1. Resist Heating Dependence on Subfield Scheduling in 50kV Electron-beam Maskmaking
S. Babin*, A.B. Kahng, I.I. Mandoiu, S. Muddu CSE & ECE Depts., University of California at San Diego
*Soft Services

2. Abstract
Resist 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

3. Motivation Mask Writing Schedule Problem
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 Problem
Given: Beam voltage, resist parameters, threshold temperature Tmax
Find: Beam current density and subfield writing schedule such that the maximum resist temperature never exceeds Tmax

4. Variable-shaped E-beam Writing
Taxonomy of mask features
Fractures: smallest features written on the mask; dimensions in the range 0.75m -2m
Minor field: collection of fractures
Subfield: collection of minor fields; typical size of a subfield is 64m X 64m
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

5. Computational Complexity
The 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 minimized
Theorem: SA-TSP is NP-hard even for di  ti  1

6. Subfield 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 Problem
Maximize 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

7. Greedy Subfield Scheduling
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
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

8. Cost 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

9. Lagarias 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

10. Lagarias Subfield Scheduling…contd1 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
Sequential
Lagarias
Optimal
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

11. Simulation 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 2m x 2m fractures, written with a sequential writing schedule in a chessboard pattern)
Chess board pattern within critical subfield
Phase-2: detailed critical subfield simulation
Phase-1: coarse subfield ordering simulation
Critical subfield

12. 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 by
TT(s,f) = * 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 time Current 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 = A/cm2, random = 21.3 A/cm2

13. 16x16 Subfields Simulation Results
Max
48.85C
Mean
27.59C
Sequential schedule
Max
37.24C
Mean
20.37C
Lagarias schedule
Max
40.49C
Mean
16.97C
Random schedule
Max
32.68C
Mean
16.07C
Greedy schedule

14. Critical Subfield Temperature Profiles
Critical subfield temperature profiles and maximum fracture temperatures before flashing for the four subfield schedules:
Sequential: Max=105.10C
Lagarias: Max=97.15C
Random: Max=104.60C
Greedy: Max=93.70C

15. Conclusions
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