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Gate-tuneable and chirality-dependent charge-to-spin conversion in Tellurium nanowires

Chiral materials are the ideal playground for exploring the relation between symmetry, relativistic effects, and electronic transport. For instance, chiral organic molecules have been intensively studied to electrically generate spin-polarized currents in the last decade, but their poor electronic conductivity limits their potential for applications. Conversely, chiral inorganic materials such as Tellurium are excellent electrical transport materials, but have not been explored to enable the electrical control of spin polarization in devices. Here, we demonstrate the all-electrical generation, manipulation, and detection of spin polarization in chiral single-crystalline Tellurium nanowires. By recording a large (up to 7%) and chirality-dependent unidirectional magnetoresistance, we show that the orientation of the electrically generated spin polarization is determined by the nanowire handedness and uniquely follows the current direction, while its magnitude can be manipulated by an electrostatic gate. Our results pave the way for the development of magnet-free chirality-based spintronic devices.


Introduction
Charge-to-spin interconversion phenomena such as the spin Hall effect 1 or the Edelstein effect 2 enable the electrical generation of spin currents without magnetic elements 3 , a fundamental step towards the next generation of spintronic devices, such as spin-based logic 4,5 and spin-orbit torque 6 MRAM memories 7 . The Edelstein effect 2 , also called inverse spin galvanic effect 8 , emerges in materials with strong spin-orbit coupling (SOC) and broken inversion symmetry, such as strained semiconductors 9 , Rashba systems [10][11][12] and the surface of topological insulators (TIs) 13,14 . In these systems, the spins of the electrons are locked in the direction perpendicular to their momenta, so that the flow of a charge current results in a perpendicularly oriented homogeneous spin polarization. Further lowering the crystal symmetry allows the creation of spin polarizations in unconventional directions 15,16 and enables new fundamental effects 17,18 and configurations for devices [19][20][21][22][23][24] .
Chiral materials are the ultimate expression of broken symmetry, lacking inversion and mirror symmetry. Indeed, the relation between structural chirality and spin effects has been investigated for organic molecules [25][26][27][28] , which act as efficient spin filters in spite of their low electrical conductivity.
This phenomenon, which is often named chiral-induced spin selectivity (CISS), remains poorly explored in chiral inorganic crystals [29][30][31][32] . Tellurium (Te), a material that possesses strong SOC, a chiral structure 33,34 and can be synthesized 35 in nanowires (NWs) or flakes with excellent electronic conductivity 36,37 , is an ideal material for the study of unconventional chirality-related charge-to-spin conversion. Signs of current-induced spin polarization have been provided only in bulk Te through optical 38,39 and NMR measurements 40,41 , but these detection techniques and the use of millimetre-sized crystals are not suitable for integration into all-electrical nanodevices.
Here, we report a chirality-dependent and gate tuneable Edelstein effect in naturally hole-doped single-crystalline Te NWs. A net spin polarization is detected by recording a unidirectional magnetoresistance (UMR) 11,42-45 dependent on the relative orientation of the electrical current and the 3 external applied magnetic field. Our results show that, unlike Rashba systems and TIs, the charge current flow in Te leads to a spin polarization oriented along the current path and pointing in opposite directions for left-or right-handed NWs. The measured UMR is explained on the basis of a chirality dependent Edelstein effect arising from the radial spin texture at the H-point of the valence band of Te, which dominates the transport in our hole-doped Te NWs. Using a conventional UMR figure of merit 45 , we show that this effect in Te is the largest yet observed. More importantly, the electrostatic gating of the Te NWs allows us to tune its Edelstein effect leading to an electrical tuning of the UMR amplitude by a factor of 6. The all-electrical generation, control, and detection of spin polarization in chiral Te NWs opens the path to exploit chirality in the design of novel solid-state spintronic devices.

