The motion of an electron and its spin are generally not coupled. However in a onedimensional material with strong spinorbit interaction (SOI) a helical state may emerge at finite magnetic fields, where electrons of opposite spin will have opposite momentum. The existence of this helical state has applications for spin filtering and cooper pair splitter devices and is an essential ingredient for realizing topologically protected quantum computing using Majorana zero modes. Here, we report measurements of a quantum point contact in an indium antimonide nanowire. At magnetic fields exceeding 3 T, the 2 e ^{2}/h conductance plateau shows a reentrant feature toward 1 e ^{2}/h which increases linearly in width with magnetic field. Rotating the magnetic field clearly attributes this experimental signature to SOI and by comparing our observations with a numerical model we extract a spinorbit energy of approximately 6.5 meV, which is stronger than the spinorbit energy obtained by other methods.
Introduction
The spinorbit interaction (SOI) is a relativistic effect where a charged particle moving in an electric field E with momentum k and velocity v = k/m _{0}, experiences an effective magnetic field B _{SO} = (−1/m _{0} c)k × E in its rest frame. The magnetic moment of the electron spin, μ = e S/m _{0}, interacts with this effective magnetic field, resulting in a spinorbit Hamiltonian H _{SO} = −μ.B _{SO} that couples the spin to the orbital motion and electric field. In crystalline materials, the electric field arises from a symmetry breaking that is either intrinsic to the underlying crystal lattice in which the carriers move, known as the Dresselhaus SOI^{1}, or an artificially induced asymmetry in the confinement potential due to an applied electric field, or Rashba^{2} SOI. Wurtzite and certain zincblende nanowires possess a finite Dresselhaus SOI, and so the SOI is a combination of both the Rashba and Dresselhaus components. For zincblende nanowires grown along the [111] growth direction the crystal lattice is inversion symmetric, and so only a Rashba component to the spinorbit interaction is thought to remain^{3}.
Helical states^{4, 5} have been shown to emerge in the edge mode of twodimensional (2D) quantum spin hall topological insulators^{6, 7}, and in quantum wires created in GaAs cleaved edge overgrowth samples^{8}. They have also been predicted to exist in carbon nanotubes under a strong applied electric field^{9}, RKKY systems^{10}, and in InAs and InSb semiconducting nanowires where they are essential for the formation of Majorana zero modes^{11,12,13}. Although the signatures of Majoranas have been observed in nanowiresuperconductor hybrid devices^{14, 15}, explicit demonstration of the helical state in these nanowires has remained elusive. The measurement is expected to show a distinct experimental signature of the helical state—a return to 1 e ^{2}/h conductance at the 2 e ^{2}/h plateau in increasing magnetic field as different portions of the band dispersion are probed^{4, 5, 16}. While ballistic transport through nanowire quantum point contacts (QPCs) is now standard^{17, 18}, numerical simulations have shown that the visibility of this experimental signature critically depends on the exact combination of geometrical and physical device parameters^{16}.
Here, we observe a clear signature of transport through a helical state in a QPC formed in an InSb nanowire when the magnetic field has a component perpendicular to the spinorbit field. We show that the state evolves under rotation of the external magnetic field, disappearing when the magnetic field is aligned with B _{SO}. By comparing our data to a theoretical model, we extract a spinorbit energy E _{SO} = 6.5 meV, significantly stronger than that measured in InSb nanowires by other techniques.
Results
Emergence of the helical gap a quantum point contact
Figure 1a shows a schematic image of a typical QPC device. An InSb nanowire is deposited on a degenerately doped silicon wafer covered with a thin (20 nm) SiN dielectric. The QPC is formed in the nanowire channel in a region defined by the source and drain contacts spaced ~325 nm apart. The chemical potential μ in the QPC channel, which sets the subband occupation, is controlled by applying a voltage to the gate V _{g}. The electric field in the nanowire, E, generated by the backgate and the substrate that the nanowire lies on, both induce a structural inversion asymmetry that results in a finite Rashba spinorbit field. As the wire is translationally invariant along its length, the spinorbit field, B _{SO}, is perpendicular to both the electric field and the wire axis. The effective channel length, L _{QPC} ~245 nm, as well as the shape of the onset potential λ ~ 80 nm , are set by electrostatics which are influenced by both the thickness of the dielectric and the amount of electric field screening provided by the metallic contacts to the nanowire (Fig. 1b). Here, we report measurements from one device. Data from an additional device that shows the same effect, as well as control devices of different channel lengths and onset potentials, is provided in the Supplementary Figs. 5–7.
