Cécile Grezes

Aurélie Kandazoglou

Maxen Cosset-Cheneau

Luis M Vicente Arche

Paul Noël

Paolo Sgarro

Stephane Auffret

Kevin Garello

Manuel Bibes

Laurent Vila
email: laurent.vila@cea.fr

Jean-Philippe Attané
email: jean-philippe.attane@cea.fr

Non-volatile electric control of spin-orbit torques in an oxide two-dimensional electron gas

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The efficient control of the magnetization using currents is the key requirement to develop high performance spintronics devices for information and communication technology. In the last decade, current-induced magnetization switching by spin transfer torque has become a well-established technology, with nowadays major foundries and integrated device manufacturers commercializing Spin-Transfer Torque Magnetic Random Access Memories (STT-MRAM) [START_REF] Sun | Commercialization of 1Gb standalone spin-transfer torque MRAM[END_REF]. Meanwhile, spin-orbit coupling emerged as an alternative method to generate spin currents and to manipulate efficiently the magnetization. SOTs rely on the injection of an in-plane current in a non-magnetic material adjacent to a ferromagnet. The spin-orbit coupling creates a spin accumulation at the interface that induces a torque on the ferromagnet magnetization. In heavy metal (HM)/ferromagnetic (FM) heterostructures [START_REF] Liu | Spin-torque switching with the giant spin Hall effect of tantalum[END_REF] [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF], SOTs arise from the bulk spin Hall effect (SHE) and/or the Edelstein effect associated with an interfacial Rashba state. These spin-orbit torques (SOTs) have attracted increasing attention, owing to their ability to induce magnetization oscillations or switching [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF], and their potential for high speed, high endurance and low energy switching [START_REF] Prenat | Beyond STT-MRAM, spin orbit torque RAM SOT-MRAM for high speed and high reliability applications[END_REF].

In recent years, oxide two-dimensional electron gases (2DEGs) have emerged as a new promising SOT system [START_REF] Lesne | Highly efficient and tunable spin-to-charge conversion through Rashba coupling at oxide interfaces[END_REF] [7] [START_REF] Vaz | Mapping spin-charge conversion to the band structure in a topological oxide two-dimensional electron gas[END_REF]. They display an efficient spin-charge interconversion through the direct and inverse Edelstein effects, arising from their broken inversion symmetry which induces a Rashba-type spin-orbit coupling. In previous works, we showed an enhancement of the spin-to-charge conversion efficiency by two orders of magnitude in such SrTiO3-based 2DEG compared to conventional HM/FM heterostructures [START_REF] Vaz | Mapping spin-charge conversion to the band structure in a topological oxide two-dimensional electron gas[END_REF], along with a non-volatile electric-control of the spin-to-charge conversion [START_REF] Noel | Non-volatile electric control of spin-charge conversion in a SrTiO3 Rashba system[END_REF]. In this context, the non-volatile electric-control of the reciprocal charge-to-spin conversion in these 2DEGs would be of great interest for developing reconfigurable SOT-MRAM and logic gates, offering the possibility to actively manipulate the torque by electric fields, and thus to design new architectures. The direction and amplitude of the spin accumulation is the key parameter in the design of SOT applications, as it determines the direction of magnetization switching. In conventional SOT devices based on HM/FM heterostructures, however, the relative direction between the injected current and the generated spin accumulation is fixed by the material used [START_REF] Torrejon | Interface control of the magnetic chirality in CoFeB/MgO heterostructures with heavy-metal underlayers[END_REF], inhibiting any dynamical reconfiguration. This feature results in two main limitations in presentday SOT-MRAMs. Firstly, the fixed direction of the SOT torque requires an inversion of polarity for the applied current in order to switch between the two magnetization states, which requires additional MRAM circuitry [START_REF] Oboril | Evaluation of hybrid memory technologies using SOT-MRAM for on-chip cache hierarchy[END_REF]. Secondly, the fixed direction and magnitude of the SOT torque prevents the selective switching of multiple bits sharing the same write layer in conventional SOT-MRAMs, inhibiting any in-memory logic operations. Achieving the non-volatile electric control of the SOT could thus open the way to a disruptive generation of magnetic memories and logic devices, with high-density integration and dynamical reconfigurability. Several approaches have been pursued in this direction, based on the combination of spin Hall effect heavy metal/ferromagnetic heterostructures with piezoelectric [START_REF] Filianina | Electric-Field Control of Spin-Orbit Torques in Perpendicularly Magnetized W/CoFeB/MgO Films[END_REF], ferroelectric [START_REF] Cai | Electric field control of deterministic current-induced magnetization switching in a hybrid ferromagnetic/ferroelectric structure[END_REF] [14] and oxide materials [START_REF] Mishra | Electric-field control of spin accumulation direction for spin-orbit torques[END_REF] [START_REF] Baek | Complementary logic operation based on electric-field controlled spin-orbit torques[END_REF] [17], or the replacement of the heavy metal by magnetically-doped topological insulators [START_REF] Fan | Electric-field control of spin-orbit torque[END_REF] (TIs). However, none of the above methods provide non-volatile control of SOT together with efficient dynamical inversion of the SOT direction in a structure compatible with magnetic tunnel junction integration.