Results
A hydrothermal process in the presence of a reducing agent was employed to grow single crystalline Te NWs which are tens-of-micrometres long and a few hundred nanometres wide (see Methods section and Fig. 1a) 37 . Scanning transmission electron microscopy (STEM) was employed to confirm that the Te NWs are single crystals displaying one of the two enantiomorphic chiral space groups, P3121 and P3221 (ref. 33,34 ). Figure 1b displays a 3D sketch of the right-and left-handed Te chains arranged in their characteristic helical structure that twists around the chiral c axis. Van der Waals forces keep together adjacent atomic chains, which possess the same helicity, providing a defined chirality to the Te crystal structure. Figure 1c shows a STEM image of a NW cross section perpendicular to the c-axis. Individual chains appear as triangles stemming from the projection of superimposed Te atoms. This atomic arrangement (see Fig. 1b) indicates that the long axis of the Te NWs is oriented along the chain direction (c-axis). While the image in Fig. 1c shows the high quality of the NW (see also Supplementary Section 1), it cannot be used to identify its chirality. However, it is possible to unambiguously distinguish between right-and left-handed crystals through a comparison between the atomic arrangement in different crystalline planes 33,34 . In Fig. 1d, we observe that the STEM images of two NWs with opposite handedness match with the models for the space groups P3121 and P3221.  Figure 1e shows a zoom of the valence bands around this point calculated from an effective model (see Methods) 39 . The iso-energy contour corresponding to a cut of the valence band at  = −0.1 eV below the band edge is characterized by a radial spin texture with spins pointing inward/outward in right-/left-handed crystals, as shown in Fig. 1f. Recent spectroscopic observations confirmed the link between the handedness of Te and the direction of the radial spin texture 46,47 . This is analogous to spin momentum locking observed in bulk Rashba systems 11,12 , but with a chirality-dependent radial spin texture instead of the conventional helical configuration 3 .
For the electrical characterization, we transferred Te NWs onto Si/SiO2 substrates using a Langmuir-Schaefer approach. Individual NWs were subsequently selected and contacted using Pt contacts defined by standard lithography methods (see Methods). Figure 2a shows the optical image of a contacted Te NW, with a sketch of the four-probe configuration used for the transport experiments.
We redefine a cartesian coordinate system to describe the directions of currents and magnetic fields applied, where z is in the same direction as the c crystallographic axis and x(y) are orthogonal to z and . This way, we obtain the equivalent of the 1 st harmonic response in a.c. transport measurements 44 , excluding current-dependent contributions to the resistance. The temperature (T) dependence of the four-probe resistance shows a monotonic decrease of R with decreasing T (Fig. 2b). From the transfer characteristics, we observed that the Te NWs are hole-doped, with a field effect mobility that ranges from ~ 500 cm 2 /Vs at 300 K to ~ 2500 cm 2 /Vs at 10 K (see Supplementary Section 4). This is in agreement with previous reports highlighting that Te vacancies cause hole-doping 48 . As a consequence of this intrinsic doping, the resistivity at room temperature is ρ = 0.02 Ω cm, one order of magnitude lower than the value reported for undoped bulk crystals 49 .  Figure 2c displays the magnetoresistance curves for in-plane magnetic fields B parallel (0º) or perpendicular (90º) to the applied current Iz, at temperatures ranging from 2 to 100 K. When B is parallel to Iz (orange curves in Fig. 2c), we observe a monotonical decrease of the resistance when increasing the magnetic field. The negative longitudinal magnetoresistance (NLMR), which is unusual 7 for non-magnetic materials, reaches a maximum value of −15% at T = 2 K and B = 9 T. This peculiar behaviour was already reported for bulk Te 48 , and it has been attributed to the presence of Weyl Fermions contributing to the transport, although other effects may also be responsible for such MR (see supplementary section 5).
