Stationary solutions to hybrid quantum hydrodynamical model of semiconductors in bounded domain

In this paper we study the behaviour of a micro-sized semiconductor device by means of a hybrid model of hydrodynamic equations. First of all, taking into account the quantum effects in the semiconductor device, we derive a new model called the hybrid quantum hydrodynamic model (H-QHD) coupled with the Poisson equation for electric potential. In particular, we write the Bohm potential in a revised form. This new potential is derived heuristically by assuming that the energy of the electrons depends on the charge density n and on ∇n just in a restricted part of the device domain, whereas the remaining parts are modeled classically. Namely, the device is designed with some parts that feel the quantum effects and some parts do not. The main target is to investigate the existence of the stationary solutions for the hybrid quantum hydrodynamic model. Since the quantum effect is regionally degenerate, this will also makes the working equation regionally degenerate regarding its ellipticity, and the corresponding solutions are weak only. This paper seems the first framework to treat the equation with regionally degenerate ellipticity. In order to prove the existence of such weak solutions, we first construct a sequence of smooth QHD solutions, then prove that such a sequence weakly converges and its limit is just our desired weak solution for the hybrid QHD problem. The Debye length limit is also studied. Indeed, we prove that the weak solutions of the hybrid QHD weakly converge to their targets as the spacial Debye length vanishes. Finally, we carry out some numerical simulations for a simple device, which also confirm our theoretical results.

In this paper we study the behaviour of a micro-sized semiconductor device by means of a hybrid model of hydrodynamic equations. First of all, taking into account the quantum effects in the semiconductor device, we derive a new model called the hybrid quantum hydrodynamic model (H-QHD) coupled with the Poisson equation for electric potential. In particular, we write the Bohm potential in a revised form. This new potential is derived heuristically by assuming that the energy of the electrons depends on the charge density n and on del n just in a restricted part of the device domain, whereas the remaining parts are modeled classically. Namely, the device is designed with some parts that feel the quantum effects and some parts do not. The main target is to investigate the existence of the stationary solutions for the hybrid quantum hydrodynamic model. Since the quantum effect is regionally degenerate, this will also makes the working equation regionally degenerate regarding its ellipticity, and the corresponding solutions are weak only. This paper seems the first framework to treat the equation with regionally degenerate ellipticity. In order to prove the existence of such weak solutions, we first construct a sequence of smooth QHD solutions, then prove that such a sequence weakly converges and its limit is just our desired weak solution for the hybrid QHD problem. The Debye length limit is also studied. Indeed, we prove that the weak solutions of the hybrid QHD weakly converge to their targets as the spacial Debye length vanishes. Finally, we carry out some numerical simulations for a simple device, which also confirm our theoretical results.