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The universal definition of current

Z. An, F. Q. Liu, Y. Lin & C. Liu

SUBJECT AREAS: Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, and School of and Technology, Wuhan MAGNETIC MATERIALS University, Wuhan 430072, China. AND DEVICES ELECTRONIC MATERIALS AND DEVICES The spin current, angular current and total current in a form QUANTUM PHYSICS have been universally defined according to the . Their conservation quantities and the continuity equations have been discussed in different cases. Non-relativistic approximation forms are MODELLING AND THEORY deduced in order to explain their physical meanings, and to analyze some experimental results. The spin current of helical edge states in HgTe/CdTe quantum wells is calculated to demonstrate the properties of the spin current of the two dimensional quantum spin-Hall system. A generalized spin-orbit term in Received the semiconducting media is deduced based on the theory of the electrodynamics in the moving media. It is 23 January 2012 recommended to use the effective total angular momentum current instead of the pure spin current to describe the distribution of and the transport properties in . Accepted 7 March 2012 Published pintronics1,2, a new sub-disciplinary of condensed physics, has been regarded as bringing hope 4 May 2012 for a new generation of electronic devices. The advantages of spintronic devices include reducing the power S consumption and overcoming the velocity limit of electric charge1. The two degrees of freedom of the spin enable to transmit more information in quantum computation and . In the past decade, Correspondence and many interesting phenomena emerged, moving the study of spintronics forward. The spin- predicts an requests for materials efficient spin injection without the need of metallic ferromagnets3, and generates a substantial amount of dis- 4 should be addressed to sipationless quantum spin current in a semiconductor . All these provide the fundamental on designing spin- 5 C.L. (cliu@acc-lab. tronic devices, such as spin transistors that were predicted several years ago . Experimental progresses have also been made in recent years6,7. whu.edu.cn; chang. Since Rashba stated problems inherent in the theory of transport spin currents driven by external fields and [email protected]) 8 gave his definition on the spin current tensor Jij , there were several works on how to define the spin current in different cases. Sun et al. suggested that there was no need to modify the traditional definition on the spin current, but an additional term which describes the spin should be included in the previously common-accepted definition9,10. A modified definition given by Shi et al.12 solved the conservation problem of the traditional spin current in the spin-orbit coupled system. His definition ensured an equilibrium thermodynamics theory built on spintronics, in accordance with other traditional transport theory, for instance, the Onsager relation. Jin, Li and Zhang11 first gave the continuity-like equation of the spin current in SU(2)3U(1) unified theory. The non- conservation of the spin current was due to the non-Abelian feature of the Yang-Mills field, and an angular momentum was intentionally introduced to cancel the non-conservation effect. They made an analogical deriva- tion on the non-relativistic Schroedinger Equation and did not use the Noether theorem. Thus, it is difficult to perform an exact analysis on the continuity of the spin current, and the result can not be used in the systems in which the relativity should be considered (the behavior obeys the ). Spin-Hall effect, a vital phenomenon induced by spin-orbit coupling, has been extensively studied for years, although the microscopic origins of the effect are still being argued. Hirsch et al.13 referred that anisotropic scattering by impurities led to the spin-Hall effect, while an intrinsic cause of spin-Hall effect was proposed by Sinova et al.3. Both theoretical and experimental work reported recently demonstrated the achievements of in semiconductors14–16. In this study, the spin current Js, orbit angular momentum (OAM) current JL and the total angular momentum (TAM) current JJ, as well as the corresponding continuity equations have been delivered. In our tensor form expressions, the velocity a and the spin operator S can well display the physical meaning of the spin current. In addition, the non-relativistic approximation (NRA) expressions have been derived and the quantum effects have been predicted, which can not be deduced from previous definitions. Its vital effect on the finite size effect of the spin current is calculated in Hg/CdTe system. It is recommended to use the effective TAM and its current to replace the traditional spin and spin current in spintronics.

