Definition of the spin current: The angular spin current and its physical consequences

Definition of the spin current: The angular spin current and its physical consequences

Qing-feng Sun ^1, * and X. C. Xie ^2,3

1

Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

2

Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA

3

International Center for Quantum Structures, Chinese Academy of Sciences, Beijing 100080, China

兲

“Ampere law.” When at large distance r, this induced electric field Eជ scales as 1/r², whereas the Eជ field generated from the linear spin current goes as 1 / r³.

DOI:

10.1103/PhysRevB.72.245305

I. INTRODUCTION

Recently, a new subdiscipline of condensed matter phys- ics, spintronics, is emerging rapidly and generating great interests. ^1,2 The spin current, the most important physical quantity in spintronics, has been extensively studied. Many interesting and fundamental phenomena, such as the spin Hall effect ^3–6 and the spin precession ^7,8 in systems with spin- orbit coupling, have been discovered and are under further study.

As for the charge current, the definition of the local charge current density j ជ

_{共r,t兲=Re关⌿} ^† _共r,t兲ev ជ ⌿共r,t兲兴 and its continuity equation 共d/dt兲 ␳

共r,t兲+⵱· j ជ

共r,t兲=0 is well known in physics. Here ⌿共r,t兲 is the electronic wave func- tion, v ជ = r˙ is the velocity operator, and ␳

共r,t兲=e⌿ ^† ⌿ is the charge density. This continuity equation is the consequence of charge invariance, i.e., when an electron moves from one place to another, its charge remains the same. However, in the spin transport, there are still a lot of debates over what is the correct definition for spin current. ^9,10 The problem stems from that the spin s ជ is no invariant quantity in the spin trans- port, so that the conventional defining of the spin current 具v ជ ^s ជ 典 is no conservative. Recently, some studies have begun investigation in this direction, ^11–13 e.g., a semiclassical de- scription of the spin continuity equation has been proposed, ^11,12 as well as studies introducing a conserved spin current under special circumstances. ⁹

In this paper, we study the definition of local spin current density. We find that due to the spin is vector and it has the translational and rotational motion, one has to use two quan- tities, the linear spin current and the angular spin current, to completely describe the spin transport. Here the linear spin current describe the translational motion of a spin, and the angular spin current is for the rotational motion. The defini- tion of two spin current densities are given and they appear naturally in the quantum spin continuity equation. Moreover, we predict that the angular spin current can generate an elec-

tric field similar as with the linear spin current, and thus contains physical consequences.

The paper is organized as follows. In Sec. II, we first discuss the flow of a classical vector. The flow of a quantum spin is investigated in Sec. III. In Sec. IV and Sec. V, we study the problem of an induced electric field and the heat produced by spin currents, respectively. Finally, a brief sum- mary is given in Sec. VI.

II. THE FLOW OF A CLASSICAL VECTOR Before studying the spin current in a quantum system, we first consider the classical case. Consider a classical particle having a vector m ជ 共e.g., the classical magnetic moment, etc.兲 with its magnitude 兩m ជ 兩 fixed under the particle motion. To completely describe this vector flow 关see Fig. 1共c兲兴, in addi- tion to the local vector density M ជ _共r,t兲= ␳ 共r,t兲m ជ 共r,t兲, one needs two quantities: the linear velocity v ជ 共r,t兲 and the angu- lar velocity ␻ជ 共r,t兲. Here ␳ 共r,t兲 is the particle density, and v ជ and ␻ជ describe the translational and rotational motions, re-

FIG. 1.共Color online兲 共a兲 and 共b兲 are the schematic diagram for the translational motion and the rotational motion of the classic vector mជ , respectively. 共c兲 Schematic diagram for a classic vector flow.

spectively 关see Figs. 1共a兲 and 1共b兲兴. In contrast with the flow of a scalar quantity in which one only needs one quantity, namely, the local velocity v ជ 共r,t兲, to describe the translational motion, it is essential to use two quantities v ជ 共r,t兲 and ␻ជ 共r,t兲 for the vector flow. We emphasize that it is impossible to use one vector to describe both translational and rotational mo- tions altogether.

Since 兩m ជ 兩 is a constant, the change of the local vector M ជ 共r,t兲 in the volume element ⌬V=⌬x⌬y⌬z with the time from t to t + dt can be obtained 关see Fig. 1共c兲兴:

d关M ជ _{共r,t兲⌬V兴 =} 兺

i=x,y,z

冋 ^⌬V ⌬i v

共r,t兲dtM ជ _共r,t兲

− ⌬V

⌬i v

共r + ⌬iˆ,t兲dtM ជ 共r + ⌬iˆ,t兲 册

+ ␻ជ 共r,t兲 ⫻ M ជ _{共r,t兲⌬Vdt. 共1兲} The first and the second terms on the right describe the clas- sical vector flowing in or out the volume element ⌬V and its rotational motion respectively, and both can cause a change in the local vector density. When ⌬V goes to zero, we have the vector continuity equation

d

dt M ជ 共r,t兲 = − ⵱ · v ជ 共r,t兲M ជ _{共r,t兲 +} ␻ជ 共r,t兲 ⫻ M ជ _{共r,t兲, 共2兲} where v ជ ^M ជ is a tensor, and its element 共v ជ ^M ជ _兲

