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

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1 Definition of the spin current: The angular spin current an its physical consequences Qing-feng Sun 1, * an X. C. Xie 2,3 1 Beijing National Lab for Conense Matter Physics an Institute of Physics, Chinese Acaemy of Sciences, Beijing , China 2 Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA 3 International Center for Quantum Structures, Chinese Acaemy of Sciences, Beijing , China Receive 14 June 2005; revise manuscript receive 22 September 2005; publishe 2 December 2005 We fin that in orer to completely escribe the spin transport, apart from spin current or linear spin current, one has to introuce the angular spin current. The two spin currents, respectively, escribe the translational an rotational motion precession of a spin. The efinitions of these spin current ensities are given an their physical properties are iscusse. Both spin current ensities appear naturally in the spin continuity equation. In particular, we preict that the angular spin current or the spin torque as calle in previous works, similar to the linear spin current, can also inuce an electric fiel E. The formula for the inuce electric fiel E by the angular spin current element is erive, playing the role of Biot-Savart law or Ampere law. When at large istance r, this inuce electric fiel E scales as 1/r 2, whereas the E fiel generate from the linear spin current goes as 1/r 3. DOI: /PhysRevB PACS number s : b, b, I. INTRODUCTION Recently, a new subiscipline of conense matter physics, spintronics, is emerging rapily an generating great interests. 1,2 The spin current, the most important physical quantity in spintronics, has been extensively stuie. Many interesting an funamental phenomena, such as the spin Hall effect 3 6 an the spin precession 7,8 in systems with spinorbit coupling, have been iscovere an are uner further stuy. As for the charge current, the efinition of the local charge current ensity j e r,t =Re r,t ev r,t an its continuity equation /t e r,t + j e r,t =0 is well known in physics. Here r,t is the electronic wave function, v =ṙ is the velocity operator, an e r,t =e is the charge ensity. 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 ebates over what is the correct efinition for spin current. 9,10 The problem stems from that the spin s is no invariant quantity in the spin transport, so that the conventional efining of the spin current v s is no conservative. Recently, some stuies have begun investigation in this irection, e.g., a semiclassical escription of the spin continuity equation has been propose, 11,12 as well as stuies introucing a conserve spin current uner special circumstances. 9 In this paper, we stuy the efinition of local spin current ensity. We fin that ue to the spin is vector an it has the translational an rotational motion, one has to use two quantities, the linear spin current an the angular spin current, to completely escribe the spin transport. Here the linear spin current escribe the translational motion of a spin, an the angular spin current is for the rotational motion. The efinition of two spin current ensities are given an they appear naturally in the quantum spin continuity equation. Moreover, we preict that the angular spin current can generate an electric fiel similar as with the linear spin current, an thus contains physical consequences. The paper is organize as follows. In Sec. II, we first iscuss the flow of a classical vector. The flow of a quantum spin is investigate in Sec. III. In Sec. IV an Sec. V, we stuy the problem of an inuce electric fiel an the heat prouce by spin currents, respectively. Finally, a brief summary is given in Sec. VI. II. THE FLOW OF A CLASSICAL VECTOR Before stuying the spin current in a quantum system, we first consier the classical case. Consier a classical particle having a vector m e.g., the classical magnetic moment, etc. with its magnitue m fixe uner the particle motion. To completely escribe this vector flow see Fig. 1 c, in aition to the local vector ensity M r,t = r,t m r,t, one nees two quantities: the linear velocity v r,t an the angular velocity r,t. Here r,t is the particle ensity, an v an escribe the translational an rotational motions, re- FIG. 1. Color online a an b are the schematic iagram for the translational motion an the rotational motion of the classic vector m, respectively. c Schematic iagram for a classic vector flow /2005/72 24 / /$ The American Physical Society

