Giant non-volatile electric field control of proximity induced magnetism in the spin-orbit semimetal SrIrO3

With its potential for drastically reduced operation power of information processing devices, electric field control of magnetism has generated huge research interest. Recently, novel perspectives offered by the inherently large spin-orbit coupling of 5d transition metals have emerged. Here, we demonstrate non-volatile electrical control of the proximity induced magnetism in SrIrO3 based back-gated heterostructures. We report up to a 700 % variation of the anomalous Hall conductivity {\sigma}_AHE and Hall angle {\theta}_AHE as function of the applied gate voltage Vg. In contrast, the Curie temperature TC = 100K and magnetic anisotropy of the system remain essentially unaffected by Vg indicating a robust ferromagnetic state in SrIrO3 which strongly hints to gating-induced changes of the anomalous Berry curvature. The electric-field induced ferroelectric-like state of SrTiO3 enables non-volatile switching behavior of {\sigma}_AHE and {\theta}_AHE below 60 K. The large tunability of this system, opens new avenues towards efficient electric-field manipulation of magnetism.


Introduction
Electric field (EF) control of magnetism in materials is of central importance for the development of sustainable and low power consumption information technology [1][2][3] .It is particularly challenging to achieve in ferromagnetic (FM) metals in which EF large enough to induce sizeable modifications of the magnetic state can generally not be applied.This has triggered intense research activity in particular on multiferroic oxides, where magnetic and ferroelectric order are inherently coupled, albeit generally weakly, or on heterostructures combining FM metals with a dielectric or ferroelectric gating material.There, the mechanism for the EF control of magnetism can be based on e.g.elastic strain mediation in combination with the inverse magnetostrictive effect, on voltage control of exchange coupling [1] , or a modulation of the charge carrier density.
Latter approach is particularly relevant when magnetic phase, -moment, and -anisotropy depend noticeably on the density of states near the Fermi energy EF but is generally not very efficient in good FM metals.There, the EF-induced modulation of the carrier density n is limited by the Thomas-Fermi screening length TF  (1/n) 1/6 which only amounts to about 1 Å [4] .This has proven more effective in semiconductors [5] or in transition metal oxides (TMOs) [1] known for exhibiting significantly smaller n.
In heavier TMOs, such as the 4d TMO SrRuO3, electrostatic modulation of n may not only result in changes of the magnetization and its anisotropy [6] but can also affect the integral of the Berry curvature (BC), thereby leading to changes of the anomalous Hall effect (AHE) -a fingerprint for the FM state in conductive materials [7] .AHE originates from the spin-orbit coupling (SOC) which is naturally present in heavy metals and enables, through an asymmetry in the scattering of spin-polarized electrons [8] , an interplay between spin and charge on the electronic transport at the heart of the emerging field of spin-orbitronics.
SOC is particularly strong in the iridium-based 5d TMOs of the Ruddlesden-Popper series Srn+1IrnO3n+1 in which, however, in contrast to archetypical correlated 3d TMOs, the electronelectron correlation strength is generally too small to host ferromagnetism.The iridates display SOC which is on a similar energy scale than that of the electron correlation or electronic bandwidth [9] .Therefore, they are at the verge of a magnetic ground state and may display AFM or FM properties as well, depending on the details of the Hubbard interaction U and SOC [9] .The rather large Ir5d and O2p orbital hybridization in SrIrO3 (SIO) (n = ∞) results in a semimetallic paramagnetic state [9][10][11] .We have recently shown that a FM state with large and positive AHE can be induced in SIO by proximity effect when putting it in direct contact with a FM insulator, LaCoO3 (LCO) [12] .Recent first principle calculations on SIO/LCO heterostructures indicate that it originates from unconventional topology of the electronic bandstructure of FM SIO [13] .
In this work we demonstrate the EF control of magnetism in SIO heterostructures, evidenced through manifold increases of the anomalous Hall conductivity   , the Hall angle Θ  and the magnetoresistance.We further show that a non-volatile EF switching behavior is enabled by the EF-induced ferroelectric state of STO.The effects are discussed in terms of Rashba effect at the SIO/LCO interface and topological BC features of the SIO band structure, and we argue that the latter are more likely to account for the experimental observation.