_{= v}

_M

_{. Note} that this vector continuity equation is from the kinematics and the invariance of 兩m ជ 兩, and it is independent of the dy- namic laws. It is well known that the scalar 共e.g., charge e兲 continuity equation 共d/dt兲 ␳

⁺ ⵱· j ជ

= 0 is from the kinematics and the invariance of the charge e. It is independent of the external force F as well dynamic laws. In other words, even if the acceleration a ⫽F/m, the continuity equation still sur- vives. It is complete same with the vector continuity Eqs. 共2兲 or 共3兲. It is also from the kinematics and the invariance of 兩m ជ 兩. In particular, it is independent of the external force and the torque acting on the vector, as well as the dynamic laws.

Introducing j

共r,t兲=v ជ 共r,t兲M ជ 共r,t兲 and j ជ _{␻ 共r,t兲=} ␻ជ 共r,t兲

⫻M ជ 共r,t兲, then Eq. 共2兲 reduces to d

dt M ជ 共r,t兲 = − ⵜ · j

共r,t兲 + j ជ _{␻ 共r,t兲. 共3兲} Here j

= v ជ ^M ជ is from the translational motion of the classical vector m ជ , and j ជ _{␻ =} ␻ជ ⫻M ជ describes its rotational motion.

Since v ជ and ␻ជ are called as the linear velocity and the angu- lar velocity respectively, it is natural to name j

and j ជ _{␻ as the} linear and the angular current densities. Notice although the unit of j ជ _␻ is different from that of the linear spin current.

However, j ជ _␻ is indeed the current in the angular space.

It is worth to mention the following two points. 共1兲 If to consider that there are many particles in the volume element

⌬V 共i.e., the volume element ⌬V is very small macroscopi- cally but very large microscopically兲, the vector direction m ជ

, the velocity v ជ

, and the angular velocity ␻ជ

for each particle may be different, however, the vector continuity

equation 共3兲 is still valid: 共d/dt兲M ជ _{= −⵱·j}

_{+ j} ជ _{␻ , with j}

_共r,t兲

⬅具v ជ M ជ _{典=lim ⌬V→0 共兺}

_v ជ

m ជ

/ ⌬V兲 and ជ j _{␻ 共r,t兲⬅具} ␻ជ ⫻M ជ _典

= lim _⌬V→0 共兺

␻ជ

⫻m ជ

/ ⌬V兲. Here j

共r,t兲 and j ជ _␻ 共r,t兲 still de- scribe the translational and the rotational motions of the clas- sical vector. 共2兲 If one adds an arbitrary curl ⵜ⫻A ជ _{arb to the} current j

= v ជ M ជ , the continuity equation does not change.

Does this imply that the linear spin current density can be defined as j

= v ជ M ជ _{+ 共ⵜ⫻A} ជ _{arb 兲C} ជ with a constant vector C ជ _{? In} our opinion, this does not because the local spin current den- sity has physical meanings. This reason is completely same with the charge current density that cannot be redefined as ជ j

_{= ev} ជ + ⵜ⫻A ជ _{arb .}

In order to describe a scalar 共e.g., charge兲 flow, one local current j ជ

共r,t兲 is sufficient. Why is it required to introduce two quantities instead of one to describe a vector flow? The reason is that the scalar quantity only has the translational motion, but the vector quantity has two kinds of motion, the translational and the rotational. So one has to use two quan- tities, the velocity v ជ and the angular velocity ␻ជ , to describe the motion of a single vector. Correspondingly, two quanti- ties j

= v ជ M ជ _{and j} ជ _{␻ =} ␻ជ ⫻M ជ are necessary to describe the vec- tor flow.

In the steady state case, the scalar 共e.g., charge兲 continuity equation reduces into ⵱· j ជ

= 0, so the scalar current j ជ

is a conserved quantity. But for a vector flow, the linear vector current j

is not conserved since ⵱·j

= j ជ _␻ . Whether it is pos- sible to have a conserved vector current through redefinition?

Of course, this redefined vector current should have a clear physical meaning and is measurable. In our opinion, this is almost impossible in the three-dimensional space. The rea- sons are as follows. 共i兲 One cannot use a three-dimensional vector to combine both v ជ ^and ␻ជ . Therefore one can also not use a three-dimensional tensor to combine both j

= v ជ ^M ជ _and ជ j _{␻ =} ␻ជ ⫻M ជ _. 共ii兲 Consider an example, as shown in Fig. 3共a兲, a one-dimensional classical vector flowing along the x axis.

