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Subfield Scheduling for Througput Subfield Scheduling for Througput Maximization in Electron-beam Maximization in Electron-beam

Photomask FabricationPhotomask FabricationS. Babin*, A.B. Kahng, I.I. Mandoiu, S. Muddu

CSE & ECE Depts., University of California, San Diego*Soft Services

Resist heating is one of the largest contributors to critical dimension (CD) variation in electron beam photomask fabrication. Previous methods for reducing CD variation caused by resist heating include lower beam currents, increased delays between electron flashes, and multiple writing passes. However, all these methods lower mask writing throughput. This leads to higher mask-making costs, which are increasingly becoming a major limiting factor to semiconductor industry productivity.

In this work, we investigate a new degree of freedom for mitigating CD variability caused by resist heating. By optimizing the order in which subfield are being written, it is possible to reduce CD variability caused by heating, without significantly decreasing mask writing throughput.

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MotivationMotivation• Using higher energy electron beams to decrease mask writing time is

limited by resist heating effects, such as Critical Dimension (CD) distortion and irreversible chemical changes in the resist

• “Multi-pass” sequential writing and higher delays between electron flashes decrease maximum resist temperature but significantly increase writing time, thus decreasing mask writer throughput

• Scheduling of subfields provides unexplored opportunities for decreasing maximum resist temperature without increasing writing time significantly

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

Mask Writing Schedule ProblemGiven: Beam and resist parameters, threshold temperature Tmax

Find: fracture and subfield writing schedule with minimum total writing time such that the maximum resist temperature never exceeds Tmax

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Variable-shaped E-beam WritingVariable-shaped E-beam WritingTaxonomy of mask features• Fractures: smallest features written on the

mask; dimensions in the range 0.75m -2m• Minor field: collection of fractures• Subfield: collection of minor fields; typical

size of a subfield is 64m X 64m• Major field or cell: collection of subfields

E-beam writing technology context• High power densities (as much as 1GW/c.c.)

needed to meet SIA Roadmap requirements • These power densities induce excessive

local heating causing significant critical dimension (CD) distortion and irreversible changes in resist sensitivity

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

• Scheduling of subfields incurs very low overhead, and is still effective in reducing excessive heating effects

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Self-Avoiding Self-Avoiding TSP TSP FormulationFormulationThe blocked set for a given time slot is defined as the set of regions which, if written during the time slot, will exceed the threshold temperature Tmax. In above definition, regions can be either fractures or subfields, depending on the granularity at which scheduling is performed. Using blocked sets, the mask 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 being written at the same time, i.e., if ti tj, i j, then ti + wi tj

(3) no region is being written while blocked, i.e., if Ri Bi then tj + di ti or tjti

(4) the completion time, maxi(ti + wi), is minimized

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Subfield SchedulingSubfield 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 Ri become Euclidean balls of radius R centered at each minor field

• Feasible schedules are similar to well-spaced labelings 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

Subfield Scheduling ProblemMaximize ball radius R subject to feasibility of a writing schedule without idle time. In other words, find a subfield schedule in which the distance between every two consecutively written subfields is at least R, where R is as large as possible

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Lagarias Subfield SchedulingLagarias Subfield SchedulingFor a M1 x M2 grid with both M1 and M2 even, the Lagarias schedule writes in the mth step the subfield located at

row and column where

m = lG*l* + iL* + j, with 0 j L*, 0 i G*, 0 l H*

G* = gcd (M1,M2), H* = and L* = lcm(M1,M2) / H*

Reference: J.C. Lagarias, SIAM J. Discr. Math. 13, 2001, pp. 521-534

)M (mod2

2Mji 1

1 )M (mod

2

2Mji 2

2

odd is /4MM if 2

even is /4MM if 1

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21

1 2 3 4

8 7 6 5

9 10 11 12

16 15 14 13

1 5 9 13

14 2 6 10

11 15 3 7

8 12 16 4

1 13 5 9

6 10 2 14

3 15 7 11

8 12 4 16

OptimalLagariasSequential

Schedules for 4x4 subfields:

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Simulation SetupSimulation Setup• Resist heating simulations were performed using the commercially

available TEMPTATION tool (S. Babin and I. Kuzmin, J. Vac. Sci. Technol., B16, 1998, p. 3241)

• Simulated subfield scheduling strategies:– Sequential schedule– Spiral schedule (Englestad, personal communication)– Random schedule– Lagarias schedule

• Simulated pattern:– 512 fractures per subfield, fracture size = 2m x 2m– Chessboard fracture layout inside each subfield, giving 50% coverage– Sequential writing schedule followed for writing fractures within a subfield– 16 x 16 subfields; subfield size = 64m x 64m

• E-beam parameters:– Acceleration voltage = 50kV– Current density = 40A/cm2

– Fracture flash time = 1 sec

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16x16 Subfields Simulation Results16x16 Subfields Simulation Results

Sequential schedule

Max48.85CMean27.59C

Random schedule

Max38.88CMean16.34C

Spiral schedule

Max47.64CMean22.59C

Lagarias schedule

Max35.23CMean19.08C

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Conclusions and Ongoing WorkConclusions and Ongoing Work

Sequential Max=105.10C

Spiral Max=110.70C

Random Max=94.77C

Lagarias Max=90.78C

Critical subfield temperature profiles and maximum fracture temperatures before flashing for the four subfield schedules:

We proposed a new subfield scheduling approach to throughput maximi-zation, which complements recently explored optimizations of such para-meters as beam current density, flash size, and number of passes. Simulation results show that excessive resist heating can be significantly reduced by avoiding successive writing of subfields that are close to each other. The lower resist temperature enables the use of a higher beam current density. Depending on the particular parameters of the writer, this can reduce total writing time and hence increase throughput while keeping CD distortion within acceptable limits. Ongoing work explores simulta-neous optimization of beam current density and subfield schedule.