Coupling Theory of Emergent Spin Electromagnetic Fieldand Electromagnetic Field

Hideo Kawaguchi1+ and Gen Tatara2

1Graduate School of Science and Engineering, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

(Received March 14, 2014; accepted April 30, 2014; published online June 19, 2014)

In ferromagnetic metals, an effective electromagnetic field that couples to conduction electron spins is induced by thesd exchange interaction. We investigate how this effective field, namely, the spin electromagnetic field, interacts with theordinary electromagnetic field by deriving an effective Hamiltonian based on the path integral formalism. It turns outthat the dominant coupling term is the product of the electric field and spin gauge field. This term describes the spin-transfer effect, as was pointed out previously. The electric field couples also to the spin electric field, but this contributionis smaller than the spin-transfer contribution in the low frequency regime. The magnetic field couples to the spinmagnetic field, and this interaction suggests an intriguing intrinsic mechanism of frustration in very weak metallicferromagnets under a uniform magnetic field. We also propose a voltage generation mechanism due to a nonlinear effectof non-monochromatic spin-wave excitations.

1. Introduction

Spin-transfer torque induced by an applied electric currentin ferromagnetic metals is a crucially important effect inspintronics. The idea was first proposed theoretically byBerger1) in the case of a domain wall motion and bySlonczewski2) and Berger3) in the case of the uniformmagnetization of thin films. The spin-transfer effect arisesfrom the transfer of spin angular momentum from conductionelectrons to localized spins which induce the magnetization.The effect is caused by the sd exchange interaction, and theangular momentum transfer occurs owing to the angularmomentum conservation.2) The interaction Hamiltoniandescribing the spin-transfer effect is

Hst ¼Z

d3rh�P

2eð1� cos �Þðj � rÞ�; ð1Þ

where ª and º are the polar coordinates representing thelocalized spin direction, j denotes the applied electric currentdensity, P is the spin polarization of the conduction electron,and e is the electron charge. The interaction is representedas a gauge coupling to a spin gauge field Az

s,4,5) Hst ¼R

d3rðjs � AzsÞ, where js � Pj and Az

s ¼ h�2e ð1� cos �Þr�. The

interaction is thus expressed as Hst ¼Rd3r P�BðE � Az

sÞ,where �B is the Boltzmann conductivity. This expressionclearly shows that the spin-transfer effect is due to a couplingof two gauge fields, the conventional electromagnetic field ofthe electric charge, and the gauge field acting on the electronspin. The aim of this work is to study the coupling betweenthe two gauge fields by calculating an effective Hamiltonian.We shall show that the effective Hamiltonian in the case of aslowly varying magnetization is made up of three contribu-tions, one representing the spin-transfer torque and the othersdescribing the couplings between the electric and magneticfields.

In the case of charge electromagnetism coupled torelativistic charged particles, the effective Lagrangianinduced by the particles is always written in a relativisticallyinvariant form as

P�� F��F

��, where F�� is the field strengthand ® and ¯ are indices representing x, y, z, and t. The onlyterms allowed in the relativistic case are thus proportional

to either jEj2 or jBj2. In ferromagnetic metals, conductionelectrons interact with two gauge fields, A acting on thecharge and As acting on the spin, and the total electric andmagnetic fields become Eþ Es and Bþ Bs, where Es andBs are the effective spin electric and spin magnetic fields,respectively. If the system is relativistic, we would thusexpect to have interactions in the form of E � Es and B � Bs

arising from ðEþ EsÞ2 and ðBþ BsÞ2. In reality, there areother contributions in ferromagnetic metals since theelectrons are not relativistic and they have a finite lifetimeof elastic scattering. We shall demonstrate that a couplingterm proportional to E � Az

s arises as the dominant contribu-tion. This term derived first in Ref. 6 represents the spin-transfer effect, as was discussed there. We also investigateother coupling terms, E � Es and B � Bs.

