By using the projection operator technique it is observed that the strong Hund's rule coupling and s-d interaction in transition metal elements may lead to an effective coupling between the spin current and spin chirality. As a result, the spin chirality can be regarded as a driving force to produce a spin current. The spin current may give rise to a novel type of field acting on the spins. A spin battery is designed based on the interactions between the spin current and spin chirality.published_or_final_versio

Shen, SQ

Xie, XC

arXiv

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Title Spin current in the Kondo lattice modelAuthor(s) Shen, SQ; Xie, XCCitation Physical Review B - Condensed Matter And Materials Physics,2003, v. 67 n. 14, p. 1444231-1444235Issued Date 2003URL http://hdl.handle.net/10722/43376Rights Physical Review B (Condensed Matter and Materials Physics).Copyright © American Physical Society.PHYSICAL REVIEW B 67, 144423 ~2003!Spin current in the Kondo lattice modelShun-Qing Shen1 and X. C. Xie21Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China2Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078and International Center for Quantum Structure, Chinese Academy of Sciences, Beijing, China~Received 6 January 2003; revised manuscript received 18 February 2003; published 28 April 2003!By using the projection operator technique it is observed that the strong Hund’s rule coupling and s-dinteraction in transition metal elements may lead to an effective coupling between the spin current and spinchirality. As a result, the spin chirality can be regarded as a driving force to produce a spin current. The spincurrent may give rise to a novel type of field acting on the spins. A spin battery is designed based on theinteractions between the spin current and spin chirality.DOI: 10.1103/PhysRevB.67.144423 PACS number~s!: 75.10.Lp, 72.15.GdSpin-dependent effects, such as giant or colossal magne-toresistance and magnetization switching, arise from the in-teraction between the spin of the charge carrier and an ap-plied external magnetic field. In spintronics, instead ofcharge, electron spin carries information and it is possiblethat capability and performance can be enhanced in spintron-ics devices.1–3 Possible applications and the fundamental sci-ence involved make the study of spin-dependent transport anintensive field in condensed matter physics. Recently, severalways to produce a pure spin current were proposed, such asthe anomalous Hall effect in ferromagnetic metals,4,5 ferro-magnetic resonance,6,7 and the spin current in a spin spiralstate.8,9 In this paper, we study spin current in the Kondolattice model, a simplified model containing essential physicsfor many ferromagnetic metals or semiconductors. We find atype of interaction between the spin current and spin chirality~defined later! in the strong Hund coupling limit. This indi-cates that the spin chirality can be regarded as a driving forceto produce spin current, similar to what an electric field doesto the electric current. Furthermore, the spin current inducesa type of effective magnetic field. Based on the interactionbetween spin current and spin chirality, we propose a schemeto design a spin battery. As a demonstration, we study thespin transport in the well known spiral state to bring out theabove mentioned spin-current effects.The Hund’s rule coupling and s-d exchange interactionare crucial in transition metal elements. They play a key rolein the formation of metallic ferromagnetism in colossal mag-netoresistance materials10–12 and diluted magnetic semi-conductors.13,14 In this paper we focus on doped transitionmetal oxides and/or ferromagnetic semiconductors. Theminimal model for these materials is the Kondo latticemodel,H52t (^n ,m& ,scn ,s† cm ,s2J2 (n ,ss8Sntss8cn ,s† cn ,s8 , ~1!where cn ,s† and cn ,s are the creation and annihilation opera-tor for conduction electrons, Sn is the spin operators for thelocalized spin, t is the Pauli matrix, and t0 is the 232identity matrix. The summation over ^n ,m& runs for thenearest neighbor pairs of lattice sites. In this paper we willset \5c51, and the lattice space is a and is set to unit.150163-1829/2003/67~14!