In this work, we propose to use 2DEGs to obtain the non-volatile gate electric-field control of SOTs. An electric modulation of the 2DEG at a STO/Ta interface is achieved, providing two remanent and switchable resistivity states of the devices, with a large contrast of 1064%. We report the observation of sizable SOTs in a SrTiO3/Metal system, and we further show that the SOT strength in this system can be modulated by the gate voltage with non-volatility, an inverted hysteresis being observed in the SOT dependence with the gate-electric-field. This non-volatile control of the SOT occurs over several orders of magnitude, down to full extinction and sign inversion. We also demonstrate the dynamical control of the SOT direction by the application of voltage pulses. The modulation of the SOT is found to originate from the combination of the gate dependence of the 2DEG band structure with the electric modulation of the current injection in the 2DEG.

Magnetotransport properties

Experiments were carried out on Ta(0.9 nm)\CoFeB(0.9 nm)\MgO(1.8 nm)\Ta(1 nm) stacks deposited onto 500 m (001)oriented SrTiO3 substrates (cf. § Methods for details on the sample preparation), allowing the creation of a 2DEG at the SrTiO3/Ta interface [START_REF] Vicente-Arche | Metal/SrTiO3 two-dimensional electron gases for spin-to-charge conversion[END_REF]. The deposit of a CoFeB\MgO stack on SrTiO3 has been developed to take advantage of the large interfacial magnetic anisotropy of the CoFeB\MgO interface [START_REF] Oh | Interfacial perpendicular magnetic anisotropy in CoFeB/MgO structure with various underlayers[END_REF]. This provides perpendicular magnetization to the CoFeB ferromagnetic layer, and ensures the compatibility of such multilayers for future integration in magnetic tunnel junctions [START_REF] Ikeda | Perpendicular-anisotropy CoFeB-MgO based magnetic tunnel junctions scaling down to 1X nm[END_REF]. The stack was then patterned into 200 nm -1 m wide and 2 -10 m long Hall cross-bars, as shown in Fig. 1a. To characterize the magnetic and electrical properties, an input current of magnitude up to 600 A was injected along the x direction (Fig. 1b), and the longitudinal Rs and transverse Hall resistances RH were measured as a function of the magnetic field, which was applied along a direction defined by the polar and azimuthal coordinates θH and φH. Hereafter, we present results obtained from 1 μm wide Hall bars, and define the injected linear current density as 𝑗 = 𝐼/𝑤 (where w is the width of the Hall bar). The temperature dependence of the sheet resistance of the device is shown in Fig. 1c for j = 0.1 A.cm -1 . A reduction of the resistance by 380% is observed as the temperature decreases, which is characteristic of 2DEG conduction [START_REF] Vaz | Mapping spin-charge conversion to the band structure in a topological oxide two-dimensional electron gas[END_REF]. As shown in Fig. 1d, the anomalous Hall resistance follows a square-shaped magnetic hysteresis loop with applied out-of-plane magnetic field, indicating that the CoFeB has a perpendicular magnetization with 100 % remanence. Furthermore, a reversible decrease of the Hall resistance is observed when increasing the in-plane field, indicating a coherent rotation of the CoFeB magnetization towards the hard plane direction. From the Hall resistance R H = R AHE cosθ + R PHE sin 2 θ sin2φ, where θ and φ are the polar and azimuthal angles of the magnetization in spherical coordinates, we determine the anomalous Hall RAHE = 25.3 Ω and planar Hall RPHE = 1.7 Ω resistances in the ungated state (see Supplementary Section S1). Note that an anomalous Hall resistance significantly larger than that of standard SiO2/Ta/CoFeB/MgO SOT devices [22] [23] is achieved in our system, taking advantage of the use of ultra-thin Tantalum layer, as previously observed by T. Zhu et al [START_REF] Zhu | Giant linear anomalous Hall effect in the perpendicular CoFeB thin films[END_REF].

