Vector Imaging Model Research Articles

Method for optical proximity correction based ona vector imaging model.

Optical proximity correction (OPC) has become an indispensable step in integrated circuit manufacturing. It requires a huge amount of calculation to obtain a sufficiently accurate OPC model and implement pattern correction. In this paper, the authors proposed an edge-based OPC method built on a vector imaging model, where the analytical correlation between the cost function and movement of each edge segment is established by the chain rule. First, the mask pattern is segmented and downsampled to get the mask image in order to reduce the total data. Second, the aerial image, various parameters on each evaluating point, and the final cost value are obtained in proper sequence. In each part of the OPC process, the procedures of solution and derivation are both recorded. After obtaining the cost value, the chain rule is applied, by which the differential relation between the cost value and movement of each segment is built. According to this differential relation, the next movement of each segment is decided under a quasi-Newton method. All results obtained by the proposed method are compared with results from commercial software. The comparison shows that the proposed OPC method has good OPC accuracy in few iterations.

Level-set based mask synthesis with a vector imaging model.

With continuous shrinking of critical dimension (CD) and the application of immersion lithography system to technology nodes 22nm and beyond, the vector nature of electromagnetic fields propagating from mask to wafer plane cannot be ignored, rendering mask synthesis under scalar imaging model inadequate. In this paper, we develop a level-set based optimization framework for mask synthesis with a vector imaging model. The forward model of vector image formation is established, and then the photomask synthesis is addressed as an inverse imaging problem whose variational level-set reformulation is represented by a stable time-dependent model, which is solved by employing conjugate gradient methods of the cost function and readily available finite-difference schemes. Experimental results demonstrate pronounced performance in terms of pattern fidelity and edge placement error, together with notable computation acceleration and better convergence performance.

Wavefront aberration measurement method for a hyper-NA lithographic projection lens based on principal component analysis of an aerial image.

A wavefront aberration measurement method for a hyper-NA lithographic projection lens by use of an aerial image based on principal component analysis is proposed. Aerial images of the hyper-NA lithographic projection lens are expressed accurately by using polarized light and a vector imaging model, as well as by considering the polarization properties. As a result, the wavefront aberrations of the hyper-NA lithographic projection lens are measured accurately. The lithographic simulator PROLITH is used to validate the accuracies of the wavefront aberration measurement and analyze the impact of the polarization rotation of illumination on the accuracy of the wavefront aberration measurement, as well as the degree of polarized light and the sample interval of aerial images. The result shows that the proposed method can retrieve 33 terms of Zernike coefficients (Z5-Z37) with a maximum error of less than 0.00085λ.

Gradient-based joint source polarization mask optimization for optical lithography

Source and mask optimization (SMO) has emerged as a key resolution enhancement technique for advanced optical lithography. Current SMO, however, keeps the polarization state fixed, thus limiting the degrees of freedom during the optimization procedure. To overcome this limitation, pixelated gradient-based joint source polarization mask optimization (SPMO) approaches, which effectively extend the solution space of the SMO problem by introducing polarization variables, are developed. First, the SPMO framework is formulated using an integrative and analytic vector imaging model that is capable of explicitly incorporating the polarization angles. Subsequently, two optimization methods, namely simultaneous SPMO (SISPMO) and sequential SPMO (SESPMO) are developed, both of which exploit gradient-based algorithms to solve for the optimization problem. In addition, a postprocessing method is applied to reduce the complexity of the optimized polarization angle pattern for improving its manufacturability. Illustrative simulations are presented to validate the effectiveness of the proposed algorithms. The simulations also demonstrate the superiority of the SESPMO over SISPMO in computational efficiency and improvement of image fidelity.

Parametric source-mask-numerical aperture co-optimization for immersion lithography

Source mask optimization (SMO) is a leading resolution enhancement technique in immersion lithography at the 45-nm node and beyond. Current SMO approaches, however, fix the numerical aperture (NA), which has a strong impact on the depth of focus (DOF). A higher NA could realize a higher resolution but reduce the DOF; it is very important to balance the requirements of NA between resolution and the DOF. In addition, current SMO methods usually result in complicated source and mask patterns that are expensive or difficult to fabricate. This paper proposes a parametric source-mask-NA co-optimization (SMNO) method to improve the pattern fidelity, extend the DOF, and reduce the complexity of the source and mask. An analytic cost function is first composed based on an integrative vector imaging model, in which a differentiable function is applied to formulate the source and mask patterns. Then, the derivative of the cost function is deduced and a gradient-based algorithm is used to solve the SMNO problem. Simulation results show that the proposed SMNO can achieve the optimum combination of parametric source, mask, and NA to maintain high pattern fidelity within a large DOF. In addition, the complexities of the source and mask are effectively reduced after optimization.