2 Q.-F. SUN AND X. C. XIE spectively see Figs. 1 a an 1 b. In contrast with the flow of a scalar quantity in which one only nees one quantity, namely, the local velocity v r,t, to escribe the translational motion, it is essential to use two quantities v r,t an r,t for the vector flow. We emphasize that it is impossible to use one vector to escribe both translational an rotational motions altogether. Since m is a constant, the change of the local vector M r,t in the volume element V= x y z with the time from t to t+t can be obtaine see Fig. 1 c : M r,t V = i=x,y,z V i v i r,t tm r,t V i v i r + î,t tm r + î,t + r,t M r,t Vt. 1 The first an the secon terms on the right escribe the classical vector flowing in or out the volume element V an its rotational motion respectively, an both can cause a change in the local vector ensity. When V goes to zero, we have the vector continuity equation t M r,t = v r,t M r,t + r,t M r,t, 2 where v M is a tensor, an its element v M ij =v i M j. Note that this vector continuity equation is from the kinematics an the invariance of m, an it is inepenent of the ynamic laws. It is well known that the scalar e.g., charge e continuity equation /t e + j e =0 is from the kinematics an the invariance of the charge e. It is inepenent of the external force F as well ynamic laws. In other wors, even if the acceleration a F/m, the continuity equation still survives. It is complete same with the vector continuity Eqs. 2 or 3. It is also from the kinematics an the invariance of m. In particular, it is inepenent of the external force an the torque acting on the vector, as well as the ynamic laws. Introucing j s r,t =v r,t M r,t an j r,t = r,t M r,t, then Eq. 2 reuces to t M r,t = j s r,t + j r,t. 3 Here j s =v M is from the translational motion of the classical vector m, an j = M escribes its rotational motion. Since v an are calle as the linear velocity an the angular velocity respectively, it is natural to name j s an j as the linear an the angular current ensities. Notice although the unit of j is ifferent from that of the linear spin current. However, j is inee the current in the angular space. It is worth to mention the following two points. 1 If to consier that there are many particles in the volume element V i.e., the volume element V is very small macroscopically but very large microscopically, the vector irection m i, the velocity v i, an the angular velocity i for each particle may be ifferent, however, the vector continuity equation 3 is still vali: /t M = j s + j, with j s r,t v M =lim V 0 i v i m i / V an j r,t M =lim V 0 i i m i / V. Here j s r,t an j r,t still escribe the translational an the rotational motions of the classical vector. 2 If one as an arbitrary curl A arb to the current j s =v M, the continuity equation oes not change. Does this imply that the linear spin current ensity can be efine as j s =v M + A arb C with a constant vector C?In our opinion, this oes not because the local spin current ensity has physical meanings. This reason is completely same with the charge current ensity that cannot be reefine as j e =ev + A arb. In orer to escribe a scalar e.g., charge flow, one local current j e r,t is sufficient. Why is it require to introuce two quantities instea of one to escribe a vector flow? The reason is that the scalar quantity only has the translational motion, but the vector quantity has two kins of motion, the translational an the rotational. So one has to use two quantities, the velocity v an the angular velocity, to escribe the motion of a single vector. Corresponingly, two quantities j s =v M an j = M are necessary to escribe the vector flow. In the steay state case, the scalar e.g., charge continuity