Non-volatile electric-control of the 2DEG properties

To modulate the 2DEG properties, an electric-field Eg is applied across the STO substrate using a back-gate. Fig. 2a shows the sheet resistance Rs as a function of the applied gate electric-field, at the temperature of 10 K and after initializing the ferromagnet in the up magnetization state. A hysteresis is observed in the sheet resistance, with two switchable and remanent high and low resistivity states of the device. The Rs contrast, defined as (Rs,max-Rs,min)/Rs,min, shows a value of 1064%, with 615% remanent contrast at Eg = 0 kV.cm -1 between the high and low remanent resistivity states. The hysteresis is inverted, i.e. it is anticlockwise, which is a typical characteristic of charge trapping effects [START_REF] Georgiou | Ferroelectricity in Polar Polymer-Based FETs: A Hysteresis Analysis[END_REF]. It contrasts with previous results on 2DEGs at the SrTiO3\\Al interface, showing no hysteresis for 500 µm-thick SrTiO3 substrates [START_REF] Vaz | Mapping spin-charge conversion to the band structure in a topological oxide two-dimensional electron gas[END_REF], and a clockwise loop compatible with field-induced ferroelectricity for thinner SrTiO3\\Al substrates [START_REF] Noel | Non-volatile electric control of spin-charge conversion in a SrTiO3 Rashba system[END_REF] in which larger electric fields can be applied, as well as for ferroelectric Ca:SrTiO3\\Al system [START_REF] Bréhin | Switchable two-dimensional electron gas based on ferroelectric Ca: SrTiO3[END_REF]. Here, the inverted loop rather points towards charge trapping, and leads to a larger resistance contrast. Similar anticlockwise hysteresis were observed in multiple SrTiO3\\Metal\CoFeB\MgO samples with different metal layers (see Supplementary Section S2) inducing the 2DEG formation [START_REF] Vicente-Arche | Metal/SrTiO3 two-dimensional electron gases for spin-to-charge conversion[END_REF]. During this process and the further annealing of the stack at 300°C, the metallic layer (tantalum) creates oxygen vacancies at the surface of the TiO2-terminated SrTiO3 substrate, and electron injection in the 2DEG occurs via tunneling from a few nanometers depth in the SrTiO3 substrate [START_REF] Biscaras | Limit of the electrostatic doping in two-dimensional electron gases of LaXO3(X=Al, Ti)/SrTiO3[END_REF]. The inverted hysteresis loop can then be understood by considering charge trapping. Large negative electric fields induce the appearance of a distribution of trapped charge near the interface. As the gate electric-field is swept from negative to positive, the 2DEG appearance is facilitated by these trapped charges [START_REF] Ojha | Electron Trapping and Detrapping in an Oxide Two-Dimensional Electron Gas: The Role of Ferroelastic Twin Walls[END_REF] [START_REF] Md | On the Characterization and Separation of Trapping and Ferroelectric Behavior in HfZrO FET[END_REF], which explains the change of resistance towards the low resistance state at negative fields. At higher fields, beyond the applied zerogate electric-field, the electric field induces charge detrapping, leading to the symmetric behavior when sweeping the field from positive to negative values. As seen in Fig. 2b, the Rs contrast decreases upon increasing the temperature until full extinction of the Rs contrast at about 105 K, corresponding to the antiferrodistortive transition from cubic to tetragonal phase of SrTiO3 [START_REF] Heifets | Calculations for antiferrodistortive phase of SrTiO3 perovskite: hybrid density functional study[END_REF], in agreement with others charge trapping studies in SrTiO3-based 2DEG [START_REF] Ojha | Electron Trapping and Detrapping in an Oxide Two-Dimensional Electron Gas: The Role of Ferroelastic Twin Walls[END_REF]. The dynamic switching between the high and low resistivity states was studied using a sequence of pulsed gate electric-field. The endurance of the device after application of successive positive and negative pulses is shown in Fig. 2c. The switching between stable high and low resistivity states shows a constant contrast, and no sign of fatigue within the 10 5 cycle attempts. This confirms the technological potential of the charge-trapping-related hysteretic effect observed here to control the 2DEG properties.

