Anomalous Hall effect derived from multiple Weyl nodes in high-mobility EuTiO₃ films

Experiments

Single-crystalline films of La-doped EuTiO₃ (ETO) were grown at 900°C by MOMBE on cubic LSAT (001) substrates with nominal La concentrations varied from 0 to 2.2 × 10²⁰ cm⁻³. The thickness of all the films was fixed at around 25 nm. Figure 1B shows a 2 θ-θ scan of an ETO film around the (002) peaks. A sharp (002) film peak and clear Laue’s fringes can be seen, indicating the high crystalline quality. The in-plane lattice of the films is coherently matched with that of the substrate a = 3.868 Å. The c-axis lattice constant is deduced as 3.931 Å (c/a = 1.016), which is slightly smaller than that of PLD films (3.940 Å, c/a = 1.019) as previously reported. The strain ε=((a||−a0)a0) is estimated as −0.95%, where a₀ is the lattice constant of bulk single-crystal EuTiO₃ (3.905 Å), and a_|| is the in-plane lattice constant of a EuTiO₃ film coherently locked with an LSAT substrate (3.868 Å). As in the cases of the SrTiO₃ and BaTiO₃ film grown by the same method, the lattice constant deviates from the proper value when the nonstoichiometry exists (21). With varying the beam flux ratio between titanium tetra isopropoxide (TTIP) as Ti precursor and Eu, the ratio was optimized, as shown in figs. S1 and S2. Eventually, we found that MBE films discussed here are stoichiometric, but PLD films are not. This small difference in c-axis lattice constant and tetragonality should cause the difference in the energy splitting between the xy orbital and the yz/zx orbitals. An atomic force microscope (AFM) image of the film is shown in Fig. 1C, exhibiting a step-and-terrace structure. The step height is approximately 4 Å, corresponding to the height of a single ETO unit cell. Figure 1D shows the temperature dependence of resistivity for the ETO films. As La doping increases, resistivity systematically decreases. At around 5.5 K, kink structures are observed. The temperature dependences of magnetization also show kink structures at around 5.5 K, corresponding to the Néel temperature, as shown in fig. S3. This indicates that the spin ordering of Eu correlates with the scattering of the conduction electrons. The relationship between the carrier density n and the La concentration is shown in Fig. 1E; n is determined from the slope of the ordinary Hall effect term at 2 K, and the La density is estimated by the flux measured by a quartz crystal microbalance. As mentioned in Materials and Methods, all the films grown under the reductive condition contain oxygen defects. Thus, the nondoped film is not an insulator but exhibits metallic behavior with electron carriers (n ~ 2.4 × 10¹⁹ cm⁻³), and carrier density n can be controlled by La doping at higher doping levels than 2.4 × 10¹⁹ cm⁻³ as the dashed line in Fig. 1E indicates. We note that the c-axis lattice constants of all the films are almost identical as 3.931 ± 0.002 Å, suggesting that similar amounts of oxygen vacancies are incorporated in the films regardless of the La doping level (22). Figure 1F shows the magnetization curves at 2 K. The magnetic field is applied along in-plane and out-of-plane axes. As previously reported, the absence of remanent magnetization at zero field indicates an antiferromagnetic ordering. The magnetization increases and saturates at 7 μ_B per Eu by applying a magnetic field of B = 1.8 T along an in-plane axis and B = 3 T along an out-of-plane axis. The saturation moment 7 μ_B per Eu coincides with the full spin moment of Eu²⁺ (4f ⁷), and such anisotropy can be explained by a demagnetizing field effect of the thin-film form.

