Effective-mass theory of collapsed carbon nanotubes

Band structure is theoretically studied in partially flattened carbon nanotubes within an effective-mass scheme. Effects of interwall interactions are shown to be important in nonchiral nanotubes such as zigzag and armchair and can essentially be neglected in chiral nanotubes except in the close vicinity of nonchiral tubes. In fact, interwall interactions significantly modify states depending on relative displacement in the flattened region in nonchiral tubes and can convert semiconducting tubes into metallic and vice versa. They diminish rapidly when the chiral angle deviates from that of the zigzag or armchair tube, although the decay is slower in the vicinity of armchair tubes.

Figure 1

(a) A schematic illustration of a collapsed carbon nanotube and (b) its development map on a graphene sheet. In panel (b), the coordinate system $(x^{\prime },y^{\prime })$ and origin $\mathrm{O}{}^{\prime }$ are fixed onto the graphene sheet and the coordinate system $(x,y)$ and origin O vary depending on the structure of the nanotube. The flattened region is denoted by the shaded area.

Figure 2

The lattice structure of graphene (a) and the first Brillouin zone (b). The primitive translation vectors are denoted by $\mathbf{a}$ and $\mathbf{b} \left(\left|\mathbf{a}\right|=\left|\mathbf{b}\right|=a\right)$ and the vectors connecting nearest-neighbor atoms are denoted by ${\vec{\tau }}_1,{\vec{\tau }}_2$, and ${\vec{\tau }}_3$ in panel (a). The reciprocal lattice vectors are denoted by ${\mathbf{a}}^{*}$ and ${\mathbf{b}}^{*}$, and $\mathbf{K}=(2\pi /a)(1/3,1/\sqrt{3})$ and ${\mathbf{K}}^{\prime }=(2\pi /a)(2/3,0)$ in panel (b).

Figure 3

Schematic illustration of energy bands for (a) an armchair tube $k_{\mu }\ne 0$ and (b) metallic and (c) semiconducting zigzag nanotubes with $k_{\mu }=0$.

Figure 4

Some examples of the structure of the double-wall region of a flattened zigzag nanotube. The $A$ and $B$ sites remain the same, but the $K$ and $K^{\prime }$ points are exchanged between the upper (red) region and the lower (green) region. (a) $\zeta =na$ corresponding to an $AA$-stacked bilayer, (b) $na<\zeta (n+\frac{1}{4})a$, (c) $\zeta =(n+\frac{1}{4})a$, and (d) $\zeta =(n+\frac{1}{2})a$ corresponding to another $AA$-stacked bilayer.

Figure 5

Some examples of the structure of the double-wall region of a flattened armchair nanotube. The $K$ and $K^{\prime }$ points remain the same, but $A$ and $B$ sites are exchanged between the upper (red) region and the lower (green) region. (a) $2\zeta =3nb$ corresponding to the $AB$-stacked bilayer, (b) $3nb<2\zeta <(3n+\frac{1}{2})b$, (c) $2\zeta =(3n+\frac{1}{2})b$ corresponding to $AB$ stacking, and (d) $2\zeta =(3n+1)b$ corresponding to $AA$ stacking, with distance $b=a/\sqrt{3}$ between neighboring carbon atoms within the plane.

Figure 6

Some examples of the potential amplitude in the dominant-term approximation as the function of chiral angle $\eta$ with family index $f=144$ defined in Eq. (74), i.e., $(n_a,n_b)=(72,0)$, (73,2), $\cdots$, (96,48). The average of the absolute values of each element of $\widehat{V}$ are shown for ${\widehat{V}}^{KK}$ and ${\widehat{V}}^{K^{\prime }{K}^{\prime }}$ by (blue) solid lines and for ${\widehat{V}}^{K{K}^{\prime }}$ and ${\widehat{V}}^{K^{\prime }K}$ by (red) dotted lines. The size of the one-dimensional unit cell $T$ is also shown in units of $L$ by (green) dashed lines. We chose relative displacement $\zeta /a=0$, 1/4, and $1/\sqrt{3}=0.5774...$ defined in Fig. 1 for three panels and a parameter ${\Delta }_{\mathrm{edge}}$ is defined in Eq. (55).

Figure 7

Calculated band structure of collapsed tubes with family index $f=144$ having zigzag structure (a), its neighboring structures (b), armchair structure (f), and its neighboring structures (e), (d), and (c). Parameters are defined in Eqs. (13), (16), and (20) and are illustrated in Fig. 1.

Figure 8

Calculated band structure of collapsed tubes with $f=144$ having zigzag and armchair structures. (a) and (b) $\zeta /a=1/4$. (c) and (d) $\zeta /a=1/\sqrt{3}$.

Figure 9

Calculated band structure of collapsed zigzag tubes with $f=144$ and $\eta =0$. With the increase of $L_F/L$, the band structure gradually approaches that of an $AA$-stacked bilayer with appropriately discretized wave vectors in the circumference direction.

Figure 10

Calculated band structure of collapsed armchair tubes with $f=144$ and $\eta =\pi /6$. With the increase of $L_F/L$, the band structure gradually approaches that of an $AB$-stacked bilayer with appropriately discretized wave vectors in the circumference direction. The tube becomes semiconducting due to interwall interaction in such a way that the bottom of the conduction band is fixed at zero energy, while the top of the valence band is lowered, forming a band gap.

Figure 11

Calculated band structure of collapsed tubes with $f=142$ (semiconducting) having (a) zigzag structure and (b) its neighboring structure, and (c) and (d) near-armchair structure. The zigzag tube turns into metallic from semiconducting due to collapse as in panel (a).