**Magnetization Dynamics–Induced Charge and Spin Transport on the Surface of a Topological Insulator Subjected to Magnetism**

Katsuhisa Taguchi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62531

#### **Abstract**

We theoretically show spin and charge transport on the disordered surface of a three‐ dimensional topological insulator with a magnetic insulator when localized spin of the magnetic insulator depends on time and space. To ascertain the transports, we use a low‐energy effective Hamiltonian on the surface of a topological insulator using the exchange interaction and calculate analytically using Green's function techniques within the linear response to the exchange interaction. As a result, the time‐depend‐ ent localized spin induces the charge and spin current. These currents are detected from change in the half‐width value of the ferromagnetic resonance of the localized spin when the magnetic resonance of the localized spin is realized in the attached magnetic insulator. We also show spin and charge current generation in a three‐dimensional Weyl–Dirac semimetal, which has massless Dirac fermions with helicity degrees of freedoms. The time‐dependent localized spin drives the charge and spin current in the system. The charge current as well as the spin current in the Weyl–Dirac system are slightly different from those on the surface of the topological insulator.

**Keywords:** Spin pumping, Spin–momentum locking, Surface of topological insulator, Weyl–Dirac semimetal, Massless Dirac fermions

## **1. Introduction**

A crucial issue in spintronics is the generation and manipulation of a charge and spin current by magnetism, since these mechanisms can be applicable to magnetic devices. One way to generate charge and spin flow is called "spin pumping," which pumps from the angular

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

momentum of a magnetization's localized spin into that of electrons through the dynamics of magnetization as well as spin-orbitinteractions [1, 2]. No other way of doing this has so far been discussed in the field of metallic spintronics.

Ever since the discovery of a topological insulator (TI) [3–6], spintronics using topology has been studied. A TI has a gapless surface, its bulk is insulating, but its surface is metallic as a result of two‐dimensional massless Dirac fermions on the surface [4–6]. Because of spin–orbit interactions on the surface the spin and momentum of Dirac fermions are perfectly linked to each other. The relation between a TI's spin and momentum is dubbed "spin–momentum locking," and the direction in which they travel is perpendicular to each other. Because of spin– momentum locking, unconventional spin‐related phenomena—such as magnetoresistance [7– 15], the magnetoelectric effect [16–21], diffusive charge–spin transport [22–25], and the spin pumping effect [26–31]—have so far been the only phenomena theoretically and experimen‐ tally studied.

Of the unconventional phenomena on the surface of a TI, spin pumping is one of the most interesting when it comes to spintronics. Here spin pumping on the surface is different from that in metals. As a result of spin–momentum locking on the surface the localized spin plays the role of a vector potential, whereas time‐differential localized spin effectively plays the role of an electric field acting on electrons on the surface of the TI [16–19]. As a result, even in the absence of an applied electric field, the charge current is generated by time‐dependent localized spin as shown in **Figure 1**. The induced charge current flows along *z*×∂<sup>t</sup> *S* [28–31]. Actually, this spin pumping effect on the surface has been experimentally demonstrated in the junction of the TI by attaching magnetic materials [29–31] by changing the half‐width value when ferromagnetic resonance is realized in the attached magnetic material [32, 33].

Recently, it has been reported that localized spin on the surface of a TI subject to magnetism depends on space and its spin texture seems to be a magnetic domain wall [34]. It is predicted that in the presence of a spatial‐dependent spin structure the charge current, which reflects the spin structure, is induced [28]. Moreover, the spin current as well as the charge density are induced when an inhomogeneous spin structure exists on the surface. Detailed results are shown in Section 2. This study may help the study of spin pumping on the surface of a TI with inhomogeneous spin textures [35, 36].

Recently, the next generation of spintronics has been theoretically and experimentally studied in Weyl–Dirac semimetals. A Weyl–Dirac semimetal possessing three‐dimensional massless Dirac fermions has attracted much attention in condensed matter physics [37–39]. Such a semimetal has been experimentally demonstrated [40–45]. In addition, Weyl–Dirac semimetals have been theoretically predicted in a superlattice heterostructure based on the TI. Such a heterostructure has been realized in the GeTe/Sb2Te3 superlattice [46].

Spin‐momentum locking occurs in a Weyl–Dirac semimetal, but the locking is slightly different from that on the surface of the TI. As a result of spin–momentum locking, the spin polarization (charge density) and the charge current (spin current) are linked to each other. Moreover, Dirac fermions have helicity degrees of freedom, which are decomposed into left‐ and right‐handed fermions. Note that the total charge flow of Dirac fermions of left‐ and right‐handed Weyl fermions is preserved. In a Weyl–Dirac semimetal the anomaly‐related effect, which is discussed in the field of relativistic high‐energy physics, has also been discussed in condensed matter physics. Studies up to the moment have asserted that the charge current is generated by magnetic properties with helicity degrees of freedoms [47–54]. Our goal is to introduce the helicity‐dependent spin pumping effect, one of the characteristic properties of Dirac fermions (as shown in Section 3).

**Figure 1.** Schematic of spin pumping on the surface of a topological insulator with localized spins of the attached mag‐ netic insulator. Here, the localized spin depends on positions on the surface. The dynamics of the localized spin (dash‐ ed line) drives the charge current (orange arrow) on the surface. Then, the charge current is also polarized because of spin**–**momentum locking on the surface.

## **2. Spin and charge transport due to spin pumping on the surface of a topological insulator**

#### **2.1. Model**

*S* [28–31].

momentum of a magnetization's localized spin into that of electrons through the dynamics of magnetization as well as spin-orbitinteractions [1, 2]. No other way of doing this has so far been

Ever since the discovery of a topological insulator (TI) [3–6], spintronics using topology has been studied. A TI has a gapless surface, its bulk is insulating, but its surface is metallic as a result of two‐dimensional massless Dirac fermions on the surface [4–6]. Because of spin–orbit interactions on the surface the spin and momentum of Dirac fermions are perfectly linked to each other. The relation between a TI's spin and momentum is dubbed "spin–momentum locking," and the direction in which they travel is perpendicular to each other. Because of spin– momentum locking, unconventional spin‐related phenomena—such as magnetoresistance [7– 15], the magnetoelectric effect [16–21], diffusive charge–spin transport [22–25], and the spin pumping effect [26–31]—have so far been the only phenomena theoretically and experimen‐

Of the unconventional phenomena on the surface of a TI, spin pumping is one of the most interesting when it comes to spintronics. Here spin pumping on the surface is different from that in metals. As a result of spin–momentum locking on the surface the localized spin plays the role of a vector potential, whereas time‐differential localized spin effectively plays the role of an electric field acting on electrons on the surface of the TI [16–19]. As a result, even in the absence of an applied electric field, the charge current is generated by time‐dependent

Actually, this spin pumping effect on the surface has been experimentally demonstrated in the junction of the TI by attaching magnetic materials [29–31] by changing the half‐width value

Recently, it has been reported that localized spin on the surface of a TI subject to magnetism depends on space and its spin texture seems to be a magnetic domain wall [34]. It is predicted that in the presence of a spatial‐dependent spin structure the charge current, which reflects the spin structure, is induced [28]. Moreover, the spin current as well as the charge density are induced when an inhomogeneous spin structure exists on the surface. Detailed results are shown in Section 2. This study may help the study of spin pumping on the surface of a TI with

Recently, the next generation of spintronics has been theoretically and experimentally studied in Weyl–Dirac semimetals. A Weyl–Dirac semimetal possessing three‐dimensional massless Dirac fermions has attracted much attention in condensed matter physics [37–39]. Such a semimetal has been experimentally demonstrated [40–45]. In addition, Weyl–Dirac semimetals have been theoretically predicted in a superlattice heterostructure based on the TI. Such a

Spin‐momentum locking occurs in a Weyl–Dirac semimetal, but the locking is slightly different from that on the surface of the TI. As a result of spin–momentum locking, the spin polarization (charge density) and the charge current (spin current) are linked to each other. Moreover, Dirac fermions have helicity degrees of freedom, which are decomposed into left‐ and right‐handed fermions. Note that the total charge flow of Dirac fermions of left‐ and right‐handed Weyl

heterostructure has been realized in the GeTe/Sb2Te3 superlattice [46].

localized spin as shown in **Figure 1**. The induced charge current flows along *z*×∂<sup>t</sup>

when ferromagnetic resonance is realized in the attached magnetic material [32, 33].

discussed in the field of metallic spintronics.

inhomogeneous spin textures [35, 36].

tally studied.

72 Magnetic Materials

We will calculate the charge and spin current due to spin pumping on the surface of a TI with an attached magnetic insulator (MI) (as illustrated in **Figure 1**). To do so, we consider the following low‐energy effective Hamiltonian, which describes the surface of the TI with localized spin of the MI [4]:

$$H = H\_{\text{Tl-aurtac}} + H\_{\text{ox}} + H\_{\text{lmp}},\tag{1}$$

where *H*TI‐surface, *H*ex, and *H*imp are the low‐energy effective Hamiltonian on the surface of the TI, the spin–exchange interaction between the localized spin of the attached MI and the spin of conduction electrons, and the nonmagnetic impurity scattering on the surface of the TI, respectively. *H*TI‐surface is given by

$$\sum\_{\tilde{k}} \psi\_{\tilde{k}}^{\dagger} [-\hbar \nu\_{F,\mathfrak{k}} (\tilde{k} \times \tilde{\sigma})\_{\varepsilon} - E\_{F}] \psi\_{\tilde{k}} \tag{2}$$

where *v*F,B, *E*F, *k*, and *σ* are the velocity of bare electrons, the Fermi energy, momentum on the surface of the TI, and the Pauli matrix in the spin space, respectively; and *Ψ*k = *<sup>t</sup>* (*Ψ*↑, *Ψ*↓) and *Ψ*† are the annihilation and creation operators of the electrons with up‐spin and down‐spin index (↑, ↓). *H*ex is represented by

$$H\_{ex} = -\overline{J}\_{ex} \int d\mathbf{x}^2 \,\psi^\dagger(\vec{S}\cdot\vec{\sigma})\psi,\tag{3}$$

where *J*ex is the coefficient of the exchange interaction; and *S* is the localized spin of the MI on the surface. Here, we take into account the space‐ and time‐dependence of *S*. *H*imp is given by

$$H\_{\rm imp} = \int d\mathbf{x}^2 \,\psi^\dagger \boldsymbol{u}\_{\rm imp} \boldsymbol{\psi} \tag{4}$$

Nonmagnetic impurity scattering is taken into account for a delta function type [19, 21–24] and is considered within the Born approximation. Because of impurity scattering the Fermi velocity of bare Dirac fermions in Eq. (2) is modified by *v*F,B → *v*F [23, 24, 28]. We use *v*<sup>F</sup> in what follows.

