Rashba-induced spin electromagnetic fields

# Rashba-induced spin electromagnetic fields in a strong sd coupling regime

## Abstract

Spin electromagnetic fields driven by the Rashba spin-orbit interaction, or Rashba-induced spin Berry’s phase, in ferromagnetic metals is theoretically studied based on the Keldysh Green’s function method. Considering a limit of strong sd coupling without spin relaxation (adiabatic limit), the spin electric and magnetic fields are determined by calculating transport properties. The spin electromagnetic fields turn out to be expressed in terms of a Rashba-induced effective vector potential, and thus they satisfy the Maxwell’s equation. In contrast to the conventional spin Berry’s phase, the Rashba-induced one is linear in the gradient of magnetization profile, and thus can be extremely large even for slowly varying structures. We show that the Rashba-induced spin Berry’s phase exerts the Lorentz force on spin resulting in a giant spin Hall effect in magnetic thin films in the presence of magnetization structures. Rashba-induced spin magnetic field would be useful to distinguish between topologically equivalent magnetic structures.

## 1 Introduction

Electromagnetism is one of the most important physical phenomena which our modern technologies are based on. Precise design of electric devices is possible owing to the Maxwell’s equation, which describes a mathematical structure of the electromagnetism. Such a structure is not unique to the electromagnetism in the vacuum; it rather arises whenever there is a U(1) gauge symmetry. In solids, several possibilities of U(1) gauge symmetry are known to emerge. A simple example is a metallic magnet. A magnet is an ensemble of many spins, which are governed quantum mechanically by a non-commutative algebra of SU(2). When a magnet has a non uniform magnetization texture, a motion of conduction electrons is equivalent to that of a particle in a curved space; To put mathematically, it is described by a gauge field having SU(2) gauge symmetry. When an interaction between the magnetization and conduction electron (sd interaction) is strong, as is the case in most metallic ferromagnets, symmetry breaking of spin space occurs, and only one component of the SU(2) gauge field survives, resulting in an emergent U(1) gauge field or emergent spin electromagnetism [1].

By definition, spin electric field drives spin up and down conduction electrons in opposite directions, inducing a spin current, (Fig. 1). In ferromagnetic metals, driven spin current is usually associated with a charge current, given by , where is a parameter representing spin polarization of carrier. Therefore, the spin electric field or spin motive force can be detected as a voltage, as was observed in the case of motion of a domain wall, vortex and skyrmions [2, 3, 4]. It is notable that a voltage generation by a moving domain wall was predicted by Berger [5] based on a phenomenological argument earlier than a gauge field argument by Volovik. In the same manner as spin electric field, the spin magnetic field, conventionally called the spin Berry’s phase, exerts the spin-dependent Lorentz force on the conduction electrons, inducing spin Hall effect, and the spin Hall effect is detected as an anomalous Hall effect [6]. To understand the whole structure of the emergent spin electromagnetic field is of essential importance in spintronics.

When spin-orbit interaction is present, SU(2) gauge symmetry is affected resulting in novel contributions to spin electromagnetic fields. The spin-orbit correction to the spin electric field has been theoretically studied in [7, 8, 9]. Of particular current interest is the effect of Rashba spin-orbit interaction arising from the breaking of inversion symmetry at surfaces and interfaces. Takeuchi et al. investigated weak sd coupling regime and found a spin electric field proportional to [10, 9], where is the Rashba electric field and is a unit vector representing the local magnetization direction, while Kim et al. obtained a different form of in a strong sd coupling regime [8]. It was shown later in [11] that the contribution found in [10] arises in the strong sd limit when spin relaxation is included. The two contributions thus coexist in general. A unique feature of the Rashba-induced spin electric field is that it arises even from spatially uniform magnetization, in contrast to the conventional one induced by inhomogeneous magnetization textures.

