Magnetic and transport properties of electron doped EuTiO₃ thin films with La³⁺ (4f) or Gd³⁺ (4f⁷) donors grown by gas source molecular beam epitaxy

Spintronics has attracted much attention due to the demand for device application in information technology.¹ To design spintronic devices, magnetic semiconductors are regarded as one of the most promising materials.^2–4 The semiconducting nature of these materials enables an electrical and/or chemical control of the carrier density, while the spin degrees of freedom of carriers can be managed by the exchange coupling with magnetic elements. Such duality can offer versatile functionalities that may help realize novel spintronics devices. Here, we focus on EuTiO₃ (ETO) as a magnetic oxide semiconductor, with peculiar characteristics, that can offer unique opportunities for the basic understanding of physical properties.

ETO is an antiferromagnetic insulator with a Néel temperature of T_N = 5.5 K.⁵ Figure 1(a) shows the crystal structure of perovskite ETO. Eu²⁺ has 4f⁷ electron configuration, exhibiting the largest possible magnetic moment of 7 μ_B per atom. The anti-ferromagnetic order can be easily forced to the ferromagnetic one by an external magnetic field.⁶ The electrical ground state of ETO is quantum paraelectric, where the quantum fluctuation suppresses the ferroelectric transition at low temperatures, resulting in a very large dielectric constant. Katsufuji and Takagi reported that the large dielectric constant of ETO is controlled by an external magnetic field through the Eu moments, indicating the strong coupling between the magnetism and dielectric properties.⁷ Due to such strong coupling, the lattice strain induced in the epitaxial films has an impact on both dielectric and magnetic properties. For example, Lee et al. demonstrated that the tensile strained ETO films grown on DyScO₃ substrates generate both ferromagnetic and ferroelectric characters, i.e., a multi-ferroic state.⁸ 

In addition to such an interplay among magnetic and quantum paraelectric properties, as well as lattice strain, ETO has attracted considerable attention as a host of spin polarized electrons with high mobility. Generally, the mobility of carriers in magnetic semiconductors is rather low. For example, in the case of the most intensively studied magnetic semiconductor, Mn doped GaAs,⁹ magnetic dopants of Mn partially substitute the Ga site to induce the exchange interaction, while they disconnect the conduction network composed of sp³ hybridized orbitals in GaAs to give strong scattering factors, resulting in low mobility. Thus, it is quite hard to investigate quantum transport phenomena of spin polarized electrons in conventional magnetic semiconductors. In the case of ETO, the magnetic element of Eu has a nominal 100% occupancy on the A-site in perovskite oxide ABO₃, while its conduction and valence bands are composed of the hybridized B site with Ti 3d and O 2p orbitals, respectively, making them nearly free from disorder. When low-density electron carriers are doped into the 3d conduction band by dilute chemical substitution of Eu²⁺ with trivalent ions, the doping can be achieved by keeping the conduction and valence bands (Ti 3d and O 2p networks) intact. Simultaneously, due to exchange coupling between localized Eu 4f and Ti 3d orbital bands, the conduction electrons can be easily spin-polarized.^10–12 

In addition to the novel situation of ETO, related to the origin of magnetism and the crystal structure as described above, the quantum paraelectric character is quite an important advantage for hosting the high-mobility electron system. ETO exhibits a large dielectric response especially at low temperatures (ε_r ∼ 400) as mentioned above,⁷ resulting in effective screening of the scattering potentials. In the case of the well-known quantum paraelectric perovskite, SrTiO₃, with much larger dielectric constant at low temperatures (ε_r ∼ 23 000), the mobility exceeds 30 000 cm² V⁻¹ s⁻¹ at 2 K in a thick La-doped film,¹³ and quantum Hall effect is readily observed in a delta doped structure with two dimensional confinement.¹⁴ In fact, previous studies of La doped ETO demonstrated that the maximum mobility reaches 3200 cm² V⁻¹ s⁻¹ at 2 K for high crystalline quality and strain-free films on SrTiO₃ substrates grown by gas source MBE.¹² In such high mobility films, clear Shubnikov–de Haas (SdH) oscillations are observed at the forced ferromagnetic states under a large magnetic field (>3 T). Besides such quantum transport phenomena, this high mobility ETO is an excellent platform to study the anomalous Hall effect (AHE) originating from the topology of the band structure. The sign of AHE in the forced ferromagnetic state was successfully controlled by the Fermi energy tuned around the band crossing points (Weyl nodes) composed of Ti t_2g bands that are crystal-field split due to the compressive strain induced by the LSAT substrate.^10,11 Unconventional AHE, which is not proportional to magnetization, was also found during the spin canting process below saturation field only for the high-mobility films.¹¹ These peculiar phenomena on AHE have been predicted and well-explained by the Berry phase theory, which is one of the hot topics of the recent condensed matter physics.¹⁵ 

