Subfield Scheduling for Througput Maximization in Electron-beam Photomask Fabrication

1. Subfield Scheduling for Througput Maximization in Electron-beam Photomask Fabrication 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. S. Babin*, A.B. Kahng, I.I. Mandoiu, S. MudduCSE & ECE Depts., University of California, San Diego *Soft Services

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 subfieldsto decrease the maximum resist temperature Mask Writing Schedule Problem Given: 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

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 limitationsof current e-beam writers • Scheduling ofsubfieldsincurs very low overhead, and is still effective in reducing excessive heating effects

4. Self-Avoiding TSP Formulation Theblocked setfor 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 Problem 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., ifti  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

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 timeswi and dibecome the same for all minor fields, and blocked sets Ribecome Euclideanballs of radius R centered at each minor field • Feasible schedules are similar to well-spaced labelings of gridsstudied by J.C. Lagarias, except that well-spaced labelings use rectilinear balls instead of Euclidean balls • Lagariasgivesexplicit solutionsguaranteed to be within anadditive factor of 2from the optimum under rectilinear metric, and within a multiplicative factor of 2/2from 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. Lagarias Subfield Scheduling 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*, 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 Schedules for 4x4 subfields: Optimal Sequential Lagarias

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/cm2 • Fracture flash time = 1 sec

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

9. Conclusions and Ongoing Work 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. Critical subfield temperature profiles and maximum fracture temperatures before flashing for the four subfield schedules: Lagarias Max=90.78C Sequential Max=105.10C Spiral Max=110.70C Random Max=94.77C