Research ArticleCONDENSED MATTER PHYSICS

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

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Science Advances  20 Jul 2018:
Vol. 4, no. 7, eaar7880
DOI: 10.1126/sciadv.aar7880
  • Fig. 1 Structure and basic properties of EuTiO3 (ETO) films.

    (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.

  • Fig. 2 Magnetotransport properties of EuTiO3 films.

    (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 (Embedded Image) and the longitudinal conductivity (σxx) at 2 K for ETO films. The polyline depicts the universal relation of anomalous Hall conductivity |Embedded Image| with σxx: The AHE is caused by the Berry phase mechanism regime suppressed by the disorder (|Embedded Image| ∝ σα=1.6xx), the so-called intrinsic AHE (|Embedded Image| = constant), and the skew-type scattering, namely, extrinsic AHE (|Embedded Image| ∝ σα=1.0xx). Red (blue) symbols are for those grown by MBE in this study [PLD in a previous study (14)]. Solid (open) symbols stand for positive (negative).

  • Fig. 3 Carrier density dependence of magnetotransport properties of EuTiO3 films.

    (A) Unconventional term of the AHE (ΔρAHE) defined as ΔρAHE = ρAHERsM 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 ρAHEsAHE) (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.

  • Fig. 4 Anomalous Hall conductivity of the effective Hamiltonian.

    (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 (/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.

  • Fig. 5 Relation between the band structure and anomalous Hall conductivity.

    (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.

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.

  • Supplementary Materials

    This PDF file includes:

    • 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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