Effects of disorder and momentum relaxation on the intertube transport of incommensurate carbon nanotube ropes and multiwall nanotubes

We study theoretically the electrical transport between aligned carbon nanotubes in nanotube ropes, and between shells in multiwall carbon nanotubes. We focus on transport between two metallic nanotubes (or shells) of different chiralities with mismatched Fermi momenta and incommensurate periodicities. We perform numerical calculations of the transport properties of such systems within a tight-binding formalism. For clean (disorder-free) nanotubes the intertube transport is strongly suppressed as a result of momentum conservation. For clean nanotubes, the intertube transport is typically dominated by the loss of momentum conservation at the contacts. We discuss in detail the effects of disorder, which also breaks momentum conservation, and calculate the effects of localized scatterers of various types. We show that physically relevant disorder potentials lead to very dramatic enhancements of the intertube conductance. We show that recent experimental measurements of the intershell transport in multiwall nanotubes are consistent with our theoretical results for a model of short-ranged correlated disorder.

Figure 1

(Color online) Schematic diagrams of the lattice structures (the tube axes run horizontally) and band structures of the metallic bands for armchair and zigzag nanotubes. The repeat distances along the tube axes differ by a factor of $\sqrt{3}$, so the tubes are incommensurate. Intertube tunneling will be suppressed for Fermi energies close to $E=0$, owing to the lack of energy-momentum conserving processes.

Figure 2

Geometry of the system that we study, illustrating the labeling of the leads and the length $L$ of the central section where intertube tunneling can occur. The length $L_N$, which includes the leads, is used for normalization and is assumed large.

Figure 3

(Color online) Two-terminal conductance of a $(5,5)$ armchair nanotube with a single vacancy (solid line) and a (9,0) nanotube with a single vacancy (dashed line). The tube length $(L=31a)$ used in these calculations is sufficiently long that the results are characteristic of the bulk behavior.

Figure 4

Intertube conductance between $(10,10)$ and $(18,0)$ tubes, in the rope geometry with an intertube distance of $b=3.4\mathrm{\AA }$. The total length of the carbon nanotubes is $L=87a$, and intertube tunneling is introduced smoothly from zero over a length $L_c=2a$. The peaks at $E∕t=\pm 0.25$ correspond to energy-momentum conserving processes described in the text.

Figure 5

(Color online) Length dependence of the intertube conductance for $(10,10)$ and $(18,0)$ nanotube in a rope geometry a distance $b=3.4\mathrm{\AA }$ apart. Results are shown as a function of the total length, at $E=-E_0,0,+E_0$ (solid, dotted, dashed, respectively). The main figure, which plots $G∕(La{)}^2$, indicates clearly that $G\sim L^2$ for $E=E_0$. The inset, which plots $G$, indicates that $G\sim L^2$ also for $E=-E_0$ but not for $E=0$. These results are consistent with momentum conserving tunneling processes at $E=\pm E_0$ but none at $E=0$. (A contact length of $L_c=2a$ has been chosen to minimize effects of momentum relaxation at the contacts.)

Figure 6

(Color online) Mean intertube conductance at $E=0$ between $(10,10)$ and $(18,0)$ carbon nanotubes in the rope geometry a distance $b=3.4\mathrm{\AA }$ apart, as a function of the contact length $L_c$. The strong dependence on $L_c$ indicates that the conductance is dominated by momentum relaxation at the contacts. The fit (dashed line) is to the functional form of Eq. (20).

Figure 7

(Color online) Mean intertube conductance at $E=0$ between $(7,7)$ and $(21,0)$ carbon nanotubes in a multiwall geometry as a function of the contact length $L_c$. The strong dependence on $L_c$ indicates that the conductance is dominated by momentum relaxation at the contacts. The fit (dashed line) is to the functional form of Eq. (20).

Figure 8

Schematic picture of a telescopic junction.

Figure 9

Conductance of a telescopic junction between two incommensurate $(6,6)$ to $(18,0)$ carbon nanotubes, as a function of the overlap length, ${\ell }_0$. By our classification of the rotational symmetry the coupling is “strong.” The two tubes are strongly coupled even for small overlap lengths.

Figure 10

Conductance of incommensurate $(10,10)$ to $(9,0)$ carbon nanotubes for a telescopic junction, as a function of the overlap length ${\ell }_0$. By our classification of the rotational symmetry the coupling is “intermediate.” The two tubes remain weakly coupled even for large overlap lengths.

Figure 11

(Color online) Intertube conductance between a $(10,10)$ armchair and $(18,0)$ zigzag in the rope geometry a distance $b=3.4\mathrm{\AA }$ apart. Results are shown both for clean carbon nanotubes (dashed line) and with a single vacancy placed on a site halfway along the armchair tube (solid line). The inset shows the dependence of the conductance at $E=0$ on the location of the vacancy at the atomic sites around the circumference of the armchair tube.

Figure 12

Intershell conductances for DWNTs in the presence of a single vacancy (placed on the inner tube), for the three types of coupling determined by the angular symmetry of Sec. 2D. The solid line is a $(3,3)$ to $(15,0)$ system (zero coupling); the dashed line is a $(5,5)$ to $(18,0)$ system (intermediate coupling); and the dotted line is a $(7,7)$ to $(21,0)$ (strong coupling).

Figure 13

(Color online) The dimensionless measures of backscattering, $R_s$ (solid line) and intertube scattering, $I_s$ (dashed line) for a (13,13) to (12,0) DWNT with a single potential defect, of varying strengths $V_0$ and correlation length $\xi$ [see Eq. (24)]. The total length of the system is $L=87a$ with tunneling introduced smoothly via a contact length of $L_c=2a$, so boundary effects are minimal and the results are characteristic of a single defect in the bulk.

Figure 14

(Color online) The dimensionless measures of backscattering, $R_s$ (solid lines) and intertube scattering, $I_s$ (dashed lines) for a (13,13) to (12,0) DWNT with a single dent in the outer shell as a function of the displacement $d_0$ and correlation length $\xi$ [see Eq. (25)]. Results are shown for the set of correlation lengths $\xi =0.00$, 1.76, 3.53, $5.29\mathrm{\AA }$, which may be assigned to the graphs by noting that both $R_s$ and $I_s$ increase with increasing $\xi$ for $d_0\sim 1\mathrm{\AA }$.

Figure 15

Combining two identical scattering sections.