For deriving an expression for spin electromagnetic fields, a gauge field argument based on a U(1) gauge invariance is useful [1] if in the absence of spin relaxation. It can be also calculated directly by evaluating a force acting on spin, which is proportional to the time-derivative of the current density [8, 11]. Spin electromagnetic field is also accessible by a transport calculation. In fact, in [10], the fields were identified by calculating the induced electric current density and then comparing the result with a general expression resulting from the Maxwell’s equation, , where is spin conductivity, is diffusion constant for spin and is spin density. This approach is highly useful to study the weak coupling regime, where adiabatic component of spin gauge field can not be defined. As noted in [10, 9], the information in the pumped current, however, is not enough to completely determine the two fields and effective permittivity and permeability, and additional information is necessary. Besides, it is not clear whether all of the contributions to the current expressed as rotation of certain vectors are to be interpreted always as due to an effective magnetic field. This issue is answered by investigating the Hall effect; If the contribution is really induced by an effective magnetic field, the field should exert the Lorentz force on the electron spin and induce spin Hall effect. The aim of this paper is to determine a spin magnetic field uniquely by calculating the spin Hall effect, and explore structure of the spin electromagnetism induced by the Rashba interaction and dynamic magnetization structures. We shall show that the whole contribution pumped current that is written as a rotation of a certain vector is indeed to be identified as an effective magnetic field, at least in the present system of strong sd coupling limit with the Rashba interaction.

The spin electromagnetic fields studied here are generalized spin Berry’s phase including the Rashba effects. In contrast to the conventional spin Berry’s phase, the Rashba-induced one is linear in the gradient of magnetization profile, and thus it would dominate in the case of slowly varying magnetization. We show that the spin electric and magnetic fields are estimated to be extremely strong like 2.5 kV/m and 2.5 kT, respectively, for a strong Rashba interaction induced at surfaces [12] when the frequency and the length scale of magnetization profile are 1GHz and 1nm, respectively.

We demonstrate that the Rashba-induced effective spin electromagnetic fields in the strong sd coupling limit without spin relaxation are described totally by an effective U(1) gauge field, ( and are the electron mass and charge, respectively), at the linear order in the Rashba interaction. There is therefore no monopole in the present system, in contrast to what was observed in [10] in the weak sd coupling regime. In the light of the present result and that of [10], spin relaxation seems to be essential for an emergence of monopole, as claimed in [10]. In fact, as will show below, the spin electric field in the absence of spin relaxation is proportional to , where is the Rashba-field, in agreement with the result of [8]. Its rotation, , is thus written as a time derivative of , and as we will show in the present paper, it is in fact written totally by a time-derivative of spin magnetic field, , i.e., . The two fields satisfy thus the conventional Faraday’s law and there is no monopole term. In contrast, the spin electric field found in [10, 11] in the presence of spin relaxation is , and cannot be written by a time derivative of any local quantity. It therefore follows that is non-vanishing for any local function , resulting in a finite current of spin damping monopole [10], . The spin damping monopole was argued to be essential in spin-charge conversion in dynamic magnetization structures such as in the inverse spin Hall effect [13].

## 2 Calculation of electric current

In this section, we derive an expression for spin electromagnetic fields by calculating an electric current induced by magnetization dynamics and the Rashba interaction in a metallic ferromagnet. The Lagrangian of the system is

 L= ∫d3rc†[\rmiℏ∂t+(ℏ22m∇2+ϵF)+Δsd(n⋅σ)−\rmi2αR⋅(↔∇×σ)−vi]c, (1)

where and are annihilation and creation operators of conduction electron respectively, represents the strength and the direction of the Rashba interaction, is the unit vector parallel to local magnetization, is the random potential caused by spin-independent impurities, is a Pauli matrix vector, and is the strength of sd exchange interaction. We consider the limit where is large, and perform a local spin gauge transformation (rotation in spin space) defined by , where is the annihilation operator in a rotated space and where , ( and are polar coordinates of ), is the rotation matrix in spin space [14]. This transformation diagonalize the sd interaction. (1) then reads

 L= ∫d3r{\rmiℏa†∂ta−ℏa†As,ta+a†(ℏ22m∇2+ϵF)a+\rmiℏ22mAℓs,j(a†↔∇jσℓa)−ℏ22ma†A2sa +Δsda†σza−\rmi2αR,jϵjkℓRℓn[(a†σn↔∇ka)+2\rmiAns,ka†a]−via†a}, (2)

where is the spin SU(2) gauge field and is a rotation matrix element. Summation is assumed for repeated indices ().

