Theoretical study of extrinsic spin current generation in ferromagnets induced by anisotropic spin-flip scattering

The spin Hall effect (SHE) is responsible for electrical spin current generation, which is a key concept of modern spintronics. We present a theoretical study of an extrinsic mechanism of SHE arising from a spin-dependent s-d scattering in ferromagnets. In order to investigate the spin conductivity in a ferromagnetic $3d$ alloy model, we employ a microscopic transport theory based on the Kubo formula and the averaged $T$-matrix approximation. From the model, we derived an extrinsic mechanism that contributes to both the SHE and the time-reversal odd SHE known as the magnetic SHE. This mechanism can be understood as the contribution from anisotropic (spatial-dependent) spin-flip scattering due to the combination of the orbital-dependent anisotropic shape of s-d hybridization and spin flipping, with the orbital shift caused by spin-orbit interaction with the $d$ orbitals. We also show that this mechanism is valid under crystal-field splitting among the $d$ orbitals in either the cubic or tetragonal symmetry.

Figure 1

(a) Real-space view of a spin-current induced by anisotropic spin-flip (ASF) scattering, s-d scattering involving an orbital-selective momentum-dependent spin-flip process. The blue spheres with arrows represent $s$ electrons with their spins moving in the direction of the black arrows under the applied electric field $E_x$ and the magnetization $\widehat{\mathit{m}}\parallel \widehat{\mathit{x}}$. The black spheres with red and blue ellipses represent an impurity with a $d$ orbital. The colors correspond to the sign of the phase. For example, an electron moving in the $[1,1,0]$ direction is more likely to receive a net spin angular momentum $+s_y$ from a momentum-dependent spin flip, and vice versa for the electrons moving in the $[1,-1,0]$ direction. Accordingly, not only is the longitudinal spin current $J_x^x$ associated with the spin-polarized current, but also the transverse spin current $J_y^y$ polarized perpendicular to the magnetization direction. (b) Schematic representation of a representative ASF scattering event. From a microscopic point of view, a momentum-dependent spin flip can be realized by a combination of the spin-orbit interaction (SOI) in the $d$ orbitals and the selection rule for the s-d transition that connects the momentum and $d$ orbital of the $s$ electron. This means that the momentum and spin of an electron are coupled via intermediate $d$ orbitals.

Figure 2

Schematics of the coordination system of the stationary coordinate with the basis $\{\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}}\}$ and the rotational coordinate with respect to the magnetization direction $\widehat{\mathit{m}}$ with the basis $\{\widehat{\mathit{\theta }},\widehat{\mathit{\varphi }},\widehat{\mathit{m}}\}$.

Figure 3

Representative terms of the diagrams excluded from this work. (a) Higher-order Born terms of s-s scattering. (b) Crossing diagrams corresponding to quantum interference between multiple impurities. (c) On-site interference between s-s and s-d scattering terms.

Figure 4

Diagrammatic representations of (a) Dyson's equation for Green's function and (b) the self-energy of the conduction electron. (a) The bold lines represent the clothed Green's function, and the thin line represents the bare one. (b) Self-energy approximated by the configuration-averaged s-d and s-s scattering $T$ matrix. The double-dashed line represents the s-d scattering, and the dotted line represents the s-s scattering. The point marked by a cross represents a coherent scattering event by a single impurity, whereas the point marked by a diamond represents the scattering by the renormalized effective potential, including all single-site scattering processes.

Figure 5

Diagrammatic representation of (a) the single-site s-d scattering $T$ matrix in the dilute limit averaged $T$-matrix approximation, (b) the current–spin-current correlation function for the Fermi-surface term in the Kubo-Streda formula, and (c) the Bethe-Salpeter equation (BSE) of the current vertex function. Here, ${\tilde{g}}^{\mathrm{d}}\equiv {(E{\tilde{\sigma }}_0-{\tilde{H}}_{\mathrm{d}})}^{-1}$ represents the bare Green's function of the impurity system, and ${\tilde{G}}^{\mathrm{d}}\equiv (E{\tilde{\sigma }}_0-{\tilde{H}}_{\mathrm{d}}-{\tilde{\Sigma }}_{\mathrm{d}})$ represents the clothed function. The second and third terms on the right-hand side of (c) vanish owing to the symmetry consideration of the s-d and s-s scattering as ${\mathrm{\Gamma }}_{l,l^{\prime }}(-\mathit{k})=-{\mathrm{\Gamma }}_{l,l^{\prime }}\left(\mathit{k}\right)$, which leads to ${\sum }_{\mathit{k}}{\mathrm{\Gamma }}_{l,l^{\prime }}\left(\mathit{k}\right)f\left(k\right)=0$. Therefore, the current vertex function can be replaced by a bare current operator. For this reason, although the $T$ matrix in (a) includes higher-order terms than Born approximation, the skew-scattering terms contribute neither charge- nor spin-conduction in this work.

Figure 6

Diagrammatic representation of an s-d scattering process contributing both to the magnetic and the spin Hall effect, which corresponds directly to Fig. 1. The solid and double lines represent the electrons of the conduction bands and impurity $d$-orbital states, respectively. The gray points represent s-d hybridization, and the white point represents the spin-orbit interaction on the $d$-orbital states.

