C. A. D. Theme

Annual Review

June 2002

The Self-Avoiding Traveling Salesman Problem: A Formulation for Reduction of Resist Heating Effects in Mask Production

A.B. Kahng, I. Mandoiu and S. Muddu -- UCSD

AbstractAt the design-manufacturing interface, mask creation and variability costs increasingly limit semiconductor industry productivity. In this work, we seek to mitigate one component of mask feature variability. We introduce a new self-avoiding traveling salesman problem (SA-TSP) writing strategy for fabrication of photomasks using variable shaped electron beam (e-beam) mask writers. Our approach mitigates resist heating effects, such as CD distortion and irreversible chemical changes, that are becoming a limiting factor for the sequential-write strategy currently used by industrial e-beam writers. Our formulation requires that (a) the temperature at any mask location be kept below a given threshold and (b) the total mask writing time be minimized. Objective (a) is met by avoiding consecutive writing of mask locations that are too close to each other, while objective (b) is ensured by avoiding idle times and minimizing beam movement. Our results are as follows. (1) We show that finding the optimal writing strategy is NP-hard in general, but can be done efficiently under some assumptions. (2) We give provably good approximations based on well-spaced labelings of grids recently introduced by J.C. Lagarias. (3) We give simulation results validating the SA-TSP approach. Average temperature for SA-TSP writing is ~10% lower than for sequential writing, with over 75% of mask locations having smaller temperature.

REFERENCESJ.C. Lagarias, “Well-Space Labelings of Points in Rectangular Grids”, SIAM J. DISCRETE MATH, Vol. 13, No. 4, 2001, pp. 521-534.Sergey Babin, “Measurement of resist heating in photomask fabrication”, J. Vac. Sci. Technol. B 15(6), Nov/Dec 1997, pp. 2209-2213.Ronald Gould, “Graph Theory”, Benjamin-Cummings, 1988. Chapter 5.Alexander C. Wei et. al, “Localized resist heating due to electron-beam patterning during photomask fabrication”, Proceedings of SPIE, Vol. 4186(2001), pp. 482-493.

Motivation• In future technology nodes, mask writing time becomes a major bottleneck for fabrication of VLSI integrated circuits

• Using higher energy electron beams to decrease mask write time is limited by resist heating effects, such as Critical Dimension (CD) distortion and irreversible chemical changes in the resist

• “Multi-pass” sequential writing decreases maximum resist temperature but significantly increases writing time, thus decreasing mask writer throughput

• All current e-beam writers use sequential writing strategies, but are able to write patterns non-sequentially

• Scheduling of subfields provides enough opportunity for decreasing maximum resist temperature without increasing writing time significantly

• Proposed solution: use non-sequential scheduling of subfields to decrease the maximum resist temperature

• Non-sequential scheduling of subfields leads to reduced CD variability and helps calibrating the effective change in resist sensitivity and feature distortion due to resist heating.

Writing schedule problem: given threshold temperature Tmax find a writing schedule with minimum writing time such that the maximum resist temperature never exceeds Tmax

Variable Shaped E-Beam WritingVariable Shaped E-Beam mask writing is a hierarchical technique used for high throughput mask fabrication of VLSI integrated circuits

Taxonomy of mask features• Fractures are the smallest features written on the mask with dimensions in the range 0.75m -2m

• A minor field is a collection of fractures• A subfield is a collection of minor fields; the typical size of a subfield is 64m X 64m

• A major field or cell is a collection of subfields

E-beam writing technology context• Higher energy electron beams decrease mask write time, increase CD distortion, and cause irreversible chemical changes in the resist

• Scheduling of fractures incurs large positioning overheads due to technological limitations ofcurrent e-beam writers

• Scheduling of subfields incurs very low overhead

Mask writing

WaferFabrication

General SA-TSP FormulationDefine a blocked set for a given time slot as the set of regions which, if written during the same time slot, will exceed the threshold temperature Tmax. Using blocked sets, the writing schedule problem can be reformulated as follows:

Self-Avoiding Traveling Salesman ProblemGiven: n non-overlapping regions R1, R2,. . ., Rn in the plane, where for each region Ri we are given its writing time wi , blocked set Bi {R1, R2,. . ., Rn }, and blocking duration di.Find: writing start times ti for each region such that(1) writing starts at time t = 0(2) no two regions are written at the same time, i.e., if ti tj, i j, then ti + wi tj(3) no region is written while blocked, i.e., if Ri Bi then tj + di ti or tjti(4) the completion time, maxi(ti + wi), is minimized

Hardness Result : The SA-TSP problem is NP-hard even when wi di 1. (Proof by reduction from the Hamiltonian Path problem.)

