Abstract
The nontrivial topology of threedimensional topological insulators dictates the appearance of gapless Dirac surface states. Intriguingly, when made into a nanowire, quantum confinement leads to a peculiar gapped Dirac subband structure. This gap is useful for, e.g., future Majorana qubits based on TIs. Furthermore, these subbands can be manipulated by a magnetic flux and are an ideal platform for generating stable Majorana zero modes, playing a key role in topological quantum computing. However, direct evidence for the Dirac subbands in TI nanowires has not been reported so far. Here, using devices fabricated from thin bulkinsulating (Bi_{1−x}Sb_{x})_{2}Te_{3} nanowires we show that nonequidistant resistance peaks, observed upon gatetuning the chemical potential across the Dirac point, are the unique signatures of the quantized subbands. These TI nanowires open the way to address the topological mesoscopic physics, and eventually the Majorana physics when proximitized by an swave superconductor.
Introduction
In topological insulator (TI) nanowires^{1,2,3}, the quantum confinement of the electron motion along the circumferential direction is described by the angularmomentum quantum number ℓ. In zero magnetic field, this quantization leads to the gap opening at the Dirac point, and the subbands become doublydegenerate (see Fig. 1a). When a magnetic flux Φ threads along the wire, the energy spectrum is modified in a nontrivial way as described by the following formula (under the simplified assumption of a circular wire crosssection):
Here, v_{F} is the Fermi velocity, R_{w} is the wire radius, and Φ_{0} = hc/e is the flux quantum; note that ℓ takes halfinteger values \(\pm \!\frac{1}{2},\pm \!\frac{3}{2},\ldots\!\) due to a Berry phase arising from the spinmomentum locking of the TI surface states^{1}. Interestingly, a spinnondegenerate gapless spectrum is restored when Φ is a halfinteger multiple of Φ_{0}; the spinmomentum locking in this gapless subband leads to the appearance of Majorana zero modes (MZMs) when the wire is proximitized by an swave superconductor^{3,4}. The tunability of the spinmomentum locking with Φ makes the subbands described by Eq. (1) a particularly interesting platform for topological mesoscopic physics.
In experiments, to elucidate the peculiar quantization effects, the TI nanowire should be bulkinsulating and as narrow as possible, preferably less than ~ 100 nm. Past efforts for TI nanowires^{5,6,7,8,9,10,11,12,13,14,15,16} have only been able to indirectly probe the quantized Dirac subbands, although bulkinsulating TI nanowires have been occasionally reported^{12,16,17,18,19,20,21}. In this work, we employed the vapor–liquid–solid (VLS) method using Au nanoparticles as catalysts^{5} and applied the concept of compensation, which has been useful for achieving bulkinsulation in bulk crystals^{22,23}. Specifically, we tuned the Bi/Sb ratio of \({({{\rm{Bi}}}_{1{\rm{x}}}{{\rm{Sb}}}_{{\rm{x}}})}_{2}{{\rm{Te}}}_{3}\) nanowires to a value that yields the most insulating properties. Fabrication of gatetunable fourterminal devices allows us to bring the chemical potential across the Dirac point, upon which we discovered unusual oscillatory behavior in the resistance near the Dirac point in very thin wires. This feature turns out to be the signature of the quantized Dirac subbands in TI nanowires as our theoretical calculations show.
Results
Structural and chemical analysis
During the VLS growth, the catalysts form a constantly oversaturated liquid alloy with the absorbed source materials, which then precipitate and form a crystal underneath. Using nominally20nmdiameter Au nanoparticles as catalysts, we obtain nanowires with a constant diameter between 20 and 100 nm, with a length of up to several μm (Fig. 1b). By using transmission electron microscopy (TEM) and energydispersive Xray (EDX) analysis (Fig. 1c), we identify the Au catalyst at the tip of most of the analyzed nanowires. The wires are found to be surrounded by a ~4nmthick amorphous oxide shell. The selectedarea diffraction patterns (SAED, Fig. 1c inset) indicate a high crystalline quality. We found hexagonal symmetry for a direction perpendicular the nanowire axis, which allows us to identify the growth direction to be \(\langle 11\bar{2}0\rangle\)type. The compositional analysis using EDX along the wire shows a constant stoichiometry (Bi_{0.68}Sb_{0.32})_{2}Te_{3} in the nanowire core and no incorporation of Au was detected (See Supplementary Note 2).