When x⬍0, the vector’s direction is in the +x axis. At 0 ⬍x⬍L, the vector rotates in accompany with its transla- tional motion. When x⬎L, its direction is along the +y axis.

Since for x ⬍0 and x⬎L the vector has no rotational motion, the definition of the vector current is unambiguous, and the nonzero element is j

for x⬍0, and j

for x ⬎L. Therefore, the vector current is obviously different for x ⬍0 and x⬎L, and the vector current is nonconservative.

In our opinion, j

= v ជ M ជ _{and j} ជ _{␻ =} ␻ជ ⫻M ជ already have clear physical meanings. They also completely and sufficiently de- scribe a vector flow, and they can determine any physical effects caused by the vector flow 共see Secs. IV and V兲. One may not need to enforce a conserved current. In particular, as shown in the example of Fig. 3共a兲, sometimes it is impos- sible to introduce a conserved current. ¹⁴

III. THE FLOW OF A QUANTUM SPIN

Now we study the electronic spin s ជ in the quantum case.

Consider an arbitrary wave function ⌿共r,t兲. The local spin

density s ជ at the position r and time t is s ជ 共r,t兲

= ⌿ ^† 共r,t兲s ជ ˆ⌿共r,t兲, where s ជ ˆ =共ប/2兲 ␴ជ ^{ˆ with} ␴ជ ˆ being the Pauli matrices. The time-derivative of s ជ 共r,t兲 is

d

dt s ជ 共r,t兲 = ប

2 再 ^{冋 dt d ⌿ † 册 ␴ជ ˆ ⌿ + ⌿ † ␴ជ ˆ d dt ⌿} 冎 ^{. 共4兲}

From the Schrödinger equation, we have 共d/dt兲⌿共r,t兲

= 共1/iប兲H⌿共r,t兲 and 共d/dt兲⌿ ^† 共r,t兲=共1/−iប兲关H⌿共r,t兲兴 ^† . Notice here the transposition in the symbol † only acts on the spin indexes. By using the above two equations, Eq. 共4兲 changes into

共d/dt兲s ជ 共r,t兲 = 关⌿ ^† ␴ជ ˆ H⌿ − 共H⌿兲 ^† ␴ជ ˆ ⌿兴/2i. 共5兲 In the derivation below, we use the following Hamiltonian:

H = p ជ ²

2m + V 共r兲 + ␴ជ ^{ˆ · B} ជ ₊ ␣

ប zˆ · 共 ␴ជ ˆ ⫻ p ជ 兲. 共6兲 Note that our results are independent of this specific choice of the Hamiltonian. In Eq. 共6兲 the first and second terms are the kinetic energy and potential energy. The third term is the Zeeman energy due to a magnetic field, and the last term is the Rashba spin-orbit coupling, ^15,16 which has been exten- sively studied recently. ^4,7,8 Next we substitute the Hamil- tonian 共6兲 into Eq. 共5兲, and Eq. 共5兲 reduces to

d dt s ជ ^{= − ប}

2 ⵱ · Re 再 ^{⌿ † 冋 m p ជ + ␣ ប 共zˆ ⫻ ␴ជ ˆ 兲 册 ␴ជ ˆ ⌿} 冎

+ Re 再 ^{⌿ † 冋 B ជ + ␣ ប p ជ ⫻ zˆ 册 ⫻ ␴ជ ˆ ⌿} 冎 ^{. 共7兲}

Introducing a tensor j

共r,t兲 and a vector j ជ _{␻ 共r,t兲:}

j

共r,t兲 = Re 再 ^{⌿ † 冋 m p ជ + ␣ ប 共zˆ ⫻ ␴ជ ˆ 兲 册 s ជ ˆ⌿} 冎 ^{, 共8兲}

ជ j _{␻ 共r,t兲 = Re} 再 ^{⌿ † 2 ប 冋 B ជ + ␣ ប 共p ជ ⫻ zˆ兲 册 ⫻ s ជ ˆ⌿} 冎 ^{, 共9兲}

then Eq. 共7兲 reduces to d

dt s ជ 共r,t兲 = − ⵜ · j

共r,t兲 + j ជ _{␻ 共r,t兲, 共10兲} or it can also be rewritten in the integral form

d

dt 冕冕冕

s ជ dV = − 冖

dS ជ _{· j}

₊ 冕冕冕

ជ j _{␻ dV. 共11兲} Due to the fact that v ជ ˆ =共d/dt兲r=p ជ ^{/ m +} 共 ␣ ^/ ប兲共zˆ⫻ ␴ជ ˆ 兲 and 共d/dt兲 ␴ជ ˆ =共1/iប兲关 ␴ជ ˆ ,H兴=共2/ប兲关B ជ _{+ 共} ␣ ^/ ប兲p ជ ⫻zˆ兴⫻ ␴ជ ˆ , Eqs. 共8兲 and 共9兲 become

