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A 1D conductor with strong SOC is predicted1,2,8 to represent a viable host for Majorana bound states. These zero-energy states feature characteristic non-Abelian exchange statistics8 and can be created by mimicking spinless p-wave Cooper pairing using a semiconductor nanowire with a helical state and inducing s-wave superconductivity. InAs and InSb nanowires are promising host materials to explore the existence and nature of Majorana bound states9,10. To this end, it is essential to both establish transport in 1D subbands and induce a helical state in the nanowire. The usual mechanism that is considered to open a helical gap involves an external Zeeman field oriented perpendicular to the uniaxial spin–orbit field4. The magnitude of the spin–orbit energy relative to the Zeeman energy is partly responsible for the size of the topological energy gap that will protect the zero-energy Majorana modes11. However, Oreg et al.2,12 and Stoudenmire et al.13 have pointed out that such an energy gap can also result from strong electronic correlations. Several mechanisms have been proposed along these lines: for example, spin-flipping two-particle backscattering7 and hyperfine interaction between nuclear spins and a Luttinger liquid14, both of which can open a gap. The latter mechanism has been invoked to explain a conductance reduction by a factor of two at low temperatures in a GaAs quantum wire15, but no re-entrant behaviour is predicted within this framework.

Other than Quay et al.3, we report on a re-entrant conductance feature in the lowest subbands of InAs nanowire quantum point contacts (QPCs), which offer the desired strong SOC (see Supplementary Section 1). Moreover, our proposed spin-mixing mechanism does not necessarily rely on external time-reversal symmetry-breaking terms: while the effect is pronounced in the presence of an external magnetic field, it persists also in its absence. Guided by the observation16 of the Landé g factor enhancement for the lowest subband17 and by signatures of the 0.7 anomaly18, we identify the important role of exchange interaction. Following ref. 7, we ascribe the re-entrant behaviour at zero magnetic field to the combined presence of 1D confinement, Rashba spin–orbit coupling and Coulomb interaction.

The experimental set-up and the main results are summarized in Fig. 1. A scanning electron microscopy (SEM) image of the investigated device is depicted in Fig. 1a. The contact separation is 2.94 μm and the wire diameter is 100 nm. Top-gate fingers of 180 nm width can be used to form local QPCs in the nanowire. Enhanced gate coupling is enabled via a high-k dielectric (LaLuO3). The Si/SiO2 back gate can induce an additional electric field in the QPCs. The conductance G in Fig. 1b is quantized in integer steps of 2e2/h, while at large magnetic field B half-integer steps emerge, reflecting the Zeeman splitting of the first subband, which is determined by a g factor of ∼7.016. A single re-entrant conductance feature appears reproducibly on the first quantized conductance plateau (G = 2e2/h) for magnetic fields smaller than 5 T. It exhibits a non-monotonous behaviour, where the conductance drops by up to a factor of two and increases again to the integer quantized conductance value at higher energies, as depicted in Fig. 1b and in the inset. The energy spectrum of the QPCs is reflected in the transconductance measurement as a function of the out-of-plane magnetic field. As shown in Fig. 1c, the subband edges rise in energy for larger magnetic fields and eventually converge towards Landau levels16,19. The riser at the right edge of the re-entrant conductance feature is related to the high-transconductance line marked by the green dot in Fig. 1c, which continuously evolves into the riser that separates the first two Zeeman-split plateaus at large magnetic fields. This behaviour is expected for a pseudogap that develops into a generic Zeeman splitting. At the single-particle level the physics is rather simple. Owing to the inversion asymmetry of the device, Rashba SOC gives rise to a spin-dependent shift of the subbands in momentum-space (Fig. 2a). For an electric field E perpendicular to the substrate plane, the spin–orbit field Bso is aligned perpendicular to the nanowire and in the plane of the substrate (see Fig. 2d, inset). The spin–orbit energy Eso = m∗αR2/2ℏ2 can be calculated from the Rashba coefficient αR and it represents the energy difference between the degeneracy point at wave number k = 0 and the band minima, where m∗ is the effective electron mass (see Fig. 2a). In the presence of a strong uniaxial spin–orbit field, a perpendicular magnetic field is expected to open a partial Zeeman gap, giving rise to quasi-helical transport for EZ = gμBB ≃ Eso4 (see Fig. 2b). The formation of a helical state becomes manifest by the appearance of a re-entrant conductance plateau at e2/h inside the larger 2e2/h-plateau related to the opening of the pseudogap (Fig. 2d, top panel). In our experiment, the gap widens roughly proportional to gμBB (see Fig. 3). This aspect of the re-entrant feature is in accordance with the simple single-particle picture just described. Moreover, with increasing magnetic field, the gap evolves towards the generic Zeeman splitting of a spin-degenerate band (see Fig. 2c), which is reflected by the emergence of a plateau at a conductance of e2/h (Fig. 2d, bottom panel). However, the experimental observation of the pseudogap feature down to B = 0 T (see Fig. 2e) reveals the need to go beyond the single-particle picture. The effect we propose to cause the zero-field gap is the combination of Coulomb interaction and the breaking of axial spin symmetry, which can be induced, for instance, by the joint effect of spin–orbit coupling and quantum confinement20. Since spin is not a conserved quantity, the effective interaction term arising in this framework is correlated two-particle backscattering, which is resonant at k = 0, similar to the single-particle backscattering caused by the magnetic field. An estimation for the corresponding gap is (see Supplementary Section 2)

