Nonequilibrium Tunneling Spectroscopy in Carbon Nanotubes

We report measurements of the nonequilibrium electron energy distribution in carbon nanotubes. Using tunneling spectroscopy via a superconducting probe, we study the shape of the local electron distribution functions, and hence energy relaxation rates, in nanotubes that have bias voltages applied between their ends. At low temperatures, electrons interact weakly in nanotubes of a few microns channel length, independent of end-to-end-conductance values. Surprisingly, the energy relaxation rate can increase substantially when the temperature is raised to only 1.5 K.

Figure 1

(a) SEM image of a nanotube device and the measurement setup (see text). (b) Schematic diagram of an electron tunneling from a nanotube to a superconductor. The density of states (DOS) of the nanotube shows a modulation with single-particle energy spacing, as expected for an open quantum dot, while the superconducting DOS exhibits a BCS-like gap of $2\Delta$.

Figure 2

End-to-end differential conductance at $U=1\mathrm{mV}$ across samples (a) A and (b) B, respectively, as a function of gate voltage. (c) Tunneling differential conductance, $dI/dV$ of sample A from the superconductor into the nanotube, as a function of $V$ at $V_g=8.285\mathrm{V}$. The blue arrow indicates the Pb superconducting gap size. Additional peaks at $V\sim -4.9$, $-2.8$, 3.4 mV are resonant tunneling peaks through the open quantum dot defined by the nanotube leads. (a) and (c) were taken at $T=1.3\mathrm{K}$ and (b) was at 1.5 K.

Figure 3

Tunneling differential conductance $dI/dV$ vs $V$ at multiple values of bias $U$ across the tube ends. (a) Sample A at $T=1.3\mathrm{K}$, $V_g=8.660\mathrm{V}$. The peaks marked by black arrows are the superconducting peaks at $V=\pm \Delta /e$; the blue peaks marked by blue arrows are the superconducting peaks at $V=\pm \Delta /e+U$ (in this case $U=1.5\mathrm{mV}$), (b) sample A at $T=1.3\mathrm{K}$, $V_g=8.285\mathrm{V}$, (c) Sample A at $T=1.3\mathrm{K}$, $V_g=8.070\mathrm{V}$, (d) sample B at $T=1.5\mathrm{K}$, $V_g=-4.824\mathrm{V}$, (e) sample B at $T=53\mathrm{mK}$, $V_g=-4.915\mathrm{V}$. The red peaks marked by red arrows are the superconducting peaks at $V=\pm \Delta /e$ and $V=\pm \Delta /e+U$ (in this case $U=0.5\mathrm{mV}$), (f) sample B at $T=53\mathrm{mK}$, $V_g=-4.460\mathrm{V}$. (a),(b), and (c) have the same legend, and (e) and (f) have the same legend.

Figure 4

Electron energy distributions calculated from the $dI/dV(V)$ data in Fig. 3. Two-step functions (a), (e), (f) imply limited $e\mathrm{\text{−}}e$ scattering, while broadened single-step functions (b), (d) imply strong $e\mathrm{\text{−}}e$ scattering. The dotted lines are noninteracting distribution functions $f_0(E)$ with $U=1.0\mathrm{mV}$, $T=1.3\mathrm{K}$, $r=0.5$ in (a), $U=1.0\mathrm{mV}$, $T=1.3\mathrm{K}$, $r=0.4$ in (c), and $U=0.5\mathrm{mV}$, $T=125\mathrm{mK}$, $r=0.5$ in (f). Insets in (e) and (f): Zero-bias differential conductance across the nanotube as a function of $V_g$ [blue arrows indicate where $dI/dV(V)$ were taken].