A Review on the Quantum Drift Diffusion Model

Abstract

We consider the quantum drift diffusion model for semiconductor devices and collect recent results on the stationary and transient equations. The stationary model including generation-recombination terms is studied for bipolar devices and the transient equations are considered in the unipolar case. We cover several topics, such as existence and uniqueness of solutions, asymptotic limits and convergence of a nonlinear iteration scheme in the stationary case as well convergence of a positivity preserving semidiscretization of the transient equations and the linear stability of stationary states.

A sixth-order parabolic system in semiconductors

“…Quantum drift-diffusion model, quantum hydrodynamic model and quantum energy transport model are three quantum macroscopic models, which were introduced to simulate the quantum effects in miniaturized semiconductor devices. Some derivations of these models could be found in [14,15,26] etc., and some mathematical results of them were given in a series of works [1-13, 16-24, 29] etc. While for the sixth-order parabolic system (1.1), up to the authors' knowledge, no theoretic results were shown.…”

The isentropic quantum drift-diffusion model in two or three space dimensions

“…Recently, a series of quantum macroscopic models, namely quantum drift-diffusion model (QDDM), quantum hydrodynamic model and quantum energy transport model were introduced to simulate the quantum effects in miniaturized semiconductor devices. Some derivation of these models could be found in [16,17,25] etc. Quantum hydrodynamic model and quantum energy transport model were studied mathematically in the literature recently, see [18,4,12] etc.…”

Existence of weak solution and semiclassical limit for quantum drift-diffusion model

“…For the complete quantum drift diffusion system, Jüngel and Pinnau in [13,14] established a positivity preserving scheme for this system and got the existence of the solution in a weak sense for fixed ε and λ in one dimension, or up to three dimension with additional assumptions on the solution and big enough lattice temperature. For more details on the quantum drift-diffusion model, see [17].…”