So far, magnetotransport properties have been investigated for electron doped ETO by chemical substitution of several trivalent rare earth ions such as La³⁺, Sm³⁺, and Gd³⁺ for the Eu²⁺ site.^{10–12,16–21} However, there are quite a few reports examining the dopant dependence of the physical properties in detail. For bulk crystals, it has been elucidated that more than 7% doping of La or Gd into ETO induces ferromagnetic ordering because of a ferromagnetic exchange interaction mediated by conduction electrons,^16,17 while any obvious changes of the magnetic property are not found up to 3% La doping.¹⁷ Here, we pose a question whether magnetic disturbance on the physical properties is unavoidable in the case of La substitution, taking into account the spin-less feature of La³⁺ (4f⁰) ions embedded in the spin-full (4f⁷) Eu²⁺ sublattice. From this viewpoint, Gd³⁺ substitution provides an appropriate reference because of the spin-full state of Gd³⁺ (4f⁷) being isoelectronic with Eu²⁺.

In this paper, the transport and magnetic properties are studied for La and Gd doped ETO, high-quality thin films grown on STO substrates by gas source MBE. As for the magnetic properties, two distinct properties can be seen. La doping induces a suppression of T_N from 5.5 to 3.7 K with 5% doping, while T_N stays constant for Gd doping. The magnetic field to induce a forced ferromagnetic state, B_saturation, is also suppressed for La doping, while it stays constant for Gd doping. These behaviors should originate from the introduction of the vacancy of magnetic moment by La doping. As for the transport properties, the electron mobility at low temperatures is lower for Gd doping than for La doping. Therefore, rather than spin scattering due to the vacancy of magnetic moment, local lattice distortion due to the smaller ionic radius of Gd³⁺ is thought to dominate the scattering. The magnetotransport properties such as magnetoresistance and AHE are basically similar for both dopants. As was the case for compressively strained ETO films with lower electron mobility,¹¹ AHE, with the origin of Berry curvature in momentum space, is still present, verifying the inherent and intrinsic nature of this unconventional AHE in this system.

Epitaxial thin films of doped ETO, namely, Eu_1-xGd_xTiO₃ (EGTO) and Eu_1-xLa_xTiO₃ (ELTO), were grown on (001) SrTiO₃ (STO) substrates (a = 3.905 Å), employing a gas source MBE system. Ti was supplied by evaporating a metal–organic precursor—titanium tetra isopropoxide (TTIP)—while Eu, Gd, and La were supplied by evaporating each pure metal from effusion cells. The flux of TTIP was measured with a beam flux monitor and those of Eu, Gd, and La were measured with a quartz crystal microbalance. The prescribed doping ratio of Gd or La was determined from equivalent beam flux density ratios [Gd/(Eu + Gd) or La/(Eu + La)]. During the deposition, oxidation gas was not injected intentionally into the chamber to prevent Eu₂Ti₂O₇ phase formation, because the Eu³⁺ state in Eu₂Ti₂O₇ is more stable than the Eu²⁺ state in EuTiO₃ under an oxidative environment. However, such a reductive environment inevitably generates oxygen deficiency in STO substrates, making them highly conductive, to prevent the precise measurement of the transport property in electron doped ETO epitaxial films. Thus, we oxidized the samples for 2 h at 550 °C in 2 × 10⁻⁶ Torr pure ozone gas in situ after depositing a capping STO layer on the doped ETO films.²² The temperature of the STO substrate was heated up to 1200 °C with a semiconductor-laser heating system for the deposition of doped ETO and capping STO layers, and then it was cooled down to 550 °C for ozone annealing. The thickness of doped ETO and capping STO layers was designed to be 100 nm each. The structural properties of the fabricated thin films were analyzed with an atomic force microscope (AFM) and an x-ray diffractometer (XRD). The magnetotransport properties were characterized under a magnetic field perpendicular to the film plane using the Quantum Design Physical Property Measurement System for the samples with electrical contacts fabricated by direct aluminum-wire bonding. The magnetic properties were measured using the Quantum Design Magnetic Property Measurement System by applying a perpendicular magnetic field as well.