equation reuces into j e =0, so the scalar current j e is a conserve quantity. But for a vector flow, the linear vector current j s is not conserve since j s = j. Whether it is possible to have a conserve vector current through reefinition? Of course, this reefine vector current shoul have a clear physical meaning an is measurable. In our opinion, this is almost impossible in the three-imensional space. The reasons are as follows. i One cannot use a three-imensional vector to combine both v an. Therefore one can also not use a three-imensional tensor to combine both j s =v M an j = M. ii Consier an example, as shown in Fig. 3 a, a one-imensional classical vector flowing along the x axis. When x 0, the vector s irection is in the +x axis. At 0 x L, the vector rotates in accompany with its translational motion. When x L, its irection is along the +y axis. Since for x 0 an x L the vector has no rotational motion, the efinition of the vector current is unambiguous, an the nonzero element is j xx for x 0, an j xy for x L. Therefore, the vector current is obviously ifferent for x 0 an x L, an the vector current is nonconservative. In our opinion, j s =v M an j = M alreay have clear physical meanings. They also completely an sufficiently escribe a vector flow, an they can etermine any physical effects cause by the vector flow see Secs. IV an V. One may not nee to enforce a conserve current. In particular, as shown in the example of Fig. 3 a, sometimes it is impossible to introuce a conserve current. 14 III. THE FLOW OF A QUANTUM SPIN Now we stuy the electronic spin s in the quantum case. Consier an arbitrary wave function r,t. The local spin ensity s at the position r an time t is s r,t

3 DEFINITION OF THE SPIN CURRENT: THE ANGULAR = r,t s ˆ r,t, where s ˆ = /2 ˆ with ˆ being the Pauli matrices. The time-erivative of s r,t is t 2 s r,t = ˆ + ˆ t t. From the Schröinger equation, we have /t r,t = 1/i H r,t an /t r,t = 1/ i H r,t. Notice here the transposition in the symbol only acts on the spin inexes. By using the above two equations, Eq. 4 changes into 4 are vali in general. They are inepenent of the special choice of Hamiltonian 6. For example, in the case with a vector potential A, the general spin-orbit coupling ˆ p V r, an so on, 17 the results still hol. Notice that for the Hamiltonian 6, one has Re vˆ iŝ j =Re ŝ j vˆ i an Re ˆ s ˆ = Re s ˆ ˆ. If Re vˆ iŝ j Re ŝ j vˆ i an Re ˆ s ˆ Re s ˆ ˆ for certain Hamiltonians, Eqs. 12 an 13 will change to /t s r,t = ˆ H H ˆ /2i. 5 In the erivation below, we use the following Hamiltonian: j s r,t =Re 1 2 v ˆs ˆ + s ˆv ˆ T, 14 H = p 2 2m + V r + ˆ B + ẑ ˆ p. 6 Note that our results are inepenent of this specific choice of the Hamiltonian. In Eq. 6 the first an secon terms are the kinetic energy an potential energy. The thir term is the Zeeman energy ue to a magnetic fiel, an the last term is the Rashba spin-orbit coupling, 15,16 which has been extensively stuie recently. 4,7,8 Next we substitute the Hamiltonian 6 into Eq. 5, an Eq. 5 reuces to t s = 2 Re p m + ẑ ˆ ˆ +Re B + p ẑ ˆ. Introucing a tensor j s r,t an a vector j r,t : j s r,t =Re p m + ẑ ˆ s ˆ, j r,t =Re 2 B + p ẑ s ˆ, then Eq. 7 reuces to t s r,t = j s r,t + j r,t, or it can also be rewritten in the integral form s V S t V = S j s j V. + V Due to the fact that v ˆ = /t r= p /m+ / ẑ ˆ an /t ˆ = 1/i ˆ,H = 2/ B + / p ẑ ˆ, Eqs. 8 an 9 become j s r,t =Re r,t v ˆs ˆ r,t, j r,t =Re s ˆ/t =Re ˆ s ˆ, where ˆ 2/ B + / p ẑ is the angular velocity operator. We emphasize that those results, 10, 12, an 13, j r,t =Re s ˆ/t =Re 1 2 ˆ s ˆ s ˆ ˆ. 15 Equation 10 is the quantum spin continuity equation, which is the same with the classic vector continuity equation 3 although the erivation process is very ifferent. In some previous works, this equation has also been obtaine in the semiclassical case. 11,12 Here we emphasize that this spin continuity equation 10 is the consequence of invariance of the spin magnitue s, i.e., when an electron makes a motion, either translation or rotation, its spin magnitue s = /2 remains a constant. Equation 10 shoul also be inepenent with the force