Numerical approximation of the viscous quantum hydrodynamic model for semiconductors

The viscous quantum hydrodynamic equations for semiconductors with constant temperature are numerically studied. The model consists of the one-dimensional Euler equations for the electron density and current density, including a quantum correction and viscous terms, coupled to the Poisson equation for the electrostatic potential. The equations can be derived formally from a Wigner–Fokker–Planck model by a moment method. Two different numerical techniques are used: a hyperbolic relaxation scheme and a central finite-difference method. By simulating a ballistic diode and a resonant tunneling diode, it is shown that numerical or physical viscosity changes significantly the behavior of the solutions. Moreover, the current-voltage characteristics show multiple regions of negative differential resistance and hysteresis effects.