j

共r,t兲 = Re兵⌿ ^† 共r,t兲v ជ ˆ s ជ ˆ⌿共r,t兲其, 共12兲

ជ j _␻ 共r,t兲 = Re兵⌿ ^† 共ds ជ ˆ/dt兲⌿其 = Re兵⌿ ^† ␻ជ ˆ ⫻ s ជ ˆ⌿其, 共13兲 where ␻ជ ˆ ⬅共2/ប兲关B ជ _{+ 共} ␣ / ប兲共p ជ ⫻zˆ兲兴 is the angular velocity op- erator. We emphasize that those results, 共10兲, 共12兲, and 共13兲,

are valid in general. They are independent of the special choice of Hamiltonian 共6兲. For example, in the case with a vector potential A ជ _{, the general spin-orbit coupling}

␣␴ជ ˆ ·关p ជ ⫻ⵜV共r兲兴, and so on, ¹⁷ the results still hold. Notice that for the Hamiltonian 共6兲, one has Re兵⌿ ^† vˆ

sˆ

⌿其

= Re兵⌿ ^† sˆ

vˆ

⌿其 and Re兵⌿ ^† ␻ជ ˆ ⫻s ជ ˆ⌿其=−Re兵⌿ ^† s ជ ˆ ⫻ ␻ជ ˆ ⌿其.

If Re兵⌿ ^† vˆ

sˆ

⌿其⫽Re兵⌿ ^† sˆ

vˆ

⌿其 and Re兵⌿ ^† ␻ជ ˆ ⫻s ជ ˆ⌿其

⫽−Re兵⌿ ^† s ជ ˆ ⫻ ␻ជ ˆ ⌿其 for certain Hamiltonians, Eqs. 共12兲 and 共13兲 will change to

j

共r,t兲 = Re 再 ^{⌿ † 1 2 关v ជ ˆ s ជ ˆ + 共s ជ ˆv ជ ˆ 兲}

^兴⌿ 冎 ^{, 共14兲}

ជ j _␻ 共r,t兲 = Re兵⌿ ^† 共ds ជ ˆ/dt兲⌿其 = Re 再 ^{⌿ † 1 2 关 ␻ជ ˆ ⫻ s ជ ˆ − s ជ ˆ ⫻ ␻ជ ˆ 兴⌿} 冎 ^.

共15兲 Equation 共10兲 is the quantum spin continuity equation, which is the same with the classic vector continuity equation 共3兲 although the derivation process is very different. In some previous works, this equation has also been obtained in the semiclassical case. ^11,12 Here we emphasize that this spin con- tinuity equation 共10兲 is the consequence of invariance of the spin magnitude 兩s ជ 兩, i.e., when an electron makes a motion, either translation or rotation, its spin magnitude 兩s ជ 兩 =ប/2 re- mains a constant. Equation 共10兲 should also be independent with the force 共i.e., the potential兲 and the torque, as well the the dynamic law. The two quantities j

共r,t兲 and j ជ _{␻ 共r,t兲 in} Eq. 共10兲, which are defined in Eqs. 共12兲 and 共13兲 respec- tively, describe the translational and rotational motion 共pre- cession兲 of a spin at the location r and the time t. They will be named the linear and the angular spin current densities accordingly, similar as v ជ ^and ␻ជ are called the linear and the angular velocities. In fact, j ជ _␻ is called the spin torque in a recent work. ¹¹ We consider both j ជ _{␻ and j}

describing the motion of a spin, capable of inducing an electric field 共see Sec. IV 兲, and so on. Namely, both of them play a parallel role in contributing to a physical quantity. Therefore, it is better to name j

and j ជ _␻ both as spin currents. Otherwise, when one calculates the contribution of spin current to a given physical quantity, one may forget to include the contribution by the angular spin current.

The linear spin current j

共r,t兲 is identical with the conventional spin current investigated in recent studies. ⁴ From j

共r,t兲, the total linear spin current along i direction 共i=x,y,z兲 is I ជ

共i,t兲=兰兰dSiˆ·j

共r,t兲. To assume I ជ

共i,t兲 independent on t and i 共e.g., in the case of the steady state and without spin flip 兲, one has I ជ

₌ 共1/L兲兰兰兰dViˆ·j

共r,t兲=共1/L兲兰兰兰dVRe⌿ ^† vˆ

s ជ ˆ⌿

= 共1/L兲兰兰兰dV⌿ ^{† 1} ₂ 共vˆ

s ជ ^{ˆ +s} ជ ^ˆvˆ

兲⌿= 具 ¹ ₂ 共vˆ

s ជ ^{ˆ +s} ជ ^ˆvˆ

兲 典 , where L is sample length in the i direction. This definition is the same as in recent publications. ⁴

Next, we discuss certain properties of j

共r,t兲 and j ជ _{␻ 共r,t兲.}

Notice that j ជ _␻ 共r,t兲 which describe the rotational motion 共pre-

cession 兲 of the spin plays a parallel role in comparison with

the conventional linear spin current j

共r,t兲 for the spin trans- port. 共1兲 Similar to the classical case, it is necessary to intro- duce the two quantities j

共r,t兲 and j ជ _␻ 共r,t兲 to completely de- scribe the motion of a quantum spin. 共2兲 The linear spin current is a tensor. Its element, e.g., j

, represents an elec- tron moving along the x direction with its spin in the y di- rection 关see Fig. 2共a兲兴. The angular spin current j ជ _␻ is a vector.