/144423~5!/$20.00 67 1444For the Hund’s rule coupling, J is always positive. Themodel has been studied extensively.16 The spin current op-erator along the a axis in the tight binding approximation isdefined as t\Inn1as, andInn1as 52i2 (s ,s8~cn ,s† tss8cn1a ,s82cn1a ,s† tss8cn ,s8!,~2!where a(5x ,y ,z) are vectors pointing to the nearest neigh-bor sites, which have one longitudinal component and twotransverse components. The spin polarized currents and elec-tric current become equal in a fully polarized ferromagneticsystem.Before we discuss the spin transport of Eq. ~1! in an ar-bitrary dimension, let us first illustrate the physics by con-sidering the well known spin spiral state in a one-dimensional system. In general, the spin spiral state may notbe the ground state of Eq. ~1!; however, it can be stabilizedby external environments, such as by applying externalfields along different directions at the ends of the system.17Our purpose here is to demonstrate the consequence ofthe spin spiral state. We parametrize Sn by the polar anglesun and wn , i.e., making the classical spin approxima-tion. The spin spiral state is defined by Sn /S5(sin u cos nw, sin u sin nw, cos u). The spin chirality definedas the cross times between the two neighboring spins: ^Sn3Sn11&z5S2 sin2u sin w. In the k space, we introduce aspinor operator F†(k)5(cka1w/2,↑† ,cka2w/2,↓† ). The meanfield Hamiltonian is written asH5(kF†~k !H~k !F~k !, ~3!where H(k)52g0t02gxtx2gztz with g05m12t cos(w/2)cos ka, gx5(J/2)sin u, and gz5(J/2)cos u22t sin(w/2)sin ka. m is the chemical potential. The singleparticle Green function in a 232 matrix form is given byG~k ,ivn!5@ ivnt02H~k !#215@~ ivn1g0!t02gxtx2gztz#~ ivn2v11m!~ ivn2v21m!, ~4!©2003 The American Physical Society23-1S. Q. SHEN AND X. C. XIE PHYSICAL REVIEW B 67, 144423 ~2003!where vn5(2n11)pkT with an integer n and T is the tem-perature. The two branches of spectra arev6~k !522t cosw2 cos ka6F S J2 cos u22t sin w2 sin ka D21J24 sin2uG1/2. ~5!In the k space, the expectation value of the longitudinal spincurrent operator is written in terms of the Green’s function,^Izs&51N (k @A1nF~v12m!2A2nF~v22m!# , ~6!whereA6~k !56sinw2cos ka1t sin w sin2ka2J2cos u cosw2sin kaAS J2 cos u22t sin w2 sin ka D21J24sin2u.We also calculated the electric current Izc and found that Izc50. Thus in the spiral state there exists a spin current with-out a charge current. In Fig. 1 we plot the spin current as afunction of the electron density in the state with a u5p/2,and w50.2p for different J. We see that a larger J producesa higher spin current. For a finite J the current may changedirection with change of density. In Fig. 2 we plot the rela-tion between the spin current and spin chirality at the densityof 0.2. The spin current increases with the spin chirality.In the large J limit, the problem can be simplified. Anequivalent Hamiltonian of Eq. ~1! or so-called double ex-change model in the large J limit is given by18Hde52 (^n ,m&tnman1am , ~7!FIG. 1. Spin current via the density of the conduction electron inthe spin spiral state of u5p/2 and w50.2p .14442withtnm5tFcos un2 cos um2 1sin un2 sin um2 e2i(wn2wm)G ,and an5cos(un/2)(12nn↓)cn↑1exp@iwn#sin(un/2)(12nn↑)cn↓ and satisfies the anticommutation relation,@ak ,ak8†#5dk ,k8 . The energy excitations and their eigen-states can be obtained from the diagonalized HamiltonianHde522t( Fcos2 u2 cos ka1sin2 u2 cos~ka1w!Gak†ak ,~8!where ak is the Fourier transform of an . In the case whenthe spins of conduction electrons are frozen by the localizedspins, and the quasiparticles become spinless fermions. Thetwo Fermi momenta are k6F a52k06rp where k05arctan@sin2u sin w/(cos2u1sin2u cos w)#, and r is the ratioof the number of conduction electrons to the number of totallattice sites and r/a is the density of conduction electrons. k0is nonzero when the order parameter for the spin spiralling isnonzero. Its lowest energy state is uGS&5) kF2<k<kF1ak†u0&.The spin current in the state can be calculated from Izs5(n^GSuInn11s uGS&/N . We have an analytical expressionfor the spin current:Izs5t\sin rppsin2u sin wA12sin2u sin2~w/2!. ~9!The result is equal to what we obtain by taking the large limitin Eq. ~6!. In the point of view of the Berry phase the spinspiral state acquires a constant Berry phase in the spin-dependent renormalized factor in the Hamiltonian @Eq. ~7!