The energymomentum diagrams in Fig. 1c–e show the dispersion from the onedimensional (1D) nanowire model of refs. ^{4,} ^{5} including both SOI with strength α and Zeeman splitting E _{Z} = gμ _{B} B, where g is the gfactor, μ _{B} the Bohr magneton and B the magnetic field strength. These dispersion relations explain how the helical gap can be detected: without magnetic field, the SOI causes the first two spin degenerate subbands to be shifted laterally in momentum space by ±k _{SO} = m ^{∗} α/ħ ^{2} with m ^{∗} the effective electron mass, as electrons with opposite spins carry opposite momentum, as shown in Fig. 1c. The corresponding spinorbit energy is given by \({E_{{\rm{SO}}}} = {\hbar ^2}k_{{\rm{SO}}}^2/2{m}\) ^{∗}. However, here Kramers degeneracy is preserved and hence the plateaus in conductance occur at integer values of G _{0} = 2 e ^{2}/h, as for a system without SOI. Applying a magnetic field perpendicular to B _{SO} the spin bands hybridize and a helical gap, of size E _{Z} opens as shown in Fig. 1d. When the chemical potential μ is tuned by the external gate voltage, it first aligns with the bottom of both bands resulting in conductance at 1·G _{0} before reducing from 1·G _{0} to 0.5·G _{0} when μ is positioned inside the gap. This conductance reduction with a width scaling linearly with increasing Zeeman energy, is a hallmark of transport through a helical state. When the magnetic field is orientated at an angle θ to B _{SO}, the size of the helical gap decreases as it is governed by the component of the magnetic field perpendicular to B _{SO}, as shown in Fig. 1e. Additionally, the two subband bottoms also experience a spin splitting giving rise to an additional Zeeman gap. For a general angle θ, the QPC conductance thus first rises from 0 to 0.5·G _{0,} then to 1·G _{0}, before dropping to 0.5·G _{0} again and the helical gap takes the form of a reentrant 0.5·G _{0} conductance feature. By comparing to a 1D nanowire model, we can extract both the size of the helical gap \({E_{{\rm{helical}}}} \approx {E_{\rm{Z}}}\sin \theta \) and the Zeeman shift \({E_{{\rm{Zeeman}}}} \approx {E_{\rm{Z}}}\cos \theta \) (Supplementary Note 2 and Supplementary Figs. 1, 9 and 10). This angle dependency is a unique feature of SOI and can be used as a decisive test for its presence in the experimental data.
Magnetic field dependence of the helical gap
Figure 2 shows the differential conductance dI/dV of our device at zero sourcedrain bias as a function of gate and magnetic field. Here and in the data shown in Fig. 3, the magnetic field B is offset at a small angle θ = 17° from B _{SO} in the x–y plane (Fig. 2a). We determine that our device has this orientation from the angle dependence of the magnetic field, by clearly resolving the 1·G _{0} plateau before the reentrant conductance feature, which is reduced at larger angles (Fig. 4 and Supplementary Note 2 and Supplementary Fig. 10). For low magnetic fields, we observe conductance plateaus quantized in steps of 0.5·G _{0}, as typical for a QPC in a spin polarizing Bfield with or without SOI. Above B = 3 T, the 1·G _{0} plateau shows a conductance dip to 0.5·G _{0}. This reentrant conductance feature evolves continuously as a function of magnetic field, before fully enveloping the 0.5·G _{0} plateau for magnetic fields larger than around 5.5 T. Line traces corresponding to the colored arrows in Fig. 2b are shown in Fig. 2d. The feature is robust at higher temperatures up to 1 K, as well across multiple thermal cycles (Supplementary Fig. 3).
Using the 1D nanowire model with θ = 17°, we find that the helical gap feature vanishes into a continuous 0.5·G _{0} plateau when E _{Z}=2.4E _{SO}. Using the extracted gfactor g = 38 of our device (Fig. 3 and Supplementary Note 2), we find a lower bound for the spinorbit energy E _{SO} = 5.5 meV, corresponding to a spinorbit length l _{SO} = 1/k _{SO} ≈ 22 nm. For a second device, we extract a similar value E _{SO} = 5.2 meV (Supplementary Figs. 5 and 6). Recently, it has been highlighted that the visibility of the helical gap feature depends crucially on the shape of the QPC potential^{16}. To verify that our observation is compatible with SOI in this respect, we perform selfconsistent simulations of the Poisson equation in Thomas–Fermi approximation for our device geometry. The resulting electrostatic potential is then mapped to an effective 1D QPC potential for a quantum transport simulation using parameters for InSb (for details, see Supplementary Note 1 and Supplementary Fig. 1). These numerical simulations, shown in Fig. 2c, fit best for l _{SO} = 20 nm (E _{SO} = 6.5 meV) and agree well with the experimental observation, corroborating our interpretation of the reentrant conductance feature as the helical gap.
