Download Definition of the spin current: The angular spin current and its

PHYSICAL REVIEW B 72, 245305 共2005兲
Definition of the spin current: The angular spin current and its physical consequences
Qing-feng Sun1,* and X. C. Xie2,3
1Beijing
National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
2Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
3
International Center for Quantum Structures, Chinese Academy of Sciences, Beijing 100080, China
共Received 14 June 2005; revised manuscript received 22 September 2005; published 2 December 2005兲
We find that in order to completely describe the spin transport, apart from spin current 共or linear spin
current兲, one has to introduce the angular spin current. The two spin currents, respectively, describe the
translational and rotational motion 共precession兲 of a spin. The definitions of these spin current densities are
given and their physical properties are discussed. Both spin current densities appear naturally in the spin
continuity equation. In particular, we predict that the angular spin current 共or the spin torque as called in
ជ . The formula for the
previous works兲, similar to the linear spin current, can also induce an electric field E
ជ
induced electric field E by the angular spin current element is derived, playing the role of “Biot-Savart law” or
ជ scales as 1 / r2, whereas the Eជ field
“Ampere law.” When at large distance r, this induced electric field E
3
generated from the linear spin current goes as 1 / r .
DOI: 10.1103/PhysRevB.72.245305
PACS number共s兲: 73.23.⫺b, 72.25.⫺b, 85.75.⫺d
I. INTRODUCTION
Recently, a new subdiscipline of condensed matter physics, 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 effect3–6 and the spin precession7,8 in systems with spinorbit coupling, have been discovered and are under further
study.
As for the charge current, the definition of the local
charge current density ជj e共r , t兲 = Re关⌿†共r , t兲evជ ⌿共r , t兲兴 and its
continuity equation 共d / dt兲␳e共r , t兲 + ⵱ · ជj e共r , t兲 = 0 is well
known in physics. Here ⌿共r , t兲 is the electronic wave function, vជ = ṙ is the velocity operator, and ␳e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 sជ is no invariant quantity in the spin transport, so that the conventional defining of the spin current
具vជ sជ典 is no conservative. Recently, some studies have begun
investigation in this direction,11–13 e.g., a semiclassical description 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 quantities, 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 definition 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 elec1098-0121/2005/72共24兲/245305共7兲/$23.00
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 summary 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
ជ 共e.g., the classical magnetic moment, etc.兲
having a vector m
ជ 兩 fixed under the particle motion. To
with its magnitude 兩m
completely describe this vector flow 关see Fig. 1共c兲兴, in addiជ 共r , t兲 = ␳共r , t兲m
ជ 共r , t兲, one
tion to the local vector density M
needs two quantities: the linear velocity vជ 共r , t兲 and the anguជ 共r , t兲. Here ␳共r , t兲 is the particle density, and vជ
lar velocity ␻
ជ describe the translational and rotational motions, reand ␻
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
ជ , respectively. 共c兲 Schematic diagram for a classic vector
vector m
flow.
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PHYSICAL REVIEW B 72, 245305 共2005兲
Q.-F. SUN AND X. C. XIE
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 vជ 共r , t兲, to describe the translational
ជ 共r , t兲
motion, it is essential to use two quantities vជ 共r , t兲 and ␻
for the vector flow. We emphasize that it is impossible to use
one vector to describe both translational and rotational motions altogether.
ជ 兩 is a constant, the change of the local vector
Since 兩m
ជ
M 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:
ជ 共r,t兲⌬V兴 =
d关M
兺
i=x,y,z
−
冋
⌬V
ជ 共r,t兲
vi共r,t兲dtM
⌬i
⌬V
ជ 共r + ⌬î,t兲
vi共r + ⌬î,t兲dtM
⌬i
册
ជ 共r,t兲⌬Vdt.
ជ 共r,t兲 ⫻ M
+␻
共1兲
The first and the second terms on the right describe the classical 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 ជ
ជ 共r,t兲 + ␻
ជ 共r,t兲,
ជ 共r,t兲 ⫻ M
M 共r,t兲 = − ⵱ · vជ 共r,t兲M
dt
共2兲
ជ is a tensor, and its element 共vជ M
ជ 兲 = v M . Note
where vជ M
ij
i j
that this vector continuity equation is from the kinematics
ជ 兩, and it is independent of the dyand the invariance of 兩m
namic laws. It is well known that the scalar 共e.g., charge e兲
continuity equation 共d / dt兲␳e + ⵱ · ជj e = 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 survives. 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
ជ 兩. In particular, it is independent of the external force and
兩m
the torque acting on the vector, as well as the dynamic laws.
