Aharonov-Bohm Oscillations in a Quasi-Ballistic 3D Topological Insulator Nanowire

In three-dimensional topological insulators (3D TI) nanowires, transport occurs via gapless surface states where the spin is fixed perpendicular to the momentum[1-6]. Carriers encircling the surface thus acquire a \pi Berry phase, which is predicted to open up a gap in the lowest-energy 1D surface subband. Inserting a magnetic flux ({\Phi}) of h/2e through the nanowire should cancel the Berry phase and restore the gapless 1D mode[7-8]. However, this signature has been missing in transport experiments reported to date[9-11]. Here, we report measurements of mechanically-exfoliated 3D TI nanowires which exhibit Aharonov-Bohm oscillations consistent with topological surface transport. The use of low-doped, quasi-ballistic devices allows us to observe a minimum conductance at {\Phi} = 0 and a maximum conductance reaching e^2/h at {\Phi} = h/2e near the lowest subband (i.e. the Dirac point), as well as the carrier density dependence of the transport.

The behavior of the AB oscillations should also vary with carrier density 16 , which can be tuned by an external voltage. In Fig. 1b, a schematic mapping shows the number of modes at various Fermi energies, E F . At | | , the number of modes at Φ/Φ 0 = 0 and 0.5 should be 0 and 1 respectively. Therefore, in a ballistic transport regime, the conductance should exhibit a minimum at Φ/Φ 0 = 0 and a maximum ≈ e 2 /h at Φ/Φ 0 = 0.5. Away from the Dirac point the phase of the conductance alternates: when E F is located at one of the green(red)-dotted lines in Fig. 1b, the conductance should have a minimum(maximum) at Φ/Φ 0 = 0 and a maximum(minimum) at Φ/Φ 0 = 0.5, as depicted in Fig. 1c.
Previous experiments have demonstrated AB oscillations in 3D TI nanowires, indicating surface transport [9][10][11] However, the predicted behavior close to the Dirac point has not been . While chemical or mechanical etching allow effective production of nanowires from layered thin-films, these processes typically introduce additional damage and defects along the edges [18][19] . We fabricated two four-terminal devices having channel lengths of L 1 = 200 nm (device 1) and L 2 = 350 nm (device 2) (see Fig. 2). Device 1 has width = 110 nm and thickness = 15 nm (S 1 = 1.65  10 -15 m 2 ), while device 2 has width = 100 nm and thickness = 16 nm (S 2 = 1.60  10 -15 m 2 ). The typical mean free path l m ≈ 100 nm of the nanowires is estimated from the gate-dependent conductivity of the surface electrons using and ℏ ( ) .
Thus, both devices are in the quasi-ballistic regime where L/l m ~ 2-3 20 . Samples were measured using standard four-terminal lock-in techniques in a dilution refrigerator at the base temperature of 16 mK. Figure 2c shows the location of the Dirac point for device 1 at V g ≈ -15V, and demonstrates that the carrier density can be tuned through the Dirac point with a backgate voltage. It has been shown that when the initial doping level is low in nanostructured TI devices, the Fermi energies of the top and bottom surfaces could be simultaneously tuned with a single gate-electrode, due to the large inter-surface capacitance 1 .  The conductance maxima near the Dirac point deviate slightly from e 2 /h, which can be explained by the fact that, when current is flowing in a 3D TI nanowire (and thus time-reversal symmetry is broken by the non-equilibrium state), the conductance is not guaranteed to be quantized 2,4 . The deviation may also be partly due to the inevitable invasiveness of voltage probes in 1D systems 20 . The data also show finite conductance (≈ 0.5 e 2 /h in Fig. 3a) at Φ/Φ 0 = 0 in the gapped nanowire near the Dirac point. This could occur if chemical potential fluctuations due to strong disorder were much larger than the gap size [7][8] . However, in that case, h/2e AAS oscillations would dominate over h/e AB oscillations, which is inconsistent with our observations. Rather, the finite conductance at Φ/Φ 0 = 0 is likely due to small deviations of E F from the Dirac point, as discussed below. The data further show that the phase of the AB oscillations becomes less regular at large magnetic field, which is likely due to the Zeeman energy. At low magnetic field, the Zeeman energy is negligible compared to the energy spacing of subbands: where g ≈ 23 is the g-factor. However, at large B (Φ/Φ 0 > 2) the Zeeman energy is comparable to .
The magneto-conductance data can be better understood by comparing it to full 3D simulations of TI nanowires coupled to metallic contacts, calculated within the non-equilibrium Green's function formalism 24

Methods
Three-dimensional TI crystals (Bi 1.33 Sb 0.67 Se 3 ) were grown using a modified floating zone method 16 . TI nanowires were obtained by mechanical exfoliation ("scotch tape method") of bulk (Bi 1.33 Sb 0.67 )Se 3 crystals on 300nm SiO 2 /highly n-doped Si substrates 27 . The nanowires were identified via optical and atomic force microscopy (AFM) 1,28,29 . Nanowire thickness was determined by AFM, while the width was more accurately determined by scanning probe microscopy after all electrical measurements were completed. After the nanowires were located on the SiO 2 /Si chips, electron beam lithography was performed to define four-point electrodes. Subsequently, the surface was cleaned with brief ion milling and Ti(2.5nm)/Au(50nm) were deposited at a base pressure ≈ 1x10 -9 Torr. Immediately after lift-off, ~ 14 nm of F4-TCNQ (Sigma-Aldrich) was deposited via thermal evaporation. Finally, the devices were wire-bonded and cooled down in a commercial dilution refrigerator. All the electrical measurements were performed at a base temperature ≈ 16 mK by using standard ac lock-in techniques.      However, the fact that few AAS oscillations appear in magneto-conductance oscillations away from the Dirac point (Fig. S3d) may indicate that the transport regime is closer to a quasiballistic regime than to a diffusive one.  where tunneling rate 1 results in transparent contacts with the nanowire. The contacts are connected to the left-most 1 and right-most faces of the nanowire, such that the size of each self-energy matrix is (4 for two sublattice and two spin degrees of freedom).
The magnetic field effect can be introduced to the Hamiltonian in terms of a nonzero vector potential, ⃗ ⃗ , which gives rise to an additional phase factor relative to the real-space Hamiltonian with no vector potential. This is added via a Peierls substitution 7 , where is the path from to ⃗⃗ . We focus on the effect of a homogeneous magnetic field lying in the longitudinal direction, ⃗ ⃗ ̂ , which in the Landau gauge yields a vector potential ⃗ ⃗ − ̂ that is only nonzero in the ̂ direction. Lattice hopping in the width direction is thus the only component in the Hamiltonian that accrues an additional phase, where is the magnetic flux threaded through each cross-sectional plaquette of area , and ⁄ is the magnetic flux quantum. Total magnetic flux through the nanowire is . Zeeman splitting is roughly an order of magnitude smaller than surface state sub band gap in the experimental regime on which we focus ⁄ 1 , and as such is neglected in the simulations.
This model thus provides a framework for studying the Aharanov-Bohm effect in TI nanowires by calculating the transmission (i.e. the differenctial conductance) from the left metallic contact to the right metallic contact, as a function of both chemical potential and