Abstract
EuTiO3, a magnetic semiconductor with a simple band structure, is one of the ideal systems to control the anomalous Hall effect (AHE) by tuning the Fermi level. The electrons in the conduction bands of La-doped EuTiO3 are subject to the spin-orbit interaction and Zeeman field from the spontaneous magnetization, which generates rich structures in the electron band such as Weyl nodes. This unique property makes EuTiO3 a relatively simple multiband system with its Berry curvature being controlled by electron doping and magnetic field. We report a nonmonotonic magnetic field dependence of the anomalous Hall resistivity, which is ascribed to the change of electronic bands induced by the Zeeman splitting during the magnetization process. The anomalous Hall resistivity measurement in high-mobility films grown by gas source molecular beam epitaxy shows additional terms in the AHE during the magnetization process, which is not proportional to the magnetization. Our theoretical calculation indicates that the change of Zeeman field in the process of canting the magnetic moments causes the type II Weyl nodes in the conduction band to move, resulting in a peculiar magnetic field dependence of the AHE; this is revealed by the high-quality films with a long scattering lifetime of conduction electrons.
INTRODUCTION
Transport properties of spin-polarized electrons receive considerable interest for their importance in basic science and for their potential in technological applications (1). One of the peculiar phenomena of the conduction of spin-polarized electrons in magnetic metals that contains rich physics is the anomalous Hall effect (AHE). Many different mechanisms contribute to the AHE, such as the Berry phase of the electronic bands (2), impurity scattering (3, 4), the spin Berry phase (5), etc. Among them, one well-studied mechanism is the intrinsic mechanism that is related to the Berry curvature of electronic bands (2, 6), and many experimental results are ascribed to this mechanism (7–13). In magnetic oxides with complex band structures, the band crossings often affect the Berry curvature, resulting in a complicated distribution of the Berry curvature. In these cases, the AHE is very sensitive to the Fermi level shift around the band crossing points, giving an opportunity to control the AHE by Fermi level tuning such as by carrier doping. In a pioneering work, the sign change of the AHE as a function of temperature was discovered in SrRuO3. This is explained by the change of the total Berry curvature due to Fermi level shifting around the crossing points through the temperature scan (11). However, for usual ferromagnetic metals such as SrRuO3, the electronic structure near the Fermi energy is so complicated that theoretical predictions of the trends in the experiment as in SrRuO3 are rare. In contrast, in a recent report, it was discussed that the magnetic semiconductor EuTiO3 (ETO) has a relatively simple band structure with band crossings (14). In addition, the magnetism is dominated by the exchange interaction between the nearest neighbors of Eu. This interaction should be rigid against the tiny amount of substitution of Eu by La. Therefore, the Fermi level can be tuned around the band crossing points while keeping the magnetization almost constant, whereas the Fermi level of SrRuO3 is hardly controlled by carrier doping because of the large amount of carriers. Hence, ETO was considered as an ideal material to study the control of Berry phase and the AHE by tuning the Fermi level (14).
As shown in Fig. 1A, ETO is a band insulator with a cubic perovskite structure. The Ti ions are in the tetravalent state, and the Eu ions are in a divalent state; the Eu ions have 4f 7 (S =
(A) A schematic of the crystal structure. (B) 2θ-θ x-ray diffraction pattern around (002) peaks for an ETO thin film on an LSAT substrate. (C) An AFM image (2 × 2 μm2) for the surface of an ETO film. (D) Temperature dependence of resistivity for ETO films with various carrier densities (n). (E) Relation between the carrier density n and nominal La concentration for ETO films. (F) Magnetization curves at 2 K for an ETO film with n = 2.4 × 1019 cm−3. The magnetic field is applied along in-plane [100] (red) and out-of-plane [001] (blue) axes.