ជ 共r , t兲 and ជj 共r , t兲 = ␻
ជ 共r , t兲
Introducing js共r , t兲 = vជ 共r , t兲M
␻
ជ
⫻ M 共r , t兲, then Eq. 共2兲 reduces to
d ជ
M 共r,t兲 = − ⵜ · js共r,t兲 + ជj ␻共r,t兲.
dt
共3兲
ជ is from the translational motion of the classical
Here js = vជ M
ជ describes its rotational motion.
ជ , and ជj␻ = ␻
ជ ⫻M
vector m
ជ are called as the linear velocity and the anguSince vជ and ␻
lar velocity respectively, it is natural to name js and ជj ␻ as the
linear and the angular current densities. Notice although the
unit of ជj ␻ is different from that of the linear spin current.
However, ជj ␻ 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 macroscopiជ i,
cally but very large microscopically兲, the vector direction m
ជ i for each
the velocity vជ i, and the angular velocity ␻
particle may be different, however, the vector continuity
ជ = −⵱ · j + ជj , with j 共r , t兲
equation 共3兲 is still valid: 共d / dt兲M
s
␻
s
ជ 典 = lim
ជ典
ជ
ជ
ជ
ជ
⬅ 具vជ M
共兺
m
/
⌬V兲
and
j
共r
,
t兲
⬅
具
␻
⫻M
v
⌬V→0
i i i
␻
ជ
ជ i⫻m
ជ i / ⌬V兲. Here js共r , t兲 and j␻共r , t兲 still de= lim⌬V→0共兺i␻
scribe the translational and the rotational motions of the clasជ to the
sical vector. 共2兲 If one adds an arbitrary curl ⵜ ⫻ A
arb
ជ
current js = vជ M , the continuity equation does not change.
Does this imply that the linear spin current density can be
ជ + 共ⵜ ⫻ Aជ 兲C
ជ with a constant vector C
ជ ? In
defined as js = vជ M
arb
our opinion, this does not because the local spin current density has physical meanings. This reason is completely same
with the charge current density that cannot be redefined as
ជje = evជ + ⵜ ⫻ Aជ arb.
In order to describe a scalar 共e.g., charge兲 flow, one local
current ជj e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 quanជ , to describe
tities, the velocity vជ and the angular velocity ␻
the motion of a single vector. Correspondingly, two quantiជ and ជj = ␻
ជ
ties js = vជ M
␻ ជ ⫻ M are necessary to describe the vector flow.
In the steady state case, the scalar 共e.g., charge兲 continuity
equation reduces into ⵱ · ជj e = 0, so the scalar current ជj e is a
conserved quantity. But for a vector flow, the linear vector
current js is not conserved since ⵱ · js = ជj ␻. Whether it is possible 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 reasons are as follows. 共i兲 One cannot use a three-dimensional
ជ . Therefore one can also not
vector to combine both vជ and ␻
ជ and
use a three-dimensional tensor to combine both js = vជ M
ជ . 共ii兲 Consider an example, as shown in Fig. 3共a兲, a
ជj␻ = ␻
ជ ⫻M
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 translational 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 jxx for x ⬍ 0, and jxy for x ⬎ L. Therefore,
the vector current is obviously different for x ⬍ 0 and x ⬎ L,
and the vector current is nonconservative.
ជ and ជj = ␻
ជ
In our opinion, js = vជ M
␻ ជ ⫻ M already have clear
physical meanings. They also completely and sufficiently describe 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 impossible to introduce a conserved current.14
III. THE FLOW OF A QUANTUM SPIN
Now we study the electronic spin sជ in the quantum case.
