Signatures of interaction-induced helical gaps in nanowire quantum point contacts

TL;DR

This paper reports the first direct experimental observation of a helical liquid in InAs nanowires, revealing interaction-induced helical gaps that are prominent even without magnetic fields, with implications for topological quantum computing.

Contribution

It provides the first direct evidence of interaction-induced helical gaps in nanowire quantum point contacts, highlighting the role of exchange interactions and spin-flipping backscattering.

Findings

Observation of reentrant conductance feature indicating helical liquid

Helical gaps are prominent without magnetic fields

Exchange interactions significantly affect transport properties

Abstract

Spin-momentum locking in a semiconductor device with strong spin-orbit coupling (SOC) is a fundamental goal of nanoscale spintronics and an important prerequisite for the formation of Majorana bound states. Such a helical state is predicted in one-dimensional (1D) nanowires subject to strong Rashba SOC and spin-mixing, its hallmark being a characteristic reentrant behaviour in the conductance. Here, we report the first direct experimental observations of the reentrant conductance feature, which reveals the formation of a helical liquid, in the lowest 1D subband of an InAs nanowire. Surprisingly, the feature is very prominent also in the absence of magnetic fields. This behaviour suggests that exchange interaction exhibits substantial impact on transport in our device. We attribute the opening of the pseudogap to spin-flipping two-particle backscattering. The all-electric origin of the ideal helical transport bears momentous implications for topological quantum computing.

Figures

Equations

$${\Delta }_{\mathrm{hel}}=\frac{m^{\ast 4}{\left({\alpha }_{\mathrm{R}}/\hslash \right)}^7{e}^{2}d}{{\hslash }^4{\omega }_0^3{\epsilon }_0{\epsilon }_{\mathrm{r}}\sqrt{\hslash /m^{\ast }{\omega }_0}}.$$

$$\Delta G=-\frac{e^2}{h}\frac{1}{L}$$

$$-3{\left(\frac{1}{l_{\phi }^2}+\frac{4}{3l_{\mathrm{so}}^2}+\frac{d}{l_e^2}+\frac{1}{l_B^2}\right)}^{-\frac{1}{2}}+{\left(\frac{1}{l_{\phi }^2}+\frac{d}{l_e^2}+\frac{1}{l_B^2}\right)}^{-\frac{1}{2}}],$$

$$l_B=\sqrt{\frac{C_m}{d}\frac{l_m^4}{w^{{\gamma }_m}{l}_e^{2-{\gamma }_m}}}$$

$$l_{\mathrm{so}}=\sqrt{\frac{C_s}{d}\frac{L_{\mathrm{so}}^4}{w^{{\gamma }_s}{l}_e^{2-{\gamma }_s}}}.$$

$${\Delta }_{\mathrm{so}}=\frac{E_{\mathrm{Z}}}{\sqrt{2}}\frac{r_e{m}^{\ast }{\alpha }_{\mathrm{R}}}{{\hslash }^2},$$

$$r_e\approx \sqrt{{\hslash }^2/m^{\ast }{\Delta }_{\mathrm{exc}}}.$$

$$H_{\mathrm{soc}}=\frac{1}{2\pi }\int dx\sum _{\nu =\rho ,\sigma }{v}_{\nu }{K}_{\nu }{\left({\partial }_x{\theta }_{\nu }\right)}^2+\frac{v_{\nu }}{K_{\nu }}{\left({\partial }_x{\varphi }_{\nu }\right)}^2.$$

$$H_B=\frac{g{\mu }_{B}B}{2\pi a}\int dx\cos \left[\sqrt{2}\left({\varphi }_{\rho }+{\theta }_{\sigma }\right)\right],$$

$$H_{2p}=g_{2p}\int dx\cos \left[2\sqrt{2}\left({\varphi }_{\rho }+{\theta }_{\sigma }\right)\right]$$

$$H_w=H_0+H_1+V.$$

$$H_0=\sum _{n,k,s}{ϵ}_{n,k,s}{c}_{n,k,s}^{†}{c}_{n,k,s},$$

$$H_1=-i{\alpha }_{\mathrm{R}}s\sqrt{\frac{m^{\ast }\hslash \omega }{2}}\sqrt{n+1}\left(c_{n,k,s}^{†}{c}_{n+1,k,-s}-h.c.\right)$$

$$V=\sum _{n_1,\dots ,n_4}\sum _{s_1,s_2}\sum _{k,k^{\prime },q}{U}_{n_1,n_2,n_3,n_4}\left(q\right)c_{n_1,k+q,s}^{†}{c}_{n_2,k^{\prime }-q,s^{\prime }}^{†}{c}_{n_3,k^{\prime },s^{\prime }}{c}_{n_4,k,s},$$