Voltage bias spectroscopy, as shown in Fig. 4a confirms that this state evolves as a constant energy feature. By analyzing the size of conductance triangles in voltage bias spectroscopy data at a range of magnetic fields, we directly convert the development of the initial 0.5·G _{0} plateau, as well as the reentrant conductance feature to energy (Fig. 4c). From the evolution of the width of the first 0.5·G _{0} plateau, we can calculate the gfactor of the first subband g = 38 ± 1. This number is consistent with the recent experiments, which reported gfactors of 35–50^{19, 20}. Comparing the slopes of the Zeeman gap and the helical gap \({E_{\rm{h}}}/{E_{\rm{Z}}} \approx \tan \theta \) provides an alternative way to determine the offset angle θ. We find θ = 13° ± 2° which is in reasonable agreement with the angle determined by magnetic field rotation.
Angle dependence of the helical gap
To confirm that the reentrant conductance feature agrees with spinorbit theory, we rotate the magnetic field in the plane of the substrate at a constant magnitude B = 3.3 T, as shown in Figs. 4a, b. When the field is rotated towards being parallel to B _{SO}, the conductance dip closes, while when it is rotated away from B _{SO}, the dip increases in width and depth. In contrast, when the magnetic field is rotated the same amount around the y–z plane, which is largely perpendicular to B _{SO}, there is little change in the reentrant conductance feature, as shown in Figs. 4c, d. Rotating through a larger angle in the x–y plane (Figs. 4e, f) shows that this feature clearly evolves with what is expected for spin orbit. Our numerical simulations in Fig. 4g agree well with the observed experimental data. The small difference in the angle evolution between the numerical simulations and experimental data can be attributed to imperfect alignment of the substrate with the x–y plane.
Discussion
The extracted SO energy of 6.5 meV is significantly larger than that obtained via other techniques, such as weak antilocalization (WAL) measurements^{21}, and quantum dot spectroscopy^{20}. This is not entirely unexpected, due to the differing geometry for this device and different conductance regime it is operated in. Quantum dot measurements require strong confinement, and so the Rashba SOI is modified by the local electrostatic gates used to define the quantum dot. Weak antilocalization measurements are performed in an open conductance regime, however they assume transport through a diffusive, rather than a ballistic channel. Neither of these measurements explicitly probe the spinorbit interaction where exactly one mode is transmitting in the nanowire, the ideal regime for Majoranas, and so the spinorbit parameters extracted from QPC measurements offer a more accurate measurement of the SOI experienced by the Majorana zero mode. Also, the SOI in a nanowire can be different for every subband, and it is expected that the lowest mode has the strongest SOI due to a smaller confinement energy^{3}. Additionally, the finite diameter of the nanowire, together with impurities within the InSb crystal lattice^{23} both break the internal symmetry of the crystal lattice which can modify the SOI and may contribute a nonzero Dresselhaus component to the spinorbit energy that has not been previously considered.
While high quality quantized conductance measurements have been previously achieved in short channel devices^{17} (L ~ 150 nm), the channel lengths required for observing the helical gap are at the experimental limit of observable conductance quantization. As shown in Supplementary Figs. 1, 8 and 10, small changes in the QPC channel length, spinorbit strength or the QPC potential profile are enough to obscure the helical gap, particularly for wires with weaker SOI. We have fabricated and measured a range of QPCs with different length and potential profiles, and only two devices of L ~ 300 nm showed unambiguous signatures of a helical gap. Possibly some of the other devices did not show clear signatures because they had weaker SOI.
Several phenomena have been reported to result in anomalous conductance features in a device such as this. Oscillations in conductance due to Fabry–Perot resonances are a common feature in clean QPCs. Typically the first oscillation at the front of each plateau is the strongest and the oscillations monotonically decrease in strength further along each plateau^{16, 22}. In our second device, we clearly observe Fabry–Perot conductance oscillations at the beginning of each plateau, however these oscillations are significantly weaker than the subsequent conductance dip. Furthermore we observe Fabry–Perot oscillations at each conductance plateau, while the reentrant conductance feature is only present at the 1·G _{0} plateau. Additionally, the width of the Fabry–Perot oscillations does not change with increasing magnetic field, unlike the observed reentrant conductance feature. A local quantum dot in the Coulomb or Kondo regimes can lead to conductance suppression, which increases in magnetic field^{24}. However both effects should be stronger in the lower conductance region, and exists at zero magnetic field, unlike the feature in our data. Additionally, a Kondo resonance should scale with bias voltage V _{sd} = ±gμ _{B} B/e as a function of external magnetic field, decreasing instead of increasing the width of the region of suppressed conductance. Given the gfactor measured in InSb quantum dots, and its variation with the angle of applied magnetic field g = 35–50^{20}, we can exclude both these effects. Similarly the Fano effect and disorder can also induce a conductance dip, but these effects should not increase linearly with magnetic field. The 0.7 anomaly occurs at the beginning of the plateau, and numerical studies have shown it does not drastically affect the observation of the helical gap^{25}.
