Lattice-dynamical model
for bundles of single-walled carbon nanotubes (SWNTs) and isolated multiwalled carbon nanotubes (MWNTs)
The phonon
dispersion of bundles of SWNTs and isolated MWNTs cannot take advantage of the
symmetry of the separate single-wall layers. Two approaches have been followed.
The first one considers the atoms of these structures and their intralayer
interactions described by valence force-fields and interlayer interactions
described by pair potentials. The dynamical problem is made tractable by
imposing translational symmetry of the structures, however thus limiting the
layer types to either armchair, or zigzag ones [1,2,3].
The
dynamical model is the standard one for one-dimensional systems with
translational symmetry. Let us label the κth atom of the lth unit cell of the structure by the
pair of integers (lκ),
where κ = 1, 2,…, N, N is the number of atoms in the unit
cell. Then, the position vector of the (lκ)th
atom is x(lκ) = R(lκ) + u(lκ), where R(lκ) is the equilibrium
position vector and u(lκ) is
the atomic displacement relative to the latter.
For small
displacements u(lκ) of the
atoms from their equilibrium positions, the harmonic approximation can be used
for the potential energy of the structure. The resulting Lagrangian
is quadratic in the atomic displacements and velocities. The equation of motion
is then readily derived in the form
where 
are the force constants.
The
solution of the equation of motion is
where T is the translation period.
Substituting this expression in the equation of motion, one obtains the matrix
eigenvalue problem
where Dακ,βκ’ is
the dynamical matrix. The solutions of this system of equations are the phonon
eigenvalues 
and eigenvectors
.
In the second
approach, the layers are considered as elastic continuum cylinders with
circular cross-section interacting by means of pair potentials [2,3,4,5]. The predictions of this model are limited to the
breathing-like phonon modes of the structures.
The
equation of motion is constructed in the same way as in the atomistic approach,
the only difference being that the Lagrangian of the
system depends on the radial expansions of the layers rather than on the atomic
displacements.
References
1.
V. N.
Popov and L. Henrard, Phys. Rev. B
63 (2001) 233407.
2.
L. Henrard,
V. N. Popov, and A. Rubio, Phys.
Rev. B 64 (2001) 205403
3.
V. N.
Popov and L. Henrard, Phys. Rev. B
65 (2002) 235415.
4.
R. Pfeiffer, Ch. Kramberger,
F. Simon, H. Kuzmany, V. N. Popov, and H. Kataura, Eur, Phys. J B 42 (2004) 345.
5.
R. Pfeiffer, F. Simon, H. Kuzmany, and V. N. Popov, Phys. Rev. B 72 (2005) 161404(R).
Valentin Popov