#### **2.2. Charge and spin current due to localized spin dynamics**

We will calculate the charge and spin current as well as the charge and spin density due to magnetization dynamics on the surface of the TI. They are given using the Keldysh–Green function and the lesser component of Green's function [48] as

$$\begin{split} \langle \dot{\jmath}^{\dagger}(\mathbf{x},t) \rangle &= e \upsilon\_{\sf F} \langle \jmath \nu^{\dagger}(\mathbf{x},t) \sigma^{\dagger} \wp(\mathbf{x},t) \rangle = -i \frac{\hbar}{2\pi} e \upsilon\_{\sf F} \text{tr} [\sigma^{\dagger} G^{<}(\mathbf{x},t;\mathbf{x},t)] = \frac{e \upsilon\_{\sf F}}{2} \langle \mathbf{s}^{\dagger} \rangle, \\ \langle \dot{\jmath}\_{s,a}^{\dagger}(\mathbf{x},t) \rangle &= \frac{1}{2} \upsilon\_{\sf F} \varepsilon^{zal} \langle \jmath \nu^{\dagger}(\mathbf{x},t) \wp(\mathbf{x},t) \rangle = -\frac{i\hbar}{\hbar} \upsilon\_{\sf F} \varepsilon^{zal} \text{tr} [G^{<}(\mathbf{x},t;\mathbf{x},t)] = \frac{\upsilon\_{\sf F}}{2\varepsilon} \varepsilon^{zal} \langle \rho \rangle. \end{split} \tag{5}$$

As a result of spin–momentum locking the charge current <*j i* > and spin current <*j i s,α*> are proportional to the spin density <*si* > and charge density <*ρ*>, respectively, where < > = denotes the expectation value in *H* and *e*(<0) is the charge of Dirac fermions. These relations are derived from the Heisenberg equation. The superscript and subscripts in <*j i* s,α> show the direction of flow and spin of the spin current, respectively. In what follows the charge and spin current are considered within the linear response to *H*ex. This assumption can be a good approximation because the energy scale of the exchange interaction is smaller than that of the bandwidth of the low‐energy effective Hamiltonian on the surface of the TI [28, 31]. We also assume the TI has a metallic surface of and that a finite Fermi surface exists on the surface; that is, *h*/(*E*F*τ*) ≪ 1 where *h* is the Planck constant and *τ* is the relaxation time of nonmagnetic impurity scatterings on the surface.

#### Magnetization Dynamics–Induced Charge and Spin Transport on the Surface of a Topological Insulator Subjected to Magnetism http://dx.doi.org/10.5772/62531 75

The charge and spin current are represented by

(2)

(3)

(5)

*i s,α*> are

s,α> show the direction of

(*Ψ*↑, *Ψ*↓) and *Ψ*†

where *v*F,B, *E*F, *k*, and *σ* are the velocity of bare electrons, the Fermi energy, momentum on the

are the annihilation and creation operators of the electrons with up‐spin and down‐spin index

where *J*ex is the coefficient of the exchange interaction; and *S* is the localized spin of the MI on the surface. Here, we take into account the space‐ and time‐dependence of *S*. *H*imp is given by

Nonmagnetic impurity scattering is taken into account for a delta function type [19, 21–24] and is considered within the Born approximation. Because of impurity scattering the Fermi velocity of bare Dirac fermions in Eq. (2) is modified by *v*F,B → *v*F [23, 24, 28]. We use *v*<sup>F</sup> in what follows.

We will calculate the charge and spin current as well as the charge and spin density due to magnetization dynamics on the surface of the TI. They are given using the Keldysh–Green

(,) (,) (,) [ ( , ; , )] , <sup>2</sup>

h

á ñ= á ñ=- = áñ

*i z i z i F z i*

2

á ñ= á ñ=- = áñ p

*ev j x t ev x t x t i ev G x t x t s*

*i v j xt v xt xt v G xtxt <sup>h</sup> <sup>e</sup>*

*i i ii*

<sup>1</sup> (,) (,) (,) [ ( , ; , )] . 2 2

h

the expectation value in *H* and *e*(<0) is the charge of Dirac fermions. These relations are derived

flow and spin of the spin current, respectively. In what follows the charge and spin current are considered within the linear response to *H*ex. This assumption can be a good approximation because the energy scale of the exchange interaction is smaller than that of the bandwidth of the low‐energy effective Hamiltonian on the surface of the TI [28, 31]. We also assume the TI has a metallic surface of and that a finite Fermi surface exists on the surface; that is, *h*/(*E*F*τ*) ≪ 1 where *h* is the Planck constant and *τ* is the relaxation time of nonmagnetic impurity

tr

 e

tr

s

<

<

aa

*i*

> and charge density <*ρ*>, respectively, where < > = denotes

*i*

e r

> and spin current <*j*

F

 y

imp (4)

2 † *H dx u* imp = ò y

**2.2. Charge and spin current due to localized spin dynamics**

function and the lesser component of Green's function [48] as

†

s F F

 y

from the Heisenberg equation. The superscript and subscripts in <*j*

As a result of spin–momentum locking the charge current <*j*

F F

 sy

†

y

proportional to the spin density <*si*

a

ey

,

a

scatterings on the surface.

surface of the TI, and the Pauli matrix in the spin space, respectively; and *Ψ*k = *<sup>t</sup>*

(↑, ↓). *H*ex is represented by

74 Magnetic Materials

$$\begin{split} \mathbf{j}^i(\mathbf{x},t) &= \frac{i\epsilon\hbar\nu\_F J\_{\rm ex}}{L^2} \sum\_{\stackrel{\circ}{q},\Omega} e^{i(\Omega^\circ - q \cdot \mathbf{x})} \text{tr}[\boldsymbol{\sigma}^i \boldsymbol{\Pi}^j(\boldsymbol{q}, \boldsymbol{\Omega})] \boldsymbol{\Sigma} \boldsymbol{r}^{\boldsymbol{\prime}}\_{\boldsymbol{q},\Omega} \,, \\ \mathbf{j}^i\_{s,a}(\mathbf{x},t) &= \frac{i\hbar\nu\_F J\_{\rm ex}}{2L^2} \boldsymbol{\varepsilon}^{\boldsymbol{\prime},a\boldsymbol{\prime}} \sum\_{\stackrel{\circ}{q},\Omega}^{\boldsymbol{\prime}} e^{i(\Omega^\circ - q \cdot \mathbf{x})} \text{tr}[\boldsymbol{\Pi}^j(\boldsymbol{q}, \boldsymbol{\Omega})] \boldsymbol{\Sigma} \boldsymbol{r}^{\boldsymbol{\prime}}\_{\boldsymbol{q},\Omega} \,, \end{split} \tag{6}$$

where *q* and Ω are the momentum and frequency of the localized spin *Sj* **q**,Ω, respectively; *L*<sup>2</sup> is the area of the surface; and Π*<sup>j</sup>* is the response function. These currents are obtained from Π*<sup>j</sup>* :

$$
\Pi^{\backslash}(\vec{q}, \Omega) = \sum\_{\boldsymbol{\alpha}, \vec{k}} (f\_{\boldsymbol{\alpha} \leftrightarrow \Omega/2} - f\_{\boldsymbol{\alpha} \to \Omega/2}) \mathcal{g}^{\boldsymbol{R}}\_{\boldsymbol{\alpha} \leftrightarrow \Omega/2, \vec{k} - \vec{q}/2} \Lambda^{\backslash}(\vec{q}, \Omega) \mathcal{g}^{\boldsymbol{A}}\_{\boldsymbol{\alpha} \leftrightarrow \Omega/2, \vec{k} + \vec{q}/2}. \tag{7}$$

where *f*ω, *g*ω,k R = ([*g*ω,kA] † ) = [*hw*/(2π) + *E*F – *hv*F/(2π)(*k* × *σ*)z + *ih*/(4*πτ*)]‐1, and Λ*<sup>j</sup>* are the Fermi distribution function, the retarded (advanced) Green function, and the vertex correction of the ladder diagram, respectively. The above functions are estimated in the regime *Ωτ* ≪ 1 and *qv*Fτ ≪ 1, which are satisfied when the dynamics of the localized spin is lower than the terahertz regime and the spatial gradient of the localized spin is slow compared with the electron mean free path. Then, by expanding *Ω* and *q* within *Ωτ* ≪ 1 and *qv*F*τ* ≪ 1, the vertex function Λ*<sup>j</sup>* is represented by Λ*<sup>j</sup>* = [Γ + Γ<sup>2</sup> + Γ<sup>3</sup> + …]*jn*σ*<sup>n</sup>* = [ (1 – Γ)‐1] *jn*σ*<sup>n</sup>* with

$$
\Gamma^{\vdash} \equiv \mathfrak{n}\_c \mathfrak{u}\_l^2 \sum\_{a \sim \Omega} \operatorname{\mathcal{g}}^{\mathbb{R}}\_{a \sim \Omega \cap 2, \bar{k} \sim \bar{\mathfrak{q}}/2} \sigma^{\flat} \operatorname{\mathcal{g}}^{\mathcal{A}}\_{a \ast \Omega \cap 2, \bar{k} \ast \bar{\mathfrak{q}}/2} = \Gamma^{\flat \mu} \sigma^{\mu}. \tag{8}
$$

where Γ*jn* (||Γ*jn*|| < 1) is the 3 × 3 matrix (*j*, *n* = 0*, x, y*). As a result, Π*<sup>j</sup>* is given by

$$
\begin{pmatrix} \Pi^{0} \\ \Pi^{x} \\ \Pi^{y} \end{pmatrix} = -\frac{\Omega}{2\pi} \begin{pmatrix} l & i \mid q\_{\boldsymbol{\gamma}} & \boldsymbol{\Gamma} \mid q\_{\boldsymbol{\gamma}} \\ \hline \boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} & q^{2} \right|^{2} + i\Omega\boldsymbol{\tau} & \boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} \\ \hline \boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} & l - \frac{\boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} \right|^{2}}{2(\boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} \right|)} & \frac{q\_{\boldsymbol{\gamma}} \left| \boldsymbol{q}^{\boldsymbol{\gamma}} \right|^{2}}{\boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} \right|} \\ \hline \boldsymbol{\frac{-i}{\boldsymbol{q}} \left| \boldsymbol{q}\_{\boldsymbol{x}} & \boldsymbol{q}\_{\boldsymbol{x}} \boldsymbol{q}\_{\boldsymbol{\gamma}} \right|^{2} & l - \frac{\boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} \right|^{2}}{2(\boldsymbol{q}^{2} \left| \boldsymbol{\gamma}^{2} + i\Omega\boldsymbol{\tau} \right|)} \end{pmatrix} \begin{pmatrix} \boldsymbol{\sigma}^{0} \\ \hline \boldsymbol{\sigma}^{\boldsymbol{x}} \\ \hline \boldsymbol{\sigma}^{\boldsymbol{x}} \end{pmatrix} \tag{9}
$$

