High-mobility magnetic two-dimensional electron gas in engineered oxide interfaces

The engineered interfaces of complex oxides have abundant physical properties and provide a powerful platform for the exploration of fundamental physics and emergent phenomena. In particular, research on the two-dimensional magnetic systems with high mobility remains a long-standing challenge for the discovery of quantum phase and spintronic applications. Here, we introduce a few atomic layers of delta doping layer at LaAlO3/SrTiO3 interfaces through elaborately-controllable epitaxial growth of SrRuO3. After inserting a SrRuO3buffer layer, the interfaces exhibit a well-dened anomalous Hall effect up to 100 K and their mobility is enhanced by 3 orders of magnitude at low temperatures. More intriguingly, a large unsaturated positive magnetoresistance is created at interfaces. Combining with the density functional theory calculation, we attribute our �ndings to the electron transfer at interfaces and the magnetic moment of Ru4+ 4dbands. The results pave a way for further research of two-dimensional ferromagnetism and quantum transport in all-oxide systems.

And, it also provides a powerful platform for the exploration of fundamental physics and applied research.The magnetic ground state in a 2DEG at interfaces of non-magnetic oxides has been drawing more and more attention, which is centrally important for spintronic applications and novel quantum phases.However, the interfacial magnetism is commonly tiny and its present temperature is very low because of nonmagnetic oxides.Usually, two ways are adopted to realize the interfacial magnetism.One is to use magnetic lms or substrates to produce the magnetism through the magnetic proximity effect.
For examples, the spin-polarized 2DEG at high temperatures was obtained by the Fe-doped STO substrates 6 or a magnetic EuO lms 7 .The magnetic order and a hysteresis loop were achieved at LAO/STO by introducing a magnetic dopant 8,9 .The ultrathin STO layer sandwiched between two magnetic GdTiO 3 layers was also magnetized although no anomalous Hall effect (AHE) was observed 10 .
The AHE effect was reported through the double-doping at interfaces 11 .The other way is to induce the magnetism by arti cially inserting a magnetic lm between two oxides.For instances, the LAO/EuTiO 3 /STO structure by Stornaiuolo et al. 12 and the LAO/La 7/8 Sr 1/8 MnO 3 /STO structure by Zhang et al. 13 were designed and their Curie temperatures were below ∼30 K.The LAO/LaCoO 3 /STO heterointerfaces also exhibited very weak magnetic properties at 2 K 14 .Therefore, exploring the spinpolarized 2DEG remains a pressing challenge.On the other hand, the SrRuO 3 (SRO) is an itinerant ferromagnetic oxide and its Curie temperature is about ∼140 K. Theoretical studies demonstrated that SRO had multiple band crossings consisting of different bands around Fermi level (E F ) 15,16 .Thus, AHE 17,18 , topological Hall effect 19 and Weyl fermions 16 were reported, igniting a renewed interest in the quantum transport and two-dimensional ferromagnetism.In most theoretical and experimental studies 20- 23 , however, the SRO lm with the thickness of less than 3 unit cells (u.c.) usually showed the insulating and antiferromagnetic state due to some defects.Thereby, creating ultrathin metallic high-quality SRO lms has always been an urgent issue.In this work, considering that the SRO with a cubic lattice parameter of 3.93 Å perfectly matches with STO 24 , we arti cially insert the atomic-scale SRO layer at LAO/STO interfaces through elaborately controlling the epitaxial growth of SRO layer.Interestingly, a 2DEG with high mobility, well-de ned AHE and a large positive magnetoresistance (MR) effect of ~803% is obtained T <100 K.More importantly, the SROs with the thickness of 1 and 2 u.c.retain the metallicity and magnetism owing to the interface effect, offering a way for further research of the two-dimensional ferromagnetism.As revealed by the results of density functional theory (DFT) calculations, the contribution of electron transfer and the electron of Ru 4+ 4d bands are the primary cause.