The lattice constant of ETO (3.905 Å) is larger than that of a LSAT (3.868 Å) substrate. Therefore, in the ETO films on LSAT, the compressive strain is induced in the ETO films because of the difference of lattice constant between ETO and LSAT. Because of this epitaxial compressive strain, two band crossing points between dxy and dyz/zx of t2g orbital bands (they are mixed by the spin orbit interaction (SOI) to form Jeff =
Recently, high–crystalline quality complex oxide films were grown by metalorganic (MO) gas-source molecular beam epitaxy (MBE). For example, a La-doped SrTiO3 (STO) film shows very high mobility exceeding the maximum mobility of a bulk single crystal (22,000 cm2 V−1s−1) (19), and quantum Hall effect was successfully observed in the two-dimensional (2D) electron gas of δ-doped STO grown at high temperature by MOMBE (20). Therefore, EuTiO3 films with higher mobility than pulsed laser deposition (PLD) films are expected by the MOMBE. In the case of PLD films, the large energy broadening of conduction bands due to the short scattering time might mask the fine structure of the bands, which would contribute to the AHE. Therefore, the MOMBE films with longer scattering time may reveal the additional character of magnetotransport, which was hidden in PLD films. Thus, it is worth investigating the transport properties of the high-mobility ETO films that can be fabricated by MOMBE.
RESULTS
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
Next, we discuss the transport properties in detail. Figure 2A shows the Hall resistivity (ρH) for the ETO film as a function of magnetic field at 2 K. ρH is expressed as
(A) Magnetic field dependence of Hall resistivity (ρH) for the ETO film with n = 3.0 × 1019 cm−3. The dashed line is the linear fit with the slope of ρH above 3 T (the saturation magnetic field), corresponding to the ordinary Hall effect term (RHB). (B) AHE at 2 K for the ETO film as a function of the magnetic field after subtraction of ordinary Hall effect term. (C) MR at 2 K for the ETO film as a function of magnetic field. (D) Relationship between the magnitude of anomalous Hall conductivity (
To discuss the peculiar anomalous Hall term, the additional component of the AHE (ΔρAHE) is defined as ΔρAHE = ρAHE − RsM, where Rs is determined to be ΔρAHE = 0 above 3 T with saturated magnetization. For example, as mentioned above, the ΔρAHE 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 ΔρAHE and MR
(A) Unconventional term of the AHE (ΔρAHE) defined as ΔρAHE = ρAHE − RsM and (B) MR at 2 K as a function of magnetic field. Vertical bars indicate saturation field defined by the derivative of MR. Curves are vertically offset as denoted by horizontal bars. Carrier density dependence of mobility (C), saturation value of ρAHE (ρsAHE) (D), maximum value of ΔρAHE (E), MR at each saturation field, and (F) at 2 K for ETO films. The lines are merely guides to the eye.
To shed light on the carrier density dependence of magnetotransport properties, we show in Fig. 3 (C to F) the mobility, ρsAHE,
Theory
The above observations suggest that the unconventional behavior of the anomalous Hall resistivity ΔρAHE 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 t2g and eg orbitals because of the crystal field; the electrons in t2g 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 t2g orbitals at the Γ point into fourfold J =
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 σAHE. The blue shade shows the region with negative σAHE, and the position of the Weyl nodes at h = 0 and h = hc is shown by the red arrows. In the Luttinger model, the two pairs of Weyl nodes are at energies
(A) Field dependence of the anomalous Hall conductivity σxy for θ = −π/4 with different chemical potential μ (the corresponding carrier density n is also shown). The relaxation time is fixed to τ = 2.2 × 10− 13 s (ℏ/τ = 3 meV). We assumed that the Zeeman splitting Δ of the band is proportional to the magnetic field Δ(h) = 24h/hc meV while it is a constant (Δ = 24 meV) above hc. Curves are vertically offset, as denoted by horizontal bars. (B) Relaxation time dependence of σxy at μ = 24 meV. (C) Three-dimensional plot of σxy for different chemical potentials and fields h. The solid lines show the path that corresponds to the curves in (A), and the red arrows are the positions of the band crossings at h = 0 and at h = hc. The right two figures show the carrier density n dependence of Hall conductivity: (D) σAHE at h = hc and (E) the maximum value of ΔσAHE. Green dots are the results of theoretical calculation, and the red dots are experimental results for corresponding carrier density n. The lines are merely guides to the eye. All results are for t = 300 meV, δt = −100 meV, Vtetra = 45 meV, and θ = − π/4.θ = −π/4.