where *σ*<sup>0</sup> is the identity matrix; and *l* is the mean free path on the surface. From Eqs. (5–9) the spin and charge current are given by

$$\begin{split} \left\{ \boldsymbol{j}^{i} \right\} &= -2\boldsymbol{\epsilon}\boldsymbol{\nu}\_{\mathrm{e}}^{\mathrm{T}} \boldsymbol{J}\_{\mathrm{ex}} \boldsymbol{\nu}\_{\mathrm{e}} \,\mathrm{r} \Big( (\boldsymbol{\tilde{z}} \times \boldsymbol{\mathcal{o}}\_{\mathrm{i}} \boldsymbol{\bar{\mathcal{S}}}^{\mathrm{i}})^{\mathrm{i}} - \boldsymbol{\ell}^{2} [\boldsymbol{\tilde{z}} \times (\boldsymbol{\tilde{z}} \times \boldsymbol{\bar{\mathcal{V}}})] \, [\, \boldsymbol{\bar{\mathcal{V}}} \times \boldsymbol{\mathcal{o}}\_{\mathrm{i}} \,\mathrm{\{\bar{\mathcal{S}}^{\mathrm{i}}\}\_{\mathrm{D}}\,\mathrm{\}}]\_{\mathrm{x}} \Big), \\ \left\{ \boldsymbol{j}\_{\mathrm{s},a}^{\mathrm{i}} \right\} &= \boldsymbol{\varepsilon}^{\mathrm{z}a} \boldsymbol{\nu}\_{\mathrm{e}}^{\mathrm{T}} \boldsymbol{J}\_{\mathrm{ex}} \boldsymbol{\nu}\_{\mathrm{e}} \,\mathrm{\{\boldsymbol{\bar{\mathcal{V}}} \}} \, \mathrm{\,}^{\mathrm{2}} [\boldsymbol{\bar{\nabla}} \times \boldsymbol{\mathcal{o}}\_{\mathrm{i}} \,\mathrm{\{\bar{\boldsymbol{\bar{\mathcal{S}}}} \}]\_{\mathrm{D}} \,\mathrm{I}\_{\mathrm{z}} \end{split} \tag{10}$$

where *ν*TIe is the density of state on the surface of the TI; <*S*||>D is the convolution between *S*|| and the diffusion propagator *D* on the surface; and *S*|| is the in‐plane localized spin *S*||= *S* – *Sz z* as

$$\begin{split} \langle \vec{S}^{\parallel} \rangle\_{\rm D} &= \frac{1}{\tau} \int\_{-\alpha}^{\alpha} dt' \Big[ dx' D(\mathbf{x} - \mathbf{x}', t - t') \vec{S}^{\parallel}(\mathbf{x}', t'), \\ D(\mathbf{x} - \mathbf{x}', t - t') &= \frac{1}{L^2} \sum\_{\vec{q}, \Omega} \frac{1}{q^2 \upsilon\_{\rm F} \tau + i\Omega} e^{i(\Omega t - \vec{q} \cdot \vec{\mathbf{x}})}. \end{split} \tag{11}$$

Eq. (10) shows that time‐dependent localized spin induces the charge and spin current and that they can be decomposed into local and nonlocal contributions. The first term in Eq. (10) is the charge current due to spin dynamics at that position on the surface; its direction is along the *z* × *∂*<sup>t</sup> *S||* direction [28–31]. On the other hand, the second term in Eq. (10) is the charge current due to both time‐ and spatial‐dependent localized spin; its direction is along the [*z* × (*z* × *∂*)] [*∂* × *∂*<sup>t</sup> <*S*||>D]z direction, where <*S*||<D indicates the nonlocal contribution from the localized spin. Moreover, this term is zero when the spin texture is spatially uniform. There‐ fore, this charge current is caused by diffusion with the dynamics of the spatial inhomogeneous spin texture. It is noted that this second term is also described by the spatial gradient of the charge density.

Because of spin–momentum locking on the surface of the TI, spin polarization is given by <*j*> = 2*ev*F(*z* × <*s*>). Thus, the properties of spin polarization are similar to those of the charge current.

Eq. (11) shows that the spin current is induced by the time‐ and spatial‐dependent *S||*. Then, the charge density is also generated because of spin–momentum locking. In addition, the second term of the charge current <*j*>nonlocal = 2*ev*F*ν*TIe*J*ex*τl*<sup>2</sup> [*z* × (*z* × *∂*)] [*∂* × (*∂*<sup>t</sup> <*S||*>D) ]z is given by the spatial gradient of the charge density, whereas <*j*>nonlocal is proportional to the spatial gradient of the spin current [28] as

$$\left\langle j^{i} \right\rangle\_{\text{nonlocal}} = -e \,\ell \,\varepsilon^{zan} \,\nabla^{i} \left\langle j\_{s,a}^{n} \right\rangle. \tag{12}$$

This result shows that the relation between the charge and spin current is different from that in the metallic spintronics system [2]. The spin current is proportional to the charge current and spin current flow is perpendicular to the charge flow and its spin polarization.

Note that no out‐of‐plane localized spin *Sz* **z** contributes in a dominant way to charge and spin current generation. The reason is *σ<sup>z</sup>* does not couple with momentum *p* in Eq. (2). Therefore,

we believe a warping effect [1–4] on the surface of the TI is likely. Spin polarization along the *z*‐direction is also generated by spin dynamics.

#### **2.3. Spin torque**

( ) TI || 2 || e ex F <sup>t</sup> <sup>t</sup> <sup>z</sup> <sup>D</sup>

r r r r r rr <sup>l</sup>

¶

( )

W- ×

*i t qx*

*S*

(10)

(11)

2 ( ) [ ( )] [ )] ,

where *ν*TIe is the density of state on the surface of the TI; <*S*||>D is the convolution between *S*|| and the diffusion propagator *D* on the surface; and *S*|| is the in‐plane localized spin *S*||= *S* –

<sup>1</sup> ' ' ( ', ') ( ', '),

*dt dx D x x t t x t*

t

å r r

*S||* direction [28–31]. On the other hand, the second term in Eq. (10) is the charge

<*S*||>D]z direction, where <*S*||<D indicates the nonlocal contribution from the

[*z* × (*z* × *∂*)] [*∂* × (*∂*<sup>t</sup>

**z** contributes in a dominant way to charge and spin

*<sup>i</sup> zn i n j ej* (12)

<*S||*>D) ]z is given

*i*

TI 2 || s, e ex F t z <sup>D</sup>

t

D

second term of the charge current <*j*>nonlocal = 2*ev*F*ν*TIe*J*ex*τl*<sup>2</sup>

gradient of the spin current [28] as

Note that no out‐of‐plane localized spin *Sz*

*S*

¥


ò

 t¶

= Ñ ´

*j Jv S*

*i i*

a

en

n

a

*Sz z* as

76 Magnetic Materials

the *z* × *∂*<sup>t</sup>

(*z* × *∂*)] [*∂* × *∂*<sup>t</sup>

charge density.

current.

*i zi*

[ ],


r r


*Dx x t t e L*

áñº ò - -

2 2 ,

r

W

q F

*q v*

Eq. (10) shows that time‐dependent localized spin induces the charge and spin current and that they can be decomposed into local and nonlocal contributions. The first term in Eq. (10) is the charge current due to spin dynamics at that position on the surface; its direction is along

current due to both time‐ and spatial‐dependent localized spin; its direction is along the [*z* ×

localized spin. Moreover, this term is zero when the spin texture is spatially uniform. There‐ fore, this charge current is caused by diffusion with the dynamics of the spatial inhomogeneous spin texture. It is noted that this second term is also described by the spatial gradient of the

Because of spin–momentum locking on the surface of the TI, spin polarization is given by <*j*> = 2*ev*F(*z* × <*s*>). Thus, the properties of spin polarization are similar to those of the charge

Eq. (11) shows that the spin current is induced by the time‐ and spatial‐dependent *S||*. Then, the charge density is also generated because of spin–momentum locking. In addition, the

by the spatial gradient of the charge density, whereas <*j*>nonlocal is proportional to the spatial

s, nonlocal . a

and spin current flow is perpendicular to the charge flow and its spin polarization.

 =- Ñ le

This result shows that the relation between the charge and spin current is different from that in the metallic spintronics system [2]. The spin current is proportional to the charge current

current generation. The reason is *σ<sup>z</sup>* does not couple with momentum *p* in Eq. (2). Therefore,

a

1 1 ( ', ') .

= - ´ - ´ ´Ñ Ñ´

*j e Jv z S z z S*

 ¶

<sup>r</sup> <sup>r</sup> <sup>l</sup>

Based on these results, we look at localized spin dynamics after generation of the charge current and spin polarization on the surface [24]. We assume there is an external static and AC magnetic field on the surface—as shown in **Figure 2(a)**. The static magnetic field arranges the localized spin texture and the AC magnetic field triggers its spin dynamics. The propagation direction of the microwave is parallel to the static magnetic field, which is along the *y*‐axis. The texture is called the "longitudinal conical spin order" [56].

After applying the microwave the dynamics of the localized spin of the MI is induced by the in‐plane AC magnetic field of the microwave. The dynamics of the spin induces the charge and spin current. Then, from <*j*> = 2*ev*F(*z* × <*s*>), Eq. (10), and spin–momentum locking, spin polarization is also generated. Note that spin polarization induced in this way acts on the localized spin as an effective exchange field for localized spin. This contribution is given from the Landau–Lifshitz–Gilbert (LLG) equation of motion [32, 33] on the surface of the TI:

$$\vec{\partial}\_{\text{t}}\vec{M} \equiv -\gamma\mu(\vec{M}\times\vec{H}) + \frac{\alpha\_{\text{C}}}{M}(\vec{M}\times\partial\_{\text{t}}\vec{M}) + \vec{T}\_{\text{e}} \tag{13}$$

**Figure 2.** (a) Schematic of the setup used to detect spin and charge current generation. The texture of the magnetization *M* depends on space. Such magnetization is along the applied static magnetic field *H*0, and the magnetization dynam‐ ics is induced by the applied magnetic field of the microwave. (b) Schematic of the spin torque acting on *M*.

where *M* = -*M*(*S*/*S*) is magnetization of the MI; *γ* is the gyromagnetic ratio; *μ* is permeability; *α*G is a Gilbert damping constant; *H* = *H*0 + *H*AC are the applied magnetic fields on the surface; and *H*0 and *H*ac denote the static and AC magnetic fields, respectively. The first term of Eq. (13) denotes the field‐like torque that drives the dynamics of *M*. The second term is the damping torque that prevents its dynamics—these terms are schematically illustrated in **Figure 2(b)**. The third term *T*<sup>e</sup> is the spin torque due to spin polarization of Dirac fermions on the surface of the TI (<*s>*). It is given by *T*e =2*J*ex*a*<sup>2</sup> (*M ×* <*s>*), where *a* is the lattice constant on the surface.

