3. Effective inter-shell coupling in DWNTs 8
3.4. Energy spectrum at the Fermi level
The details of the spectrum at the Fermi level depend on the form of both intra- and inter-shell interaction, most notably on whether the curvature of the nanotube is taken into account or not.
Among the effects of curvature in SWNTs are the rehybridization ofσ andπ bonds and varying angle betweenπ orbitals [44,45]. They result in variations in the bond length and bond angle between the lattice atoms, which can cause the opening of a diameter-dependent gap at the Fermi level in metallic SWNTs. Moreover, the band structure at the Fermi level depends on the relative
1
Figure 5. Possible couplings between momentum states in commensurate and incommensurate finite chains. Solid black lines mark the first Brillouin zones,ba
andbb are the reciprocal lattice generators on chainaandb, respectively. Filled dots stand for states in the first Brillouin zones, open dots for the states in the rest of the reciprocal space. Dashed gray rectangles mark the regions in which the momenta on different chains match underδ˜. Black dashed arrows connect states which are coupled after the translational equivalence has been taken into account.
(a)Commensurate chains with unit cell ratio 2/3. The couplings between states 1a,1b and 2a,2b are the result of direct matching of momenta. The state 3a has the same momentum as 3b, but 3b is equivalent to 1b under the translation by bb, which means that 3a and 1b are also coupled. Similar situation occurs for 4a
and 4b, which are equivalent to 1a and 2b, and so forth. In the end, all states are coupled, although the coupling may be weak. It is possible to define a common Brillouin zone, with the length 2π/3aa =2π/2ab. (b) Incommensurate chains with unit cell ratio1/√
3. The lengths of the chains cannot match—here they are chosen as three unit cells of chaina and two unit cells of chainb. The momenta in chain a are shifted with respect to those in chainb, but this mismatch is not large and the same couplings as in the case a) occur.
position of the shells, as was found to be the case in a (5,5)@(10,10) DWNT studied in [17,27].
When the (5,5)@(10,10) DWNT is in a configuration of maximum symmetry D5h [18, 46], the only effect of the inter-shell hopping is a uniform split and shift of the Fermi sub-bands, resulting in the presence of four sub-band crossings. If the symmetry of the system is lowered, four pseudogaps (the largest of the order of 0.1 eV) open in the spectrum [17].
When, as in this work, curvature effects are neglected, only a sub-band shift is observed (see figure 7)—in other words, our nanotube is always in the configuration of maximum symmetry. Due to the small size of the curvature-induced gap, we think that our model still yields a reliable description also of band features near the Fermi energy.
The presence of a uniform shift between the sub-bands of a (5,5)@(10,10) DWNT can be understood by considering just the coupling between the Fermi sub-bands. The general Hamiltonian (14) for ka and kb becomes a 4×4 matrix if all other couplings are ignored.
In the sub-lattice basis, it has the form
H(ka,kb)=
where γβ(kβ)=γ0
P3
j=1exp(ikβ·dj)=:|γβ|exp(iθβ), tab is the coupling amplitude between ka,kb from (22) and ϕνν0 =iGa·τaν−iGb·τbν0 is the phase associated with hopping between different sub-lattices. It is clear thatϕB B=ϕA B+ϕB A. When this Hamiltonian is expressed with the help of (17) in the valence/conduction basis, it becomes
H(ka,kb)=
The elements of the coupling matrix in this basis are T˜++ =tab
In order to analyze the nature of the sub-band split at the Fermi level, we need to evaluate γ (kβ), tab, ϕA B and ϕB A. Let us begin with the in-shell part. For Fermi sub-bands the angular momentum is l⊥a= −5 and l⊥b= −10. We can rewrite γ (kβ) as a function of the distance between longitudinal momentum and the Fermi point,1k=kk−kF. Both sub-bands have the same position in the reciprocal cell of the armchair nanotube, thereforeγa(1k)=γb(1k). The examination ofγ (1k)reveals that its phase has only two values:
γ (1k)= In the inter-shell part we have to perform a sum over reciprocal lattice vectors as in (15). The vectors which give the dominant contribution to the coupling areGa=b1,Gb=b1. The phases associated with hopping between sub-lattices are then
ϕA A=0, ϕA B =2π
k
kF
kF
(b)
Conduction
Valence
Conduction
Valence
k
(a)
T˜++>0 T˜++= 0 T˜−−>0 T˜−−= 0
Figure 6. Schematic plot of the band crossing at the Fermi level in a (5,5)@(10,10) DWNT. (a) The coupling between sub-bands atk<kFandk>kF. Only the conduction band is affected in the former, only the valence band in the latter case. (b) Resulting shift of the energy levels close to the Fermi point.
The parts with negative slope are split, the parts with positive slope remain degenerate.
The coupling does not mix bands and moreover affects only the conduction band fork<kFand the valence band fork>kF, as shown in figure6. In consequence, the negative slope parts of the two sub-bands−5a and−10b are split evenly on both sides of the Fermi point. The results of our calculation show that there is also a smaller uniform split of the part with positive slope (see figure7), which is due to the smaller couplings between−5a and−5b and 5b. This asymmetry is explained in [47] in real space terms as the result of different phases of the wavefunctions on sub-lattices Aand B. The wavefunctions belonging to the negative slope parts of the sub-bands have constant phase on the whole circumference, whereas the wavefunctions belonging to the unshifted parts have different phases on sub-lattices Aand B, therefore they cannot hybridize so well.