Implicit function characterization of the curvilinear mask to realize parametric optical proximity correction with a neighborhood parallel tabu search.

The curvilinear mask structures provide significant benefits in improving lithographic resolution. Curvilinear masks, as opposed to rectilinear masks, have a wider range of structure types that can be used precisely to correct the contour of diffraction at sharp technological nodes. However, the curvilinear structure also makes the inverse design of mask in optical proximity correction (OPC) flow difficult. The current OPC of curvilinear masks uses pixel-based inverse optimization, which is extremely computationally intensive and takes up a lot of design data storage space. This paper proposes an implicit function to represent a large number of curve types with a small number of parameters to reduce computational complexity and the R&D cycle. Therefore, the ultra-high dimensional pixel-based OPC problem is transformed into a low-dimensional parameter search problem in the critical diffraction area of the mask pattern. The tabu search algorithm and neighborhood parallel computing strategy are then used to quickly search for optimal characterized parameters. The results of the simulation show that the parametric curvilinear OPC method achieves a higher image fidelity than that of rectilinear OPC. At the same time, it addresses the shortcomings of the traditional pixelated curvilinear mask OPC method, including high computational complexity, low manufacturability, and storage space occupancy.

Effective multi-objective inverse lithography technology at full-field and full-chip levels with a hybrid dynamic priority algorithm.

Inverse lithography technology (ILT), such as source mask optimization (SMO), is used to improve lithography performance. Usually, a single objective cost function is selected in ILT, and an optimal structure for one field point is achieved. The optimal structure is not the case for other images at full field points where the aberrations of the lithography system are different, even in high-quality lithography tools. The optimal structure that must match the high-performance images at the full field is urgently required for extreme ultraviolet lithography (EUVL). In contrast, multi-objective optimization algorithms (MOAs) limit the application of multi-objective ILT. Assigning target priority is incomplete in current MOAs, which results in the over-optimization of some targets and under-optimization of others. In this study, multi-objective ILT and a hybrid dynamic priority (HDP) algorithm were investigated and developed. High-performance images with high fidelity and high uniformity were obtained at multi-field and multi-clip areas across the die. A hybrid criterion was developed for the completion and reasonable prioritization of each target to ensure sufficient improvement. Compared to the current MOAs, the uniformity of images at full-field points was improved by up to 31.1% by the HDP algorithm in the case of multi-field wavefront error-aware SMO. The multi-clip source optimization (SO) problem showed the universality of the HDP algorithm to deal with different ILT problems. It acquired higher imaging uniformity than existing MOAs, which indicated that the HDP is more qualified for multi-objective ILT optimization than existing MOAs.

Spatial interpolation of regional PM2.5 concentrations in China during COVID-19 incorporating multivariate data.

During specific periods when the PM2.5 variation pattern is unusual, such as during the coronavirus disease 2019 (COVID-19) outbreak, epidemic PM2.5 regional interpolation models have been relatively little investigated, and little consideration has been given to the residuals of optimized models and changes in model interpolation accuracy for the PM2.5 concentration under the influence of epidemic phenomena. Therefore, this paper mainly introduces four interpolation methods (kriging, empirical Bayesian kriging, tensor spline function and complete regular spline function), constructs geographically weighted regression (GWR) models of the PM2.5 concentration in Chinese regions for the periods from January-June 2019 and January-June 2020 by considering multiple factors, and optimizes the GWR regression residuals using these four interpolation methods, thus achieving the purpose of enhancing the model accuracy. The PM2.5 concentrations in many regions of China showed a downward trend during the same period before and after the COVID-19 outbreak. Atmospheric pollutants, meteorological factors, elevation, zenith wet delay (ZWD), normalized difference vegetation index (NDVI) and population maintained a certain relationship with the PM2.5 concentration in terms of linear spatial relationships, which could explain why the PM2.5 concentration changed to a certain extent. By evaluating the model accuracy from two perspectives, i.e., the overall interpolation effect and the validation set interpolation effect, the results showed that all four interpolation methods could improve the numerical accuracy of GWR to different degrees, among which the tensor spline function and the fully regular spline function achieved the most stable effect on the correction of GWR residuals, followed by kriging and empirical Bayesian kriging.

Sampling-based imaging model for fast source and mask optimization in immersion lithography.

