1. 1 Subfield Scheduling for Througput Maximization in Electron-beam Photomask Fabrication S. 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.

2. 2 Motivation 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 Problem Given: Beam and resist parameters, threshold temperature T max Find: fracture and subfield writing schedule with minimum total writing time such that the maximum resist temperature never exceeds T max

3. 3 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 (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

4. 4 Self-Avoiding TSP Formulation The 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 T max. 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 Problem Given: n non-overlapping regions R 1, R 2,..., R n in the plane, where for each region R i we are given its writing time w i, blocked set B i  {R 1, R 2,..., R n }, and blocking duration d i. Find: writing start times t i 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 t i  t j, i  j, then t i + w i  t j (3) no region is being written while blocked, i.e., if R i  B i then t j + d i  t i or t j  t i (4) the completion time, max i (t i + w i ), is minimized

5. 5 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 w i and d i become the same for all minor fields, and blocked sets R i 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 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

6. 6 Lagarias Subfield Scheduling For a M1 x M2 grid with both M1 and M2 even, the Lagarias schedule writes in the m th 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 (M 1,M 2 ), H* = and L* = lcm(M 1,M 2 ) / H* Reference: J.C. Lagarias, SIAM J. Discr. Math. 13, 2001, pp. 521-534 1234 8765 9101112 16151413 159 142610 111537 812164 11359 610214 315711 812416 Optimal LagariasSequential Schedules for 4x4 subfields:

7. 7 Simulation 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 = 2  m x 2  m –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 = 64  m x 64  m E-beam parameters: –Acceleration voltage = 50kV –Current density = 40A/cm 2 –Fracture flash time = 1  sec

8. 8 16x16 Subfields Simulation Results Sequential schedule Max 48.85  C Mean 27.59  C Random schedule Max 38.88  C Mean 16.34  C Spiral schedule Max 47.64  C Mean 22.59  C Lagarias schedule Max 35.23  C Mean 19.08  C

9. 9 Conclusions and Ongoing Work Sequential Max=105.10  C Spiral Max=110.70  C Random Max=94.77  C Lagarias Max=90.78  C 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.