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Origin of the magnetic spin Hall effect: Spin current vorticity in the Fermi sea

Alexander Mook, Robin R. Neumann, Annika Johansson, Jürgen Henk, and Ingrid Mertig
Phys. Rev. Research 2, 023065 – Published 22 April 2020
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  • FROM THE CONVENTIONAL TO THE MAGNETIC…
  • CHIRAL VORTICES OF SPIN CURRENTS…
  • LINEAR-RESPONSE THEORY OF THE MAGNETIC…
  • EXAMPLES
  • DISCUSSION
  • CONCLUDING REMARKS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The interplay of spin-orbit coupling (SOC) and magnetism gives rise to a plethora of charge-to-spin conversion phenomena that harbor great potential for spintronics applications. In addition to the spin Hall effect, magnets may exhibit a magnetic spin Hall effect (MSHE), as was recently discovered [M. Kimata et al., Nature (London) 565, 627 (2019)]. To date, the MSHE is still awaiting its intuitive explanation. Here, we relate the MSHE to the vorticity of spin currents in the Fermi sea, which explains pictorially the origin of the MSHE. For all magnetic Laue groups that allow for nonzero spin current vorticities the related tensor elements of the MSH conductivity are given. Minimal requirements for the occurrence of a MSHE are compatibility with either a magnetization or a magnetic toroidal quadrupole. This finding implies in particular that the MSHE is expected in all ferromagnets with sufficiently large SOC. To substantiate our symmetry analysis, we present various models, in particular a two-dimensional magnetized Rashba electron gas, that corroborate an interpretation by means of spin current vortices. Considering thermally induced spin transport and the magnetic spin Nernst effect in magnetic insulators, which are brought about by magnons, our findings for electron transport can be carried over to the realm of spin caloritronics, heat-to-spin conversion, and energy harvesting.

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    • Received 4 November 2019
    • Accepted 4 March 2020

    DOI:https://doi.org/10.1103/PhysRevResearch.2.023065

    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)

    1. Research Areas
    Dzyaloshinskii-Moriya interactionElectrical generation of spin carriersMagnetic orderMagnetismMagnonsRashba couplingSpin Hall effectSpin accumulationSpin caloritronicsSpin currentSpin generationSpin polarizationSpin wavesSpin-orbit couplingSpintronicsTransport phenomena
    1. Physical Systems
    AntiferromagnetsChiral magnetsFerromagnetsKagome latticeMagnetic insulatorsMagnetic semiconductorsMagnetic systemsNoncollinear magnetsPyrochloresTwo-dimensional electron gas
    1. Techniques
    Bloch wave theoryDiscrete symmetries in condensed matterHeisenberg modelHolstein-Primakoff methodMethods in transportSymmetries in condensed matterTight-binding model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Alexander Mook1,2, Robin R. Neumann1, Annika Johansson1, Jürgen Henk1, and Ingrid Mertig1,3

    • 1Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany
    • 2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
    • 3Max-Planck-Institut für Mikrostrukturphysik, D-06120 Halle (Saale), Germany

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    Vol. 2, Iss. 2 — April - June 2020

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    • Figure 1
      Figure 1

      Conventional and magnetic spin Hall effect in ferromagnets with magnetization M along (a) the z direction and (b) the −z direction (e.g., MLG 4/mm′m′). Upon application of an electric field E in the z direction (and a charge current in the same direction) spin accumulates at the boundaries of the sample. Those associated with the conventional spin Hall effect (SHE) are indicated by blue arrows, those with the MSHE by red arrows. Magnetization reversal acts only on the MSHE accumulations.

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    • Figure 2
      Figure 2

      Vorticities of spin current vector fields in reciprocal space. Red/white/blue color indicates positive/zero/blue vorticity in a region about the origin. The integral over the vorticity within the Fermi surface, indicated by black circles, is proportional to the magnetic spin Hall conductivity. (a) The irrotational source field has zero vorticity. (b) The quadrupolar field has locally nonzero vorticity but zero integral. (c) A general vortex with vorticity of varying sign.

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    • Figure 3
      Figure 3

      MSHE in the pyrochlore lattice for (a) ferromagnetic (MLG 4/mm′m′) and (b) antiferromagnetic textures (MLG 4′/mm′m). The pyrochlore lattice is projected onto the xy plane, such that the tetrahedra appear as squares. (c) and (d) depict Fermi surfaces with arrows indicating Jkx and the color scale depicts the y component of v̂k×Jkx (blue/white/red indicates negative/zero/positive values; the value range is symmetric about zero). (e), (f) As (c) and (d) but for Jky and the x component of v̂k×Jky.

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    • Figure 4
      Figure 4

      Two-dimensional electron gas with Rashba SOC and in-plane magnetic field along x [(a) and (b)] or y [(c) and (d)]. ɛd and ɛs are the energy of the degeneracy point and of the saddle point. The color scales represent spin current vorticities ωk,zx [(a) and (c)] and ωk,zy [(b) and (d); blue/white/red color indicates negative/zero/positive values]. For details, see text.

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    • Figure 5
      Figure 5

      Charge conductivity [(a)] and MSHE [(b)–(d)] in a Rashba system with in-plane magnetic field B and hexagonal warping. Model parameters as for Fig. 4, except for the warping strength λ=18eVÅ3. The constant relaxation time is (2Γ)−1=26fs [82]. For λ=18eVÅ3, the model is applicable only for ɛ<0.2eV. For comparison, systems without (λ=0) and with reduced warping (λ=5eVÅ3) are considered. Circles (squares) for B∥x̂(B∥ŷ); σ̲γ,(a) multiplied by 2eℏ to better compare with the charge conductivity.

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    • Figure 6
      Figure 6

      Spin current vorticity (red/blue color scale) and isoenergy lines (colored lines) of the lower band of Rashba systems with and without hexagonal warping in the presence of an in-plane magnetic field B∥x̂. Model parameters as in Fig. 5.

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    • Figure 7
      Figure 7

      As Fig. 6 but for B∥ŷ.

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    • Figure 8
      Figure 8

      Magnon SCVs for two antiferromagnetic textures on the kagome lattice. Top row: MLG 2′/m′. (a) Colored arrows at the vertices and black arrows at the bond centers indicate the spin texture and the DMI vectors for counterclockwise circulation, respectively. (b), (c) Dispersion relation of the lowest band in the vicinity of the Brillouin zone center. Color scales indicate the value of the SCVs (b) ωk,zx and (c) ωk,zy (red/white/blue stands for positive/zero/negative SCV). Bottom row: as top row but for the MLG 2/m. (e), (f) Depict ωk,zx and ωk,zy, respectively.

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