Hybrid source mask optimization for robust immersion lithography

To keep pace with the shrinkage of critical dimension, source and mask optimization (SMO) has emerged as a promising resolution enhancement technique to push the resolution of 193 nm argon fluoride immersion lithography systems. However, most current pixelated SMO approaches relied on scalar imaging models that are no longer accurate for immersion lithography systems with hyper-NA (NA>1). This paper develops a robust hybrid SMO (HSMO) algorithm based on a vector imaging model capable of effectively improving the robustness of immersion lithography systems to defocus and dose variations. The proposed HSMO algorithm includes two steps. First, the individual source optimization approach is carried out to rapidly reduce the cost function. Subsequently, the simultaneous SMO approach is applied to further improve the process robustness by exploiting the synergy in the joint optimization of source and mask patterns. The conjugate gradient method is used to update the source and mask pixels. In addition, a source regularization approach and source postprocessing are both used to improve the manufacturability of the optimized source patterns. Compared to the mask optimization method, the HSMO algorithm achieves larger process windows, i.e., extends the depth of focus and exposure latitude, thus more effectively improving the process robustness of 45 nm immersion lithography systems.

Block-based mask optimization for optical lithography

Pixel-based optical proximity correction (PBOPC) methods have been developed as a leading-edge resolution enhancement technique (RET) for integrated circuit fabrication. PBOPC independently modulates each pixel on the reticle, which tremendously increases the mask's complexity and, at the same time, deteriorates its manufacturability. Most current PBOPC algorithms recur to regularization methods or a mask manufacturing rule check (MRC) to improve the mask manufacturability. Typically, these approaches either fail to satisfy manufacturing constraints on the practical product line, or lead to suboptimal mask patterns that may degrade the lithographic performance. This paper develops a block-based optical proximity correction (BBOPC) algorithm to pursue the optimal masks with manufacturability compliance, where the mask is shaped by a set of overlapped basis blocks rather than pixels. BBOPC optimization is formulated based on a vector imaging model, which is adequate for both dry lithography with lower numerical aperture (NA), and immersion lithography with hyper-NA. The BBOPC algorithm successively optimizes the main features (MF) and subresolution assist features (SRAF) based on a modified conjugate gradient method. It is effective at smoothing any unmanufacturable jogs along edges. A weight matrix is introduced in the cost function to preserve the edge fidelity of the printed images. Simulations show that the BBOPC algorithm can improve lithographic imaging performance while maintaining mask manufacturing constraints.

Pixelated source and mask optimization for immersion lithography

Immersion lithography systems with hyper-numerical aperture (hyper-NA) (NA>1) have become indispensable in nanolithography for technology nodes of 45nm and beyond. Source and mask optimization (SMO) has emerged as a key technique used to further improve the imaging performance of immersion lithography. Recently, a set of pixelated gradient-based SMO approaches were proposed under the scalar imaging models, which are inaccurate for hyper-NA settings. This paper focuses on developing pixelated gradient-based SMO algorithms based on a vector imaging model that is accurate for current immersion lithography. To achieve this goal, an integrative and analytic vector imaging model is first used to formulate the simultaneous SMO (SISMO) and sequential SMO (SESMO) frameworks. A gradient-based algorithm is then exploited to jointly optimize the source and mask. Subsequently, this paper studies and compares the performance of individual source optimization (SO), individual mask optimization (MO), SISMO, and SESMO. Finally, a hybrid SMO (HSMO) approach is proposed to take full advantage of SO, SISMO, and MO, consequently achieving superior performance.

Vectorial mask optimization methods for robust optical lithography

Continuous shrinkage of critical dimension in an integrated circuit impels the development of resolution enhancement techniques for low k1 lithography. Recently, several pixelated optical proximity correction (OPC) and phase-shifting mask (PSM) approaches were developed under scalar imaging models to account for the process variations. However, the lithography systems with larger-NA (NA>0.6) are predominant for current technology nodes, rendering the scalar models inadequate to describe the vector nature of the electromagnetic field that propagates through the optical lithography system. In addition, OPC and PSM algorithms based on scalar models can compensate for wavefront aberrations, but are incapable of mitigating polarization aberrations in practical lithography systems, which can only be dealt with under the vector model. To this end, we focus on developing robust pixelated gradient-based OPC and PSM optimization algorithms aimed at canceling defocus, dose variation, wavefront and polarization aberrations under a vector model. First, an integrative and analytic vector imaging model is applied to formulate the optimization problem, where the effects of process variations are explicitly incorporated in the optimization framework. A steepest descent algorithm is then used to iteratively optimize the mask patterns. Simulations show that the proposed algorithms can effectively improve the process windows of the optical lithography systems.

Mask optimization approaches in optical lithography based on a vector imaging model

Recently, a set of gradient-based optical proximity correction (OPC) and phase-shifting mask (PSM) optimization methods has been developed to solve for the inverse lithography problem under scalar imaging models, which are only accurate for numerical apertures (NAs) of less than approximately 0.4. However, as lithography technology enters the 45 nm realm, immersion lithography systems with hyper-NA (NA>1) are now extensively used in the semiconductor industry. For the hyper-NA lithography systems, the vector nature of the electromagnetic field must be taken into account, leading to the vector imaging models. Thus, the OPC and PSM optimization approaches developed under the scalar imaging models are inadequate to enhance the resolution in immersion lithography systems. This paper focuses on developing pixelated gradient-based OPC and PSM optimization algorithms under a vector imaging model. We first formulate the mask optimization framework, in which the imaging process of the optical lithography system is represented by an integrative and analytic vector imaging model. A gradient-based algorithm is then used to optimize the mask iteratively. Subsequently, a generalized wavelet penalty is proposed to keep a balance between the mask complexity and convergence errors. Finally, a set of methods is exploited to speed up the proposed algorithms.