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 1 www..com/scientificreports

Results In relativistic quantum , the physical meanings of the The angular momentum current in a tensor form. According to the velocity operator a^ has been clearly described. Also, it should be quantum electrodynamics(QED) theory17, the Lagrangian pointed out that, there is a relationship between the electric current 1 L ~L zL zL ð1Þ and the spin current in order (which is shown in the Discussion). QED Dirac Maxwell int c The spin-orbit coupling effect demands to replace the momentum ~ m { 2 { 1 2 { m y icc Lm mc y Fmn yec yAm ð2Þ ^ 4 operator ^p or P with the operator a^. Deriving the expression of the OAM current JL and the TAM JJ is 1 ~ m { 2 { 2 similar to that of spin current Js: y icc Dm mc y Fmn ð3Þ 4 ~ ðÞJL mn iamLn ð10Þ can be represented in two terms m 2 J ~J zJ ð11Þ Le~y icc Dm{mc y, ð4Þ J s L ~ ab 1 where OAM operator Lc c xaPb. L ~{ F2 , c 4 mv Angular momentum current of . Generating and and the corresponding Hamiltonian of Le is well-known as manipulating the polarization of is vital for spintronics. H^ ~c ~a:P~ zbmc2zV: ð5Þ The main method is by letting the electron absorb or emit photons, in order to change its spin state. According of the Noether theorem, one can derive the following The corresponding terms to describe the ’s spin current, equation the OAM current and the TAMhi current for the Maxwell field are ~ p~ ~~ |~ LmðÞJJ m 0: ð6Þ Js +A A, ð12Þ while the corresponding Noether current is p~ | JL ~r T, ð13Þ JJ ~JszJL, p~ pz p JJ Js JL ð14Þ where the spin current density Js is expressed as m ~ 1 m z m 1 ðÞJS ab yc ab abc y, ð7Þ respectively. Here T ~ d ðÞE E zH H {E E {H H . Obviously, 4 ij 2 ij i i i i i j i j z p only the TAM current JJ JJ meet the continuity equation and the OAM current JL m ~ mz m L p p ðÞJL xaT xbTa ~Jz~J z+: J zJ ~0: ab b Lt J J 1 with T ~ yc D y. Here c is the Dirac Matrix, and By choosing the TAM current JJ (without the photon field) or mn 2 m n m z p JJ JJ (in the general occasion), one can keep the traditional theory i s ~ c ,c . The deduction details of Eq.(7) are shown in unchanged, like the Onsager relation and the conservation law, ab 2 a b which are built on the equilibrium state theory. Methods. The Lorentz invariance of the Lagrangian ensures the conser- The NRA expression. In order to easily discuss and describe the physical meanings of the current expression, it is necessary to have vation of TAM current JJ of electrons. Eq.(6) shows that the spin current alone is not conserved, unless the orbital angular momentum a non-relativistic form of the spin current. After some tedious is fixed. simplifications (shown in Methods), we derive the non-relativistic expression of the spin current, the OAM current and the TAM The tensor form in three dimensional space. It is necessary to current. ~ ~ ~ ~ bridge the definition of the spin current JS with the traditional Js~ P z Pzi |P zi P| ð15Þ s 0 descriptions in spintronics. Using the operator S^ ~ i and i s 0 i ~ ~ ~ ~ 0 si JL~ P~Lz~LPzi |P ~Lzi~L P| ð16Þ a^i~ , the Eq.(7) turns as follows (see details in Methods) si 0 ^ ~ i { ^ JS y a^mSn y, ð8Þ ~ ~~z~~z |~ ~z~ ~| mn 2 JJ PJ JP i P J iJ P ð17Þ thus the spin current operator is where two important relations i J ~ a^ S^ ð9Þ ðÞs:P 1 ðÞs:P S 2 m n x~ w~ 1{ y 2mc 8m2c2 2mc where a^ and S^ are the velocity operator and the spin operator in :~A :~B ~~A:~Bzi : ~A|~B Dirac equation, respectively. 9 1 In the traditional definition , the spin current density operator are used. The result is shown to be completely equivalent to ðÞ^u^sz^s^u means the! carriers with a spin ^s flowing at a speed of 1 2 Eq.(9),(10) and (11) up to the order of . Obviously, not only the ^p P^ c ^u ^u~ or ^u~ . However, the traditional definition based on m m traditional term of the spin current, but also the other term an analogy of the classical current can not accurately describe the i |P~ zi P~| ð18Þ spin current, because the spin is an intrinsic physical character in quantum theory. contributes to the spin current in the same order.

