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Volume 41, Issue 5
A Positivity-Preserving Finite Element Method for Quantum Drift-Diffusion Model

Pengcong Mu & Weiying Zheng

J. Comp. Math., 41 (2023), pp. 909-932.

Published online: 2023-05

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  • Abstract

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.

  • AMS Subject Headings

35J60, 35Q81, 49M15, 49M37, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mupengcong@lsec.cc.ac.cn (Pengcong Mu)

zwy@lsec.cc.ac.cn (Weiying Zheng)

  • BibTex
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  • TXT
@Article{JCM-41-909, author = {Mu , Pengcong and Zheng , Weiying}, title = {A Positivity-Preserving Finite Element Method for Quantum Drift-Diffusion Model}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {909--932}, abstract = {

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2206-m2021-0353}, url = {http://global-sci.org/intro/article_detail/jcm/21679.html} }
TY - JOUR T1 - A Positivity-Preserving Finite Element Method for Quantum Drift-Diffusion Model AU - Mu , Pengcong AU - Zheng , Weiying JO - Journal of Computational Mathematics VL - 5 SP - 909 EP - 932 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2206-m2021-0353 UR - https://global-sci.org/intro/article_detail/jcm/21679.html KW - Quantum drift-diffusion model, Positivity-preserving finite element method, Newton method, FinFET device, High bias voltage. AB -

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.

Pengcong Mu & Weiying Zheng. (2023). A Positivity-Preserving Finite Element Method for Quantum Drift-Diffusion Model. Journal of Computational Mathematics. 41 (5). 909-932. doi:10.4208/jcm.2206-m2021-0353
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