i.e., the potential an the torque, as well the the ynamic law. The two quantities j s r,t an j r,t in Eq. 10, which are efine in Eqs. 12 an 13 respectively, escribe the translational an rotational motion precession of a spin at the location r an the time t. They will be name the linear an the angular spin current ensities accoringly, similar as v an are calle the linear an the angular velocities. In fact, j is calle the spin torque in a recent work. 11 We consier both j an j s escribing the motion of a spin, capable of inucing an electric fiel see Sec. IV, an so on. Namely, both of them play a parallel role in contributing to a physical quantity. Therefore, it is better to name j s an j both as spin currents. Otherwise, when one calculates the contribution of spin current to a given physical quantity, one may forget to inclue the contribution by the angular spin current. The linear spin current j s r,t is ientical with the conventional spin current investigate in recent stuies. 4 From j s r,t, the total linear spin current along i irection i=x,y,z is I si i,t = Sî j s r,t. To assume I si i,t inepenent on t an i e.g., in the case of the steay state an without spin flip, one has I si= 1/L Vî j s r,t = 1/L VRe vˆ is ˆ = 1/L V 1 2 vˆ is ˆ +s ˆvˆ i = 1 2 vˆ is ˆ +s ˆvˆ i, where L is sample length in the i irection. This efinition is the same as in recent publications. 4 Next, we iscuss certain properties of j s r,t an j r,t. Notice that j r,t which escribe the rotational motion precession of the spin plays a parallel role in comparison with

4 Q.-F. SUN AND X. C. XIE FIG. 2. Color online a The linear spin current element j s,xy. b an c The angular spin current element j,x. The spin current in a quasi 1D quantum wire. e The currents of two magnetic charges that are equivalent to a angular MM current. the conventional linear spin current j s r,t for the spin transport. 1 Similar to the classical case, it is necessary to introuce the two quantities j s r,t an j r,t to completely escribe the motion of a quantum spin. 2 The linear spin current is a tensor. Its element, e.g., j s,xy, represents an electron moving along the x irection with its spin in the y irection see Fig. 2 a. The angular spin current j is a vector. In Fig. 2 b, its element j,x escribes the rotational motion of the spin in the y irection an the angular velocity in the z irection. 3 From the linear spin current ensity j s r,t, one can calculate or say how much the linear spin current I s flowing through a surface S see Fig. 2 : I s S = S S j s. However, the behavior for the angular spin current is ifferent. From the ensity j r,t, it is meaningless to etermine how much the angular spin current flowing through a surface S because j is the current in the angular space. On the other han, one can calculate the total angular spin current I V in a volume V from j : I V = V j r,t V. 4 It is easy to prove that the spin currents in the present efinitions of Eqs. 12 an 13 are invariant uner a space coorinate transformation as well the gauge transformation. 5 If the system is in a steay state, j s an j are inepenent of the time t, an /t s r,t =0. Then the spin continuity equation 10 reuces to j s = j or S S j s = V j V. This means that the total linear spin current flowing out of a close surface is equal to the total angular spin current enclose. If to further consier a quasi-one-imensional 1D system see Fig. 2, then one has I S s I s S =I V. 6 The linear spin current ensity j s =Re v ˆs ˆ gives both the spin irection an the irection of spin movement, so it completely escribes the translational motion. However, the angular spin current ensity, j =Re s ˆ /t =Re ˆ s ˆ involves the vector prouct of ˆ s ˆ, not the tensor ˆ s ˆ. Is it correct or sufficient to escribe the rotational motion? For example, the rotational motion of Fig. 2 b with the spin s in the y irection an the angular velocity in the z irection is ifferent from the one in Fig. 2 c in which s is in the z irection an FIG. 3. Color online a Schematic iagram for the spin moving along the x axis, with the spin precession rotational motion in the x-y