Sagnac Interference in Carbon Nanotube Loops

In this Letter we study electron interference in nanotube loops. The conductance as a function of the applied voltage is shown to oscillate due to interference between electron beams traversing the loop in two opposite directions, with slightly different velocities. The period of these oscillations with respect to the gate voltage, as well as the temperatures required for the effect to appear, are shown to be much larger than those of the related Fabry-Perot interference. We calculate interaction effects on the period of the oscillations, and show that even though interactions destroy much of the near degeneracy of velocities in the symmetric spin channel, the slow interference effects survive.

Figure 1

(a) In a nanotube loop, electrons can tunnel (dashed line) from one branch ($X$) to the other ($X^{\prime }$) [33, 34]. Electrons impinging on point $X$ either proceed through the loop moving counterclockwise, or tunnel to $X^{\prime }$, and proceed clockwise. Velocity difference between right and left movers produces Sagnac interference between the counterpropagating beams. (b) Conductance vs gate voltage of a nanoloop device. From top to bottom: $T=64$, 48, 32, 24, 16, 12, 8, and 4 K. Strong conductance fluctuations, consistent with Sagnac interference, appear already at $T=64\mathrm{K}$ with period $\delta V_G\sim 20\mathrm{V}$. At lower temperatures Fabry-Perot interference appears as well with short periods $\delta V_g\sim 0.15\mathrm{V}$, 0.3 V. Left inset shows the device; data from both loops are qualitatively the same. Right inset shows the suspected Sagnac envelope at $T=4\mathrm{K}$ and $V_{\mathrm{sd}}=30\mathrm{mV}$ for wider range of $V_g$. Note that the asymmetry in $V_g$ is not understood.

Figure 2

(a) Dispersion of an armchair nanotube. When the Fermi surface is away from the nodes, right and left movers in each node have different velocities, $v_{R/L}=v_F\pm u$ ($a_0$ is the lattice constant), which leads to two Sagnac interference effects: (b) Within one node (say node 1), a beam entering the loop from the left (short black arrow) splits by partially tunneling between points $X$ and $X^{\prime }$ to two counterpropagating beams (black and gray), in region $b$. They then recombine at point $X^{\prime }$. Long black arrows represent the two chiral electronic modes near node 1. The regions $a$, $b$, and $c$ correspond to those indicated in Fig. 1. (c) A beam impinging on point $X$ in node 1 (2) partially scatters to node 2 (1). Both traverse the loop in the same direction (short black and gray arrows), and recombine at point $X^{\prime }$. This effect is similar to the slow conductance oscillation due to impurities propounded in Ref. 35, and in the presence of axial magnetic field in Ref. 36.