Determination of the Intershell Conductance in Multiwalled Carbon Nanotubes

We report on the intershell electron transport in multiwalled carbon nanotubes (MWNTs). To do this, local and nonlocal four-point measurements are used to study the current path through the different shells of a MWNT. For short electrode separations $\lesssim 1\mu \mathrm{m}$ the current mainly flows through the two outer shells, described by a resistive transmission line with an intershell conductance per length of $\sim (10\mathrm{k}\Omega {)}^{-1}/\mu \mathrm{m}$. The intershell transport is tunnel type and the transmission is consistent with the estimate based on the overlap between $\pi$ orbitals of neighboring shells.

Figure 1

Description of multiwalled nanotube. (a) A MWNT which consists of four shells. (b) A graphene sheet which gives a MWNT shell when rolled up. Also shown is the slowly varying transverse part $\propto \exp (ikr)$ of the electronic wave function. ${\Psi }_0$ corresponds to the two crossing subbands at the $K$ point and ${\Psi }_1$ and ${\Psi }_2$ to the next successive subbands. (c) A hexagon of carbon $\pi$ orbitals. The two hexagons belong to two nearby incommensurate shells. The intershell resistance depends on the orbital overlap. (d) Current pathways. The three outermost shells are represented by lines of different colors. When current is externally applied, the current enters and exits the MWNT through the outermost shell. Importantly, part of $I$ flows through to the second shell, so that the current pathway penetrates beside the region lying between the bias electrodes. (e) Resistive transmission line, which models the two outermost shells.

Figure 2

Nonlocal voltage measurements on a MWNT electrically addressed by 11 electrodes. The schematics show the $5\mu \mathrm{m}$ long MWNT. The diameter is 17 nm, which corresponds to about 20 shells. The MWNT, synthesized by arc-discharge evaporation and carefully purified [28], was dispersed onto a 500 nm oxidized Si wafer from a dispersion in dichloroethane. $\mathrm{C}\mathrm{r}/\mathrm{A}\mathrm{u}$ electrodes were patterned above the tube by electron beam lithography. (a) $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ as a function of the separation between the current and the voltage electrodes. $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}/I$ is plotted on a logarithmic scale. The straight line corresponds to a decay length of $0.94\mu \mathrm{m}$. Data are taken at 250 K. $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}/I$ is measured in the linear regime with $eV=eI{R}_{2P}$ below $kT$, $R_{2P}$ being the two-point resistance. (b) Nonlocal voltage as a function of the tube length. The voltage reference is taken at the extremity of the electrode series. The continuous curve is an exponential decay fit with a decay length of $0.92\mu \mathrm{m}$. (c) $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ as a function of the separation between the current electrodes. The continuous curve is proportional to $1-\exp (-L/L_a)$ with $L_a=0.94\mu \mathrm{m}$.

Figure 3

Local voltage as a function of separation between the current biased electrodes. $L$ is symmetrically increased with respect to the center between the voltage electrodes. The continuous curve is exponential with a decay length equal to $2\mu \mathrm{m}$.

Figure 4

Nonlocal voltage at low temperature. (a) $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ as a function of temperature measured from 8 to 280 K. The MWNT diameter is 7 nm and the electrode separations are 450 nm. $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ is averaged over the gate voltage. (b) $\Delta V_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ as a function of gate voltage.