Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Abstract

Inverse mask synthesis is achieved by minimizing a cost function on the difference between the output and desired patterns. Such a minimization problem can be solved by a level-set method where the boundary of the pattern is iteratively evolved. However, this evolution is timeconsuming in practice and usually converges to a local minimum. The velocity of the boundary evolution and the size of the evolution step, also known as the descent direction and the step size in optimization theory, have a dramatic influence on the convergence properties. This paper focuses on developing a more efficient algorithm with faster convergence and improved performance such as smaller pattern error, lower mean edge placement error, wider defocus band, and higher normalized image log slope. These improvements are accomplished by employing the conjugate gradient of the cost function as the evolution velocity, and by introducing an optimal time step for each iteration of the boundary evolution. The latter is obtained from an extended Euler time range by using a line search method. The authors present simulations demonstrating the efficacy of these two improvements.

Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography

“…18,19 Subsequently, the optimization framework was further generalized for phase-shifting masks 20,21 and partially coherent imaging systems, [22][23][24][25][26] and the optimization algorithm was improved with an active set method 21 and with an augmented Lagrangian method. 27 Most recently, Lv et al further improved the computational efficiency of the mask synthesis by using the conjugate gradient and an optimal iterative step, 28 and by introducing a mask filtering technique to enhance the regularity of the synthesized mask pattern. 29,30 On the other hand, with ever-shrinking feature size, the printed feature becomes increasingly sensitive to the fluctuation of the fabrication process, which limits the yield in semiconductor industry.…”

“…Also, most recently we developed a metric called edge distance error (EDE) to guide mask synthesis. 29,30 Compared to the commonly used metric pattern error, [16][17][18][19][20][21][22][23][24][25][26][27][28] the metric EDE has a dimension of length and is independent of the simulation grid size. The analytical circle-sampling technique and EDE are both independent of grid size.…”

Mask-filtering-based inverse lithography

“…Various computation techniques have been proposed to deal with this inverse problem in the literature, such as the level-set method, [5][6][7][8] the discrete cosine transform (DCT)-based method, 9 and the gradient-based method. [10][11][12][13][14][15] The level-set method treats a mask as a sophisticated continuum, [5][6][7][8] and consequently, the boundary of the mask is iteratively evolved according to an optimization algorithm. The DCT-based method transforms a mask to the frequency space using a two-dimensional DCT.…”

“…The gradient-based method considers a mask as a raster image constituted by pixels directly, where it is synthesized pixel-by-pixel in an iterative direction of the steepest descent, conjugate gradient, and so on. 8,[10][11][12][13][14][15] It is probably the most popular technique in the literature due to its high flexibility, ease of understanding, and implementation.…”

“…[6][7][8][10][11][12][13][14][15] The pattern error employs an approximated and continuous resist model, and it is defined as a square of the L 2 norm of the difference between the output pattern of the input mask and the desired feature, which causes pattern error to be continuous and differentiable with respect to the input mask explicitly. 10,11 However, pattern error is a dimensionless quantity and highly depends on mask feature and simulation parameters, such as simulation area and simulation resolution.…”