Using m∗ = 0.026 me (me being the free electron mass), the relative permittivity ɛr = 15.15, the confinement energy ℏω0 = 7 meV (ref. 16) and αR = 1.2 eV Å, which is discussed below, only the screening length d remains unknown. Assuming d = 1 nm, compatible with the upper bound for the electron density in the QPC segments, and considering the expected exchange-mediated renormalization of the gap by a factor of up to 3 (ref. 14), we estimate a pseudogap of the order of Δhel ≈ 2.5 meV, which is compatible with the experimental findings. In fact, using the gate lever arm (≈0.04 eV V−1), we can estimate the energy width of the exchange-mediated gap of Δhel ≈ 1.1 meV at T = 100 mK (for example, see Fig. 2e).

Figure 1: Quantized conductance and the pseudogap feature.
figure 1

a, Top-view SEM image of the InAs nanowire covered with a layer of the high-k dielectric LaLuO3. Each of the top-gate (TG) electrodes can be used to deplete the channel and control the number of 1D subbands. b, Re-entrant conductance behaviour on the first plateau for QPC-IV (d.c.-bias voltage Vd.c. = 4 mV) at T = 100 mK for various magnetic fields B. Inset: zero-bias conductance at T = 4 K for a different QPC (QPC-II) for B increasing from 0 to 10 T (ΔB = 1 T) with the re-entrant conductance feature on the 2 e2/h-plateau. Here, the back-gate voltage is fixed at VBG = 2 V and the curves are offset for clarity. Black arrows indicate the feature location. c, QPC-IV transconductance as a function of B for Vd.c. = 3 mV and VBG = 2 V at T = 3.5 K. The high-transconductance line marked by the green dot indicates the evolution of the riser at the edge of the dip feature into the Zeeman-split subband edge at large B.

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Figure 2: Helical energy dispersion and bias-dependent re-entrant conductance feature.
figure 2

a–c, Dispersion relations for αR = 0.8 eVÅ at different magnetic fields B (nanowire radius r = 50 nm). a, The two spinful subbands are lowered in energy by Eso = m∗αR2/2ℏ2 and shifted in k-space by ±m∗αR/ℏ2 (B = 0 T). b, At B = 2 T the two spin bands are mixed by the magnetic field, which is perpendicular to the spin–orbit field Bso, leading to an avoided crossing at k = 0. Red and blue denote spin orientations along the spin–orbit field direction, while green and orange denote spins along the magnetic field direction. c, At B = 10 T the Zeeman energy dominates over Eso. d, Calculated conductance for the conditions in b (upper panel) and c (lower panel). e, Re-entrant feature on the first d.c. conductance plateau for QPC-IV (T = 100 mK) as a function of the d.c.-bias voltage at B = 0 T.

