The Quantum Hydrodynamic Model for Semiconductor Devices

The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections. These quantum corrections are $O( {\hbar ^2 } )$. The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation. The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms. These quantum terms allow particles to tunnel through potential barriers and to build up in potential wells. The three-dimensional QHD transport equations are mathematically classified as having two Schrödinger modes, two hyperbolic modes, and one parabolic mode. The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind method. Simulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-voltage curve. These are the first simulations of the full QHD equations to show NDR in the resonant tunneling diode. The computed current-voltage curve agrees quantitatively with experimental measurements. NDR is interpreted in terms of the time spent by electrons in the quantum well.