Electronic transport in carbon nanotubes: Diffusive and localized regimes

A fully self-consistent analysis of Boltzmann’s equations for electrons and phonons is used to study how the resistance of single-walled carbon nanotubes evolves as a function of its length. We demonstrate that the population of hot optical phonons controls the electronic transport of short nanotubes, whereas acoustical phonons take the leading role when the nanotube is very long. In this limit of long tubes, we also analyze the interplay between the diffusive and localized transport regimes when the electron mean-free path and the localization length due to impurities are comparable.

Figure 1

(Color online) Schematic picture showing the discretized electron levels in the nanotube. In our model, electrons can emit (or absorb) optical phonons of energy $h{\nu }^{\text{opt}}$, and emit acoustical phonons of energy $h{\nu }^{\text{ac}}$. All these processes yield, in a self-consistent solution, the electron distribution function, $n\left(x,E\right)$ and the optical phonon distribution function, $n_B^{\text{opt}}\left(x\right)$. For a nanotube, all electrons move with a constant velocity, $v_F$, even if their energy w.r.t. the local chemical potential, $E-\mu \left(x\right)$, changes.

Figure 2

(Color online) Qualitative behavior of ${\lambda }_{\text{ac}}^{\text{eff}}$ as a function of the electron energy, $\hslash \omega =E-\mu \left(x\right)$. The inset shows one electron of energy $\hslash \omega$ being backscattered by acoustical phonons of energy $h{\nu }^{\text{ac}}$ and momentum $\hslash q^{\text{ac}}$.

Figure 3

Resistance (a) and differential resistance, $dV/dI$, in panel (b) versus $L$ (in microns). Four different values of $V$ are studied: $V=0.4\text{V}$ (full lines), $V=0.7\text{V}$ (dashed lines), $V=1.0\text{V}$ (dash-dotted lines) and $V=1.5\text{V}$ (dotted lines). Dots corresponds to experimental data as reported in Ref. 17.

Figure 4

(Color) Contour plots rendering $n^{+}\left(x,E\right)$ at $V=1.0\text{V}$ for two different $L$’s [$L=0.5\mu \text{m}$ in (a) and $L=4\mu \text{m}$ in (b)]. (c) Mean value of $n_B\left(x\right)$ for the four voltages analyzed $V=0.4\text{V}$ (blue full line), $V=0.7\text{V}$ (black dashed line), $V=1.0\text{V}$ (red dotted line), and $V=1.5\text{V}$ (green dash-dotted line) as a function of the length of the nanotube. Inset shows $n_B\left(x\right)$ for two voltages ($V=0.4\text{V}$: blue curves and $V=1.0\text{V}$: red curves) for three different $L$’s ($L=0.1\mu \text{m}$: full lines; $L=0.5\mu \text{m}$: dashed lines; and $L=4.0\mu \text{m}$: dash-dotted lines.

Figure 5

Theoretical results for $R$ as obtained from Eq. (13) for three different $T$’s: $T=1.65\text{K}$ (full line), $T=5\text{K}$ (dashed line), and $T=10\text{K}$ (dotted line). Dots correspond to the experimental data as reported in Ref. 16. Inset displays the behavior of ${\lambda }_{\text{ac}}^{\text{eff}}$ as a function of $L$ for the three $T$’s analyzed in the main panel.