## 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 momentum of a magnetization's localized spin into that of electrons through the dynamics of magnetization as well as spin-orbit interactions [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 experimentally 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**×∂

_{t}

**[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].**

*S*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/Sb_{2}Te_{3} 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).

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

### 2.1. Model

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]:

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

where *v*_{F},_{B}, *E*_{F}, * k*, and

**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**

*σ**Ψ*

_{k}=

*(*

^{t}*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

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*_{F} 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

(5) |

As a result of spin–momentum locking the charge current <*j ^{i}*> and spin current <

*j*> are proportional to the spin density <

^{i}_{s,α}*s*> and charge density <

^{i}*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.

The charge and spin current are represented by

(6) |

where ** q** and Ω are the momentum and frequency of the localized spin

*S*

^{j}_{q,}

_{Ω}, respectively;

*L*

^{2}is the area of the surface; and Π

*is the response function. These currents are obtained from Π*

^{j}*:*

^{j}where *f*_{ω}, *g*_{ω},_{k}^{R} = ([*g*_{ω,k}^{A}]^{†}) = [*hw*/(2π) + *E*_{F} – *hv*_{F}/(2π)(** k** ×

**)**

*σ*_{z}+

*ih*/(4

*πτ*)]

^{‐1}, and Λ

^{j}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*

_{F}τ ≪ 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

^{j}is represented by Λ

^{j}= [Γ + Γ

^{2}+ Γ

^{3}+ …]

^{jn}σ

^{n}= [ (1 – Γ)

^{‐1}]

^{jn}σ

^{n}with

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

(9) |

where *σ*^{0} 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

(10) |

where *ν*^{TI}_{e} 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** –

*S*as

^{z}**z**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** ×

*∂*_{t}

*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 [*

**S**^{||}**× (**

*z***×**

*z***)] [**

*∂***×**

*∂*

*∂*_{t}<

*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. Therefore, 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***>). Thus, the properties of spin polarization are similar to those of the charge current.**

*s* 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*

^{TI}

_{e}

*J*

_{ex}

*τl*

^{2}[

**× (**

*z***×**

*z***)] [**

*∂***× (**

*∂*

*∂*_{t}<

*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

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 *S ^{z}*

**z**contributes in a dominant way to charge and spin current generation. The reason is

*σ*does not couple with momentum

^{z}**in Eq. (2). Therefore, we believe a warping effect [1–4] on the surface of the TI is likely. Spin polarization along the**

*p**z*‐direction is also generated by spin dynamics.

### 2.3. Spin torque

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***>), 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:**

*s*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

**. The second term is the damping torque that prevents its dynamics—these terms are schematically illustrated in**

*M***Figure 2(b)**. The third term

*T*_{e}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*

^{2}(

**<**

*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*_{e}

^{L}) and nonlocal spin torque (

*T*_{e}

^{D}) terms as

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<*_{e}

*∼*

^{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:

We now consider magnetization when it is given by ** M** = (

*M*

_{x},

*M*

_{y},

*M*

_{z}) = [

*m*cos(

**– Ω**

*q.x**t*),

*M*,

_{y}*m*sin(

**– Ω**

*q.x**t*)] with

**= (**

*q**q*

_{x},

*q*

_{y}),

*m*

^{2}≪

*M*

_{y},

*M*

_{y}∼

*M*, and ∂

_{t}

*M*

_{y}∼ 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*

_{z}≪

*H*

_{0}. Then, substituting this

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

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

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

(18) |

The coefficients *κ '_{q}_{,}*

_{Ω}and

*ζ*

_{q},_{Ω}are given by

Hence, *κ '_{q}_{,}*

_{Ω}and

*ζ*

_{q,}_{Ω}depend on

*q*and Ω. If

*q*= 0,

*κ*

**'**_{q}_{,}_{Ω}and

*ζ*

_{q,}_{Ω}go to

*κ*

**'**_{q}_{,}_{Ω}→

*κ*and

*ζ*

_{q,}_{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*

_{H}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*=

*q*where

_{0}l,*q*

_{0}satisfies

*q*

_{0}

^{2}

*l*

^{2}∼ Ω

*τ*. In 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*

_{H}=2π

*f*

_{H}frequencies when realistic parameters on the surface of the TI are taken into account [31]. Change in the normalized half‐width value ΔΩ

*/ω*(

_{H}*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*_{e}^{L}**+ T**_{e}* ^{D}* is proportional to the induced charge current [29–31]. In addition, the half‐width value ΔΩ of

*T*_{e}

*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*

^{D}*ql*, ΔΩ includes the contribution from the whole of the spin torque

*T*_{e}and the contribution from the nonlocal term

*T*_{e}

*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*

^{D}*ql*decreases and reaches zero. Then, the half‐width value ΔΩ(

*T*_{e}

*= 0) has no contribution from*

^{D}

*T*_{e}

*. The difference ΔΩ(*

^{D}

*T*_{e}

^{D}≠ 0) – ΔΩ(

*T*_{e}

*= 0) includes a pure contribution from*

^{D}

*T*_{e}

*.*

^{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]

