Time-Dependent Numerical Methods for a Regularized Quantum Hydrodynamic Model

Abstract

Numerical methods are developed for the time-dependent smooth quantum hydrodynamic (QHD) model by using a mixture of hyperbolic, parabolic, and elliptic partial differential equation methods: (i) the underlying hyperbolic gas dynamical part of the transport equations is solved with a third-order weighted essentially non-oscillatory (WENO) method, treating the electric field, scattering, and quantum terms as source terms; (ii) the parabolic heat conduction term is incorporated with the trapezoidal rule/backward difference formula second-order (TRBDF2) method, and (iii) the elliptic Poisson equation is solved with a standard (sparse direct or modern iterative) elliptic solver. For the time-dependent simulations, a regularization of the smooth QHD model equations to prevent an unstable growing mode is implemented. The regularization involves replacing the spatial derivative of the electron density on the right-hand side of the momentum conservation equation by using a quantum generalization of the semiclassical Boltzmann distribution for electron density that captures in a simple way essential effects of quantum tunneling and resonance. Time-dependent simulations of the resonant tunneling diode to steady state using the smooth QHD model are presented, which show realistic negative differential resistance (NDR) (the experimental signal of quantum resonance) in the current-voltage curve. These are the first time-dependent simulations of the smooth QHD model. The simulations match fully quantum mechanical simulations of the resonant tunneling diode much better than any other QHD simulations to date.

Keywords

Quantum hydrodynamic (QHD) model / Time-dependent numerical methods / Weighted essentially non-oscillatory (WENO) / Resonant tunneling diode / 65M06 / 76Y05

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