Consider an arbitrary wave function ⌿共r , t兲. The local spin
density sជ at the position r and time t is sជ共r , t兲
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DEFINITION OF THE SPIN CURRENT: THE ANGULAR…
ជ̂ with ␴ជ̂ being the Pauli
= ⌿†共r , t兲sជ̂⌿共r , t兲, where sជ̂ = 共ប / 2兲␴
matrices. The time-derivative of sជ共r , t兲 is
d
ប
sជ共r,t兲 =
dt
2
再冋 册
冎
d
d †
ជ̂ ⌿ + ⌿†␴ជ̂ ⌿ .
⌿ ␴
dt
dt
共4兲
are valid in general. They are independent of the special
choice of Hamiltonian 共6兲. For example, in the case with a
ជ , the general spin-orbit coupling
vector potential A
␣␴ជ̂ · 关pជ ⫻ 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⌿†v̂iŝ j⌿其
ជ̂ ⫻ sជ̂⌿其 = −Re兵⌿†sជ̂ ⫻ ␻
ជ̂ ⌿其.
and
Re兵⌿†␻
= Re兵⌿†ŝ jv̂i⌿其
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
ជ̂ ⫻ sជ̂⌿其
and
Re兵⌿†␻
Re兵⌿†v̂iŝ j⌿其 ⫽ Re兵⌿†ŝ jv̂i⌿其
†ជ̂
ជ̂ ⌿其 for certain Hamiltonians, Eqs. 共12兲 and
⫽ −Re兵⌿ s ⫻ ␻
共13兲 will change to
ជ̂ H⌿ − 共H⌿兲†␴ជ̂ ⌿兴/2i.
共d/dt兲sជ共r,t兲 = 关⌿†␴
1
js共r,t兲 = Re ⌿† 关vជ̂ sជ̂ + 共sជ̂vជ̂ 兲T兴⌿ ,
2
If
再
共5兲
In the derivation below, we use the following Hamiltonian:
H=
␣
pជ 2
ជ̂ · Bជ + ẑ · 共␴ជ̂ ⫻ pជ 兲.
+ V共r兲 + ␴
2m
ប
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 extensively studied recently.4,7,8 Next we substitute the Hamiltonian 共6兲 into Eq. 共5兲, and Eq. 共5兲 reduces to
再冋
册 冎
d
pជ ␣
ប
ជ̂ 兲 ␴ជ̂ ⌿
sជ = − ⵱ · Re ⌿†
+ 共ẑ ⫻ ␴
dt
2
m ប
再冋
册 冎
ជ + ␣ pជ ⫻ ẑ ⫻ ␴ជ̂ ⌿ .
+ Re ⌿† B
ប
共7兲
Introducing a tensor js共r , t兲 and a vector ជj ␻共r , t兲:
再冋
js共r,t兲 = Re ⌿†
再 冋
册 冎
册 冎
pជ ␣
ជ̂ 兲 sជ̂⌿ ,
+ 共ẑ ⫻ ␴
m ប
ជj␻共r,t兲 = Re ⌿† 2 Bជ + ␣ 共pជ ⫻ ẑ兲 ⫻ sជ̂⌿ ,
ប
ប
共8兲
共9兲
then Eq. 共7兲 reduces to
d
sជ共r,t兲 = − ⵜ · js共r,t兲 + ជj ␻共r,t兲,
dt
共10兲
or it can also be rewritten in the integral form
d
dt
冕冕 冕
V
sជdV = −
冖
S
dSជ · js +
冕冕 冕
ជj␻dV.
共11兲
V
ជ̂ 兲 and
Due to the fact that vជ̂ = 共d / dt兲r = pជ / m + 共␣ / ប兲共ẑ ⫻ ␴
ជ̂ = 共1 / iប兲关␴ជ̂ , H兴 = 共2 / ប兲关Bជ + 共␣ / ប兲pជ ⫻ ẑ兴 ⫻ ␴ជ̂ , Eqs. 共8兲
共d / dt兲␴
and 共9兲 become
js共r,t兲 = Re兵⌿†共r,t兲vជ̂ sជ̂⌿共r,t兲其,
共12兲
ជj␻共r,t兲 = Re兵⌿†共dsជ̂/dt兲⌿其 = Re兵⌿†␻
ជ̂ ⫻ sជ̂⌿其,
共13兲
ជ̂ ⬅ 共2 / ប兲关Bជ + 共␣ / ប兲共pជ ⫻ ẑ兲兴 is the angular velocity opwhere ␻
erator. We emphasize that those results, 共10兲, 共12兲, and 共13兲,
再
冎
共14兲
冎
ជj␻共r,t兲 = Re兵⌿†共dsជ̂/dt兲⌿其 = Re ⌿† 1 关␻
ជ̂ ⫻ sជ̂ − sជ̂ ⫻ ␻
ជ̂ 兴⌿ .