Figure 7

Polar plot for the nonequilibrium charge distribution function $f_0$ with respect to ${\varphi }_{\mathit{k}}$ for the different magnetization directions: (a) ${\varphi }_{\mathit{m}}=0^{\circ }$, (b) ${\varphi }_{\mathit{m}}={45}^{\circ }$, and (c) ${\varphi }_{\mathit{m}}={90}^{\circ }$, where ${\varphi }_{\mathit{k}}$ and ${\varphi }_{\mathit{m}}$ are the azimuthal angles of $\widehat{\mathit{k}}$ and $\widehat{\mathit{m}}$ on the stationary coordinate system. Each line corresponds to the projection of ${\theta }_{\mathit{k}}$ and the polar angle of $\widehat{\mathit{k}}$. The bold arrow represents $\widehat{\mathit{m}}$, and the thin red arrow represents the anisotropic part of the charge current induced by the direction-dependent distribution.

Figure 8

Spherical plot for the isosurface of a nonequilibrium spin distribution function $\delta f_{\theta }$ with respect to ${\varphi }_{\mathit{k}}$ for the different magnetization directions: (a) ${\varphi }_{\mathit{m}}=0^{\circ }$, (b) ${\varphi }_{\mathit{m}}={45}^{\circ }$, and (c) ${\varphi }_{\mathit{m}}={90}^{\circ }$. The colored area represents the sign of the distribution function, with red and blue denoting positive and negative signs, respectively. The $\uparrow$ spins are more likely to flip to $\downarrow$ for the red area, and vice versa for the blue area, which corresponds to the illustration in Fig. 1. Note that $\delta f_{\varphi }$ shows the same isosurface as $\delta f_{\theta }$.

Figure 9

Schematics of the projected density of states (PDOS) of the conduction bands and the 3$d$ level for $E_{\mathrm{imp}}=0.45$ and ${\mathrm{\Delta }}_{\mathrm{s}}={\mathrm{\Delta }}_{\mathrm{d}}=0.5$. The solid gray and solid black lines represent the PDOS of the conduction band and the PDOS of the 3$d$ states for each spin, respectively. The dotted red and blue lines represent the PDOS of each splitting 3$d$ level due to the cubic and tetragonal crystal fields, respectively. The hatched area represents a half-metallic region, where a single spin conduction band is present. To improve visualization, the spin-orbit interaction is ignored, and the spectral widths are adjusted using different values from those in the numerical calculations.

Figure 10

Spin Hall angles ${\vartheta }_{yx}^{\mu }\equiv -(2e/\hslash ){\sigma }_{yx}^{\mu }/{\sigma }_{xx}^0$ as functions of the in-plane magnetization direction ${\varphi }_{\mathit{m}}$.

Figure 11

Fermi energy dependencies of the spin Hall angles for (a) ${\sigma }_{\parallel }^{\left(\mathrm{odd}\right)}$ and (b) ${\sigma }_{\perp }^{\left(\mathrm{even}\right)}, {\sigma }_{\perp }^{\left(\mathrm{odd}\right)}$. The vertical red line represents the position of each 3$d$ level. The insets show the intensities of each spin conductivity before being normalized by the longitudinal conductivity. The planar Hall angle is also plotted in (a). The hatched area represents the half-metallic region shown in Fig. 9.

Figure 12

Spin Hall angles ${\tilde{\sigma }}_{yx}^{\mu }\equiv -(2e/\hslash ){\sigma }_{yx}^{\mu }/{\sigma }_{xx}^0$ as functions of the in-plane magnetization direction ${\varphi }_{\mathit{m}}$ with (a) the cubic field splitting ${\mathrm{\Delta }}_{\mathrm{C}}=0.15,{\mathrm{\Delta }}_{\mathrm{T}}=0.0$ and (b) the tetragonal field splitting ${\mathrm{\Delta }}_{\mathrm{C}}=0.15,{\mathrm{\Delta }}_{\mathrm{T}}=0.05$.

Figure 13

Fermi energy dependencies of spin Hall angles for (a) ${\sigma }_{\perp }^{\left(\mathrm{even}\right)}$ and (b) ${\sigma }_{\perp }^{\left(\mathrm{odd}\right)}$ for various values of the cubic crystal field ${\mathrm{\Delta }}_{\mathrm{C}}=0.0,0.15,0.3$. The arrows at the bottom represent the positions of the corresponding energy levels.

Figure 14

Fermi energy dependencies of spin Hall angles for (a) ${\sigma }_{\perp }^{\left(\mathrm{even}\right)}$ and (b) ${\sigma }_{\perp }^{\left(\mathrm{odd}\right)}$ for various values of the tetragonal distortion ${\mathrm{\Delta }}_{\mathrm{T}}=-0.1,-0.05,0.0,0.05,0.1$ and the cubic crystal field ${\mathrm{\Delta }}_{\mathrm{C}}=0.15$ (fixed).