For estimating the effective field, we calculate the electric current induced by the Rashba interaction following the approach of [10]. The electric current density written in terms of and is

 ji= \rmieℏ2m⟨a†↔∇ia⟩−eℏmAℓs,i⟨a†σℓa⟩−eℏϵijkαR,jRkℓ⟨a†σℓa⟩, (3)

where denotes the expectation value in the ground state. We calculate (3) at the first order of Rashba interaction. Generally, electric current pumped by dynamic spins is a sum of local terms and diffusion terms [15, 16], but we here look into the local terms only, which represents the local effect of the effective fields. The leading contributions of local electric current density are diagramatically depicted in Fig.2. The contribution represented by the first two diagrams in figure 2 reads

 j12i(r,t)= −\rmieℏ2mαR,ℓϵℓmn∑k,p∑ω,¯Ω\rme−\rmip⋅r+\rmi¯ΩtRno(p,¯Ω)\tr[kikmgk−p2,ω−¯Ω2σogk+p2,ω+¯Ω2+δimmℏ2σogk,ω]<, (4)

where is the counter ordered Green’s function of conduntion electron with wave vector and angular frequency , and represents a lesser component. It includes the elastic lifetime due to the impurities, , and is a matrix in spin space. The contribution of remaining diagrams in figure 2 reads

 j3−6i= −\rmieℏ2mαR,ℓϵℓmn∑k,q,p∑ω,Ω,¯Ω\rme−\rmi(q+p)⋅r+\rmi(Ω+¯Ω)tRno(p,¯Ω) ×\tr[ki(k+q2)mℏJμ(k−p2)Ajs,μ(q,Ω)gk−q2−p2,ω−Ω2−¯Ω2σjgk+q2−p2,ω+Ω2−¯Ω2σogk+q2+p2,ω+Ω2+¯Ω2+c.c. +kiAos,m(q,Ω)gk−q2−p2,ω−Ω2−¯Ω2gk+q2+p2,ω+Ω2+¯Ω2+Ajs,i(q,Ω)kmσjgk−p2,ω−¯Ω2σogk+p2,ω+¯Ω2 +δimmℏ2ℏJμ(k)Ajs,μ(q,Ω)σogk−q2,ω−Ω2σjgk+q2,ω+Ω2]<, (5)

where .

First, we culculate (4). The lesser components in (4) are given as

 (gk−p2,ω−¯Ω2σogk+p2,ω+¯Ω2)<= fω−¯Ω2(gak−p2,ω−¯Ω2−gak+p2,ω+¯Ω2)σogak+p2,ω+¯Ω2 +fω+¯Ω2grk−p2,ω−¯Ω2σ0(gak+p2,ω+¯Ω2−grk+p2,ω+¯Ω2) (6) g

where and are retarded and advaiced Green’s function and is the Fermi distribution function ( is the inverse temperature). Therefore is calculated as

 j12i= ∑p,¯Ω\rme−\rmip⋅r+i¯Ωtξij(αR×n(p,¯Ω))j (8)

where

 ξij ≡\rmieℏ2m∑k,ω∑σ=±σ[fω−¯Ω2kikj(gak−p2,ω−¯Ω2,σ−grk−p2,ω−¯Ω2,σ)gak+p2,ω+¯Ω2,σ +fω+¯Ω2kikjgrk−p2,ω−¯Ω2,σ(gak+p2,ω+¯Ω2,σ−grk+p2,ω+¯Ω2,σ)+δijmℏ2fω(gak,ω,σ−grk,ω,σ)], (9)

and ( is spin index). Expanding with respect to and , assuming a slowly varying magnetization structure, we obtain

 ξij= \rmieℏ2m∑k,ω∑σσ{fω[kikj((gak,ω,σ)2−(grk,ω,σ)2)+δijmℏ2(gak,ω,σ−grk,ω,σ)] +fωkikj[ℏ24mp2((gak,ω,σ)3−(grk,ω,σ)3)+(ℏ22mk⋅p)2((gak,ω,σ)4−(grk,ω,σ)4)] Missing or unrecognized delimiter for \Bigr (10)

Assuming a rotational symmetry in space and carrying out an integraton by parts with respect to and , (10) becomes

 ξij= \rmieℏ2m∑k,ω∑σσf′ω[−112ℏ(δijp2−pipj)(gak,ω,σ−grk,ω,σ)−¯Ω6δijk2(gak,ω,σ−grk,ω,σ)2]+\Or(p3,Ω2). (11)

Using and , ( and ) the coefficient is obtained as

 ξij≃ ∑σ[(δijp2−pipj)eℏνσ12m−δij¯Ω2\rmieϵF,σνστσ3ℏ], (12)

where is the density of states of electron. The result of is

 j12= Missing or unrecognized delimiter for \Bigr (13)