2
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 continuity equation 共10兲 is the consequence of invariance of the
spin magnitude 兩sជ兩, i.e., when an electron makes a motion,
either translation or rotation, its spin magnitude 兩sជ 兩 = ប / 2 remains 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 js共r , t兲 and ជj ␻共r , t兲 in
Eq. 共10兲, which are defined in Eqs. 共12兲 and 共13兲 respectively, describe the translational and rotational motion 共precession兲 of a spin at the location r and the time t. They will
be named the linear and the angular spin current densities
ជ are called the linear and the
accordingly, similar as vជ and ␻
ជ
angular velocities. In fact, j ␻ is called the spin torque in a
recent work.11 We consider both ជj ␻ and js 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 js and ជj ␻ 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 js共r , t兲 is identical with the
conventional spin current investigated in recent
studies.4 From js共r , t兲, the total linear spin current
along i direction 共i = x , y , z兲 is ជIsi共i , t兲 = 兰兰dSî · js共r , t兲. To
assume ជIsi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 ជIsi = 共1 / L兲 兰 兰兰dVî · js共r , t兲 = 共1 / L兲 兰 兰兰dVRe⌿†v̂isជ̂⌿
= 共1 / L兲 兰 兰兰dV⌿† 21 共v̂isជ̂ + sជ̂v̂i兲⌿ = 具 21 共v̂isជ̂ + sជ̂v̂i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 js共r , t兲 and ជj ␻共r , t兲.
Notice that ជj ␻共r , t兲 which describe the rotational motion 共precession兲 of the spin plays a parallel role in comparison with
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Q.-F. SUN AND X. C. XIE
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 magnetic charges that are equivalent to a angular MM current.
the conventional linear spin current js共r , t兲 for the spin transport. 共1兲 Similar to the classical case, it is necessary to introduce the two quantities js共r , t兲 and ជj ␻共r , t兲 to completely describe the motion of a quantum spin. 共2兲 The linear spin
current is a tensor. Its element, e.g., js,xy, represents an electron moving along the x direction with its spin in the y direction 关see Fig. 2共a兲兴. The angular spin current ជj ␻ is a vector.
In Fig. 2共b兲, its element j␻,x describes the rotational motion
ជ in the
of the spin in the y direction and the angular velocity ␻
−z direction. 共3兲 From the linear spin current density js共r , t兲,
one can calculate 共or say how much兲 the linear spin current ជIs
flowing through a surface S 关see Fig. 2共d兲兴: ជIsS = 兰兰SdSជ · js.
However, the behavior for the angular spin current is different. From the density ជj ␻共r , t兲, it is meaningless to determine
how much the angular spin current flowing through a surface
S because ជj ␻ is the current in the angular space. On the other
hand, one can calculate the total angular spin current ជI␻V in a
volume V from ជj ␻: ជI␻V = 兰兰兰Vជj ␻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 transformation as well the gauge transformation. 共5兲 If the system is in
a steady state, js and ជj ␻ are independent of the time t, and
共d / dt兲sជ共r , t兲 = 0. Then the spin continuity equation 共10兲 reduces to ⵱ · js = ជj ␻ or 养SdSជ · js = 兰兰兰Vជj ␻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 ជIsS⬘ − ជIsS = ជI␻V . 共6兲 The linear spin current
density js = Re兵⌿†vជ̂ sជ̂⌿其 gives both the spin direction and the
direction of spin movement, so it completely describes the
translational motion. However, the angular spin current denជ̂ ⫻ sជ̂⌿其 involves the vecsity, ជj ␻ = Re兵⌿†共dsជ̂ / dt兲⌿其 = Re兵⌿†␻
ជ̂ ⫻ sជ̂, not the tensor ␻
ជ̂ sជ̂. Is it correct or suffitor product of ␻
cient to describe the rotational motion? For example, the
rotational motion of Fig. 2共b兲 with the spin sជ in the y direcជ in the −z direction is different
tion and the angular velocity ␻
from the one in Fig. 2共c兲 in which sជ is in the z direction and
FIG. 3. 共Color online兲 共a兲 Schematic diagram for the spin moving 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 moment pជ e. This configuration will generate an electric field equivalent
to the field from the spin currents in 共a兲.