Since *T*e is proportional to <*s*>, this torque is decomposed into local (*T*<sup>e</sup> L) and nonlocal spin torque (*T*<sup>e</sup> D) terms as

$$\begin{aligned} \vec{T}\_{\text{c}}^{\perp} &= \frac{\boldsymbol{\kappa}}{M^{\parallel}} \vec{M} \times \boldsymbol{\mathcal{O}}\_{\text{t}} \vec{M}^{\parallel}, \\\\ \vec{T}\_{\text{c}}^{\text{D}} &= \frac{\boldsymbol{\kappa}}{M^{\parallel}} \ell^{2} \vec{M} \times (\boldsymbol{z} \times \vec{\nabla}) (\vec{\nabla} \times \boldsymbol{\mathcal{O}}\_{\text{t}} \left\langle \vec{M}^{\parallel} \right\rangle\_{\text{D}})\_{\text{x}}, \end{aligned} \tag{14}$$

where *M*|| = -*M*(*S||*/*S*) is in‐plane magnetization; and *κ* is the dimensionless coefficient.

As a result of spin–momentum locking the torque is *T*e ∼ *M ×* <*j>*. This relation is useful as a way of detecting nonzero <*j>* [29–31]. Moreover, nonlocal spin torque can be described by *T<*<sup>e</sup> *D* ∼ *M ×* <*j>*nonlocal . Eq. (12) shows this torque may well be affected by spin current contribution, whereas the relation between spin torque and spin current can be used as a way to detecting <*j i* s,α> on the surface:

$$
\vec{T}\_{\circ}^{D} = \frac{J\_{\alpha \circ} \pi a^2}{\hbar} \varepsilon^{z \alpha i} \vec{M} \times \left[ (z \times \vec{\nabla}) (\vec{\nabla} \times \langle j\_{s,i}^a \rangle) \right]. \tag{15}
$$

We now consider magnetization when it is given by *M* = ( *M*x, *M*y, *M*z) = [*m* cos(*q.x* – Ω*t*), *My*, *m* sin(*q.x* – Ω*t*)] with *q* = (*q*x, *q*y), *m*<sup>2</sup> ≪ *M*y, *M*<sup>y</sup> ∼ *M*, and ∂*<sup>t</sup> My* ∼ 0. Such a time‐dependent magnetic structure is assumed to be realized when the magnitude of the static magnetic field is larger than that of the AC magnetic field *H*AC=(*h*x, 0, *h*z); that is, *h*x, *h*<sup>z</sup> ≪ *H*0. Then, substituting this *M* into Eqs. (13) and (14), the LLG equation becomes a linear approximation:

$$\begin{aligned} \left\| \left\| \left\| \boldsymbol{m}\_{\mathrm{x}} = \boldsymbol{\alpha}\_{\mathrm{H}} \boldsymbol{m}\_{\mathrm{x}} - \boldsymbol{\alpha}\_{\mathrm{M}} \boldsymbol{h}\_{\mathrm{x}} + \boldsymbol{\alpha}\_{\mathrm{G}} \boldsymbol{\partial}\_{\mathrm{i}} \boldsymbol{m}\_{\mathrm{x}}, \\ \left\| \left\| \boldsymbol{\alpha}\_{\mathrm{i}} = - \boldsymbol{\alpha}\_{\mathrm{H}} \boldsymbol{m}\_{\mathrm{x}} + \boldsymbol{\alpha}\_{\mathrm{M}} \boldsymbol{h}\_{\mathrm{x}} - (\boldsymbol{\alpha}\_{\mathrm{G}} + \boldsymbol{\kappa}) \boldsymbol{\partial}\_{\mathrm{i}} \boldsymbol{m}\_{\mathrm{x}} + \boldsymbol{\kappa} \boldsymbol{q}\_{\mathrm{y}}^{2} \right\|^{2} \right\| \left\| \left\langle \boldsymbol{M}^{\mathrm{l},\mathrm{x}} \right\rangle\_{\mathrm{D}}, \end{aligned} \tag{16}$$

where *ω*H = *γμH* and *ω*M = *γμM* are the angular frequencies of *H*0 and *M*, respectively; and *κq*<sup>y</sup> 2 *l* 2 ∂*t* <*M*||,*<sup>x</sup>* >D indicates the contribution from *T*<sup>e</sup> D. Substituting <*M*||,*<sup>x</sup>* >D into Eq. (13), we obtain magnetic permeability around the surface:

$$
\begin{pmatrix} m\_{\times} \\ m\_{\times} \end{pmatrix} = \begin{pmatrix} \mathcal{Z}\_{\times \times} & \mathcal{Z}\_{\times \times} \\ \mathcal{Z}\_{\times \times} & \mathcal{Z}\_{\times \times} \end{pmatrix} \begin{pmatrix} h\_{\times} \\ h\_{\times} \end{pmatrix} \tag{17}
$$

where *χ*xx and *χ*zx denote the longitudinal and transverse magnetic permeability as

#### Magnetization Dynamics–Induced Charge and Spin Transport on the Surface of a Topological Insulator Subjected to Magnetism 79

$$\mathtt{http://doi.org/10.5772/62531}{\$}$$

$$\begin{split} \chi\_{\infty}(q,\Omega) &= \frac{(o\_{\mathbb{H}} + i\alpha\Omega)o\_{\mathbb{M}}}{[o\_{\mathbb{H}} + i\alpha\Omega][o\_{\mathbb{H}} + i(\alpha\tilde{k}\_{q,\Omega})\Omega] - \zeta\_{q,\alpha}\Omega^2}, \\ \chi\_{\infty}(q,\Omega) &= \frac{o\_{\mathbb{H}} + i(\alpha + \tilde{k}\_{q,\Omega})\Omega}{o\_{\mathbb{H}} + i\alpha\Omega}\chi\_{\infty}(q,\Omega), \\ \chi\_{\infty}(q,\Omega) &= -\zeta\_{q,\Omega}\chi\_{\infty}(q,\Omega) = -\chi\_{\infty}. \end{split} \tag{18}$$

The coefficients *κ'q,*Ω and *ζq,*Ω are given by

The third term *T*<sup>e</sup> is the spin torque due to spin polarization of Dirac fermions on the surface

Since *T*e is proportional to <*s*>, this torque is decomposed into local (*T*<sup>e</sup>

e t ||

*T MM*

= ´

k

*M*

r

*L*

*D*

k ||

¶

r r

= ´ ´Ñ Ñ´

*<sup>z</sup> <sup>M</sup>*

2

a

e

=

´ ´Ñ á ñÑ´ r r r r

e s i

this *M* into Eqs. (13) and (14), the LLG equation becomes a linear approximation:

 ak

c c

æ öæ öæ ö ç ÷ç ÷ <sup>=</sup> ç ÷ ç ÷ è øè øè ø x xx xz x z zx zz y *m h*

c c

where *χ*xx and *χ*zx denote the longitudinal and transverse magnetic permeability as

¶ =- + - + ¶ + ¶ á ñ **|**

t z H x Mx G t z y t D

t

h ex

*m* sin(*q.x* – Ω*t*)] with *q* = (*q*x, *q*y), *m*<sup>2</sup> ≪ *M*y, *M*<sup>y</sup> ∼ *M*, and ∂*<sup>t</sup>*

 wa

>D indicates the contribution from *T*<sup>e</sup>

t x H z Mz G t z

*mmh m*

¶ = - +¶

 w

w

magnetic permeability around the surface:

w

rr r <sup>l</sup>

*TM M*

,

where *M*|| = -*M*(*S||*/*S*) is in‐plane magnetization; and *κ* is the dimensionless coefficient.

2 || <sup>e</sup> || t z <sup>D</sup>

r r

As a result of spin–momentum locking the torque is *T*e ∼ *M ×* <*j>*. This relation is useful as a way of detecting nonzero <*j>* [29–31]. Moreover, nonlocal spin torque can be described by *T<*<sup>e</sup>

∼ *M ×* <*j>*nonlocal . Eq. (12) shows this torque may well be affected by spin current contribution, whereas the relation between spin torque and spin current can be used as a way to detecting

We now consider magnetization when it is given by *M* = ( *M*x, *M*y, *M*z) = [*m* cos(*q.x* – Ω*t*), *My*,

magnetic structure is assumed to be realized when the magnitude of the static magnetic field is larger than that of the AC magnetic field *H*AC=(*h*x, 0, *h*z); that is, *h*x, *h*<sup>z</sup> ≪ *H*0. Then, substituting

,

where *ω*H = *γμH* and *ω*M = *γμM* are the angular frequencies of *H*0 and *M*, respectively; and

( )( ) ,

¶

, [( )( ].

 a

*<sup>D</sup> z i J a T Mz j* (15)

2 2 ||,

*<sup>m</sup> <sup>h</sup>* (17)

*x*

( ) ,

*m m h mq M* (16)

D. Substituting <*M*||,*<sup>x</sup>*

 k

(*M ×* <*s>*), where *a* is the lattice constant on the surface.

L) and nonlocal spin

*My* ∼ 0. Such a time‐dependent

>D into Eq. (13), we obtain

(14)

*D*

of the TI (<*s>*). It is given by *T*e =2*J*ex*a*<sup>2</sup>

D) terms as

torque (*T*<sup>e</sup>

78 Magnetic Materials

<*j i*

*κq*<sup>y</sup> 2 *l* 2 ∂*t* <*M*||,*<sup>x</sup>*

s,α> on the surface:

$$\kappa^{\prime}\_{\,\,q,\Omega} = \kappa \left( \mathbf{l} - \frac{q^4}{q^4 \left| \right|^4 + \Omega^2 \tau^2} \right), \qquad \zeta\_{\,\,q,\Omega} = \mathbf{l} + \kappa \frac{q^2 \left| \right|^2 \Omega \tau}{q^4 \left| \right|^4 + \Omega^2 \tau^2}. \tag{19}$$

Hence, *κ'q,*Ω and *ζq,*Ω depend on *q* and Ω. If *q* = 0, *κ'q,*Ω and *ζq,*Ω go to *κ'q,*Ω → *κ* and *ζq,*Ω → 1, respectively. Then, the magnetic permeability is isotropic. In *q* = 0, *χ*xx and *χ*zz go to the same value and *χ*xz = –*χ*xx is satisfied. On the other hand, in nonzero *q* the magnetic permeability is anisotropic: *χ*xx ≠ *χ*zz.

