The Aharonov-Bohm Interference and Beating in Single-Walled Carbon Nanotube Interferometers

A hallmark of the Aharonov-Bohm (AB) effect is conductance oscillations of metallic rings or cylinders as a function of enclosed magnetic flux with a period on the order of the flux quantum \textit{$\Phi $}$_{0}=h/e$ due to quantum interference. Carbon nanotubes are chemically derived cylinders with atomically well-defined structures. Multi-walled nanotubes (MWNT) have radius $r \quad \sim $ 10 nm and in magnetic fields parallel to the tube axis, conductance modulations with a period of $B_{0}$\textit{=$\Phi $}$_{0}$\textit{ /$\pi $ r }$^{2} \quad \sim $ 10T in magnetic field have been seen. Single-walled nanotubes (SWNT) are ultra-small with $r \quad \sim $ 1 nm and the magnetic field needed to approach 1\textit{$\Phi $}$_{0}$ flux through the nanotube cross section is $B_{0} \quad \sim $ 1000T, far beyond reach by experiments. We show here that in the Fabry-Perot interference regime, beating in the AB-interference between two modes of spiraling electrons with non-degenerate wave-vectors causes conductance modulations under fields much smaller than that needed to reach 1\textit{$\Phi $}$_{0.}$ Single-walled nanotubes hence represent the smallest cylinders exhibiting the AB effect with rich interference and beating phenomena arising from well-defined molecular orbitals reflective of the nanotube chirality. The observation of quantum beats for the AB effect is to our knowledge unprecedented in mesoscopic systems and is a result of well-defined molecular orbitals of nanotubes in magnetic fields.