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Figure 3: Re-entrant conductance feature at higher temperatures.
figure 3

Conductance for QPC-IV at T = 6.1 K (offset in VTG−IV) (a) and T = 8.9 K (b) for different magnetic fields. With increasing magnetic field, the dip feature on the first quantized conductance plateau broadens and the shoulder below the dip disappears at larger B. For B > 4 T the Zeeman energy contribution dominates over the spin–orbit energy.

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With regard to the estimate of αR, we observe weak antilocalization in the open, unconfined regime and an avoided crossing in the magnetic field evolution of spin states in quantum dots formed in this device. Both findings substantiate the sizeable SOC in the nanowire (see Supplementary Section 1). Accounting for the gate lever arm, we derive a spin–orbit energy on the order of 2.4 meV from the gate voltage position of the centre of the re-entrant conductance region (see Fig. 2e). Thus, αR ≈ 1.2 eV Å, which is a factor of four larger than αR derived with conventional methods (see Supplementary Section 1). It is, however, of similar magnitude to the Rashba parameters found in InSb nanowires via weak antilocalization, which go up to 1 eV Å (ref. 21). It should be noted that those methods probably underestimate Eso, since they consider spin relaxation only in the weakly confined multi-mode regime.

Despite the conceptual simplicity, the visibility of the re-entrant behaviour in the conductance for any value of the magnetic field is not to be taken for granted: the unambiguous identification of the SOC-induced conductance feature is generally obstructed by the non-optimal gate potential shape forming the QPC22 (see Supplementary Section 3). Moreover, the helical gap can be obscured by Fabry–Pérot resonances that are superimposed on the quantized zero-bias conductance plateaus at low temperatures16,23. Also the shape of the constriction24 as well as local potential fluctuations25 can have a critical impact on the transmission. However, at T > 6 K the feature can still be observed (see Fig. 3a), while Fabry–Pérot oscillations largely disappear in this regime. This rules out phase-coherent interference as the origin of the effect. At high temperatures (T ≈ 9 K) the re-entrant conductance feature is less pronounced, but it broadens with increasing magnetic field and evolves into the first e2/h-plateau for B > 4 T (see Fig. 3b). Another process that could induce a re-entrant behaviour is reflection by impurities. Similar re-entrant conductance features have, however, been observed for all investigated QPCs. The reproducibility of the feature position for different QPCs indicates that resonant reflections due to backscattering at impurities in the constrictions26 are unlikely to explain the observed effect. An additional consistency check is provided by the fact that the re-entrant conductance behaviour appears as long as the Zeeman energy is of the order of or smaller than Eso, which is a requirement for the existence of the helical gap.

Further validation of the helical nature of the re-entrant behaviour is provided by the analysis of the conductance as a function of Rashba SOC. The magnitude of spin–orbit coupling in the QPC is determined by the strong electric field from the top gate that creates the confinement potential of the constriction. A positive back-gate voltage VBG does not only increase the conductance of the system, it also enhances the Rashba coefficient at the QPC21. As depicted in Fig. 4, at B = 1.5 T the conductance dip is a well-defined singular feature at VBG ≫ 0. It can be seen that for decreasing VBG the feature becomes less pronounced, and finally develops into a double-plateau, which is characteristic of conventional Zeeman splitting. The modified SOC strength is expected to manifest in a changed visibility of the re-entrant conductance feature. For Eso ≲ gμBB + Δhel the shoulder below the dip disappears and a single e2/h-plateau remains.

