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Conference on Metrology, Inspection, and Process Control for Microlithography XVIII

作者： Opsal, J Chu, H Leng, J Therma Wave Inc Fremont CA 94539 USA

ISBN: (纸本)0819452882

In real-time optical CD applications of shallow trench isolation (STI), shallow trench removal (STR), deep trench isolation (DTI), and deep trench removal (DTR), a single recipe is required for each type of application to accommodate wide ranges of process windows by monitoring parameters such as bottom CD (BCD), middle CD (MCD), top CD (TCD) and side wall angle (SWA). The modeling of the grating profiles of silicon trenches with nitride caps requires a large number of slices (> 10) to generate smooth shapes for top rounding of the nitride, curvature of the silicon trench waist, and the silicon trench footing or undercut. The number of orders for Fourier expansion is also high (larger than 13 in the best case). With these requirements we found that the rigorous coupled wave analysis (RCWA) algorithm is generally too slow to calculate the CD profiles from the raw scatterometry spectra. In this paper we present a finite difference (FD) algorithm and its applications to real-time CD scatterometry. The mathematical analysis of the FD algorithm was published elsewhere(1). We demonstrate that the FD algorithm has an advantage over RCWA in terms of calculation speed (up to a factor of 10 improvement), better capture of profile shapes in comparison with cross sectional SEM (X-SEM) and more robust in terms of numerical stability. Details of comparisons between FD and RCWA will be shown for the applications of STR and DTR.

关键词： finite difference algorithm real-time critical dimension (RT/CD) shallow trench isolation (STI) shallow trench removal (STR) deep trench isolation (DTI) and deep trench removal (DTR)

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Absorbing boundary conditions in a fourth-order accurate SH-wave staggered grid finite difference algorithm

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ACTA GEOPHYSICA 2008年第4期56卷 1090-1108页

作者： Kumar, Sanjay Narayan, Jay Prakash Indian Inst Technol Dept Earthquake Engn Roorkee Uttar Pradesh India

This article presents the implementation of two well known absorbing boundary conditions in a fourth-order accurate staggered grid SH-wave finite difference (FD) algorithm with variable grid size, in a very simplified manner. Based on simulated results, it was confirmed that the Clayton and Engquist absorbing boundary condition causes edge-reflections in case of larger angle of incidence of body waves on the model edges. The results of various numerical experiments revealed that the Israeli and Orszag sponge boundary condition is efficient enough to avoid edge-reflections for any angle of incidence of the body. We recommend the use of both the Clayton and Engquist and Israeli and Orszag absorbing boundary conditions simultaneously to avoid any edge-reflections.

关键词： SH-wave finite difference algorithm fourth-order spatial accuracy boundary condition edge-reflections

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Accuracy of the subsurface damage parameters calculated by the finite difference algorithm

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JOURNAL OF THE EUROPEAN OPTICAL SOCIETY-RAPID PUBLICATIONS 2015年 10卷

作者： Wang, H. Fu, G. Xiao, L. Chen, H. Xiao, H. State Key Laboratory for Manufacturing Systems Engineering Mechanical Engineering School Xi’an Jiaotong UniversityXi’an Shaanxi710049 China

An important approach to characterize the full three-dimensional information of subsurface damage is to simulate the etching process of a sample reversely. The simulation starts from the morphology of the sample after the subsurface damage micro cracks being opened totally. During the etching experiment, it is possible for us to get the surface morphology at any moment. This paper presents a finite difference algorithm to simulate the morphology evolution during the etching process and then the surface's morphology of the sample at a specific time can be obtained. Comparison between the simulated morphology and the measured one provides the clue of improving the precision of the finite difference algorithm. This method is kind of the fast calculation. In addition, the accuracy of this calculation of the corrosion model needs to be ensured. In order to improve the precision of calculation, the time interval should be set as the appropriate value by comparison and analysis. In this paper, the accuracy can be calculated through comparing the simulated result with the experimental result, and the maximum error of this method can be gained.