Next, we discuss the transport properties in detail. Figure 2A shows the Hall resistivity (ρ_H) for the ETO film as a function of magnetic field at 2 K. ρ_H is expressed asρH=RHB+ρAHEwhere R_H is the ordinary Hall coefficient and ρ_AHE is the anomalous Hall resistivity, which would have been proportional to the magnetization in conventional magnetic systems. The dashed line is the linear fit with the slope of ρ_H above 3 T (the saturation magnetic field), corresponding to the ordinary term (R_HB). By subtracting the linear ordinary term from the ρ_H, the anomalous term was extracted, as shown in Fig. 2B. The carrier density for each film is determined using the R_H. To be precise, the part of the ρ_H curve above 3 T in Fig. 2A is not perfectly linear. The fitting error is about 5%, resulting in the same order of that for the carrier density. Presumably, the tiny deviation from the linear function is related to the magnetic field dependence of the magnetic moment derived from 3d conduction electrons or to the multiband nature of the t_2g orbitals. The magnetization curve is also plotted on the right axis to scale both the values of the magnetization and ρ_AHE above 3 T in saturation. For the following discussions, the saturation value of ρ_AHE is defined as ρ^s_AHE. Note that there is a large additional component of the AHE shaded by blue and orange colors deviating from the curve of magnetization below saturation field, which will be discussed later. We could find such an unconventional component below the saturation field in the anomalous Hall conductivity σ_AHE(B) curves deduced from ρ_AHE(B) and ρ_xx(B) as well. Typical data for four carrier density samples are shown in fig. S4. Note that the nonmonotonic behavior of the AHE is observed not only in La-doped films but also in the film that is doped solely by the oxygen vacancy. Figure 2C shows the magnetoresistance (MR) of the film. During the magnetization process from 0 to 3 T, a large negative MR appears presumably because of the suppression of the scattering through the spin alignment of Eu. Figure 2D displays σAHEs versus σ_xx by a log-log plot. Here, σ_xx and σAHEs are calculated as σxx=ρxx(B=0)(ρAHEs)2+ρxx(B=0)2 and σAHEs=ρAHEs(ρAHEs)2+ρxx(B=0)2. Positive (negative) σAHEs is plotted as solid (open) circles. It is known that there exist three regimes in the scaling behavior of the AHE depending on the range of σ_xx, which are categorized by a scaling parameter, α, in σAHEs ∝ σ^α_xx (6, 23, 24). (i) α ~ 1.6, when σ_xx is below 1 × 10⁴ ohm⁻¹ cm⁻¹, where the anomalous Hall velocity induced by the Berry curvature of conduction bands is disturbed and suppressed by the disorder; (ii) α ~ 1.0 for large σ_xx above 1 × 10⁵ ohm⁻¹ cm⁻¹, where the AHE is caused by the skew-type scattering, namely, the extrinsic AHE; and (iii) α ~ 0 for the intermediate σ_xx between (i) and (ii), the so-called intrinsic AHE due to Berry curvature. These scaling laws are denoted as polylines in Fig. 2D. For the ETO films discussed in this study, the carrier density is designed between 2 × 10¹⁹ and 1.5 × 10²¹ cm⁻³, and all the data obey the scaling σAHEs ∝ σ^α=1.6_xx, suggesting that the Berry phase mechanism regime of the AHE is suppressed by the disorder.

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To discuss the peculiar anomalous Hall term, the additional component of the AHE (Δρ_AHE) is defined as Δρ_AHE = ρ_AHE − R_sM, where R_s is determined to be Δρ_AHE = 0 above 3 T with saturated magnetization. For example, as mentioned above, the Δρ_AHE for n = 3.0 × 10¹⁹ cm⁻³ corresponds to the shaded component in Fig. 2B. Figure 3 (A and B) shows the magnetic field dependence of Δρ_AHE and MR [MR (%)=ρ(B)−ρ(B=0)ρ(B=0)] for ETO films with various carrier densities. Although the magnetization versus magnetic field behaviors are almost the same in this range of carrier density, the saturation magnetic field slightly decreases as the carrier density increases, as indicated by vertical bars in Fig. 3 (A and B). Each bar denotes the saturation magnetic field that is determined by the differential curve of MR. The existence of Δρ_AHE at the magnetization process reminds us that a spin texture such as a skyrmion lattice causes a fictitious magnetic field and topological Hall effect, which are realized in B20 compounds (25, 26). Those skyrmion lattices are stabilized by the Dzyaloshinskii-Moriya (DM) interaction arising in noncentrosymmetric crystal structures and interfaces or the competition of long-range dipolar energy in a thin-film geometry and domain wall energy (27, 28). Given that skyrmions exist in the ETO film on the LSAT substrate, those may be caused by DM interaction due to the broken inversion symmetry at the interface. If Δρ_AHE were a consequence of the skyrmion formation based on the interface DM interaction, then the sign of the topological Hall effect would be determined by the sign of the normal Hall effect (29). Thus, the sign change of topological Hall effect, as observed, could be hardly reconciled with the constant sign (negative) of the charge carrier. In our experiment, however, the sign of Δρ_AHE changes twice as the carrier density increases. These sign changes as a function of carrier density cannot be explained by conventional topological Hall effect scenarios. The MR also shows an interesting behavior by changing the carrier density. Although low–carrier density films below 1.4 × 10²⁰ cm⁻³ exhibit negative MR for the magnetization process between 0 and 3 T, positive MR behaviors are seen for high–carrier density films. As mentioned above, the negative MR between the antiferromagnetic state at 0 T and the ferromagnetic spin ordering above 3 T seems to be understood as follows. The scattering time of conduction electrons in ferromagnetic spin texture is longer than that in the antiferromagnetic one. However, such negative to positive change in MR by increasing the carrier density cannot be explained by the conventional scattering time difference. The theoretical approach to the longitudinal MR (R_xx) behavior is much more complicated than that to the transverse (Hall) resistance (R_xy). Therefore, although we expect that the positive MR might be also related to the Weyl nodes, the origin of the positive MR remains unclear to be studied in future. Here, we focus on the behavior of the Hall effect to which the Berry phase theory can be directly applied.