Non-volatile electrostatic gating of SrIrO3
Three terminal back-gating devices consisting of epitaxial SIO/LCO heterostructures were prepared by pulsed laser deposition and photolithography, see Methods and Supporting Information (SI).SIO and LCO film thicknesses of 10 monolayers demonstrate stable and reproducible proximity induced ferromagnetism in SIO.A scheme of the device layout is shown in Figure 1a.The high degree of crystallinity at the SIO/LCO interface is documented by the high-resolution scanning transmission electron microscopy (HR-STEM) micrograph in Fig. 1b.A gate-voltage Vg > 0 (< 0) usually results in an electron accumulation (electron depletion) in the SIO channel.The expected charge carrier modulation n induced by electrostatic gating can be estimated by assuming a parallel-plate capacitor model to be of the order of 0.35 % (SI).The modulation thickness of the SIO channel is in principle limited by the Thomas-Fermi screening length [4] , TF , which can be significantly larger in semimetals than in good metals due to increased dielectric permittivity (SI).In addition, charge carrier localization may increase TF considerably as well.In LCO/SIO heterostructures, the first 6 SIO layers show insulating behavior [12] .Therefore, even for the rather short TF ≈ 0.92 ML as deduced from experiment, a distinct electric field can be expected at the SIO/LCO interface (SI).V g (V) In Fig. 1c the relative change of the SIO resistance, R/R, is shown for Vg-sweeps at different temperatures T. R/R increases with increasing Vg.The positive field coefficient (dR/dVg > 0) suggests a hole-dominated conductivity in contrast to the generally observed negative Hall coefficient which hints to an electron-type transport [14][15][16] .This can be accounted for by the different mobilities of the electron and hole charge carriers demonstrated by magnetotransport [16] and electronic Raman scattering [17] .Note that electron-and hole-like pockets of semimetallic SIO sensitively depends on the structural properties (epitaxial strain) [11,14] .
The increase of R/R at any Vg with decreasing T is consistent with the increase of the dielectric constant of STO, g.We obtain R/R ≈ 15% at T = 2 K for Vg = 50 V, which is larger than what is expected from simple electrostatics consideration and indicates decrease of electron mobility for electron accumulation in SIO/LCO (SI).Weak charge carrier localization also likely contributes to the monotonous increase of R with decreasing T [12] , and explains the distinct increase of TF.
At low temperatures, R/R displays a non-linear and hysteretic behavior upon sweeping Vg, akin to gating effects reported in other STO heterostructures [18] .As such, the resistive state for Vg = 0 depends on the history (sweeping direction).The effect is only seen for T ≤ 60 K, the temperature below which STO undergoes a phase transition to a ferroelectric-like state under electric fields E ≥ 2 kV/cm [19,20] .The applied field strength (Eg = +/-5 kV/cm) here is large enough to induce such transition and is therefore likely responsible for the observed and wellreproducible non-volatile switching behavior (Fig. 1d).