关键词： Subsurface damage finite difference algorithm simulation accuracy

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State of charge estimation by finite difference extended Kalman filter with HPPC parameters identification

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Science China(Technological Sciences) 2020年第3期63卷 410-421页

作者： HE Lin HU MinKang WEI YuJiang LIU BingJiao SHI Qin Automotive Research Institute Hefei University of TechnologyHefei 230009China School of Automotive and Transportation Engineering Hefei University of TechnologyHefei 230009China

State of charge(SOC) is a key parameter of lithium-ion battery. In this paper, a finite difference extended Kalman filter(FDEKF)with Hybrid Pulse Power Characterization(HPPC) parameters identification is proposed to estimate the SOC. The finite difference(FD) algorithm is benefit to compute the partial derivative of nonlinear function, which can reduce the linearization error generated by the extended Kalman filter(EKF). The FDEKF algorithm can reduce the computational load of controller in engineering practice without solving the Jacobian matrix. It is simple of dynamic model of lithium-ion battery to adopt a secondorder resistor-capacitor(2 RC) network, the parameters of which are identified by the HPPC. Two conditions, both constant current discharge(CCD) and urban dynamometer driving schedule(UDDS), are utilized to validate the FDEKF *** convergence rate and accuracy between the FDEKF and the EKF algorithm, it can be seen that the former is a better candidate to estimate the SOC.

关键词： state of charge lithium-ion battery parameters identification finite difference algorithm extended Kalman filter

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Application of finite difference to study morphology evolution during etching the optical surface with subsurface damage

Application of finite difference to study morphology evoluti...

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International Conference on Photonics and Optical Engineering (icPOE)

作者： Wang, Hairong Xiao, Lihui Chen, Hongfeng Jiang, Zhuangde State Key Lab Mfg Syst Engn Xian Shaanxi Peoples R China Xi An Jiao Tong Univ Xian 710049 Shaanxi Peoples R China

ISBN: (纸本)9781628415650

One important approach to characterize full three-dimensional information is to simulate the etching process of a sample with subsurface damage reversely. The simulation starts from the morphology of the sample at a certain time when the subsurface damage can be opened totally. In the etching experiment, it is possible for us to get the surface morphology at any time. The paper presents a finite difference algorithm to simulate the morphology evolution in an etching process and by the finite difference algorithm the morphology of the sample at a specific time can be given. Comparison of the simulated morphology and measured one provides us the clue of improving the finite difference algorithm. In this paper, the accuracy can be calculated through comparing the simulation with experimental result, and the maximum error of subsurface damage will be calculated.

关键词： subsurface damage finite difference algorithm simulation accuracy

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An efficient high-order algorithm for solving systems of 3-D reaction-diffusion equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第1期155卷 1-17页

作者： Gu, YX Liao, WY Zhu, JP Univ Akron Dept Theoret & Appl Mech Akron OH 44325 USA Dalian Univ Technol State Key Lab Struct Anal Ind Equip Dalian 116023 Peoples R China Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA

We discuss an efficient higher order finite difference algorithm for solving systems of 3-D reaction-diffusion equations with nonlinear reaction terms. The algorithm is fourth-order accurate in both the temporal and spatial dimensions. It requires only a regular seven-point difference stencil similar to that used in the standard second-order algorithms, such as the Crank-Nicolson algorithm. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. (C) 2003 Elsevier Science B.V. All rights reserved.

关键词： high-order algorithms approximate factorization reaction-diffusion equations finite difference algorithm

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Numerical study on radiative MHD flow of viscoelastic fluids with distributed-order and variable-order space fractional operators

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MATHEMATICS AND COMPUTERS IN SIMULATION 2024年 215卷 291-305页

作者： Li, Nan Wang, Xiaoping Xu, Huanying Qi, Haitao Shandong Univ Sch Math & Stat Weihai 264209 Shandong Peoples R China

The magnetohydrodynamic (MHD) flow has been concerned widely for its widespread adoption in the field of astrophysics, electronics and many other industries over the years. The purpose of this article is to introduce the variable and distributed order space fractional models to characterize the MHD flow and heat transfer of heterogeneous viscoelastic fluids in a parallel plates. Based on the central difference approximation of Riesz space fractional derivative, the Crank-Nicolson difference scheme for the governing equations is established, and the effectiveness of the algorithm is verified by two numerical examples. We examine the effects of fractional-order model parameters on the velocity and temperature, our investigation indicates that for the constant fractional model, the larger the fractional order parameter, the smaller the velocity and temperature. The variable space fractional method can be used to describe dynamic behavior with time and space dependence, while the distributed space fractional model can describe various phenomena in which the number of differential orders varies over a certain range, characterizing their complex processes over space, and it is also more suitable for simulating the fluid flow and thermal behavior of complex viscoelastic magnetic fluid. (c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Distributed-order space fractional model Variable-order space fractional model Magnetic field Heat transfer finite difference algorithm