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To shed light on the carrier density dependence of magnetotransport properties, we show in Fig. 3 (C to F) the mobility, ρ^s_AHE, ΔρAHEmax, and MR at 2 K for MBE and PLD films as a function of carrier density, where ΔρAHEmax is the maximum value of Δρ_AHE at a magnetic field in Fig. 3A. All data for PLD films measured in the previous report were reanalyzed. Because of the higher crystalline quality of MBE films than that of PLD films, the mobility of electrons in MBE films is one order of magnitude higher than that of PLD films, as shown in Fig. 3C; the maximum value reaches 300 cm² V⁻¹ s⁻¹ at 2 K. Thus, the scattering time of MBE films becomes longer, and the energy broadening of conduction bands must be smaller than that of PLD films. In Fig. 3D, the sign change of ρ^s_AHE from negative to positive occurs at around 1 × 10²⁰ cm⁻³ for the MBE films, whereas it occurs around 2 × 10²⁰ cm⁻³ for PLD films. Such a sign change for PLD films was already explained in the previous paper (14). The sign change occurs at the carrier density where the Fermi energy is around the band crossing point that is formed by the orbital band lifting because of the tetragonal distortion in the epitaxially strained film. As mentioned above, the tetragonal lattice distortion of MBE films is smaller than that of PLD films. This might cause the smaller energy splitting of t_2g orbitals for MBE films, resulting in the lower energy where the band crossing occurs in k-space, as shown by inset schematics. Accordingly, the carrier density dependence of ρ^s_AHE for MBE films shifts to the lower carrier density side than that for PLD films. The sign of ΔρAHEmax for MBE films changes twice from 2 × 10¹⁹ to 5 × 10²⁰ cm⁻³. For MR of MBE and PLD films, the sign change can be seen around 1.6 × 10²⁰ cm⁻³ from negative to positive as the carrier density increases. These behaviors of ρ^s_AHE, ΔρAHEmax, and MR as a function of carrier density seem to correlate with each other.