Electric field control of magnetism
We now turn to magnetotransport of the thin SIO layer.The insulating character of LCO confines electric transport solely to SIO and allows unambiguous selective characterization of the proximity induced FM state in SIO [12] .The AHE and the anisotropic magnetoresistance (AMR), two hallmarks of a FM metal, are related to the structure and ordering temperature of the FM state and are therefore useful quantities to characterize the magnetic properties of the material.The Hall resistivity xy(0H) and the magnetoresistance  = [  ( 0 ) −   0 ]   0 ⁄ (  0 =   ( 0  = 0)) with magnetic field 0H applied perpendicular to the film surface are displayed in Figure 2 for different Vg at T = 20 K, i. e., in the FM state of SIO (TC ≈ 100 K).For other temperatures, see SI.
The total Hall resistivity of SIO, xy(0H) can be decomposed in two components, respectively ordinary (OHE) and anomalous (AHE).OHE is caused by Lorentz force and varies linearly with 0H within the investigated field range.The large hysteresis seen in xy (Fig. 2a) results from the anomalous contribution, typical for a FM metal [21] .In conventional magnetic systems, AHE is proportional to the perpendicular magnetization M, AHE = R A ×M, where R A depends on the longitudinal conductivity xx [22] .On this basis, we can well describe our data using the empirical formula xy = R O ×H + R A ×M, where M=(Ms×tanh(h×(H±Hc))) is modelled using a modified Heaviside-step function.Here Ms, h and Hc are the saturation value, the slope at Hc and the coercive field, respectively.AHE is obtained after subtraction of the field-linear part contribution to xy(0H) and is shown in Fig. 2b (for OHE see SI).
Remarkably, we observe a strong dependence of AHE with Vg, from which we extract a relative increase of Ms: (Ms(Vg)-Ms(0))/Ms(0) by a factor of 7 when going from Vg = -50 V to +50 V.A significant increase is also observed for Hc and the saturation field Hs.
The MR obtained from longitudinal resistivity xx(0H) at 20K (see Fig. 2c) exhibits strong dependence with Vg alike.It is also well described by the sum of two contributions, the classical Lorentz scattering (MR  H 2 ) [23] resulting in a positive contribution and spin-flip scattering (MR  -M 2 ), which contributes negatively to MR in the FM state [24] .For Vg = +50 V MR is dominated by the negative hysteretic contribution confirming strong FM character, in strong contrast to Vg = -50 V where MR is bestridden by the classical positive contribution.For 0H = 14 T and Vg = +(-)50V MR amounts to -0.2% (0.4%).
As for the Hall effect, we can extract M from the MR data using a Heaviside step-function to model M and fit the data, albeit only reliably for Vg > 0. The effective magnetization M perpendicular to the plane obtained this way is very similar to that obtained from AHE, as shown on a normalized scale in Fig. 2d.It is worth noting that this is not a trivial result as many different mechanismsintrinsic (integral of the Berry curvature over occupied states [25] ) as well as extrinsic (side-jump-and skew-impurity scattering [26] ) can contribute to the AHE.For our SIO/LCO heterostructures the AHE was found to be intrinsic [12] .
-  increases by more than one order of magnitude.To the best of our knowledge, and in contrast to reports in e.g.4d TMO SrRuO3 and SrRuO3/SIO heterostructures, the dependence of the effective magnetization with Vg reported here is remarkably large [7,27,28] .Note that the unipolar electrostatic gating asymmetry of the effect for    which is less affected for Vg < 0 (in contrast to    ) than for Vg > 0 might be of practical interest for the realization of spintronic devices.(Wcm) Another quantity which is also highly relevant for spintronic purposes is the ratio between the saturated anomalous Hall conductivity /  0 .Generally, due to the large SOC,   is much larger in the 5d iridates compared to 3d TMOs [22,29] .In Fig. 3c,   is shown versus Vg for T = 20 K.The gate voltage dependence is very similar to that of    , indicating a much stronger influence of Vg on    than on   0 .This is in full agreement with the intrinsic nature of the AHE in SIO [12] , where AHE does not depend on xx.Sweeping Vg from -50V to +50 V, increases   by more than 500%.The temperature dependence of M expressed by    () is shown in Fig. 3d for different Vg.Although    () is significantly enhanced for positive Vg, the onset of the effect TC ≈ 100 K is not much affected by Vg.

Anomalous magnetoresistance and magnetic anisotropy
Manipulation of the magnetic anisotropy or even switching of the magnetic easy-axis from into out-of-plane is of special practical interest.In the following, we discuss the EF-dependence of the magnetic anisotropy.
The angle-dependence of the two-fold component is similar to that of the classical AMR() [30] , the spin-Hall magnetoresistance, SMR() [31] , or the spin-orbit magnetoresistance, SOMR() [32] .However, the amplitude of ~ 0.04% is about two orders of magnitude larger compared to values generally reported for SMR() and SOMR() [31,32] and more typical for the normal AMR(), where the amplitude depends on the effective magnetization of the sample.This is confirmed by the strong increase of <C2> by a factor of 5 with increasing Vg, strongly reminiscent of the behavior of    (so of M).
The four-fold component, on the other hand, does not depend on current direction and is thus related to the magnetocrystalline anisotropy.The minima positions correspond to reduced spinflip scattering and indicate an in-plane <110> magnetic easy-axis direction [12] .In comparison, <C4> changes only by a factor of 1.6 when Vg is increased which indicates slightly enhanced <110>-easy-axis behavior.The minima positions of the four-fold magnetocrystalline component are obviously not affected by Vg.The measurements suggest negligible influence of Vg on the symmetry or strength of the magnetocrystalline anisotropy.