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 instance, the induced electric field by them is identical since a
spin sជ has only the direction but no size 共see detail discussion
below兲. Thus, the vector ជj ␻ is sufficient to describe the rotational 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 coupling, and its Hamiltonian is
H=
ប2kR2
␴z
pជ 2
+ V共y,z兲 + 关␣共x兲px + px␣共x兲兴 +
, 共16兲
2m
2ប
2m
where kR共x兲 ⬅ ␣共x兲m / ប2. ␣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␴x pz
is neglected because the z direction is quantized.7 Let ⌿ be a
stationary wave function
⌿共r兲 =
冑2
2
e
ikx
冉
x
e−i兰0kR共x兲dx
x
ei兰0kR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 confined 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 densities of the wave function ⌿共r兲 are easily obtained. There
are only two nonzero elements of js共r兲:
jsxx共r兲 =
ប 2k
兩␸共y,z兲兩2 cos 2␾共x兲,
2m
共18兲
jsxy共r兲 =
ប 2k
兩␸共y,z兲兩2 sin 2␾共x兲.
2m
共19兲
The nonzero elements of ជj ␻共r兲 are
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PHYSICAL REVIEW B 72, 245305 共2005兲
DEFINITION OF THE SPIN CURRENT: THE ANGULAR…
j␻x共r兲 = −
j␻y共r兲 =
ប2kkR共x兲
兩␸共y,z兲兩2 sin 2␾共x兲,
m
共20兲
ប2kkR
兩␸共y,z兲兩2 cos 2␾共x兲,
m
共21兲
where ␾共x兲 = 兰x0kR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 = kR共x兲 = 0, and ␴ˆ is a good quantum number, hence,
ជj␻ = 0. In this case, the definition of the spin current js 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 example, 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
js = Re兵⌿†vជ̂ sជ̂⌿其 and the angular spin current ជj ␻
ជ̂ ⫻ sជ̂⌿其 have clear physical meanings, representing
= Re兵⌿†␻
the translational motion and the rotational motion 共precession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 js and ជj ␻.
IV. SPIN CURRENTS INDUCED ELECTRIC FIELDS
Recently, theoretic studies have suggested that the 共linear兲
ជ .18–20 Can the anspin current can induce an electric field E
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 ជj ␻dV at the origin. Associated with
ជ = g␮B␴ជ
the spin sជ, there is a magnetic moment 共MM兲 m
= 共2g␮B / ប兲sជ where ␮B is the Bohr magneton. Thus, corresponding to ជj ␻, there is also a angular MM current ជj m␻dV
= 共2g␮B / ប兲jជ␻dV. From above discussions, we already know
ជ
that ជj m␻ 共or ជj ␻兲 comes from the rotational motion of a MM m
ជ
ជ
ជ
ជ
共or s兲 关see Figs. 2共b兲 and 2共c兲兴, and jm␻ = ␻ ⫻ m 共or
ជj␻ = ␻
ជ ⫻ sជ兲. Under the method of equivalent magnetic charge,
ជ is equivalent to two magnetic charges: one with
the MM m
magnetic charge +q located at ␦n̂m and the other with −q at
ជ and ␦ is a tiny
−␦n̂m 关see Fig. 2共e兲兴. n̂m is the unit vector of m
ជ
length. The angular MM current j m␻ is equivalent to two
ជ 兩sin ␪ at the lomagnetic charge currents: one is ជj +q = n̂ jq␦兩␻
ជ
ជ 兩sin ␪ at −␦n̂m 关see Fig.
