Efficient Calculation of Charging Effects in Electron Beam Lithography Using the SA-AMG

Abstract

When simulating charging effects in electron beam lithography (EBL) using the drift-diffusion recombination (DDR) model, traditional numerical iterative solvers face issues of slow computational speed and limited scalability. To solve this problem, we employ the Smoothed aggregation algebraic multigrid (SA-AMG) solver based on bi-conjugate gradient stabilized (BICGSTAB) and preconditioned conjugate gradient (PCG) to accelerate the computation of the DDR and Poisson equations separately. Compared to the Gauss-Seidel solver with a time complexity of ${O}\text {(}{n}^{{2}}\text {)}$ and the PCG, BICGSTAB solvers with a time complexity of O ( ${n}^{\text {3/ {2}}}\text {)}$ , the SA-AMG solver reduces the time complexity to ${O}\text {(}{n}\text {)}$ . This implies that at a charging time of $1~\mu $ s, the computation time for simulating charging effects with SA-AMG is 93, 296, and 646 times faster than the Gauss-Seidel solver when the grid point count of ${n} = 10^{{6}}$ , $10^{{7}}$ , $10^{{8}}$ , respectively. Furthermore, compared to the combined PCG and BICGSTAB solvers, the SA-AMG solver is 4, 7, and 13 times faster for the same grid point counts.