Next, we calculate , (5). Expanding with respect to and , (5) reduces to

 j3−6i= −\rmieℏ2mαR,ℓϵℓmn∑k,q,p∑ω,Ω,¯Ω\rme−\rmi(q+p)⋅r+\rmi(Ω+¯Ω)tRno(p,¯Ω) ×f(ω)ℏ22mk23[(q+p)mAjs,i(q,Ω)−δim(q+p)pAjs,p(q,Ω)] ×2Re\tr[σjgak,ωσo(gak,ω)2−σogak,ωσj(gak,ω)2] +O(q3,Ω1). (14)

(14) is calculated by use of the following identities,

 ∑k,ωf(ω)ϵRe\tr[σjgak,ωσo(gak,ω)2−σogak,ωσj(gak,ω)2] = −25ℏ∑k,ωf′(ω)ϵ2Re\tr[σjgak,ωσo(gak,ω)2−σogak,ωσj(gak,ω)2], (15)

obtaied by an integration by parts, and

 2ϵjozAjs,iRno= ∇inn, (16) \tr[σjAσoB−σoAσjB]= 2iϵjoz∑σσA−σBσ, (17)

where and are any 22 diagonal matrices. The leading contribution of (5) then reads

 j3−6= η∇×[∇×(αR×n)], (18)

where the coefficient is

 η= 2eℏ15πm∑kIm[∑σσϵ2gak,−σ(gak,σ)2] = Missing or unrecognized delimiter for \Bigl (19)

and we used .

Therefore the total local electric current density is

 j= (ξ1+η){∇×[∇×(αR×n)]}+ξ2(αR×˙n), (20)

where , .

Electric current driven by effective electromagnetic fields is generally written in diffusive regime as follows:

 j= σsEs+1μs∇×Bs−Ds∇ρs, (21)

where and represent driving fields, is the conductivity for spin, is the magnetic permeability of spin magnetic field and the last term is a diffusive contribution ( is the diffusion constant for spin and is the density of electron spin). Comparing our result, (20), to (21), we see that

 σsEs± =ξ2(αR×˙n) Bsμs =(ξ1+η)[∇×(αR×n)]. (22)

We know that spin conductivity is given by

 σs= ∑σ=±σe2ℏ23m2k2Fσνστσ, (23)

and thus the spin electric field reads

 Es =−meℏαR×˙n. (24)

This result agrees with result of direct estimate of spin motive force [8, 11]. In contrast, to identify spin magnetic field from (22), we need additional information on the permeability. This is accomplished by analyzing the spin Hall effect, which is carried out in the next section.

## 3 The spin Hall effect induced by spin magnetic filed

To determine the effective spin magnetic field, (22) is not sufficient. In this section we calculate the Hall effect caused by the spin magnetic field when an electric field, , is applied and determine the spin magnetic field uniquely. Since the Hall effect studied here drives electron spin, the effect is spin Hall effect. The Lagrangian has now the following additional terms coming from an applied vector potential, (),

 δL= ∫d3r[\rmieℏ2mAj(a†↔∇ja)−e22m(a†A2a)−eℏm(a†AjAs,ja)+eℏαR,jϵjklRlnAk(a†σna)]. (25)

The electric current density is also modified to be , where

 δji= −e2mAi⟨a†a⟩. (26)

We calculate the Hall current induced by the effective magnetic field with applied electric field which is spatially homogeneous.

The leading contribution to the Hall current density, , described by the Feynmann diagrams in figure 3, reads

 jhalli =−\rmie2ℏm∑k,p,ω,Ω\rme−\rmip⋅r+\rmiΩtαR,jϵjkl ×\tr{ℏ2mkikkAm(Ω)Rln(p)[(k−p2)mgk−p2,ω−Ω2gk−p2,ω+Ω2σngk+p2,ω+Ω2 +(k+p2)mgk−p2,ω−Ω2σngk+p2,ω−Ω2gk+p2,ω+Ω2] −kiAk(Ω)Rln(p)gk−p2,ω−Ω2σngk+p2,ω+Ω2+δikkmAm(Ω)Rln(p)σngk,ω−Ω2gk,ω+Ω2 −kkAi(Ω)Rln(p)gk−p2,ωσngk+p2,ω}<, (27)

