Transcript (.ppt)

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
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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
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Motivation
• 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 nonsequential 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
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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
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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
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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 nonsequential 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
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
• We propose (1) Greedy and (2) Lagarias subfield scheduling to order the
writing of subfields
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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
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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:
Tf  TS
Trise (s)  c
d(s, f) 2
– 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
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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
l j
M1  2
(mod M1)
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and column i  j
M2  2
(mod M2) where
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m = lG*L* + iL* + j, with 0  j  L*-1, 0  i  G*-1, 0  l  H*-1
1 if M1M2/4 is even
G* = gcd (M1,M2), H* = 
and L* = lcm(M1,M2) / H*
2 if M1M2/4 is odd
gcd: Greatest Common Divisor, lcm: Least Common Multiple
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Lagarias Subfield Scheduling…contd
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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
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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-1: coarse subfield
ordering simulation
Phase-2: detailed critical
subfield simulation
Critical subfield
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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) = 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 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 = 21.8 A/cm2, random = 21.3 A/cm2
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16x16 Subfields Simulation Results
Max
48.85C
Mean
27.59C
Max
37.24C
Mean
20.37C
Sequential schedule
Lagarias schedule
Max
32.68C
Mean
16.07C
Max
40.49C
Mean
16.97C
Random schedule
Greedy schedule
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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
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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
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