The strongly confined {S}chrödinger-{P}oisson system for the transport of electrons in a nanowire

Naoufel Ben Abdallah (1) , François Castella (2, 3) , Fanny Delebecque (3) , Florian Méhats (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes

We study the limit of the three-dimensional Schrödinger-Poisson system with a singular perturbation, to model a quantum electron gas that is strongly confined near an axis. For well-prepared data, which are polarized on the ground space of the transversal Hamiltonian, the resulting model is the cubic defocusing nonlinear Schrödinger equation. Our main tool is a refined analysis of the Poisson kernel when acting on strongly confined densities. In that direction, an appropriate scaling of the initial data is required, to avoid divergent integrals when the gas concentrates on the axis.

Mots clés: Schrödinger-Poisson system; asymptotic analysis; singular perturbation; quantum transport; nanowire; nanoelectronics

Domaines: Equations aux dérivées partielles [math.AP]

Naoufel Ben Abdallah, François Castella, Fanny Delebecque, Florian Méhats. The strongly confined {S}chrödinger-{P}oisson system for the transport of electrons in a nanowire. SIAM Journal on Applied Mathematics, 2009, 69 (4), pp.1162--1173. ⟨10.1137/080715950⟩. ⟨hal-00777432⟩