Quantum Interferences in Ultraclean Carbon Nanotubes

Electronic analogs of optical interferences are powerful tools to investigate quantum phenomena in condensed matter. In carbon nanotubes (CNTs), it is well established that an electronic Fabry-Perot interferometer can be realized. Other types of quantum interferences should also arise in CNTs, but have proven challenging to realize. In particular, CNTs have been identified as a system to realize the electronic analog of a Sagnac interferometer—the most sensitive optical interferometer. To realize this Sagnac effect, interference between nonidentical transmission channels in a single CNT must be observed. Here, we use suspended, ultraclean CNTs of known chiral index to study both Fabry-Perot and Sagnac electron interferences. We verify theoretical predictions for the behavior of Sagnac oscillations and the persistence of the Sagnac oscillations at high temperatures. As suggested by existing theoretical studies, our results show that these quantum interferences may be used for electronic structure characterization of carbon nanotubes and the study of many-body effects in these model one-dimensional systems.

Figure 1

(a) Scanning electron microscope image of a suspended CNT; scale bar is $1\mu \mathrm{m}$. (b) Energy spectrum of left-moving $(L)$ and right-moving $(R)$ carriers in two subbands. Geometries for realizing (c) Fabry-Perot and (d) Sagnac interferences in a CNT waveguide. Red (blue) arrows represent an electron traveling in the $K(K^{\prime })$ subband. (e) Conductance $(G)$ versus $V_g$ at $T=1.5\mathrm{K}$ for $D1$ with $\sim 2\mu \mathrm{m}$ length.

Figure 2

(a) Conductance ($G$) versus $V_g$ at $T=1.5\mathrm{K}$ for the whole (black) and half-length (red) of $D2$. Gray scale plot of differential conductance versus $V_g$ and source-drain bias $V_{\mathrm{SD}}$ for (b) the whole and (c) half-length of $D2$.

Figure 3

(a) Photocurrent spectrum of $D3$ (25, 10) and fit to the excitonic model. (b) Experimental data (black) and fit functions (red) of conductance versus $V_g$. The measured chiral index is noted for each device. (c) Comparison of measured (dots) and calculated (red line) of $\gamma /(2L{e}^2{\alpha }^2)$ as a function of $\theta$ for the six CNTs.

Figure 4

(a) Temperature dependence of the conductance of $D2$ as a function of $V_g$. Black arrows show the extremum points (Ext) of the slow oscillation. Averaged amplitudes of (b) fast and (c) slow oscillations over the whole range of $V_g$ calculated using FFT over a sliding window of 0.75 V as a function of the temperature. Dashed blue line is a fit to Eq. (6) in the text, and the red line is an extrapolation to $T=0\mathrm{K}$. Insets: amplitude of (b) fast and (c) slow oscillations at Ext of slow oscillation vs temperature. (d) $T^{*}$ of fast (red) and slow (black) oscillations at Ext of slow oscillation. Here, $T^{*}$ is a temperature at which $A(T^{*})=\exp (-1)\times A(T=0)$.