半导体器件数值模拟中电子连续性方程的求解算法

doi:10.19596/j.cnki.1001-246x.8656

摘要/Abstract

摘要：

针对电子连续性方程的离散代数方程组, 对离散线性系统的矩阵进行分析, 得到矩阵的三类特点; 针对大规模电子连续性方程的离散方程组, 采用预处理Krylov子空间方法进行求解, 并比较和分析几类预处理方法的效果。结果表明: 代数多重网格(AMG)预处理Krylov子空间方法在求解离散电子连续性方程方面非常有效。开展AMG预处理Krylov子空间方法求解离散电子连续性方程的大规模并行可扩展性测试, 比较和分析了AMG方法中三类关键算法参数的选取。

关键词: 电子连续性方程, 迭代方法, 代数多重网格, 预处理, Krylov方法

Abstract:

For solving the discretized electronic continuity equation, two aspects of work are carried out. Firstly, the matrix of the discretized linear system is analyzed, and three types of characteristics of the matrix are obtained. Secondly, based on the characteristics of the matrix, the discretized electronic continuity equation is solved by preconditioned Krylov subspace methods, and the effectiveness of several types of preconditioned methods is compared and analyzed. The results show that the algebraic multigrid (AMG) preconditioned Krylov subspace method is very effective for solving discretized electronic continuity equations. A large-scale parallel scalability test of the AMG preconditioned Krylov subspace method for solving discretized electronic continuity equations is carried out, and the selection of three key algorithm parameters in the AMG method is compared and analyzed.

Key words: electron continuity equation, iteration method, algebraic multigrid, preconditioned method, Krylov subspace method

胡毅, 安恒斌. 半导体器件数值模拟中电子连续性方程的求解算法[J]. 计算物理, 2023, 40(5): 570-582.

Yi HU, Hengbin AN. Algorithms for Solving Electronic Continuity Equation in Numerical Simulation of Semiconductor Devices[J]. Chinese Journal of Computational Physics, 2023, 40(5): 570-582.