The contributions containing spin gauge field at the linear order are neglected, because those are smaller than the ones in (27) by the order of . Expanding with respect to external wave vector and frequency, and , we obtain

 jhalli =−\rmie2ℏm∑k,p,ω,Ω\rme−\rmip⋅r+\rmiΩtαR,jϵjkl ×\tr{ℏ22mkikkpmAm(Ω)Rln(p)[−gk−p2,ω−Ω2gk−p2,ω+Ω2σngk+p2,ω+Ω2+gk−p2,ω−Ω2σngk+p2,ω−Ω2gk+p2,ω+Ω2] −ki(k⋅p)Ak(Ω)Rln(p)gk−p2,ω−Ω2σngk+p2,ω+Ω2/(k⋅p)}<+\Or(Ω2,p2). (28)

The lesser components of (27) are calculated as

 [gk−p2,ω−Ω2gk−p2,ω+Ω2σngk+p2,ω+Ω2]< = (fω+Ω2−fω−Ω2)grk−p2,ω−Ω2gak−p2,ω+Ω2σngak+p2,ω+Ω2 +fω−Ω2gak−p2,ω−Ω2gak−p2,ω+Ω2σngak+p2,ω+Ω2+fω+Ω2grk−p2,ω−Ω2grk−p2,ω+Ω2σngrk+p2,ω+Ω2, (29) [gk−p2,ω−Ω2σngk+p2,ω+Ω2]< = (fω+Ω2−fω−Ω2)grk−p2,ω−Ω2σngak+p2,ω+Ω2 +fω−Ω2gak−p2,ω−Ω2σngak+p2,ω+Ω2+fω+Ω2grk−p2,ω−Ω2σngrk+p2,ω+Ω2. (30)

Assuming a rotational symmetry (i.e., using ), Hall current density reduces to

 jhalli Missing or unrecognized delimiter for \bigr (31)

where

 Γ1 ≡∑ω,σσ[ℏfω((gak,ω,σ)4−(grk,ω,σ)4)−f′ω(grk,ω,σ(gak,ω,σ)2−(grk,ω,σ)2gak,ω,σ)]. (32)

The summation about and is evaluated as follows;

 ∑kϵΓ1 =13∑k,ω,σσf′ωϵ(gak,ω,σ−grk,ω,σ)3 ≃2iℏ2∑σσϵFσνστ2σ. (33)

The Hall current is finally obtained as

 jhall =−∑±(±)eτ±mσB±meℏ(E×(∇×(α×n))), (34)

where is the spin-resolved Boltzmann conductivity. Thus the Hall effect is described by a standard expression of

 jhall=σH(E×Bs), (35)

where and

 Bs≡meℏ∇×(αR×n). (36)

In terms of Hall electric field, (34) is written as

 Ehall=−1nej×Bs, (37)

where is electron density, is the magnitude of longitudinal electric current driven by applied electric field. (35) and (37) are the central results of the present paper, indicating that the field exerts the Lorentz force of for electron spin and that certainly play a role as a magnetic field for conduction electron.

## 4 Rashba-induced effective gauge field

In the preceeding sections, we have demonstrated that the effective spin electromagnetic fields are determined by calculating transport properties. Our results, (24) and (36), indicate that the Rashba-induced effective field is written as

 Es =−˙AR Bs =∇×AR, (38)

with

 AR ≡meℏ(αR×n). (39)

The present spin electromagnetic fields are therefore explained by a standard U(1) gauge theory, as was argued in [8]. This fact is not obvious, since the combination of the Rashba spin-orbit interaction and sd interaction does not necessarily lead to an emergence of U(1) gauge symmetry. Neverthelss a gauge field scenario holds as far as non-linear effects of the Rashba interaction are neglected. In fact, the Rashba interaction has the same effect as an SU(2) gauge field if non-linear effects are neglected. This is easily seen from (2), which indicates that the Rashba interaction in a rotated frame is written as

 ∫d3r\rmiℏ22mAℓR⋅(a†σℓ↔∇a), (40)

where . The kinetic term of (2) thus is written by an SU(2) gauge field defined as as

 ∫d3rℏ22ma†(∇+\rmi~AR)2a+\Or((AR)2). (41)

In the strong sd coupling limit, therefore, acts as an effective U(1) gauge field and effective electromagnetic fields emerge according to and , where