Disorder-induced transverse delocalization in ropes of carbon nanotubes

A rope of carbon nanotubes consists in an array of parallel single wall nanotubes with nearly identical diameters. In most cases, the individual nanotubes within a rope have different helicities and 1/3 of them are metallic. In the absence of disorder within the tubes, the intertube electronic transfer is negligible because of the longitudinal wave vector mismatch between neighboring tubes of different helicities. The rope can then be considered as a number of parallel independent ballistic nanotubes. On the other hand, the presence of disorder within the tubes favors the intertube electronic transfer. This is first shown by using a very simple model wherein disorder is perturbatively treated as inspired by the work of Maarouf et al. [Phys. Rev. B 61, 11156 (2000)]. We then present numerical simulations of a tight binding model of a rope. A disorder induced transverse delocalization shows up as an increase (by typically 1 order of magnitude of the sensitivity to the transverse boundary conditions in the presence of small disorder). This is accompanied by an increase in the longitudinal localization length. The implications on the nature of electronic transport within a rope of carbon nanotubes are discussed.

Figure 1

Top: Brillouin zones of two metallic tubes of different helicities. Bottom: Dispersion relation as function of the longitudinal component of the wave vector of the lowest energy bands for these two tubes for which $k_{a\parallel }^0\ne k_{b\parallel }^0$ at the Fermi energy illustrating the second order coupling, $⟨{\Psi }_a|H_{\perp }|{\Psi }_b⟩=2t_{\perp }{\left(a,b\right)}^2/\delta ϵ\left(a,b\right)$, via the higher energy state at energy ${ϵ}_F+\delta ϵ\left(a,b\right)/2$ for which $k_{a\parallel }=k_{b\parallel }$.

Figure 2

Histogram of possible values of the coefficients $t_{\perp }\left(a,b\right)$ and $\frac{t_{\perp }\left(a,b\right)}{\Delta ϵ\left(a,b\right)}$ for all couple of helicities $\left(a,b\right)$ of metallic tubes with a diameter between 1.2 and 1.5 nm. The average yields $3e_{\text{bot}}=0.09$.

Figure 3

Different transport regimes depending on the amplitude of disorder compared to a rope of length $L=1{\mu }_m$ as discussed in the text. Region 1— 1D ballistic; region 2—3D ballistic; and region 3—3D diffusive. The arrows indicate the direction of increasing time. In region 2, the transport corresponds to a 3D quasiballistic regime on the whole length of the tube. On the other hand, in region 3, the transport is 3D ballistic at short time scales but becomes diffusive before reaching the end of the tube.

Figure 4

Left: Schematic of a rope of nanotubes by $N_m$ slightly coupled, tight binding chains of different longitudinal energies $t_1\ne t_2\ne t_3\cdots$. Right: The chains lie on a cylinder threaded by a fictitious Aharonov–Bohm flux, which enables the investigation of transverse transport.

Figure 5

Transverse delocalization by disorder: evolution of $\delta {ϵ}_{\text{perp}}$ with the amplitude of disorder for 10 coupled chains of 100 sites, $W$ corresponds either to on site disorder (black dots) or to bond disorder (open circles). The situation of identical values of $t_n$ corresponding to $\delta t_{\parallel }=0$ (upper curves) exhibits expected disorder induced localization, whereas the situation with different values of $t_n$ where $\delta t_{\parallel }/t_{\parallel }=0.2$ shows a regime of disorder induced transverse delocalization. The transverse hopping energy is chosen to be $t_{\perp }=0.06t_{\parallel }$. $\nu$ is the density of states (inverse nearest level spacing).

Figure 6

(Color online) Results obtained by diagonalization of a system of $N_m=10$ chains of $N_s=1013$ sites. Continuous curves: Average energy difference $\delta {ϵ}_{\text{perp}}$ as a function of on site disorder amplitude. From top to bottom: $t_{\perp }/t_{\parallel }=0.1$, 0.07, 0.03, 0.02, and 0.01. Note the dependence in $t_{\perp }^{10}$ at low disorder. Dashed curve: Power law fits approximation of those curves with 2.5 and $-9$ exponents (left and right parts).

Figure 7

(Color online) Results obtained by diagonalization of a system of $N_m=10$ chains of $N_s=1013$ sites. Average typical energy difference $\delta {\epsilon }_{\text{par}}$ between the symmetric and/or antisymmetric boundary conditions in the longitudinal direction as a function of $t_{\perp }/t_{\parallel }$. From top to bottom $W/t_{\parallel }=0.55$, 0.9, 1.1, 1.3, and 1.5.

Figure 8

Size distribution of the number of tubes $n$ in a percolating cluster of metallic tubes for ropes containing $N=100$ and $N=400$ tubes among which 1/3 on average are metallic.