**Figure 3(a)** shows the dependence of Ω on the imaginary part of the longitudinal magnetic permeability for several *q*. The resonance frequency Ω*r* and the half‐width value ΔΩ are slightly changed by the nonzero *q*. **Figure 3(b)** shows change in the resonance frequency for several *f*<sup>H</sup> in greater detail—*f*H is the frequency described by the static magnetic field *H*0. The resonance frequency decreases with increasing *ql* from *ql* = 0 into *ql* = *q0l,* where *q*0 satisfies *q*<sup>0</sup> 2 *l* 2 ∼ Ω*τ*. In

**Figure 3.** (a) Frequency dependence of the imaginary part on the longitudinal magnetic permeability of several *ql*. The half‐width value and resonance frequency change according to *ql.* (b) Dependence of *ql* on the resonance frequency for several frequencies of the applied static magnetic field. (c) Dependence of *ql* on the half‐width of several frequencies of the static magnetic field [49].

addition, when *ql* takes a large value the resonance frequency returns to that frequency at *ql*=0. **Figure 3(c)** shows the momentum dependence of the half‐width value on renormalization by the angular frequency of the applied magnetic field for several *ω*<sup>H</sup> =2π*f*H frequencies when realistic parameters on the surface of the TI are taken into account [31]. Change in the nor‐ malized half‐width value ΔΩ*/ωH*(*ql*) decreases from ΔΩ/*ω*H(*ql* = 0) = 0.038 to ΔΩ/*ω*H(*ql* ∼ 0.1) = 0.030 with increasing *ql*.

Any change in the half‐width value ΔΩ indicates an induced charge current on the surface of the TI because *T*<sup>e</sup> *<sup>L</sup>* **+ T**<sup>e</sup> *<sup>D</sup>* is proportional to the induced charge current [29–31]. In addition, the half‐width value ΔΩ of *T*<sup>e</sup> *<sup>D</sup>* represents a contribution from the spatial gradient of the spin current. These relations can be useful for the detection of spin current. For example, let us suppose that in a finite *ql*, ΔΩ includes the contribution from the whole of the spin torque *T*<sup>e</sup> and the contribution from the nonlocal term *T*<sup>e</sup> *<sup>D</sup>* is reduced by the large magnetic field. In a higher magnetic field the inhomogeneous spin texture would be expected to be aligned along the magnetic field direction and the spin texture to be spatially uniform; hence, the value of *ql* decreases and reaches zero. Then, the half‐width value ΔΩ(*T*<sup>e</sup> *<sup>D</sup>* = 0) has no contribution from *T*e *<sup>D</sup>*. The difference ΔΩ(*T*<sup>e</sup> <sup>D</sup> ≠ 0) – ΔΩ(*T*<sup>e</sup> *<sup>D</sup>* = 0) includes a pure contribution from *T*<sup>e</sup> *D*.

Any contribution from the applied magnetic field is of course a concern. Note that an in‐plane static magnetic field contributes no finite charge current generation or spin polarization [49], whereas the contribution from the applied magnetic field is negligible.

## **3. Spin pumping in a Weyl–Dirac semimetal**

## **3.1. Model**

We now consider spin pumping in a Weyl–Dirac semimetal subjected to magnetism. To do this, a Weyl–Dirac semimetal subjected to spin and momentum locking, such as a supperlattice hetrod structure constructed from a TI/normal insulator/TI [39, 46 – 48], is considered. The low‐energy effective Hamiltonian describing the Weyl–Dirac semimetals that have a spin– exchange interaction is given by [50–52]

$$H = H\_{\text{Weyl}} + H\_{\text{ox}} + H\_{\text{imp}}.\tag{20}$$

The first term takes the form

$$H\_{W\eta l} = \sum\_{k} \boldsymbol{\nu}\_{\boldsymbol{\epsilon}}^{\prime} [\boldsymbol{\hbar} \boldsymbol{\nu}\_{\boldsymbol{\mu}} \boldsymbol{\pi}^{\prime} \otimes (\boldsymbol{\vec{k}} - \boldsymbol{\tau}^{\prime} \boldsymbol{\vec{b}}) \cdot \boldsymbol{\sigma} - \mu \boldsymbol{\pi}^{0} \otimes \boldsymbol{\sigma}^{0} - \mu^{\prime} \boldsymbol{\pi}^{\prime} \otimes \boldsymbol{\sigma}^{0}] \boldsymbol{\nu}\_{\boldsymbol{\epsilon}},\tag{21}$$

where *Ψk* = *<sup>t</sup>* (*Ψ*↑, +*Ψ*↓, +*Ψ*↑, –*Ψ*↓,–) is the annihilation operator of an electron with spin (↑, ↓) and helicity τ(+, –) degrees of freedom; *τα*=0,*x*,*y*,*<sup>z</sup>* and *σ*α=0,*x*,*y*,*<sup>z</sup>* are Pauli matrices of the helicity and spin, respectively; *v*<sup>F</sup> is the Fermi velocity; *μ* is the chemical potential of the Weyl–Dirac semimetal; and *b* and *μ*<sup>5</sup> denote the difference in position between each Weyl cone in momentum and energy space, respectively. In Dirac semimetals, we set *b* = 0 and *μ*<sup>5</sup> = 0.

The second term of Eq. (20) indicates the spin–exchange interaction:

addition, when *ql* takes a large value the resonance frequency returns to that frequency at *ql*=0. **Figure 3(c)** shows the momentum dependence of the half‐width value on renormalization by the angular frequency of the applied magnetic field for several *ω*<sup>H</sup> =2π*f*H frequencies when realistic parameters on the surface of the TI are taken into account [31]. Change in the nor‐ malized half‐width value ΔΩ*/ωH*(*ql*) decreases from ΔΩ/*ω*H(*ql* = 0) = 0.038 to ΔΩ/*ω*H(*ql* ∼ 0.1) =

Any change in the half‐width value ΔΩ indicates an induced charge current on the surface of

current. These relations can be useful for the detection of spin current. For example, let us suppose that in a finite *ql*, ΔΩ includes the contribution from the whole of the spin torque *T*<sup>e</sup>

higher magnetic field the inhomogeneous spin texture would be expected to be aligned along the magnetic field direction and the spin texture to be spatially uniform; hence, the value of *ql*

Any contribution from the applied magnetic field is of course a concern. Note that an in‐plane static magnetic field contributes no finite charge current generation or spin polarization [49],

We now consider spin pumping in a Weyl–Dirac semimetal subjected to magnetism. To do this, a Weyl–Dirac semimetal subjected to spin and momentum locking, such as a supperlattice hetrod structure constructed from a TI/normal insulator/TI [39, 46 – 48], is considered. The low‐energy effective Hamiltonian describing the Weyl–Dirac semimetals that have a spin–

*<sup>D</sup>* is proportional to the induced charge current [29–31]. In addition, the

*<sup>D</sup>* represents a contribution from the spatial gradient of the spin

*<sup>D</sup>* = 0) includes a pure contribution from *T*<sup>e</sup>

= ++ Weyl ex mp. *HH H Hi* (20)

are Pauli matrices of the helicity and spin,

(*Ψ*↑, +*Ψ*↓, +*Ψ*↑, –*Ψ*↓,–) is the annihilation operator of an electron with spin (↑, ↓) and

and *σ*α=0,*x*,*y*,*<sup>z</sup>*

respectively; *v*<sup>F</sup> is the Fermi velocity; *μ* is the chemical potential of the Weyl–Dirac semimetal;

*<sup>D</sup>* is reduced by the large magnetic field. In a

*<sup>D</sup>* = 0) has no contribution from

*D*.

(21)

0.030 with increasing *ql*.

half‐width value ΔΩ of *T*<sup>e</sup>

*<sup>D</sup>*. The difference ΔΩ(*T*<sup>e</sup>

*<sup>L</sup>* **+ T**<sup>e</sup>

and the contribution from the nonlocal term *T*<sup>e</sup>

decreases and reaches zero. Then, the half‐width value ΔΩ(*T*<sup>e</sup>

**3. Spin pumping in a Weyl–Dirac semimetal**

exchange interaction is given by [50–52]

helicity τ(+, –) degrees of freedom; *τα*=0,*x*,*y*,*<sup>z</sup>*

The first term takes the form

<sup>D</sup> ≠ 0) – ΔΩ(*T*<sup>e</sup>

whereas the contribution from the applied magnetic field is negligible.

the TI because *T*<sup>e</sup>

80 Magnetic Materials

*T*e

**3.1. Model**

where *Ψk* = *<sup>t</sup>*

$$
\delta H\_{\epsilon\epsilon} = -J\_{\epsilon\epsilon} \int d\mathbf{x} \,\Psi^{\dagger} [\mathbf{r}^{0} \otimes \vec{S} \cdot \vec{\sigma}] \Psi,\tag{22}
$$

where *J*ex is the coefficient of the exchange interaction; and *S* is the localized spin that depends on space and time in the Weyl semimetal. This exchange interaction is independent of the helicity index. *H*imp shows the nonmagnetic impurity scattering in the Weyl–Dirac semimetal:

$$H\_{\rm imp} = \int d\mathbf{x} \,\,\mu\_{\rm imp} \,\mu^{\dagger} [\mathbf{r}^{0} \otimes \sigma^{0}] \mathcal{W}. \tag{23}$$

This Hamiltonian is similar to that on the surface of the TI—see Eq. (4). In the following calculations, *H*ex and *H*imp are treated as perturbations within the same formalism as laid out in Section 2.2. This treatment is allowed when the energy scale of the exchange interaction is smaller than the bandwidth of Eq. (21). Then, the low‐energy effective Hamiltonian in Eq. (21) gives a good approximation.

#### **3.2. Response function within the linear response to the exchange interaction**

To calculate the charge and spin current within the linear response to *H*ex, we use Green's functions. From Eq. (20) the charge current <*j i* > (*i* = *x, y, z*) can be defined by

$$
\langle \dot{f}^{\dagger}(\mathbf{x},t) \rangle = \epsilon v\_{\text{F}} \langle \psi^{\dagger}(\mathbf{x},t)(\boldsymbol{\tau}^{z} \otimes \boldsymbol{\sigma}^{\prime}) \psi(\mathbf{x},t) \rangle. \tag{24}
$$

where < > denotes the expectation value in Eq. (20). Such a charge current can be decomposed as <*j*> = <*j*+> + <*j*–>, where <*j*±> = ±*ev*F<*Ψ*† ±*σ*Ψ±> and *Ψ*±=*<sup>t</sup>* (*Ψ*↑, ±*Ψ*↓,±) is the annihilation operator around each helicity *τ* = ±1. Since there is no mixing between *Ψ*† ↓,+ and *Ψ*↑,– in the Hamiltonian in Eq. (20) the charge current around each helicity can be calculated separately. Because of spin–momentum locking in the Weyl–Dirac semimetal the charge current links to the spin polarization. The charge current of each helicity is proportional to the spin polarization of each helicity as

$$
\langle \dot{j}\_{\pm}^{\dagger}(\mathbf{x}, t) \rangle = \pm \frac{e\upsilon\_{\text{F}}}{2} \langle s\_{\pm}^{\dagger}(\mathbf{x}, t) \rangle. \tag{25}
$$

The spin current in the Weyl‐Dirac semimetal can be defined from the Heisenberg equation for the spin operator:

$$
\partial\_t \mathbf{s}\_a + \partial\_t \mathbf{j}\_{s,a}^\prime = T\_{\mathbf{r},a}, \tag{26}
$$

where *j*s,<sup>α</sup> *i* is the spin current operator; and *T*r,α is the spin relaxation term. Spin current density can also be decomposed into the spin current density of each helicity and can be calculated separately:

$$
\left< \left< j\_{s\alpha,\pi}^{\iota} \right> - \pm \frac{\hbar \nu\_F}{2} \delta\_\alpha^{\iota} \left< \psi\_\pi^{\dagger} \psi\_\pi \right> - \pm \frac{\hbar \nu\_F}{2e} \delta\_\alpha^{\iota} \left< \rho\_\pi \right>. \tag{27}
$$

The superscript and subscripts of *j*s,<sup>α</sup> *<sup>i</sup>* denote the direction of flow and spin of the spin current of each helicity, respectively. The direction of spin is perfectly parallel to that of the flow, and the spin current density is proportional to the charge density of each helicity.

