Kondo-like zero-bias conductance anomaly in a three-dimensional topological insulator nanowire

Zero-bias anomalies in topological nanowires have recently captured significant attention, as they are possible signatures of Majorana modes. Yet there are many other possible origins of zero-bias peaks in nanowires—for example, weak localization, Andreev bound states, or the Kondo effect. Here, we discuss observations of differential-conductance peaks at zero-bias voltage in non-superconducting electronic transport through a 3D topological insulator (Bi1.33Sb0.67)Se3 nanowire. The zero-bias conductance peaks show logarithmic temperature dependence and often linear splitting with magnetic fields, both of which are signatures of the Kondo effect in quantum dots. We characterize the zero-bias peaks and discuss their origin.

The three dimensional topological insulator (3D TI) is a new class of material having metallic surface states inside a bulk band gap [1][2][3] . The topological surface states are characterized by gapless Dirac dispersions and novel properties such as momentum-spin locking, which were confirmed by angle-resolved photoemission spectroscopy (ARPES) [4][5][6] , scanning tunneling spectroscopy (STS) [7][8][9][10][11] and electrical transport measurements [12][13][14][15][16][17][18][19] . 3D TI nanowires with an insulating bulk, which can be described as a hollow metallic cylinder, have shown Aharonov-Bohm oscillations when a magnetic flux is threaded through the axis 20,21 and Coulomb blockade behavior when connected to metal electrodes through ultrathin TI tunnel barriers 22 . Recently, TI nanowires in the proximity of s-wave superconductors have been predicted to harbor Majorana bound states 23,24 , a transport signature of which is a zero-bias tunneling-conductance peak. Similar proximity-coupled topological nanowire systems of InSb 25 and Fe 26 have recently demonstrated Majorana-like zero-bias anomalies, leading to a worldwide effort to better understand the origin and behavior of zero-bias anomalies in topological wires. Zero-bias conductance peaks, not observed yet in 3D TI nanowires, are novel features which may be related to various physical origins such as weak antilocalization, Andreev bound states, and Kondo effect besides Majorana bound states [27][28][29] . Here we report the first observation of zero-bias anomalies in non-superconducting electronic transport through a 3D TI nanowire contacted by two metal electrodes. We also observed a logarithmic temperature dependence and linear splitting with magnetic fields, both of which imply that the zero-bias peaks result from Kondo-like effect.

Results
We have grown bulk (Bi 1.33 Sb 0.67 )Se 3 crystals and confirmed the existence of the surface states inside the bulk band gap by ARPES, as described elsewhere 30 . Using the "scotch tape method", we obtained naturally cleaved topological insulator nanowires on 300 nm SiO 2 /highly n-doped Si substrates. Subsequently, we characterized and chose nanowires having widths ≤ 100 nm and thickness > 12 nm by atomic force microscope, to avoid unwanted wavefunction hybridizations of top and bottom surfaces 31 . Widths of nanowires were measured again, after all the electrical measurements were done, by scanning electron microscopy (SEM). Immediately after we chose nanowires and identified their locations on a SiO 2 /Si chip, we spun an electron-beam resist (Microchem Corp. PMMA 950 A4) double layer at 4000 rpm to avoid possible contamination or excessive doping of the nanowire surfaces by long exposure to air 32 . Subsequently, we performed electron beam lithography, developed to remove the resist in the regions of source/drain electrodes, and finally deposited Ti/Al (2.5 nm /150 nm) with an Au 10 nm capping following a brief ion milling at low power. The devices were wire-bonded and cooled-down in a commercial dilution refrigerator immediately following lift-off in acetone for 1 hour. Figure 1 completed device, where a small section of the nanowire having dimensions of width ~90 nm, length ~90 nm, and thickness ~13 nm is contacted by a source and a drain electrode.
Here, we report our two-probe differential conductance measurements made between Ti/Al source-drain electrodes driven normal (non-superconducting). To do this, we applied small perpendicular magnetic fields of 30 − 200 mT, which is above the critical field of the Al electrodes ~12 mT (see Supplementary Info.). Figure 1(b) shows that as gate voltage decreased, differential conductance (dI/dV) at zero-bias voltage decreased due to the decrease in the density of states, and finally as gate voltages exceeded V g = − 80 V, the conductance reached its minimum implying that the chemical potential is tuned near the charge neutrality point and therefore a minimum density of states. Due to the low electron doping of our Sb-doped crystals and exfoliated devices as reported previously 21,30 , we are able to tune the chemical potential effectively below the bottom of the bulk conduction band using a back-gate through SiO 2 in our nanowire device. In all range of gate voltages, we observed reproducible conductance fluctuations reminiscent of Coulomb charging effect or phase coherent interference effects such as Fabry-Perot and universal conductance fluctuations. Figure 1(c) shows differential conductance spectroscopy, a two-dimensional plot of G(V sd ,V g ) measured while varying dc source-drain bias-voltages V sd at different gate voltages V g . The transport spectroscopy unexpectedly showed enhanced zero-bias conductance persisting with gate voltages at V g < − 50 V where the background conductance ≤ 5 e 2 /h. Figure 1(d) shows differential conductance as a function of bias voltages at V g = − 53.7 V. We have further performed differential conductance spectroscopy measurements in wider ranges of gate voltages as shown in Fig. 2. Zero-bias conductance peaks having amplitudes ranging from 0.05 to 0.7 e 2 /h were observed in a wide range of gate voltages. Often, zero-bias conductance dips appeared in certain regions of gate voltages as shown in Fig. 2(a,c). In the rest of this letter, we discuss the possible origin of the observed zero-bias conductance anomaly.

