Aharonov–Bohm oscillations in carbon nanotubes

Abstract

When electrons pass through a cylindrical electrical conductor aligned in a magnetic ®eld, their wave-like nature manifests itself as a periodic oscillation in the electrical resistance as a function of the enclosed magnetic¯ux 1 . This phenomenon re¯ects the dependence of the phase of the electron wave on the magnetic ®eld, known as the Aharonov±Bohm effect 2 , which causes a phase difference, and hence interference, between partial waves encircling the conductor in opposite directions. Such oscillations have been observed in micrometre-sized thin-walled metallic cylinders 3±5 and lithographically fabricated rings 6±8 . Carbon nanotubes 9,10 are composed of individual graphene sheets rolled into seamless hollow cylinders with diameters ranging from 1 nm to about 20 nm. They are able to act as conducting molecular wires 11±18 , making them ideally suited for the investigation of quantum interference at the single-molecule level caused by the Aharonov±Bohm effect. Here we report magnetoresistance measurements on individual multi-walled nanotubes, which display pronounced resistance oscillations as a function of magnetic ux. We ®nd that the oscillations are in good agreement with theoretical predictions for the Aharonov±Bohm effect in a hollow conductor with a diameter equal to that of the outermost shell of the nanotubes. In some nanotubes we also observe shorter-period oscillations, which might result from anisotropic electron currents caused by defects in the nanotube lattice.In a diffusive and thin-walled metallic cylinder, a prominent periodic quantum correction to the resistance arises from the interference of closed electron trajectories that encircle the cylinder once. The phase difference Df between each such trajectory ¡ and the time-reversed counter-propagating trajectory ¡9 (Fig. 1a) is solely determined by the magnetic¯ux © enclosed: Df 2p2e=h©, where e and h are the electron charge and Planck's constant, respectively. Consequently, the electrical resistance has an oscillating contribution with period h/2e, known as the Altshuler± Aronov±Spivak (AAS) effect 1 . For zero magnetic¯ux, these interference terms add up constructively, increasing electron backscattering and thereby the electrical resistance, an effect known as weak localization 19 . For a thin metallic ®lm in a perpendicular magnetic ®eld, the weak-localization resistance correction monotonically disappears in higher ®elds (negative magnetoresistance, MR). In contrast, for a cylinder in a parallel magnetic ®eld, weak localization is periodically modulated with a magnetic-®eld period given by DB h=2e=r 2 p, where r is the radius of the cylinder.We have carried out electric-transport measurements on multiwalled carbon nanotubes (MWNTs) composed of multiple coaxial graphene cylinders, in a magnetic ®eld parallel to the axis of the nanotubes. An example is shown in Fig. 1b. Coulomb blockade can strongly affect electrical transport in small structures, and we therefore use samples containing only a single contacted nanotube with low contact resis...

Observation of quantum oscillations in FIB fabricated nanowires of topological insulator (Bi2Se3)

“…Note that the AB and AAS effects can be observed in non-topological insulator material also if the electron transport in investigated system maintains the quantum coherency which gives rise to conductance oscillations. The earlier work on Bismuth nanowires [24], multiwall carbon nanotube [25], Au nanorings [26], magnesium film evaporated onto a quartz filament [27], graphene ring [28] and Dirac semimetal nanowires [29] have shown quantum interference of surface states indicating the AB oscillations. We assume these periodic oscillations could be coming from AB effects and gallium doping and or material deformation also affecting oscillation period and phase coherence length of the electrons.…”

Aharonov–Bohm oscillations in a quasi-ballistic three-dimensional topological insulator nanowire

“…Thus, the topological nature of the 1D surface sub-bands can be observed via the behaviour of the AB oscillation maxima and minima as a function of F/F 0 : for the lowest-energy mode (at the Dirac point), a minimum in conductance should occur at F/F 0 ¼ 0, while a maximum in conductance should occur at F/F 0 ¼ 0.5, as the non-degenerate gapless mode reappears 10,11 . This mode contains massless Dirac-like excitations that follow a 1D linear energy-momentum dispersion, and is not observed in non-topological systems [7][8][9]12 .…”

“…When a magnetic field (B) is applied along the nanowire axis, the surface electrons encircling the wire pick up a phase of 2pF/F 0 , where F ¼ BS is the magnetic flux through cross-sectional area S and F 0 ¼ h/e is the magnetic flux quantum, where h is Planck's constant and e the electron charge. The energymomentum relation is, thus, periodic in F/F 0 , leading to h/e Aharonov-Bohm (AB) oscillations [7][8][9] . In topological insulators, the spin-momentum locking causes carriers encircling the wire to also acquire a p Berry phase, which is predicted to open up a gap in the lowest-energy one-dimensional (1D) surface sub-band.…”

Conductance properties of nanotubes coupled to superconducting leads: signatures of Andreev states dynamics

“…The theory provides a good fitting of the experimental results. Carbon nanotubes connected to metallic electrodes allow to study many different regimes in mesoscopic electron transport exhibiting either ballistic or diffusive behavior [1,2]; Luttinger liquid features [3], etc. Several experiments [4] have demonstrated that carbon nanotubes weakly coupled to normal leads can also exhibit Coulomb blockade and Kondo effect, i.e.…”