The relaxation term can also be decomposed by <*T*r>= <*T*r,+> + <*T*r,–> and can be given by <*T*r,+>:

$$
\langle \left\langle T^{a}\_{\nu,a} \right\rangle - \mp \frac{i\hbar \nu\_{\rm F}}{2} \varepsilon^{a\#} \left\langle \psi\_{\rm a}^{\dagger} \sigma^{\theta} (\partial^{\prime} \psi\_{\rm a}) - (\partial^{\prime} \psi\_{\rm a}^{\dagger}) \sigma^{\theta} \psi\_{\rm a} \right\rangle. \tag{28}
$$

The charge and spin current can be obtained by calculating the response function (**Figure 4**):

**Figure 4.** Diagrammatic representation of the charge current of each helicity within the linear response to the exchange interaction. The bold line is Green's function; *vi* is the velocity operator; and *S<sup>j</sup>* is the localized spin (wavy lines). (a) A bubble diagram without taking impurity scattering from the exchange interactions into account. (b1) and (b2) Bubble diagrams that take impurity scattering within the ladder approximation into account.

$$\begin{split} \left< j\_{\pm}^{i} \right> &= \pm \frac{i e \hbar \nu\_{\rm F} J\_{\rm ex}}{L^{3}} \sum\_{\vec{q}, \Omega} e^{i(\Omega - \vec{q} \cdot \vec{x})} tr[\sigma^{i} \Pi\_{\pm}^{\prime}(\vec{q}, \Omega)] S\_{\frac{\vec{q}}{\vec{q}, \Omega}}^{\prime}, \\ \left< j\_{s, a \pm}^{i} \right> &= \mp \frac{i \hbar \nu\_{\rm F} J\_{\rm ex}}{2L^{3}} \delta\_{a}^{\prime} \sum\_{\vec{q}, \Omega} e^{i(\Omega - \vec{q} \cdot \vec{x})} tr[\Pi\_{\pm}^{\prime}(\vec{q}, \Omega)] S\_{\frac{\vec{q}}{\vec{q}, \Omega}}^{\prime}, \end{split} \tag{29}$$

where *q* and Ω are the momentum and frequency of the localized spin, respectively; *L*<sup>3</sup> is the volume of the system; and Π*<sup>j</sup>* ± is the response function within the helicity index:

$$\Gamma^{\prime}\_{\pm} \equiv n\_{\text{c}} \mu\_{\text{i}}^{2} \sum\_{\alpha, \mathbf{k}} g\_{\alpha - \alpha \mathbf{1}/2, \tilde{\mathbf{k}} - \mathbf{q}/2, \pm}^{\mathbf{R}} \sigma^{\prime} g\_{\alpha - \mathbf{1}/2, \tilde{\mathbf{k}} - \mathbf{q}/2, \pm}^{\mathbf{A}} = \Gamma^{\prime \mu}\_{\pm} \sigma^{\prime}. \tag{30}$$

#### Magnetization Dynamics–Induced Charge and Spin Transport on the Surface of a Topological Insulator Subjected to Magnetism http://dx.doi.org/10.5772/62531 83

where *fω*, *g<sup>ω</sup>*,*k,*<sup>±</sup> *<sup>R</sup>*= ([*g<sup>ω</sup>*,*k,*<sup>±</sup> *A*] † ) = [*hw*/(2*π*) + *μ*± – *hvFk.σ*/(2*π*) + *ih*/(4*πτ*±)]–1, and Λ*<sup>j</sup>* ± are the Fermi distribution function, the retarded (advanced) Green's function, and the vertex correction of the ladder diagram at each helicity ±, respectively; and *τ*± is the relaxation time of nonmagnetic impurities in the material, which depends on the helicity, because it depends on the chemical potential *μ*±. When Ω*τ*e,+ ≪ 1 and *qv*F*τ*e,+ ≪ 1 are satisfied the vertex function Λ*<sup>j</sup>* <sup>±</sup> is given by Λ*<sup>j</sup>* <sup>±</sup>= [Γ± + Γ<sup>±</sup> 2 + Γ<sup>±</sup> 3 + …]*jnσ<sup>n</sup>* = [(1 – Γ±) –1] *jnσ<sup>n</sup>* with

$$\Gamma^{'}\_{\pm} \equiv \mathfrak{u}\_{c} \mathfrak{u}\_{l}^{'} \sum\_{a \ll \Omega} \operatorname{\mathfrak{g}}^{R}\_{a \sim \Omega/2, \stackrel{\circ}{\mathbf{k}} - \stackrel{\circ}{\mathbf{q}}/2, \pm} \mathfrak{o}^{'} \mathfrak{g}^{A}\_{a \approx \Omega/2, \stackrel{\circ}{\mathbf{k}} \stackrel{\circ}{\mathbf{q}}/2, \pm} = \Gamma^{' \mu}\_{\pm} \mathfrak{o}^{\mu} \tag{31}$$

where Γ*jn* ± (||Γ*jn* ±|| < 1) is a 4 × 4 matrix (*j*, *n* = 0*, x, y, z*). As a result, Π*<sup>j</sup>* ± can be represented by

$$
\begin{pmatrix}
\Pi^{0}\_{\pm} \\
\Pi^{x}\_{\pm} \\
\Pi^{y}\_{\pm} \\
\Pi^{y}\_{\pm} \\
\Pi^{z}\_{\pm}
\end{pmatrix} = \begin{pmatrix}
\Pi^{00}\_{\pm} & \Pi^{0x}\_{\pm} & \Pi^{0y}\_{\pm} & \Pi^{0z}\_{\pm} \\
\Pi^{x0}\_{\pm} & \Pi^{xx}\_{\pm} & \Pi^{xy}\_{\pm} & \Pi^{xz}\_{\pm} \\
\Pi^{y0}\_{\pm} & \Pi^{yx}\_{\pm} & \Pi^{yy}\_{\pm} & \Pi^{yz}\_{\pm} \\
\Pi^{z0}\_{\pm} & \Pi^{zx}\_{\pm} & \Pi^{zy}\_{\pm} & \Pi^{zz}\_{\pm}
\end{pmatrix} \begin{pmatrix}
\sigma^{0} \\
\sigma^{x} \\
\sigma^{y} \\
\sigma^{z}
\end{pmatrix}.
\tag{32}
$$

where the above matrix component (*m*, *m*'= *x, y, z*) is

, , ,

can also be decomposed into the spin current density of each helicity and can be calculated

of each helicity, respectively. The direction of spin is perfectly parallel to that of the flow, and

The relaxation term can also be decomposed by <*T*r>= <*T*r,+> + <*T*r,–> and can be given by <*T*r,+>:

The charge and spin current can be obtained by calculating the response function (**Figure 4**):

**Figure 4.** Diagrammatic representation of the charge current of each helicity within the linear response to the exchange

bubble diagram without taking impurity scattering from the exchange interactions into account. (b1) and (b2) Bubble

<sup>3</sup> , q,

W- ×

s

<sup>q</sup> <sup>q</sup>

W W

[ ( , )] , <sup>2</sup>

*t x jq*

r r

q

r

.

 s

r

[ ( , )] ,

r

*q*

± is the response function within the helicity index:

, /2,

*<sup>j</sup> RA n <sup>j</sup> jn gn gu* (30)

r

*q*

diagrams that take impurity scattering within the ladder approximation into account.

h

h m

2

,

k i

w

w

*L*

*L*

F ex ( )

å

*r*

 a

± ±

r

d

F ex ( ) s, . <sup>3</sup> , ,

*i vJ <sup>j</sup> e tr S*

å

= P W

where *q* and Ω are the momentum and frequency of the localized spin, respectively; *L*<sup>3</sup>

/2, /2, / 2

± ± s


 w

*<sup>j</sup> <sup>i</sup> i i*

W- × ± ± <sup>W</sup> <sup>W</sup>

*<sup>j</sup> <sup>i</sup> it x i j*

r

*q*

r

= ± P W

*ie v J <sup>j</sup> e tr <sup>S</sup>*

is the velocity operator; and *S<sup>j</sup>*

is the spin current operator; and *T*r,α is the spin relaxation term. Spin current density

*<sup>i</sup> s jT* (26)

*<sup>i</sup>* denote the direction of flow and spin of the spin current

(27)

(28)

(29)

is the

is the localized spin (wavy lines). (a) A

 aa

¶ +¶ = t is r

a

the spin current density is proportional to the charge density of each helicity.

where *j*s,<sup>α</sup>

82 Magnetic Materials

separately:

*i*

The superscript and subscripts of *j*s,<sup>α</sup>

interaction. The bold line is Green's function; *vi*

volume of the system; and Π*<sup>j</sup>*

a

$$\begin{split} \Pi^{00}\_{\pm} &= -\frac{\Omega}{4\pi} \frac{2 + i\Omega\tau\_{\rm e,\pm}}{q^{2} v\_{\rm r}^{2} \tau\_{\rm e,\pm} / 2 + i\Omega}, \\ \Pi^{\rm m0}\_{\pm} = \Pi^{\rm m}\_{\pm} &= \mp \frac{\Omega}{4\pi} \frac{i\nu\_{\rm F} q^{\rm m}}{q^{2} v\_{\rm r}^{2} \tau\_{\rm e,\pm} / 2 + i\Omega} \\ \Pi^{\rm mw}\_{\pm} &= -\frac{\Omega}{4\pi} \bigg( 1 - \frac{v\_{\rm r}^{2} \tau\_{\rm e,\pm}^{2} q^{\rm w} q^{\rm w}}{q^{2} v\_{\rm r}^{2} \tau\_{\rm e,\pm} / 2 + i\Omega} \bigg). \end{split} \tag{33}$$