Current source and mask optimization (SMO) research tends to focus on advanced inverse optimization algorithms to accelerate SMO procedures. However, innovations of forward imaging models currently attract little attention, which impacts computational efficiency more significantly. A sampling-based imaging model is established with the innovation of an inverse point spread function to reduce computational dimensions, which can provide an advanced framework for fast inverse lithography. Simulations show that the proposed SMO method with the help of the proposed model can further speed up the algorithm-accelerated SMO procedure by a factor of 3.

Exposure latitude aware source and mask optimization for extreme ultraviolet lithography.

Extreme ultraviolet (EUV) lithography is a new generation of integrated circuit manufacturing technology with great development prospects. EUV lithography has more significant demand for high exposure latitude (EL) due to greater requirements for the stability of the light source. Source and mask optimization (SMO) technology is widely used to compensate for imaging distortion. In this paper, we propose an EL-aware SMO (ELASMO) method that uses a low-resist threshold sensitivity (LRS) penalty function to improve the EL in EUV lithography. Compared to conventional SMO, the proposed ELASMO method can significantly enhance the aerial image contrast, improve the EL, and enlarge the process window while ensuring high imaging fidelity.

Vectorial pupil optimization to compensate for a polarization effect at full exposure field in lithography.

Recently, a single vectorial pupil optimization (VPO) was proposed to compensate for the polarization effect induced by thick mask and image optics at one field point in a lithography system, which does not work at full field points. In this paper, we propose a multi-objective VPO (MOVPO) method to obtain a universal vectorial pupil that can compensate for the polarization aberration at full field points. A novel multi-objective cost function, to the best of our knowledge, is built and includes uneven image pattern errors causing by polarization aberration (PA) at full field points in the MOVPO method. Comprehensive simulations demonstrate that the proposed MOVPO method can effectively improve the consistency of imaging and enlarge the overlapped process window at full field points.

Ingestion of GNSS-Derived ZTD and PWV for Spatial Interpolation of PM2.5 Concentration in Central and Southern China.

With the increasing application of global navigation satellite system (GNSS) technology in the field of meteorology, satellite-derived zenith tropospheric delay (ZTD) and precipitable water vapor (PWV) data have been used to explore the spatial coverage pattern of PM2.5 concentrations. In this study, the PM2.5 concentration data obtained from 340 PM2.5 ground stations in south-central China were used to analyze the variation patterns of PM2.5 in south-central China at different time periods, and six PM2.5 interpolation models were developed in the region. The spatial and temporal PM2.5 variation patterns in central and southern China were analyzed from the perspectives of time series variations and spatial distribution characteristics, and six types of interpolation models were established in central and southern China. (1) Through correlation analysis, and exploratory regression and geographical detector methods, the correlation analysis of PM2.5-related variables showed that the GNSS-derived PWV and ZTD were negatively correlated with PM2.5, and that their significances and contributions to the spatial analysis were good. (2) Three types of suitable variable combinations were selected for modeling through a collinearity diagnosis, and six types of models (geographically weighted regression (GWR), geographically weighted regression kriging (GWRK), geographically weighted regression-empirical bayesian kriging (GWR-EBK), multiscale geographically weighted regression (MGWR), multiscale geographically weighted regression kriging (MGWRK), and multiscale geographically weighted regression-empirical bayesian kriging (MGWR-EBK)) were constructed. The overall R2 of the GWR-EBK model construction was the best (annual: 0.962, winter: 0.966, spring: 0.926, summer: 0.873, and autumn: 0.908), and the interpolation accuracy of the GWR-EBK model constructed by inputting ZTD was the best overall, with an average RMSE of 3.22 µg/m3 recorded, while the GWR-EBK model constructed by inputting PWV had the highest interpolation accuracy in winter, with an RMSE of 4.5 µg/m3 recorded; these values were 2.17% and 4.26% higher than the RMSE values of the other two types of models (ZTD and temperature) in winter, respectively. (3) The introduction of the empirical Bayesian kriging method to interpolate the residuals of the models (GWR and MGWR) and to then correct the original interpolation results of the models was the most effective, and the accuracy improvement percentage was better than that of the ordinary kriging method. The average improvement ratios of the GWRK and GWR-EBK models compared with that of the GWR model were 5.04% and 14.74%, respectively, and the average improvement ratios of the MGWRK and MGWR-EBK models compared with that of the MGWR model were 2.79% and 12.66%, respectively. (4) Elevation intervals and provinces were classified, and the influence of the elevation and the spatial distribution of the plane on the accuracy of the PM2.5 regional model was discussed. The experiments showed that the accuracy of the constructed regional model decreased as the elevation increased. The accuracies of the models in representing Henan, Hubei and Hunan provinces were lower than those of the models in representing Guangdong and Guangxi provinces.

