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Oxygen vacancy-induced anomalous Hall effect in a nominally non-magnetic oxide

The anomalous Hall effect, a hallmark of broken time-reversal symmetry and spin-orbit coupling, is frequently observed in magnetically polarized systems. Its realization in non-magnetic systems, however, remains elusive. Here, we report on the observation of anomalous Hall effect in nominally non-magnetic KTaO3. Anomalous Hall effect emerges in reduced KTaO3 and shows an extrinsic to intrinsic crossover. A paramagnetic behavior is observed in reduced samples using first principles calculations and quantitative magnetometry. The observed anomalous Hall effect follows the oxygen vacancy-induced magnetization response, suggesting that the localized magnetic moments of the oxygen vacancies scatter conduction electrons asymmetrically and give rise to anomalous Hall effect. The anomalous Hall conductivity becomes insensitive to scattering rate in the low temperature limit (T<5 K), implying that the Berry curvature of the electrons on the Fermi surface controls the anomalous Hall effect. Our observations describe a detailed picture of many-body interactions, triggering anomalous Hall effect in a non-magnetic system.

The description of the anomalous Hall effect (AHE) of magnetically polarized systems 1 in terms of the Berry curvature of the quasiparticles on the Fermi surface is one of the major accomplishments of modern condensed matter theory [2][3][4][5][6][7] . The mobile charge carriers gain a transverse momentum in systems with a magnetic polarization and spin-orbit coupling. The builtup Hall voltage, i.e. anomalous Hall component, can be observed as a distinct, additional contribution in the Hall measurements, superposed on the ordinary Hall effect [8][9][10][11][12][13] . The AHE originates from spin-orbit coupling and is proportional to the magnetization, where relates magnetization to the anomalous Hall component 14 . Theoretical [15][16][17] and experimental [18][19][20] observations of defect-induced magnetization were previously reported in nonmagnetic oxides. Experimental realization of anomalous Hall effect, however, remains elusive in these systems.
The degree of disorder within a system determines the nature of the AHE response. The intrinsic AHE occurs in the moderately dirty regime in which anomalous Hall conductivity becomes nondissipative, i.e. independent of the scattering rate 21 . In this regime, the Berry phase of the quasiparticles on the Fermi surface acts as an effective magnetic field generating a transverse momentum. In the dirty regime, however, the extrinsic anomalous Hall conductivity, , is directly proportional to the longitudinal conductivity, ∝ 1.6 , and asymmetric scattering of the conduction electrons gives rise to AHE. The extrinsic to intrinsic crossover occurs when the scattering rate becomes comparable to Fermi energy 22,23 . In the clean regime, an extrinsic anomalous Hall response emerges due to skew scattering of charge carriers and shows a linear relationship with longitudinal conductivity, ∝ .
KTaO3 is an incipient ferroelectric 24  Here, we report on the observation of an anomalous Hall signal in reduced KTaO3. An extrinsic AHE emerges at low temperature (T<60 K). Our observations suggest that oxygen vacancies play the role of paramagnetic point defects, scattering the conduction electrons and giving rise to AHE.
The anomalous Hall conductivity becomes independent of the scattering rate below 5 K, suggesting an extrinsic to intrinsic AHE crossover.
Oxygen vacancies were introduced to KTaO3 (100) single crystal substrates in ultra-high vacuum using a molecular beam epitaxy system. A 10 nm TiOx layer was deposited at 800 ℃ and 1×10 -9 Torr on the surface of KTaO3. Here, the TiOx layer acts as an oxygen getter and capping layer during annealing. The substrate became dark due to introduction of oxygen vacancies after annealing. The temperature dependent magneto-transport measurements were performed using a Oxygen vacancies introduce itinerant electrons to Ta 5d derived states. Figure 1(a) shows resistance with temperature (300-0.03 K). The sheet carrier density is 2.5×10 15 cm -2 at 60 K and reveals carrier freeze out below 5 K, reaching 4×10 14 cm -2 at 30 mK, Fig. 1    The AHE saturation resistance ( . ) does not change systematically with the temperature at higher temperatures (5-60 K) and shows an abrupt enhancement below 5 K, peaking at 30 mK ). The non-vanishing AHE suggests that the mechanism(s) 6 leading to the AHE remain down to low temperatures. The abrupt enhancement of AHE coincides with a carrier freeze out ( Fig. 1(b)) and suggests a potential change in the AHE mechanism.   speculate that oxygen vacancies act as paramagnetic centers with unpaired spins. We normalized the magnetization and AHE to their saturation value at high field to compare their trend. Figure 3 shows that the normalized magnetization and AHE are in close agreement, confirming that the magnetic nature of oxygen vacancies gives rise to AHE.

