Clustering Simulated Annealing based Packing Output 2D-Stencil 10

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

12 1D-OSP Writing Time Comparison Shot Number for 1D cases 1,000, , , , , , , , , ,000 0 Greedy in [TCAD 12] [TCAD 12] E BLOW 1D 1 1D 2 1D 3 1D 4 1M 1 1M 2 t For 1D cases, greedy algorithm introduces 47% more wafer writing time, and [TCAD 12] introduces 19% more wafer writing time. 12 1M 3 1M 4 1M 5 1M 6 1M 7 1M 8

13 2D-OSP Writing Time Comparison 1,400,000 1,200,000 Greedy in [TCAD 12] [TCAD 12] E BLOW Shot Number for 2D cases 1,000, , , , , D 1 2D 2 2D 3 2D 4 2M 1 2M 2 2M 3 t For 2D cases, greedy introduces 30% more wafer writing time, while [TCAD 12] introduces 14% more wafer writing time. 13 2M 4 2M 5 2M 6 2M 7 2M 8

14 CPU Runtime Comparison 10,000 1,000 [TCAD 12] E BLOW Runtime (s) M 2 2M 3 2M 4 2M 5 2M 6 2M 7 2M 8 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 t Compared with [TCAD 12], E-BLOW can reduce 34.3% of runtime for 1D cases, while 2.8 speedup for 2D cases. 14

15 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 15

16 t Thank You 16