Figure 1 shows the structural characterization of doped ETO thin films. A typical AFM image is shown in Fig. 1(b) for a thin film [cap-STO/EGTO/STO(001) substrate] with 0.25% Gd doping, which shows smooth surface morphology and a step-and-terrace structure, with a step height of 4 Å (=one unit cell). XRD 2θ-θ scans for EGTO thin films with 0.25% and 9% Gd doping, as well as ELTO with 10% La doping, are presented in Fig. 1(c). The (002) peak of 0.25% EGTO overlaps with that of the STO substrate due to the almost identical lattice constant (a = 3.905 Å), but clear Laue fringes can be seen, indicating the abrupt interface between the EGTO and STO layers. Films with maximum doping levels in this study—9% EGTO and the 10% ELTO—have (002) peaks, well separated from those of STO substrates. Due to the smaller (larger) cell volume of GdTiO₃ (LaTiO₃)^23–25 compared to ETO, the (002) peak of the 9% EGTO (10% ELTO) film shifted to high (lower) 2θ angle, with the smaller (larger) out-of-plane lattice constant, c = 3.895 Å (3.910 Å). All the films in this study have a pseudomorphic structure, with the same in-plane lattice constant as that of STO (a = 3.905 Å), as verified by reciprocal space mapping of XRD (not shown). The tetragonality, defined as 100 × (c/a − 1) (%), is −0.26% (+0.13%) for the EGTO (ELTO) film.

Next, we discuss the fundamental transport properties. Figures 2(a) and 2(b) show the temperature dependence of longitudinal resistivity (ρ_xx) for the EGTO and ELTO thin films, respectively. As the carrier density increases, resistivity systematically decreases. Around 5 K, all ρ_xx-T curves show kink-like structures, which correspond to the magnetic transitions from paramagnetic to antiferromagnetic or ferromagnetic orderings. ρ_xx at 2 and 300 K for EGTO and ELTO films is shown as a function of carrier density in Fig. 2(c). At 300 K, the dominant scattering factor is the electron–phonon scattering, and the resistivity for EGTO and the resistivity for ELTO almost coincide with each other. ρ_xx is proportional to 1/n (a dotted line), indicating that the mobility is constant at around 10 cm²/Vs at 300 K, regardless of the dopant elements or doping concentration. On the other hand, at 2 K, the residual resistivity for ELTO is systematically smaller than that of EGTO, indicating the larger mobility at low temperatures. The carrier density as a function of the prescribed dopant (La or Gd) concentration is shown in Fig. 2(d). Above Gd (La) concentration of 2 × 10²⁰ cm⁻³ (2 × 10¹⁹ cm⁻³), the donor activation ratio for EGTO (ELTO) thin films is close to 100% (denoted as a dotted line). However, the activation ratio for EGTO thin films begins to drop below [Gd³⁺] = 1 × 10²⁰ cm⁻³, and, finally, the film turns into an insulator at [Gd³⁺] = 2 × 10¹⁹ cm⁻³. On the other hand, even with [La³⁺] = 2 × 10¹⁹ cm⁻³, ELTO thin films are metallic, and the activation ratio is still close to 100%. Below [La³⁺] = 8 × 10¹⁸ cm⁻³, the activation ratio begins to decrease, and some films become insulators. Such different thresholds for deactivation of the dopant can be understood from the viewpoint of smaller (larger) ionic radius for Gd³⁺ (La³⁺) than Eu²⁺, where the tolerance factor of ETO is close to unity. In the case of smaller Gd³⁺, considerable buckling of TiO₆ octahedrons around the Gd³⁺ is expected to enhance the scattering or reduce the electron transfer interaction, compared to the case of La³⁺, though the identification of the microscopic origin of the scattering is unknown and is to be discussed in a future study.