Spin-orbit-torques characterization

We used the harmonic Hall voltage measurements method to quantify the spin-orbit torques in our systems [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF] [31] [START_REF] Avci | Interplay of spin-orbit torque and thermoelectric effects in ferromagnet/normal-metal bilayers[END_REF]. An AC current of frequency /2 = 60 Hz and amplitude I 0 =400 µA (j = 4 A.cm -1 ) is injected along the x direction to induce small oscillations of the magnetization around its equilibrium. These oscillations generate first and second-harmonic contributions to the Hall resistance R H (ω) = V H (ω)/I 0 = R H,ω + R H,2ω , providing a sensitive way to measure the currentinduced fields. The first harmonic R H,ω corresponds to the Hall resistance measured in DC, while the second harmonic term R H,2ω includes modulation of the Hall resistance by the SOT effective fields, R SOT,2ω , as well as magneto thermal effect due to unintentional Joule heating, R T,2ω . After subtracting longitudinal and perpendicular thermal effects contribution to R H,2ω , SOT effective fields can be determined by sweeping the magnetic field along the x (y) direction. In the small angles approximation, one measures a longitudinal (transverse) SOT effective fields ΔHx (ΔHy) respectively [START_REF] Hayashi | Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements[END_REF]:

ΔH x (y) = ( dR SOT,2ω dH x (y) ) / ( d 2 R H,ω dH x (y) 2 ) (1)
By defining 𝜉 = 𝑅 𝑃𝐻𝐸 /𝑅 𝐴𝐻𝐸 , the anti-damping effective field HAD and field-like effective field HFL are given by [START_REF] Hayashi | Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements[END_REF]:

H AD (FL) = -2 ΔH x (y) + 2𝜉ΔH y (x) 1 -4𝜉 2 (2)
Measurements were performed with a magnetic field µ0Hext applied close to the in-plane direction (θH = 85°, H = 0, 90°) and swept between ± 1.1 T. Fig. 3a,b shows the second harmonic Hall resistances 𝑅 𝑆𝑂𝑇,2𝜔 at φH = 0° and 90°, after subtraction of the thermal effects and experimental resistance offsets (cf. Supplementary Section S3), for positive and negative gate electricfields of ±3.2 kV.cm -1 . Typical symmetric (antisymmetric) contributions are observed for the magnetic field perpendicular (parallel) to the current, corresponding to the field like-torques and anti-damping like torques, respectively. Remarkably, we observe a sign change of 𝑅 𝑆𝑂𝑇,2𝜔 for high and low resistivity states. When using Eqs. (1-2), we find µ0HAD = +6.2 mT (-0.72 mT) and µ0HFL = +1.35 mT (-0.39 mT) for the low (high) resistivity states measured at Eg = +3.2 kV.cm -1 (-3.2 kV.cm -1 ), respectively. The sign change and gate dependence of SOT are discussed afterwards.

Non-volatile electric-field control of the spin-orbit torques

We now repeat SOT measurements described in the previous section while modulating the 2DEG properties with gate-electricfields ranging between ± 2.6 kV.cm -1 . Fig. 4a shows the measured SOT anti-damping like effective field per linear current density µ0HAD/j, as a function of the gate electric-field. The results show a remanent modulation of the SOT-AD effective field, with inversion of the HAD sign for opposite maximum gate-electric-fields. The dependence shows a hysteresis that is inverted, similarly to the resistance hysteresis shown in Fig. 2a.

Let us discuss the origin of this SOT hysteresis. The main contribution to this loop is the nonvolatile modulation of the 2DEG resistivity, and thus of the current injection in the 2DEG. Fig. 4b shows the measured modulation of the current injection in the 2DEG. The 2DEG and the Ta layer constitute two parallel channels of conduction, with two separate SOT contributions. At maximum negative electric-field (A), the device is in a high resistivity state which corresponds to the resistivity of the partially oxidized CoFeB/Ta bilayer (see Supplementary Section S4), indicating that the 2DEG is largely depleted and has an ultra-low conductance. Hence, there is no in the 2DEG at this negative extrema. Inversely, the 2DEG resistivity reaches 0.26 kΩ.sq -1 for the maximum positive electric-field (C). The device is then in the low resistivity state, which means that 92% of the applied current is flowing in the 2DEG.