Results And Discussion
Characteristics of sample.The surface morphology of d = 2 sample is shown in Fig. 1a, where the atomically at surfaces with clear steps are observed and its root mean square (RSM) is about 167 pm, indicating the layer-by-layer growth.The HRTEM image of d = 2 sample and its corresponding FFT pattern are shown in Fig. 1b, which reveals a clear lattice structure corresponding to the (001) planes of lm and substrate.Figure 1c is the annular dark eld image of cross-section and the corresponding EDS elemental mapping images.From the results of elemental mapping, we get the atomic distribution at interfaces as shown in the inset of Fig. 1c.The SrO is rst bonded with the TiO 2 layer of STO, and then the RuO 2 layer is on top of it.After growing the 2 u.c.SRO lm, LaO is preferentially grown on the RuO 2 layer.Overall, these results provide strong evidence to the high quality of epitaxial crystalline lms.
Transport properties.Figure 2a shows the temperature dependence of sheet resistance.It can be seen that all heterointerfaces show a metallic state, meaning that we break through the SRO thickness limitation and obtain the metallic state in its ultra-thin layer by the interface effect.At room temperature, the sheet resistance increases with increasing the SRO thickness.When the temperature is cooled down to 2 K, the sheet resistance of LAO/SRO/STO is reduced by more than three orders of magnitude, demonstrating the high mobility.We measured Hall resistance of the heterointerfaces in the temperature ranges of 2-300 K.The LAO/STO interface shows an ordinary Hall effect (OHE) in the Supplementary Fig. 1.After inserting SRO, the samples exhibit a nonlinear Hall effect below 100 K as shown in Fig. 2b and Supplementary Fig. 2a although the role of OHE is dominative.For further analysis, the relationship between the derivation of R xy to H and the magnetic eld is acquired, as shown in Fig. 2c and Supplementary Fig. 2b.We can see that the dR xy /dH forms an inverse peak when the magnetic eld is around 0 T, which is the signature of obvious AHE with long range magnetic order.With increasing the magnetic eld, the Hall coe cient becomes a constant, revealing only one type of carrier at interfaces.