Discussions
Recently, zero-bias peaks observed in 1D topological superconductors, i.e., a strong spin-orbit coupled semiconducting nanowires in the proximity of superconductors under a parallel magnetic field, captured significant attention. The zero-bias anomaly in those systems have been explained as a signature of Majorana zero modes 23 . There have been similar theoretical proposals on producing Majorana zero modes in topological insulator (Bi 2 Se 3 ) nanowires proximity-coupled to s-wave superconductors under a parallel magnetic field 23,24 . However, the absence of superconductivity in a magnetic field higher than the critical field of the Al electrodes in our setup excludes the Majorana zero modes among possible explanations for our zero-bias peaks. Due to the lack of superconductivity, Andreev states bound to superconductors also do not provide an explanation for our observed zero-bias conductance peaks.
To understand the origin of the zero-bias conductance peaks, we have performed the transport measurements at different magnetic fields. Figure 3 shows two different ways in which peaks evolve with magnetic fields at fixed gate voltages (see Supplementary Fig. S4 for 2D G(V sd ,V g ) plot). Peaks having large amplitude (> 0.4 e 2 /h, such as the one) measured at V g = − 69 V, split with magnetic fields (Fig. 3(a,b)). The formation of zero-bias peaks and the splitting of the peaks with a magnetic field are reminiscent of the Kondo effect in quantum dots. A characteristic feature of the Kondo effect in quantum dots is that the zero-bias peak splits with magnetic field at = ± µ g V B B (the Zeeman splitting), where μ B is the Bohr magneton. Assuming the relation of the zero-bias peaks to Kondo effect, we estimate the Zeeman g-factor of the surface states in our topological insulator (Bi 1.33 Sb 0.67 )Se 3 nanowire is ~15 from the blue line in Fig. 3(b), which is roughly half the reported g factor value of the bulk states in Bi 2 Se 3 33 .
On the other hand, peaks of relatively small amplitude (< 0.1 e 2 /h), such as that observed at V g = − 65 V, switched to conductance dips without splitting as the magnetic field was increased (Fig. 3(c)). This collapse of the zero-bias peaks having small peak amplitudes (< 0.1 e 2 /h) with a magnetic field, without being splitted, and having large background conductance is most likely due to lower Kondo temperatures (T ~ T K ) at these gate voltages 34 .
To further investigate the observed Kondo-like anomaly at zero-bias voltage, we measured temperature dependence of the zero-bias peaks. Figure 4 shows that conductance of peaks having both small and large Each gate voltage whose differential conductance was plotted is marked with yellow dotted-lines in the two- dimensional plots (a,d,f).
amplitudes increases logarithmically with decreasing temperature and eventually saturated to a constant conductance value as T → 0 35 (see also Supplementary Fig. S4 for 2D G(V sd ,V g ) plot). In the high-temperature limit, Small peaks with large background conductance vanished at relatively low temperatures < 1 K as shown in Fig 4(a), and the large peaks with relatively small conductance background did not vanish up to the temperature limit (~1.8 K) in our dilution refrigerator. Both the logarithmic temperature dependence and the peak splitting with magnetic field suggest that the zero-bias peaks are Kondo-like effects in a quantum dot. We obtain rough estimates of the Kondo temperatures, T K, which ranges from 300 mK to 5 K, by equating the full-width at half-maximum of different zero-bias conductance peaks at the base temperature to 2k B T K /e 35,36 .
The Kondo effect describes the upturn of the resistance of metals at low temperatures when magnetic impurities are added 37 . A similar Kondo effect in semiconducting nanostructures results from a bound state formed between a local spin in a quantum dot and the electrons in the reservoir of source/drain electrodes. We find that the overall background conductance values in our experiments are relatively high (3e 2 /h < G < 5e 2 /h) compared to the conductance values (G < 2e 2 /h) previously reported in quantum dots showing the Kondo effect 35,36 , and that our topological insulator nanowire device behaves as an open quantum dot. Coulomb charging is required in order for the Kondo effect to be observed in quantum dots; this is usually observed in a system having low dot-electrode transmission probabilities and conductance less than 2e 2 /h. However, Coulomb charging effects have often been observed in open quantum dots where 2 e 2 /h < G < 6 e 2 /h [38][39][40] . Therefore, high overall conductance values of 3e 2 /h < G < 5e 2 /h does not necessarily exclude the possibility of Coulomb charging and the Kondo effect in our open quantum dot device. The large, oval regions of low-conductance shown in the 2D transport spectroscopy G(V sd , V g ) of Fig. 2(a) at − 54 < V g < − 53, − 52 < V g < − 51and − 50.5 < V g < − 49.5 are consistent with a diamond-shaped Coulomb blockade in the presence of high lead-dot transparency. Although we do not clearly observe the even-odd parity behavior typical for the Kondo effect in quantum dots, several experiments of Kondo resonances have reported the absence of even-odd parity behavior [41][42][43] , due to either the formation of higher spin states (spin-triplet Kondo resonances) or correlation effects due to electron-electron interaction dominating over the confinement effect in quantum dots. We often observe zero-bias peaks that increased with increasing magnetic field (see Supplementary Info.). This has previously been explained by singlet-triplet transitions of electron spin states in a quantum dot 27 . The magnetic-field-induced zero-bias peaks persist up to magnetic fields as large as B = 630 mT. We currently do not understand the physical mechanism of these magnetic-field-induced zero-bias peaks, although it is possible that they are related to the unique spin-momentum locking on the surface of topological insulators. In this case, it may be more energetically favorable to create a multi-particle triplet state than a single-particle spin-1/2 state. To our knowledge, no theoretical and experimental research has been reported on the subject and further studies are required to understand the physics of possible singlet-triplet transitions in topological insulator nanowires. Pikulin et al. pointed out in their simulation studies that weak antilocalization by disorder can also be a source of zero-bias conductance peaks at non-zero magnetic fields which break time-reversal symmetry 29 . However, this scenario is only possible when the 1D system has particle-hole symmetry resulting from superconductivity. Without the particle-hole symmetry, weak antilocalization effects should disappear when a magnetic field breaks time-reversal symmetry. The behavior of magnetic-field-induced zero-bias peaks persisting up to B = 630 mT in our device cannot be explained by a weak antilocalization effect, considering the absence of superconductivity. Moreover, lowering background conductance by increasing the tunnel barriers between electrodes and nanowires by a gate voltage is expected to suppress the zero-bias peaks originating from weak antilocalization. Our observation is opposite to this scenario: zero-bias peaks are absent in the gate voltage regions of V g ≥ − 40 V where conductance is higher, but more prominent as the gate voltage drops below − 50 V and conductance decreases. This observation implies that weak antilocalization is not the origin of the zero-bias peaks in our device.

