Algorithms for Solving Electronic Continuity Equation in Numerical Simulation of Semiconductor Devices

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

Abstract

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

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.

Figures/Tables 9

References 35

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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