Microscopic origin of the anomalous Hall effect in noncollinear kagome magnets

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Microscopic origin of the anomalous Hall effect in noncollinear kagome magnets

Oliver Busch, Börge Göbel, and Ingrid Mertig

Phys. Rev. Research 2, 033112 – Published 21 July 2020

Abstract

The anomalous Hall effect is commonly considered a signature of ferromagnetism. However, recently, an enormous anomalous Hall conductivity was measured in the compensated kagome magnets ${Mn}_{3} Sn$ and ${Mn}_{3} Ge$ . The occurrence of this effect is allowed by the magnetic point group of these materials; however, its emergence is still lacking a microscopic explanation. Herein we show that the spin-orbit coupling and an out-of-plane tilting of the texture are equivalent for several kagome magnets. Consequently, a coplanar system with spin-orbit coupling behaves as if it were virtually noncoplanar. We show via tight-binding model calculations that the Hall effect can mainly be interpreted as a topological Hall effect generated by the opening angle of the virtually tilted texture. Furthermore, upon tilting the fixed texture out of the kagome plane, we find a critical tilting angle for which the Hall conductivity vanishes for all energies. In this case, the Hamiltonian is invariant under a combined time-reversal and mirror symmetry, because the virtual texture is coplanar.

Received 28 May 2020
Accepted 1 July 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033112

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anomalous Hall effect Magnetic texture Spin-orbit coupling Topological Hall effect

Kagome lattice

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Oliver Busch , Börge Göbel ^*, and Ingrid Mertig

Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany

^*Corresponding author: boerge.goebel@physik.uni-halle.de

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Issue

Vol. 2, Iss. 3 — July - September 2020

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Images

Figure 1
Coplanar magnetic textures on a kagome lattice. (a) The three basis atoms have four nearest neighbors each. For each of these bonds the spin-orbit vector $n_{i j} = - n_{j i}$ is indicated (red; the black arrow indicates the direction $i j$ versus $j i$ ). (b) The three basis atoms and magnetic moments $m_{i}$ (indicated by arrows) in the coplanar radial configuration. The unit cell has three symmetry planes indicating a combined time-reversal and mirror symmetry (red). The texture is characterized by a positive vector spin chirality $κ = + 1$ and an in-plane offset $Δ Φ = 0^{\circ}$ , as indicated. (c) The toroidal phase. All moments are locally rotated by $90^{\circ}$ compared to (b) giving $Δ Φ = 90^{\circ}$ . This magnetic texture has three mirror symmetry planes. (d), (e) Textures with a negative vector chirality are shown. They only have one symmetry plane each.
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Figure 2
Band structure and anomalous Hall conductivity for the coplanar configurations, $θ = 90^{\circ}$ . (a) The band structure of the radial configuration, $Δ Φ = 0$ , when spin-orbit interaction is not taken into account. The out-of-plane spin expectation value $〈 s_{z} 〉$ of all bands and $k$ points vanishes (gray). (b) Similar to (a) but the spin-orbit interaction is now taken into account. Degeneracies are lifted and the spin expectation value now has a finite out-of-plane component (red, positive; blue, negative). (c) The anomalous Hall conductivity corresponding to the band structure in (b). (d) shows the anomalous Hall conductivity for various Fermi energies [colored lines indicated in (c)] upon locally rotating each moment around the local $z$ axis. Cartoons of the texture for several angles $Δ Φ$ are shown below. (e) The spin expectation value for the energetically lowest band in (b). The color encodes the out-of-plane component. Below, the atom-resolved spin expectation value is shown.
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Figure 3
Anomalous Hall conductivity under out-of-plane tilting of the fixed magnetic texture. In (a)–(e) spin-orbit coupling (SOC) has not been considered, whereas in (f)–(j) it is taken into account. (a) A noncollinear and noncoplanar configuration. Black arrows indicate the magnetic moments $m_{i}$ . (b) Hall conductivity versus azimuthal angle $θ$ . All four curves (different colors correspond to different Fermi energies) are antisymmetric with respect to $θ = 90^{\circ}$ which is the coplanar configuration as indicated by the cartoons below. (c)–(e) Band structures with out-of-plane components of the spin expectation value (red, positive; blue, negative) for different azimuthal angles $θ$ as indicated. (f) The texture from (a) in black. Additionally, the virtual magnetic texture ${{\tilde{m}}_{i}}$ is shown in red. This texture arises due to the spin-orbit coupling, as explained in the main text, and it is characterized by a reduced azimuthal angle $θ - Δ θ_{c}$ . (g) Same as (b), but spin-orbit interaction is taken into account. Here the curves are antisymmetric with respect to $θ_{c} = 90^{\circ} + Δ θ_{c} \approx 103^{\circ}$ . (h)–(j) Same as (c)–(e), but spin-orbit interaction was considered as in panel (g).
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Figure 4
Different contributions to the Hall effect for different magnetic textures. (a) The Hall conductivity for a collinear magnetic texture is shown under variation of the azimuthal angle $θ$ . Only a conventional anomalous Hall effect proportional to $m_{z}$ of the fixed magnetic texture is present when spin-orbit interaction is considered. (b) The toroidal texture is considered. Upon variation of $θ$ the scalar spin chirality $χ_{s}$ is finite and an additional topological Hall effect arises that is much larger than the anomalous Hall effect. (c) The radial texture is considered, as discussed above. The curve from the toroidal texture is mainly shifted by $Δ θ_{c} = 13^{\circ}$ which characterizes the nature of the “new” anomalous Hall effect in this material class. All calculations were performed at a constant occupation number of $n_{occ} = 3.4$ .
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Figure 5
Results for the phase of negative vector spin chirality, $κ = - 1$ and $Δ Φ = 180^{\circ}$ . This is the texture that has been experimentally investigated in ${Mn}_{3} Ge$ [17]. (a)–(e) show the results without spin-orbit coupling. (a) The band structure, (b) the energy-resolved conductivity for the coplanar ( $θ = 90^{\circ}$ ) texture, (c) the Hall conductivity for different constant Fermi energies [colors as indicated in (a), (b)] in dependence on the azimuthal angle $θ$ . (d) shows a magnification of (c) near $θ = 90^{\circ}$ . (e) The spin texture of the energetically lowest band for $θ = 90^{\circ}$ . (f)–(j) show the same quantities when the spin-orbit interaction $λ = 0.2 t$ is taken into account. Similarly to the result of the radial phase (as extensively presented in the paper), the virtual texture ${{\tilde{m}}_{i}}$ , determined by the Hamiltonian, is tilted due to the spin-orbit coupling. This also affects the reciprocal-space spin texture shown in (j).
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Figure 6
Local rotation of the coordinate system as unitary transformation. The dashed coordinate system at each lattice cite $i, j, k$ is rotated about the same angle $α$ in the plane spanned by $z$ and the local magnetic moment $m_{i}, m_{j}, m_{k}$ , respectively. The $z$ axes of the reoriented, site-dependent coordinate systems are visualized by the solid arrows ${\tilde{z}}_{i}, {\tilde{z}}_{j}, {\tilde{z}}_{k}$ . In these coordinate systems the magnetic moments are characterized by a different azimuthal angle $θ - α$ ; i.e., the texture is virtually tilted.
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