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Resist Heating Dependence on Resist Heating Dependence on Subfield Scheduling in 50kV Subfield Scheduling in 50kV Electron-beam MaskmakingElectron-beam Maskmaking

S. Babin*, A.B. Kahng, I.I. Mandoiu, S. MudduCSE & ECE Depts., University of California at San Diego

*Soft Services

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AbstractAbstractResist heating is one of the largest contributors to critical dimension (CD) variation in electron beam photomask fabrication. Previous works on reducing CD variation caused by resist heating have explored the optimization of such parameters as beam current density, flash size, number of passes, and subfield writing order. A common drawback of these optimizations is that the decreases in resist temperature are obtained at the expense of increasing mask writing time and cost.

In this work, we propose a new method for minimizing CD distortion caused by resist heating. Our method performs simultaneous optimization of beam current density and subfield writing order, and is the first to result in decreased resist heating with unchanged mask writing throughput. Simulation experiments using the commercially available TEMPTATION tool show that non-sequential writing of subfields allows for effective dissipation of heat, and leads to overall reductions in resist temperature.

REFERENCES • S. Babin and I. Kuzmin, “Optimization of throughput of electron-beam lithography in photomask

fabrication regarding acceptable accuracy of critical dimensions”, Proc. BACUS Photomask 2001

• S.V. Babin, A.B. Kahng, I.I. Mandoiu, and S. Muddu, “Subfield scheduling for throughput maximization in electron-beam photomask fabrication”, Emerging Lithographic Technologies VII, R.L. Engelstad (ed.), Proc. SPIE #5037, 2003, to appear

• J.C. Lagarias, “Well-Spaced Labelings of Points in Rectangular Grids”, SIAM J. Discr. Math. 13(4), 2001, p. 521

• S. Babin and I. Kuzmin, “Experimental Verification of TEMPTATION software tool”, J. Vac. Sci. Technol. B 16(6), 1998, p. 3241

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MotivationMotivation• The use of higher energy electron beams is limited by resist heating effects,

such as Critical Dimension (CD) distortion and irreversible chemical changes in the resist

• Resist temperature can be reduced by using lower beam current density, insertion of delays between electron flashes, “multi-pass” sequential writing, and non-sequential writing of subfields. However, all these techniques result in increased mask writing time, i.e., reduced mask writing throughput

• In this work we use simultaneous optimization of beam current density and subfield writing order for minimizing CD distortion caused by resist heating. To reduce excessive resist heating, we avoid successive writing of subfields that are close to each other. To maintain mask writing throughput, we increase beam current density so that resulting reduction in dwell time compensates for the increased travel and settling time caused by non-sequential writing of subfields

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

Find: Beam current density and subfield writing schedule 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 (up to 1GW/c.c.) will be

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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Computational ComplexityComputational ComplexityThe blocked set for a given time slot is defined as the set of regions (e.g., subfields) which, if written during the time slot, will exceed the threshold temperature Tmax. Using blocked sets, the mask writing schedule problem can be reformulated as follows:

Self-Avoiding Traveling Salesman Problem (SA-TSP)Given: 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 tj ti

(4) the completion time, maxi(ti + wi), is minimizedTheorem: SA-TSP is NP-hard even for di ti 1

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Subfield SchedulingSubfield Scheduling• Key observation: scheduling of subfields provides enough opportunity for

decreasing maximum resist temperature• Non-sequential writing throughput overhead due to beam settling time• To maintain througput, we equalize mask write times by increasing beam

current density (higher current density leads to small dwell times)• Rise in temperature due to increased current density is offset by non-

sequential writing schedule• 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

• We propose (1) Greedy and (2) Lagarias subfield scheduling to order the writing of subfields

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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Greedy Subfield SchedulingGreedy Subfield Scheduling

• The greedy algorithm starts from a random ordering of subfields and iteratively modifies the ordering by swapping pairs of subfields

• Evaluating the cost function takes O(n2) time, and thus the greedy algorithm requires O(n4) time per improving swap, where n is the number of subfields in a main deflection field

• Our implementation evaluates only pairs (i,j) in which i is a subfield with max temperature; this reduces runtime to O(n3) per improving swap

Greedy scheduling

1. Start with random subfield order 2. Repeat forever

– For all pairs (i,j) of subfields, compute cost of with i and j swapped– If there exists at least one cost improving swap, then modify by applying a swap with highest cost gain– Else exit repeat

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Cost Function ComputationCost Function Computation• The cost of a subfield order is Tmax + (1-)Tavg where Tmax and Tavg

are the maximum, respectively average subfield temperature before writing. In our experiments we use = 0.5

• Tmax corresponds to CD distortion due to resist heating, while Tavg corresponds to increase in mask write time