Theorem: An optimum SA-TSP schedule can be found in O(n2) time if the cardinality of each blocked set is at most n/2

An algorithm follows from the proof of Dirac’s theorem: A graph with n 3 vertices in which every vertex has at least n/2 neighbors has a Hamiltonian circuit

Subfield Scheduling• Key observation: Scheduling of subfields provides enough opportunity for decreasing maximum resist temperature without increasing writing time significantly

• For subfield scheduling the SA-TSP graph becomes a grid graph, writing and blocking times wi and di become the same for all minor fields, and blocked sets Ribecome Euclidean balls of radius R centered at each minor field

Subfield Scheduling Problem: Maximize ball radius R subject to feasibility of a writing schedule without idle time (i.e., schedule for which completion time equals the sum of writing times)

• Feasible schedules are similar to well-spaced labeling of grids studied by J.C. Lagarias, except that well-spaced labelings use rectilinear balls instead of Euclidean balls

• Lagarias gives explicit solutions guaranteed to be within an additive factor of 2 from the optimum under rectilinear metric, and within a multiplicative factor of 2/2 from the optimum under Euclidean metric

Lagarias subfield schedulingFor an M1 X M2 grid with both M1 and M2 even, the Lagarias schedule writes in the

mth time step the subfield located at and column

where

Scheduling over 16 subfields

Sequential schedule Lagarias schedule Optimal schedule

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13951

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142106

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Experim ental ResultsComparison of fracture temperature distributions for sequentia l and Lagarias subfie ld scheduling over a mask of size 0.512m m X 0.512m m

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0-15 16-30 31-45 46-60 61-75 76-90 91-105

106-120

121-135

136-150

151-165

166-180

181-195

196-210

211-225

226-240

241-255

256-270

271-285

286-300

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Lagarias

S equential

Conclusions

Future plans

• We proposed a new self-avoiding writing strategy to mitigate the resist heating effects that lead to CD distortion in photomask fabrication.

• Study the effect of scheduling on effective change of resist dose and its impact on CD distortion for different current densities and resist sensitivities

• Study the effect of scheduling fractures, by incorporating the positioning and timing overheads

• Study the effects of fracturing on the change in effective resist dose during variable shaped e-beam mask writing.

• The timing of the subfields in the mask is sequenced to overcome the cumulative resist heating effect

• Simulation of the temperature evolution over different mask models using TEMPTATION revealed the effectiveness of scheduling. 75% of the fractures on the mask model have lower average temperature as a result of scheduling

Temperature evolutionTem perature h istory o f a fracture located at (160um , 96um ) in a 4 X 4 subfie ld m odel.

0

50

100

150

200

250

300

1 32

63

94

12

5

15

6

18

7

21

8

24

9

28

0

31

1

34

2

37

3

40

4

43

5

46

6

49

7

52

8

55

9

59

0

62

1

65

2

68

3

71

4

74

5

77

6

80

7

83

8

86

9

90

0

93

1

96

2

99

3

Flash numbe r/8

Te

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era

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C

Lagarias

Sequential

Temperature profile of a fracture in 4 x 4 subfield model. With Lagarias scheduling, peak temperature is reduced. Rapid temperature decay in case of Lagarias can overcome exposure bake, leading to lower CD distortion.

3-D plot showing the pre-flash temperature difference between Sequential and Lagarias schedules for a single subfield. Only 25% of the fractures in the subfield are at higher temperature in Lagarias than those in Sequential schedule.