Pixel-based OPC optimization based on conjugate gradients

Abstract

Optical proximity correction (OPC) methods are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In pixel-based OPC (PBOPC), the mask is divided into small pixels, each of which is modified during the optimization process. Two critical issues in PBOPC are the required computational complexity of the optimization process, and the manufacturability of the optimized mask. Most current OPC optimization methods apply the steepest descent (SD) algorithm to improve image fidelity augmented by regularization penalties to reduce the complexity of the mask. Although simple to implement, the SD algorithm converges slowly. The existing regularization penalties, however, fall short in meeting the mask rule check (MRC) requirements often used in semiconductor manufacturing. This paper focuses on developing OPC optimization algorithms based on the conjugate gradient (CG) method which exhibits much faster convergence than the SD algorithm. The imaging formation process is represented by the Fourier series expansion model which approximates the partially coherent system as a sum of coherent systems. In order to obtain more desirable manufacturability properties of the mask pattern, a MRC penalty is proposed to enlarge the linear size of the sub-resolution assistant features (SRAFs), as well as the distances between the SRAFs and the main body of the mask. Finally, a projection method is developed to further reduce the complexity of the optimized mask pattern.

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.…”