A Positivity-Preserving Finite Element Method for Quantum Drift-Diffusion Model (Journal of Computational Mathematics, 2023)
In this paper, we propose a positivity-preserving finite element method for solving thethree-dimensional quantum drift-diffusion model. The model consists of five nonlinearelliptic equations, and two of them describe quantum corrections for quasi-Fermi levels.We propose an interpolated-exponential finite element (IEFE) method for solving the twoquantum-correction equations. The IEFE method always yields positive carrier densitiesand preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equationswith the edge-averaged finite element (EAFE) method to reduce numerical oscillations ofquasi-Fermi levels. The Poisson equation of electrical potential is solved with standardLagrangian finite elements. We prove the existence of solution to the nonlinear discreteproblem 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 theNewton method is robust for source-to-gate bias voltages up to 9V and source-to-drainbias voltages up to 10V.
Keywords: Quantum drift-diffusion model; Positivity-preserving finite element method; Newton method; FinFET device; High bias voltage.