A negative SOT-AD effective field of -0.23 mT.A.cm -1 is observed at the negative end of the sweep (A) when there is no current flowing in the 2DEG. This suggests that the SOT is then due to a residual contribution from the Ta layer, this interpretation being in good agreement, both in sign and magnitude, with previous observations on SiO2/Ta/CoFeB/MgO [START_REF] Avci | Fieldlike and antidamping spin-orbit torques in as-grown and annealed Ta/CoFeB/MgO layers[END_REF]. The highest SOT efficiency is achieved at the positive end of the electric-field sweep (C), when the 2DEG is in the low resistivity state, reaching +1.6 mT.A.cm -1 . Notably, the SOT-AD effective field at this extrema is positive, of opposite sign compared to that of the Ta/CoFeB/MgO system [START_REF] Avci | Fieldlike and antidamping spin-orbit torques in as-grown and annealed Ta/CoFeB/MgO layers[END_REF] (see Supplementary Section S4). This confirms that when the device is in the low resistivity state, the SOT arises from the 2DEG via the Edelstein effect, rather than being due to a residual Spin Hall effect in the Tantalum layer.

If the nonvolatile modulation of the 2DEG resistivity was the sole origin of the SOT hysteresis, one would expect to have similar hysteresis loops for the SOT (Fig. 4a) and for the resistance (Fig. 4b). However, the SOT hysteresis of Fig. 4a does not simply reflect the charge current redistribution within the stack. The differences between the loops of Figs. 4a and4b are very probably due to the dependence of the spin-orbit conversion with the position of the Fermi level, which is linked to specific points of the k space and which has been previously studied in SrTiO3\Al [START_REF] Vaz | Mapping spin-charge conversion to the band structure in a topological oxide two-dimensional electron gas[END_REF] and SrTiO3\LAO [START_REF] Lesne | Highly efficient and tunable spin-to-charge conversion through Rashba coupling at oxide interfaces[END_REF] samples. For a given current density flowing in the 2DEG, the produced SOT is expected to vary with the applied electric field. The SOT hysteresis has thus to be understood as resulting from both the current redistribution in the stack when varying the 2DEG resistivity, and the variations of the conversion with the 2DEG Fermi level position.

The dynamical control of the SOT-AD effective field is further evidenced in Fig. 4c, which displays the measured normalized second harmonic Hall resistance at shifted electrical remanence Eg,r, after application of negative and positive 200 ms-long gate electric-fields pulses of ±2.6 kV.cm -1 . A gate electric-field pulse is first applied to the device, then the first and second harmonic Hall resistances are simultaneously measured at Eg,r. The reproducible inversion of the SOT torque sign is demonstrated, achieving µ0HAD/j = +0.30 ± 0.1 mT.A.cm -1 and -0.19 ± 0.1 mT.A.cm -1 after negative and positive voltage pulses, respectively. The control of the SOT shows a deviation of less than ±0.1 mT.A.cm -1 over successive trials.

For applications, it would be interesting to have a large remanence of the SOT at zero electric field. Here the experiment was not performed at zero electric field due to the small SOT difference induced by intrinsic electric-field dependence of the band structure in our SrTiO3/Ta system, but at Eg,r, where the points B and D of Fig. 4a are well splitted. Note however that the intrinsic electric-field dependence of the band structure strongly dependent on the stack materials, as observed by L.M Vicente-Arche et al. [START_REF] Vicente-Arche | Metal/SrTiO3 two-dimensional electron gases for spin-to-charge conversion[END_REF] for different SrTiO3/Metal system. Hence, further material engineering can be made to ensure a large contrast at zero-field remanence for applications.

