Waves of spin-current in magnetized dielectrics
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Waves of spin-current in magnetized dielectrics
P. A. Andreev ∗ Department of General Physics, Faculty of Physics,Moscow State University, Moscow, Russian Federation.
L. S. Kuz’menkov † Department of Theoretical Physics, Physics Faculty,Moscow State University, Moscow, Russian Federation. (Dated: November 6, 2018)Spin-current is an important physical quantity in present day spintronics and it might be very usefullin the physics of quantum plasma of spinning particles. Thus it is important to have an equation of thespin-current evolution. This equation naturally appears as a part of a set of the quantum hydrodynamics(QHD) equations. Consequently, we present the set of the QHD equations derived from the many-particlemicroscopic Schrodinger equation, which consists of the continuity equation, the Euler equation, theBloch equation and equation of the spin-current evolution. We use these equations to study dispersionof the collective excitations in the three dimensional samples of the magnetized dielectrics. We show thatdynamics of the spin-current leads to formation of new type of the collective excitations in the magnetizeddielectrics, which we called spin-current waves.
I. INTRODUCTION
The spin-current is a very important characteristic forvarious physical systems. For instance it is useful for de-scription of such processes as spin injection [1] and otherprocess, where spin transport is involved. Most of them areaccumulated in a grate field of physics: spintronics [2], [3].There are different methods of the spin-current generation,such as spin pumping by sound waves [4], optical spininjection [5], and at using of junctions as Polarizers [6],spin flip in the result of electron interaction with electro-magnetic wave [7]. Spin-polarized currents are used inspin lasers development [8] and in spin-diode structures[9]. It is interesting to admit that in some devises the spin-current exhibits the sine wave-like behavior [10]. Newmodification of spin-field effect transistors have been sug-gested (see for example [11], [12]), one of the spintronicdevices utilize the electron’s spin properties in addition toits charge properties. Effects in junction play key role inspintronic devises. The spin-dependent Peltier effect wasobserved experimentally [13]. It is based on the ability ofthe spin-up and spin-down channels to transport heat inde-pendently. The use of graphene has been involved in thisfield either [14], [15]. Processes of spin transport and re-laxation in graphene have also been considered [16]. Ithas been expected that silicon spintronics has potential tochange information technology, this possibility discussedin Ref. [17].We have described a number of examples there the spin-current plays important role. Thus, it is necessary to haveanalytical definition of the spin-current and equation of thespin-current evolution. In Refs. [18]- [22] authors dis-cussed the definition of the spin-current in terms of the one-particle wave function describing the quantum state of thespinning particle. Knowledge of the wave function allowsto calculate the spin-current and make conclusions on the system behavior. We keep the spin-current as an indepen-dent macroscopic variable defined via the many-particlewave function governs system behavior. We use the spin-current along with the particle concentration, the velocityfield, and the magnetization. We consider the spin-currentevolution due to interparticle interaction. For this purposewe derive the spin-current evolution equation as a part ofthe set of quantum hydrodynamic equations. Total angularmomentum current was presented in Ref. [18] as a gener-alization of the spin current, which conserve even when thespin-current does not conserve. In our paper we considerthe spin-current only, since the spin-current has appeared inthe quantum hydrodynamic equations. It is not necessaryto have a conserved quantity for the spin-current. We studythe spin-current evolution due to interparticle interaction.It can be expected that the spin-current evolution alsogives influence on the spin waves, which are well-knownphenomenon in many different physical systems, first of allthey exist in the ferromagnetics and