Rigorous imaging-based measurement method of polarization aberration in hyper-numerical aperture projection optics.

Imaging-based measurement methods of polarization aberration (PA) are indispensable in hyper-numerical aperture projection optics for advanced lithography. However, the current methods are derived from the Kirchhoff model and ignore the 3D mask effect of the test mask, which will impact the measurement accuracy. In this paper, a novel imaging-based measurement method of PA is proposed based on a rigorous imaging model to improve the measurement accuracy. Through the quantitative description of the 3D mask effect, a rigorous imaging-based measurement model of PA is established. A synchronous orientation measurement method is designed to effectively reduce the cost of establishing the overdetermined equations. A deep neural network is used to retrieve the PA accurately. Simulations show that the proposed method effectively eliminates the impact of the 3D mask effect of test mask on PA measurement, and the measurement error is reduced by 72% compared with the measurement method based on the Kirchhoff model.

Vectorial pupil optimization to compensate polarization distortion in immersion lithography system.

Polarization distortion innately exists in hyper numerical aperture immersion lithography system. Polarization distortion, mainly including polarization aberration (PA) of lithography projection optics and thick mask induced polarization distortion, would seriously impact on lithography imaging quality. Some computational lithography technologies, such as robust optical proximity correction and robust source and mask optimization, have been introduced and developed to reduce the impact of polarization distortion on lithography imaging. In this paper, we innovate a vectorial pupil optimization (VPO) method to further extend degrees of freedom for pupil optimization and compensate polarization distortion for immersion lithography system. An analytical relationship between lithography imaging and active vectorial pupil, and the gradient-based algorithm is adopted to effectively solve VPO. Extensive simulations demonstrate the VPO method simultaneously compensate the PA of projection optics and the thick mask induced polarization distortion sufficiently. Based on PA-aware source mask optimization, the VPO method can further reduce the impact of polarization distortion on lithography imaging. Compared to current pupil wavefront optimization, the proposed VPO effectively reduces the pattern error by 37.2%, which demonstrates the VPO method can improve lithography pattern fidelity.

Fast lithographic source optimization method of certain contour sampling-Bayesian compressive sensing for high fidelity patterning.

Fast source optimization (SO) is in demand urgently for holistic lithography on-line at 14-5 nm nodes. Our earlier works of fast compressive sensing (CS) SO methods adopted randomly sampling monitoring pixels on layout patterns, consequently resulting in failure of SO sometimes and poor image fidelity compared to gradient-based SO with complete sampling (SD-SO). This paper proposes a novel certain contour sampling-Bayesian compressive sensing SO (CCS-BCS-SO) method to achieve the goals of fast SO and high fidelity patterns simultaneously. The CCS assures the optimized source uniquely and reduces the computational complexity significantly. The BCS theory, to our best knowledge, is for the first time applied to resolution enhancement techniques (RETs) in lithography systems to ensure high fidelity patterns. The results demonstrate that CCS-BCS-SO simultaneously achieves fast SO like CS-SO and high fidelity patterns like SD-SO.

Multiple-field-point pupil wavefront optimization in computational lithography.

Some pupil wavefront optimization (PWO) approaches were studied to compensate the thick mask effects considering only a field point, and these PWO methods neglect the inherent wave aberration in a realistic lithography system. Particularly, the wave aberration of lithography projection optics is exposure field dependent, and the wave aberrations at different fields of view (FOVs) would seriously and unevenly impact the results and effects of PWO. The current PWO method for single FOV cannot match full FOV. In this paper, we propose a multiple-field-point PWO (MPWO) method to improve lithography imaging quality for full FOV. A multiple-field-point cost function is built including the uneven impact of multiple aberrations on lithography imaging at full FOV. The comprehensive simulations demonstrate that the proposed MPWO method can effectively improve consistency of lithography imaging and enlarge the overlapped process window for full FOV. The most important point is that the optimized wavefront attained by MPWO can be realized via pupil wavefront manipulator FlexWave in lithography equipment, which is significant in holistic lithography for the next technology node.