1.1 Spin electromagnetic fieldAn effective electromagnetic field arises from the sd

exchange interaction described by

Hsd ¼ ��sd

Zd3r n � se; ð2Þ

where �sd is the exchange energy, n is a unit vectorrepresenting the direction of the localized spin, and se isthe direction of the conduction electron spin. When thisexchange interaction is strong, the conduction electron spin isaligned parallel to the localized spin direction, and thiseffect results in a quantum mechanical phase attached to theelectron spin when the electron moves (see Ref. 7 for detailsof derivation). The spin part of the electron wave functionwith the expectation value along n is jni ¼ cos �

2j"i þ

sin �2ei�j#i, where ª and º are the polar coordinates of n and

j"i and j#i denote the spin states.8) When the electron hopsover a small distance dr to a nearby site where the localizedspin is along n0, the overlap of the wave functions iscalculated as hn0jni ’ e

ih�eAz

s�dr, where

Azs ¼

h�

2eð1� cos �Þr�; ð3Þ

and the factor of 12is due to the magnitude of the electron

spin. The field Azs is an effective vector potential or an

Journal of the Physical Society of Japan 83, 074710 (2014)

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074710-1 ©2014 The Physical Society of Japan

effective gauge field. When the electron’s path is finite, thephase becomes ’ ¼ e

h�

RC dr � Az

s. The existence of the phasemeans that there is an effective magnetic field Bs, as seen byrewriting the integral over a closed path using the Stokestheorem as ’ ¼ e

h�

RS dS � Bs, where Bs � r � Az

s. The timederivative of the phase is equivalent to a voltage, and thus,we have an effective electric field defined by _’ ¼� e

h�

RC dr � Es, where Es � � _Az

s. These two fields satisfyFaraday’s law, r � Es þ _Bs ¼ 0. We therefore have effectiveelectromagnetic fields that couple to the conduction electronspin as a result of the sd exchange interaction. We call thefield a spin electromagnetic field.9) Using the explicit form ofthe effective gauge field, Eq. (3), we see that the emergentspin electromagnetic fields are

Es;i ¼ � h�

2en � ð _n�rinÞ;

Bs;i ¼ h�

4e

Xjk

�ijkn � ðrjn�rknÞ: ð4Þ

The magnetic component Bs is the spin Berry’s curvature10)

or scalar chirality. The electric component Es, called the spinmotive force, is a chirality in the space-time, which ariseswhen the localized spin structure n is time-dependent.

The expression Eq. (4) was derived by Volovik in 1987.11)

Originally, the emergence of the effective electric field Es

from moving magnetic structures was found in 1986 byBerger, where a voltage generated by canting a movingdomain wall was calculated.1) Stern discussed the motiveforce in the context of the spin Berry’s phase and theAharonov–Bohm effect in a ring, and showed similarity toFaraday’s law.12) The spin motive force was rederived inRef. 13 in the case of the domain wall motion, and discussedin the context of topological pumping in Ref. 14. Those worksconsider only the adiabatic limit, i.e., in the case of a strong sdexchange interaction and in the absence of spin-dependentscattering. The idea of the spin motive force has recently beenextended to include the spin–orbit interaction,9,15–20) and itwas shown that the spin–orbit interaction modifies the spinelectric field. It was also shown that the spin electromagneticfield arises even in the limit of a weak sd interaction.21,22) Thecase of the Rashba spin–orbit interaction has been studied indetail recently. It was shown that the spin electric field in thiscase emerges even from a uniform precession of magnetiza-tion.9,20) This fact suggests that the Rashba interaction atinterfaces would be useful in controlling the spin–chargeconversion. The Rashba-induced spin electric field induces avoltage in the same direction as in the inverse spin Hall andinverse Edelstein effects23,24) driven by the spin pumpingeffect.25) It was also pointed out that the spin electromagneticfields in the presence of spin relaxation satisfy Maxwell’sequations with spin magnetic monopoles that are drivendynamically.21) The coupling between the spin magnetic fieldand the helicity of light was theoretically studied in thecontext of the topological inverse Faraday effect, which is anonlinear effect with respect to the incident electric field.26)

Experimentally, the spin magnetic field (the spin Berry’scurvature) has been observed using the anomalous Hall effectin frustrated ferromagnets.27,28) The spin electric field hasbeen measured in the motion of various ferromagneticstructures such as domain walls,29) magnetic vortices,30) andskyrmions.31)

2. Derivation of Effective Hamiltonian

The effective Hamiltonian is calculated in the imaginary-time (denoted by ¸) path integral formalism.32) In this section,we set h� ¼ 1. The system we consider is a ferromagneticmetal, where conduction electrons, represented by two-component annihilation and creation fields, cðr; �Þ and �cðr; �Þ,interact with localized spins, described by the vector fieldnðr; �Þ, via the sd exchange interaction. The Hamiltonian thusreads

