Carbon atoms possess four valence electrons. Three are used for molecular bonding with nearest neighbours atoms, and the last one is a conduction electron. Among its many allotropes, arrangement of 2-dimensional carbon atoms in an hexagonal lattice cristals, called Graphene, has revealed to be a remarkable material, both from fundamental and technological points of view, due to its electronic properties [7, 9, 13].
One of the technological challenge is to turn this semimetal into a semiconductor. Remind that the energy gap is the defining concept for semiconductor materials and essential for controlling the conductivity by electronic means. The possibility of predicting and being able to tune the size of the gap has very attractive applications in novel technologies, like building graphene-based chips.
Five years ago, a group of physicists [15, 16] proposed a “simple” technological method in order to turn graphene into a semiconductor by opening a gap at the Fermi level. It consisted in producing periodic small but macroscopic perforations in the graphene sample. These experiments have since been theoretically analyzed through extensive numerical studies based on a continuous model of a two dimensional massless Dirac operator that models the dynamics of the low-energy charge carriers for the tightbinding model for graphene on the honeycomb lattice. A rigorous mathematical result on this problem has been the subject of an article by the principal investigators (see [1]) with most progress done during a REB program at CIRM in 2015.
The mathematical analysis of Dirac operators, and more recently of two dimensional massless Dirac operators, has long been the subject of intensive studies (see the textbook [18] and references therein), like e.g. spectral properties of Hamiltonian operators, or the study of propagation of the solutions of the associated evolution equation, yielding spectral and dynamical properties for various two-dimensional Dirac operators [10, 14, 6, 12, 11, 2, 3, 4].
Our research program is dedicated to the proof of existence of a gap in the spectrum of the hamiltonian for a Graphene model with periodic perturbations. There are two possible approaches. The first consists of periodically perturbating the initial discrete model, and the second is to consider periodic perturbation of the 2-dimensional Dirac operator (model on L2 (R2 , C2)) associated to Pristine Graphene. Both cases yield to the study of Dirac operators.

We shall focus on the following two items during the REB corresponding to the two above mentionned approaches:
a) Bonds sparse removal – discrete model. 
b) Periodically perturbed Dirac Hamiltonians.

Participants

Jean-Marie Barbaroux (Université de Toulon)
Horia Cornean (Aalborg University, Denmark)
Edgardo Stockmeyer (Universidad Católica de Chile, Santiago)

Sponsor