Conclusion
In conclusion, we have observed zero-bias conductance peaks in non-superconducting transport through a topological insulator nanowire contacted by source-drain electrodes. The logarithmic temperature dependence and splitting of the peaks with magnetic fields strongly imply that the zero-bias peaks occur from Kondo-like origins in a quantum dot. Additional features differing from typical Kondo effects in quantum dots such as high background conductance (> 2 e 2 /h) and absence of even-odd parity behavior were observed, which may be consistent with a singlet-triplet Kondo effect and related to the topological nature of the nanowires. Methods Device fabrication and measurement. Topological insulator nanowires were obtained by mechanical exfoliation ('scotch tape method') from bulk crystals of Bi 1.33 Sb 0.67 Se 3 , which were grown by a modified floating zone method 27 . After mechanical exfoliations of bulk crystals onto 300 nm SiO 2 /highly n-doped Si substrates, the nanowires were found under optical microscope. The dimensions of nanowires were measured by Atomic Force Microscopy and Scanning Electron Microscopy. Electron beam lithography and metal (Ti/Al/ Au = 2.5 nm/150 nm/10 nm) deposition were used to pattern two-terminal devices on the nanowires. Completed devices were wire-bonded and cooled-down in a commercial dilution refrigerator (base temperature = 16 mK). The electrical measurements were performed using standard ac lock-in techniques.