Temperature and gatedependent device resistance
In the following, we report five representative devices 1–5. The scanning electron microscope (SEM) picture of device 5 is shown in Fig. 1d, with its schematic depicted in Fig. 1e. The singlecrystalline nanowires are most likely of hexagonal shape (see Supplementary Note 1). The resistance R vs. temperature T curves shown in Fig. 1f present both insulating and metallic behavior; nevertheless, all these samples were bulkinsulating, which can be seen in their gatevoltage V_{G} dependences of R (Fig. 2a–c and Supplementary Fig. 4) showing a clear maximum, indicating that the Dirac point is crossed. The difference in the R(T) dependence is most likely explained by a slightly different electron density n of the samples in the absence of gating (n ≈ 0.38, 0.14, 0.42, − 0.2, 0.5 nm^{−1} relative to the Dirac point, according to our analysis described later).
In the R(V_{G}) traces, we found a hierarchy of fluctuation features. We observe semioscillatory features in the V_{G} dependence with the amplitude A_{I} ≈ 3, 2 and 1 kΩ for devices 1, 2, and 3, respectively. We will show these features (type I) to be the signature of subband crossings. They are not universal conductance fluctuations (UCF) whose main fingerprint would be a strong change in magnetic field and the lack of any clear periodicity and a random amplitude. In contrast, the present features of type I occur in a regular fashion, i.e., at regularly spaced gate voltages and with a largely uniform amplitude. Further, they are robust in small applied magnetic fields (for a detailed discussion on UCF see Supplementary Note 3). The other type of fluctuations have the amplitude A_{II} ≈ 0.5 kΩ (type II) and were changing with time (Supplementary Fig. 3). We speculate that they arise from timedependent conductance fluctuations due to charge traps or mobile scattering centers, similar to those observed in metallic nanowires of similar mesoscopic size^{24}, but they may also be affected by the presence of electronhole puddles^{25,26,27}. Averaging over several gatevoltage sweeps suppresses type II fluctuations while type I fluctuations remain unaffected, see Supplementary Note 4.
Model of gatevoltage dependent surface conduction
We now discuss the main observation of this work, that is, the reproducible semioscillatory feature in the R(V_{G}) curves. Due to the 1D nature of the energy bands in the nanowire, the density of states (DOS) diverges as \(1/\sqrt{E}\) at each of the subband’s edges as shown in Fig. 1a. This causes a subband crossing to have two contrasting effects on R: (i) The opening of a new conductance channel can decrease the resistivity as more charge can be transported. It can, however, also (ii) increase the resistivity by opening a new channel where electrons from other bands can scatter into. Thus, we have performed a straightforward theoretical calculation using an idealized model based on the surface state of a circular TI nanowire. (Small anisotropy effects arising both from the hexagonal shape of the wire and the anisotropic electrostatic environment are discussed in the Supplementary Notes 13–16). The effects of local impurities are taken into account using the Tmatrix formalism and we find that the experimental data is best described by weak impurities (see Supplementary Note 11). In Fig. 3, we schematically show how different subbands contribute to the conductivity: When a new channel is added (Fig. 3b), all other channels scatter efficiently into the new channel and, as a result, the conductivity contribution of each channel drops. This is by far the dominant effect and leads to pronounced peaks in R even when several channels are present. The diverging density of states of the newly added channel (Fig. 1a) is the main reason why this effect is so large, but it is further enhanced by a matrixelement effect originating in the topological protection of the surface states (Supplementary Note 11).
Hence, our calculations show that the resistance is expected to show a peak, each time a subband is crossed. This leads to equidistant peaks in Fig. 3, at μ = ℓℏv_{F}/R_{w}, when the conductivity is plotted as function of the chemical potential μ. In the experiment, however, the gate voltage V_{G}, rather than μ, is varied and we observe a superlinear dependence of the spacings of the main peaks (neglecting features of type II). This originates from the fact that the effective capacitance of the nanowire devices (which dictates the V_{G} dependence of the accumulated charge) must be computed from their quantum capacitance C_{Q} and geometric (or galvanic) capacitances C_{G} in series where C_{Q} is proportional to the DOS^{16}. In our experiment, C_{G} strongly dominates and the gate voltage directly controls the electron density n (n ≈ C_{G}ΔV_{G}/e, with ΔV_{G} measured from the Dirac point), rather than μ. This relation is used for the theory plots in Fig. 2. It also determines the peak positions indicated by dashed lines. We label the position of the peaks identified in the experimental data by the angular momentum quantum number ℓ of the added channel. The influence of the flat bottom gate geometry on the charge homogeneity around the wire is negligible, since it does not affect the position of peaks due to Kleintunneling physics^{16} (see Supplementary Note 13). When the chemical potential reaches the bottom of the first electron or the top of the first hole band (\(\ell =\!\pm\! \frac{1}{2}\)), the charge density is approximately zero in both cases and therefore there is only a single peak in the center for \(\ell =\!\pm \!\frac{1}{2}\). For large ℓ, the peak position scales with ℓ^{2}, which is peculiar to the subbands of Dirac origin, where the charge density grows as μ^{2} due to the 2D nature of the TI surface.
