The continual shrinkage of minimum feature size in integrated circuit fabrication increases the difficulty of the optical lithography process, for the desired circuit patterns printed are distorted by diffraction. To push the limit further, resolution enhancement techniques (RETs) must be used to modify the masks so that the distortion caused by optical diffraction is reduced. While traditional RETs focuses on local pattern matches, we consider inverse lithography, a method that treats the mask design process as an inverse optimization problem. Through numerical algorithms, optimal masks can be designed.
In inverse lithography, the mask design process is modeled as an inverse problem. That is, given a target pattern, how do we design a mask so that it can compensate the distortion? Inverse lithography is a global minimization problem, with optimization variables being either 1 or 0. Our method to solve this discrete minimization is to relax the discrete problem to a continuous one and introduce penalty functions to force the binary characteristic of the solution.
Figure 2: Comparison of before and after using inverse lithography method. Top: Without inverse lithography, the output pattern is highly distorted by the diffraction. Bottom: With inverse lithography, the output pattern is much more similar to the target (two rectangles in the center, the pattern shown in the un-designed mask). [Chan, Wong, Lam 2007]
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Compared to convention chrome-on-glass (COG) masks, phase shifting masks offer higher resolution by utilizing the interference between two anti-phase light rays. Design of phase shifting masks can also be formulated in the framework of inverse lithography. However, due to non-linear property of the optimization, a careful initialization of the search should be made. In this research we proposed a simple initialization scheme that improves the contrast of the output pattern with moderate computational cost.
Figure 3: An illustration of phase shifting masks.
Figure 4: Comparisons of before applying the initialization scheme and after the initialization scheme. Left: Without the initialization, the contrast in dense areas is low (e.g. the bottom right corner of the L pattern). Right: With the initialization, the contrast is much higher. Higher contrast implies robustness in the resist process (i.e., the pattern is less likely to be corrupted during the resist process). [Chan, Wong, Lam 2008], [Chan, Lam 2008]