(A to D) The band structure of the Luttinger model at h = 0 (A), h = hc/3 (B), h = 2hc/3 (C), and h = hc (D). The solid circles (open squares) denote the position of the Weyl (double Weyl) nodes; the red (blue) dots are for nodes with positive (negative) chirality. (E) Field dependence of anomalous Hall conductivity for chemical potential μ = 24 meV. All results are for t = 300 meV, δt = −100 meV, Vtetra = 45 meV, and θ = −π/4.
We next turn to the relaxation time dependence of σAHE; 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 σAHE 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,
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, kz) 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
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).
DISCUSSION
To conclude, we have investigated magnetotransport properties of La-doped ETO films on LSAT substrates. The mobilities of MBE films are one order of magnitude higher than those of PLD films. The carrier density dependence of the AHE is explained by the band crossing points induced by the epitaxial compressive strain from the substrate. In the AHE, an additional term that is not proportional to the magnetization curve can always be observed for MOMBE films. Such a nonlinear term could be rarely found for PLD films. Model calculations reveal that the change of Zeeman energy splitting during the magnetization process causes the nonmonotonic behavior of the AHE as a function of magnetic field. Because of the long scattering time of high-mobility MOMBE films, the effect of the Zeeman energy splitting on the AHE can be observed. These new findings strongly suggest that the electron-doped ETO film with high mobility is an ideal magnetic semiconductor to explore magnetotransport phenomena attributed to the Berry curvature of the simple band structure near k = 0.
Note that in preparing this article, we noticed a related paper by Ahadi et al. (34). They attribute the origin of the additional anomalous Hall resistivity to the topological Hall effect, which is different from our conclusion.
MATERIALS AND METHODS
Sample fabrication
Eu and La were provided from conventional effusion cells with each pure elemental source. For the Ti source, TTIP (99.9999%) was supplied from an MO container kept around 100°C without any carrier gas. Eu flux was kept at a beam equivalent pressure of 8 × 10−8 torr, and the rate of TTIP was varied to optimize the TTIP/Eu ratio. La flux was controlled by the temperature of the effusion cell that was calibrated by a quartz crystal microbalance flux monitor. Eu2Ti2O7 with Eu3+ phase was more stable than perovskite EuTiO3 with Eu2+ phase (35). To avoid oxidation to the Eu2Ti2O7 phase, the films were grown in a base pressure of 2 × 10−8 torr. EuTiO3 can be grown while supplying no intentional oxidation gas because of the four oxygen atoms incorporated per Ti in TTIP and epitaxial stabilization from the perovskite substrate LSAT. However, such insufficient oxidation conditions create a certain amount of oxygen vacancies in the films, doping electron carriers (~2 × 1019 cm−3) even without La doping, as discussed in Fig. 1E.
Effective Hamiltonian
As the effective Hamiltonian for the Ti electrons, we considered the Luttinger model used in a precedent work (14). As the coupling term between the Eu moments and Ti electrons, we considered the couplings to the ferromagnetic moment
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/7/eaar7880/DC1
Section S1. Optimization of the growth condition of EuTiO3 films by changing the flux ratio between TTIP and Eu
Section S2. Temperature dependence of magnetization
Section S3. Anomalous Hall conductivity σxy for La-doped ETO films
Section S4. Berry curvature distribution of the Luttinger model
Fig. S1. Structure characterizations of EuTiO3 films on LSAT (001) substrates grown at various TTIP/Eu ratios.
Fig. S2. Tetragonal distortion of EuTiO3 film on LSAT (001) substrate.
Fig. S3. Magnetization property.
Fig. S4. Magnetic field dependence of anomalous Hall conductivity.
Fig. S5. Band structure and Berry curvature.
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REFERENCES AND NOTES
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