Correlation of Electron Tunneling and Plasmon Propagation in a Luttinger Liquid

Quantum-confined electrons in one dimension behave as a Luttinger liquid. However, unambiguous demonstration of Luttinger liquid phenomena in single-walled carbon nanotubes (SWNTs) has been challenging. Here we investigate well-defined SWNT cross junctions with a point contact between two Luttinger liquids and combine electrical transport and optical nanoscopy measurements to correlate completely different physical properties (i.e., the electron tunneling and the plasmon propagation) in the same Luttinger liquid system. The suppressed electron tunneling at SWNT junctions exhibits a power-law scaling, which yields a Luttinger liquid interaction parameter that agrees quantitatively with that independently determined from the plasmon velocity based on the near-field optical nanoscopy.

Figure 1

AFM topography image of a representative SWNT cross junction sample (indicated by the arrow) on $h\text{−}\mathrm{BN}$.

Figure 2

Near-field optical nanoscopy characterizations on SWNTs on $h\text{−}\mathrm{BN}$. (a) Near-field optical nanoscopy image of an individual metallic SWNT with a diameter of $\sim 1\mathrm{nm}$. (Inset) Corresponding topography image that is simultaneously recorded. (b) Experimental intensity profile (in black) of Luttinger liquid plasmon oscillations and the corresponding theoretical fitting (in red) with the damped oscillator form $e^{-2\pi x/(Q·{\lambda }_P)}\sin [(4\pi x)/{\lambda }_p]$ along the tube axis between two bars in (a), where ${\lambda }_p$ is the plasmon wavelength and $Q$ is the quality factor. Luttinger parameter $g\sim 0.31$ [shown in (a)] is directly obtained from the measured plasmon wavelength ${\lambda }_p$. (c) AFM topography image of an individual metallic SWNT (tube $A$) and an individual semiconducting SWNT (tube $B$) on $h\text{−}\mathrm{BN}$ with similar diameters $\sim 0.8\mathrm{nm}$. (d) The corresponding near-field optical nanoscopy image of (c). The metallic SWNT (tube $A$) exhibits prominent Luttinger liquid plasmon oscillations, whereas the semiconducting SWNT (tube $B$) barely shows any near-field optical response owing to the finite band gap.

Figure 3

Correlation of electron tunneling and plasmon propagation in a Luttinger liquid (device no. 1). (a) Near-field optical nanoscopy characterization on a metallic SWNT cross junction. Luttinger parameters are determined to be $g\sim 0.32$ (tube 1-3) and $g\sim 0.31$ (tube 2-4) for each of two nanotubes from the measured Luttinger liquid plasmons. Metal contacts are denoted by numbers. (b) $dI/dV$ measurement of the electron tunneling probability across the Luttinger liquid junction as a function of electrical bias ($Vsd$) at 15 K (through contacts 1 and 4). The SWNT junction dominates the total resistance and a two probe measurement is carried out. A power function fitting (blue line) yields the power index $\alpha \sim 0.43$, which corresponds to $g\sim 0.29$ by using Eq. (1). (c) Power-law scaling behavior on electrical bias at 15 K at a different backgate voltage with respect to (b), which yields $g\sim 0.33$. (d) The corresponding temperature-dependent electron tunneling data (zero $Vsd$) with the same backgate voltage as in (b), which yields $g\sim 0.29$. (e) Scaled conductance $(dI/dV)/T^{\alpha }$ as a function of $eV/k_{B}T$ at different temperatures, where $\alpha$ is the power component with bias scaling at each temperature. All data collapse onto a single curve reasonably well, which provides an independent verification of Luttinger liquid behavior.

Figure 4

Correlation of electron tunneling and plasmon propagation in a Luttinger liquid (device no. 3). (a) Near-field optical nanoscopy characterization on a metallic SWNT cross junction. Luttinger parameters are determined to be $g\sim 0.30$ (tube 1-3) and $g\sim 0.31$ (tube 2-4), respectively. Metal contacts are denoted by numbers. (b) Differential conductance ($dIx/dVx$) measurement of the electron tunneling probability across the Luttinger liquid junction as a function of voltage drop across the junction ($Vx$) at 15 K. Measurements are carried out in a four probe configuration where the electric current is forced to flow through contacts 1 and 2 and voltage drop is measured through contacts 3 and 4. A power function fitting (blue line) yields $g\sim 0.24$ by using Eq. (1). (c) $dIx/dVx$ at 15 K at a different backgate voltage with respect to (b), which yields $g\sim 0.33$. (d) The corresponding temperature-dependent electron tunneling data (zero $Vx$) with the same backgate voltage as in (b), which yields a best fit of $g\sim 0.26$.