Figure 4: Impact of the back-gate voltage on the re-entrant conductance feature.
figure 4

Conductance for QPC-IV at T = 4 K and B = 1.5 T as a function of the back-gate voltage. VBG changes the Fermi level and also tunes Rashba SOC. The black arrow points at the re-entrant feature. The characteristic re-entrant behaviour is most pronounced for VBG ≫ 0, as can be seen in the blue line cut for VBG = 2 V. The corresponding energy dispersion is illustrated to the right of the line cut. The conductance shoulder below the dip feature shrinks for decreasing VBG. For VBG ≪ 0 the feature disappears and a double-step develops, as depicted in the orange line cut for VBG = −1.5 V in agreement with the energy dispersion where Eso ≲ EZ + Δhel. Right: cross-section of the dual-gate device.

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In summary, a robust re-entrant conductance feature at the 2e2/h-plateau is observed for all investigated QPCs along the nanowire. The variation with magnetic field, bias voltage, temperature and back-gate voltage reproduces the signatures of a helical gap. We suggest a spin-mixing mechanism based on strong exchange interaction to explain the gap opening at zero magnetic field. The observed spin–orbit energy of 2.4 meV is attributed to a strong electric field that generates the distinct confinement in the QPC and enhances Rashba SOC. The all-electric nature of the helical gap preserves time-reversal symmetry and, in the presence of an induced superconducting gap, offers the desired conditions for fractional excitations that give rise to parafermionic quasiparticles27, which are promising building blocks in the context of topologically protected quantum computing28.

Methods

Device fabrication.

InAs nanowires are grown via gold-catalysed metal–organic vapour phase epitaxy on GaAs (111)B substrates29. The field-effect mobility was found to be 25,000 cm2 Vs−1 (ref. 16) and the electron concentration is about 1.0 × 1017 cm−3 (ref. 19). Nanowires are mechanically transferred onto a Si substrate with a 200-nm-thick SiO2 layer that enables back-gate functionality. LaLuO3 dielectric30 is deposited onto the nanowire via pulsed laser deposition at room temperature and using a lift-off technique. The LaLuO3 layer is 100 nm thick and the relative permittivity is ɛr = 26.9. Ti/Au top-gate electrodes with 180 nm width and 30 nm pitch are fabricated. Following in situ Ar+ sputtering of the nanowire, 120 nm of Ti/Au electrodes are evaporated.

Measurements.

All measurements have been carried out in a 3He/4He dilution refrigerator with a superconducting single-axis magnet. All conductance measurements at zero d.c.-bias voltage are performed at an a.c. excitation voltage of Va.c. = 80 μVrms. The bias voltage is applied symmetrically with respect to ground from source to drain electrode and the current is measured simultaneously at both ends of the nanowire. The actual voltage across the QPCs is significantly smaller than the applied voltage, considering the voltage drop across the series resistance related to non-ballistic nanowire segments. Typical values of the subtracted series resistance including contributions from the measurement set-up are of the order of 20 kΩ, chosen such that the conductance plateaus match with integer multiples of 2e2/h.

Model.

The re-entrant feature in the absence of applied magnetic fields is explained by means of a Luttinger liquid model in which an electron–electron interaction term that couples the different spin species is introduced. The relevant process is dubbed correlated two-particle backscattering. It emerges as an effective interaction process due to the hybridization of different confinement subbands of the nanowire. We identify Rashba spin–orbit coupling as the most plausible physical mechanism responsible for the hybridization. This model is rather distinct from the description of the system as a (zig-zag) Wigner molecule31, which could be an alternative explanation for a variation of the conductance as a function of gate voltage. As we explain in Supplementary Section 2, the experimental signatures that we would expect from the formation of a Wigner molecule are incompatible with the observed features. We also discuss in Supplementary Section 2 that our model and its observable consequences are distinct from the 0.7 anomaly.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional Information

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