In the case of B perpendicular to Iz (purple curves in Fig. 2c), the magnetoresistance at low T changes from negative to positive when increasing the field, showing a relative maximum at B = 0 T and two minima around B = ± 1 T, followed by a change of slope at around B = 3 T. The minima slowly flatten out with increasing T, up to T = 50 K, where they disappear (see section 5 of SI for more details.). In Fig. 2d, we show the angular dependence of the magnetoresistance at T = 10 K as a function of different magnetic fields. Here and in the following experiments, we set the angle between the magnetic field and the NWs to be α = 90º, when the field is orthogonal to the NW's z-axis and to the current applied, and at α = 0º, when the field is parallel to the NW's z-axis and to the positive current direction (see Fig. 2a). The curves are characterized by sharp minima at around α = 0º and α = 180º, i.e., for B // Iz, and maxima at around α = ±90º, for B ⊥ Iz. We highlight that, for our study, we focused our attention on the regime at B > 3 T where, for every field, the resistance measured at ±90º (0º and 180º) is higher (lower) than the resistance measured at B = 0 T, which is shown as a dashed line. The appearance of the maxima and minima is a direct consequence of the MR curves shown in Fig. 2c, which transit from positive at 90º to negative at 0º. Additionally, we notice that, at 10 K and for B > 3 T, the dependence of the resistance on the field is almost linear for both the transverse and the longitudinal MR, but with an opposite slope (Fig. 2c). Consequently, the difference between the maximum and the minimum value in the angular dependence (|A avg |) increases linearly with the applied field (see inset in Fig. 2d).
We now focus on the detection of the chirality-dependent current-induced spin polarization in the Te NWs. Figure 3a,b displays a sketch of the radial spin texture of Te for left-and right-handed crystals.
An electric field applied along the z-direction generates a current density jz and causes a redistribution of states along the kz direction, which is depicted as a Δkz shift of the Fermi contour. Due to the radial spin texture of Te, Δkz induces a homogeneous spin density that is parallel or antiparallel to jz, 8 depending on the chirality of the NWs. Unlike Rashba systems, the electrical current in Te is expected to acquire a net spin polarization oriented along jz, which is a peculiar manifestation of the Te symmetries (see Supplementary Section 6). This current induced spin polarization can be detected in magnetotransport measurements since it introduces a dependence of the resistance on the mutual orientation of current and magnetic field. This effect, also known as UMR or non-reciprocal charge transport, allows to map the spin texture of a material 11,42,44,45 .
Using the configurations of current and magnetic field illustrated in Fig. 3c, we analysed the angledependent magnetoresistance of the NWs, measured for opposite current directions along z (±Iz). By changing the current direction and varying the angle α between the NWs at a fixed magnetic field, we reverse the mutual orientation of Iz and B. A parallel alignment between field and current is obtained at 0° for +Iz and at 180° for −Iz (see Fig. 3c).
Figures 3d,e show the resistance measured for +Iz and −Iz as a function of the angle α between the NW and a magnetic field (B = 9 T), for the two NWs with opposite chirality (cross-sections of these two same devices are shown in Fig. 1d). For both the right-and left-handed NW, we find a strong dependence of the resistance on the relative alignment between Iz and B, which is the hallmark of UMR, and it demonstrates the presence of a net spin polarization in the Te NWs. In particular, the left-handed NW shows a significantly higher (lower) resistance when the current is parallel (antiparallel) to the external magnetic field (Fig. 3d). This non-reciprocal effect can also be observed as a shift in the MR traces measured for opposite current directions (see Supplementary Section 5, Fig. S5). Conversely, the right-handed NW displays lower (higher) resistance for Iz parallel (antiparallel) to B (Fig. 3e), indicating that the UMR of Te is mirrored in NWs with opposite chirality.
Hence, the current induced spin polarization is reversed for opposite NW handedness. To analyse these data, we calculate the half difference between the resistance measured applying +Iz and −Iz (R diff = [R(+Iz) − R(−Iz)]/2) obtaining the equivalent of the 2 nd harmonic signal in a.c. transport measurements 44 , providing a direct signature of UMR. In Fig. 3f-i, the angular dependence R diff (α) presents sharp peaks for collinear current and magnetic field (at 0º and 180º), corresponding to the asymmetries in Fig. 3d,e. This indicates that, in agreement with the Edelstein mechanism (Fig. 3a,b), the spin polarization is oriented along the direction of the current, parallel to the chiral axis z. The deviation of R diff (α) from the sin dependence typically observed for UMR signals can be explained by analysing the presence of additional UMR components not related to the chirality and the dependence of UMR on the peculiar MR recorded in our Te NWs (see Supplementary Section 5).
