Massively Parallel Algorithms for Scattering in Optical Lithography

A massively parallel implementation of the Time-Domain Finite Difference (TDFD) method for simulation of wafer scattering effects in optical lithography is described. Tradeoffs in algorithms and boundary conditions which impact the accuracy and processor loading efficiency are discussed. Optical alignment and optical patterning involve complex interactions of incident electromagnetic waves with nonplanar topography and with materials which dynamically change during illumination. The simulation of the interactions is, however, a substantial undertaking as the fields must be found simultaneously at over 10,000 nodes for a variety of illumination conditions and the optical parameters must be allowed change as bleaching takes place. Large vectorized computers have been utilized to simulate this class of problems using both frequency-domain [1] and time-domain [2] approaches. Recently, a massively parallel implementation of a time-domain approach on a parallel Connection Machine CM2 indicated that nearly the full 2 GFLOP computational capability of this architecture could be realized in these lithographic applications [3]. In the Connection Machine implementation, a time-domain approach was chosen because the solution of a large indefinite system of linear equations, typical of frequency-domain techniques, is not required. The steady-state solution is achieved as the transient phenomena die out. Coupled first-order equations for the electric and magnetic field components were solved on a staggered grid shown in Figure 1. This allowed the use of high-efficiency boundary conditions developed for first order equations. Tapered material boundaries were easilv included bv suitable averaging of the different dielectric constants. Referring to Figure 2, the following boundary conditions (b.c.) have been implemented and tested successfully: along side DC, the modeling of the incident field must be accounted for and the scattered field must be "absorbed", along boundary CB periodic b.c. are required when a periodic structure is studied, symmetric b.c. when a single half of a structure is modeled and absorbing when an isolated feature is considered, along side AB, absorbing b.c. have been implemented in a lossy material (silicon), and along AD absorbing or periodic b.c. are available. The starting of the sinusoidal wave has been smoothed out to avoid spurious noise produced by the discontinuity of the incident field. Off-angle, converging and diverging incident waves are now under study. The accuracy and efficiency of the program has been explored using structures and incident fields which have previously been studied to a high precision using integral techniques. Figure 3 shows the performance for symmetric gratings: these results show that second order boundary conditions offer higher precision than first order b.c., however this occurs at the expense of a lower processor utilization. The suitability of this algorithm for photolithographic problems has been assessed by considering three different applications. The simplest is the image of resist coated alignment marks to explore asymmetry effects. A second is imaging in resist which requires the modeling of dynamic bleaching through about 15 successive changes in refractive index. The most difficult is reflective notching from an isolated feature which also requires the modeling of dynamic bleaching. A 20/spl lambda/ by 10 /spl lambda/ alignment structure requires about 75 seconds on a 1K processors system with projected performance of less than 5 seconds on a fully configured 32K (or larger) machine. Additional CPU times are shown in Figure 4. An example of the results produced by the modeling of the exposure of photoresist in a reflective notching structure are shown in Figure 5-a and the profiles developed by processing this output with SAMPLE axe given in Figure 5-b. The massively-parallel time-domain approach is very promising as it is not difficult to implement, computationally efficient and is very effective in investigating physical phenomena in alignment, lithography and metrology.