plane while 0 x L. b A 1D wire of electric ipole moment p e. This configuration will generate an electric fiel equivalent to the fiel from the spin currents in a. is in the y irection, but their angular spin currents are completely the same. Shall we istinguish them? It turns out that the physical results prouce by the above two rotational motions Figs. 2 b an 2 c are inee the same. For instance, the inuce electric fiel by them is ientical since a spin s has only the irection but no size see etail iscussion below. Thus, the vector j is sufficient to escribe the rotational motion, an no tensor is necessary. Now we give an example of applying the above formulas, 12 an 13, to calculate the spin currents. Let us consier a quasi-1d quantum wire having the Rashba spin orbit coupling, an its Hamiltonian is H = p 2 2m + V y,z + z 2 x p x + p x x k R 2m, 16 where k R x x m/ 2. x =0 for x 0 an x L, an x 0 while 0 x L. The other Rashba term / x p z is neglecte because the z irection is quantize. 7 Let be a stationary wave function r = 2 2 x e i 0 eikx kr x x y,z, e i 0 x k R x x 17 where y,z is the boun state wave function in the confine y an z irections. r represents the spin motion as shown in Fig. 3 a, 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. 12 an 13, the spin current ensities of the wave function r are easily obtaine. There are only two nonzero elements of j s r : j sxx r = 2 k 2m y,z 2 cos 2 x, j sxy r = 2 k 2m y,z 2 sin 2 x. The nonzero elements of j r are

5 DEFINITION OF THE SPIN CURRENT: THE ANGULAR j x r = 2 kk R x y,z 2 sin 2 x, m 20 j y r = 2 kk R m y,z 2 cos 2 x, 21 where x = x 0 k R x x. Those spin current ensities confirm with the intuitive picture of an electron motion, precession in the x-y plane in 0 x L an movement in the x irection see Fig. 3 a. In particular, in the region of x 0 an x L, x =k R x =0, an ˆ is a goo quantum number, hence, j =0. In this case, the efinition of the spin current j s is unambiguous. However, the spin currents are ifferent in x 0 an x L except for L =n n=0,±1,±2,.... This is clearly seen from Fig. 3 a. Therefore, through this example, one can conclue that it is sometimes impossible to efine a conserve spin current. 14 The example of Fig. 3 a inee exists an has been stuie before. 7,8 Above iscussion shows that the linear spin current j s =Re v ˆs ˆ an the angular spin current j =Re ˆ s ˆ have clear physical meanings, representing the translational motion an the rotational motion precession respectively. They completely escribe the flow of a quantum spin. Any physical effects of the spin currents, such as the inuce electric fiel, can be expresse by j s an j. IV. SPIN CURRENTS INDUCED ELECTRIC FIELDS Recently, theoretic stuies have suggeste that the linear spin current can inuce an electric fiel E Can the angular spin current also inuce an electric fiel? If so, this gives a way of etecting the angular spin current. Following, we stuy this question by using the metho of equivalent magnetic charge. 21 Let us consier a steay-state angular spin current element j V at the origin. Associate with the spin s, there is a magnetic moment MM m =g B = 2g B / s where B is the Bohr magneton. Thus, corresponing to j, there is also a angular MM current j m V = 2g B / j V. From above iscussions, we alreay know that j m or j comes from the rotational motion of a MM m or s see Figs. 2 b an 2 c, an j m = m or j = s. Uner the metho of equivalent magnetic charge, the MM m is equivalent to two magnetic charges: one with magnetic charge +q locate at nˆ m an the other with q at nˆ m see Fig. 2 e. nˆ m is the unit vector of m an is a tiny length. The angular MM current j m is equivalent to two magnetic charge currents: one is j +q =nˆ jq sin at the location nˆ m, the other is j q =nˆ jq sin at nˆ m see Fig. 2 e, with nˆ j being the unit vector of j m an the angle between an m. In our previous work, 20 by using the relativistic theory, we have arrive at the formulae for the inuce electric fiel by a magnetic charge current. The electric fiel inuce by j m V can be calculate by aing the contributions from the two magnetic charge