other structures withstrong magnetization. For example quantum dots show in-teresting spin waves behavior [23]- [27]. Dynamics of themagnetic moments in the quantum plasma has also beenstudied. In Ref. [28] existence of the self-consistentspin waves in the magnetized plasma was shown. Thusnew branches of the wave dispersion appear in the magne-tized plasma due to the dynamics of magnetic moment ofelectrons and ions. It was demonstrated by means of thequantum hydrodynamics (QHD) method [29]- [32]. Later,the generalization of the Vlasov equation for the plasmaof spinning charged particles was used to study the sameproblem [33]. Magnetic moment dynamics also leads toexistence of the effect of resonances interaction of the neu-tron beam with the magnetized plasma, which gives newmethod of wave generation in the magnetized plasma [34],[35], [36]. Method of the QHD has become very popularand powerful method of studying influence of the magneticmoment dynamics on various processes in the magnetizedplasma [34], [37]- [41]. The method of the QHD canbe used for systems of the neutral particles with the mag-netic moment. Such systems are more preferable to demon-strate results given by the QHD method at description ofthe physical effects caused by evolution of the magneticmoments. The QHD description of the self-consistent spinwaves in a system of neutral particles in the one-, two-, andthree-dimensional dielectrics was described in Ref. [42].Usually the set of QHD equations for the spinning chargedparticles consists of the three equations for evolution ofthe material fields, these are the continuity equation for theparticles concentration, the Euler equation for the velocityfield, and the generalized Bloch equation for the magneticmoment evolution, and the Maxwell equations for the elec-tromagnetic field description.The spin-current and an equation of its evolution arein the center of attention of this paper. The spin-currentfor many-particle systems of charged spinning particlesappears in the QHD equations [34], [43]. Usually weneed to find approximate relation between the spin-currentand other hydrodynamic variables to get closed set of theQHD equations. In this paper we go further, we derivean equation for the spin-current evolution by means of theQHD method. Recently, an analogous equation was de-rived for graphene electrons which have also been obtainedby the QHD method [44]. The spin-current definition andcorresponding equations appears in the QHD in the semi-relativistic approximation. The spin-current have been uni-versally defined according to the quantum electrodynamicsin Ref. [18].The first step in the development of the QHD was madeafter the Schrodinger equation had been suggested. E.Madelung represented the Schrodinger equation for the onecharged particle in an external field as a set of two equa-tions [45]. These equations are the continuity equationand the Euler equation for the eddy-free motion. Later T.Takabayasi considered the QHD representation of the Pauliequation [46].It is well-known that the one-particle Pauli equation canbe represented as the set of the QHD equations. This set ofequations is almost equivalent to the first three equationsof the chain of the many-particle QHD equations takenin the self-consistent field approximation. The differencesare the follows. One-particle equations do not contain thethermal pressure and the thermal contribution in the spin-current. Hence, the one-particle approximation may beused in semiconductors at low temperatures. However, thehydrodynamical equations should be coupled with the setof Maxwell equations.If we derive the QHD equations from the Pauli equationfor the one-particle in an external field we find three equa-tion of material field evolution mentioned above. However,if we have deal with many-particle system the set of theQHD equations contains some new functions, for examplethe kinetic pressure p αβ caused by the thermal motion of particles, and the spin-current J αβ . Two-particle macro-scopic functions also appear in the terms describing inter-particle interaction. In the self-consistent field approxima-tion, containing no contribution of the exchange interac-tion, the terms describing interaction have same form asthe terms caused by the external fields.To