As a result, the charge current and spin current can be obtained by

$$
\langle \zeta\_{\pm}^{\dagger} \rangle = \pm \frac{e \upsilon\_{\text{F}} J\_{\text{ox}} \nu\_{\pm}^{\text{WS}} \tau\_{a,\pm}}{2} \partial\_{\text{i}} \vec{S} - \frac{1}{6} \upsilon\_{\text{F}}^{2} \tau\_{\pm} \vec{\nabla} \langle \rho\_{\pm} \rangle,\tag{34}
$$

$$
\langle \dot{j}\_{s,\ a,\pm}^{\parallel} \rangle = \pm \frac{\hbar v\_{\text{F}}}{\mathfrak{D}e} \delta\_{\alpha}^{\parallel} \langle \rho\_{\pm} \rangle,\tag{35}
$$

where *νWS*± and <*ρ*±> are the charge density of state and the induced charge density of each helicity, respectively:

$$\langle \rho\_{\pm} \rangle = \mp \frac{1}{2} e \upsilon\_{\mathbb{H}} J\_{\text{ox}} \nu\_{\pm}^{\text{WS}} \tau\_{\text{e},\pm} \left( \vec{\nabla} \cdot \vec{\mathcal{O}}\_{\text{t}} \langle \vec{S} \rangle\_{\text{D},\pm} \right) \tag{36}$$

where <*S*>D,± is the convolution between the localized spin and diffusive propagation of each helicity. <*S*>D,± can be defined by

$$\begin{split} \langle \vec{S} \rangle\_{\rm D,\pm} &\equiv \frac{1}{\tau\_{e,\pm}} \Big| \int\_{\rm \mathbb{R}} dt' \Big[ \mathrm{d}x' D\_{\mathrm{W},\pm}(\mathbf{x} - \mathbf{x}', t - t') \vec{S}(\mathbf{x}', t'), \\ \langle D\_{\mathrm{W},\pm}(\mathbf{x} - \mathbf{x}', t - t') \rangle &= \frac{1}{L^{\dagger}} \sum\_{\vec{q}, \Omega} \frac{1}{q^{2} v\_{\mathrm{F}} \tau\_{e,\pm} / \sqrt{2 + i\Omega}} e^{i(\Omega t - \vec{q} \cdot \vec{x})} . \end{split} \tag{37}$$

Note that the above results are obtained when *b* = 0. These results are easily generalized when *b* = 0. Because *b* behaves like a static Zeeman field acting on the whole of the band of *H*Weyl, it can shift as a result of Pauli paramagnetism. However, *b* cannot drive a net current because *b* is static [49]. On the other hand, the dynamics of the localized spin are only effective near the Fermi surface, the structure of which does not depend on *b*. Therefore, we obtain the same charge and spin current in Eqs. (33)–(35) even when *b* ≠ 0.

#### **3.3. Charge and spin current due to spin pumping effects**

From the above results the total charge current can be given by

$$
\langle \dot{j}^i \rangle = \frac{\varepsilon \upsilon\_{\rm F} I\_{\rm ex}}{2} (\nu\_{\rm \ast}^{\rm WS} \tau\_{\rm e,+} - \nu\_{-}^{\rm WS} \tau\_{\rm e,-}) \hat{\mathcal{O}}\_i S^i + \frac{\varepsilon \upsilon\_{\rm F}^3 I\_{\rm ex}}{12} \nabla^i \left[ \vec{\nabla} \cdot \hat{\mathcal{O}}\_i \left( \nu\_{+}^{\rm WS} \tau\_{\rm e,+}^2 \langle \vec{S} \rangle\_{\rm D,+} - \nu\_{-}^{\rm WS} \tau\_{\rm e,-}^2 \langle \vec{S} \rangle\_{\rm D,-} \right) \right]. \tag{38}
$$

The charge current is triggered by the dynamics of localized spin. The first term indicates the local term of the dynamics of localized spin. Its direction is parallel to ∂<sup>t</sup> **S**. The second term shows the nonlocal term, which is generated by the convolution between localized spin and the diffusion propagator. A nonzero nonlocal term is given when the spin texture of localized spin depends on space. Thus, the driving force needed to induce the charge current is the same as localized spin dynamics, which plays the role of driving the charge current on the surface of the TI. Note that the charge current in a Weyl–Dirac semimetal depends strongly on the valley index. When the difference between *τ*+ and *τ*– as well as between *νWS*+ and *νWS*– is realized, there is a population imbalance between two of the bands of each helicity. Then, nonzero <*j*> is induced when *νWS*+*τ*e,+ ≠ *νWS*–*τ*e,– or *μ*<sup>5</sup> = 0.

Note that the property of the charge current at each helicity links to that of spin polarization because of spin–momentum locking. However, after summation of the indices of helicity the relation between total charge current and total spin polarization is changed. As a result, even in the absence of population imbalance, nonzero spin polarization can be given by Eq. (25) as Magnetization Dynamics–Induced Charge and Spin Transport on the Surface of a Topological Insulator Subjected to Magnetism http://dx.doi.org/10.5772/62531 85

$$
\langle \mathbf{s}' \rangle = \frac{J\_{\infty}}{2} (\boldsymbol{\nu}\_{\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{\text{WS}}}} \boldsymbol{\tau}\_{\mathrm{\boldsymbol{a}},\mathrm{\boldsymbol{s}}} + \boldsymbol{\nu}\_{\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{\text{WS}}}} \boldsymbol{\tau}\_{\mathrm{\boldsymbol{a}},\mathrm{\boldsymbol{s}}}) \hat{\boldsymbol{\alpha}}\_{\mathrm{i}}^{\mathrm{\boldsymbol{s}}} + \frac{\boldsymbol{\nu}\_{\mathrm{\boldsymbol{s}}}^{2} \boldsymbol{I}\_{\mathrm{ox}}}{24} \nabla \left[ \vec{\boldsymbol{\nabla}} \cdot \boldsymbol{\mathcal{o}}\_{\mathrm{i}} \left( \boldsymbol{\nu}\_{\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{\text{WS}}}} \boldsymbol{\tau}\_{\mathrm{\boldsymbol{a}},\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{s}}} \boldsymbol{\langle \boldsymbol{\bar{S}} \rangle\_{\mathrm{D},\mathrm{s}}} + \boldsymbol{\nu}\_{\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{s}} \mathrm{\boldsymbol{S}}} \boldsymbol{\tau}\_{\mathrm{\boldsymbol{a}},\mathrm{\boldsymbol{s}}}^{\mathrm{\boldsymbol{s}}} \boldsymbol{\langle \bar{\boldsymbol{S}} \rangle\_{\mathrm{D},\mathrm{s}}} \right) \right]. \tag{39}
$$

Local spin polarization is along *∂*<sup>t</sup> *S* and nonlocal spin polarization is along the spatial gradient of *∂*<sup>t</sup> **(***∂i Si* ).

Total spin current can be represented from Eqs. (35) and (36) as

$$
\langle \dot{j}\_{s,\
u}^{\flat} \rangle = -\frac{\hbar v\_{\rm F}^{2} I\_{\rm ex}}{4} \boldsymbol{\delta}\_{\rm a}^{\flat} \vec{\nabla} \cdot \boldsymbol{\mathcal{O}}\_{\rm t} \Big[ \boldsymbol{\nu}\_{+}^{\mathrm{WS}} \boldsymbol{\tau}\_{\rm e,+} \langle \vec{\boldsymbol{S}} \rangle\_{\rm D,+} + \boldsymbol{\nu}\_{-}^{\mathrm{WS}} \boldsymbol{\tau}\_{\rm e,-} \langle \vec{\boldsymbol{S}} \rangle\_{\rm D,-} \Big]. \tag{40}
$$

Total spin current can be generated by spatial divergence of localized spin dynamics, where localized spin is the convolution with diffusion. As a result, the spin current can be regarded as a nonlocal spin current. Note that a nonzero spin current is generated when localized spin depends on space. Such a spin current becomes nonzero even in the absence of population imbalance.

The nonlocal spin current in this case is obtained from the diffusive motion of spin density, which is driven by the dynamics of localized spin, where localized spin depends on space. The spin diffusive motion of each helicity can be given from Eqs. (25), (34), (37) as

$$
\left(\partial\_{\mathbf{t}} - \frac{1}{2} \boldsymbol{\upsilon}\_{\mathrm{F}}^{2} \boldsymbol{\tau}\_{\mathrm{e},\pm}\right) \langle \boldsymbol{s}\_{\pm}^{\prime} \rangle = \frac{\boldsymbol{e} \boldsymbol{v}\_{\mathrm{F}}^{\mathrm{\top}} \boldsymbol{\tau}\_{\mathrm{e},\pm}^{2} \boldsymbol{J}\_{\mathrm{e}\mathbf{x}}}{2} \boldsymbol{\nu}\_{\pm}^{\mathrm{\prime}\mathrm{NS}} \boldsymbol{\partial}\_{\mathrm{t}} \nabla(\nabla \cdot \mathbf{S}).\tag{41}
$$

As a result, the diffusive motion of total spin becomes:

$$
\left[\hat{\mathcal{O}}\_t\left(\langle \mathbf{s}\_+^{\langle}\rangle + \langle \mathbf{s}\_-^{\langle}\rangle\right) - \frac{1}{2}\sigma\_\mathbf{F}^2\Big(\tau\_{\mathbf{e},\*}\langle \mathbf{s}\_+^{\langle}\rangle + \tau\_{\mathbf{e},\*}\langle \mathbf{s}\_-^{\langle}\rangle\right)\right] = \frac{e\upsilon\_\mathbf{F}^{\dagger}l\_\mathrm{ox}}{2} (\tau\_+^2 \nu\_{\mathbf{e},\*}^{\mathrm{WS}} + \tau\_-^2 \nu\_{\mathbf{e},-}^{\mathrm{WS}}) \hat{\mathcal{O}}\_t \nabla(\nabla \cdot \mathbf{S})\tag{42}
$$

Time‐dependent and spatial‐dependent localized spin, *∂*[*∂*<sup>t</sup> (*∂i Si* )], triggers the diffusive motion of total spin density. When there is no population imbalance between each helicity, diffusive motion can accompany total spin density without any charge flow. Hence, a pure spin current can be generated.

