Quantum hydrodynamic equations and quantum-hierarchy decoupling scheme

There is a need to extract the relevant device physics from exact quantum transport formulations, and hence to reduce the complexity of quantum transport simulations for practical applications. We use a Hermite polynomial expansion to drastically reduce the number of degrees of freedom associated with the momentum variables. The result is a quantum hierarchy in real space. We also give a general procedure for the quantum-hierarchy decoupling scheme to derive the quantum hydrodynamic (QHD) and quantum drift-diffusion transport equations. We present some numerical results for the quantum hierarchy. A rigorous foundation of a decoupling procedure is given whereby the lower-order equations are renormalized in terms of a self-consistent effective potential, quantum diffusion coefficient, and moments, endowed with all the quantum corrections to order ${ħ}^2.$ Our decoupling scheme is based on the general expression of $\mathrm{Tr}{\mathcal{H}}^n$ to order ${ħ}^2,$ valid at all temperatures without the need for expansion in terms of the small parameter and high temperature assumption. This is very important conceptually since existing QHD formulations, using expansion to order ${ħ}^2,$ are based on a Boltzmann distribution with the restrictive assumption of a small parameter, which is not valid in abrupt heterojunction semiconductor devices. They also fail to account for important quantum nonlinearity in the form of nonequilibrium quantum corrections. These nonequilibrium quantum corrections are expected to play a major role in approximating the coherence manifested by the highly nonlinear current-voltage characteristics of resonant tunneling structures.