Current and temperature dependence of the SOTs

The SOT characterizations were repeated for linear current densities ranging from 1 to 6 A.cm -1 , for the device in the high (Eg = -2.6 kV.cm -1 ) and low (Eg = +2.6 kV.cm -1 ) resistivity states. Fig. 5a shows that the amplitude of the SOT-AD effective field scales linearly with the linear current density, with a positive (negative) slope for the low (high) resistivity states, respectively. The linear scaling behavior confirms that Joule heating has a negligible impact on magnetic properties within the current range used for characterization [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF]. This result further demonstrates that the SOT sign inversion is reproducible, with a difference as large as 8.7 mT at a current density of 6 A.cm -1 .

Finally, the temperature dependence of µ0HAD/j in the high and low resistivity states is shown in Fig. 5b. In the high resistivity state (Eg = -2.6 kV.cm -1 ), the result shows a constant AD-SOT torque-to-current ratio of -0.23 mT. A. cm -1 , with a less than ±0.01 mT. A. cm -1 deviation in the 10 K to 80 K temperature range. This is in good agreement with our hypothesis of a current injected only in the Ta layer, generating a SOT which is expected to be independent of the temperature [START_REF] Avci | Fieldlike and antidamping spin-orbit torques in as-grown and annealed Ta/CoFeB/MgO layers[END_REF].

At the opposite gate electric-field (Eg = -2.6 kV.cm -1 ), µ0HAD/j is found constant from 10 K to 30 K range, at around +1.35 ± 0.10 mT. A. cm -1 , followed by a decrease with increasing temperatures, reaching -0.23 mT. A. cm -1 at 80 K. Notably, the AD-SOT torque-to-current ratio at this temperature of 80 K is identical for Eg = ±2.6 kV.cm -1 , suggesting a common origin. This result is in good agreement with the measured contrast displayed in Fig. 2b, in which we observed a merging of the high and low resistivity states above 80 K. Above this temperature range, the conductivity in these 2DEGs is known to disappear [START_REF] Vicente-Arche | Metal/SrTiO3 two-dimensional electron gases for spin-to-charge conversion[END_REF], and thus the current mostly flows in the Ta layer, which becomes the only SOT generator.

To conclude, the non-volatile electric control of the spin-orbit torques in 2DEGs could open the way to a new generation of spin-orbit torques devices. Concerning memory applications, the additional functionality provided by the electriccontrol can be used for building reconfigurable SOT-MRAM, in novel architectures suitable for efficient and fast operation. While efforts remains to be made to bring this technology to room temperature, the development of CoFeB/MgO heterostructures on SrTiO3 with strong CoFeB perpendicular magnetization ensures the compatibility for future integration in magnetic tunnel junctions. Beyond memory applications, this non-volatile control of SOTs offers an additional way to manipulate skyrmions, domain walls or magnons by permitting a local control of the SOT along circuits. It could also be used in SOT oscillators, to allow developing logic architectures and agile terahertz emitters. a, Anti-damping like effective field µ0HAD as a function of the applied 2D current density for positive (blue) and negative (green) gate electricfield of ± 2.6 kV.cm -1 at the temperature of 10K. Dashed and dotted lines are linear fits, yielding SOT effective fields µ0HAD = +1.36 mT. A. cm -1 (-0.18 mT. A. cm -1 ) for positive (negative) gate electric-field. b, Temperature dependence of the SOT anti-damping like effective field µ0HAD at j = 2 A. cm -1 for positive (blue) and negative (green) gate electric-field of ± 2.6kV.cm -1 . All data have been measured with a nearly in-plane applied magnetic field (θH = 85°, φH = 45°).

Methods

The Ta(0.9 nm)/CoFeB(0.9 nm)/MgO(1.8 nm)/Ta(1 nm) samples have been deposited by DC magnetron sputtering on TiO2-terminated (001)-oriented STO substrates of 500 μm thickness (from SurfaceNet). TiO2-termination was achieved through a chemical treatment, where the substrate was subsequently submerged in H2O for 10 minutes and an acid solution (HCl 3 : HNO3 1 : H2O 16) for 20 minutes. The Hall bar devices were patterned by electron-beam lithography into 200-1000 nm-wide crosses, and subsequent deposition and lift off of the stack. After deposition, the stack was annealed at 300°C during 10 minutes to crystallize the MgO. A sample-wide back-gate of Ti(10 nm)/Au(100 nm) was then added by evaporation. The Hall voltage measurements were performed by using an AC current with an amplitude of 200 to 600 μA, modulated at ω/2π = 60 Hz. VH,ω and VH,2ω were recorded simultaneously using 2 lock-ins during sweeps of the external magnetic field for 6 s at each field step.

