Impacts of cost functions on inverse lithography patterning

Abstract

For advanced CMOS processes, inverse lithography promises better patterning fidelity than conventional mask correction techniques due to a more complete exploration of the solution space. However, the success of inverse lithography relies highly on customized cost functions whose design and know-how have rarely been discussed. In this paper, we investigate the impacts of various objective functions and their superposition for inverse lithography patterning using a generic gradient descent approach. We investigate the most commonly used objective functions, which are the resist and aerial images, and also present a derivation for the aerial image contrast. We then discuss the resulting pattern fidelity and final mask characteristics for simple layouts with a single isolated contact and two nested contacts. We show that a cost function composed of a dominant resist-image component and a minor aerial-image or image-contrast component can achieve a good mask correction and contour targets when using inverse lithography patterning.

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.…”

“…[10][11][12][13][14][15] To address these problems, regularization approaches are introduced to guarantee the synthesized mask to be binary and less complex. In the literature, almost all the regularization approaches take the regularization terms as penalty functions incorporated into a cost function with corresponding weighted parameters [10][11][12][13][14][15] as…”

“…Generally, regularization terms can be classified into two types: one is related to the manufacturability, such as the quadratic penalty term, 11,12 the total variation penalty term, 11,12 and the wavelet penalty term; 13,15 the other type is related to the fabrication process, such as the image slope term 14,16 and the mask error enhancement factor (MEEF). 17,18 λ i is the weight of the corresponding regularization term R i ðMÞ.…”

“…[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.…”