Incommensurate double-walled carbon nanotubes as one-dimensional moiré crystals

The cylindrical multishell structure is one of the prevalent atomic arrangements in nanowires. Being multishell, the well-defined atomic periodicity is hardly realized in it because the periodic units of individual shells therein generally do not match except for very few cases, posing a challenge to understanding its physical properties. Here we show that moiré patterns generated by superimposing atomic lattices of individual shells are decisive in determining its electronic structures. Double-walled carbon nanotubes, as an example, are shown to have spectacular variations in their electronic properties from metallic to semiconducting and further to insulating states depending on their moiré patterns, even when they are composed of only semiconducting nanotubes with almost similar energy gaps and diameters. Thus, aperiodic multishell nanowires can be classified into one-dimensional moiré crystals with distinct electronic structures.

Figure 1

Step-by-step operations for atomic structure mapping from graphene bilayer structure to DWNT. (a) Two parallelograms on upper and lower layers are drawn for unfolded single-walled carbon nanotubes (SWNTs) with different chiral vectors $\mathbf{C}$ and ${\mathbf{C}}^{\prime }$, respectively. (b) Rotation $\left(\mathcal{R}\right)$ and (c) subsequent contraction $\left(\mathcal{M}\right)$ of lower layer to align the axial direction of two tubes and then to match their widths. Specific examples for operations shown in (b) and (c) are displayed in (d), (e), and (f). Here we use $\mathbf{C}=8{\mathbf{a}}_1+2{\mathbf{a}}_2$ and ${\mathbf{C}}^{\prime }=14{\mathbf{a}}_1-10{\mathbf{a}}_2$ for illustration where the common in-plane primitive vectors ${\mathbf{a}}_1=a(1,0)$ and ${\mathbf{a}}_2=a(1/2,\sqrt{3}/2)$ with the lattice constant $a\approx 0.246\mathrm{nm}$ are used to label the atomic positions of both layers. In (d), thin solid lines perpendicular to chiral vectors correspond to parallelograms in (a). A usual two-dimensional moiré pattern of TBLG is shown in (e) and distorted TBLG with typical moiré lattice for DWNT is in (f). Reverse-mapping operations from a DWNT to double layer graphene nanoribbon are shown from (g) to (i). Unfolding (h) and then subsequent contraction of the lower nanoribbon (i) maps onto the modified TBLG structure [(c)and (f)] exactly.

Figure 2

After mapping operations in Fig. 1, two-dimensional atomic lattices corresponding to (35,19)@(40,24), (35,19)@(47,15), and (26,3)@(35,3) DWNTs are drawn in (a), (b), and (c), respectively. Blue arrows are chiral vectors (C) for each one. Normal (black) and distorted (red) hexagonal Brillouin zones (BZs) for three examples are presented in (d), (e), and (f), respectively. Upper (lower) Dirac points are indicated by $\mathbf{K}$ $\left(\tilde{\mathbf{K}}\right)$. Enlarged BZs near around its corner are shown in the right panels. In enlarged panels, thin slant lines are one-dimensional BZs separated by $2\pi /\left|\mathbf{C}\right|$, and ${\mathbf{G}}_{\mathrm{i}}^{\mathrm{M}}$ $\left(i=1,2\right)$ is a reciprocal lattice vector corresponding to the distorted moiré lattice.

Figure 3

Electronic energy band diagrams obtained by the atomic structure mapping and effective theory are drawn for commensurate DWNTs. (a) Energy bands of decoupled (10,10) and (15,15) SWNTs are drawn in the left panel. Red (blue) dotted lines are for inner (outer) tubes. In the right panel, the energy bands for coupled tubes, i.e., (10,10)@(15,15) DWNT, are drawn. Same band diagrams for (15,0)@(24,0) and (16,0)@(25,0) DWNTs are drawn in (b) and (c), respectively. The former case is metallic and the latter semiconducting zigzag DWNT. In all band diagrams hereafter, the origin on the $x$ axis for one-dimensional crystal momentum is taken at the $K$ point for metallic tubes and at the closest point to the $K$ point for semiconducting tubes, respectively.

Figure 4

(a) Two-dimensional distance map of $|\Delta {\mathbf{K}}_{\xi }-\xi {\mathbf{G}}_1^{\mathrm{M}}|$ by varying ${\mathbf{C}}^{\prime }$ while $\mathbf{C}$ is fixed to (35,19) SWNT. Here the darker color indicates smaller distance and semicircle lines indicate same radius of SWNTs. (b) Two-dimensional map of length of ${\mathbf{G}}_2^{\mathrm{M}}$ as a function of ${\mathbf{C}}^{\prime }$ with $\mathbf{C}$ fixed to (26,3) SWNT, where the darker color indicates smaller length.

Figure 5

Energy-momentum dispersion relations for incommensurate DWNTs are drawn with the projected extend band plot method (see Appendix pp3). (a) Energy bands of decoupled (35,19) and (40,24) SWNTs are drawn in the left panel. When the intertube coupling turns on, the dispersion of (35,19)@(40,24) DWNT is shown in the middle panel. The corresponding density of states (DOS) is drawn in the right panel. The dotted red (blue) lines are DOSs for inner (outer) nanotubes and solid black line for the coupled DWNT. Same dispersion diagrams and DOS plots for (35,19)@(47,15) and (26,3)@(35,3) DWNTs are drawn in (b) and (c), respectively. The atomic lattice structures with distorted moiré lattices and corresponding BZs for all three cases are shown in Fig. 2. For both figures, a pair of panels in the same row belongs to specific coupling conditions: from top to bottom rows, strong, weak, and localized coupling cases.

Figure 6

(a) Energy bands of decoupled (18,15) and (23,20) SWNTs are drawn in the left panel. When the intertube coupling is turned on, the dispersion of (18,15)@(23,20) DWNT is shown in the middle panel. The corresponding density of states (DOS) is drawn in the right panel. The dotted red (blue) lines are DOSs for inner (outer) nanotubes and solid black line for the coupled DWNT. Same dispersion diagrams and DOS plots for (27,3)@(36,3) and (27,3)@(35,3) DWNTs are drawn in (b) and (c), respectively.

Figure 7

Dependence of $u_0$ on interlayer spacing $d$.

Figure 8

Geometry to explain the strong-coupling condition (see the text).

Figure 9

Band structures near ${\mathbf{K}}_{+}$ valley of DWNTs with (a) (35,19)@(40,24) (strong coupling), (b) (35,19)@(47,15) (weak coupling), and (c) (26,3)@(35,3) (flat band). The solid (black) lines represent the energy bands of the DWNT in the presence of interlayer coupling, while the dotted red (blue) lines are those of inner (outer) SWNTs. We do not show the subgroups which contain no energy bands in the given range.

Figure 10

Lowest energy bands of DWNT in strong-coupling condition. The surface plot shows the energy dispersion of the modified TBLG, Eq. (D5). The black lines show the one-dimensional dispersion Eq. (D8) along the quantization line of semiconducting DWNT (see text).

Figure 11

(a) Numerically calculated band dispersions in the extended zone scheme for (35,19)@(40,24). (b) Band dispersions of the same DWNT calculated by the analytic expression [Eq. (D8)].