#### **4. Conclusion**

, , ( ) <sup>1</sup>

where <*S*>D,± is the convolution between the localized spin and diffusive propagation of each

, ,

W ±

Note that the above results are obtained when *b* = 0. These results are easily generalized when *b* = 0. Because *b* behaves like a static Zeeman field acting on the whole of the band of *H*Weyl, it can shift as a result of Pauli paramagnetism. However, *b* cannot drive a net current because *b* is static [49]. On the other hand, the dynamics of the localized spin are only effective near the Fermi surface, the structure of which does not depend on *b*. Therefore, we obtain the same

, , ,, ,, ( ) . <sup>2</sup> <sup>12</sup>

The charge current is triggered by the dynamics of localized spin. The first term indicates the

shows the nonlocal term, which is generated by the convolution between localized spin and the diffusion propagator. A nonzero nonlocal term is given when the spin texture of localized spin depends on space. Thus, the driving force needed to induce the charge current is the same as localized spin dynamics, which plays the role of driving the charge current on the surface of the TI. Note that the charge current in a Weyl–Dirac semimetal depends strongly on the valley index. When the difference between *τ*+ and *τ*– as well as between *νWS*+ and *νWS*– is realized, there is a population imbalance between two of the bands of each helicity. Then, nonzero <*j*>

Note that the property of the charge current at each helicity links to that of spin polarization because of spin–momentum locking. However, after summation of the indices of helicity the relation between total charge current and total spin polarization is changed. As a result, even in the absence of population imbalance, nonzero spin polarization can be given by Eq. (25) as

e e eD eD

*ev J ev J <sup>j</sup> <sup>S</sup> S S* (38)

 ++-- ++ +-- - é ù á ñ= - ¶ + Ñ Ñ×¶ á ñ - á ñ ë û r r r F ex F ex

*i WS WS i i WS WS t t*

local term of the dynamics of localized spin. Its direction is parallel to ∂<sup>t</sup>

= 0.

t

*L qv i*

å r r

1 1 ( ', ') . / 2

q F e


r

*D x xt t e*

*S dt dx D x x t t S x t*

<sup>1</sup> ' ' ( ', ') ( ', '),

*<sup>e</sup> ev J S* (36)

( )

(37)

W- ×

*i t qx*

( ) <sup>3</sup>

nt

2 2

 nt

**S**. The second term

2

, , ,

áñ º ò - -

r r

¥ ± ± ± -¥

ò

D W e

, 3 2

 nt <sup>±</sup> ± ± <sup>±</sup> á ñ= Ñ×¶ á ñ <sup>r</sup> <sup>r</sup> <sup>m</sup> F ex t D *WS*

r

t

charge and spin current in Eqs. (33)–(35) even when *b* ≠ 0.

**3.3. Charge and spin current due to spin pumping effects**

From the above results the total charge current can be given by

±

W

nt

is induced when *νWS*+*τ*e,+ ≠ *νWS*–*τ*e,– or *μ*<sup>5</sup>

 nt

helicity. <*S*>D,± can be defined by

84 Magnetic Materials

Our results on the charge and spin current due to spin pumping on the surface of a three‐ dimensional TI (Section 2) [28] and in the bulk of a three‐dimensional Weyl–Dirac semimetal (Section 3) [51] are summarized in this chapter.

Section 2 summarizes our results on spin pumping on the surface of a TI attached to an MI. The results are calculated using the standard Keldysh–Green function method within the linear response to the exchange interaction between the conduction spin and localized spin of an MI. The purpose of this work is to derive charge and spin current generation due to localized spin dynamics on the disordered surface of a TI; in particular, when the localized spin depends on space on the surface. The main results on the surface of a TI are summarized in **Table 1**. Time‐ dependent localized spin on the surface is a prerequisite to obtaining nonzero charge and spin current generation. Moreover, **Table 1** shows that when the spin texture is spatially inhomo‐ geneous, not only the local charge but also the nonlocal charge and spin current are generated by time‐dependent localized spin. The flow and spin polarization of the spin current are perfectly perpendicular to each other because of spin–momentum locking. The magnitude of the spin current is proportional to the charge density, which is induced by divergence between time‐dependent localized spin and the diffusive propagator on the surface—see Eqs. (10) and (11). Such pumping effects are caused by time‐dependent localized spin, which plays a role in driving the charge current and can be regarded as an effective electric field *E*<sup>s</sup> TI.


**Table 1.** Brief summary of the charge current, spin current, spin, and charge density induced by localized spin dynamics on the disordered surface of a TI. The charge current and spin density have both a local and nonlocal contribution. The spin current and charge density are described by the nonlocal contribution.

Recently, it has been reported that the localized spin texture at the junction of the TI/MI is spatially inhomogeneous [34]. We suppose that the spin current we have obtained at the junction is generated when the spin texture moves temporally.

On the basis of these results, in Section 2.3 we discussed a way of detecting the charge current and spin current induced on the surface of a TI attached to an MI by using ferromagnetic resonance. We assume that the dynamics of localized spin is triggered by the applied static and AC magnetic field of the microwave. The dynamics of the localized spin induced both the charge current and the spin current. Such induced currents are related to spin density, and spin polarization acts on the localized spin in much the same way as spin torque—see Eq. (14). Hence, the half‐width of ferromagnetic resonance changes as shown in **Figure 3**.

Spin pumping in a Weyl–Dirac semimetal hosting massless Weyl–Dirac fermions is summar‐ ized in section 3. The results are obtained within the same formalism as laid out in Sections 2.1 and 2.2. The charge and spin current as well as the charge and spin density are given in **Table 2**. Semimetals are subject to spin–momentum locking. The spin direction of Weyl–Dirac fermions brought about by spin–momentum locking is perfectly parallel/antiparallel to its momentum and its locking is determined by the helicity degrees of freedom of Weyl–Dirac fermions. As a result, the charge current and spin polarization induced depend on the helicity indices. Eqs. (38) and (39) show that localized spin dynamics induces the charge current and spin polarization, respectively; hence, localized spin plays the role of an effective electric field *E*s,±WDS acting on electrons [51]. Moreover, *E*s,±WDS depends on the helicity index, whereas total charge current is proportional to *μ*<sup>5</sup> . As a result, a nonzero charge current is generated when there is population imbalance between each helicity. On the other hand, localized spin dynamics also drives the spin current when localized spin depends on time and position in a Weyl–Dirac semimetal. The spin current is finite even in the absence of population imbalance. Then, the spin current does not accompany charge flow. These results may be of use to next‐ generation spintronics devices based on Weyl–Dirac semimetals.


**Table 2.** Brief summary on charge and spin current generation by spin pumping in a Weyl–Dirac semimetal with the dynamics of localized spin, where ± denotes the helicity index. Because of spin–momentum locking the charge (spin) current is proportional to the spin (charge) density with each helicity. Time‐dependent localized spin drives the local and nonlocal charge and spin current. The nonzero population imbalance between each helicity *μ*<sup>5</sup> is a prerequisite to obtaining the nonzero charge current. The nonzero spin current (charge density) is triggered by time‐dependent and spatial‐dependent localized spin dynamics even when *μ*<sup>5</sup> = 0.

## **Acknowledgements**

Section 2 summarizes our results on spin pumping on the surface of a TI attached to an MI. The results are calculated using the standard Keldysh–Green function method within the linear response to the exchange interaction between the conduction spin and localized spin of an MI. The purpose of this work is to derive charge and spin current generation due to localized spin dynamics on the disordered surface of a TI; in particular, when the localized spin depends on space on the surface. The main results on the surface of a TI are summarized in **Table 1**. Time‐ dependent localized spin on the surface is a prerequisite to obtaining nonzero charge and spin current generation. Moreover, **Table 1** shows that when the spin texture is spatially inhomo‐ geneous, not only the local charge but also the nonlocal charge and spin current are generated by time‐dependent localized spin. The flow and spin polarization of the spin current are perfectly perpendicular to each other because of spin–momentum locking. The magnitude of the spin current is proportional to the charge density, which is induced by divergence between time‐dependent localized spin and the diffusive propagator on the surface—see Eqs. (10) and (11). Such pumping effects are caused by time‐dependent localized spin, which plays a role in

driving the charge current and can be regarded as an effective electric field *E*<sup>s</sup>

*S*׀׀*∂* <sup>t</sup>

∇ ×*ε<sup>s</sup>*

contribution. The spin current and charge density are described by the nonlocal contribution.

junction is generated when the spin texture moves temporally.

׀׀*S*

**Table 1.** Brief summary of the charge current, spin current, spin, and charge density induced by localized spin dynamics on the disordered surface of a TI. The charge current and spin density have both a local and nonlocal

<sup>D</sup>]<sup>z</sup> ∇[∇ × *∂*<sup>t</sup>

TI*<sup>D</sup> <sup>z</sup> <sup>z</sup>* <sup>×</sup>*ε<sup>s</sup>*

Recently, it has been reported that the localized spin texture at the junction of the TI/MI is spatially inhomogeneous [34]. We suppose that the spin current we have obtained at the

On the basis of these results, in Section 2.3 we discussed a way of detecting the charge current and spin current induced on the surface of a TI attached to an MI by using ferromagnetic resonance. We assume that the dynamics of localized spin is triggered by the applied static and AC magnetic field of the microwave. The dynamics of the localized spin induced both the charge current and the spin current. Such induced currents are related to spin density, and spin polarization acts on the localized spin in much the same way as spin torque—see Eq. (14). Hence, the half‐width of ferromagnetic resonance changes as shown in **Figure 3**.

<sup>D</sup>]<sup>z</sup> *∈*zαi[∇ × *∂*<sup>t</sup>

〈*j*〉 *js***,***<sup>α</sup>*

׀׀*S*

TI D z

*z* ×*ε<sup>s</sup>* TI

∇ ∇ × *ε*<sup>s</sup>

Local z × *∂*<sup>t</sup>

*evF* ∂*<sup>t</sup>* s ||

Driving force

86 Magnetic Materials

*εs* TI≡ − *<sup>J</sup>* ex

Nonlocal ∇[∇ × *∂*<sup>t</sup>

TI.

<sup>D</sup>]<sup>z</sup> This work [28]

TI*<sup>D</sup> <sup>z</sup>* This work [28]

*i* 〈*s*〉 〈*p*〉 **Ref.**

׀׀*S*

TI D Z

TI

∇ ∇ × *ε<sup>s</sup>*

[31‐29] ׀׀*S*

׀׀*S*

∇ ×*ε<sup>s</sup>*

<sup>D</sup>]<sup>z</sup> [∇ × *∂*<sup>t</sup>

This work was supported by a Grant‐in‐Aid for JSPS Fellows (Grant No. 13J03141), by a Grant‐ in‐Aid for Challenging Exploratory Research (Grant No. 15K13498), and by the Core Research for Evolutional Science and Technology (CREST) of Japan Science.

## **Author details**

Katsuhisa Taguchi

Address all correspondence to: taguchi@rover.nuap.nagoya-u.ac.jp

Department of Applied Physics, Nagoya University, Nagoya, Japan

## **References**


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**Author details**

88 Magnetic Materials

Katsuhisa Taguchi

**References**

10.1063/1.2199473

10.7566/JPSJ.82.102001

nature08234

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10.1063/1.4751255

Address all correspondence to: taguchi@rover.nuap.nagoya-u.ac.jp

Department of Applied Physics, Nagoya University, Nagoya, Japan

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0.034, and 0.030, respectively. Figure 3(c) in this chapter is revised.]

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111.166801

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108.187201

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10.1103/PhysRevB.66.104413

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