$$U_{n_1,n_2,n_3,n_4}\left(q\right)$$

$$U\left(q,q_y\right)$$

$${\Gamma }_{n_1,n_2,n_3,n_4}\left(q_y\right)$$

$$U\left(x,y\right)=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}\left(\frac{1}{\sqrt{x^2+y^2}}-\frac{1}{\sqrt{x^2+y^2+d^2}}\right),$$

$$S=-\left[\sum _{n,k,s}\left(\frac{\sqrt{n+1}{\alpha }_{\mathrm{R}}\sqrt{\frac{m^{\ast }\hslash \omega }{2}}}{{ϵ}_{n,k,s}-{ϵ}_{n+1,k,-s}}{c}_{n+1,k,-s}^{†}{c}_{n,k,s}\right)-h.c.\right].$$

Full text

Signatures of interaction-induced helical gaps in nanowire quantum point contacts

S. Heedt

Current address: QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands

Peter Grünberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany

  

N. Traverso Ziani

Institute of Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany

  

F. Crépin

Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 place Jussieu, 75252 Paris Cedex 05, France

  

W. Prost

Solid State Electronics Department, University of Duisburg-Essen, 47057 Duisburg, Germany

  

St. Trellenkamp

Peter Grünberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany

  

J. Schubert

Peter Grünberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany

  

D. Grützmacher

Peter Grünberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany

  

B. Trauzettel

Institute of Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany

  

Th. Schäpers

th.schaepers@fz-juelich.de

Peter Grünberg Institut (PGI-9) and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich, 52425 Jülich, Germany

**Spin-momentum locking in a semiconductor device with strong spin-orbit coupling (SOC) is a fundamental goal of nanoscale spintronics and an important prerequisite for the formation of Majorana bound states Lutchyn2010; Oreg2010; Quay2010. Such a helical state is predicted in one-dimensional (1D) nanowires subject to strong Rashba SOC and spin-mixing Streda2003, its hallmark being a characteristic reentrant behaviour in the conductance. Here, we report the first direct experimental observations of the reentrant conductance feature, which reveals the formation of a helical liquid, in the lowest 1D subband of an InAs nanowire. Surprisingly, the feature is very prominent also in the absence of magnetic fields. This behaviour suggests that exchange interaction exhibits substantial impact on transport in our device. We attribute the opening of the pseudogap to spin-flipping two-particle backscattering Wu2006; Xu2006; Pedder2016. The all-electric origin of the ideal helical transport bears momentous implications for topological quantum computing.

**A 1D conductor with strong SOC is predicted Lutchyn2010; Oreg2010; Alicea2011 to represent a viable host for Majorana bound states. These zero-energy states feature characteristic non-Abelian exchange statistics Alicea2011 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 states Mourik2012; Das2012. 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 field Streda2003. 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 modes Sau2012. However, Oreg et al. Oreg2010; Oreg2014 and Stoudenmire et al. Stoudenmire2011 have pointed out that such an energy gap can also result from strong electronic correlations. Several mechanisms have been proposed along these lines: e.g. spin-flipping two-particle backscattering Pedder2016 and hyperfine interaction between nuclear spins and a Luttinger liquid Braunecker2009, 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 wire Scheller2014, but no reentrant behaviour is predicted within this framework.

Other than Quay et al. Quay2010, we report on a reentrant 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 observation Heedt2016 of the Landé $g$ factor enhancement for the lowest subband Martin2008 and by signatures of the $0.7$ anomaly Micolich2011, we identify the important role of exchange interaction. Following Ref.  Pedder2016, we ascribe the reentrant behaviour at zero magnetic field to the combined presence of 1D confinement, Rashba spin-orbit coupling and Coulomb interaction.

The experimental setup 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 $2.94\,\mu$m 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 $2e^{2}/h$, 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 $\sim 7.0$ Heedt2016. A single reentrant conductance feature appears reproducibly on the first quantized conductance plateau ($G=2e^{2}/h$) 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 levels Heedt2016; Heedt2016b. The riser at the right edge of the reentrant 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 $\mathbf{E}$ perpendicular to the substrate plane, the spin-orbit field $\mathbf{B_{\mathrm{so}}}$ is aligned perpendicular to the nanowire and in the plane of the substrate (cf. Fig. 2d, inset). The spin-orbit energy $E_{\mathrm{so}}=m^{*}\alpha_{\mathrm{R}}^{2}/2\hbar^{2}$ can be calculated from the Rashba coefficient $\alpha_{\mathrm{R}}$ 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 $g\mu_{B}B\simeq E_{\mathrm{so}}$ Streda2003 (cf. Fig. 2b). The formation of a helical state becomes manifest by the appearance of a reentrant conductance plateau at $e^{2}/h$ inside the larger $2e^{2}/h$-plateau related to the opening of the pseudogap (Fig. 2d, top panel). In our experiment, the gap widens roughly proportional to $g\mu_{\mathrm{B}}B$ (see Fig. 3). This aspect of the reentrant 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 (cf. Fig. 2c), which is reflected by the emergence of a plateau at a conductance of $e^{2}/h$ (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 confinement Governale2002. 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)