cation ␦n̂m, the other is j −q = n̂ jq␦兩␻
2共e兲兴, with n̂ j being the unit vector of ជj m␻ and ␪ the angle
ជ and m
ជ . In our previous work,20 by using the relabetween ␻
tivistic theory, we have arrived at the formulae for the induced electric field by a magnetic charge current. The electric field induced by ជj m␻dV can be calculated by adding the
contributions from the two magnetic charge currents. Let
ជ 兩兩m
ជ 兩 and 兩␻
ជ 兩sin ␪ = 兩jជm␻兩, we ob␦ → 0, and note that 2q␦ → 兩m
ជ
tain the electric field E␻ generated by an element of the angular spin current ជj ␻dV:
ជ = − ␮0
E
␻
4␲
冕
ជjm␻dV ⫻ r − ␮0g␮B
=
r3
h
冕
ជj␻dV ⫻ r .
r3
共22兲
ជ generated by an element
We also rewrite the electric field E
s
of the linear spin current using the tensor js:20
ជ = − ␮ 0g ␮ B ⵱ ⫻
E
s
h
冕
jsdV ·
r
.
r3
共23兲
Below we emphasize three points. 共i兲 In the large r case, the
electric field Eជ ␻ decays as 1 / r2. Note that the field from a
linear spin current Eជ s goes as 1 / r3. In fact, in terms of generating an electric field, the angular spin current is as effective as a magnetic charge current. 共ii兲 In the steady-state
ជ = Eជ + Eជ contains the property
case, the total electric field E
T
␻
s
ជ
ជ
养CET · dl = 0, where C is an arbitrary close contour not passជ
ing through the region of spin current. However, for each E
␻
ជ
ជ
ជ
ជ
ជ
or Es, 养CE␻ · dl or 养CEs · dl can be nonzero. 共iii兲 As mentioned
ជ
above, an angular spin current ជj ␻ may consist of different ␻
and sជ 关see Figs. 2共b兲 and 2共c兲兴. However, the resulting elecជ ⫻ sជ. This is because a spin
tric field only depends on ជj␻ = ␻
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 ជj ±q reduce to
ជ ⫻m
ជ / 2 at the origin. Therefore, the overall effect of the
␻
ជ ⫻m
ជ , not separately on
rotational motion is only related to ␻
ជ and m
ជ . Hence it is enough to describe the spin rotational
␻
ជ ⫻ sជ, instead of a tensor ␻
ជ sជ.
motion by using a vector ␻
Due to the fact that the direction of sជ can change during
the particle motion, the linear spin current density js is not a
conserved quantity. It is always interesting to uncover a conserved physical quantity from both theoretical and experimental points of view. Let us apply ⵜ· acting on two sides of
Eq. 共10兲, we have
d
共⵱ · sជ兲 + ⵱ · 共⵱⬘ · js − ជj ␻兲 = 0,
dt
共24兲
where ⵜ⬘ · js means that ⵱⬘ acts on the second index of js,
i.e., 共⵱⬘ · js兲i = 兺 j共d / dj兲js,ij with i , j 苸 共x , y , z兲. To introduce
ជj⵱·sជ = ⵱⬘ · js − ជj␻. Note it is different from ⵱ · j − ជj␻. In a steady
state, ⵱ · j − ជj ␻ = 0, however ជj ⵱·sជ = ⵱⬘ · j − ជj ␻ is usually nonzero.
By using ជj ⵱·sជ, the above equation reduces to
d
共⵱ · sជ兲 + ⵱ · ជj ⵱·sជ = 0.
dt
共25兲
This means that the current ជj ⵱·sជ of the spin divergence is a
conserved quantity in the steady state case. In fact,
−⵱ · sជ共r , t兲 represents an equivalent magnetic charge, so ជj ⵱·sជ
can also be named the magnetic charge current density.