Discussion
We have presented a series of experimental results demonstrating efficient EF-control of the magnetic behavior of SIO-based heterostructures.Very large changes of AHE (×7) and QAHE (×5) when sweeping the gate voltage from Vg =-50V to +50 V are observed, with a strong asymmetry of the effect, which is essentially occurring for Vg > 0. In contrast, the magnetic anisotropy and TC are rather unaffected.Note that in a Stoner description of SIO ferromagnetism, one could expect TC to be sensitive to changes of the charge carrier density n upon gating.Given the weakness of the electrostatic modulation (n/n ≈ 0.35%, see SI) in this system, the variation of TC might simply be too small to be detected.
How to understand in this context the huge EF-induced changes of the AHE?Two natural options are the EF-dependence of the Rashba effect at the SIO/LCO interface [33][34][35] [ [33][34][35][36] on the one hand and topological features of SIO band structure on the other hand.
The strength of the Rashba effect can be controlled by an applied strong EF through the linear dependence of the Rashba coefficient R on E for free charge carriers [37] .In order to obtain significant effects, large electric field strengths of the order of V/nm have to be applied [33] .Such a large EF may also result in a strong coupling to electronic structure via orbital deformation and anomalous band splitting [38] .The resulting momentum dependent 'Rashba equivalent' magnetic field is expected to affect AHE quadratically (so symmetrically) on EF or Vg [33] , in strong contrast with our experimental observation, ruling out Rashba origin as the main source for the EF tunability of the AHE reported here.Note, electric field strength applied here is only of the order 10 -4 V/nm.Topological band properties of SIO may also contribute to the AHE.The scatteringindependent intrinsic contribution to the AHE comes from the Berry phase supported anomalous velocity.An interesting aspect of the intrinsic contribution to the AHE is that the Hall conductivity   is given as an integral of the BC, Ω  , over all occupied states below the Fermi energy [39] : Here, f(k) is the Fermi-Dirac distribution function.
A direct connection between BC and magnetotransport has been recently reported for topological insulator Bi2Se3 [40] in which gating-induced upshift of EF was found to increase the contribution of conduction electrons to BC and to increase the spin Hall conductivity.In magnetic oxides with complex band structures, the intrinsic mechanism for the AHE and the spin Hall effect (SHE) is the same [22] and depends on the detailed properties of the momentumspace BC.The presence of band crossing points close to EF can affect the BC and even result in a sign-change of the AHE [41] .
For SIO, the large intrinsic SHE previously reported [42,43] supports the existence of BC anomalies.Furthermore, recent first principle calculations on SIO/LCO heterostructures unraveled a ferromagnetic band structure for the tetragonal structured SIO exhibiting non-trivial topological features (double Weyl points above and below EF) responsible for a large AHE [13] .Their location in the Brillouin zone of the tetragonal distorted FM SIO is shown in Fig. 5b.
They indeed contribute positively to the AHE and the integral BC results in AHE = 7.5 W -1 cm - 1 , well comparable to the experimental value of 3 W -1 cm -1 for SIO/LCO below 30 K [12] .Weyl points with positive (green) and negative (grey) chirality are highlighted.Data were taken from Ref. [13] .c Integral of the BC as a function of EF.Weyl crossing points occurring This allows us to sketch a scenario for the EF control of magnetism in SIO reported here: since double Weyl points are located only 10 meV above EF, an EF-induced charge carrier accumulation (Vg > 0) may shift EF across those band-crossing points (indicated by the dashed black lines in Fig. 5c).The shift of EF can be estimated by the electron doping n = 3.5×10 18 cm -3 for Vg = +50V (see SI) and the density of states (DOS) at EF (see SI of Ref. [13] ).Considering the integrated density of states, a shift of 0.01 eV by n and therefore access of the second Weyl point by electric gating is very likely.When this occurs, the integral BC increases abruptly and so do AHE or QAHE [44,45] .The minor influence of Vg on the magnetic ordering temperature TC and anisotropy tend to favor such topology-based scenario [45] .
Many properties such as the orbital magnetization and anomalous Hall conductivity can be expressed in terms of Berry phases, connections, and curvatures and are therefore directly related to each other.Hence, xx(0H) and MR will naturally be affected by the AHE alike [46] .As such, the direct relation between (0H) and MR emphasized earlier (Fig. 2d) also favors the topological scenario.