Co-optimization method to reduce the pattern distortion caused by polarization aberration in anamorphic EUV lithography.

Extreme ultraviolet lithography is regarded as the most attractive technology to achieve 7 nm node and below. A new high-numerical-aperture anamorphic objective lens is designed to extend the single exposure resolution limit. However, the polarization aberrations (PAs) induced by the multilayer coatings on mirrors cause pattern distortions that cannot be neglected. In this paper, a source, mask, and process parameter co-optimization method is developed to compensate for the pattern distortions caused by PAs and increase the process window (PW). We first present an asymmetric source represented by the superposition of Zernike polynomials to reduce the pattern placement error (PPE). Then, a weighted cost function that incorporates the influences of PAs is innovated. Finally, a gradient-based statistical optimization method is adopted to minimize the cost function by optimizing the lithography system parameters alternately. Simulations at the 7 nm node of the 1D mask pattern indicate that for the system with a PA of marginal field, compared with our earlier work, the critical dimension error and PPE of the proposed method are reduced by 75.0% and 82.4%, respectively, and the PW is increased by 97.4%.

Multi-objective lithographic source mask optimization to reduce the uneven impact of polarization aberration at full exposure field.

Source and mask optimization (SMO) technology based on vectorial image model is indispensable in immersion lithography process at advanced technology node. Many kinds of algorithms have achieved successes in aspect of fast and robust SMO without accounting polarization aberration (PA). However, because the PA arising from immersion projection optics unevenly impacts on imaging performance, the conventional SMO would not be applicable in real lithography system. In this paper, we first investigate the serious impact of PA on SMO in details. The SMO accounting the assigned PA of one field point is not applicable to other field points, where the pattern fidelity is fiercely worse and the pattern error (PAE) is nearly doubled. Then, we innovate a MOSMO method to reduce the uneven impact of PA on lithography imaging at full exposure field. Compared to the assigned PA aware SMO, the proposed MOSMO reduces the standard deviation of PAE distribution by 53.3% and enlarges the maximum exposure latitude from 4% to 6.7%, which demonstrates the MOSMO is very significant to balance imaging quality and improve process robustness at full exposure field.

Fast nonlinear compressive sensing lithographic source and mask optimization method using Newton-IHTs algorithm.

Source and mask optimization (SMO) is an important method to improve lithography imaging fidelity. However, constrained by the computational inefficiency, the current SMO method can be used only in clip level applications. In this paper, to our best knowledge, the fast nonlinear compressive sensing (CS) theory is for the first time applied to solve the nonlinear inverse reconstruction problem in SMO. The proposed method simultaneously downsamples the layout pattern in the SMO procedure, which can effectively reduce the computation complexity. The space basis and two-dimensional (2D) discrete cosine transform (DCT) basis are selected to sparsely represent the source pattern and mask pattern, respectively. Based on the sparsity assumption of source and mask pattern, the SMO can be formulated as a nonlinear CS reconstruction problem. A Newton-iteration hard thresholding (Newton-IHTs) algorithm, by taking into account the second derivative of the cost function to accelerate convergence, is innovated to realize nonlinear CS-SMO with high imaging fidelity. Simulation results show the proposed method can significantly accelerate the SMO procedure over a traditional gradient-based method and IHTs-based method by a factor of 9.31 and 7.39, respectively.

A nonlinear measurement method of polarization aberration in immersion projection optics by spectrum analysis of aerial image.

Polarization aberrations (PA) can be presented by Jones pupil and can also impact the imaging performance of immersion projection optics significantly. Precise PA measurement is most important for resolution enhancement technology and holistic lithography at 7nm node and below, in order to improve the pattern fidelity and processing stability. However, the current imaging-based measurement method of PA by linear approximation has not taken the coupling effect of the PA coefficients into account. This paper proposes a nonlinear measurement method of PA based on a rigorous nonlinear model to improve the measurement accuracy significantly. In this invention, the new spectrum modulation theory is developed to establish a rigorous quadratic form of PA and aerial image spectrum. A hybrid genetic algorithm is developed to solve the quadratic form inversely to obtain the PA accurately. An overall simulation validates that this method provides a superior quality of PA measurement with very high precision of 10-4λ.