In conclusion, we have observed a return to 1 e ^{2}/h conductance at the 2 e ^{2}/h plateau in a QPC in an InSb nanowire. The continuous evolution in increasing magnetic field and the strong angle dependence in magnetic field rotations agree with a SOI related origin of this feature and distinguish it from Fabry–Perot oscillations and other gfactor related phenomena. Additional confirmation is given by numerical simulations of an emerging helical gap in InSb nanowires. The extracted spinorbit energy of 6.5 meV is significantly larger than what has been found by other techniques, and more accurately represents the true spinorbit energy in the first conduction mode. Such a large spinorbit energy reduces the requirements on nanowire disorder for reaching the topological regime^{26}, and offers promise for using InSb nanowires for the creation of topologically protected quantum computing devices.
Methods
Device fabrication
The InSb nanowires were grown using the metalorganic vapor phase epitaxy (MOVPE) process, and are grown along the [111] growth direction in a zincblende crystal structure^{27}. The InSb nanowires were deposited using a deterministic deposition method on a degenerately doped silicon wafer. The wafer covered with 20 nm of low stress LPCVD SiN which is used as a high quality dielectric. Electrical contacts (Cr/Au, 10 nm/110 nm) defined using ebeam lithography were then evaporated at the ends of the wire. Before evaporation the wire was exposed to an ammonium polysulfide surface treatment and short helium ion etch to remove the surface oxide and to dope the nanowire underneath the contacts^{17}.
Measurements
Measurements are performed in a dilution refrigerator with base temperature ~20 mK fitted with a 3axis vector magnet, which allowed for the external magnetic field to be rotated insitu. The sample is mounted with the substrate in the x–y plane with the wire orientated at a small offset angle θ = 17° from the xaxis. We measure the differential conductance G = dI/dV using standard lockin techniques with an excitation voltage of 60 μV and frequency f = 83 Hz. Additional resistances due to filtering are subtracted to give the true conductance through the device. The helical gap, Zeeman gap, and subband spacing reported in Fig. 3c were extracted from analysis of the full voltage bias conductance diamonds shown in Fig. 3a and Supplementary Fig. 2. The subband spacing was extracted by summing the widths of the 0.5 and 1 plateaus, the helical gap and Zeeman gap from their respective conductance diamonds.
Numerical transport simulations
We use the method of finite differences to discretize the 1D nanowire model of ref. ^{5}. In order to obtain a 1D QPC potential, we solve the Poisson equation selfconsistently for the full threedimensional (3D) device structure treating the charge density in the nanowire in Thomas–Fermi approximation. To this end, we use a finite element method, using the software FEniCS^{28}. The resulting 3D potential is then projected onto the lowest nanowire subband and interpolated using the QPC potential model of ref. ^{16}. Transport in the resulting tightbinding model is calculated using the software Kwant^{29}.
Code availability
All code used for the simulations in this study is available from the 4TU.ResearchData repository at doi:10.4121/uuid:f82b6a24201f4de794cbafc95ad1adea (http://doi.org/10.4121/uuid:f82b6a24201f4de794cbafc95ad1adea).
Data availability
All data underlying this study are available from the 4TU.ResearchData repository at doi:10.4121/uuid:686925fd017c49dfa92b3dc84138c513 (http://doi.org/10.4121/uuid:686925fd017c49dfa92b3dc84138c513).
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Acknowledgements
We gratefully acknowledge D. Xu, S. Goswami, D. van Woerkom and R.N. Schouten for their technical assistance and helpful discussions. This work has been supported by funding from the Netherlands Foundation for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO/OCW), the Office of Naval Research, Microsoft Corporation Station Q, the European Research Council (ERC) and an EU MarieCurie ITN.
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J.K. and F.P. fabricated the samples, J.K., M.C.C., and F.P. performed the measurements with input from M.W. Ö.G. contributed to the experiments. M.W., M.N., and A.V. developed the theoretical model and performed simulations. D.C., S.R.P., and E.P.A.M.B. grew the InSb nanowires. All authors discussed the data and contributed to the manuscript.
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Kammhuber, J., Cassidy, M.C., Pei, F. et al. Conductance through a helical state in an Indium antimonide nanowire. Nat Commun 8, 478 (2017). https://doi.org/10.1038/s4146701700315y
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