In Fig. 2共b兲, its element j _␻,x describes the rotational motion of the spin in the y direction and the angular velocity ␻ជ ^{in the}

−z direction. 共3兲 From the linear spin current density j

共r,t兲, one can calculate 共or say how much兲 the linear spin current I ជ

flowing through a surface S 关see Fig. 2共d兲兴: I ជ

_{= 兰兰}

_dS ជ _{· j}

_. However, the behavior for the angular spin current is differ- ent. From the density j ជ _␻ 共r,t兲, it is meaningless to determine how much the angular spin current flowing through a surface S because j ជ _␻ is the current in the angular space. On the other hand, one can calculate the total angular spin current I ជ _␻

_{in a} volume V from j ជ _{␻ : I} ជ _␻

_{= 兰兰兰}

ជ _{j ␻} 共r,t兲dV. 共4兲 It is easy to prove that the spin currents in the present definitions of Eqs. 共12兲 and 共13兲 are invariant under a space coordinate transforma- tion as well the gauge transformation. 共5兲 If the system is in a steady state, j

and j ជ _␻ are independent of the time t, and 共d/dt兲s ជ 共r,t兲=0. Then the spin continuity equation 共10兲 re- duces to ⵱·j

= j ជ _{␻ or 养}

_dS ជ _{· j}

_{= 兰兰兰}

ជ _{j ␻} dV. This means that the total linear spin current flowing out of a closed surface is equal to the total angular spin current enclosed. If to further consider a quasi-one-dimensional 共1D兲 system 关see Fig.

2共d兲兴, then one has I ជ

− I ជ

_{= I} ជ _␻

_. 共6兲 The linear spin current density j

= Re兵⌿ ^† v ជ ˆ s ជ ˆ⌿其 gives both the spin direction and the direction of spin movement, so it completely describes the translational motion. However, the angular spin current den- sity, j ជ _{␻ = Re兵⌿} ^† _共ds ជ ˆ /dt兲⌿其=Re兵⌿ ^† ␻ជ ˆ ⫻s ជ ˆ⌿其 involves the vec- tor product of ␻ជ ˆ ⫻s ជ ˆ, not the tensor ␻ជ ^{ˆ s} ជ ˆ. Is it correct or suffi- cient to describe the rotational motion? For example, the rotational motion of Fig. 2共b兲 with the spin s ជ in the y direc- tion and the angular velocity ␻ជ in the −z direction is different from the one in Fig. 2共c兲 in which s ជ is in the z direction and

␻ជ is in the y direction, but their angular spin currents are completely the same. Shall we distinguish them? It turns out that the physical results produced by the above two rotational motions 共Figs. 2共b兲 and 2共c兲兲 are indeed the same. For in- stance, the induced electric field by them is identical since a spin s ជ has only the direction but no size 共see detail discussion below兲. Thus, the vector j ជ _␻ is sufficient to describe the rota- tional motion, and no tensor is necessary.

Now we give an example of applying the above formulas, 共12兲 and 共13兲, to calculate the spin currents. Let us consider a quasi-1D quantum wire having the Rashba spin orbit cou- pling, and its Hamiltonian is

H = p ជ ²

2m + V共y,z兲 + ␴

2ប 关 ␣ 共x兲p

+ p

␣ 共x兲兴 + ប ² k

² 2m , 共16兲 where k

共x兲⬅ ␣ 共x兲m/ប ² . ␣ 共x兲=0 for x⬍0 and x⬎L, and

␣ 共x兲⫽0 while 0⬍x⬍L. The other Rashba term −共 ␣ /ប兲 ␴

p

is neglected because the z direction is quantized. ⁷ Let ⌿ be a stationary wave function

⌿共r兲 =

2

2 e

冉 ^{e e −i}

^{兰 兰}

^{共x兲dx 共x兲dx} 冊 ^{␸ 共y,z兲, 共17兲}

where ␸ 共y,z兲 is the bound state wave function in the con- fined y and z directions. ⌿共r兲 represents the spin motion as shown in Fig. 3共a兲, in which the spin moves along the x axis, as well the spin precession in the x-y plane in the region 0 ⬍x⬍L. ^7,8 Using Eqs. 共12兲 and 共13兲, the spin current den- sities of the wave function ⌿共r兲 are easily obtained. There are only two nonzero elements of j

共r兲:

j

共r兲 = ប ² k

2m 兩 ␸ 共y,z兲兩 ² cos 2 ␾ 共x兲, 共18兲 j

共r兲 = ប ² k

2m 兩 ␸ 共y,z兲兩 ² sin 2 ␾ 共x兲. 共19兲 The nonzero elements of j ជ _{␻ 共r兲 are}

FIG. 2. 共Color online兲 共a兲 The linear spin current element js,xy. 共b兲 and 共c兲 The angular spin current element j␻,x. 共d兲 The spin current in a quasi 1D quantum wire.共e兲 The currents of two mag- netic charges that are equivalent to a angular MM current.