Figure 2(e) shows the carrier mobility at 2 K as a function of electron density for EGTO and ELTO films on STO substrates, as well as the ELTO films on LSAT substrates reported in the previous study.¹¹ In the low carrier density side, the dominant scattering source is the ionized impurity potential, and, hence, the mobility is enhanced as carrier density increases because of the enhanced screening. A too large concentration of donor impurities also works as a strong scattering factor that lowers the mobility. As a result of these two contributions, all three μ-n curves show an upward convex with peaks. The ELTO thin films on STO substrates exhibit the highest mobility, the second is EGTO films on STO substrates, followed by ELTO films on LSAT substrates. Because the lattice constant of the LSAT substrate (a = 3.868 Å) is 0.95% smaller than that of ETO, compressive strain is induced, resulting in +1.6% tetragonal distortion of the films. Such heavy distortion gives rise to a modification of the band structure and quantum paraelectric property to reduce the mobility.¹⁰ In addition, the deposition temperature of ETO on LSAT substrates is limited to 1000 °C, much lower than 1200 °C in the present study, for preserving a coherent growth on such a lattice mismatched substrate, causing the inferior crystalline quality. Thus, the electron mobility for ETO thin films on STO substrates overwhelms that on LSAT substrates. As for the comparison of EGTO and ELTO on STO substrates, the lower mobility in EGTO can be understood in the same way as the deactivation threshold discussed for Fig. 2(d). The smaller ionic radius of Gd³⁺ compared to La³⁺ makes a larger local lattice distortion around the donor. One of our naive expectations for EGTO was that spin scattering may be mitigated for Gd³⁺ with a full magnetic moment of 4f⁷ configuration that is the same as that for Eu²⁺. However, it turns out that the dominant factor for limiting the mobility in doped ETO is not the vacancy of the magnetic moment, but the local lattice distortion due to the smaller ionic radius of the dopants.

We now discuss the magnetic properties of EGTO and ELTO thin films, highlighting the importance of presence (absence) of magnetic moment on the Gd³⁺ (La³⁺) dopant site. Figure 3(a) shows the temperature dependence of the magnetization measured while cooling under a magnetic field of 1000 Oe for EGTO thin films on the STO substrate with [Gd³⁺] = 4.1, 79, and 150 × 10¹⁹ cm⁻³, with0.25%, 5%, and 9% doping ratio, respectively. The curves of EGTO films with [Gd³⁺] = 4.1 and 79 × 10¹⁹ cm⁻³ show kink structures around 5.5 K, indicating the anti-ferromagnetic transition from the paramagnetic state, and that with [Gd³⁺] = 150 × 10¹⁹ cm⁻³ shows a sharp increase of magnetization around 5 K, implying the transition from paramagnetic state to ferromagnetic state. Similar behaviors were also observed for ELTO thin films on the STO substrate. Figure 3(b) shows magnetization curves for the ELTO thin films with [La³⁺] = 2.1, 84, and 170 × 10¹⁹ cm⁻³ (0.13%, 5% and 10%, respectively). The curves of ELTO thin films with [La³⁺] = 2.1 and 84 × 10¹⁹ cm⁻³ also show kink structures, indicating antiferromagnetic transitions at T_N of 5.5 and 3.7 K, respectively, the latter of which being considerably lower than that of EGTO (5.5 K) with similar doping concentration. The film with [La³⁺] = 170 × 10¹⁹ cm⁻³ shows a ferromagnetic transition at T_C = 7 K. Such a magnetic transition from antiferromagnetic to ferromagnetic ground state in carrier doped ETO is due to ferromagnetic interaction between Eu 4f spins mediated by itinerant Ti 3d electrons, which is analogous to the Ruderman–Kittel–Kasuya–Yoshida (RKKY) interaction.^16,17 Magnetic transition temperatures (T_C and T_N) are plotted as a function of the Gd or La ratio in Fig. 3(c). T_N for the EGTO thin films (red solid circles) remains unchanged around 5.5 K with varying the Gd concentration, while T_N for the ELTO thin films (blue solid circles) decrease systematically as La concentration increases, and, finally, 10% La doped ETO exhibits ferromagnetism (blue solid square). This behavior for ELTO is consistent with the previous reports for ELTO bulk crystals^16,17 (plotted as open circles for T_N and open squares for T_C). These results indicate that non-magnetic La doping weakens the long-range anti-ferromagnetic ordering in ETO, while this antiferromagnetic ordering can be well preserved in the case of Gd³⁺ doping, plausibly due to the same 4f⁷ magnetic moment with Eu²⁺.