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 2 www.nature.com/scientificreports

Figure 2 | the spin current of k 5 0 nm21: the spin up, spin down, and the sum. 21 Figure 1 | the spin current of Y"1 (kx, y)atk 5 0.01 nm ,Y#2 (2kx, y) 21 at k 520.01 nm , the spin up, spin down,and the sum. equation is L ~Sz+:J ~0: ð19Þ In quantum physics, there are some quantum effects that can not Lt S be analogized with the classical theory. The term (18) can only be described as ‘‘similar’’ as a kind of quantum rotation. In Sun’s work9, When the OAM is not frozen (suitable for most spintronic systems), the extra term vs is used to describe the spin rotation, because a the continuity equation (19) turns into complete description of a vector current should include translation L ~Jz+:J ~0: and rotation as the classical theory shows. Here, the term Lt J (18) which is accurately deduced yields two important conclusions as The spin-orbit coupling effect makes the spin not a good quantum follows: Firstly, the traditional definition of spin current can not number any more. Because of the TAM J is a good , make the spin conserved, which has been widely accepted. one can only choose the TAM ^J and its corresponding current J to Secondly, the term (18) causes the so-called quantum rotation, indu- J describe the transport phenomena. The QED theory points out that cing the nonconservation property of the spin current, which is the electron’s TAM can not stay in conservation in the external field. mentioned in Sun’s paper9 and in Jin’s paper11. The of the system’s Lagrangian gives out the More importantly, because the term (18), with an ‘‘i’’ in its coef- continuity equation ficient, stands for its quantum effect that can not be analogized L classically, it does not only contribute to the magnitude of the spin ~z~p z : z p ~ J J + JJ JJ 0 ð20Þ current in the same order compared with the traditional definition, Lt but also predict some important effects, such as the . Eq.(20) shows that the TAM of the system (the electrons and the photons) stays in conservation. It can be written in another form Helical edge states in Quantum SHE system. We choose Kane L L model for semiconductors confining in a heterojunction of HgTe/ ~z : ~{ ~pz : p J + JJ J + JJ CdTe. The parameters are adopted from the reference18. Lt Lt Fig. 1 shows the spin current of our definition. The wave functions The existence of Lint enables the electrons and the photons to Y(kx, y) are the edge states for L 5 200 nm. It is shown that the exchange angular momentum by some specific rules. This is exactly current exists not only in the bulk, but also on both edges (dependent the theoretical support on the experiments, namely by absorbing and on the spatial distribution parameters of the wave functions l1, l2 18 emitting the photons, the electron’s TAM can be changed. Since the and kinetic momentum k in reference ), while, no spin current exists spin current itself is not conserved, its rate equation can be derived according to the traditional definition using the Heisenberg equation of (shown in Methods). 1 ^Jyz~ ^vy^szz^sz^vy 2 The TAM in semiconductors. The NRA of the Dirac equation Eq.(5) can be written ~{ with ^vy Ly. H~H zH ð21Þ me 1 2 When k 5 0, the spin current still exists on the surface, as shown in Fig. 2. This distinctive character other than the traditional electric where 1,3,4 e 2 current has been discussed in previous papers . ~P{ ~A ~3 c P e e It should be pointed out that the surface effect of the spin current H1~ { zeA0{ +|~A z DA0ð22Þ can be much more enhanced due to the existence of the term (18). 