FIG. 3. 共Color online兲 共a兲 Schematic diagram for the spin mov- ing along the x axis, with the spin precession共rotational motion兲 in the x-y plane while 0⬍x⬍L. 共b兲 A 1D wire of electric dipole mo- ment pជe. This configuration will generate an electric field equivalent to the field from the spin currents in共a兲.

j _␻x 共r兲 = − ប ² kk

共x兲

m 兩 ␸ 共y,z兲兩 ² sin 2 ␾ 共x兲, 共20兲 j _␻y 共r兲 = ប ² kk

m 兩 ␸ 共y,z兲兩 ² cos 2 ␾ 共x兲, 共21兲 where ␾ 共x兲=兰 0

x

k

共x兲dx. Those spin current densities confirm with the intuitive picture of an electron motion, precession in the x-y plane in 0⬍x⬍L and movement in the x direction 关see Fig. 3共a兲兴.

In particular, in the region of x⬍0 and x⬎L,

␣ 共x兲=k

共x兲=0, and ␴ ˆ is a good quantum number, hence, ជ j _␻ = 0. In this case, the definition of the spin current j

is unambiguous. However, the spin currents are different in x⬍0 and x⬎L except for ␾ 共L兲=n ␲ 共n=0, ±1, ±2, ...兲. This is clearly seen from Fig. 3共a兲. Therefore, through this ex- ample, one can conclude that it is sometimes impossible to define a conserved spin current. ¹⁴ The example of Fig. 3 共a兲 indeed exists and has been studied before. ^7,8

Above discussion shows that the linear spin current j

= Re兵⌿ ^† v ជ ^{ˆ s} ជ ˆ⌿其 and the angular spin current j ជ _␻

= Re兵⌿ ^† ␻ជ ˆ ⫻s ជ ˆ⌿其 have clear physical meanings, representing the translational motion and the rotational motion 共preces- sion 兲 respectively. They completely describe the flow of a quantum spin. Any physical effects of the spin currents, such as the induced electric field, can be expressed by j

and j ជ _{␻ .}

IV. SPIN CURRENTS INDUCED ELECTRIC FIELDS Recently, theoretic studies have suggested that the 共linear兲 spin current can induce an electric field E ជ _. ^18–20 Can the an- gular spin current also induce an electric field? If so, this gives a way of detecting the angular spin current. Following, we study this question by using the method of equivalent magnetic charge. ²¹ Let us consider a steady-state angular spin current element j ជ _␻ dV at the origin. Associated with the spin s ជ , there is a magnetic moment 共MM兲 m ជ ^{= g} ␮

␴ជ

= 共2g ␮

/ ប兲s ជ where ␮

is the Bohr magneton. Thus, corre- sponding to j ជ _␻ , there is also a angular MM current j ជ

_{␻ dV}

= 共2g ␮

/ ប兲j ជ _␻ dV. From above discussions, we already know that j ជ

_{␻ 共or j} ជ _␻ 兲 comes from the rotational motion of a MM m ជ 共or s ជ 兲 关see Figs. 2共b兲 and 2共c兲兴, and j ជ

_{␻ =} ␻ជ ⫻m ជ 共or ជ j _{␻ =} ␻ជ ⫻s ជ 兲. Under the method of equivalent magnetic charge, the MM m ជ is equivalent to two magnetic charges: one with magnetic charge +q located at ␦ ^nˆ

and the other with −q at

− ␦ ^nˆ

关see Fig. 2共e兲兴. nˆ

is the unit vector of m ជ and ␦ ^{is a tiny} length. The angular MM current j ជ

_␻ is equivalent to two magnetic charge currents: one is j ជ _+q = nˆ

q ␦ 兩 ␻ជ 兩sin ␪ ^{at the lo-} cation ␦ ^nˆ

, the other is j ជ _{−q = nˆ}

_q ␦ 兩 ␻ជ 兩sin ␪ at − ␦ ^nˆ

关see Fig.