Moreover, the trend of the magnetoresponse (measured in right-and left-handed NWs) is specular since it is directly related to the chirality of the NWs. In this regard, the orientation of spin polarization is determined by the Te handedness, in a similar way the spin polarization in a ferromagnetic metal is determined by its magnetization. By matching the transport measurements with the STEM analysis in Fig. 1, we can distinguish the NWs handedness through their magnetotransport behaviour. Figure 3f,g shows the variation of R diff (α) measured at B = 9 T and different current Iz. A larger magnetoresponse is measured at higher currents, with the signal amplitude |A| diff = | R(0º) − R(180º) | increasing linearly with I (see insets). This dependence can be understood considering that the spin density generated by the Edelstein effect increases linearly with the displacement Δkz induced by the current. According to the model in Fig. 3a,b, a higher current induces a larger spin density, resulting in more spins coupling to the magnetic field. Figure 3h,i shows R diff (α) measured for fixed current and different magnetic fields. Even in this case, we observe a linear increase of |A| diff with B, which indicates that the UMR recorded in our Te NWs possesses a bilinear response to current and magnetic field. This bilinearity, which has been observed in spin-momentum locked states in Rashba systems 42 and in TIs 44 , originates from the B dependent relaxation processes at scalar and spin-orbit defects, respectively 42,43 . 11 Besides the fundamental connection between the chirality of Te and its magnetotransport, we highlight that the UMR amplitude in our NWs is very substantial. For the NW shown in Fig. 3d,f,h, the maximum |A| diff measured at 9 T and 5 µA amounts to 800 Ω, which corresponds to a 7% variation with respect to the current-averaged resistance measured at the same field and at the same angle, R avg (0º, 9 T) = 10.8 kΩ. To the best of our knowledge, in previous reports of UMR, the relative variation in resistance was always < 1% (Ref. 11,42,44,45,50 ). To further compare our results with previous works, we use the figure of merit η defined as η=RUMR/(R0jB) in Ref. 45 , where in our case, due to the different symmetries of Te, we use |A diff |/2 as RUMR and R(B = 0 T) as R0, while j is the current density (j~10 7 A/m 2 , see Supplementary Section 1) and B the magnetic field applied. As a result, we obtained an η ~ 5 × 10 -6 cm 2 /(AT), which is one order of magnitude larger than the highest value reported for Ge(111) at 15 K (η ~ 4.2 × 10 -7 cm 2 /(AT)) in ref. 45 and at least three orders of magnitude larger than observed in SrTiO3 (ref. 50 ), Bi2Se3 (ref. 44 ) and α-GeTe (ref. 11 ). The large UMR magnitude can be explained by the strong Edelstein effect, as detailed below.
Finally, we focus on the control of the UMR by a gate voltage (VG). We will show how the shifts in energy of the Fermi level (F) of the Te NWs leads to the tuning the Edelstein effect. In Fig. 4a, we show the magnetoresponse evolution under changes of carrier density, driven by VG modulation. To account for the variation of resistance with the gate (see Supplementary Section 7), we normalize R diff (α) measured at different VG with respect to the value R avg (90º). For negative VG, i.e., lowering the Fermi level and increasing the hole concentration, the magnetoresponse trend is relatively flat except for the two prominent features at α = 0º and α = 180º, indicating that the electrical current induces a significant spin polarization only in the z-direction. Increasing VG, i.e., decreasing the hole concentration and bringing the Fermi level closer to the band gap, we observe an increase in the amplitude of the UMR (up to a factor of 6, see Fig. 4b) and the appearance of shoulders close to the peaks at α = 0º and α = 180º. Similar shoulders were observed for VG = 0 V in some NWs, including those shown in Fig. 3f-i. Their relative intensity was found to change from sample to sample (see Supplementary Section 8), probably due to the slightly different doping in different NWs. The increase of the magnetoresponse at positive VG can be correlated with the Te band structure.