currents. Let 0, an note that 2q m an m sin = j m,weobtain the electric fiel E generate by an element of the angular spin current j V: E = 0 j m V r 4 r 3 = 0g B j V r h r We also rewrite the electric fiel E s generate by an element of the linear spin current using the tensor j s : 20 E s = 0g B h j s V r r Below we emphasize three points. i In the large r case, the electric fiel E ecays as 1/r 2. Note that the fiel from a linear spin current E s goes as 1/r 3. In fact, in terms of generating an electric fiel, the angular spin current is as effective as a magnetic charge current. ii In the steay-state case, the total electric fiel E T=E +E s contains the property C E T l =0, where C is an arbitrary close contour not passing through the region of spin current. However, for each E or E s, C E l or C E s l can be nonzero. iii As mentione above, an angular spin current j may consist of ifferent an s see Figs. 2 b an 2 c. However, the resulting electric fiel only epens on j = s. This is because a spin vector contains only a irection an a magnitue, but not a spatial size i.e., the istance approaches to zero. Inthe limit 0, both magnetic charge currents j ±q reuce to m /2 at the origin. Therefore, the overall effect of the rotational motion is only relate to m, not separately on an m. Hence it is enough to escribe the spin rotational motion by using a vector s, instea of a tensor s. Due to the fact that the irection of s can change uring the particle motion, the linear spin current ensity j s is not a conserve quantity. It is always interesting to uncover a conserve physical quantity from both theoretical an experimental points of view. Let us apply acting on two sies of Eq. 10, we have t s + j s j =0, 24 where j s means that acts on the secon inex of j s, i.e., j s i = j /j j s,ij with i, j x,y,z. To introuce j s = j s j. Note it is ifferent from j j. In a steay state, j j =0, however j s = j j is usually nonzero. By using j s, the above equation reuces to t s + j s =0. 25 This means that the current j s of the spin ivergence is a conserve quantity in the steay state case. In fact, s r,t represents an equivalent magnetic charge, so j s can also be name the magnetic charge current ensity

6 Q.-F. SUN AND X. C. XIE Moreover, the total electric fiel prouce by j s an j can be rewritten as E T = E s + E = 0g B h j s j V r r 3 = 0g B h j s V r r So the total electric fiel E T only epens on the current j s of the spin ivergence. Note that E T can be measure experimentally in principle. Through the measurement of E T r, j s can be uniquely obtaine. In the following, let us calculate the inuce electric fiels at the location r= x,y,z by the spin currents in the example of Fig. 3 a. Substituting the spin currents of Eqs into Eqs. 22 an 23 an assuming the transverse sizes of the 1D wire are much smaller than y 2 +z 2, the inuce fiels E an E s can be obtaine straightforwarly. Then the total electric fiel E T=E +E s is E T = a k m = a V S z sin 2 x x x 2 + y 2 + z 2 3/2x r r r r 3x, 27 where V = k/m,0,0, S =(cos 2 x,sin 2 x,0), r = x,0,0, the constant a= 0 g B s /4, an s is the linear ensity of moving electrons uner the bias of an external voltage. The total electric fiel E T represents the one generate by a 1D wire of electric ipole moment p e =(0,0,c sin 2 x ) at the x axis see Fig. 3 b, where c is a constant. It is obvious that E T=0, i.e., C E T l =0. However, in general C E l an C E s l are separately nonzero. Finally, we estimate the magnitue of E T. We use parameters consistent with realistic experimental samples. Take the Rashba parameter = ev m corresponing to k R =1/100 nm for m=0.036m e, s =10 6 /m i.e., one moving electron per 1000 nm in length, an k=k F =10 8 /m. The electric potential ifference between the two points A an B see Fig. 3 b is about 0.01 V, where the positions of A an B are 1/2k R /2,0,0.01 an 1/2k R /2,0, This value of the potential is measurable with toay s technology. 