Subband crossings observed in experiment
It is striking that in Figs. 2a–c the theory can reproduce the essential features of our experiment, in particular the locations of the peaks in the averaged R(V_{G}) curves. While for devices 1 and 2 every peak can be indexed, type II features arising from disorder are more pronounced in device 3, such that some of the subband crossing features are not discernible despite averaging. To visualize the agreement between theory and experiment, we plot in Fig. 2d the rescaled gate voltage values of the peaks, ΔV_{G}/V_{0}, vs the subband index ℓ, and compare it to the theoretically calculated electron density n at the peak position (in units of 2π/R_{w}). In these units the rescaling factor is given by \({V}_{0}=\frac{2\pi e}{{R}_{{\rm{w}}}C}\), where C is the capacitance per length of the wire. The superlinear behavior in the V_{G}dependent subband crossings and the excellent agreement of theory and experiment is a direct signature of the quantumconfined Dirac surface states, which is observed here for the first time.
It is prudent to mention that the quantumconfined subband structure of TI nanowires have been indirectly inferred^{5,6,7,8,9,10,11,12,13,14,15,16} from the Aharonov–Bohm (AB)like oscillations of R as a function of the axial magnetic flux Φ, which is due^{1} to a periodic change in the number of occupied subbands at a given μ. In particular, the observation by Cho et al.^{12} that R at Φ = 0 takes a maximum when μ is near the Dirac point and changes to a minimum at some other μ was consistent with the gapped Dirac cone; however, the V_{G} dependence was not very systematic nor convincing in ref. ^{12}. A relatively systematic V_{G} dependence of R was recently reported for HgTe nanowires and was carefully analyzed^{16}; unfortunately, the Dirac point of HgTe is buried in the bulk valence band, hindering the characteristic superlinear behavior in the ΔV_{G} vs ℓ relation from observation.
Discussion
The realization of very thin, bulkinsulating TI nanowires and the observation of the quantumconfined Dirac subband structure reported here is crucial for exploring the mesosocpic physics associated with the topological surface states, not to mention their potential for future studies of MZMs. For example, the dependence of the spin degeneracy on the magnetic flux along the nanowires will give us a new tuning knob for mesoscopic transport phenomena, in which the spinmomentum locking can be varied. Also, it is an interesting insight that the charge inhomogeneity induced by gating on TI nanowires will not affect the energy locations of the subband crossings due to Kleintunneling physics. Therefore, the newgeneration TI nanowires realized here will open vast opportunities for future studies of topological mesoscopic physics including MZMs.
Methods
Nanowire synthesis
The \({({{\rm{Bi}}}_{1{\rm{x}}}{{\rm{Sb}}}_{{\rm{x}}})}_{2}{{\rm{Te}}}_{3}\) nanowires were synthesized by the VLS method using powders of Bi_{2}Te_{3} and Sb_{2}Te_{3} as starting materials in a twozone 50mm tubefurnace under a constant Ar flow. The Si/SiO_{2} substrates were first decorated with suspended 20nm Aunanoparticles with the help of PolyLLysine solution and then placed between the two zones (set to temperatures T_{1} and T_{2}) of the furnace. The temperature was first ramped to T_{1} = 500–510 ^{∘}C and T_{2} = 280 ^{∘}C within 60 min, kept at these values for 60 min, and finally reduced back to ambient temperature in roughly 4 h, while keeping a constant Ar flow of 600 SCCM.