Figures 4(a) and 4(b) show magnetization as a function of the magnetic field, applied perpendicular to the films. For the EGTO films with [Gd³⁺] = 4.1–79 × 10¹⁹ cm⁻³, magnetization increases almost linearly with the applied magnetic field due to the spin canting process and saturates around 2.3 T. For the [Gd³⁺] = 150 × 10¹⁹ cm⁻³ (9% doped Gd) sample, the magnetization curve appears like a soft ferromagnetic material. The transition from antiferromagnetic to ferromagnetic states is also observed for the ELTO thin films, but a striking difference can be found in the saturation magnetic field, B_saturation, between EGTO and ELTO. B_saturation for EGTO and ELTO thin films is plotted as a function of carrier density at 2 K in Fig. 4(c) to clarify the difference. The drop in B_saturation for ELTO thin films with the increase in doping concentration plausibly indicates that the anti-ferromagnetic order is weakened by the doping of La³⁺ (4f⁰), which causes the defects in the antiferromagnetic ordering lattice, as schematically depicted in the bottom right. It is notable that B_saturation for EGTO thin films stays around 2.3 T irrespective of the doping concentration, which indicates that the anti-ferromagnetic network of ETO can be maintained despite the doping Gd³⁺ (4f⁷).

To clarify the relation between transport and magnetic properties, we plot the magnetoresistance ratio (MRR), with the magnetic field B applied perpendicular to the film plane (current direction), defined as

and magnetization (M) as a function of the perpendicular magnetic field in Fig. 5 for EGTO (a) and ELTO (b) films, respectively. To compare MRR with the magnetization process, B_saturation is indicated as a vertical dotted line for each sample. All the films show the sign change in the slope of MRR from positive through negative to positive again with increasing the magnetic field. Peaks in MRR appear close to B_saturation for the higher donor concentration films; they shift to the smaller magnetic field side as donor concentration decreases, and valleys in MRR appear at B_saturation for lower donor concentration films. The dependence of MRR on the magnetic field and carrier density is qualitatively similar between EGTO and ELTO thin films, which indicates that the vacancies of magnetic moment at A site below 10% doping do not affect much the magnetotransport property, in contrast to the magnetic property discussed in Figs. 3 and 4. Note that the Shubnikov–de Haas (SdH) oscillation was observed in the high-mobility EGTO thin film with μ = 1500 cm² V⁻¹ s⁻¹, as well as the ELTO thin films discussed in detail in the previous report.¹² 

Figure 6 shows the magnetic field dependences of MRR, the magnetization and their derivatives for representative EGTO and ELTO films with low and high donor concentrations, [Gd³⁺] = (a) 4.1 and (b) 40, and [La³⁺] = (c) 2.1 and (d) 22 × 10¹⁹ cm⁻³. The differences in magnetotransport can be seen more clearly between samples with low and high doping concentrations, whereas the difference is not clear between dopants of Gd and La, as mentioned above. For the films with low donor concentrations [Figs. 6(a) and 6(c)], the MRR curve first shows a peak at B = 0.5 T that coincides with the peak in the derivative of the M-H curve. Thus, such a positive peak in MRR in the lower field region is attributed to the spin–flop transition of the Eu²⁺ magnetic moment.⁶ In the spin canting process after spin–flop transition, MRR shows a negative slope, which can be ascribed to the suppression of spin scattering in the spin canting process. Above saturation field of B = 2.4 T, positive MRR, driven by Lorentz force, was observed. In contrast, for the films with higher donor concentrations [Figs. 6(b) and 6(d)], positive slope in MRR can be seen for the lower field region up to the saturation field, and, above the field, negative slope in MRR was observed. This positive slope in MRR cannot be explained solely by the spin–flop transition because a peak structure that indicates the same flop as that in low concentration samples can be observed in dM/dB at B = 0.5 T, and a corresponding peak is also observed in d(MRR)/dB. A similar positive MRR, with a peak at B_saturation, is also observed in Sm-doped ETO thin films on the LSAT substrate,^19,20 and it may be associated with the Weyl nodes of the Ti t_2g band, which contribute to the unconventional Hall responses (shown later), but further study is needed to identify its origin. The negative MRR slope observed above the saturation field is presumably ascribed to the suppression of the residual spin disorder.