245305-5
PHYSICAL REVIEW B 72, 245305 共2005兲
Q.-F. SUN AND X. C. XIE
Moreover, the total electric field produced by js and ជj ␻ can
be rewritten as
ជ + Eជ
Eជ T = E
s
␻
␮ 0g ␮ B
=
h
=
␮ 0g ␮ B
h
冕
冕
V. HEAT PRODUCED BY SPIN CURRENTS
r
共⵱⬘ · js − ជj ␻兲dV ⫻ 3
r
ជj⵱·sជdV ⫻ r .
r3
共26兲
ជ only depends on the current ជj ជ
So the total electric field E
T
⵱·s
ជ can be measured experiof the spin divergence. Note that E
T
ជ 共r兲,
mentally in principle. Through the measurement of E
T
ជj⵱·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 冑y 2 + z2, the induced fields
ជ can be obtained straightforwardly. Then the total
Eជ ␻ and E
s
ជ = Eជ + Eជ is
electric field E
T
␻
s
ជ = a បk ⵱
E
T
m
=aⵜ
冕
冕
z sin 2␾共x⬘兲
dx⬘
关共x − x⬘兲2 + y 2 + z2兴3/2
ជ ⫻ Sជ 兲 · r − r⬘ dx⬘ ,
共V
兩r − r⬘兩3
Here we consider another physical effect, the heat produced by the spin currents. Assume a uniform isotropic conductor having a linear spin current js and a charge current ជj e,
and considering the simple case that there exists no spin flip
process 共i.e., sជ is conserved兲 so that ជj ␻ = 0. Then the produced
Joule heat Q in unit volume and in unit time is
Q=
兺
i=x,y,z
=␳
冉兺
ij
2␳ ជ
关共兩j si兩 + jei兲2 + 共兩jជsi兩 − jei兲2兴
4
冊
j2s,ij + 兺 j2ei ⬅ ␳共js2 + ជj 2e 兲,
i
where ␳ is the resistivity. The term ␳ជj 2e is the Joule heat from
the charge current which is well known, and the other term
␳js2 is the heat produced by the linear spin current. So the
produced heat can indeed be expressed by js 共for the case of
ជj␻ = 0兲. Also note the produced heat Q depends on j2,
s
whereas the induced electric field depends on ⵱⬘ · js − ជj ␻.
VI. CONCLUSION
共27兲
ជ = 共បk / m , 0 , 0兲,
Sជ = (cos 2␾共x⬘兲 , sin 2␾共x⬘兲 , 0),
where
V
r⬘ = 共x⬘ , 0 , 0兲, the constant a = ␮0g␮B␳s / 4␲, and ␳s is the linear density of moving electrons under the bias of an external
ជ represents the one genervoltage. The total electric field E
T
ated by a 1D wire of electric dipole moment
pជ e = (0 , 0 , c sin 2␾共x兲) at the x axis 关see Fig. 3共b兲兴, where c is
ជ = 0, i.e., 养 Eជ · dlជ = 0.
a constant. It is obvious that ⵱ ⫻ E
T
C T
ជ · dlជ and 养 Eជ · dlជ are separately
However, in general 养CE
␻
C s
nonzero.
ជ . We use paramFinally, we estimate the magnitude of E
T
eters consistent with realistic experimental samples. Take the
Rashba parameter ␣ = 3 ⫻ 10−11 eV m 共corresponding to kR
= 1 / 100 nm for m = 0.036me兲, ␳s = 106 / m 共i.e., one moving
electron per 1000 nm in length兲, and k = kF = 108 / 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 / 2kR兲共␲ / 2 , 0 , 0.01兲 and 共1 / 2kR兲共␲ / 2 , 0 , −0.01兲.
This value of the potential is measurable with today’s
*Electronic address: [email protected]
1
technology.19 Furthermore, with the above parameters the
ជ at A or B is about 5 V / m which is rather
electric field E
T
large.
S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton,
S. V. Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M.
Treger, Science 294, 1488 共2001兲; G. A. Prinz, ibid. 282, 1660
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 current兲, one has to introduce another quantity, the angular spin
current. The angular spin current describes the rotational motion of the spin, and it plays a parallel role in comparison
with the conventional linear spin current for the spin translational 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
ជ is derived and Eជ scales as 1 / r2 at
generated electric field E
␻
␻
large r. In addition, a conserved quantity, the current ជj ⵱·sជ of
the spin divergence, is discovered, and the total electric field
only depends on ជj ⵱·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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14
Of course, in some special cases, one may be able to introduce a
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17
Recently, Ref. 10 has investigated the definition of the spin current in the insulating Heisenberg model. In our opinion, our
definition is valid in the Heisenberg model, for which the linear
spin current js is always zero since particles do not move in this
insulating model.
18
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11
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