Double-exposure mask synthesis using inverse lithography

Abstract

Inverse lithography mask design and double-exposure lithography are two technologies that have gained a lot of attention in the recent past. Inverse lithography consists of synthesizing the input mask that leads to the desired output wafer pattern by inverting the mathematical forward model from mask to wafer. Double-exposure lithography uses two pairs of mask and exposure to print a single ͑desired͒ wafer pattern. It usually involves splitting the latter into two parts. In this work, we present some preliminary results in our unique effort to combine the previous two powerful techniques. The goal is to use the inverse imaging approach to automatically synthesize the masks required to print the desired wafer pattern employing double-exposure lithography. We employ the pixel-based mask representation, analytically calculate the gradient, and use a cyclic coordinate descent optimization algorithm to synthesize the two masks. We present results for chromeless phase-shift masks for an idealized case of a coherent imaging system ͑ =0͒ using the Kirchhoff approximation. The results indicate that our algorithm automatically splits the target pattern into two ͑overlapping͒ parts, which are used separately during the individual exposures. Furthermore, the proposed approach is also capable of resolving the phase conflicts. The comparison with a single-exposure case indicates a superior contrast and no hot-spots.

Inverse lithography technology: 30 years from concept to practical, full-chip reality

“…He developed an optimization framework for inverse lithography based on a pixel-based, continuous-function formulation, well-suited for the gradient-based search algorithms. [34][35][36] While still at Luminescent, the author gave many talks at various conferences in China 37-41 that also stimulated ILT research there. Many papers were published by Yang and Shen from Zhejiang University.…”

Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis

“…The level-set method has also been explored as an effective approach [9], and an analysis on its numerical formulation and details of its implementation have been published recently [10]. Poonawala and Milanfar formulated the mask synthesis problem as a steepest descent optimization [11], based on which additional work has been done to refine its implementation [12][13][14][15][16]. Zhang et al proposed an innovative iterative method recently, flipping pixels guided by the first-and second-order derivatives of the objective function [17].…”

“…As mentioned in section 2, our previous investigation can be seen as an application of batch gradient descent, where we transform the stochastic problem to a deterministic one. A series of defocus values are sampled to approximate the expectation described in equation (13). Still using the above test patterns, we compare it with the work in this paper.…”

“…I A (x, y; β i ) = |M(x, y) * H (x, y; β i )| 2 = [M(x, y) * H (x, y; β i )][M(x, y) * H (x, y; β i )] * = [M(x, y) * H (x, y; β i )][M(x, y) * H * (x, y; β i )] (A.3) since M(x, y) is real.Therefore, the partial derivative of the aerial image with respect to the variable θ is∂ I A (x, y; β i ) ∂θ( p, q) = [M(x, y) * H (x, y; β i )][M(x, y) * H * (x, y; β i )] ∂ M( p, q) × ∂ M( p, q) ∂θ( p, q) , = {[M(x, y) * H * (x, y; β i )]H (x − p, y − q; β i ) + [M(x, y) * H (x, y; β i )]H * (x − p, y − q; β i )} × − sin θ( p, q) 2 , (A.4) because M( p, q) = (1 + cos θ( p, q))/2.Since Î (x, y), I (x, y; β i ), I A (x, y; β i ), H (x, y; β i ), and M(x, y) are matrices of many variables, we use their short forms Î , I , I A , H and M for convenience in the following derivations. Given the cost function derivations (equations (A.1)-(A.4)), the gradient∇ F is therefore ∂ F ∂θ( p, q) − Î )sig(I A )[1 − sig(I A )] ∂ I A ∂θ( p, q) I − Î )sig(I A )[1 − sig(I A )][(M * H * ) × H (x − p, y − q; β i ) + (M * H )H * (x − p, y − q; β i )] sin θ( p, q) 2 = −α i η i {H (β i ) * [(I − Î ) I (1 − I ) (M * H * (β i ))] + H * (β i ) * [(I − Î ) I (1 − I ) (M * H (β i ))]} sin θ, (A.6)which is the expression in equation(13).…”

Wavefront-based pixel inversion algorithm for generation of subresolution assist features

“…18,19 The inverse problems have been solved by linear programming, quadratic programming, or nonlinear formulations based on the assumptions made for the optical systems. [20][21][22][23] Recently, the traditional pixel-flipping techniques have been replaced with elaborated algorithms such as gradient descent methods 24,25 and levelset 26,27 alorithms, which have also been demonstrated on large-scale layouts. Although pixel-flipping techniques could also be very efficient for inverse lithography, the techniques exhibit some known issues.…”