• To find an ordering, we can associate different weightings to Tmax, Tavg • The computation of subfield temperatures before writing for a given

subfield order is done using the following simplified model:– Subfield writing time is assumed to be negligible– The temperature rise of a subfield s due to the writing of subfield f

depends on the distance between s and f, the energy deposited while writing f, and the thermal properties of resist:

– The temperature of each subfield decays exponentially between flashes

• With this model, evaluating the cost function for a given subfield order requires O(n2) time

2Sf

rise f)d(s, TT

(s)T

c

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Lagarias Subfield SchedulingLagarias Subfield Scheduling• Subfield scheduling is similar to a well-spaced labeling proposed by Lagarias (SIAM J. Disc. Math, 2001)• Well-spaced labeling originally proposed for increasing fault tolerance of

flash memories• Lagarias “well-spaced” labeling scheme allocates integers to grid cells such that adjacent labels are

far apart in the grid (in Manhattan metric)• We apply the “well-spaced” labeling scheme to find the ordering of subfields For 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*-1, 0 i G*-1, 0 l H*-1

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

gcd: Greatest Common Divisor, lcm: Least Common Multiple)M (mod

2

2Mjl 1

1 )M (mod

2

2Mji 2

2

odd is /4MM if 2

even is /4MM if 1

21

21

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Lagarias Subfield Scheduling…contdLagarias Subfield Scheduling…contd

• Subfield schedules for 4x4 subfield orderings

• Lagarias scheduling is based on Manhattan distances between subfields. However, heat dissipation phenomenon is Euclidean

• Lagarias ordering does not differentiate between subfields on edges and subfield inside the mask pattern

• Greedy subfield ordering searches over all possible combinations of subfields and yields optimum schedule and hence performs better than Lagarias ordering

1 2 3 4

8 7 6 5

9 10 11 12

16 15 14 13

Sequential

1 5 9 13

14 2 6 10

11 15 3 7

8 12 16 4

Lagarias

1 13 5 9

6 10 2 14

3 15 7 11

8 12 4 16

Optimal

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Simulation SetupSimulation Setup• Resist heating simulations were performed using TEMPTATION• Simulated subfield scheduling strategies:

– Sequential schedule– Greedy schedule– Lagarias schedule– Random schedule

• A two-phase simulation setup was used to simulate 16 x 16 subfields– Phase I: Every subfield is flashed using 4 coarse flashes with total

dose equal to that of detailed fracture flashes– Phase II: The simulation is repeated with the “critical” subfield (i.e., the

subfield with maximum temperature before writing in phase I) flashed using detailed fracture flashes (512 2m x 2m fractures, written with a sequential writing schedule in a chessboard pattern)

Phase-1: coarse subfield ordering simulation Critical subfield

Phase-2: detailed critical subfield simulation

Chess board pattern within critical subfield

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Simulation Setup…contd.Simulation Setup…contd.• In both phases, delays are added between flashes to simulate beam

traveling times. Beam current density is also adjusted for each scheduling strategy to ensure equal throughput

• In TEMPTATION, beam travel times are specified as delays between flashings of subfields

• The minimum and maximum travel times between subfields are 25 sec and 100 sec respectively

• The travel times between any two subfields s and f is determined byTT(s,f) = 25 + 5 * max ( (sx – fx),(sy – fy) )

• We consider distance between subfields in Chebyshev norm• Beam current density is computed from total write time

Total write time = dwell time + travel timeCurrent density = Exposure dose x Total write time

• E-beam parameters used in the simulation:– Dwell time of each subfield = 512 sec– Travel time between subfields = 25 sec – 100 sec– Acceleration voltage = 50kV, Fracture flash time = 1 sec– Current density: sequential = 21A/cm2, greedy = 21.5 A/cm2,

Lagarias = 21.8 A/cm2, random = 21.3 A/cm2

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16x16 Subfields Simulation Results16x16 Subfields Simulation ResultsMax48.85CMean27.59C

Sequential schedule

Max32.68CMean16.07C

Greedy schedule

Max40.49CMean16.97C

Random schedule

Max37.24CMean20.37C

Lagarias schedule

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

Sequential: Max=105.10C

Greedy: Max=93.70CRandom: Max=104.60C

Lagarias: Max=97.15C

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ConclusionsConclusions• We proposed a new subfield scheduling approach to simultaneously

optimize subfield ordering and beam current density

• Subfield ordering reduces the maximum temperature of resist by spacing successive writings

• To normalize the throughput due to scheduling, we decrease the dwell time of each subfield by increasing the 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

• Using Lagarias scheduling, beam current density can be increased by a factor of 1.6 without increase of temperature. This gives a throughput gain of 30%

• With greedy scheduling, beam current density can be increased by a factor of 1.8 without increase of temperature. This translates to a throughput increase of 40%

• Simulation results show that excessive resist heating can be significantly reduced by avoiding successive writing of subfields that are close to each other