图/表 9

参考文献 35

1	SHOCKLEY W . The theory of p-n junctions in semiconductors and p-n junction transistors[J]. Bell System Technical Journal, 1949, 28 (3): 435- 489. DOI
2	GUMMEL H . A self-consistent iterative scheme for one-dimensional steady state transistor calculations[J]. IEEE Transactions on Electron Devices, 1964, 11 (10): 455- 465. DOI
3	SELBERHERR S . Analysis and simulation of semiconductor devices[M]. New York: Springer Science & Business Media, 1984.
4	BANK R , ROSE D , FICHTNER W . Numerical methods for semiconductor device simulation[J]. SIAM Journal on Scientific and Statistical Computing, 1983, 4 (3): 416- 435. DOI
5	JIA X , AN H , HU Y . A physical-based iterative initial value strategy for solving drift-diffusion equations[J]. Computers & Mathematics with Applications, 2022,
6	SCHARFETTER D , GUMMEL H . Large-signal analysis of a silicon read diode oscillator[J]. IEEE Transactions on Electron Devices, 1969, 16 (1): 64- 77. DOI
7	MULTIPHYSICS C . Introduction to comsol multiphysics[J]. COMSOL Multiphysics, Burlington, MA, Accessed Feb, 1998, 9, 32.
8	CHENG J , ZHANG L . A scalable parallel algorithm for three-dimensional semiconductor device simulation[J]. Chinese Journal of Computational Physic, 2012, 29 (3): 439- 448.
9	YUAN Y . Recent progress in numerical methods for semiconductor devices[J]. Chinese Journal of Computational Physic, 2009, 26 (3): 317- 324.
10	GONG D , WANG J , ZHANG D , et al. A general-purpose two-dimensional semiconductor simulator[J]. Chinese Journal of Computational Physic, 2007, 24 (2): 247- 252.
11	王芹, 马召灿, 白石阳, 等. 三维半导体器件漂移扩散模型的并行有限元方法研究[J]. 数值计算与计算机应用, 2020, 41 (2): 85- 104.
12	LIN P , SHADID J , TUMINARO R , et al. A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift-diffusion, flow/transport/reaction, resistive MHD[J]. International Journal for Numerical Methods in Fluids, 2010, 64 (10): 1148- 1179.
13	LIN P . Improving multigrid performance for unstructured mesh drift-diffusion simulations on 147 000 cores[J]. International Journal for Numerical Methods in Engineering, 2012, 91 (9): 971- 989. DOI
14	LIN P , SHADID J , SALA M , et al. Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling[J]. Journal of Computational Physics, 2009, 228 (17): 6250- 6267. DOI
15	LIN P , SHADID J , TUMINARO R , et al. Performance of a Petrov-Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift-diffusion equations[J]. International Journal for Numerical Methods in Engineering, 2010, 84 (4): 448- 469.
16	FARRELL P , PESCHKA D . Nonlinear diffusion, boundary layers and nonsmoothness: Analysis of challenges in drift-diffusion semiconductor simulations[J]. Computers & Mathematics with Applications, 2019, 78 (12): 3731- 3747.
17	FARRELL P , KAYSER S , ROTUNDO N . Modeling and simulation of the lateral photovoltage scanning method[J]. Computers & Mathematics with Applications, 2021, 102, 248- 260.
18	BRIGGS W, HENSON V, McORMICK S. A multigrid tutorial[M]. SIAM, 2000.
19	BOUSQUET A , HU X , METTI M S , et al. Newton solvers for drift-diffusion and electrokinetic equations[J]. SIAM Journal on Scientific Computing, 2018, 40 (3): B982- B1006. DOI
20	CAI X . Additive Schwarz algorithms for parabolic convection-diffusion equations[J]. Numerische Mathematik, 1991, 60 (1): 41- 61. DOI
21	贾晓伟. 面向半导体器件数值模拟的漂移扩散方程高效求解方法研究[D]. 北京: 中国工程物理研究院, 2022.
22	BANK R , COUGHRAN W , COWSAR L C . The finite volume Scharfetter-Gummel method for steady convection diffusion equations[J]. Computing and Visualization in Science, 1998, 1 (3): 123- 136. DOI
23	HEINREICHSBERGER O , SELBERHERR S , STIFTINGER M , et al. Fast iterative solution of carrier continuity equations for three-dimensional device simulation[J]. SIAM Journal on Scientific and Statistical Computing, 1992, 13 (1): 289- 306. DOI
24	张家驹. M矩阵的一些性质[J]. 数学年刊: 中文版, 1980, 1 (1): 46- 55.
25	POOLE G , BOULLION T . A survey on M-matrices[J]. SIAM Review, 1974, 16 (4): 419- 427. DOI
26	SAAD Y . Iterative methods for sparse linear systems[M]. 2nd ed Philadelphia: SIAM, 2003.
27	谷同祥, 安恒斌, 刘兴平, 等. 迭代方法和预处理技术(上册)[M]. 北京: 科学出版社, 2015.
28	CAI X , SARKIS M . A restricted additive Schwarz preconditioner for general sparse linear systems[J]. SIAM Journal on Scientific Computing, 1999, 21 (2): 792- 797. DOI
29	SAH C , NOYCE R , SHOCKLEY W . Carrier generation and recombination in pn junctions and pn junction characteristics[J]. Proceedings of the IRE, 1957, 45 (9): 1228- 1243. DOI
30	张志刚, 刘长军, 黄卡玛, 等. 一维PN结二极管稳态模型的数值模拟[J]. 洛阳工业高等专科学校学报, 2005, 15 (1): 28- 30.
31	SZE S M , LI Y , NG K K . Physics of semiconductor devices[M]. 4th ed New York: John Wiley & Sons, 2017.
32	HU S , XU X , ZHENG Y , et al. Algorithms in linear solver for large-scale time-harmonic Maxwell's equations in SiP applications: State-of-the-art and challenges[J]. Chinese Journal of Computational Physics, 2021, 38 (2): 131- 145.
33	LIU Q , MO Z , ZHANG A , et al. JAUMIN: A programming framework for large-scale numerical simulation on unstructured meshes[J]. CCF Transactions on High Performance Computing, 2019, 1 (1): 35- 48. DOI
34	FALGOUT R, YANG U. HYPRE: A library of high performance preconditioners[C]//International Conference on Computational Science Heidelberg: Springer, 2002: 632-641.
35	BALAY S, ABHYANKAR S, Adams M, et al. PETSc users manual[R]. Argonne National Lab (ANL), Argonne, IL(United States), 2019.