2共e兲兴, with nˆ

being the unit vector of j ជ

_{␻ and} ␪ ^{the angle} between ␻ជ ^{and m} ជ . In our previous work, ²⁰ by using the rela- tivistic theory, we have arrived at the formulae for the in- duced electric field by a magnetic charge current. The elec- tric field induced by j ជ

_␻ dV can be calculated by adding the contributions from the two magnetic charge currents. Let

␦ →0, and note that 2q ␦ →兩m ជ 兩 and 兩 ␻ជ 兩兩m ជ 兩sin ␪ ⁼ 兩j ជ

_{␻ 兩, we ob-} tain the electric field E ជ _␻ generated by an element of the an- gular spin current j ជ _{␻ dV:}

E ជ _{␻ =} ⁻ ␮ 0

4 ␲ 冕 ^{ជ j}

^{␻ dV r 3 ⫻ r = − ␮ 0 h g ␮}

冕 ^{ជ j ␻ dV ⫻ r r 3 .}

共22兲 We also rewrite the electric field E ជ

generated by an element of the linear spin current using the tensor j

: ²⁰

E ជ

₌ ⁻ ␮ 0 g ␮

h ⵱ ⫻ 冕 ^j

^{dV · r r 3 . 共23兲}

Below we emphasize three points. 共i兲 In the large r case, the electric field E ជ _␻ decays as 1 / r ² . Note that the field from a linear spin current E ជ

goes as 1 / r ³ . In fact, in terms of gen- erating an electric field, the angular spin current is as effec- tive as a magnetic charge current. 共ii兲 In the steady-state case, the total electric field E ជ

_{= E} ជ _{␻ + E} ជ

contains the property 养

E ជ

_{· dl} ជ = 0, where C is an arbitrary close contour not pass- ing through the region of spin current. However, for each E ជ _␻ or E ជ

_{, 养}

_E ជ _{␻ · dl} ជ _{or 养}

_E ជ

_{· dl} ជ can be nonzero. 共iii兲 As mentioned above, an angular spin current j ជ _␻ may consist of different ␻ជ and s ជ 关see Figs. 2共b兲 and 2共c兲兴. However, the resulting elec- tric field only depends on j ជ _␻ = ␻ជ ⫻s ជ . This is because a spin vector contains only a direction and a magnitude, but not a spatial size 共i.e., the distance ␦ approaches to zero兲. In the limit ␦ →0, both magnetic charge currents j ជ _{±q reduce to}

␻ជ ⫻m ជ / 2 at the origin. Therefore, the overall effect of the rotational motion is only related to ␻ជ ⫻m ជ , not separately on

␻ជ ^{and m} ជ . Hence it is enough to describe the spin rotational motion by using a vector ␻ជ ⫻s ជ , instead of a tensor ␻ជ ^s ជ ^.

Due to the fact that the direction of s ជ can change during the particle motion, the linear spin current density j

is not a conserved quantity. It is always interesting to uncover a con- served physical quantity from both theoretical and experi- mental points of view. Let us apply ⵜ· acting on two sides of Eq. 共10兲, we have

d

dt 共⵱ · s ជ 兲 + ⵱ · 共⵱ ⬘ ^{· j}

^{− j ជ} ␻ 兲 = 0, 共24兲 where ⵜ ⬘ ^{· j}

^{means that ⵱} ⬘ acts on the second index of j

, i.e., 共⵱ ⬘ ^{· j}

^兲

^{= 兺}

^共d/dj兲j

with i , j 苸共x,y,z兲. To introduce ជ j _{⵱·sជ = ⵱} ⬘ ^{· j}

^{− j ជ} ␻ . Note it is different from ⵱·j− j ជ _␻ . In a steady state, ⵱·j− j ជ _␻ = 0, however j ជ _{⵱·sជ = ⵱} ⬘ ^{· j − j ជ} ␻ is usually nonzero.

By using j ជ _⵱·sជ , the above equation reduces to d

dt 共⵱ · s ជ 兲 + ⵱ · j ជ _⵱·sជ = 0. 共25兲 This means that the current j ជ _⵱·sជ of the spin divergence is a conserved quantity in the steady state case. In fact,

−⵱·s ជ 共r,t兲 represents an equivalent magnetic charge, so j ជ _⵱·sជ

can also be named the magnetic charge current density.

Moreover, the total electric field produced by j

and j ជ _{␻ can} be rewritten as

E ជ

_{= E} ជ

_{+ E} ជ _␻

= ␮ 0 g ␮

h 冕 ^{共⵱ ⬘ · j}

^{− j ជ ␻ 兲dV ⫻ r r 3}

= ␮ 0 g ␮

h 冕 ^{ជ j ⵱·sជ dV ⫻ r r 3 . 共26兲}

So the total electric field E ជ

only depends on the current j ជ _⵱·sជ of the spin divergence. Note that E ជ

can be measured experi- mentally in principle. Through the measurement of E ជ

_共r兲, ជ j _⵱·sជ can be uniquely obtained.

