Conductance and coherence lengths in disordered carbon nanotubes: Role of lattice defects and phonon vibrations

We report on a theoretical study of quantum transport in carbon nanotubes in the presence of two different sources of scattering: a static short-range random potential that simulates lattice defects, superimposed onto a long-range time-dependent perturbation that mimics the phonon-induced real-space atomic displacements. In the weak-localization regime, fluctuations of the coherent length scales are shown to be driven by band-structure features, whereas the phonon-induced delocalization effect occurs in the stronger-localization regime.

Figure 1

Top panel: elastic mean free path for a metallic (10,10) disordered nanotube as a function of energy of charge carriers and for several values of the Anderson random potential $W$ given in ${\gamma }_0$ units. Note that ${\ell }_e(W=0.5)$ has been rescaled by a factor of 8. The rescaled density of states (dotted line) is also shown. Bottom panel: corresponding conductance (main frame) with elastic disorder and electron-phonon time-dependent dephasing. The staircase gives the exact number of quantum channels, $N_{\perp }$. Inset: conductance for a larger value of the disorder potential $W=0.5{\gamma }_0$ with (dotted line) or without (bold line) LA-mode displacements.

Figure 2

Top panel: conductance for the disordered (10,10) tube with $W=0.07{\gamma }_0$, taken at evolution times $t=3500\hslash ∕{\gamma }_0$ and $t=35000\hslash ∕{\gamma }_0$ (dashed curves). The bold curve gives the classical part $G_{\text{class}}\left(E\right)=G_0{N}_{\perp }{\ell }_e\left(E\right)∕L(E,t)$, whereas the dotted curve gives the quantum correction $\delta G\left(E\right)$ $(t=35000\hslash ∕{\gamma }_0)$. Bottom panel: coherence lengths deduced from $\delta G\left(E\right)=L_{\varphi }\left(E\right)∕L\left(E\right)$ computed at $t=35000\hslash ∕{\gamma }_0$ and for several values of the disorder potential. The dotted region is not addressed since $\delta G\left(E\right)\simeq 0$ at $t=35000\hslash ∕{\gamma }_0$.

Figure 3

Main frame: coherence length as a function of energy for random potential of $W=0.1{\gamma }_0$ (bold curve), $W=0.2{\gamma }_0$ (dotted curve), and $W=0.5{\gamma }_0$ (dashed curve). Inset: corresponding ${\tau }_{\varphi }\left(E\right)$ for the same parameters. In addition to the random short-range disorder, a LA-mode modulation is introduced.

Figure 4

Main frame: conductance of disordered (10,10) nanotubes in the presence of the superimposed phonon-dephasing term due to the LO phonon mode. A drawing gives the relative atomic displacements between first neighbors. Inset: $L_{\varphi }\left(E\right)$ for the same parameters at $t=35000\hslash ∕{\gamma }_0$.