Systèmes et applications des technologies de
l’information et de l’énergie
Master 2 internship proposition 2016-2017
Laboratory: SATIE (TEMA group)
Group leader: Zoubir KHATIR
Address: 25 allée des marronniers - F - 78000 Versailles.
Internship supervisors: Fadi ZAKI – Zoubir KHATIR
Resolution of the 2-D Schrödinger-Poisson coupled equations
for hetero-structure by Finite Element Method
Power transistors based on hetero-structure have attracted recently significant interest, benefit from their
excellent properties for power switching applications. Despite the existence of many advantages of these
devices, there are some limiting phenomena that hinder their proper functioning. However, to understand the
electrical behavior of these devices and especially the effect of those limiting phenomena, physics-based
modeling is essential. This type of modeling is based on semiconductor equations which mainly composed of the
Poisson-Schrodinger and the first three moments of the Boltzmann equation. Whereas physics-based analytical
modeling has reached its limits, the interest begin to grow in the direction of numerical modeling or the finite
element method can address this need. For this internship, the candidate will study only the coupled equations
Poisson-Schrodinger. The numerical solution of these two equations will enable us to calculate the local
distribution of field in 2-D and to understand some physical degradation mechanism taking into account the
quantum effects.
Actually, this numerical modeling presents many challenges, from a structural standpoint, the device is
composed of various types of materials (Aluminum Gallium Nitride (AlGaN), Gallium Nitride (GaN) and
Silicon (Si),...) with various scales of thickness (few nm to several μm), from a physical standpoint, we posed
those equations to take into consideration the physical limiting phenomena which increases the system
nonlinearity. A brief idea of the basic equations and the simplified domain are presented as following:
Non-linear Poisson equation:
∇. (𝜅(𝑥, 𝑦)∇𝜙(𝑥, 𝑦)) = 𝑞. 𝜌(𝜙, 𝜓)
Schrödinger equation for the electrons to the hetero-interface:
ℏ2
− ∇. (𝛽̃ (𝑥, 𝑦). ∇𝜓(𝑥, 𝑦)) + [𝜙(𝑥, 𝑦) + Δ𝐸𝐶 (𝑥, 𝑦)]𝜓(𝑥, 𝑦) = 𝐸𝜓(𝑥, 𝑦)
2
(1)
(2)
Where 𝜙 𝑖𝑠 𝑡ℎ𝑒 potential distribution, 𝜌(𝜙) the density of charges, 𝜅 the dielectric constant, 𝜓 the wave function, 𝛽̃ the effective mass
coefficients, ℏ the Planck's constant, Δ𝐸𝐶 the pseudopotential energy due to the band offset at the hetero-interface and 𝐸 the Energy
The intern may begin to study and analyze the variational formulation and the discrete approximation
scheme for the resolution of these two equation, which have been very well studied in the literature. Due to the
high non-linearity, an iterative method may be used for the resolution of this system. Then, a suitable way for the
discretization of the domain and the implementation may be proposed by the intern.
Required candidate:
The candidate is expected to have a strong base in solving partial differential
equation by finite elements and functional analysis. A good knowledge of an
implementation tool of the method e.g. Matlab/Free Fem will be essential.
Remuneration of the internship: Yes
Internship duration: 5 or 6 months, (preferably beginning in January)
y
To apply, please send your CV to the following addresses:
fadi.zaki@ifsttar.fr & zoubir.khatir@ifsttar.fr
x
Simplified domain
SATIE : ENS Cachan, 61 Av. du président Wilson, 94235 CACHAN CEDEX | secretariat@satie.ens-cachan.fr | www.satie.ens-cachan.fr
Groupe TEMA : IFSTTAR, Site de Versailles, 25 allée des marronniers, 78000 VERSAILLES-Satory