continue the comparison of the many-particle QHDwith the one-particle one we admit that in the one-particlecase p αβ = 0 and J αβ = M α v β , where M α is the den-sity of magnetic moments, and v β is the velocity field. Wecan also notice that in the one-particle case the kinetic en-ergy field ε easily related with the particle concentration n , magnetization M α and velocity field v β , so we have ε = mnv / . For the many-particle system we have that p αβ , J αβ and ε are independent material fields additionalto n , v α , and M α . In the many-particle system J αβ and ε are partly connected with n , v α , and M α . Let us makean example to describe the last statement. For the energywe have ε = mnv / nǫ , where the last term corre-sponds to internal energy caused by the thermal motion ofparticles [29]. A set of QHD equations including the en-ergy evolution equation and the non-zero thermal pressurein the Euler equation can be derived from a quantum kineticequation, but derivation of the quantum kinetic equationneeds some additional assumption to construct a distribu-tion function [47], [48]. Thus, to make more detailed studyof magnetic moment dynamics we are going to presentthe straightforward derivation of equation for the magneticmoment current (or the spin current) J αβ evolution fromthe many-particle Schrodinger and study its influence onthe spin wave dispersion. During several last decades a lotof different ways of the quantum kinetic equation deriva-tion were suggested, but there are a lot of open questionsin this field. Derivation of a kinetic equation via the Wignerdistribution function is the most popular and actively usedin recent publications. Moreover we keep developing themethod of many-particle quantum hydrodynamics whichis more direct way of derivation of equations for collectivequantum dynamics [34], [49], [50].Waves in systems of neutral and charged spinning parti-cles have been considered by means of many-particle quan-tum hydrodynamics with no account of the spin-currentequation. It has been assumed that the spin-current J αβ appearing in the magnetic moment evolution equation canbe approximately considered as J αβ = M α v β . In this pa-per studying waves in the magnetized dielectrics we con-sider two kind of equilibrium states. One of them corre-sponds to the case when the equilibrium spin current J αβ equals to zero, in second case we consider a non zero equi-librium spin current J αβ , but we suppose that the equilib-rium velocity field equals to zero. Such structure might berealized by means two currents (flows of neutral particles)directed in opposite directions and having opposite equilib-rium spin.We also need to accent the fact that the many-particleQHD method has been used for the different physical sys-tems, thus the sets of the QHD equations have been ob-tained for graphene [44], the neutral ultracold quantumgases [51], the Bose-Einstein condensate of excitons ingraphene [52], along with the physics of plasma describedabove.Presented here results are also important for the physicsof magnetized ultracold quantum gases. Used where mod-els are equivalent to the first three equations of the QHD,they are the continuity equation, the Euler equation, andthe Bloch equation [53], [54].This paper is organized as follows. In Sec. II we presentand describe the set of the QHD equations derives in thepaper. In Sec. III dispersion of the spin waves is consid-ered, a contribution of the spin-current in the spin waveproperties is studied. In Sec. IV brief summary of obtainedresults are presented. II. THE MODEL
The many-particle QHD equations are derived fromthe microscopic many-particle Schrodinger equation ı ~ ∂ t Ψ( R, t ) = ˆ H Ψ( R, t ) , where Ψ( R, t ) is the wavefunction of N interacting particles. Ψ( R, t ) depends oncoordinate of all particles. We present it shortly by means R , which is R = ( r , r , ..., r N ) , where r i is coordinate of i th particle. The structure of the QHD equations dependson the explicit form of the Hamiltonian of considered sys-tem of particles. We do not described here the method ofderivation of the QHD equations, a lot of paper are dedi-cated to this topic [29]- [32]. However, to be certain wepresent the Hamiltonian ˆ H = X p (cid:18) m p D p + e p ϕ extp − γ p σ αp B αp ( ext ) (cid:19) + 12 X p,n = p ( e p e n G pn − γ p γ n G αβpn σ αp σ βn ) (1)used for derivation of equations presented below, where D αp = − ı ~ ∂ αp − e p A αp,ext /c , ϕ p,ext , A αp,ext are the po-tentials of the external electromagnetic field, ∂ αp = ∇ αp is the derivatives on space variables, and G pn = 1 /r pn is the Green functions of the Coulomb interaction, G αβpn =4 πδ αβ δ ( r pn )+ ∂ αp ∂ βp (1 /r pn ) is the Green function of spin-spin interaction, γ p is the gyromagnetic ratio, σ αp is thePauli matrix, a commutation relations for them is [ σ αp , σ βn ] = 2 ıδ pn ε αβγ σ γp ,e n , m n are the charge and the mass of particle, ~ is thePlanck constant and c is the speed of light. For electrons γ p reads γ p = e p ~ / (2 m p c ) , e p = −| e | .This Hamiltonian (1) consists of two parts. The first ofthem is presented by the first three terms, which describe motion of independent particle in an external electromag-netic field. The first term is the kinetic energy, which con-tains vector potential via covariant derivative D n . There-fore, it contains action of an external magnetic field anda rotational electric field on particle charge. The secondterm in formula (1) describes interaction of particle chargewith an external potential electric field. And the third termpresents the action of an external magnetic field on themagnetic moments. The second group of terms consists ofthe two last terms. They describe the interparticle interac-tion. In this paper we consider the Coulomb and the spin-spin interactions presented by the fourth and fifth termscorrespondingly.Using explicit form of the Hamiltonian, we obtain thechain of equations, we truncate the chain of the QHD equa-tions including the four equations only. These are the evo-lution equations for the particle concentration n , the veloc-ity field v β , the density of magnetic moment or spin M α ,and the spin-current J αβ .The first step in derivation of the many-particle QHDequations is the definition of particle concentration. Whichis the first collective quantum mechanical observable in ourmodel. The particle concentration is the quantum mechan-ical average of the microscopic concentration n ( r , t ) = Z Ψ + ( R, t ) X p δ ( r − r p )Ψ( R, t ) dR, (2)where we integrate over the 3N dimensional configura-tional space, and dR = Q Np =1 d r p . The formula (2)is more than the first collective quantum mechanical ob-servable. Using of the particle concentration operator ˆ n = P p δ ( r − r p ) gives the projector of the 3N dimen-sional configurational space in the three dimensional phys-ical space. Waves propagation, the charge-current flow, thespin-current flow happen in the three dimensional physi-cal space. Consequently it is worthwhile to have a model,which explicitly describes the dynamic of quantum many-particle system in the physical space. The QHD is an ex-ample of such model.We now differentiate the particle concentration with re-spect to time and find the continuity equation, where theparticles current j = n v emerges.Hence, the first equation of the QHD set of equations isthe continuity equation ∂ t n + ∇ ( n v ) = 0 . (3)At derivation of the continuity equation (3) the explicitform of the particles current appears as j = Z X p δ ( r − r p ) 12 m p (cid:18) Ψ + ( R, t ) D p Ψ( R, t )+ h.c. (cid:19) dR, (4)where h.c. means the hermitian conjugation.Differentiating the function of current with respect totime, we obtain the momentum balance equations, thisequation is an analog of the Euler equation mn ( ∂ t + v ∇ ) v α + ∂ β p αβ − ~ m ∂ α △ n + ~ m ∂ β ∂ α n · ∂ β nn ! = enE α + ec ε αβγ nv β B γ + M β ∇ α B β , (5)where E and B are the electric and magnetic fields, M is thedensity of magnetic moments, ε αβγ is the antisymmetricsymbol (the Levi-Civita symbol), p αβ is the kinetic pres-sure tensor. The momentum balance equation (5) has usualform, we see that evolution of the velocity field caused bymomentum current on thermal velocities p αβ , the quantumBohm potential specific for quantum kinematics (two termsin the left-hand side of equation (5), which proportional to ~ ), and interaction, which is presented in the right-handside of the momentum balance equation (5). We derivethis equation for charged spinning particles, thus the forcefield contains