H ¼ H0 þHsd þHem; ð5Þ

H0 ¼Z

d3r1

2mjrcðr; �Þj2 � � �cðr; �Þcðr; �Þ

� �;

Hsd ¼ ��sd

Zd3r nðr; �Þ � ð �cðr; �Þ�cðr; �ÞÞ; ð6Þ

where ® is the chemical potential, m is the electron mass, and� is the vector of Pauli matrices. The term Hem represents theinteraction between the conduction electron and the appliedelectromagnetic field, described by a vector potential A,which reads

Hem ¼ �Z

d3rAðr; �Þ

� ie

2m�cðr; �Þ$rcðr; �Þ � e2

2mAðr; �Þ �cðr; �Þcðr; �Þ

� �; ð7Þ

where �c$rc � �cðrcÞ � ðr �cÞc and ¹e is the electron charge

(e > 0). The system we consider is a film thinner thanthe penetration depth of the electromagnetic field. TheLagrangian of the system is

L ¼Z

d3r �cðr; �Þ@�cðr; �Þ þ H; ð8Þ

and the effective Hamiltonian describing the localized spinand the gauge field is obtained by carrying out a path integralover the conduction electrons as Heff ð�; �;AÞ � �ln Z,where

Z ¼Z

D �cðr; �ÞDcðr; �Þe�R

0d�L

; ð9Þ

is the partition function and D denotes the path integral.We are interested in the case where the sd exchange

interaction is large and thus the conduction electron spin isaligned parallel to the localized spin direction n, i.e., theadiabatic limit. To describe this limit, the use of the spingauge field, which characterizes the deviation from theadiabatic limit, is convenient.5) The spin gauge field isintroduced by diagonalizing the sd interaction using a unitarytransformation, cðr; �Þ ¼ Uðr; �Þaðr; �Þ, where Uðr; �Þ is a2� 2 unitary matrix and a is a new electron field operator. Aconvenient choice of Uðr; �Þ is Uðr; �Þ ¼ mðr; �Þ � � withmðr; �Þ ¼ ðsin �

2cos�; sin �

2sin�; cos �

2Þ, where ª and º are the

polar angles of n. It is easy to confirm that Uyðn � �ÞU ¼ �z issatisfied. Because of this local unitary transformation,derivatives of the electron field become covariant derivatives@�c ¼ Uð@� þ ieAs;�Þa, where As;� � � i

e U�1@�U is the

gauge field. Since U is a 2� 2 matrix, the gauge fieldAs;� is written using Pauli matrices as As;� ¼

P A

s;��

(� ¼ x; y; z; � is a suffix for space and time and ¼ x; y; zis for spin). It is thus an SU(2) gauge field, which we call the

J. Phys. Soc. Jpn. 83, 074710 (2014) H. Kawaguchi and G. Tatara

074710-2 ©2014 The Physical Society of Japan

spin gauge field. The Lagrangian in the rotated space is thusgiven by

L � L0 þ LA; ð10Þ

L0 �Z

d3r �a @� � 1

2mr2 � ���sd�z

� �a; ð11Þ

LA �Z

d3r

"ie �aAs;�aþ

Xi

X

As;ij

s;i þ Aiji

!

þ e2

m

Xi;

AiAs;i �a�

aþ e2

2m

Xi

X

ðAs;iÞ2 þ ðAiÞ2

!�aa

#;

ð12Þwhere js;i � �ie 1

2m �að r!i � r iÞ�a and ji � �ie 12m �að r!i �

r iÞa are the spin current and charge current, respectively.The electron field a is strongly spin-polarized owing to the sdexchange interaction (the last term of L0).

We carry out the path integral with respect to the electronfield and derive the effective Hamiltonian for the twogauge fields A

s;i and Ai describing the spin and charge gaugefields, respectively. The spin gauge field is written using thelocalized spin direction, ª and º, and thus the effectiveHamiltonian can be regarded as that describing theinteraction of the localized spin and the charge electro-magnetic field.

