Propagating, evanescent, and localized states in carbon nanotube–graphene junctions

We study the electronic structure of the junctions between a single graphene layer and carbon nanotubes, using a tight-binding model and the continuum theory based on Dirac fermion fields. The latter provides a unified description of different lattice structures with curvature, which is always localized at six heptagonal carbon rings around each junction. When these are evenly spaced, we find that it is possible to curve the planar lattice into armchair $\left(6n,6n\right)$ as well as zigzag $\left(6n,0\right)$ nanotubes. We show that the junctions fall into two different classes, regarding the low-energy electronic behavior. One of them, constituted by the junctions made of the armchair nanotubes and the zigzag $\left(6n,0\right)$ geometries when $n$ is a multiple of 3, is characterized by the presence of two quasibound states at the Fermi level, which are absent for the rest of the zigzag nanotubes. These states, localized at the junction, are shown to arise from the effective gauge flux induced by the heptagonal carbon rings, which has a direct reflection in the local density of states around the junction. Furthermore, we also analyze the band structure of the arrays of junctions, finding out that they can also be classified into two different groups according to the low-energy behavior. In this regard, the arrays made of armchair and $\left(6n,0\right)$ nanotubes with $n$ equal to a multiple of 3 are characterized by the presence of a series of flat bands, whose number grows with the length of the nanotubes. We show that such flat bands have their origin in the formation of states confined to the nanotubes, with little overlap in the region between the junctions. This is explained in the continuum theory from the possibility of forming standing waves in the mentioned nanotube geometries, as a superposition of modes with opposite momenta and the same quantum numbers under the $C_{6v}$ symmetry of the junction.

Figure 1

(Color online) (a) Sketch of carbon nanotube attached to a graphene plane. The building blocks of the structure are triangles which include many carbon atoms. The orientation of the bonds of the honeycomb lattice in the basic triangle may give rise to (b) armchair or (c) zigzag nanotubes. The threefold coordination of the carbon atoms induces the existence of six heptagonal rings, schematically shown in blue at the corners of the hexagonal prism contacting the plane in (a).

Figure 2

(Color online) Sequence of local densities of states for a circular ring of atoms at the end of a (54,0) nanotube close to the junction, for the different sectors corresponding to eigenvalue $q$ under $\pi /3$ rotation equal to (a) 1, (b) $e^{\pm i\pi /3}$, (c) $e^{\pm 2i\pi /3}$, and (d) $-1$. Energy is measured in units of the transfer integral $t$.

Figure 3

(Color online) Similar sequence as in Fig. 2 for a (48,0) nanotube close to the junction, for (a) $q=1$, (b) $e^{\pm i\pi /3}$, (c) $e^{\pm 2i\pi /3}$, and (d) $-1$.

Figure 4

(Color online) Similar sequence as in Fig. 2 for a (12,12) nanotube close to the junction, for (a) $q=1$, (b) $e^{\pm i\pi /3}$, (c) $e^{\pm 2i\pi /3}$, and (d) $-1$.

Figure 5

(Color online) Sequence of local densities of states for a ring of atoms at the end of the nanotube close to the junction, for the different geometries (a) (54,0), (b) (18,0), (c) (48,0), and (d) (12,12). Energy is measured in units of the transfer integral $t$.

Figure 6

(Color online) Density of states at the bulk of the nanotube (blue, hexagons), and at the graphene plane (red, diamonds), obtained using the numerical method discussed in the text.

Figure 7

(Color online) Density of states at a graphene plane attached to a nanotube. The radius of the nanotube is $N_0=100$. The density of states is shown at position $n=10$ from the junction (red, diamonds), and $n=100$ (blue, hexagons).

Figure 8

(Color online) Low-energy bands of arrays of junctions with nanotube geometries (a) (12,12), (b) (12,0), (c) (18,0), and (d) (24,0). In all the cases the unit cell of the array is of the type shown in Fig. 1, with a side of the hexagon in the basal plane equivalent to ten carbon rings of the graphene sheet, and a nanotube height equivalent to ten unit cells of the nanotube for (a), (b), (c), and 20 nanotube unit cells for (d).