DOF	Precondition	GMRES			BiCGSTAB
DOF	Precondition	NIT	Pre/s	Total/s	NIT	Pre/s	Total/s
15 729	NONE	474		4.66	*	*	*
15 729	JAC	221	1.46	2.83	215	1.81	2.24
15 729	BJAC	217	2.02	3.42	169	2.48	2.88
15 729	ASM	215	1.98	3.34	158	2.37	2.76
15 729	AMG	12	0.31	0.57	7	0.18	0.22
62 177	NONE	851		37.84	*	*	*
62 177	JAC	440	4.65	15.57	415	5.78	6.87
62 177	BJAC	438	4.93	15.78	335	7.39	8.35
62 177	ASM	409	5.51	15.08	330	9.99	11.00
62 177	AMG	11	0.63	0.69	7	0.34	0.37
231 361	NONE	*	*	*	*	*	*
231 361	JAC	937	19.57	93.41	884	24.77	52.18
231 361	BJAC	925	22.31	94.08	792	25.12	54.29
231 361	ASM	862	24.42	85.59	599	29.34	47.75
231 361	AMG	12	1.53	1.61	7	0.72	0.80

DOF	Precondition	GMRES			BiCGSTAB
DOF	Precondition	NIT	Pre/s	Total/s	NIT	Pre/s	Total/s
15 729	NONE	474		4.66	*	*	*
15 729	JAC	221	1.46	2.83	215	1.81	2.24
15 729	BJAC	217	2.02	3.42	169	2.48	2.88
15 729	ASM	215	1.98	3.34	158	2.37	2.76
15 729	AMG	12	0.31	0.57	7	0.18	0.22
62 177	NONE	851		37.84	*	*	*
62 177	JAC	440	4.65	15.57	415	5.78	6.87
62 177	BJAC	438	4.93	15.78	335	7.39	8.35
62 177	ASM	409	5.51	15.08	330	9.99	11.00
62 177	AMG	11	0.63	0.69	7	0.34	0.37
231 361	NONE	*	*	*	*	*	*
231 361	JAC	937	19.57	93.41	884	24.77	52.18
231 361	BJAC	925	22.31	94.08	792	25.12	54.29
231 361	ASM	862	24.42	85.59	599	29.34	47.75
231 361	AMG	12	1.53	1.61	7	0.72	0.80

DOF	Precondition	GMRES			BiCGSTAB
DOF	Precondition	NIT	Pre/s	Total/s	NIT	Pre/s	Total/s
231 361	NONE	*	*	*	*	*	*
231 361	JAC	*	*	*	*	*	*
231 361	BJAC	1 128	13.13	45.98	909	24.15	30.86
231 361	ASM	1 022	12.09	44.13	878	23.70	30.19
231 361	AMG	21	0.81	0.87	16	1.03	1.08
923 521	NONE	*	*	*	*	*	*
923 521	JAC	*	*	*	*	*	*
923 521	BJAC	*	*	*	*	*	*
923 521	ASM	*	*	*	*	*	*
923 521	AMG	21	1.46	1.55	16	1.78	1.95
3 690 241	NONE	*	*	*	*	*	*
3 690 241	JAC	*	*	*	*	*	*
3 690 241	BJAC	*	*	*	*	*	*
3 690 241	ASM	*	*	*	*	*	*
3 690 241	AMG	21	3.00	3.25	16	3.92	4.05

DOF	Precondition	GMRES			BiCGSTAB
DOF	Precondition	NIT	Pre/s	Total/s	NIT	Pre/s	Total/s
231 361	NONE	*	*	*	*	*	*
231 361	JAC	*	*	*	*	*	*
231 361	BJAC	1 128	13.13	45.98	909	24.15	30.86
231 361	ASM	1 022	12.09	44.13	878	23.70	30.19
231 361	AMG	21	0.81	0.87	16	1.03	1.08
923 521	NONE	*	*	*	*	*	*
923 521	JAC	*	*	*	*	*	*
923 521	BJAC	*	*	*	*	*	*
923 521	ASM	*	*	*	*	*	*
923 521	AMG	21	1.46	1.55	16	1.78	1.95
3 690 241	NONE	*	*	*	*	*	*
3 690 241	JAC	*	*	*	*	*	*
3 690 241	BJAC	*	*	*	*	*	*
3 690 241	ASM	*	*	*	*	*	*
3 690 241	AMG	21	3.00	3.25	16	3.92	4.05

θ	GMRES			BiCGSTAB
θ	NIT	Pre/s	Total/s	NIT	Pre/s	Total/s
0.10	22	9.98	10.87	15	14.02	14.32
0.25	22	9.91	10.84	16	13.99	13.41
0.50	44	23.95	26.47	31	32.15	33.37
0.75	64	33.60	38.03	54	56.17	55.18
0.90	114	58.21	70.29	151	152.66	154.83