Theory

The above observations suggest that the unconventional behavior of the anomalous Hall resistivity Δρ_AHE is also related to the band crossings of the conduction electrons. To provide further insight into the nonmonotonic behavior of the AHE, we next turn to the theoretical analysis of the AHE. In ETO, the conduction bands consist of the 3d orbitals of Ti ions that split into the t_2g and e_g orbitals because of the crystal field; the electrons in t_2g orbitals are responsible for the conduction, which form sixfold degeneracy at the Γ point. In the La-doped ETO with the carrier density up to around 1 × 10²⁰ cm⁻³, the Fermi level is expected to be about 100 meV or less above the band bottom. Because of the low Fermi level, as we will discuss below, the fine structure of the bands due to spin-orbit interaction can give a relevant contribution. Therefore, we also take into account the spin-orbit interaction that further splits the sixfold t_2g orbitals at the Γ point into fourfold J = 32 and twofold J = 12 states (30); we consider an effective four-band model for the J = 32 orbitals as they have lower energy than the J = 12 states. For the ETO thin films synthesized by MOMBE, we find small lattice distortion (c/a = 1.016) due to the strain from the substrate. Considering the above features, the effective model for the La-doped ETO thin films is constructed by the k⋅p theory around the Γ point for the J = 32 bands (14). It is given by the Luttinger Hamiltonian with a uniaxial anisotropy term that represents the strain from the substrate (14, 31)H0(k→)=t∑αkα2(14+Jα23)+δt3∑α≠βkαkβJαJβ−Vtetra3Jz2(1)Here, J_α (α = x, y, z) are the 4 × 4 spin operators for J = 32, k_α is the α component of the crystal momentum, and V_tetra is the uniaxial anisotropy induced by the strain. We here set the band parameters in the Hamiltonian consistent with that in a preceding work (14) (t = 300 meV and δt = −100 meV), while using a smaller uniaxial distortion, V_tetra = 45 meV, as the distortion observed in MBE films is smaller than that in the PLD films (V_tetra = 60 meV). In addition to these terms, we consider the coupling of conduction electrons to the magnetic moments of Eu 4f moments via the Zeeman coupling. For the Zeeman coupling, in general, we need to consider the coupling of itinerant electrons to both uniform and the staggered magnetizations. We, however, find that the coupling to the staggered magnetization is not allowed by the symmetry (see Materials and Methods). Therefore, the Hamiltonian for the symmetry-allowed exchange coupling between the Eu 4f localized moments and Ti J = 32 itinerant electrons readsHh=Δ(h)3(Jzcos θ+Jz3sin θ)(2)Here, Δ(h) is the strength of the Zeeman coupling as a function of uniform magnetization h, and θ = −π/4 is the parameter that defines the ratio of J_z and Jz3. θ is a parameter that defines the form of Zeeman coupling (30); it is chosen such that the anomalous Hall conductivity becomes consistent with the experiment.

In Fig. 4, we show the theoretical result of anomalous Hall conductivity for the effective Hamiltonian. Figure 4C shows the Zeeman splitting h and the chemical potential μ dependence of the anomalous Hall conductivity σ_AHE. The blue shade shows the region with negative σ_AHE, and the position of the Weyl nodes at h = 0 and h = h_c is shown by the red arrows. In the Luttinger model, the two pairs of Weyl nodes are at energies EW±=23Vtetra±12h sin θ and that of double Weyl nodes at EDW±=23Vtetra∓18h(4 cos θ+5 sin θ) (see Fig. 5, A to D). For the field h ≤ h_c, all nodes are located in between 21.5 meV<EW,DW±<38.5 meV, as shown by the arrows. At h = h_c, we find a sign change of the anomalous Hall conductivity from negative to positive, consistent with the experiment in Fig. 3D; the comparison of σ_AHE to that in the experiment is shown in Fig. 4D. We also find the nonmonotonic behavior of the anomalous Hall conductivity in the wide range of electron density with respect to the external magnetic field. The field dependence of the anomalous Hall conductivity for various chemical potentials with relaxation time τ = 2.2 × 10⁻¹³ s (ℏ/τ = 3 meV) is shown in Fig. 4A; the same curve is also shown in Fig. 4C by the solid lines on the surface. In the calculation of the Hall conductivity, we used a relaxation time that was somewhat longer than in the previous paper (14), as the conductivity in MOMBE films are higher than that in PLD by the enhancement of the mobility. In addition, we here assumed that the major contribution to the Zeeman splitting comes from the coupling of the Eu moments to the Ti electrons. Hence, we assumed the Zeeman field to increase linearly [Δ(h) = 24(h/h_c) meV] up to h_c, while it remains a constant [Δ(h) = 24 meV] above h_c. In Fig. 4A, in the low carrier density region, the conductivity shows a concave curve, as shown in the result for n ~ 6.9 × 10¹⁹ cm⁻³ (the green shaded regions in the conductivity curve). In addition, in some cases, it shows sign changes of the anomalous Hall conductivity with increasing h, as in the case of n = 7.4 × 10¹⁹ cm⁻³. Our results also show sign changes of the nonlinear part with increasing carrier density, as shown by the curves for larger n. To see the n dependence of the nonlinear part of σ_AHE, we define Δσ_AHE as the maximum of the convex/concave, in a similar way to Δρ_AHE above; it is shown in Fig. 4E, along with that in the experiment. We find that the Δσ_AHE changes from negative to positive at around n ~ 9.0 × 10¹⁹ cm⁻³ (μ = 29 meV) and from positive to negative at around n ~ 1.7 × 10²⁰ cm⁻³ (μ = 45 meV). Although the characteristic carrier densities of the sign changes do not match with the experimental results precisely, the theory and the experiment agree semiquantitatively.