Figures 7(a) and 7(b) show the anomalous Hall conductivity (σ_AHE) as a function of the magnetic field, with each M-H curve normalized by the saturated magnetization as a black curve. Here, the data in Figs. 7(a) and 7(b) are σ_AHE after subtracting the ordinary Hall term derived by fitting σ_xy in the region B = 2.5–5 T with the equation

where the fitting parameters are the carrier density (n), electron mobility (μ), and saturation anomalous Hall conductivity $(σAHEs)$⁠. For conventional ferromagnetic materials, the anomalous Hall component can be easily extracted from Hall resistance data (ρ_yx) by subtracting the ordinary Hall part as a field-linear component (ρ_OHE = R_HB), namely, ρ_yx = R_HB + ρ_AHE, where R_H is the ordinary Hall coefficient. However, in a high-mobility system, such as this ETO, especially with a low carrier density regime, the approximate equation of Hall resistivity ρ_yx = ρ_OHE + ρ_AHE is not valid due to the large value of ρ_yx/ρ_xx (Hall angle). Instead of the resistivity, the Hall conductivity, which is a fundamental quantity, must be used to analyze the data precisely. Thus, we plot and analyze the σ_xy in this paper. For a clear comparison, the results analyzed with fitting of ρ_yx and σ_xy are given in the supplementary material. Owing to the long scattering time of the electrons, we can find several peculiar behaviors in AHE: (I) There exists an anomalous Hall term that is not proportional to the evolution of magnetization (M) below B_saturation, as has been discussed in previous report.¹¹ (II) The sign of σ_AHE changes, depending on the carrier density or Fermi energy position relative to the Weyl node, as in previous reports.^10,11 (III) The non-saturating anomalous Hall conductivity as a function of magnetic field appears above B_saturation.

As for (I), all EGTO or ELTO films, except the highest donor concentration (lowest electron mobility) samples in Figs. 7(a) and 7(b), respectively, show an unconventional AHE that is not proportional to the evolution of magnetization below B_saturation. In the previous report for La doped ETO on the LSAT substrate,¹¹ we concluded that this behavior can be explained by the significant change of the Berry curvature induced by the generation and shifting in energy position of the type II Weyl nodes in the Ti t_2g conduction band during the magnetization process [Figs. 8(a) and (b)]. Here, we adopt the same explanation for the AHE in ETO on STO substrate. At zero magnetic field, the Ti t_2g up and down spin bands are degenerate [Fig. 8(c)]. Under a magnetic field, Eu 4f spins start to align to the field direction, and the exchange coupling between the Eu 4f and Ti 3d orbital bands produces Zeeman splitting in this band structure [Fig. 8(d)]. During the magnetization process, enhancement of the Zeeman splitting causes the shift in energy position of the Weyl nodes. Since the Berry curvature in the vicinity of the Weyl nodes tends to be large, the Weyl nodes crossing the Fermi energy predominately contribute to the integral of the Berry curvature within the Fermi surface, resulting in the non-monotonic change in ρ_AHE or σ_AHE during the magnetization process. In the case of the highest donor concentration films with [Gd³⁺] = 79 × 10¹⁹ cm⁻³ and [La³⁺] = 47 × 10¹⁹ cm⁻³,a negligible unconventional term in ρ_AHE below B_saturation, i.e., ρ_AHE ∝ M, can be seen. This is likely either because the Fermi energy is far above the Weyl nodes, or the short lifetime of carriers smears out the unconventional (nonperturbative) contribution.¹¹ These results support the finding that the unconventional AHE below B_saturation is an inherent property in electron doped ETO systems.

