Electronic transport in carbon nanotubes: From individual nanotubes to thin and thick networks

We measure and compare the electronic transport properties of individual multiwall carbon nanotubes (MWNTs) and individual single-wall carbon nanotubes (SWNTs), and SWNT networks of varying thickness. The thinnest SWNT networks, like the individual semiconducting SWNTs, show nonlinear current-voltage $\left(I\text{−}V\right)$ characteristics at low temperatures with a current that can be modulated by a gate-source voltage. The overall temperature dependence of conductance in the transparent networks changes systematically as the thickness of the network increases and is consistent with hopping conduction. On the other hand, the thickest SWNT networks (freestanding film) show more metallic behavior: their $I\text{−}V$ characteristics are linear with no gate-voltage effect, and a large fraction of their conductivity is retained at very low temperatures, consistent with tunneling through thin barriers separating metallic regions. We make a comparison with individual MWNTs, which in some cases show even greater retention of conductance at very low temperatures, but (unlike the thickest SWNT networks) no changeover to metallic temperature dependence at higher temperatures. The temperature dependence of conductance in individual MWNTs is consistent with a model involving conduction in the two outer shells.

Figure 1

An example of TEM images of our thin SWNT networks showing the morphology of the SWNT bundles (a), and at higher magnification, the SWNTs within a bundle (b).

Figure 2

Current-voltage $(I_{\mathrm{D}}-V_{\mathrm{DS}})$ characteristics of SWNTs at a range of temperatures between room temperature and $4\mathrm{K}$: (a) m-SWNT (metallic type) and (b) s-SWNT (semiconductor type); the upper left insets (a) and (b) show examples of the dependence of the drain current $I_{\mathrm{D}}$ on gate voltage $V_{\mathrm{GS}}$, while the lower right inset (b) shows the dependence on temperature of the threshold bias voltage $V_T$ at which detectable current flows.

Figure 3

(Color online) Power laws for metallic m-SWNT. (a) Drain current $I_{\mathrm{D}}$ as a function of drain-source voltage $V_{\mathrm{DS}}$ (using logarithmic scales) for temperatures ranging from $4\mathrm{K}\text{to}296\mathrm{K}$, showing the trend to a power law with exponent approximately $\beta =1.75$ at higher voltages (measurements for increasing and decreasing voltage are shown for each temperature). (b) Conductance at zero bias as a function of temperature $T$ (using logarithmic scales), showing approximate agreement with a power law with exponent $\alpha =0.60$.

Figure 4

(Color online) Conductance $G$ at $100\mathrm{mV}$ (using a logarithmic scale) as a function of inverse temperature, showing two mechanisms: tunneling at low temperatures and activated behavior with energy $E_{\mathrm{A}}=5\mathrm{meV}$ at high temperatures.

Figure 5

(Color online) Power laws for semiconducting s-SWNT: (a) Drain current $I_{\mathrm{D}}$ as a function of drain-source voltage $V_{\mathrm{DS}}$ (using logarithmic scales) for temperatures ranging from $4\mathrm{K}\text{to}296\mathrm{K}$, showing approximate agreement with a power law with exponent $\beta =3.3$ for data above $500\mathrm{mV}$ at $4\mathrm{K}$; (b) Zero-bias conductance $G$ as a function of temperature (using logarithmic scales), showing approximate agreement with a power law with exponent $\alpha =3.9$ for data above $70\mathrm{K}$ [as indicated in Fig. 5a the zero-bias conductance becomes too small to measure at low temperatures].

Figure 6

(Color online) Conductance $G$ at $100\mathrm{mV}$ (using a logarithmic scale) as a function of inverse temperature, showing a good agreement with activated behavior with activation energy $E_{\mathrm{A}}=29\mathrm{meV}$.

Figure 7

Magnitudes of the square conductance for each sample showing the correlation with the optical transmittance at $1100{\mathrm{cm}}^{-1}$ for the transparent SWNT networks of different densities.

Figure 8

$I_{\mathrm{D}}-V_{\mathrm{DS}}$ characteristics of thin SWNT networks Net 1 (thinnest) to Net 4 at $4.3\mathrm{K}$ (a) and $291\mathrm{K}$ (b).

Figure 9

The effect of a gate voltage $V_{\mathrm{GS}}$ on the current $I_{\mathrm{D}}$ of Net 4 at temperatures of $4.3\mathrm{K}$ (a) and at $290\mathrm{K}$ (b) and of Net 1 at temperatures of $4.3\mathrm{K}$ (c) and at $290\mathrm{K}$ (d).

Figure 10

Temperature dependences of the electrical conductance of SWNT freestanding film and four thin networks normalized to their values at room temperature $\sigma \left(T\right)∕\sigma \left(292\mathrm{K}\right)$; the full lines are fits to the variable-range hopping expression Eq. (5) for the networks, and to the interrupted metallic model Eq. (4) for freestanding film.

Figure 11

(a) Temperature dependence of the electrical conductance of an individual multiwall nanotube MWNT-1 fitted to Eq. (7) for our simple model of conduction in the two outer shells (full line) and to Eq. (6) with $R_m=0$ for fluctuation-assisted tunneling (FAT) alone (dashed line); for clarity only a third of the data points are plotted. (b) $I_{\mathrm{D}}-V_{\mathrm{DS}}$ characteristics for a similar individual MWNT-2 to the tube MWNT-1 (the conductance values for MWNT-2 from this $I_{\mathrm{D}}-V_{\mathrm{DS}}$ plot are within 8% of those for MWNT-1 at all temperatures).