2m 8m3c2 2mc 8m2c2 Because the quantum rotation is much stronger at the edges, it con- and tributes much more than the traditional definition of the spin current. e e H ~{ ~E| ~p{ ~A : ð23Þ 2 4m2c2 c The conservation and the continuity equations. As pointed out, the conservation of spin current is a contradictory issue. Different H2 is called the spin-orbit coupling, which is one of fundamentals conclusions have been drawn for taking different occasions into of the spintronics. To study the transport properties, the consideration. In non-relativistic , the spin is a electromagnetic susceptibility should be taken into calculation. In conserved quantity when the OAM is frozen. The continuity the case of the media having a relative velocity respect to the

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 3 www.nature.com/scientificreports carriers, the in the polarized media interacting 1 J ~{ w{ 2 P z + s w ð31Þ with the carriers is ij 4mc ijk k i j m{1 which is proportional to the matrix element of the current density ~D~ ~Ez ~v|H~ ð24Þ c operator in the QED. Eq.(31) shows that the spin current of J is proportional to the m{1 xz ~B~mH~z ~E|~v ð25Þ charge current Jy. This result coincides with the experimental data in c Kato’s work6, which strongly supports our definition. Therefore, the where~v is the relative speed of the media in the field. By placing these theoretical approach to estimate and calculate the spin current in relations into Eq.(21) and utilizing the relation terms of the density of electronic states has been provided. Meanwhile, the relation between the spin Hall conductance and 1 ~A~ ~B|~r, the charge conductance has been formulated, laying the foundation 2 of the electrically induced electron-spin polarization in spintronics. 1 Zhang proposed a semi-classical Boltzmann-like equation to the Hamiltonian (up to o ) turns to be 20 m2 describe the distribution of the spins . The similar behaviour can also be deduced from our definition, considering the finite size ~ 0 z 0 effects. In the system, the spin up current is H H1 H2 ð26Þ ~ Y Y ~ z hiJs xz :+jjJs :+ Jt Je P^2 em ^ e2 H’ ~ { ~Lz :H~z ~A2{ 1 2m 2mc 2mc2 The Jt and Je are the traditional definition of the spin current and the ð27Þ extra term (18), respectively. As shown in Methods, J is proportional ^4 t ~P e to k , namely zeA z DA x 8m2c2 0 8m2c2 0 +~ + e Jt +Ch Ex H0 ~ ðÞ2 m{1 z2ðÞm{1 ~L : ~E|P~ : ð28Þ 2 4m2c2 But Je is independent on kx, and is only as a function of the density 0 distribution of electrons in y direction, namely The spin-orbit coupling H2 turns into a larger H2. According to the +~ + + QED theory, the spin-orbit coupling is induced by the electric field in Je Cy Ey ðÞy : which the electron moves at a speed of P~ acting on the electron’s spin. This is a similar result compared to the Eqs. (12) and (13) in Zhang’s 20 e e LV work . The spin accumulates in y direction, which is exactly the ~|P~ ~ :~ 2 2 E 2 2 L ð29Þ same as his conclusion drawn from the anomalous Hall field. The 4m c 4m c Lr spin diffusion is decided by the parameters v and D in his study. For the moving carriers, one should include the OAM into However, in our expression, the spin diffusion is determined by the 18 calculation, considering the electromagnetic polarization in the spatial distribution parameters l1 and l2 . Now Let us discuss the solid-state media under the external field. This means that not spin-Hall effect in GaAs bulk system with consideration of the spin- only the spin, but also the OAM is coupled with the electric field. orbit coupling effect. According to Eq.