Robustness of topological surface states against strong disorder observed in $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$ nanotubes

Three-dimensional topological insulators are characterized by Dirac-like conducting surface states, the existence of which has been confirmed in relatively clean metallic samples by angle-resolved photoemission spectroscopy, as well as by anomalous Aharonov-Bohm oscillations in the magnetoresistance of nanoribbons. However, a fundamental aspect of these surface states, namely, their robustness to time-reversal-invariant disorder, has remained relatively untested. In this work, we have synthesized thin nanotubes of $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$ with extremely insulating bulk at low temperatures due to disorder. Nonetheless, the magnetoresistance exhibits quantum oscillations as a function of the magnetic field along the axis of the nanotubes, with a period determined by the cross-sectional area of the outer surface. Detailed numerical simulations based on a recursive Green function method support that the resistance oscillations are arising from the topological surface states which have substantially longer localization length than that of other nontopological states. This observation demonstrates coherent transport at the surface even for highly disordered samples, thus providing a direct confirmation of the inherently topological character of surface states. The result also demonstrates a viable route for revealing the properties of topological states by suppressing the bulk conduction using disorder.

Figure 1

Microscope images of $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$ nanotube samples. (a),(b) TEM and FESEM images of typical $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$ nanotubes. The TEM image (a) shows the wall thickness. The SEM image (b) shows a cross-section view of a nanotube, with hexagonal cross-section area and empty hollow center. (c) Platinum contacts fabricated on a nanotube by FIB with a four-contact geometry for transport measurements.

Figure 2

Resistance as a function of temperature for the nanotube A. It shows a monotonic increase with decreasing temperature, indicating an insulating behavior. Inset: The temperature dependence of conductance. The solid line is a fitting curve using the Mott variable range hopping model plus a constant.

Figure 3

Magnetoconductance oscillations of the nanotube A (background subtracted) at temperature 1.8 K. Magnetic field is applied parallel to the nanotube axis. The observed period is $3.53\pm 0.20\mathrm{kOe}$. This period is consistent with $h/e$ oscillations associated with the outer surface of the nanotube which has an average diameter of 125 nm.

Figure 4

Magnetoconductance oscillations of the nanotube B (background subtracted) at temperature 2.1 K. (a) Conductance oscillations for the nanotube B. The oscillations show an $h/e$ period associated with the outer surface of the nanotube with a diameter of 95 nm. (b) Conductance oscillations for different magnetic field directions, plotted as a function of the parallel component of the applied field. Evidently, the period depends solely on the parallel component of the applied field.

Figure 5

(a) Temperature dependence of FFT amplitude of the conductance oscillations. It is seen that the data points generally follow a $T^{-1/2}$ trend. (b) Comparison of magnetoresistance in parallel and perpendicular magnetic fields. (c) Fitting of magnetoconductance with weak-antilocalization (WAL) theory. The fitting yields a very short coherence length, which may due to the irrelevance of the formula in a strongly disordered system or because the nanotube geometry is not a real 2D sheet perpendicular to the applied field.

Figure 6

(a) Band dispersions for a disorder-free system are shown for $\varphi =0$ and $\varphi ={\varphi }_0/2$. The solid blue line marks the Fermi energy at $E_f=0.6$, which is used for the calculations in (b), (c), (d), and (e). (b) Length dependence of the conductance is shown for $\varphi =0$ and $\varphi ={\varphi }_0/2$ with $W=2.8$ and $E_f=0.6$. The solid lines are linear fits, from which we can extract the localization lengths to be ${\lambda }_{\mathrm{non}}=150a$ for $\varphi =0$ and ${\lambda }_{\mathrm{top}}=4800a$ for $\varphi ={\varphi }_0/2$. (c) The schematic phase diagram for $L=100a.$ It includes three different regions, denoted as (I) ${\lambda }_{\mathrm{non}},{\lambda }_{\mathrm{top}}>L$, (II) ${\lambda }_{\mathrm{non}}<L<{\lambda }_{\mathrm{top}}$, and (III) ${\lambda }_{\mathrm{non}}<{\lambda }_{\mathrm{top}}<L$. (d) The magnetoconductance oscillations as a function of magnetic flux $\varphi$ in region II for $W=2.8,L=800a$, and $E_f=0.6$. (e) The magnetoconductance oscillations for a system with variations in the cross-sectional area along the length of the nanotube. The cross section is taken to be $9a\times 9a$ for all calculations.

Figure 7

Theoretical analysis. (a), (b), (c), and (d) correspond to the green, blue, red, and black points in the phase diagram of Fig. 6. (a) AB oscillation and its higher harmonics at $W=0.2,E_f=0.8$. (b) AB oscillation at $W=0.8,E_f=1.3$. (c) AAS oscillation at $W=2.4,E_f=0.8$. (d) No oscillation at $W=3.6,E_f=1.3$. Insets are the FFTs of corresponding conductance oscillations.

Figure 8

(a) The fitted conductance (with the VRH model plus a constant) extended to temperatures below 50 K and the measured conductance. (b) The surface channel conductance obtained by subtracting the bulk conductance (VRH) from the total conductance. Surface conductance of $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$ MBE-grown films from Ref. [22] is shown for comparison.