Tight-binding model of the
lattice dynamics of single-walled carbon nanotubes (SWNTs)
The phenomenological models of the
lattice dynamics of SWNTs do not account correctly for the electronic response
to the atomic motion. The effect of the electrons is either neglected or a
simple isotropic model of the deformation of the electronic density is adopted.
On the other hand, the deformation of the electronic density upon distortion of
the crystal lattice by a phonon can be highly anisotropic. The simplest way to
include the impact of the electrons on the atomic motion is by explicitly
including the electron-lattice interaction within a tight-binding model of the
electronic band structure. In particular, the lowest-order change of the energy
of the lattice distorted by a phonon with a wavevector q is determined by second-order electron-lattice interaction in
first-order perturbation theory and first-order electron-lattice interaction in
second-order perturbation theory. The expression for the energy change within
the symmetry-adapted non-orthogonal tight-binding model is too involved [1].
However, it is sufficient to provide here the expression for 3d periodic
systems and an orthogonal tight-binding model hoping that the reader could
easily extend it for the more complicated case of SWNTs.
The second-order change of the
nanotube energy can be written as
.
The first term in the rhs is the
band-structure contribution and the second term is the repulsive contribution.
We remind that in the tight-binding models one usually needs additional
repulsive potentials between the atoms because the band-structure energy cannot
ensure structural stability by itself. The band-structure energy change is
further given by the expression
.
Here H is the Hamiltonian matrix, 
and 
are solutions to the
matrix eigenvalue problem 
. 
and 
are the phonon-induced
first- and second-order changes of H,
respectively.
Note that in metallic tubes the
denominator in the second term can become very small. Then the second term can
become very large resulting in Kohn anomalies of the phonon dispersion and, at
low temperatures, in Peierls instability of the lattice.
The energy change can be written
also as
,
where 
is the phonon
eigenvector and 
is the dynamical
matrix. The comparison between the tight-binding expression for the energy
change and the one above allows one to determine the dynamical matrix and to
further solve the dynamical problem for any phonon wavevector, practically for
any presently synthesized SWNT.
References
1.
V. N.
Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407.
2.
V. N.
Popov and Ph. Lambin, Phys. Rev. B 82 (2010) 045406.
Valentin Popov
May 16, 2010