The anomalous Hall resistance (R xy AHE
) can be described using Langevin-type function 25 .Thereby, the total Hall resistance can be expressed as: where, n is the carrier density, e is the elementary charge, H c AHE is the critical eld at which R xy AHE saturates to the value of R sat AHE .As an example, we show the tting curve of d = 1 sample at 50 K in Fig. 2d.The tted curve and the data are well coincided, manifesting that this reasonably agrees with the AHE.As the temperature is raised to 130 K, the linear Hall resistance dawns and the AHE disappears.
By subtracting the linear background from the total R xy -H curve, we obtain the relationship between the anomalous Hall resistance (R xy AHE ) and the magnetic eld at different temperatures.As shown in Fig. 3a and Supplementary Fig. 2c, the R xy AHE increases with the magnetic eld and then saturates.The Hall coe cients are obtained by the tting, further giving the relationship of carrier density and mobility with temperature.The carrier density is decreased from 1.3×10 16 cm -2 to 7.5×10 14 cm -2 at 2 K with increasing d as shown in Supplementary Fig. 3.As expected, the SRO buffer layer has a huge effect on the mobility in Fig. 3b, which is promoted about three orders of magnitude at 2 K, from 10 cm 2 /Vs (d = 0) to 9835 We can see the following features.Firstly, the MR displays a positive effect in the whole temperature range.Secondly, the MR of d= 1 sample presents a parabola style in the high magnetic eld at 2-10 K, i.e.
MRµH 2 , which is a typical orbital effect.As the magnetic eld is less than 2 T, a minimum MR value appears around H = 0 T, manifesting the weak antilocalization (WAL).A very large positive MR value of 324% is observed at 2 K and 13 T. Thirdly, the d = 2 sample simply possesses an unsaturated linear positive MR and its value can reach ~ 803% at 2 K and 13 T.The SRO has a complex Fermi surface with both topologically trivial and nontrivial bands, strongly affecting the MR effect at interfaces.In this system, an unsaturated positive MR is clearly dominated.This means that the contribution of WAL and OMR is very limited to the net MR.In fact, various theories have been proposed to explain the unsaturated positive MR effect.The underlying mechanism includes the MR in the quantum limit conditions 26,27 and the inhomogeneity, involving electric eld inhomogeneity 28 , density inhomogeneity 29 , density uctuations 30 , and antiferromagnetic uctuations 31 .In fact, the quantum MR has been proposed by Lifshits and Peschansky 32 when only the lowest Landau band is partially lled and all others empty.Firstly, the requirements to realize the quantum limit state are ω c τ>1 and ħω c >E F 33 , where ω c =eH/m * is the cyclotron frequency (e is the elementary charge, H is the magnetic eld, and m * » 0.76m e is the electron effective mass 4 ), t=m * m/e is the carrier relaxation time, ħ is Planck constant and E F is Fermi energy.Considering ω c τ=mH and m=9835 cm 2 /Vs at 2 K, we can get ω c τ>1 when H is larger than 1 T in this work.According to E F =ħ 2 k F 2 /2m * (the Fermi wave vector k F =(2πn s ) 1/2 , n s is the carrier density), the ħω c of 1.54 meV is far larger than E F (0.24) meV at H=10 T. Consequently, this satis es ħω c >E F and the quantum transport contributes to a large unsaturated positive MR.In addition, the inhomogeneity from the disorder and/or dopants is also analyzed as shown in Supplementary Fig. 5, which is not ignored on the premise of reaching the quantum limit.
In order to analyze the WAL effect of d = 1 sample, we use the MF (Maekawa-Fukuyama) theory, which considers the orbital effect to perform a quantitative analysis of [∆σ(H)/G 0 ]-H relation.The formula is as follows: (2) where, Ψ(x) is the digamma function, which is expressed by is the conductance at H, G 0 =e 2 /πħ is the quantum conductance.The H tr , H i and H so are effective elds of elastic scattering, inelastic scattering and spin-orbit scattering, respectively.The last term of formula is Kohler term, which mainly describes the orbital MR in the vertical magnetic eld.The perfect tting curves are shown in Fig. 4c.And the parameters of H i and H so at different temperatures are obtained in Fig. 4d.
As shown, there is a steady rise of H i and H so with increasing the temperature.The inelastic scattering time (τ i ) and spin relaxation time (τ so ) satisfy the relationship with the effective eld: H i,so =ħ/4eDt i,so (D is the diffusion coe cient given by the Drude model 34 ).The results in Fig. 4e show that τ i,so is reduced with increasing the temperature.There are two mechanisms about the spin relaxation: the D'yakonov-Perel' (DP) mechanism 35,36 and the Elliott-Yafet (EY) mechanism 37,38 , which can be determined by studying the relationship between τ so and t.In the case of DP mechanism, τ so ~ 1/t, the conduction electron will experience an internal magnetic eld that is always perpendicular to their wave vector.For the EY mechanism, τ so ~ t, the spin relaxation process originates from the spin-orbit interaction of lattice ions with conduction electrons.Hence, the SOC effect of d = 1 sample conforms to the EY mechanism according to Supplementary Fig. 6.Further, we can obtain the Rashba spin splitting energy (Δ=2ak F ), where the Rashba constant a satis es the formula: t so =ħ 4 /8a 2 m e 2 D. Figure 4f shows that the Δ increases with increasing the temperature.For con rming that the behavior is caused by SRO in the 2D interfacial properties, we also measured the resistance vs. temperature curve, Hall resistance and MR of SRO lm at 6 K, as shown in Supplementary Fig. 7.It can be seen that the SRO lm behaves a paramagneticferromagnetic phase transition at ~135 K.The MR exhibits a negative value with a butter y-shaped hysteresis.Obviously, the interfacial properties are completely different from that of the SRO lm.orbitals at the LaO/RuO interface (Fig. 5d).The magnetic moment of Ru here is ~1.37 mB, resulting in an enhanced interface magnetism.
To make a quantitative comparison, we survey a broad class of oxide 2DEG systems with MR vs. mobility at low temperature and 7 T, including LAO/STO [39][40][41] , a-LAO/KTaO 3 42 , CaZrO 3 (CZO)/STO 43 , γ- 44 , LaTiO 3 (LTO)/STO 45 , bulk STO 46 and KTaO 3 47 , as shown in Fig. 6.It can be seen that the MR is linearly increased with the mobility, indicating that the high mobility is a common hint at an origin of linear positive MR.
To summarize, we obtain the two-dimensional magnetic systems by inserting an ultra-thin SRO layer into the engineered LAO/STO interfaces.Firstly, the mobility is greatly improved from 10 cm 2 /Vs of the LAO/STO interface to 9835 cm 2 /Vs of the LAO/SRO (2 u.c.)/STO heterointerface.Secondly, the MR of d = 1 sample exhibits a WAL effect and reaches the value of ~324% at 2 K and 13 T.The d = 2 sample displays an unsaturated linear positive MR with the value of ~803% at 2 K and 13 T.Meanwhile, the buffered SRO induces the interfacial magnetism at T < 100 K. Additionally, the magnetism is contributed by Ru 4+ 4d bands through DFT calculations.In this work, we use the itinerant ferromagnetism oxide to modulate the ferromagnetic 2DEG and further provide a new approach to study quantum transport in alloxide systems.

