Tunneling in Suspended Carbon Nanotubes Assisted by Longitudinal Phonons

Current-voltage characteristics of suspended single-wall carbon nanotube quantum dots show a series of steps equally spaced in voltage. The energy scale of this harmonic, low-energy excitation spectrum is consistent with that of the longitudinal low-$k$ phonon mode (stretching mode) in the nanotube. Agreement is found with a Franck-Condon-based model in which the phonon-assisted tunneling process is modeled as a coupling of electronic levels to underdamped quantum harmonic oscillators. A comparison with this model indicates a rather strong electron-phonon coupling factor of order unity.

Figure 1

(a) Schematic drawing of a suspended nanotube (NT) clamped between two $\mathrm{Cr}/\mathrm{Au}$ electrodes on top of silicon oxide. The underlying oxide is partially removed by a wet etch step leaving the nanotube suspended. The highly doped silicon plane is used as a global gate to tune the electrostatic potential of the nanotube. (b) Scanning-electron microscope micrograph of a suspended nanotube. The scale bar represents 200 nm.

Figure 2

Stability diagrams for three different suspended nanotubes with a length in between contacts of $1.2\mu \mathrm{m}$, 420 nm, and 140 nm for (a), (b), and (c), respectively. The conductance ($dI/dV$) is plotted as a function of source-drain voltage, $V$, and gate voltage, $V_G$. Blue corresponds to low and red to high conductance. Measurements have been performed at $T=300\mathrm{mK}$ except in (a), where the base temperature was 10 mK. (a) Small region of a stability diagram showing closely spaced sets of lines running parallel to the Coulomb diamond edges for two charge states. At low bias, a strong suppression of the conductance is present. Red lines indicate positive differential conductance; blue lines indicate negative differential conductance. Inset: regular diamonds that close are observed in a different cooldown at $T=300\mathrm{mK}$. (b),(c) Diamond crossings for two other samples, again showing lines parallel to the diamond edges with energy separations smaller than expected for electronic excitations.

Figure 3

Current as a function of source-drain voltage at a voltage indicated by the vertical (green) lines in Fig. 2. The drawn (red) lines represent the step heights calculated in the Franck-Condon model (see text) for an electron-phonon coupling parameter of 0.95, 1.1, and 0.5 for (a), (b), and (c), respectively. In the insets, the energy separation between the peaks or steps (lines in Fig. 2) is plotted, showing an equally spaced, harmonic spectrum. The slope of the drawn line is 140, 690, and $530\mu \mathrm{eV}$ for the insets of (a), (b), and (c), respectively.

Figure 4

Energy scales of different vibrations and electronic excitations plotted on a log scale for a nanotube with a 1.4 nm diameter. The radial breathing mode (green) does not depend on the length $L$. The bending mode vibrations (red) have a $L^{-2}$ dependence. The mean electronic level spacing (black) and the stretching mode (blue) vibrations depend inversely on the length.