Hydrodynamic theory of vorticity-induced spin transport

Gen Tatara

Phys. Rev. B 104, 184414 – Published 15 November 2021

Abstract

Electron spin transport in a disordered ferromagnetic metal is theoretically studied from the hydrodynamic viewpoint, focusing on the role of electron vorticity. The spin-resolved momentum flux density of electrons is calculated microscopically, taking account of the spin-orbit interaction and uniform magnetization, and the expression for the spin motive force is obtained as the linear response to a driving electric field. It is shown that the spin-resolved momentum flux density and motive force are characterized by troidal moments, vector products of the applied external electric field and the spin polarization or magnetization, which act as effective driving fields when the anomalous or spin Hall effects are taken into account. The spin-vorticity and magnetization-vorticity couplings are shown to arise naturally as conservative forces driven by the toroidal moments, and nonconservative forces are also found to exist. Spin accumulation induced by the electron flow is calculated and vorticity-induced torque and spin relaxation are discussed. The vorticity-induced torque, a linear effect of the spin-orbit interaction, is argued to be larger than the conventional relaxation torques acting on magnetization structures such as nonadiabatic ( $β$ ) current-induced torque. The direct and inverse spin Hall effects and spin-orbit torque are discussed in the context of spin-vorticity coupling.

Received 23 August 2021
Revised 26 October 2021
Accepted 8 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.184414

Physics Subject Headings (PhySH)

Anomalous Hall effect Electrical generation of spin carriers Spin Hall effect Spin accumulation Spin current Spin diffusion Spin relaxation Spin torque Spin-orbit coupling Spintronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Gen Tatara

RIKEN Center for Emergent Matter Science (CEMS) and RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

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Issue

Vol. 104, Iss. 18 — 1 November 2021

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Images

Figure 1
Two scenarios for the spin Hall-induced spin-orbit torque acting on magnetization $M$ induced by an applied current in a bilayer of ferromagnet (F) and nonmagnetic metal (NM). (a) Conventional picture explaining the torque in terms of spin current ( $j_{s}$ ) generation in NM. (b) In the spin-vorticity picture, spin accumulation is generated directly by the vorticity $ω_{e}$ created near the interface as a result of suppression of current $j$ . They are equivalent, as was argued in Ref. [18].
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Figure 2
Parameterization of spin vectors, $s_{⊥} \equiv s - \hat{M} (\hat{M} \cdot s)$ and $\tilde{s} \equiv s \times \hat{M}$ .
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Figure 3
A schematic picture showing spin motive forces of Eq. (24) in a thin film in the $x y$ plane with applied current along the $x$ direction. The contributions ${\tilde{ζ}}_{s}^{f, ⊥}$ and ${\tilde{ξ}}_{s}^{ω, ⊥}$ connecting vorticity and $\hat{s}$ are depicted. At the boundaries, the flow vanishes, meaning that total effective field $E$ including the applied field and friction force from the boundary vanishes due to proportionality of $j$ and $E$ . The derivative $\partial_{z} E_{x}$ is therefore finite, namely, vorticity $ω$ emerges near the boundary along the $y$ direction. The vorticity has opposite signs on the upper and lower planes, resulting in $\partial_{z} ω_{y}$ . The contribution of conventional conservative spin-vorticity coupling, the term ${\tilde{ζ}}_{s}^{f, ⊥}$ , thus induces a spin polarization (short arrows with sphere) along the $y$ direction, and motive force along the $z$ axis (blue arrows). The nonconservative contribution, ${\tilde{ξ}}_{s}^{ω, ⊥}$ , in contrast, induces a spin polarization along the $z$ direction, and motive force along the $y$ axis (magenta arrows).
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Figure 4
Feynman diagrams for the dominant contribution to the spin-resolved momentum flux density $π_{i j k}^{s, α}$ at the linear order in the spin-orbit interaction and linear response to the applied field, denoted by $\times$ at the right end (suffix $k$ represents the direction of the applied field). Solid lines represent free electron Green's functions, where upper and lower lines denote retarded ( $g_{k}^{r}$ ) and advanced ( $g_{k}^{a}$ ) Green's functions, respectively, and $k$ and $k^{'}$ are the electron wave vectors. The wave vector $q$ is that of the external field and represents the inhomogeneity of flow. Complex conjugate processes (turned upside down) are also taken into account. (a) Skew-scattering contribution $π_{i j k}^{(ss) α}$ . The left vertex with $k_{i} k_{j}$ represents the vertex for the momentum flux density, and a vertex with $λ$ denotes the spin-orbit interaction. (b) A small contribution that is neglected. (c) The contribution arising from the anomalous velocity $δ v$ in $π_{i j k}^{0 α}$ , called the side-jump contribution. (d) The anomalous contribution $δ π_{i j k}^{α}$ .
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Figure 5
(a) Schematic figure showing the current profile near the boundary (chosen as $z = 0$ ). Vorticity is finite in the length scale of $ℓ_{ω}$ , where the current density changes. (b) Schematic figure showing vorticity-induced spin relaxation mechanism due to electron diffusion. The sign denotes the direction of vorticity $ω$ . Electron diffusion is represented by a zigzag line.
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Figure 6
Feynman diagram representing the vorticity-induced inverse spin Hall effect at the second order of the spin-orbit interaction ( $λ$ ). The anomalous velocity vertex gives rise to an antisymmetric component of the momentum flux density that describes inverse spin Hall current, Eq. (71).
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