Integr. Equ. Oper. Theory 89 (2017), 631–646
Published online November 16, 2017
Springer International Publishing AG,
part of Springer Nature 2017
and Operator Theory
Spectral Gaps in Graphene Antidot Lattices
Jean-Marie Barbaroux, Horia Cornean and
Abstract. We consider the gap creation problem in an antidot graphene
lattice, i.e. a sheet of graphene with periodically distributed obstacles.
We prove several spectral results concerning the size of the gap and
its dependence on diﬀerent natural parameters related to the antidot
Mathematics Subject Classiﬁcation. Primary 81Q10; Secondary 46N50,
81Q37, 34L10, 47A10.
Graphene, a two-dimensional material made of carbon atoms arranged in a
honeycomb structure, has risen a lot of attention due to its many unique
properties. Remarkably, charge carriers close to the Fermi energy behave
as massless Dirac fermions. This is due to its energy band structure which
exhibits two bands crossing at the Fermi level making graphene a gapless
semimetal . Many eﬀorts have been carried out for the possibility of tuning
an energy gap in graphene .
The main physical motivation of our work is related to the so-called
antidot graphene lattice , which consists of a regular sheet of graphene
having a periodic array of obstacles well separated from each other. These
obstacles can be thought, for instance, as actual holes in the graphene layer
. More generally, substrate induced obstacles  or those created by dop-
ing or by mechanical defects have also been considered in the literature (see
e.g.,  and references therein). It has been observed both experimentally
and numerically that such an array causes a band gap to open up around the
Fermi level, turning graphene from a semimetal into a gapped semiconductor
(see e.g., [3,9], and ).
In  and  there are given several proposals concerning the modelling
of this phenomenon. In one of these proposals the authors replace the usual
Edgardo Stockmeyer has been partially funded by Fondecyt (Chile) project # 114–1008
and Iniciativa Cient´ıﬁca Milenio (Chile) through the Nucleus RC120002.