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TWO-DIMENSIONAL PROFILES OF IMPURITY DIFFUSION IN SEMICONDUCTORS

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ELECTRONICS LETTERS 1984年第22期20卷 909-910页

作者： BUONOMO, A DIBELLO, C UNIV NAPOLI DIPARTIMENTO ELETTRONI-80125 NAPOLIITALY

The letter shows how two-dimensional impurity profiles in semiconductors can be calculated by applying a simple and efficient finite difference algorithm. The diffusion coefficient employed in the model depends on the impurity concentrations in accordance with the recent theory of nonlinear diffusion in semiconductors. The limit of validity of the more simple linear diffusion model is also determined.

关键词： Impurity concentration, distribution, and gradients Diffusion, migration, and displacement of impurities in solids finite difference algorithm diffusion in solids nonlinear diffusion two-dimensional impurity profiles linear diffusion model impurity diffusion Semiconductor theory, materials and properties doping profiles difference equations Semiconductor doping diffusion coefficient semiconductors

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Computational analysis of turbulent flow through an eccentric annulus under different temperature conditions

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INTERNATIONAL JOURNAL OF NUMERICAL METHODS FOR HEAT & FLUID FLOW 2018年第9期28卷 2189-2207页

作者： Ulker, Erman Korkut, Sila Ovgu Sorgun, Mehmet Izmir Katip Celebi Univ Fac Engn & Architecture Dept Civil Engn Izmir Turkey Izmir Katip Celebi Univ Fac Engn & Architecture Dept Engn Sci Izmir Turkey

Purpose The purpose of this paper is to solve Navier-Stokes equations including the effects of temperature and inner pipe rotation for fully developed turbulent flow in eccentric annuli by using finite difference scheme with fixing non-linear terms. Design/methodology/approach A mathematical model is proposed for fully developed turbulent flow including the effects of temperature and inner pipe rotation in eccentric annuli. Obtained equation is solved numerically via central difference approximation. In this process, the non-linear term is frozen. In so doing, the non-linear equation can be considered as a linear one. Findings The convergence analysis is studied before using the method to the proposed momentum equation. It reflects that the method approaches to the exact solution of the equation. The numerical solution of the mathematical model shows that pressure gradient can be predicted with a good accuracy when it is compared with experimental data collected from experiments conducted at Izmir Katip Celebi University Flow Loop. Originality/value The originality of this work is that Navier-Stokes equations including temperature and inner pipe rotation effects for fully developed turbulent flow in eccentric annuli are solved numerically by a finite difference method with frozen non-linear terms.

关键词： Temperature Convergence analysis finite difference algorithm Navier-Stokes equations Annulus

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Kernel Density Estimation on a Linear Network

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SCANDINAVIAN JOURNAL OF STATISTICS 2017年第2期44卷 324-345页

作者： McSwiggan, Greg Baddeley, Adrian Nair, Gopalan Univ Western Australia Sch Math & Stat Nedlands WA Australia Curtin Univ Dept Math & Stat GPO Box U1987 Perth WA 6845 Australia

This paper develops a statistically principled approach to kernel density estimation on a network of lines, such as a road network. Existing heuristic techniques are reviewed, and their weaknesses are identified. The correct analogue of the Gaussian kernel is the heat kernel', the occupation density of Brownian motion on the network. The corresponding kernel estimator satisfies the classical time-dependent heat equation on the network. This diffusion estimator' has good statistical properties that follow from the heat equation. It is mathematically similar to an existing heuristic technique, in that both can be expressed as sums over paths in the network. However, the diffusion estimate is an infinite sum, which cannot be evaluated using existing algorithms. Instead, the diffusion estimate can be computed rapidly by numerically solving the time-dependent heat equation on the network. This also enables bandwidth selection using cross-validation. The diffusion estimate with automatically selected bandwidth is demonstrated on road accident data.

关键词： bandwidth selection diffusion equal-split kernels finite difference algorithm heat equation heat kernel intensity road accidents spatial point process

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