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We next turn to the relaxation time dependence of σ_AHE; we find that the Hall conductivity is highly sensitive to the relaxation time. In Fig. 4B, we show the results of the anomalous Hall conductivity at n ~ 6.9 × 10¹⁹ cm⁻³ (μ = 22.5 meV). The nonlinear behavior with τ = 2.2 × 10⁻¹³ s (high mobility) is almost lost in the case of τ = 6.6 × 10⁻¹⁴ s and lower (low mobility). Here, τ = 1 × 10⁻¹³ s corresponds to μ = 180 cm² V⁻¹ s⁻¹ assuming free electron mass. These results suggest that a very clean sample is necessary for the observation of the nonlinear behavior of the AHE with respect to h. Indeed, while the nonmonotonic curve of σ_AHE appears in the wide range of carrier density in the MBE films, such behavior was hardly observed in the previous samples by PLD (14).

These features found in the theoretical calculations are likely to be the consequences of multiband nature, in particular, the unusual structure of the Berry curvature due to the presence and movement of type II Weyl nodes in the conduction bands. As it is well known, the Weyl node is a drain (source) of the Berry curvature flux when it has a negative (positive) chirality, that is, the Berry curvature flows inward to (outward from) the Weyl node in the upper band when the node has negative (positive) chirality. On the other hand, the Berry curvature on the lower band has the opposite sign as the upper band, namely, b+(k→)=−b−(k→), where b+(k→) and b−(k→) are the Berry curvatures at wave number k→ for the upper and lower bands, respectively. As the texture of the Berry curvatures for the upper and lower bands have opposite signs, the contribution from the upper and lower bands close to the Fermi surface cancels when both the upper and lower bands are below the Fermi level. Therefore, the contribution to the AHE only comes from the region where only the lower band is occupied. For the Weyl node noted in blue circle on the right-hand side of Fig. 5D, the z component of the Berry curvature is negative if the Fermi level is below the Weyl node while positive if above (see also fig. S5). As the anomalous Hall conductivity is proportional to the integral of Berry curvature over the Fermi sea, the contribution from the electron states around the Weyl node gives a negative contribution when the Fermi surface is below the node (Fig. 5B), while it will be positive when above (Fig. 5D). A similar argument on the pair node (the node at −k→W, where k→W is the position of the Weyl node) shows that the pair node also gives the same contribution. As the Berry curvature in the vicinity of the Weyl nodes is very large, this contribution may become visible when the Fermi volume is small (and hence, the integral of the Berry curvature within the Fermi surface is small) such as in the La-doped semiconductors as in EuTiO₃.

This scenario may explain the nonmonotonic behavior of the Hall conductivity around μ ~ 24 meV (n = 7.4 × 10¹⁹ cm⁻³). In Fig. 5A, we show the band structure of the Luttinger Hamiltonian without the Zeeman splitting; the plot is for the (0, 0, k_z) line. Because of the uniaxial anisotropy induced by the substrate, the fourfold degeneracy at the Γ point is lifted, forming two light and one heavy electron bands (both bands are doubly degenerate); these two bands cross at E=23Vtetra=30 meV. With finite Zeeman splitting, the two bands further split into four, leaving eight band crossing points for the heavy and light bands along the (0, 0, k_z) line. Figure 5 (B to D) shows the band structures for different Zeeman fields, h = h_c/3, 2h_c/3, and h_c, respectively. The eight crossing points shown by the circles and squares in Fig. 5 (B to D) are type II Weyl nodes (32, 33); the red (blue) dots are the Weyl nodes with positive (negative) chirality, and the double Weyl nodes (the Weyl nodes with twice the charge of the ordinary Weyl nodes) with positive and negative chirality are shown by red and blue squares, respectively. As shown in Fig. 5E, the minimum of the Hall conductivity at n = 7.4 × 10¹⁹ cm⁻³ (μ = 24 meV) coincides with the h at which the Weyl nodes cross the Fermi surface, qualitatively consistent with the above argument.

For the chemical potentials above μ = 24 meV, multiple Weyl nodes and the structure of the Berry curvature around them are also expected to contribute to the Hall conductivity. Complex structures of the anomalous Hall conductivity in Fig. 4 (A and C) appear around the Weyl nodes. This is also confirmed by calculating the Berry curvature on each band; we find a complex structure of the Berry curvature especially on the second and third bands where six Weyl nodes are located close by (fig. S5).