Up to the second order with respect to the gauge fields, theeffective Hamiltonian reads

Heff ¼ �Z

0

d�

Zd3r

"2ieAz

s;�seðr; �Þ þ2e2

m

Xi

AiAzs;iseðr; �Þ

þ e2

2m

Xi

X

ðAs;iÞ2 þ ðAiÞ2

!nðr; �Þ

#

þ 1

2

Z

0

d�

Z

0

d�0Z

d3r

Zd3r0

Xij

�"X

As;iA

s;j�

ij ðr; r0; �; �0Þ

þ 2X

As;iAj�

ijðr; r0; �; �0Þ þ AiAj�ijðr; r0; �; �0Þ

#;

ð13Þwhere seðr; �Þ � 1

2h �aðr; �Þ�zaðr; �Þi, nðr; �Þ � h �aðr; �Þaðr; �Þi,

h� � �i denotes the thermal average, and

�ij ðr; r0; �; �0Þ � hjs;iðr; �Þjs;jðr0; �0Þi;�ijðr; r0; �; �0Þ � hjs;iðr; �Þjjðr0; �0Þi;

�ijðr; r0; �; �0Þ � hjiðr; �Þjjðr0; �0Þi; ð14Þare the current–current correlation functions. The spin densityse and the electron density n are calculated as se ¼12V

Pk

P�¼� �fð�k�Þ and n ¼ 1

V

Pk

P�¼� fð�k�Þ, respec-

tively, where fð�k�Þ ¼ ðe�k� þ 1Þ�1 is the Fermi–Diracdistribution function, �k� ¼ k2

2m � �� ��sd, and � ¼ � isthe spin index. The Fourier components of the correlationfunctions are

�ij ðq; i�‘Þ ¼ � e2

m2V

Xn;k

kikj tr½�Gk�q2;n�Gkþq

2;nþ‘�;

�ijðq; i�‘Þ ¼ � e2

m2V

Xn;k

kikj tr½�Gk�q2;nGkþq

2;nþ‘�;

�ijðq; i�‘Þ ¼ � e2

m2V

Xn;k

kikj tr½Gk�q2;nGkþq

2;nþ‘�: ð15Þ

Here Gk;n is defined as Gk;n � ½i!n � �k þ i2�e

sgnðnÞ��1,where sgnðnÞ ¼ 1 and ¹1 for n > 0 and n < 0, respectively,�k ¼ k2

2m � ���sd�z is the electron energy in the matrixrepresentation, �e is the electron elastic scattering lifetime,and tr denotes the trace over spin space. The Fermionicthermal frequency is represented by !n � ð2nþ1Þ� , and �‘ �2�‘ is a bosonic thermal frequency.The correlation functions are calculated by rewriting the

summation over the thermal frequency using the contourintegral (z � i!n) as

�zzijðq; i�‘Þ ¼ e2

m2V

Xk

kikjX�¼�

�ZC

dz

2�ifðzÞgk�q

2;�ðzÞgkþq

2;�ðzþ i�‘Þ;

�þ�ij ðq; i�‘Þ ¼ e2

m2V

Xk

kikjX�¼�

�ZC

dz

2�ifðzÞgk�q

2;�ðzÞgkþq

2;��ðzþ i�‘Þ;

�zijðq; i�‘Þ ¼ e2

m2V

Xk

kikjX�¼�

�ZC

dz

2�i�fðzÞgk�q

2;�ðzÞgkþq

2;�ðzþ i�‘Þ;

�ijðq; i�‘Þ ¼ e2

m2V

Xk

kikjX�¼�

�ZC

dz

2�ifðzÞgk�q

2;�ðzÞgkþq

2;�ðzþ i�‘Þ; ð16Þ

where C is an anticlockwise contour surrounding theimaginary axis33,34) and gk;�ðzÞ is defined as gk;�ðzÞ �½z� �k� þ i

2�esgnðIm zÞ��1.