In the following, let us calculate the induced electric fields at the location r = 共x,y,z兲 by the spin currents in the example of Fig. 3共a兲. Substituting the spin currents of Eqs. 共18兲–共21兲 into Eqs. 共22兲 and 共23兲 and assuming the transverse sizes of the 1D wire are much smaller than

y ² + z ² , the induced fields E ជ _{␻ and E} ជ

can be obtained straightforwardly. Then the total electric field E ជ

_{= E} ជ _{␻ + E} ជ

_is

E ជ

_{= a} ^បk

m ⵱ 冕 关共x − x ^{z sin 2} ⬘ ^{兲 2 + y ␾ 共x 2 + z ⬘ 兲 2 兴 3/2 dx} ⬘

= a ⵜ 冕 ^{共V ជ ⫻ S ជ 兲 ·} 兩r − r ^{r − r} ⬘ ^⬘ 兩 ³ dx ⬘ ^, 共27兲 where V ជ ₌ 共បk/m,0,0兲, S ជ _{= (cos 2} ␾ 共x ⬘ ^{兲,sin 2 ␾ 共x} ⬘ ^兲,0), r ⬘ ^{= 共x} ⬘ , 0 , 0兲, the constant a= ␮ 0 g ␮

␳

/ 4 ␲ ^{, and} ␳

is the lin- ear density of moving electrons under the bias of an external voltage. The total electric field E ជ

represents the one gener- ated by a 1D wire of electric dipole moment p ជ

= (0 , 0 , c sin 2 ␾ 共x兲) at the x axis 关see Fig. 3共b兲兴, where c is a constant. It is obvious that ⵱⫻E ជ

_{= 0, i.e., 养}

_E ជ

_{· dl} ជ _{= 0.}

However, in general 养

E ជ _{␻ · dl} ជ _{and 养}

_E ជ

_{· dl} ជ are separately nonzero.

Finally, we estimate the magnitude of E ជ

. We use param- eters consistent with realistic experimental samples. Take the Rashba parameter ␣ = 3 ⫻10 ⁻¹¹ eV m 共corresponding to k

= 1 / 100 nm for m = 0.036m

兲, ␳

= 10 ⁶ / m 共i.e., one moving electron per 1000 nm in length兲, and k=k

= 10 ⁸ / m. The electric potential difference between the two points A and B 关see Fig. 3共b兲兴 is about 0.01 ␮ V, where the positions of A and B are 共1/2k

兲共 ␲ / 2 , 0 , 0.01兲 and 共1/2k

兲共 ␲ / 2 , 0 , −0.01兲.

This value of the potential is measurable with today’s

technology. ¹⁹ Furthermore, with the above parameters the electric field E ជ

at A or B is about 5 V / m which is rather large.

V. HEAT PRODUCED BY SPIN CURRENTS Here we consider another physical effect, the heat pro- duced by the spin currents. Assume a uniform isotropic con- ductor having a linear spin current j

and a charge current j ជ

_, and considering the simple case that there exists no spin flip process 共i.e., s ជ is conserved 兲 so that j ជ _␻ = 0. Then the produced Joule heat Q in unit volume and in unit time is

Q = 兺

i=x,y,z

2 ␳

4 关共兩j ជ

_{兩 + j}

_兲 ² _{+ 共兩j} ជ

_{兩 − j}

_兲 ² _兴

= ␳ 冉 ^兺

^j

^{2 + 兺}

^j

² 冊 ^{⬅ ␳ 共j}

^{2 + j ជ}

^{2 兲,}

where ␳ is the resistivity. The term ␳ ជ ^j

² is the Joule heat from the charge current which is well known, and the other term

␳ ^j

2 is the heat produced by the linear spin current. So the produced heat can indeed be expressed by j

共for the case of ជ j _␻ = 0兲. Also note the produced heat Q depends on j

2 , whereas the induced electric field depends on ⵱ ⬘ ^{· j}

^{− j ជ} ␻ .

VI. CONCLUSION

In summary, we find that in order to completely describe the spin flow 共including both classic and quantum flows兲, apart from the conventional spin current 共or linear spin cur- rent兲, one has to introduce another quantity, the angular spin current. The angular spin current describes the rotational mo- tion of the spin, and it plays a parallel role in comparison with the conventional linear spin current for the spin trans- lational motion. In particular, we point out that the angular spin current 共or the spin torque as being called in other works兲 can also induce an electric field. The formula for the generated electric field E ជ _␻ is derived and E ជ _␻ scales as 1 / r ² at large r. In addition, a conserved quantity, the current j ជ _⵱·sជ of the spin divergence, is discovered, and the total electric field only depends on j ជ _⵱·sជ .

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from the Chinese Academy of Sciences and NSFC under Grant Nos.

90303016, 10474125, and 10525418. X.C.X. was supported by the US-DOE under Grant No. DE-FG02-04ER46124, NSF CCF-0524673, and NSF-MRSEC under Grant No.

DMR-0080054.

*

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