#,i.e., tnn115utnn11ue2idw. The nonzero phase dw drives aspin current.By analogy with electric field and resistance, we intro-duce the concept of spin current resistance or its reciprocal,FIG. 2. Spin current via spin spirality at J51.0 with the densityof conduction electron 0.20.3-2SPIN CURRENT IN THE KONDO LATTICE MODEL PHYSICAL REVIEW B 67, 144423 ~2003!spin conductance. The role of Sn3Sm here is analogous tothe local electric field. In the spin spiral state the spin chiral-ity is nonzero, ^Sn3Sn11&z5S2 sin2u sin w. In Eq. ~9! thecurrent is proportional to the spin chirality ^Sn3Sn11&z , andcan be written in a compact form Izs5c^Sn3Sn11&z /S2 byintroducing a spin conductancec5sin rp/@pA12sin2u sin2~w/2!# ~10!in a large J limit. The spin conductance is determined by thefilling of electrons and the magnetic structure. Just like elec-tric conductance, the spin conductance reflects intrinsic prop-erties of spin transport in materials. It depends on the spin-dependent scattering mechanism of electrons. In Eq. ~10!, forthe case of r→0 or →1, c is proportional to the density ofelectrons r or the density of holes 12r . For a small spinchirality sin2u sin2(w/2)→0, the spin conductance ap-proaches @sin rp#/p, which is proportional to the averagingkinetical energy of the charge carriers in this tight bindingtheory. From the numerical results in Figs. 1 and 2, we de-duce that the spin conductance also depends on J. It becomesmore complicated for the finite J cases. Similarly, we canalso introduce a spin voltage, which is also proportional tothe length along the spin spiraling path, Vspin5*^Sn3Sn11&dlÕS2 where dl5rn112rn is the difference of twoposition vectors of neighboring spins. Vspin is an importantparameter characterizing the ability to produce a spincurrent.Thus far we have introduced several important concepts,such as spin chirality or spin current resistance, through theexample of the classical spiral state. To determine the micro-scopic mechanism to produce a spin current, we go back tothe Kondo lattice model @Eq. ~1!# in the large Hund’s rulecoupling limit, and address the quantum spin case. The fol-lowing discussions are not limited to one-dimensional sys-tems and the conclusion is independent of the dimensional-ity. According to the Hund’s rule coupling, the energy ofempty or double occupancy of conduction electrons is zero.For a single occupancy the conduction electron and localizedspin can form a spin S11/2 state with 2JS/2 and a spin S21/2 state with 1J(S11)/2. In the large J limit the spinS21/2 state and double occupancy should be excluded asthey have much higher energy than the spin S11/2 state.This process can be realized with the help of the projectiontechnique.19 For that, we introduce the spinor fn15@(12nn ,↓)cn ,↑† ,(12nn ,↑)cn ,↓† # . The dressed operators (12nn ,2s)cn ,s† , instead of cn ,s† , are used here to avoid thedouble occupancy due to the strong coupling. The projectionoperator for the spin S11/2 state is defined asP n15fn1Pn1fn , ~11!where Pn15@Snt1(S11)t0#/(2S11). The operator P n1forces the electron spin at site n to be fully polarized with thelocalized spin at the same site. The projection operator fornonoccupancy is P ne5(12nn ,↓)(12nn ,↑). The total projec-tion operator to exclude the spin S-1/2 single occupancy and14442double occupancy is P5)n(Pne1P n1). Therefore theequivalent Hamiltonian in the strong Hund’s rule couplinglimit is expressed as12,20He f f5PHP52t(nm@fn1Pn1Pm1fm1H.c.