Looking at the band structure at different energies, a change of the Fermi contour in k-space is expected to cause a variation of the Edelstein effect efficiency 51 and, consequently, a variation of the magnetoresponse. In Fig. 4b,c, we compare the gate dependence of |A norm | = [R diff (0º) − R diff (180º)]/R avg (90º) with the calculated spins per unit cell (Sz), generated by the Edelstein effect at different F and a constant charge current density of 10 7 A/m 2 (see Methods).
In order to compare experiments and calculations, we measured the carrier density of a Te NW through ordinary Hall measurements to estimate the position of F in the band structure shown in Fig.   1e. The extracted hole density was p = 7.4×10 17 cm -3 at VG = 0 V and it could be varied in the range 7-8×10 17 cm -3 using the back-gate voltage ranging from -30 V to 10 V (see Supplementary Section 9).
Interestingly, the rather small modulation in p induces significant changes on the recorded signals, 13 highlighting that it is possible to finely tune the Edelstein effect in Te by acting on its band structure through standard electrostatic gating.
Our calculations on the charge carrier density reveal that the extracted p corresponds to an F approximately 20 meV below the band edge, which can be tuned in a 5 meV energy window by gating. Figure 4c shows that Sz, which is associated to the efficiency of the Edelstein effect creating the spin polarisation, increases when F is moved towards the valence band edge, in good agreement with the increase of |A norm | with VG displayed in Fig. 4b. Moreover, we highlight that the calculated F position implies a single band occupancy, since the band which lies closest in energy is populated at energies 130 meV below the valence band edge (see Supplementary Section 9). This situation is ideal to maximize the Edelstein effect and thus the UMR, since the occupation of the lower band, with opposite radial spin texture, would partially compensate the induced spin density (see Supplementary Section 10 and 11 for more details on the calculations and for the full Sz vs F).

Conclusion
In conclusion, we demonstrated all-electrical generation, manipulation and detection of chirality dependent spin polarization in single crystalline Te NWs. The spin polarization from chiral origin gives rise to a UMR that is one-to-several orders of magnitude larger than the reported in other nonchiral systems. The UMR is also tuneable with electrical gating, providing us with an extra knob for controlling the spin polarization. These effects are induced by an Edelstein effect emerging from the Te radial spin texture. Unlike conventional Rashba systems and similarly to the CISS observed in organic molecules, the induced spin polarization is oriented along the current direction. We think that our description this phenomenon on the basis of the Edelstein effect might be extended to other chiral systems characterized by translational symmetry. Our results put on a firm ground the fundamental interplay between structural chirality and electron spin in inorganic chiral Te NWs, enabling the design of unconventional all-electrical devices in which the spin polarization is not determined by the direction of a magnetization, but by the handedness of the crystal. 14

Chemical synthesis of Te NWs
The synthesis consists of a high-temperature reduction of a tellurium oxide in presence of hydrazine (N2H4) in a basic aqueous medium [35][36][37]52 . We followed the hydrothermal growth recipes proposed in literature 36,37 . Briefly, we mixed Na2TeO3 (104 mg) and polyvinylpyrrolidone (average Mw 360,000 -PVP360, 547.9 mg) in 33 mL of MilliQ ® water by magnetic stirring up to achieve a clear solution at room temperature. Then, NH4OH solution (3.65 mL, 25%w in water) and hydrazine hydrate (1.94 mL, 80%, w/w%) were added while stirring. The mixture is transferred to an autoclave that was sealed and heated at 180°C for 23 h. We washed the resulting material by successive centrifugeassisted precipitation (5000 rpm -5 min) and redispersion with MilliQ® water (10 steps of 4 mL water each). At this stage, we are still not able to produce selectively one or the other enantiomer and the distribution of left-and right-handed NWs is random.