(28), the can be When m~1, the coupling term (28) turns back to be Eq.(29), the written in cikiji. Y3,3, Y3,1 and Y1,1 accumulate on one edge while ? 2 2 2 2 2 2 same as the traditional spin-orbit coupling. When m 1, however, Y3,{3, Y3,{1, Y1,{1 on the other edge, namely the TAM j accumu- 2 2 2 2 2 2 the orbit angular accumulation affects the coupling term to the same lates in both edges. It is easy to find that on both sides, extent as the spin. Thus the OAM becomes crucial to describe the polarization of the system. DE X 3 3 According to the theory of the spin-Hall effect, the carriers car- YðÞr !b YðÞr ~ ,j !b ,j =0: ð32Þ J 2 z j 2 z rying different spins flow in the opposite directions. In our case, the jz carries with different angular momenta (j, jz) flow in the different 21 z{ { directions. The only difference is that the OAM is included in our The Kerr angular rotation is proportional to fab fab whose model. It should be noticed that the condition m?1 usually holds for expression is + mvab +2 most semiconductors, such as III-V compound semiconductors like f ~ P ð33Þ GaAs and GaN. Thus, ab e2 ab e : ~ ðÞ2 m{1 z2ðÞm{1 ~L ~E|P +~ Y + Y , 4m2c2 Pab ehiajjx iy b ð34Þ : ~ ~2 m gs~szgl~L ~E|P m ð30Þ where Ya is the ground state, Yb is the excitation state and vab is the B z{ { = energy gap. When fab fab 0, the Kerr rotation occurs. ~ ~ : ~|~ 2 m gjJ E P mB: The TAM j accumulation gives the same image as the traditional spin-Hall effect. Note that the spin does not accumulate actually, so According to the relation of the effective Lande g value and the the OAM plays an important role on the accumulation. Moreover, effective mass, g in Eq.(30) should be replaced by g* in the semi- because the TAM J offers more degrees of freedom, one can use it to conductors19. These imply that~j should replace the spin, as the phys- transmit more information under the same conditions. In summary, ical quantity in more general cases. the spin-orbit coupling has been regarded as the TAM j coupling with the electric field in systems with a large m. It is recommended that the TAM j current replaces the spin current to describe the Discussion motion of the carriers with different angular momenta. The physical According to Eq.(15), the non-diagonal matrix element of the spin nature of polarization accumulation and the Kerr rotation can be current that can be determined by explained using our theory.

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 4 www.nature.com/scientificreports

8 Methods <> i { ~ 1 w ðÞPi i w when i j The definition of spin current. The Lagrangian of the system17 of s~ is hiJ ~ 4mc 2 s ij > 1 m : { = L~y ic D {m y: ðÞWe choose c~1 { w 2 ijkPkz +i j w when i j m 4mc 0 i ab According to the Noether theorem, when y ~Ly~ 1{ ab y, 4 The NRA expressions of the OAM current and the TAM current are similar, except m LL m i ab i ab m for that the (s)S should be replaced by the operators L and J, respectively. dðÞJs ~ Ly~iyc { ab y{iy ab c y LDmy 4 4 The momentum current of Photons. The Lagrangian of the system of s 5 117 is m i i m ab 1 m m ab ~{iyc ab z ab