Conclusion
In summary, we have shown that electrostatic gating allows for a very large tunability of the proximity induced AHE, QAHE and MR in SIO/LCO heterostructures likely rooted by the singular band structure of SIO.The results demonstrate that the magnetic properties of FM topological materials can be very effectively controlled by electric fields, offering a promising avenue for realizing energy efficient spintronic devices.

Sample preparation
The SIO/LCO heterostructures were grown on (001) oriented SrTiO3 (STO) substrates by pulsed laser deposition.First, 10 MLs of SIO were deposited on TiO2-terminated (001)-oriented STO substrate, followed by the deposition of 10 MLs of LCO.Film-thickness and layer-bylayer growth were controlled in-situ by reflection high energy electron diffraction (RHEED).More details on film preparation are described elsewhere [12,47] .After film preparation, microbridges (40 m width and 200 m length) were patterned by standard ultraviolet photolithography and Ar-ion etching.Next, the STO substrate was thinned down from the backside to about 0.1 mm to increase possible electric field strength.The back-gate electrode was provided by Pt-sputtering, silver-paste and Al-wiring, whereas source and drain contacts to the SIO/LCO interface were done by ultrasonic Al-wire bonding.
The structural properties of the SIO/LCO heterostructures were analyzed by x-ray diffraction using a Bruker D8 DaVinci diffractometer and high-resolution transmission electron microscopy (HRTEM) as shown elsewhere [12] .

Electronic Transport and data analysis
Measurements of the electronic transport were carried out using a physical properties measurement system (PPMS) from Quantum Design.The modulated (2Hz) source-drain sample current was typically 1-10 A.A sample rotator HR-133 was used for angle-dependent magnetoresistance measurements.The gate-voltage was provided externally by a Keithley 6517B Electrometer, constrained to 50V by the PPMS.Before the measurements, the sample were kept for 12 hours in the cryostat at 2K to stabilize sample and the gate-voltage was ramped up and down several times.AMR() measurements were done starting from Vg = -50 V to Vg = +50 V.
The longitudinal and transversal resistivity were symmetrized and anti-symmetrized with respect to magnetic field to obtain xx(0H) and xy(0H), respectively.AHE(0H) was deduced from xy(0H) by subtraction of the linear part, i. e., OHE(0H).Angle-dependent xx() was corrected with respect to sample wobbling and offset resistance.All the fitting routines described in the text were carried out with MATLAB and fitting parameters are listed in the SI.

