Tight-binding electronic
band structure of single-walled carbon nanotubes (SWNTs)
|
Fig. 1. πTB band structure
(dotted line) and the four-electron orthogonal TB band structure (full line). |
The
electronic band structure of carbon nanotubes is usually obtained by
zone-folding of the π-band tight-binding band (πTB) structure of
graphene. The πTB model uses a single orbital pz (perpendicular to
the graphene sheet) and therefore a single Hamiltonian matrix element.
Therefore, the πTB calculations for the particular tube structure yield
the same result as the zone-folding of graphene band structure (Fig. 1,
dotted line). A more
complete description of the band structure is obtained by considering four
electrons, i.e., four orbitals s, px, py,
and pz,
per carbon atom. A four-electron TB model, in which the overlap between
orbitals on adjacent atoms is neglected (an orthogonal TB model), already
yields differences between the zone-folded band structure and the band
structure of the particular nanotube due to σ*-π* rehybridization effects (Fig. 1, full line). In the
four-electron TB model, the s and p atomic orbitals centered
on adjacent atoms are non-orthogonal. This non-orthogonality is introduced
through a non-unit overlap matrix. This matrix, similarly to the Hamiltonian
matrix, also contains curvature effects arising from rehybridization
of the unoccupied orbitals. The non-orthogonal TB band structure can differ
much from the orthogonal TB one in the case of narrow tubes (Fig. 2). Finally,
it is a wide-spread practice in the band structure calculations to use the
nanotube structure obtained by rolling up graphene into a cylinder
("rolled-up" nanotube structure). The rolled-up structure can
differ much from the relaxed one for narrow tubes and the resulting band
structures can show differences as well (Fig. 3). In
conclusion, a semi-quantitatively correct description of the band structure
of narrow nanotubes can be obtained within a well-tuned non-orthogonal tight-binding
model for the relaxed nanotube structure [1]. Reference 1. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. |
|
Fig. 2. Orthogonal TB band
structure (dotted line) and non-orthogonal TB band structure (full line). |
|
|
Fig. 3. Non-orthogonal TB
band structure for the rolled-up tube (dotted line) and the relaxed tube (full line). |
Valentin Popov