c y~ yc abz abc y: 4 4 4 1 n m L~{ LmA LnA : i 2 When y0~Ly~ 1{ mnx D y, 2 n m 1 Similar to s~ , dAi ~ i Aj hk. According to the Noether theorem, one gets L 2 jk m~ L ~ m { i ab { i ab m dðÞJL dy iyc xaDb y iy xbDa c y LDmy 2 2 0~ p z p hk: j JL k JS k 1 m m ab ~ y xac Dbzxbc Da y: 2 The OAM current and the spin current are Here, Jm is the current operator of the spin s, and Jm is the current operator of the s L p~ i j OAM. JL ijkx P m ~ 1 m z m ðÞJs yc ab abc y LL ab 4 p~ i JS i jk Aj LL0A ðÞJ m ~x T zx T L ab a br b ar LL i~ m{ 0i where P m A Lg . 1 LL0A where T ~ yc D y. mn 2 m n The motion equations of angular momentum currents. According to the The tensor form Heisenberg equation, we have hi L i 1 1 1 i { ~z ~z :~ z 2z ðÞJ ~ ycsabzsabc y~ y bbamsabzsabbam y ðÞJS ðÞJS ,H amSn ,c ~a P bmc V s m,ab 4 m m 4 Lt mn mn 2 ð36Þ i c i c ~ y{ a S zS a y~ y{ a S y 1 1 : 1 1 2 ab m c c m ab m c ~z amSn ,c ~a P~ z amSn ,bmc : 4 2 2 2 In Dirac representation, we have Because of the relations ~ { m~ m y y b,c iba : : : : : : ½ai,a P ~ ai,ai Pizaj Pjzak Pk ~ ai,aj Pjzak Pk cacb~dab{i abcSc ~2i PjSk{PkSj i ~ ~ sab ca,cb abcSc: : : : : : : 2 ½Si,a P ~ Si,ai Pizaj Pjzak Pk ~ Si,aj Pjzak Pk ~2i Pjak{Pkaj The NRA form of spin current. The tensor form of the spin current is i ~{ J ~ ~a~S ½a,b 2ba S 2 namely, and w ~ Hy H x 1 : 1 : 1 : amSn ,c ~a P~ ~ am,c ~a P~ Snzam Sn,c ~a P~ 2 2 2 : e ~ : s {i +{ A ðÞs P c ~{ic ðÞS|P SnzamðÞa|P where x~ w~ y. Thus, m n 2mc 2mc "#"#! ! 0 0 w 1 1 1 {i i { { 2 ~ 2 z 2 hiJs ~ y ðÞaS y ~ w x amSn ,bmc am,bmc Sn am Sn,bmc 2 2 2 2 2 0 a 0 S x ! 2 ~{bammc Sn, i w 1 ~ { { ~ { z { x w a x a w S w a x S 2 x 2 the Eq.(36) turns into S 00 e 1 0 e 1 1 : i +{ ~A : {i +{ ~A i @@ c { A { @ c A A hi ~ y y z y y L S a ~{ i 2 2mc 2mc ð35Þ ðÞJS mn ðÞJS mn,H a S L t i e { { e i 2 ~ : { ~ z : { { ~ ~{ c amðÞa|P zðÞS|P Sn {ibmc amSn : i + A y y S y a i + A y n m 4mc c a c S :~A :~B ~~A:~Bzi : ~A|~B The spin-Hall effect in the finite size effect. For the edge states Y"1 :~B ~~Bzi ~B| z {ikx x z z z z 1 J ~~cze fzzc f{,g f{zc g fz ½ðÞkx z zh:c: : t kx 1 kx 2 2 ~A ~~Azi |~A T ik x z z z z ~cze x fzzc f{,g f{zc g fz The Eq. (35) turns to be kx 1 kx 2 2 2 e ~ z z { z z z z i e { { ~ kx fz ck f{ g1 f{ ck g2 fz hi~ : +{ ~ y y z y : {i +{ A y x x Js i A S a c 4mc c a S

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 5 www.nature.com/scientificreports

4. Murakami, S., Nagaosa, N. & Zhang, S. Dissipationless quantum spin current at z {ikx x z z z z 1 J ~~cze fzzc f{,g f{zc g fz iszkysz zh:c: e kx 1 kx 2 2 room temperature. Science 301, 1348–1351 (2003). T 5. Datta, S. & Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. ik x z z z z ~cze x fzzc f{,g f{zc g fz Lett. 56, 665 (1990). kx 1 kx 2 6. Kato, Y. K., Myers, R. C., Gossard, A. C. & Awschalom, D. D. Observation of the 1 {ikx x z z z z spin hall effect in semiconductors. Science 306, 1910–1913 (2004). ~~cze fzzc f{,g f{zc g fz {szLysz zh:c: kx 1 kx 2 2 7. Matsuzaka, S., Ohno, Y. & Ohno, H. dependence of the spin hall T ik x z z z z effect in GaAs probed by scanning kerr rotation microscopy. Phys. Rev. B 80, ~cze x fzzc f{,g f{zc g fz kx 1 kx 2 241305 (2009). 8. Rashba, E. I. Spin currents in thermodynamic equilibrium: The challenge of z 0 z 0 z z z z 0 z z 0 ~ fzzc f{ f zc f z g f{zc g fz g f zc g f : kx z kx { 1 kx 2 1 { kx 2 z discerning transport currents. Phys. Rev. B 68, 241315 (2003). 