13-04-2013, 04:43 PM
Phonon Spectra of Graphene
ABSTRACT
We use a Born model to calculate the phonon dispersion of graphene by accounting for stretching
( s) and bending ( ) interactions between nearest neighbors. Our model describes four in-plane
vibrational modes to whose dispersion relations we fit experimental lattice mode frequencies, yielding
force constants s = 445N/m and = 102N/m. Our model also reasonably accounts for graphene’s
macroscopic properties, particularly sound speeds and elastic constants.
INTRODUCTION
The lattice dynamical properties of graphene form the
basis of understanding the vibrational spectra of carbonbased
allotropes of various geometries, such as graphite
or carbon nanotubes. We can use an analytical descriptions
of micromechanical behavior to better understand
the acoustic and optical properties of these materials.
In the following, we calculate the in-plane vibrational
spectrum of graphene and its contributions to macroscopic
elastic and thermodynamic quantities. We first
discuss the force parameters of our Born model and derive
a general potential to calculate the dynamical matrix
of a primitive cell. Using our dispersion relations at
high-symmetry points, we can determine the vibrational
density of states, in-plane sound velocity, and elastic constants.
We will briefly touch upon weak out-of-plane
vibrational modes and its contributions to graphene’s
macroscopic properties.
Nearest Neighbor Couplings
If you use the only nearest neighbor cou-
plings, how many force constants will your
model require for your material? How large
will the dynamical matrix be? What if you
used nearest neighbor and next-nearest neigh-
bor couplings?
Two force constants are needed to model graphene
whether a nearest neighbor coupling or nearest and next
nearest neighbor model is used, as long as the model is
two-dimensional. As mentioned before, these are αs and
α , representing restoring forces due to bond stretching
and bond bending respectively. As graphene is a two
dimensional material with a two atom basis, the dynamical
matrix will be four-by-four. A third force constant
αz is required to account for out-of-plane phonon modes.
The size of the dynamical matrix is not affected by the
number of force constants used in our model, however if
a third dimension is added to account for out of plane
vibrations then the dynamical matrix will be six-by-six.
ABSTRACT
We use a Born model to calculate the phonon dispersion of graphene by accounting for stretching
( s) and bending ( ) interactions between nearest neighbors. Our model describes four in-plane
vibrational modes to whose dispersion relations we fit experimental lattice mode frequencies, yielding
force constants s = 445N/m and = 102N/m. Our model also reasonably accounts for graphene’s
macroscopic properties, particularly sound speeds and elastic constants.
INTRODUCTION
The lattice dynamical properties of graphene form the
basis of understanding the vibrational spectra of carbonbased
allotropes of various geometries, such as graphite
or carbon nanotubes. We can use an analytical descriptions
of micromechanical behavior to better understand
the acoustic and optical properties of these materials.
In the following, we calculate the in-plane vibrational
spectrum of graphene and its contributions to macroscopic
elastic and thermodynamic quantities. We first
discuss the force parameters of our Born model and derive
a general potential to calculate the dynamical matrix
of a primitive cell. Using our dispersion relations at
high-symmetry points, we can determine the vibrational
density of states, in-plane sound velocity, and elastic constants.
We will briefly touch upon weak out-of-plane
vibrational modes and its contributions to graphene’s
macroscopic properties.
Nearest Neighbor Couplings
If you use the only nearest neighbor cou-
plings, how many force constants will your
model require for your material? How large
will the dynamical matrix be? What if you
used nearest neighbor and next-nearest neigh-
bor couplings?
Two force constants are needed to model graphene
whether a nearest neighbor coupling or nearest and next
nearest neighbor model is used, as long as the model is
two-dimensional. As mentioned before, these are αs and
α , representing restoring forces due to bond stretching
and bond bending respectively. As graphene is a two
dimensional material with a two atom basis, the dynamical
matrix will be four-by-four. A third force constant
αz is required to account for out-of-plane phonon modes.
The size of the dynamical matrix is not affected by the
number of force constants used in our model, however if
a third dimension is added to account for out of plane
vibrations then the dynamical matrix will be six-by-six.