The first term takes the form

where *Ψ _{k}* =

*(*

^{t}*Ψ*

_{↑, +}

*Ψ*

_{↓, +}

*Ψ*

_{↑, –}

*Ψ*

_{↓,–}) is the annihilation operator of an electron with spin (↑, ↓) and helicity τ(+, –) degrees of freedom;

*τ*

^{α}^{=0,x,y,z}and

*σ*

^{α=0,x,y,z}are Pauli matrices of the helicity and spin, respectively;

*v*

_{F}is the Fermi velocity;

*μ*is the chemical potential of the Weyl–Dirac semimetal; and

**and**

*b**μ*

^{5}denote the difference in position between each Weyl cone in momentum and energy space, respectively. In Dirac semimetals, we set

**= 0 and**

*b**μ*

^{5}= 0.

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

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:

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

where < > denotes the expectation value in Eq. (20). Such a charge current can be decomposed as <** j**> = <

*j*_{+}> + <

*j*_{–}>, where <

*j*_{±}> = ±

*ev*

_{F}<

*Ψ*

^{†}

_{±}

**Ψ**

*σ*_{±}> and

*Ψ*

_{±}=

*(*

^{t}*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

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

where *j*_{s},_{α}* ^{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:

The superscript and subscripts of *j*_{s},_{α}* ^{i}* 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}_{,+}>:

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

(29) |

where ** q** and Ω are the momentum and frequency of the localized spin, respectively;

*L*

^{3}is the volume of the system; and Π

^{j}_{±}is the response function within the helicity index:

where *f _{ω}*,

*g*

_{ω}_{,k,±}

^{R}= ([

*g*

_{ω}_{,k,±}

^{A}]

^{†}) = [

*hw*/(2

*π*) +

*μ*

_{±}–

*hv*/(2

_{F}**k.σ***π*) +

*ih*/(4

*πτ*

_{±})]

^{–1}, and Λ

^{j}

_{±}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

^{j}

_{±}is given by Λ

^{j}

_{±}= [Γ

_{±}+ Γ

_{±}

^{2}+ Γ

_{±}

^{3}+ …]

^{jn}

*σ*

^{n}= [(1 – Γ

_{±})

^{–1}]

^{jn}

*σ*

^{n}with

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

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

(33) |

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

where *ν ^{WS}*

_{±}and <

*ρ*

_{±}> are the charge density of state and the induced charge density of each helicity, respectively:

where <** S**>

_{D,±}is the convolution between the localized spin and diffusive propagation of each helicity. <

**>**

*S*_{D,±}can be defined by

(37) |

Note that the above results are obtained when ** b** = 0. These results are easily generalized when

**= 0. Because**

*b***behaves like a static Zeeman field acting on the whole of the band of**

*b**H*

_{Weyl}, it can shift as a result of Pauli paramagnetism. However,

**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.**

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

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

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 ∂_{t}**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 *ν ^{WS}*

_{+}and

*ν*

^{WS}_{–}is realized, there is a population imbalance between two of the bands of each helicity. Then, nonzero <

**> is induced when**

*j**ν*

^{WS}_{+}

*τ*

_{e}

_{,+}≠

*ν*

^{WS}_{–}

*τ*

_{e}

_{,–}or

*μ*

^{5}= 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

Local spin polarization is along *∂*_{t}** S** and nonlocal spin polarization is along the spatial gradient of

**∂**_{t}

**(**

*∂*).

^{i}S^{i}Total spin current can be represented from Eqs. (35) and (36) as

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

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

Time‐dependent and spatial‐dependent localized spin, ** ∂**[

*∂*_{t}(

*∂*)], 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.

^{i}S^{i}## 4. Conclusion

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 inhomogeneous, 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*_{s}^{TI}.

〈j〉 | 〈s〉 | 〈p〉 | Ref. | ||
---|---|---|---|---|---|

Local | z × ∂_{t}S^{׀׀} | ∂_{t}S^{׀׀} | [29‐31] | ||

Nonlocal | ∇[∇ × ∂_{t}S^{׀׀}_{D}]_{z} | ∈^{zαi}[∇ × ∂_{t}S^{׀׀}_{D}]_{z} | ∇[∇ × ∂_{t}S^{׀׀}_{D}]_{z} | [∇ × ∂_{t}S^{׀׀}_{D}]_{z} | This work [28] |

Driving force | This work [28] |

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 summarized 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*^{5}. 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.

〈j _{±}〉α±〈S _{±}〉 | Ref. | ||
---|---|---|---|

Local | [51] | ||

Nonlocal | δ ∇·_{i,α}∂_{t}〈s〉_{D,±} | [51] | |

Driving force | This work [51] | ||

Total | 〈j〉_{local} α μ^{5} ∂_{t S}〈 j〉_{nonlocal} α ∂_{t} ∇[∇·(〈s〉_{D,+} ‐ 〈S〉_{D,‐})] | This work |