9. Sun, Q. F. & Xie, X. C. Definition of the spin current: The angular spin current and For the edge states Y"2 its physical consequences. Phys. Rev. B 72, 245305 (2005). 10. Sun, Q. F., Xie, X. C. & Wang, J. Persistent spin current in nanodevices and { { 2 { { { 2 definition of the spin current. Phys. Rev. B 77, 035327 (2008). J ~kx f{zc fz { g fzzc g f{ t kx 2 kx 1 11. Jin, P., Li, Y. & Zhang, F. SU(2)*U(1) unified theory for charge, orbit and spin currents. J. Phys. A 39, 7115 (2006). { { 0 { 0 { { { { 0 { { 0 12. Shi, J., Zhang, P., Xiao, D. & Niu, Q. Proper definition of spin current in Spin- J ~ f{zc fz f{zc fz z g fzzc g f{ g fzzc g f{ e kx kx 2 kx 1 2 kx 1 Orbit coupled systems. Phys. Rev. Lett. 96, 076604 (2006). 13. Hirsch, J. E. Spin hall effect. Phys. Rev. Lett. 83, 1834 (1999). The Spin-Orbit coupling in media. According to the Maxwell equations in the 14. Ohno, Y. et al. Electrical spin injection in a ferromagnetic semiconductor media heterostructure. Nature 402, 790–792 (1999). 15. Tombros, N., Jozsa, C., Popinciuc, M., Jonkman, H. T. & van Wees, B. J. Electronic m{1 ~D~ ~Ez ~v|H~ spin transport and spin in single graphene layers at room temperature. c Nature 448, 571–574 (2007). 16. Valenzuela, S. & Tinkham, M. Direct electronic measurement of the spin hall m{1 ~B~mH~z ~E|~v, effect. Nature 442, 176–179 (2006). c 17. Peskin, M. E. & Schroeder, D. V. An Introduction To (Westview Press, 1994), repr. edn. the first term in the NRA of Dirac equation turns to be 18. Zhou, B., Lu, H., Chu, R., Shen, S. & Niu, Q. Finite size effects on helical edge states e 2 in a quantum Spin-Hall system. Phys. Rev. Lett. 101, 246807 (2008). ~P{ ~A ~3 19. Shen, K., Weng, M. Q. & Wu, M. W. L-valley electron g-factor in bulk GaAs and ~ c { P z { e |~ z e H1 eA0 + A DA0 AlAs. J. Appl. Phys. 104, 063719 (2008). 2m 8m3c2 2mc 8m2c2 20. Zhang, S. Spin hall effect in the presence of spin diffusion. Phys. Rev. Lett. 85,393 e 2 ~{ ~| (2000). P B ~r ~P3 e e ~ 2c { z { ~z D 21. Condon, E. & Shortley, G. H. The Theory of Atomic Spectra (Univ. Press, 3 2 eA0 B 2 2 A0 2m 8m c 2mc 8m c Cambridge, 1977), repr. edn. 2 2 3 ~P e e 2 ~P e e ~ { ~L:~B z ~B|~r { zeA { ~Bz DA 2m 2mc 8mc2 8m3c2 0 2mc 8m2c2 0 ~P2 em e2 ~P4 e ~ { ~Lz :H~z ~A2{ zeA z DA Acknowledgements 2m 2mc 2mc2 8m2c2 0 8m2c2 0 This work was supported by the NSFC (Grants Nos. 11175135, 11074192). eðÞm{1 ~ 0 { ~z ~|P~ H1 2 L E 2mc Author contributions Therefore, AZ carried out all derivation work under the guidance of LFQ and LC. LY attended the eðÞm{1 e discussion. All authors contributed to the writing of the manuscript. H { ~Lz ~E|P~ ~{ ðÞ2 m{1 z2ðÞm{1 ~L : ~E|P~ 2 2mc2 4m2c2 Additional information Competing financial interests: The authors declare no competing financial interests. 1. Zˇutic´, I., Fabian, J. & Sarma, S. D. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004). License: This work is licensed under a Creative Commons 2. Wolf, S. A. et al. Spintronics: A Spin-Based electronics vision for the future. Attribution-NonCommercial-ShareAlike 3.0 Unported License. To view a copy of this Science 294, 1488–1495 (2001). license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ 3. Sinova, J. et al. Universal intrinsic spin hall effect. Phys. Rev. Lett. 92, 126603 How to cite this article: An, Z., Liu, F.Q., Lin, Y. & Liu, C. The universal definition of spin (2004). current. Sci. Rep. 2, 388; DOI:10.1038/srep00388 (2012).

SCIENTIFIC REPORTS | 2 : 388 | DOI: 10.1038/srep00388 6