Device fabrication
Our gatetunable fourterminal devices were fabricated on degeneratelydoped Si wafers covered by 280nm thermallygrown SiO_{2} which acts as a gate dielectric. Gold contact pads and a coordinate system were predefined by optical lithography. The asgrown nanowires were transferred by gently bringing together the surfaces of the prepatterned wafer and the growth substrate, and nanowires suitable for device fabrication were identified by optical microscopy. Per device, five to seven contacts with varying distances were defined by electron beam lithography, which was performed by exposing a PMMA A4 resist layer using a Raith PIONEER Two system. The contact area was cleaned using gentle oxygen plasma treatment and a dip in dilute hydrochloric acid shortly before metallization. Subsequently, 5nmthick Pt was sputterdeposited as a wetting layer and an additional 45nmthick Au layer was deposited by thermal evaporation (devices 4 & 5) or by sputtering (devices 1, 2, and 3), resulting in the structure schematically shown in Fig. 1e. The contact resistance was well below 1 kΩ for all of the devices. Following the transport measurements, SEM was used to determine the device geometry and the nanowire diameter. The distance between the centers of the voltage contacts were 0.5, 0.8, 1.0, 1.2, and 0.7 μm, and the diameter of the nanowires were 41, 32, 41, 43, and 29 nm for devices 1–5, respectively.
TEM analysis
TEM micrographs, as well as TEM diffraction patterns, were recorded by using a JEM 2200FS (JEOL) microscope operated at an acceleration voltage of 200 kV. A carbon film supported by a standard copper grid was used as sample carrier for TEM characterization. Elemental chemical analysis of the samples was done by EnergyDispersive Xray Analysis (EDX) performed with a JEOL Dry SD100GV detector.
Measurements
Transport measurements were performed in a liquidhelium cryostat in the temperature range of 2–300 K. The wafers were glued onto copper sample holders and manually bonded with 50μm gold wires using vacuumcured silver paste. For fast measurements, we used a quantum transport measurement system (SPECS Nanonis Tramea) in the lowfrequency lockin mode with the ac current of 100 nA at the frequency f ≈ 17 Hz, while the device is configured in a conventional four(device 3, 4, & 5) or three(device 1 & 2) terminal geometry. Gatevoltage sweeps were performed at various rates from 0.0125 V/s to 0.25 V/s while monitoring the sample temperature with a dedicated thermometer using a lowpower AC resistance bridge (Lakeshore Model 370).
Theoretical calculations
We consider the surface states of a quantum wire described by the 2D Dirac equation where antiperiodic boundary conditions in the transverse direction arise from curvatureinduced Berry phase effect^{1}. Disorder is modeled by a small density of randomly located local scattering potentials, which is treated within a (nonselfconsistent) Tmatrix approximation, which can be calculated in a fully analytic way, see Supplementary Note 9. Within our approximation, qualitative features are independent of the density of impurities; however, they do depend on the amplitude of the scattering potential. The Kubo formula was used to calculate the conductivity. Vertex corrections were ignored as a previous study showed that they have only a small, purely quantitative effect^{28}. Plots of resistivities were obtained from ρ = 1/(σ_{0} + σ(μ)), where σ_{0} is mainly used to avoid the divergence of ρ when σ = 0. It describes the presence of conductance contributions (e.g., from impurity bands on the surface or in the bulk) not taken into account in our approximation. Note that, although the experimental data are shown in resistance R, the theory calculates the resistivity ρ, because the transport is assumed to be in the diffusive regime. The dependence of μ on ΔV_{G} is computed from n(μ) = C_{G}ΔV_{G}/e, where n(μ) is the electron density along the wire and C_{G} is treated as a fitting parameter. Full details of our calculations including a discussion of effects arising from deviations of the circular shape of the wire are given in Supplementary Notes 13–16.
Data availability
The experimental data that support the findings of this study are available in figshare with the identifier doi:10.6084/m9.figshare.13524050 (ref. ^{29}).
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 741121) and was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under CRC 1238—277146847 (Subprojects A04, B01, and C02) as well as under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1  390534769. O.B. acknowledges the support from the Quantum Matter and Materials Program at the University of Cologne funded by the German Excellence Initiative.
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Y.A. conceived the project. F.M., O.B., D.F., and M.R. performed the growth and device experiments. S.R. performed the TEM analysis. H.F.L., supported by A.R., developed the theory. Y.A., F.M., H.F.L., O.B., and A.R. wrote the manuscript with inputs from all authors.
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Münning, F., Breunig, O., Legg, H.F. et al. Quantum confinement of the Dirac surface states in topologicalinsulator nanowires. Nat Commun 12, 1038 (2021). https://doi.org/10.1038/s41467021212303
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DOI: https://doi.org/10.1038/s41467021212303
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