Discussion
To briefly summarize the results, our main findings are as follows: (i) Reduced KTaO3 shows a robust anomalous Hall response; (ii) AHE follows the oxygen vacancy-induced magnetization; and (iii) The anomalous Hall response becomes non-dissipative below 5 K, suggesting an extrinsic to intrinsic crossover.
The first important conclusion from these results is that KTaO3-δ, a nominally non We observe a positive magnetoresistance, which is not expected in conventional AHE systems, i.e., itinerant magnets 31,43 .

The Giovannini-Kondo (GK) model describes an extrinsic anomalous
Hall effect based on coupling between orbital motion of the conduction electrons and localized magnetic moments 32,40 . This model, interestingly, does not require the conduction electrons to be magnetically polarized and is consistent with observed positive magnetoresistance. The GK picture, however, predicts a linear scaling ( = 1) between longitudinal conductivity and AHE conductivity. As a result, the model describes the AHE in the clean limit due to skew scattering of charge carrier 44 . Here, the scaling power factor is ~1.6 at higher temperature regime (60-5 K).
Decreasing the temperature suppresses the scattering events and the extrinsic to intrinsic crossover occurs when the relaxation rate becomes comparable to the Fermi energy, ℎ⁄ ∝ , where ℎ, , and are Planck's constant, relaxation rate, and Fermi energy, respectively 22 .
In summary, our results, especially the temperature dependence of the anomalous Hall effect, should be of interest for testing different theoretical models that have been proposed in the literature that relate scattering and AHE 4,5,7,21,45 . It would also be interesting to explore whether similar behavior will be observed in KTaO3 2D electron systems, especially at superconducting interfaces.

Data availability statement
The data that support the findings of this study are available within the article.

Conflict of Interests
The authors declare that they have no conflict of interest.

Magneto-transport results
Transverse magnetoresistance was obtained at different temperatures using a Van der Pauw geometry. The temperature dependent magneto-transport measurements were performed using a The longitudinal magneto-conductivity (Fig. S2) shows a sharp peak at zero magnetic field, suggesting a weak anti-localization correction to conductivity. The coherence length of charge carriers must remain larger than conduction channel thickness, implying that only the electrons near the KTaO3 surface show weak anti-localization.

Two-band fit to Hall results
The anti-symmetrized Hall results were modeled using a two-band model (Eq. S1).
where µ is the mobility and n is the sheet carrier concentration of each band, N is the total sheet carrier concentration, e is the elementary charge, and B is the magnetic field. The fit was constrained by The 1 and µ 1 are the only fitting parameters. The best fit, shown in Figure S3, does not describe the anti-symmetrized hall data. Point defect information was managed with a point defects informatics framework. 10 These parameters, and their convergences, have been widely tested in complementary structures of SrTiO3 and BaTiO3. [11][12][13][14][15][16] The formation energy of the oxygen vacancy was calculated within the established grand canonical formalsm. 17,18 The formation energy of defect D in charge state q is given in Eq. S4.
In this expression, is the formation energy of defect in charge state , is the total energy of a supercell containing this defect in its respective charge state, is the total energy of the ideal KTaO3 bulk cell, atoms are exchanged with the chemical reservoir with chemical potential (here this is the chemical potential of oxygen), electronic charges are exchanged with the Fermi level that is used as a free parameter to plot the formation energies at 0 K and is taken relative to the valence band maximum (VBM), and is the finite size correction discussed previously. TTLs. While this reduces the magnitude of the formation energy, it is still high as compared to that in SrTiO3 and BaTiO3, indicating that it is more difficult to form oxygen vacancies in KTaO3 as compared to other perovskites. 13 This is consistent with the strength of the Ta-O bonds in KTaO3.

Magnetization response in reduced KTaO 3
The magnetic properties of the reduced KTaO3 were measured at different temperatures using a Quantum Design superconducting quantum interference device (SQUID) with a quartz holder. A diamagnetic behavior dominates the magnetic response of the sample, however, a small paramagnetic signal was found with subtracting the diamagnetic response. The paramagnetic response was obtained by removing the high field contribution where the paramagnetic response saturates. Figure S6 shows the Arrott plot at 3 K, confirming the paramagnetic nature of magnetic response at helium liquid temperature using a simple mean field theory. The pristine samples show a significantly smaller magnetization response ( Figure S7). We conclude that oxygen vacancies are the main source of paramagnetic response.