Supporting Information
where proximity effect is suppressed by a 4 ML thick STO insertion layer.The field effect devices are prepared as documented in the main text.From a back-gate capacitor model, the change in the charge carrier concentration n at Vg = +50 V amounts to n = 3.5×10 18 cm -3 , see below.For sample A (n = 1.9×10 21cm -3 ) and B (n = 1×10 21 cm -3 ) this corresponds to a relative change n/n of 0.18% and 0.35%, respectively.
The expected charge carrier modulation n induced by electrostatic gating was estimated by assuming a parallel-plate capacitor model: , where 0 and g are the permittivity of vacuum and STO, Eg the gate electric field strength, e the elementary charge and dch the channel thickness.Due to the T-dependence of g the gating effect becomes increasingly efficient with decreasing T. Assuming g ≈ 5000 for T < 40 K and Eg = 5 kV/cm (Vg ≈ 50 V), n amounts to 3.5×10 18 cm -3 .With respect to the charge carrier density n ≈ 10 21 cm -3 of the SIO channel this corresponds to a charge carrier modulation of about 0.35 %.
To probe charge carrier concentration, Hall measurements were carried out.The transversal Hall resistance Rxy of SIO films typically show linear behavior for magnetic fields B ≤ 14 T, despite the semimetallic behavior of SIO.Considering two types of charge carriers, i. e., electrons and holes, a two-band model is used to describe Rxy: which in the low field limit results in: where nh, ne, and h, e are the density and mobility of holes and electrons, respectively.In Fig. S2(a), the relative change of the SIO channel resistance of sample A is shown versus the gate voltage Vg for different T. The hysteretic behavior for T < 60 K is caused by electric-field induced ferroelectric -like state in STO, see also main text.Interestingly, R increases for Vg > 0. Generally, Vg > 0 results in an electron accumulation, i. e., in an increase of ne (ne > 0).Therefore, a decrease of the electron mobility is expected where: A change of nh and h is neglected for Vg > 0. With respect to Eq. (S3), a decrease of  with increasing n is usually observed for SIO and oxide heterostructures at low T for n > 10 19 cm -3 .
For sample A, R/R amounts to about +0.5% (-0.5%) for Vg = 50 V (Vg =-50V) and T ≤ 30 K. Assuming R  (n) -1 the relative change of R is given by: Therefore, for n/n = 0.18%, the relative decrease of  should be around / = -0.68%,which The T-dependence of the gating effect is shown in Fig. S2(b), where R/R versus T is plotted for Vg = +50 V.As expected, the T-dependence is very similar to that of the permittivity (T) of the STO gate material, i. e., a distinct increase below 100 K and saturation below about 30 K for E > 1.5 kV/cm. 3om these observations, the gating effect on the STO/SIO/STO/LCO heterostructure (sample A) is well described by simple electrostatics and the semimetallic behavior of SIO.However, for the STO/SIO/LCO heterostructure (sample B), where LCO is in direct contact with SIO, the gating effect is different.The relative change R/R for Vg = +50 V amounts to 15% at 2K and is about 30 times larger compared to sample A even though the field-induced electron accumulation should be the same in both samples.With respect to Eq. (S4) a decrease of / ≈ -15.35% is expected, which may hint towards weak charge carrier localization in SIO.This is well consistent with the R versus T behavior of both samples (see SI of Ref. [12]).The T-dependence of R/R, see Fig. S2(d), also differs significantly to that of sample A. R/R steadily increases with decreasing T down to 2K.
-50 0 50 -10 Electric screening.The effective modulation thickness of the SIO channel is in principle limited by the Thomas-Fermi screening length TF, which describes the penetration of the electric field E into the channel material as a function of distance z from the gate-channel interface: E(x) = E(0)exp(-z/TF), where TF = (0k/e 2 N(EF)) 1/2 .Here, k and N(EF) are the dielectric permittivity and density of states near the Fermi energy EF of the channel material.In the limit of a 3-dimensional electron gas this results in TF ≈ (0kℏ 2 me 2 ) 1/2 ×(/(3n)) 1/6 , with the particle mass m and density n.Assuming k = 10 (typical for semimetalslarge values of k = 30 have been reported for Sr2IrO4 48,49 and even higher values k ≈ 600 for other iridates [S1]) TF amounts to about 6 Å, which is significantly larger compared to good metals (TF ~ 1 Å), however still smaller compared to the SIO film thickness of 10 ML (≈ 40 Å).To increase TF by one order of magnitude (TF ≥ 60 Å), k has to be increased by two orders of magnitude (k ≥ 1000).