We expand the correlation functions with respect to theexternal wave vector q and frequency ³ after the analyticalcontinuation to �þ i0 � i�‘.34) The result up to the secondorder in q and ³ is

�ij ðq;�Þ ¼

e2

mfð � z zÞ ijb

þ z z½ ijnð1þ i��e � ð��eÞ2Þþ cðqiqj � q2 ijÞ�g;

�zijðq;�Þ ¼

e2

m½ ij2seð1þ i��e�ð��eÞ2Þ þ dðqiqj� q2 ijÞ�;

�ijðq;�Þ ¼ e2

m½ ijnð1þ i��e � ð��eÞ2Þ þ cðqiqj � q2 ijÞ�;

ð17Þwhere b � 1

3mV�sd

Pk

P�¼� �k

2fð�k�Þ, c � 112m

P�¼� �� ,

d � 112m

P�¼� ��� , n ¼P�¼�

k2F���3m , se ¼ 1

2

P�¼� �

k2F���3m ,

and kF� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2F þ 2m��sd

pand �� � m

32ffiffi

2p

�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�F þ ��sd

pare

the spin-dependent Fermi wave number and the density ofstates per unit volume, respectively. Vertex corrections are

J. Phys. Soc. Jpn. 83, 074710 (2014) H. Kawaguchi and G. Tatara

074710-3 ©2014 The Physical Society of Japan

irrelevant, since they are proportional to r � E, whichvanishes in metals. In this work, we do not consider surfaceeffects played by induced surface charges such as the surfaceplasmon effect. We thus obtain the effective Hamiltonian upto the order of q2 and �2 as

Heff ¼ ��2ieAz

s;�se þ1

2mðn� bÞ

Xi;q;�

Aþs;ið�q;��ÞA�s;iðq;�Þ

� e2

m2se�e

Xi;q;�

i�Azs;ið�q;��ÞAiðq;�Þ

þ e2

2m�2eXi;q;�

�2½nðAzs;ið�q;��ÞAz

s;iðq;�Þ

þ Aið�q;��ÞAiðq;�ÞÞ þ 4seAzs;ið�q;��ÞAiðq;�Þ�

� e2

2m

Xij;q;�

ðqiqj � q2 ijÞ½cðAzs;ið�q;��ÞAz

s;jðq;�Þ

þ Aið�q;��ÞAjðq;�ÞÞ þ dAzs;ið�q;��ÞAjðq;�Þ�

�;

ð18Þwhere A�s;iðq;�Þ � Ax

s;iðq;�Þ � iAys;iðq;�Þ. The terms quad-

ratic in the charge gauge field describe the electricpermittivity and magnetic permeability of the media. Weused the fact that

Rdt Ai

_Ai ¼ 0 to drop the term proportionalto �Aið��ÞAið�Þ. The terms quadratic in the spin gaugefield contribute to the renormalization of the exchangeinteraction and dissipation as shown in Ref. 35. In fact, thecontribution of the order of �0 reduces to

1

mðn� bÞ

Xi

Aþs;iA�s;i ¼ Jeff ðrnÞ2; ð19Þ

where Jeff � 14m ðn� bÞ. We treat this renormalization as a

term in the original exchange interaction and do not considerit further. The term quadratic in the spin gauge field andlinear in ³ represents a dissipation,35) but we neglect thiseffect since we are not interested in the spin dynamics wheredissipation plays an important role. Instead, we are interestedin the coupling between the two gauge fields. The effectiveHamiltonian describing the coupling reads

Hint ¼ e2

m

Zd3r 2se�eE � Az

s þ 2se�2eE � Es þ d

2B � Bs

� �;

ð20Þwhere E � � _A and B � r � A are the electric and magneticfields, respectively, and Es � � _Az

s and Bs � r � Azs are the

effective spin electric and magnetic fields, respectively.

3. Discussion

Let us discuss the effect of the coupling terms, Eq. (20).The first term indicates that Az

s is induced when an electricfield is applied. In fact, this term is the term describing thespin-transfer torque, as seen by denoting 2se

e2

m �eE ¼ Pj,where P � n"�n#

n , j ¼ �BE, and �B � e2

m n�e is the Boltzmannconductivity. As pointed out in Ref. 6, the effectiveHamiltonian method that we used thus easily reproducesthe spin-transfer effect, which is usually discussed in thecontext of the conservation law of angular momentum.Although the spin gauge field Az

s is related to the spin electricfield as Es ¼ � _Az

s, the generation of Azs does not always

imply the generation of a spin electric field. In fact, a direct

consequence of the spin-transfer torque is to drive magnet-ization textures.5) Only when the induced magnetizationdynamics creates a non-coplanarity, the spin electric fieldis induced. The emergence of the spin electric fieldthus depends in an essential way on the dynamics of themagnetization.