Now, we discuss (II), the anomalous Hall angle (θ_AHE) at B_saturation, as shown in Fig. 7(c) for EGTO and ELTO films on STO substrates (upper panel) and ELTO films on LSAT substrate (lower panel).¹¹ We note that θ_AHE for EGTO and ELTO films on STO substrates is calculated by θ_AHE = σ^s_AHE/σ_xx(B_saturatoin), whereas θ_AHE for ELTO films on LSAT substrates is estimated from the data of the previous report.¹¹ In the previous report, the sign inversion of ρ_AHE for the La or Sm doped ETO films was observed,^10,11,19,21 and it is attributed to the shift of the Fermi energy across the Weyl nodes of the t_2g band. For almost unstrained EGTO and ELTO films on STO substrates, θ_AHE shows sign inversion around 3 × 10¹⁹ cm⁻³, while θ_AHE for strained ELTO films on the LSAT substrate shows the sign inversion at much larger electron density around 1 × 10²⁰ cm⁻³. Such a difference can be attributed to the strain-induced tetragonality in the ETO films on the LSAT substrate. The crystal symmetry of the EGTO or ELTO films on STO substrates is close to cubic because ETO and STO have the same lattice constant (a = 3.905 Å). As a result, the t_2g band structure with Zeeman splitting can be described as schematics in Figs. 8(c) and 8(d). On the other hand, the LSAT substrate with a = 3.868 Å induces compressive strain to the ETO film, and it deforms from cubic to tetragonal (c/a = 1.016), which resolves the degeneracy of t_2g bands to lower energy level d_yz/d_zx bands and the higher d_xy one [Fig. 8(a)]. As a result, the Weyl nodes are lifted to higher energy levels, as shown in Fig. 8(a), and, concomitantly, the critical carrier density (n_c) of sign inversion for ELTO films on the LSAT substrate becomes larger than that on the STO substrate. The crystal field splitting arising from the lattice tetragonality is estimated as 45 meV in the previous report for La doped ETO on LSAT substrate,¹¹ and by converting this value to carrier density with the free electron model, this splitting energy corresponds to 4.4 × 10¹⁹ cm⁻³. Because all energy scales, including Zeeman splitting, are within several tens of meV in this ETO system, we successfully observed such interesting phenomena in this doping range.

Regarding (III), in the high magnetic field region above B = 5 T, σ_AHE shows non-saturating behavior, though magnetization is already saturated [Figs. 7(a) and 7(b)]. Although we have not been able to identify the origin, there could be some possible scenarios to explain the feature (III). One could be due to the multicarrier contribution of the ordinary Hall effect, where a nonlinear ordinary Hall resistance appears when carriers with different mobilities exist. The other could be the change of the AHE amplitude itself above the saturation field. As discussed above, the AHE in this system is dominated by the Berry curvature of t_2g bands hybridized with the Eu 4f band. Above the saturation field, though the Zeeman splitting by f-d exchange coupling is constant, the external magnetic field itself may make an energy shift of the Weyl nodes, causing an additional change of the AHE. Further study is needed to clarify the origin.

We have systematically studied the physical properties of ETO films doped with La³⁺ (4f ⁰) or Gd³⁺ (4f ⁷) donors grown by gas source MBE. The magnetic properties revealed that the magnetic dopant Gd³⁺ (up to 5%) maintains the anti-ferromagnetic ordering of ETO with a constant T_N of 5.5 K, while spin vacancy introduced by the nonmagnetic dopant La³⁺ destabilizes the anti-ferromagnetic ordering to decrease both the T_N and saturation magnetic field. From the transport measurements, it is revealed that ELTO exhibits higher mobility and more efficient carrier activation at the diluted limit compared to EGTO, implying that the local lattice distortion given by the smaller ionic radius Gd³⁺ induces more scattering and trapping of electron carriers. In the present high-mobility electron system hosted in nearly cubic ETO, peculiar behaviors in AHE, originated with a topology of band structure, have also been seen, such as sign change, depending on the electron density, and a non-monotonic behavior during the magnetization process that is not proportional to the magnetization. These results offer us important insight into the behavior of magnetic impurities in magnetic semiconductors. The findings of the relationship between the AHE and topology of band structure strongly suggest that the electron-doped ETO film with high mobility is a unique playground to explore the magnetotransport physics in magnetic semiconductors.

See the supplementary material for the further structural characterization, the possibility of oxygen vacancy in SrTiO₃ substrates, and the comparative analysis of the Hall response with fitting of ρ_yx and σ_xy.

We are grateful to N. Nagaosa and N. Kanazawa for fruitful discussions. This research was supported by the Japan Society for the Promotion of Science (JSPS) under Grant No. 22H04958.

The authors have no conflicts to disclose.

N. Takahara: Data curation (lead); Formal analysis (lead); Investigation (lead); Writing – original draft (lead). K. S. Takahashi: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (equal); Investigation (supporting); Project administration (equal); Resources (equal); Writing – review & editing (equal). K. Maruhashi: Data curation (equal); Formal analysis (equal); Investigation (equal). Y. Tokura: Supervision (supporting); Writing – review & editing (supporting). M. Kawasaki: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.