On the other side, in case of electron localization classical Lindhard theory and derived formula for Thomas Fermi screening length TF, which assumes uniform electron gas, does not hold any more.Localized electrons cannot move freely, and hence electric fields are not fully screened.In this case, even weak charge carrier localization such as likely present in sample B will result in a significant drop of the dynamic screening and a much larger screening length compared to the expected value of TF.We want to point out here, that the first few layers of SIO on STO do show insulating behavior.As documented in Fig. 2b of Ref. [12], the first 6 layers of SIO in the SIO/LCO heterostructures are obviously not conductive.Therefore, distinct damping of the electric field in SIO may start not before the 7th layer!
To probe the thickness dependence of the electric field effect in more detail, we have prepared a set of LCO(10 ML)/SIO(xML) heterostructures with different SIO film thickness (x = 6 -15 ML).In Fig. S3(a) we have shown the normalized resistance versus T for different x.Obviously, for x < 10 ML the resistance strongly increases with decreasing T. For x = 6, the films are insulating below 100K.The gating effect, i. e., the relative change of the resistance with gate voltage R/R = [R(+50V)-R(0)]/R(0) versus x is displayed in Fig. S3(b) for T = 10K.For x > 10 ML the gating effect drops down exponentially, as expected from classical electric field damping.Assuming charge carrier modulation n/n proportional to R/R and proportional to electric field strength E we can deduce electric field damping.From the exponential drop, TF ≈ 0.92 ML is deduced, which compares well with the estimated value of 6 Å.This result is also consistent with the assumption that the first 6 SIO layers are insulating.Figure S4 demonstrates E(x) assuming insulating behavior for the first 6 ML of SIO and TF = 0.92 ML.A distinct electric field can be expected in the magnetic active SIO layers.As already shown in Ref. [12], the magnetic proximity effect is triggered by interfacial charge transfer from Ir4+ to Co2+ and limited to the first or second SIO layer close to the SIO/LCO interface.Magnetotransport of SIO/LCO heterostructures.The normal magnetoresistance MR = (xx(0H)-xx(0))/xx(0) and the Hall resistivity xy(0H) with magnetic field 0H perpendicular to the film surface are displayed in Fig. S6 for different Vg (+50, 0, and -50 V) at various temperatures for T < TC ( T = 20, 30, 40, and 80 K).Magnetotransport reflects the FM state of SIO (TC ≈ 100 K).MR (see Fig. S6a) is well described by the sum of two contributions, the classical Lorentz scattering (MR  H 2 ) resulting in a positive contribution to MR and spin-flip scattering (MR  -M 2 ), which is effectively suppressed in the FM state and leading to a negative contribution to MR below TC.The Hall resistivity xy(0H) (see Fig. S6b) is also best described by the sum of two components, i. e., the ordinary Hall resistivity OHE caused by Lorentz force which for SIO films is usually found to be linear to 0H in that field-range, and a hysteretic anomalous part AHE, typical for a FM metal.AHE (see Fig. S6c) has been obtained by subtracting the linear contribution OHE (see Fig. S6d) from xy(0H).Magnetization obviously increases with decreasing T and positive gate voltage (Vg > 0) resulting in a continuous increase of AHE and the negative contribution to MR.
The slope of OHE versus T is always negative indicating electron-like transport.However, slope of OHE increases with increase Vg.Within a two-type carrier model, this is explained by a reduction of the electron mobility  in that way, that it overcompensates the increase of n, see Eq. ( S3).C2() shows maxima at  = 0° (current parallel to H and the <100>pc pseudo-cubic crystallographic direction) and 180°, whereas C4() displays minima at  = 45°+n×90° (n = 0, 1, 2, and 3).The amplitude <C2> increases strongly when Vg changes from -50 V to 50 V.The increase is well comparable to the increase of    and hence obviously related to the increase of the magnetization.In comparison, <C4> changes less with Vg however still indicate enhanced <110>-easy-axis behavior.The positions of the extrema are obviously not affected by Vg.For T > 40 K AMR() diminishes very much which makes distinct data analysis difficult.Fitting parameters.In the tables S1-S3 we have summarized all fitting parameters as deduced from fittings shown in the main manuscript, Fig. 2b and c, and Fig. 4a.