Figure 1 |

 1 Figure 1 | Schematic of the sample structure and its magnetotransport properties. a, 3D schematic of the sample and of the coordinate system. b, Scanning electron micrograph of a Hall cross-bar and measurement principle. c, Temperature dependence of the stack sheet resistance Rs. d, Out-of-plane (θH = 0°) and near in-plane (θH = 85°, φH = 0°) magnetic field dependence of the Hall resistance at 10 K, showing well-defined up and down perpendicular magnetization states. All data have been measured with a 2D current density of j = 0.1 A.cm -1 .

Figure 2 |

 2 Figure 2 | Non-volatile electric-field control of the 2D gas properties. a, Sheet resistance Rs as a function of the applied gate electric-field Eg across the sample, showing high and low resistivity states of the device with Rs contrast of 1064%. b, Temperature dependence of the Rs contrast. c, Endurance property of Rs at remanence (Eg = 0 kV.cm-1) after application of positive and negative gate electric-field pulses of ±4 kV.cm -1 . All data have been measured at 10K with a 2D current density of j = 0.1 A.cm -1 , and after initialization of the ferromagnet in the up magnetization state.

Figure 3 |

 3 Figure 3 | Spin-orbit torques characterization by second harmonic Hall measurement. a, Second harmonic Hall resistances 𝑅 𝑆𝑂𝑇,2𝜔 at φH = 0°, corresponding to the anti-damping like signal, for Eg = ±3.2 kV.cm-1 . b, Second harmonic Hall resistances 𝑅 𝑆𝑂𝑇,2𝜔 at φH = 90°, corresponding to the field-like signal, for Eg = ±3.2 kV.cm-1 . All data have been measured at 10K with a linear AC current density of j = 4 A.cm -1 .

Figure 4 |

 4 Figure 4 | Non-volatile electric-field control of the spin-orbit torques. a, Gate electric-field dependence of the spin-orbit torques antidamping -like effective field µ0HAD. The points A-D are examined in c. b, Measured portion p of the current injected in the 2DEG as a function of the gate electric-field Eg. c, Normalized second harmonic Hall resistance at shifted electrical remanence Eg,r after successive application of negative (blue) or positive (red) gate electric-field pulses of ±2.6 kV.cm -1 . Dashed and dotted lines are linear fits, yielding SOT effective fields µ0HAD/j = +0.29, -0.20, +0.32 and -0.18 mT. A. cm -1 , respectively. All data have been measured at 10K with a linear AC current density of j = 4 A.cm -1 and a nearly in-plane magnetic field (θH = 85°, φH = 45°).

Figure 5 |

 5 Figure 5 | Dependence of the spin-orbit torques anti-damping like components on the injected current density and on the temperature.a, Anti-damping like effective field µ0HAD as a function of the applied 2D current density for positive (blue) and negative (green) gate electricfield of ± 2.6 kV.cm -1 at the temperature of 10K. Dashed and dotted lines are linear fits, yielding SOT effective fields µ0HAD = +1.36 mT. A. cm -1 (-0.18 mT. A. cm -1 ) for positive (negative) gate electric-field. b, Temperature dependence of the SOT anti-damping like effective field µ0HAD at j = 2 A. cm -1 for positive (blue) and negative (green) gate electric-field of ± 2.6kV.cm -1 . All data have been measured with a nearly in-plane applied magnetic field (θH = 85°, φH = 45°).

Acknowledgements

This work received support from the ERC Advanced grant number 833973 "FRESCO", the H2020 ITN project SPEAR, the French Research Agency (ANR) as part of the projects OISO (ANR-17-CE24-0026-03) and CONTRABASS (ANR-20-CE24-0023) and in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02). The authors would like to thank the Institut Universitaire de France, the Plateforme Technologique Amont (PTA) for technical support, as well as members of Spintec and UMR Thales for fruitful discussions.

Author contributions

C.G., L.V., J.P.A. and M.B. planned the experiment. C.G., L.V and S.A. fabricated the samples. C.G and A.K performed the measurements. C.G., K.G., L.V. and J.P.A. analyzed the data. C.G., K.G., L.V. and J.P.A wrote the manuscript. All authors discussed the results and the manuscript.