Effective Dirac equations in honeycomb structures
A talk in the Keine Reihe series by
William Borrelli from Université Paris-Dauphine
| Abstract: | Recently, new two-dimensional materials possessing Dirac fermions low-energy excitations have been discovered, the most famous being graphene. In those materials electrons at the Fermi level have zero apparent mass and can be described using the massless Dirac equation. More generally, Schrödinger operators with honeycomb potentials generically exhibit conical intersections (the so-called Dirac points) in their dispersion bands. This leads to the appearance of Dirac as the effective operator, describing the electron dynamics around Dirac points. The large, but finite, time-scale validity of the Dirac approximation has been proved for the linear case, and for cubic nonlinearities. The latter case corresponds to the $\textit{Gross-Pitaevski equation}$, which is a fundamental model in the description of macroscopic quantum phenomena and in nonlinear optics. The cubic Dirac equation in 2D is $\textit{critical}$ for the Sobolev embedding, and this makes the existence of stationary solutions a non-trivial problem. Describing finite size samples of graphene requires to choose suitable boundary conditions for the Dirac operator. Local well-posedness for a model of electron transport in graphene has been proved, and the existence of stationary solutions has also been adressed.
In this talk I will give an overview of this results. |