• graphene
• electric field
• curvature
• flexoelectricity
• electromechanical
• polarizability
• DFT
• silicon carbide

Graphene is a monolayer of carbon atoms arranged in a honeycomb lattice, and it is
the basis for the carbon based structures, from fullerene to graphite. Its bidimensionality
gives rise to unique structural and electronic properties. The sp2 hybridization of the
carbon atoms leads to a trigonal planar geometry, with sigma bonds between atoms that give
strength and robustness to the structure. The electronic structure is characterized by
the linear dispersion of the pi bands near the K point in the Brillouin zone, called Dirac
point, where the Fermi energy lies. The linear dispersion implies that electrons near the
Dirac point are described by a Dirac-like equation, hence they behave like relativistic
particles, yet at low velocities and energies. Moreover the charge carriers exhibit a high
mobility.
Due to its properties, there is a great interest on graphene based technological appli-
cation. An important aspect for this purpose is the response of graphene to an external
electric field, in particular if it is possible to tune the structure and the electronic proper-
ties. The aim of this work is to study the electromechanical properties and the electronic
response to an orthogonal electrostatic field of graphene with the specific focus on the pos-
sibility of controlling the local curvature of the graphene sheet by means external electric
fields. The interplay between electric field and curvature is related to the flexoelectricity,
namely the polarization response to a gradient of strain.
Several production methods have been developed and one promising is the grown on
a silicon carbide (SiC) substrate. The layer grown on SiC has some differences from
the free standing one. The graphene structure displays spontaneous ripples due to the
compression of the lattice parameter caused by the mismatch with the substrate. The
SiC-graphene interface shown a double periodicity: the first is the exact periodicity corresponding to a supercell of 13x13 compared to the unit cell, the second is the periodicity
of the rippled structures on graphene, corresponding to a 4sqrt(3)x4sqrt(3)R30 supercell. Due
to the spontaneous rippling, epitaxial graphene on SiC is an ideal experimental system to
study the effect of electric field on curvature. However, up to now very a few experimental
studies of graphene embedded in electric fields were published, due to the experimental
difficulties.
The aim of this Thesis work is the theoretical study of the electronic and structural properties of a graphene system exposed to electric field, as far as possible similar to the
real one, by means of Density Functional Theory (DFT) based computer calculations
and simulations. At variance with the experiment, in computer simulations the exposure
to an uniform and static electric field, even of high intensity, is possible with only minor
additional difficulties.
However, the model system reproducing the exact symmetry is quite large, especially
when one addresses it with ab initio methods, such as DFT. For this reason, massively
parallel computational resources and high performing codes were used for calculations,
and besides the 13x13 "real" model system, including also the substrate (several thou-
sands atoms), also the smaller 4sqrt(3)x4sqrt(3)3R30 one, approximately mimicking the real
rippling periodicity, was considered. In addition, the standard unit cell was also used as
test and for comparison.
Model systems were simulated at null electric field and with fields of increasing intensity and different direction. The 4sqrt(3)x4sqrt(3)R30 graphene cell was simulated at zero
compression and with a 2% compression, in order to reproduce the ripples present in the
graphene grown on SiC. For this cell also BN doped and N doped graphene structure
were simulated. The range of considered electric fields is very large, reaching the limits
of those that can currently be practically produced.
Results are reported for the change of electronic properties (band structure, charge
distribution and density of electronic states) and structure due to the electric fields.
Directly measurable observables, such as the local DOS measured by Scanning Tunneling
Microscopy (STM), were evaluated. The ionization limit is evaluated. The change of
flexoelectric properties and the possibility of manipulating curvature is quantitatively
estimated, for bare and substituted graphene. These results are of particular interest
for technological applications in energy storage and harvesting. In addition, the model
systems mimic the real experimental ones, and results could hopefully stimulate direct
measurements with which they could be straightforwardly compared.