Figure 1 .
Figure 1.Electrostatic gating of a SIO/LCO heterostructure.a Scheme of the three terminal gating device structure.Source (S) and drain (D) contacts to the SIO/LCO interface were done by Al-wire bonding.The gate contact (G) was done by Pt-sputtering, silver paint and Al-wiring.As indicated, a positive gate voltage (Vg) usually results in a negative charging of the SIO channel.b (top) Cross-sectional HR-STEM micrograph of a typical SIO/LCO interface.Cations are indicated.The interface displays stoichiometric composition without distinct

Figure 2 .
Figure 2. Electric-field dependence of the magnetotransport.a Hall resistivity xy versus magnetic field 0H for different Vg at T = 20 K and b the extracted anomalous part of the Hall resistivity   (symbols).Fits to the data (see text) are shown by solid lines.

Figure 3 .
Figure 3.Control of the ferromagnetic state by electrostatic gating.a The saturated and b remnant anomalous Hall resistivity    and    versus Vg at T = 20 K, c The Hall-angle QAHE =    /  0 versus Vg at T = 20 K. Arrows indicate the field-sweep direction.d    versus T for different Vg.

Figure 4 .
Figure 4. Anomalous magnetoresistance of the SIO/LCO heterostructure.a The angle-dependent anisotropic magnetoresistance AMR() for different Vg at T = 20 K and 14 T. b Schematic of the pseudo-cubic crystallographicand in-plane magnetic field direction with respect to the current flow direction.c The two-fold and d the four-fold component of AMR(), shown in a. Fitting parameters are listed in SI.

Figure 5 .
Figure 5. Structure and topology of FM SIO and influence on the AHE. a Structure of the SIO/LCO heterostructure.Only the pseudo-cubic perovskite cells are indicated.In-plane lattice parameters are the same as for STO substrate.Out-of-plane lattice parameter of SIO (LCO) is larger (smaller) compared to that of STO.The (a 0 a 0 c -) octahedral rotation pattern of LCO and SIO is indicated by arrows.b The Brillouin zone of the tetragonal distorted FM SIO.High symmetric points are indicated.Weyl points with positive (green) and negative (grey) chirality are highlighted.Data were taken from Ref.[13] .c Integral of the BC as a function of EF.Weyl crossing points occurring 20 meV below EF (indicated by black dashed line) result in a strong increase of − ∫ Ω  .Similar behavior is expected when EF (indicated by red dashed line) is further increased above the second double of Weyl points located 10 meV above.An increase of Vg and n also increases EF resulting in a similar rise of AHE compared to that of − ∫ Ω  which is shown on the right.
is well in the range of what has been observed for SIO (see Manca et al., Phys.Rev. B. 97 (2018)).

Figure S2 .
Figure S2.Electric field-effect in SIO heterostructures.The electric field-effect on a STO/SIO/STO/LCO (sample A, see (a) and (b)) and a STO/SIO/LCO heterostructure (sample B, see (c) and (d)).Both samples display a layer thickness of 10 monolayer (ML) of SIO and LCO.In contrast to sample B, where SIO shows proximity induced magnetism at the SIO/LCO interface, SIO is not magnetic in sample A, where proximity effect is prevented by a 4 ML thick STO insertion layer.The relative change of the SIO channel resistance is shown versus the gate voltage

Figure
Figure S3.(a) Normalized resistivity of LCO/SIO heterostructures for different SIO layer thickness x as a function of T. For x ≤ 6 ML resistance is above measurement limit below 100K.(b) Relative change of the resistance with gating voltage Vg as a function of SIO layer thickness x at T = 10K.R/R = [R(+50V)-R(0)]/R(0).Red symbols correspond to samples with a STO-gate thickness of 0.17 mm.The black symbol corresponds to the heterostructure sample (B)see also main text-, where the STO thickness was only 0.1 mm.

Figure S4 .FieldFigure S5 .
Figure S4.Screening of the electric field in the LCO/SIO/STO heterostructures.The insulating layers and magnetic active layers in SIO are indicated by grey and blue shaded region.The screening length was chosen to be 0.92 ML as deduced from experiment (see above).Distinct residual field is expected at the SIO/LCO interface.