Energetics and electronic properties of twisted single-walled carbon nanotubes

We perform a systematic first-principles study of energetics and electronic properties of chiral carbon nanotubes (CNTs) in the density-functional theory. It is found that chiral CNTs possess slightly twisted ground-state geometries. Moderate-diameter CNTs show twisting-dependent electronic properties well classified by their chiral indices, while the electronic structures of small-diameter CNTs possess sizable but individually different twisting dependences, leading to metal-semiconductor transitions in some CNTs. The CNT having the widest fundamental gap is predicted to be the twisted (4,3) CNT.

Figure 1

Expanded planes of the (6,3) CNT (a) without and (b) with twisting. The red rhombus in each figure represents the helical unit cell without or with twisting. The blue rectangle defined by $\mathit{C}$ and $\mathit{T}$ in (a) is a translational unit cell without twisting, while the green parallelogram defined by $\mathit{C}$ and $\mathit{S}$ in (a) is not. The ${\theta }_T$ value in (b) (about 11${}^{\circ }$) corresponds to the special angle which gives a new relatively small translational unit cell defined by $\mathit{C}$ and ${\mathit{S}}^{\prime }$ (green rectangle). The parallelogram defined by $\mathit{C}$ and ${\mathit{T}}^{\prime }$ in (b) is not a unit cell any more.

Figure 2

Variations of (a) total energy and (b) band gap as a function of the twist angle. Circles, squares, and triangles represent the total energy of CNTs with mod$(n-m,3)=0$, 1, and 2, respectively. Curves in (a) represents the quadratic fitting, while lines in (b) are a guide to the eye. The chiral angles $\phi$ of (7,6), (6,5), (8,4), (9,2), (6,6), (8,3), and (7,5) CNTs are given in the margin of (b).

Figure 3

Twist angle dependence of the band gap of the small-diameter CNTs. Red (light gray) symbols connected with dotted lines, blue (gray) symbols connected with dashed-dotted lines, and black symbols connected with solid lines represent the band-gap values of mod$(n-m,3)=0$, 1, and 2 CNTs, respectively.

Figure 4

Band structures of (3,2) CNTs at several special twist angles obtained by using the translational-symmetry DFT computational code. The relative width of each figure corresponds to the relative length of the first Brillouin zone. The “$*$” state is the so-called “nearly-free-electron” state.