[TABLE]

Using $m^{*}=0.026\,m_{\mathrm{e}}$ ($m_{\mathrm{e}}$ being the free electron mass), the relative permittivity $\varepsilon_{\mathrm{r}}=15.15$, the confinement energy $\hbar\omega_{0}=7\,$meV Heedt2016 and $\alpha_{\mathrm{R}}=1.2\,$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 $3$ Braunecker2009, we estimate a pseudogap of the order of $\Delta_{\mathrm{hel}}\approx 2.5\,$meV, which is compatible with the experimental findings. In fact, using the gate lever arm ($\approx 0.04\,$eV/V), we can estimate the energy width of the exchange-mediated gap of $\Delta_{\mathrm{hel}}\approx 1.1\,$meV at $T=100\,$mK (e.g. see Fig. 2e).

With regard to the estimate of $\alpha_{R}$, 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 reentrant conductance region (see Fig. 2e). Thus, $\alpha_{\mathrm{R}}\approx 1.2\,$eV Å, which is a factor of four larger than $\alpha_{\mathrm{R}}$ derived with conventional methods (cf. Supplementary Section 1). It is however of similar magnitude as the Rashba parameters found in InSb nanowires via weak antilocalization, which go up to $1\,$eV Å vanWeperen2015. It should be noted that those methods likely underestimate $E_{\mathrm{so}}$, since they only consider spin relaxation in the weakly-confined multi-mode regime.

Despite the conceptual simplicity, the visibility of the reentrant 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 QPC Rainis2014 (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 temperatures Heedt2016. Also the shape of the constriction Tekman1989 as well as local potential fluctuations Nixon1991 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 ($T\approx 9\,$K) the reentrant conductance feature is less pronounced but it broadens with increasing magnetic field and evolves into the first $e^{2}/h$-plateau for $B>4\,$T (see Fig. 3b). Another process that could induce a reentrant behaviour is reflection by impurities. Similar reentrant 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 constrictions McEuen1990 are unlikely to explain the observed effect. An additional consistency check is provided by the fact that the reentrant conductance behaviour appears as long as the Zeeman energy is of the order of or smaller than $E_{\mathrm{so}}$, which is a requirement for the existence of the helical gap.

Further validation of the helical nature of the reentrant 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 $V_{\mathrm{BG}}$ does not only increase the conductance of the system, it also enhances the Rashba coefficient at the QPC vanWeperen2015. As depicted in Fig. 4, at $B=1.5\,$T the conductance dip is a well-defined singular feature at $V_{\mathrm{BG}}\gg 0$. It can be seen that for decreasing $V_{\mathrm{BG}}$ 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 reentrant conductance feature. For $E_{\mathrm{so}}\ll g\mu_{\mathrm{B}}B$ the shoulder below the dip disappears and a single $e^{2}/h$-plateau remains.

In summary, a robust reentrant conductance feature at the $2e^{2}/h$-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 in order to explain the gap opening at zero magnetic field. The observed spin-orbit energy of $2.4\,$meV is attributed to the 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 quasiparticles Pedder2015, which are promising building blocks in the context of topologically-protected quantum computing Hutter2016.

Methods

Device fabrication. InAs nanowires are grown via gold-catalysed metal-organic vapour phase epitaxy on GaAs (111)B substrates Sladek2012. The field-effect mobility was found to be $25000\,$cm${}^{2}/$Vs Heedt2016 and the electron concentration is about $1.0\cdot 10^{17}\,$cm*-3* Heedt2016b. Nanowires are mechanically transferred onto a Si substrate with a $200$-nm-thick SiO2 layer that enables back-gate functionality. LaLuO3 dielectric Ozben2011 is deposited onto the nanowire via pulsed laser deposition at room temperature and using lift-off technique. The LaLuO3 layer is $100\,$nm thick and the relative permittivity is $\varepsilon_{\mathrm{r}}=26.9$. 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 dc-bias voltage are performed at an ac excitation voltage of $V_{\mathrm{ac}}=80\,\mu$Vrms. 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 setup are of the order of $20\,$k$\Omega$, chosen such that the conductance plateaus match with integer multiples of $2e^{2}/h$.