# , ~12!where a constant is omitted in the last step. In the large Sapproximation the Hamiltonian of @Eq. ~12!# becomes thedouble exchange model in Eq. ~7!. Using a mathematicalidentity~Snt!~Smt!5SnSmt01i~Sn3Sm!t , ~13!the Hamiltonian can be rewritten as21H52~S11 !2t~2S11 !2 (nmknm2t~2S11 !2 (nmknmSnSm2~S11 !t~2S11 !2 (nm ~Sn1Sm!Mnm1t~2S11 !2 (nm ~Sn3Sm!Inms , ~14!where knm5fn1t0fm1fm1t0fn and Mnm5fn1tfm1fm1tfn . The first term is the kinetic part for conductionelectrons excluding double occupancy, like the t term in thet-J model. The second term contains an effectiveHeisenberg-type spin-spin exchange interaction. Approxi-mately, the effective coupling is J f erro52t^knm&/(2S11)2, proportional to the kinetic energy in the sense of themean field approximation. The kinetic energy 2t^knm& in thetight binding approximation is always negative if the elec-tron filling r,1, such that the effective coupling is ferro-magnetic. This is consistent with the double exchangepicture.22,23 The third term is the interaction between localspin Sn and the vector Mnm . For a paramagnetic state ^Mnm&is equal to zero, but in a fully polarized state its value isequal to ^knm&, i.e., proportional to the kinetic energy.Among the four terms the most important observation isfrom the last term: it contains an interaction between thespin current operator Inms and the spin chirality Sn3Sm . Byanalogy with the coupling between the electric current andelectric field it is straightforward to understand why a non-zero spin chirality induces a pure spin current. Although wecannot derive an analytic expression for a finite J, it is ex-pected that the mechanism to produce a spin current remainsthe same.24An effective field and spin battery. To clarify the physicalmeaning of the last term, we come to calculate the rate ofchange of a local spin. From classical mechanics or from theHeisenberg equation of motion the rate of change of angularmomentum Sn is equal to the torque mn3he f fn which acts onthe spin,dSn /dt52gmBSn3he f fn , ~15!where the effective field experienced by the spin is3-3S. Q. SHEN AND X. C. XIE PHYSICAL REVIEW B 67, 144423 ~2003!he f fn 5tgmB~2S11 !2(d@knn1dSn1d1~S11 !Mnn1d1Inn1ds 3Sn1d# .The first two terms, again, are related to the particle’s orbitalmotions. The third term is the torque between the spin cur-rent and the local spin. A type of force or effective field iscaused due to the interaction between the spin current andlocal spin or local magnetization.The force may be respon-sible for magnetization precession by the spin injection into amagnetic conductor25 and the anomalous Hall effect in fer-romagnetic materials.4,26 Further discussion along this linewill be published elsewhere. Our observation will lead topotential applications. For example, by means of the spinspiraling properties we can design a spin battery to produce apure spin current. The battery contains two parts as shown inFig. 3. The main part is a spin spiraling material. The spinsspiral about the axis which is perpendicular to the interfacewith a ferromagnetic metal film, such that the spin voltagealong the axis is nonzero. The spin chirality in the materialwill produce the spin accumulation at two terminals. Themagnetization of the film is also perpendicular to the inter-face. The film is a source of spins which will flow out of thebattery driven by the spin chirality. The key element in thisbattery is the spin spiraling materials. Possible realizations ofspiral states have been discussed by several authors,27–30 and14442the state may also be stabilized with the help of the externalmagnetic field and exchange coupling between localizedspins. In conclusion, we have shown that the strong couplingbetween the spins of charge carriers and local spins leadsto an effective interaction between spin current and spinchirality in the system. A nonzero spin chirality may give riseto a pure spin current, and in return a spin current can causean interesting type of force on the spins. We