Methods
A series of crystalline LAO (26 u.c.)/SRO (d u.c.)/STO heterointerfaces (d = 0, 1 and 2) with controllable interface diffusion was grown through high-throughput method.All heterointerfaces were deposited on TiO 2 -terminated STO substrates (3×3×0.5 mm) using pulsed laser deposition with a KrF excimer laser (λ = 248 nm) operating at 1 Hz.For the SRO layer, the growth temperature was xed at 750 °C in an oxygen atmosphere of 10 Pa.The LAO lm was deposited at 800 °C in an oxygen atmosphere of 1×10 -3 Pa.The surface morphology of samples was characterized by MFP-3D atomic force microscopy (AFM).The transport properties of samples were measured by four-point Van der Pauw method in a Physical Property Measurement System (CFMS-14 T) with the magnetic eld up to 14 T. High-resolution transmission Electron Microscope (HRTEM) was performed at room temperature using an Aberration Corrected Scanning Transmission Electron Microscope (STEM), and the cross-sectional cuts were prepared using a dual-beam focused ion beam (FIB) instrument.

Declarations
Figures

cm 2 /
Vs (d = 2).The temperature dependence of R sat AHE and H c AHE is shown in Figs.3c and 3d, respectively.With enhancing the temperature, the R sat AHE increases to the maximum values of 0.178 (d = 1) and 1.67 Ω (d = 2) at 50 and 75 K, respectively.Subsequently, it decreases and the AHE disappears above 130 K, which may be related with the paramagnetic-ferromagnetic transition of SRO layer.The critical magnetic eld also increases with the temperature.Large Positive Magnetoresistance.Further, we obtain the magnetic eld dependence of MR at different temperatures when the magnetic eld is perpendicular to the interface.Here, we de ne MR=[R(H)-R(H=0)×100%]/R(H=0), where R(H) is the resistance under the magnetic eld of H T. The MR curves of d = 0 sample at different temperatures are shown in Supplementary Fig. 4 and the maximum MR is about 16.3% at 2 K and 7 T.The MR curves of d = 1 and 2 samples are shown in Figs.4a and 4b, respectively.
Density functional theory calculations.To understand the effect of SRO on the interface, we performed DFT calculations.The atomic positions are completely relaxed and the lattice constants used in the model are the same as that of lms as shown in Figs.5a-b.For the d = 1 sample, the electrons provided by LAO ll up the 4d xz /d yz orbital of Ru due to the discontinuous polarization at LaO/RuO 2 interface, weakening the magnetic moment of Ru (~1.27 mB).Meanwhile, these carriers have low mobility and the high-mobility carriers further enter the 3d orbital of Ti, thus making the interface produce a higher mobility.When electrons are lled into the d xz /d yz band, the sample shows a stronger SOC effect because the d xz /d yz states have much stronger polarization than the d xy states.Additionally, we can know from the DOS of Ru that the d xy orbital is completely spin polarized (Fig. 5c).In light of the electron occupancy, it is revealed that the spin polarization is mainly contributed by the d xy orbital of RuO 2 layer.About the d = 2 sample, the t 2g orbitals are almost degenerate and the spin polarization is contributed by d xy , d xz , d yz

Figure 2 Transport
Figure 2