3.1 Spin electric field induced by domain wall motionLet us consider as an example a domain wall. A domain

wall favors a non-coplanar motion since its center of masscoordinate X and the angle of the wall plane º are canonicalconjugates to each other.5) When the spin-transfer torque dueto an electric field is applied, the wall plane starts to tilt andits angle drives the motion of the wall.36) The spin electricfield generated by this wall motion is calculated as follows.We consider a case of uniaxial anisotropy and neglect thenonadiabaticity which is represented by ¢ in Ref. 5. A planardomain wall with the magnetization changing along thex-direction at position x ¼ XðtÞ is described by cos � ¼tanh x�X

� and sin � ¼ ðcosh x�X� Þ�1 with a constant º. The

equations of motion for X and º are37)

_X � � _� ¼ a302eS

Pj;

_�þ _X

�¼ 0; ð21Þ

where ­ is the width of the wall, ¡ is the Gilbert dampingparameter, a0 is the lattice constant, and S is the magnitudeof the localized spin. The solution for Eq. (21) is _X ¼1

1þ2a30

2eS Pj and _� ¼ � 1þ2

a30

2eS� Pj. The spin electric field,Eq. (4), for a moving domain wall is calculated usingrxn ¼ � sin �

� e� and _n ¼ sin �ð _�e� þ _X� e�Þ, where e� ¼

ð�sin�; cos�; 0Þ and e� ¼ ðcos � cos �; cos � sin�;�sin �Þ.The spin electric field then arises along the x-direction andthe magnitude is

Es ¼ h�

2e

1

2�_� ¼ h�

2e2

1þ 2

a30P

4S�2�BE: ð22Þ

Let us estimate the magnitude choosing � 10�2, P �0:8,38) �B � 108 ��1m¹1, and S ¼ 1. For a field of E � 104

V/m corresponding to j ¼ 1012A/m2, we have j _Xj � 4m/sand j _�j � 4� 106 s¹1. For � � 10�8 m,39) we thus obtainjEsj � 0:1V/m. In experiments, what is measured is thevoltage due to Es. Since Es is localized at the wall, thevoltage generated by a single domain wall is Es� � 1 nV.This value is not large, but is detectable. For instance, in thecase of a domain wall driven by an external magnetic field, avoltage of 400 nV was observed for a wall speed of 150m/s.29) The conversion efficiency from the electric field tothe spin electric field is given by

� � Es

E¼ h�

2e2

1þ 2

a30P

4S�2�B � P

4ðkF�Þ2�F�eh�

� �; ð23Þ

where we approximated sea30 � P=2. A typical value of ®

is � � 10�4 for � 10�2, kF� � 100, and ð�F�eÞ=h� � 100.The conversion efficiency is larger in a thin wall, such asperpendicular anisotropy magnets and weak ferromagnets.

In contrast to the spin-transfer term, the second term inEq. (20) describes a direct coupling between the electric fieldand the spin electric field. The strength of the induced spinelectric field is determined by solving spin dynamics as we

J. Phys. Soc. Jpn. 83, 074710 (2014) H. Kawaguchi and G. Tatara

074710-4 ©2014 The Physical Society of Japan

did above, because Es is determined using the equation ofmotion for the spin (the Laudau–Lifshitz equation) and notfor the Lagrangian like ðEsÞ2 � ðBsÞ2 as in the chargeelectromagnetism. The effect is generally small, since theE � Es coupling term is smaller than the spin-transfer term bya factor of ��e, which is small in the GHz frequency wheremagnetization structures can respond [�e & 10�13 s for ametal with ð�F�eÞ=h� ¼ 100]. The term may be important in avery clean metal in the THz range.

3.2 Magnetic couplingThe third term of Eq. (20) is due to the external magnetic

field. It indicates that the spin magnetic field is induced whena uniform external magnetic field is applied. This effectindicates a novel intrinsic mechanism involving frustration,since a finite Bs denotes a non-coplanar spin structure, whilea uniform magnetic field favors the uniform magnetizationalong the magnetic field. When the magnetization structurehas a finite non-coplanarity at the scale of ­, the magnitude ofthe induced spin magnetic field is Bs � h�

e�2 . The energygain per site due to the non-coplanarity is then ðeh�m Þ2 Bs

�FB ’

eh�m B 2

ðkF�Þ2 , assuming that d � Oð 112m�FÞ. [Note that d ¼

112m ð�þ � ��Þ can be positive or negative.] In ferromagneticsystems, a non-coplanar structure costs the exchange energyof J