19 Furthermore, with the above parameters the electric fiel E T at A or B is about 5 V/m which is rather large. V. HEAT PRODUCED BY SPIN CURRENTS Here we consier another physical effect, the heat prouce by the spin currents. Assume a uniform isotropic conuctor having a linear spin current j s an a charge current j e, an consiering the simple case that there exists no spin flip process i.e., s is conserve so that j =0. Then the prouce Joule heat Q in unit volume an in unit time is 2 Q = i=x,y,z 4 j si + j ei 2 + j si j ei 2 = ij 2 j s,ij + j ei i 2 j 2 s + j 2 e, where is the resistivity. The term j e 2 is the Joule heat from the charge current which is well known, an the other term j s 2 is the heat prouce by the linear spin current. So the prouce heat can inee be expresse by j s for the case of j =0. Also note the prouce heat Q epens on j s 2, whereas the inuce electric fiel epens on j s j. VI. CONCLUSION In summary, we fin that in orer to completely escribe the spin flow incluing both classic an quantum flows, apart from the conventional spin current or linear spin current, one has to introuce another quantity, the angular spin current. The angular spin current escribes the rotational motion of the spin, an 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 or the spin torque as being calle in other works can also inuce an electric fiel. The formula for the generate electric fiel E is erive an E scales as 1/r 2 at large r. In aition, a conserve quantity, the current j s of the spin ivergence, is iscovere, an the total electric fiel only epens on j s. ACKNOWLEDGMENTS We gratefully acknowlege financial support from the Chinese Acaemy of Sciences an NSFC uner Grant Nos , , an X.C.X. was supporte by the US-DOE uner Grant No. DE-FG02-04ER46124, NSF CCF , an NSF-MRSEC uner Grant No. DMR *Electronic aress: 1 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. V. Molnar, M. L. Roukes, A. Y. Chtchelkanova, an D. M. Treger, Science 294, ; G. A. Prinz, ibi. 282, I. Zutic, J. Fabian, an S. Das Sarma, Rev. Mo. Phys. 76, S. Murakami, N. Nagaosa, an S.-C. Zhang, Science 301,

7 DEFINITION OF THE SPIN CURRENT: THE ANGULAR 2003 ; Phys. Rev. B 69, ; Z. F. Jiang, R. D. Li, S.-C. Zhang, an W. M. Liu, ibi. 72, J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, an A. H. MacDonal, Phys. Rev. Lett. 92, ; B.K.Nikolic, S. Souma, L. P. Zarbo, an J. Sinova, ibi. 95, S. Q. Shen, M. Ma, X. C. Xie, an F. C. Zhang, Phys. Rev. Lett. 92, ; S.-Q. Shen, Y.-J. Bao, M. Ma, X. C. Xie, an F. C. Zhang, Phys. Rev. B 71, ; S. Murakami, N. Nagaosa, an S.-C. Zhang, Phys. Rev. Lett. 93, ; S.-P. Kou, X.-L. Qi, an Z.-Y. Weng, Phys. Rev. B 72, G. Y. Guo, Y. Yao, an Q. Niu, Phys. Rev. Lett. 94, ; Y. Yao an Z. Fang, ibi. 95, ; X. Dai, Z. Fang, Y.-G. Yao, an F.-C. Zhang, con-mat/ unpublishe. 7 S. Datta an B. Das, Appl. Phys. Lett. 56, T. Matsuyama, C.-M. Hu, D. Grunler, G. Meier, an U. Merkt, Phys. Rev. B 65, ; F. Mireles an G. Kirczenow, ibi. 64, ; J. Wang, H. B. Sun, an D. Y. Xing, ibi. 69, P. Zhang, J. R. Shi, D. Xiao, an Q. Niu, con-mat/ unpublishe ; J. Wang, B. G. Wang, W. Ren, an H. Guo, conmat/ unpublishe. 10 F. Schätz, P. Kopietz, an M. Kollar, Eur. Phys. J. B 41, D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. Mac- Donal, an Q. Niu, Phys. Rev. Lett. 93, S. Zhang an Z. Yang, Phys. Rev. Lett. 94, A. A. Burkov, A. S. Nunez, an A. H. MacDonal, Phys. Rev. B 70, Of course, in some special cases, one may be able to introuce a conserve current, for example see Janin Splettstoesser, M. Governale, an U. Zälicke, Phys. Rev. B 68, E. I. Rashba, Fiz. Tver. Tela Leningra 2, Sov. Phys. Soli State 2, Y. A. Bychkov an E. I. Rashba, J. Phys. C 17, Recently, Ref. 10 has investigate the efinition of the spin current in the insulating Heisenberg moel. In our opinion, our efinition is vali in the Heisenberg moel, for which the linear spin current j s is always zero since particles o not move in this insulating moel. 18 J. E. Hirsch, Phys. Rev. B 42, ; 60, ; F. Meier an D. Loss, Phys. Rev. Lett. 90, F. Schütz, M. Kollar, an P. Kopietz, Phys. Rev. Lett. 91, ; Phys. Rev. B 69, Q.-F. Sun, H. Guo, an J. Wang, Phys. Rev. B 69, For example, see D. J. Griffiths, Introuction to Electroynamics Prentice-Hall, Englewoo Cliffs, NJ,

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