Study of quantum corrections to hydrodynamic-like models for semiconductors

Key words : numerical simulation; semiconductors; hydrodynamic model; quantum corrections; nonlinear boundary value problems; Schrödinger-Poisson problem; bifurcations

Project leader : Caussignac Ph.

Participation : Descloux J.

Description (goals, methods, perspectives) :

In this work, we compare several models for computing the current-voltage (I(V)) curves for the resonant tunnelling diode device. In particular, we shall be able to obtain the so called negative differential resistivity, i.e. a negative slope of the I(V)-curve in some intervals. The considered models range from semi-classical ones with quantum corrections (drift-diffusion, energy-transport, hydrodynamic) to the quantum Schrödinger-Poisson problem.
Since I(V) curves can show hysteresis phenomena (bifurcation) and because of numerical difficulties at ''high voltage'', we shall use numerical continuation techniques.

The device and a simple model

Main results during this year :

Numerical simulation results of the following models have been obtained and compared for the resonant tunnelling diode device:

hydrodynamic and drift-diffusion models with quantum corrections,

smooth quantum drift-diffusion model,

Schrödinger-Poisson model with different type of given electrostatic potential.

For the quantum hydrodynamic model, we have obtained a typical IV-curve like in the following picture, the curve coming backwards and crossing itself at some points.

This behaviour is related to the so-called hysteresis mentionned in the literature.

Then, we have studied this IV-curve as a bifurcation diagram using the software VBM. First, we have shown the absence of bifurcations; this means that the crossing points are fake bifurcations. On the next figures, one shows a part of the bifurcation diagram and a zoom of the region close to the first crossing after 0.

The red marker is on the forward (increasing V) part of the curve, the grey marker on the backward one. The qualitative behaviour of the electronic density on both parts is shown on the next pictures.

We notice that on the forward part, there is an accumulation of electrons to the right of the second barrier; on the backward part, these electrons have moved into the well, thus increasing the maximum of the electronic density.

Consequently, we introduced the maximum norm nmax of the electronic density as a new parameter and plotted I-V-nmax as a 3D diagram (the color is also given by nmax). This is shown in the next picture, from which we could conclude that the whole bifurcation diagram is connected and without bifurcation.

Publication : Ph. Caussignac, J. Descloux and A. Yamnahakki, Simulation of some quantum models for semiconductors, to be submitted to Math. Models and Methods applied Sciences.