2ðrnÞ2 ’ J

a20

1ðkF�Þ2 , where J is the exchange energy of the

localized spin, in the absence of other frustration. The totalenergy cost due to the generation of Bs is thus

�E ’ 1

ðkF�Þ2J

a20� eh�

mB

� �: ð24Þ

In common 3d ferromagnetic metals, the exchange energyJ=a20 is on the order of 1 eV, while the applied magnetic fieldof 1 T corresponds to the energy of eh�

m B ¼ 10�4 eV. The spinmagnetic field Bs is therefore not induced by simply applyinga uniform magnetic field. The situation may be differentin very weak ferromagnets like in molecular conductingferromagnets. For a system with a ferromagnetic criticaltemperature of 5K,40) a magnetic field of 5 T may besufficient to induce a finite Bs if the Fermi energy is on theorder of 1 eV. Our present study assuming a strong sdexchange interaction does not directly apply to weakferromagnets, and different approaches like in Ref. 21 areneeded to study the interplay between Bs and B. Molecularconducting ferromagnets would be unique systems in thecontext of an emergent spin electromagnetic field.

3.3 Spin electric field induced by spin waveThe spin-transfer term Hst induces a spin wave excitation

when the electric field is applied if the induced currentexceeds a threshold value.41) If the spin wave is mono-chromatic, a spin electric field is not induced, since amonochromatic plane wave, such as nx � iny ¼ ’e�iðkx��tÞ,does not have a non-coplanarity because _n and rxn areparallel to each other. It is possible to excite a spin electricfield if we use two spin waves having different wave vectorsor frequencies. Let us consider the case described by

s� ¼ 1

2

Xj¼1;2

’je�iðkjx��jtÞ; ð25Þ

where s� � 12ðsx � isyÞ is a small spin fluctuation, ’j are the

amplitudes of spin waves, and kj and �j (j ¼ 1; 2) are the

wave vector and angular frequency of the two spin-waveexcitations, respectively. The spin electric field associatedwith the spin waves then reads

Es;x ¼ � h�

2e’1’2ð�1k2 ��2k1Þ sin½ðk1 � k2Þx

� ð�1 ��2Þt�: ð26ÞNamely, a spin electric field having a wave vector (k1 � k2)and a frequency (�1 ��2) is induced by the two spin-waveexcitations. Two spin waves having the same frequency ofabout 7GHz have been generated recently using a magneticfield which is induced by applying a current throughantennas.42) The method would be applicable to create aspin electric field. For spin waves with k ¼ 0:4µm¹1 and anangular frequency of 2�� 7GHz,42) k� � 1:7� 1015ms¹1,and the spin electric field is expected to be on the order ofh�e k�’2 ’ 1:7� ’2V/m for the amplitude of spin waves ¤.The spin electric field is expected to be induced generally bythe spin wave excitation owing to a nonlinear effect (Es;x

nonlinearly depend on the spin wave amplitude), if the waveis non-monochromatic. The present method of generating aspin electric field applies to a uniform ferromagnet and wouldhave better possibilities of applications than the conventionalmethods using magnetic structures such as domain walls.

4. Summary

We have derived the effective Hamiltonian describing thecoupling between the emergent spin electromagnetic fieldand the charge electromagnetic field. The dominant termturns out to be the one corresponding to the spin-transfertorque. The coupling between the magnetic componentssuggests an interesting possibility of inducing frustrationby applying a uniform external magnetic field on weakferromagnets. We have proposed a generation mechanismof a spin electric field using a nonlinear effect of non-monochromatic spin-wave excitations. This mechanism isapplicable to the case of a uniform magnetization, and itwould have a great advantage in applications over commonsetups using non-coplanar structures. Our theoretical consid-erations call for an experimental verification of the effect.

Acknowledgments

The authors thank N. Nakabayashi, H. Kohno, H.Saarikoski, and H. Seo for valuable comments anddiscussions. This work was supported by a Grant-in-Aidfor Scientific Research (C) (Grant No. 25400344) and (A)(Grant No. 24244053) from the Japan Society for thePromotion of Science and UK–Japanese Collaboration onCurrent-Driven Domain Wall Dynamics from JST.

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