A critical examination of the basis of macroscopic quantum transport approaches

The density gradient, effective potential and smooth quantum hydrodynamic approaches have been proposed in recent years as promising candidates for the efficient simulation of quantum effects in semiconductor devices. The microscopic justifications for these three approaches are based on, in that order, very specific approximations to the equilibrium Wigner function, the average carrier energy and the equilibrium density matrix. The validity of these approximations is however very questionable in realistic devices containing material heterojunctions. For example, the density gradient method is derived from the equilibrium Wigner function for a slowly varying device potential, but it is often applied to model MOS inversion layer transport near an abrupt barrier where the approximation is invalid. Furthermore, attempts to extend it to the treatment of tunneling phenomena have been made for which it is expected to have even lesser applicability. In this work, we directly examine the microscopic basis for the density gradient and the smooth quantum hydrodynamic approaches in one dimension using the Green´s function formalism. These approaches are both predicated upon particular equilibrium relationships between the stress tensor and the local carrier gas density to close the hydrodynamic hierarchy at the current transport equation. We therefore derive the equilibrium density matrix for different barrier potentials, and then explicitly construct the stress tensor to compare it with the forms postulated in the two approaches. We show, that as expected the two forms are inaccurate near the barrier for realistic abrupt barrier heights and are thus of questionable validity as such for transport simulations in the barrier direction.