Sample preparation
Te NWs were redispersed in a dimethylformamide: CHCl3 mixture (1.  Fig. 1d (left-handed), Fig. 2 b,c,d and Fig. 3 d,f,h. Results obtained from D2 are shown in Fig. 4 a,b. Results obtained from D3 are shown in Fig. 1b,d (right-handed), Fig. 2a and Fig. 3 e,g,i. Devices D4-D7 are included in the supporting information. 15 Two devices showing opposite magneto responses were sectioned by FIB for STEM chirality characterisation. Two lamellae were prepared from every device: one perpendicular to the wire in the contact are (to validate the quality of electrical contact) and one along the wire for measuring the chirality (see Supplementary Section 1). Helios 600 DualBeam™ (ThermoFisher, USA) was used for lamellae preparation and SEM imaging, TitanG2 60-300 operated at 300kV without STEM corrector was used for STEM imaging. For chirality determination samples were imaged in either of <010> zones and then tilted ±30º resulting in mirrored images. The handedness was determined by comparison to atomic models of the known chirality. The absence of the mirror plane between the sample and digital image has been ruled out by imaging alphanumeric grid. The STEM analysis was performed after the samples were already measured electrically.

Magneto transport measurements
The devices were wire-bonded to a sample holder and installed in a physical property measurement system (PPMS, Quantum Design) for transport measurements with a temperature range of 2-400 K and maximum magnetic field of 9 T. We performed measurements of d.c. longitudinal and transversal resistance using a Keithley 6221 current source and a Keihtley 2128 nanovoltmeter. Asymmeric (I+/0 and I−/0) and symmetric (I+/I−) delta mode (16 to 32 counts) were employed to improve the signal to noise ratio (with current ranging from 50 nA to 10 μA). The average of the signals obtained from the asymmetric delta mode measurements was equivalent to the signals obtained from the symmetric delta mode and corresponds to the current-independent resistance (R avg ). The half difference of the signals obtained from the asymmetric delta mode was taken to analyse the current-dependent resistance (R diff ). For the angle dependent measurements, the magnetic field was rotated in the plane of the sample. In the current dependent angular dependences, we excluded the current that were high enough to raise the local temperature of the NWs. For the gate dependence measurements, a Keithley 2636 source meter was used to apply a constant voltage to Si doped substrate, while monitoring the leakage current through the SiO2 dielectric to be smaller than 10 nA. 16 The states relevant for electric transport are located around the H point. Therefore, we use an effective model Hamiltonian which reproduces the band effective band structure introduced in Ref. 39 as well as the radial spin texture 46,47

Calculations on band structure and Edelstein effect
Here, kz is along the H-K direction and the Pauli spin vector ̂ represents the spin degree of freedom.
We calculate the charge current density j as well as the spin density S response to an external electric field E, Here, e is the absolute value of the elementary charge, V is the sample volume, vk = ∂ε/ℏ∂k is the group velocity, <s>k is the spin expectation value, ε the energy dispersion, and fk is the distribution function. Its evolution under the influence of an external electric field E as well as scattering is where we have assumed the system to be stationary and spatially homogeneous. ∂fk/∂t|scatt is the scattering term which we approximate by the relaxation time approximation Here gk = fk − fk 0 is the nonequilibrium part of the distribution function, fk 0 is the Fermi Dirac distribution function and τk is the momentum relaxation time. The Boltzmann equation (4) = ∑ 〈 〉 ( • ) = : κ (8) In our calculations, we assume T = 0 and a constant relaxation time τ0. Hence, only states at the Fermi level contribute to the transport. Using Eqs. (7) and (8) we calculate the charge current as well as the current-induced spin density in bulk Te, which is shown as a function of the Fermi level in Fig. 4c. To calculate the induced spin density, we use Sz = κzzjz/σzz, where j and σ are the current density and the charge conductivity respectively, and the Edelstein susceptibility κ is the tensor which connects the induced spin density S to the applied electric field, S = κE (see supplementary section 10 for more details).