confirm thesefindings by studying in detail the spin transport in the spinspiral state.The work was supported by grants from the ResearchGrants Council of Hong Kong, China ~Project No.HKU7088/01P! and the U.S. DOE.FIG. 3. Schematic view of dc spin battery. Arrows in the ferro-magnetic metal as a spin source indicate the magnetization is per-perdicular to the interface. The spin spiraling material provides anonzero spin spirality to drive the spin current along the axis.1 A. A. Wolf et al., Science ~Washington, DC, U.S.! 294, 1488~2001!.2 Spin Electronics, edited by M. Ziese and M. J. Thornton~Springer, New York, 2002!.3 Semiconductor Spintronics and Quantum Computation, edited byD. D. Awschalom, D. Loss, and N. Samarth ~Springer, NewYork, 2002!.4 J. E. Hirsch, Phys. Rev. Lett. 83, 1834 ~1999!; Phys. Rev. B 60,14787 ~1999!.5 S. Zhang, Phys. Rev. Lett. 85, 393 ~2002!.6 A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin,Phys. Rev. B 66, 060404 ~2002!.7 L. Berger, Phys. Rev. B 59, 11465 ~1999!; J. Appl. Phys. 90, 4632~2001!.8 S. Q. Shen, Phys. Lett. A 235, 403 ~1997!.9 J. Konig, M. C. Bonsager, and A. H. MacDonald, Phys. Rev. Lett.87, 187202 ~2001!.10 C. Zener, Phys. Rev. 82, 403 ~1951!.11 P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 ~1955!.12 K. Kubo and N. Ohata, J. Phys. Soc. Jpn. 33, 21 ~1972!.13 H. Ohno, Science ~Washington, DC, U.S.! 281, 951 ~1998!.14 T. Dietl, O. Ohno, F. Matskura, J. Cibert, and D. Ferrand, Science~Washington, DC, U.S.! 287, 1019 ~2000!.15 D. J. Scalapino, S. R. White, and S. Zhang, Phys. Rev. B 47, 7995~1993!.16 H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. 69, 809~1997!.17 As an illustration we consider a ferromagnetic Heisenberg chainwith n spins. The directions of two spins at two terminals can becontroled by strong magnetic fields. Assume the angles betweenthe two spins are fixed to be w . In the classic spin limit theground state is obtained by minimizing the classical energy andis shown to be an spin spiral state: the angle between two nearestneighbour spins is w/(n21). The quantum case can be solvedby the Bethe ansatz and/or in the spin wave theory. Other con-ditions such as longer-range exchange coupling and spin latticestructures may also give arise a spin spiral state.18 E. Mu¨ller-Hartmann and E. Dagotto, Phys. Rev. B 54, R6819~1996!.19 P. Fulde, Electron Correlations in Molecules and Solids~Springer, New York, 1997!.20 S. Q. Shen and Z. D. Wang, Phys. Rev. B 58, R8877 ~1998!; 61,9532 ~2000!.21 The microscopic mechanism should hold in the case of an anti-ferromagnetic coupling J,0 and large S. In the strong limit theconduction electron and local spin form a spin S21/2 state. Theprojection operator for the state is Pn25@2Snt1St0#/(2S11) instead of Pn1 . The main different point is that the couplingin the third term in Eq. ~14! is antiferromagnetic instead of fer-romagnetic, and the correspondent coefficients are revisedslightly. The ferromagnetism in diluted magnetic semiconduc-tors originates from the antiferromagnetic coupling between theconduction electrons and doped magnetic elements ~Ref. 14!.22 N. Furukawa, J. Phys. Soc. Jpn. 64, 2734 ~1995!.3-4SPIN CURRENT IN THE KONDO LATTICE MODEL PHYSICAL REVIEW B 67, 144423 ~2003!23 A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett.74, 5144 ~1995!; H. Ro¨der, J. Zang, and A. R. Bishop, ibid. 76,1356 ~1996!.24 Apparently the stronger coupling favors to produce the spin cur-rent in a spiral state, as shown in Fig. 1.25 W. Weber, S. Riesen, and H. C. Siegmann, Science ~Washington,DC, U.S.! 291, 1015 ~2001!.1444226 T. Jungwirth, Qian Niu, and A. H. MacDonald, Phys. Rev. Lett.88, 207208 ~2002!.27 K. Kubo, J. Phys. Soc. Jpn. 51, 782 ~1982!.28 J. Jensen and A. R. Machintosh, Rare Earth Magnetism ~Claren-don Press, Oxford, 1991!.29 M. Korling and J. Ergon, Phys. Rev. B 54, 8293 ~1996!.30 J. Inoue and S. Maekawa, Phys. Rev. Lett. 74, 3407 ~1995!.3-5

Spin current in the Kondo lattice model

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