the density of the Lorentz force describingaction of electromagnetic field on charges presented by twofirst terms and force acting on the magnetic moment den-sity from the magnetic field presented by the last term. Wepresent the force field in the self-consistent field approxi-mation. General form of the force field appearing in theEuler equation and introducing of the self-consistent fieldapproximation are presented in Appendix A.The second-fifth terms in the left-hand side of equation(5) appear due to representation of the momentum flux Π αβ . At derivation of the Euler equation the momentumflux Π αβ emerges in the following explicit form Π αβ = Z X p δ ( r − r p ) 12 m p ×× (cid:18) Ψ ∗ ( R, t ) ˆ D βp ˆ D αp Ψ( R, t ) + h.c. (cid:19) dR. (6)To represent the momentum flux via the hydrodynamicvariables we need to introduce the velocity of a quantumparticle v αi ( R, t ) . It appears via the phase S ( R, t ) of themany-particle wave function Ψ( R, t ) = a ( R, t ) e ıS ( R,t ) .The velocity of i th particle is v αi ( R, t ) = ~ m ∇ i S ( R, t ) .We can also introduce the thermal velocity of i th particle u αi = v αi ( R, t ) − v α ( r , t ) as difference of the velocity of ith particle and the the center of mass velocity (the velocityfield) v ( r , t ) = j ( r , t ) /n ( r , t ) (for details see Refs. [29],[32]).The definition of magnetization M α ( r , t ) = Z X p δ ( r − r p )Ψ + ( R, t ) b σ α Ψ( R, t ) dR, (7)appears at derivation of the Euler equation (5)The equation of evolution of the magnetic moments ∂ t M α + ∇ β J αβ = 2 γ ~ ε αβγ M β B γ (8)is derived at differentiating of the magnetization with re-spect to time and using of the Schrodinger equation for the time derivatives of the wave function. This equation is ageneralization of the Bloch equation. From equation (8)we see that evolution of magnetic moment density causedby both the spin current J αβ and interaction of the mag-netic moments with the magnetic field. Charge of particlesgives no interference in dynamics of the magnetic momentdensity.The explicit form of spin-current in the many-particlesystem is J αβ = Z X p δ ( r − r p ) 12 m p ×× (cid:18) Ψ + ( R, t ) ˆ D βp b σ αp Ψ( R, t ) + h.c. (cid:19) dR. (9)Next equation is the equation of spin-current evolution.The spin-current appears in the Bloch equation, and if weinclude the spin-orbit and the spin-current interaction weget that the spin-current gives contribution in the force fieldin the Euler equation (5), for example see Ref. [34]. Thespin-current evolution equation for system of neutral parti-cles is ∂ t J αβ + ∂ γ ( J αβ v γ )= γ m n∂ β B α − γ ~ ε αγδ B γ J δβ . (10)for flux of the spin-current J αβγ which emerges in the sec-ond term in the left-hand side of equation (10). Its explicitform is J αβγ = Z X p δ ( r − r p ) 14 m p ×× (cid:18) Ψ + ( R, t ) ˆ D γp ˆ D βp b σ αp Ψ( R, t ) + h.c. (cid:19) dR. (11)Definition of the flux of spin-current contains two opera-tors of the long derivative ˆ D αp when the spin-current con-tains one operator of the long derivative. Since J αβγ isthe flux of the spin-current we use an approximate formula J αβγ = J αβ v γ . In general case J αβγ has more com-plex structure and contains additional contribution of boththe thermal motion and quantum kinematics as the quan-tum Bohm potential. The last one shows in the form of aterm analogous to the quantum Bohm potential. We do notconsider these contributions and pay attention to the spincurrent evolution caused by inter-particle interaction. Weshould pay special attention to equation (10) because allthis paper is dedicated to consideration of the influence ofthe spin-current evolution on dynamics of particles system.Equation (10) is presented for the chargeless spinning par-ticles, and this form will be used in the paper. However, wenow present it for the charged spinning particles ∂ t J αβ + ∂ γ ( J αβ v γ ) = em M α E β + emc γε βγδ J αγ B δ + γ m n∂ β B α − γ ~ ε αγδ B γ J δβ . (12)Equations (3)-(10) take place for each species of parti-cles. The electric E and magnetic B fields appearing inthe equations (3)-(10) are caused by charges, electric cur-rents, and magnetic moments of medium and satisfy tothe Maxwell equations. Thus the QHD equations for eachspecies of particles connect by means of the Maxwell equa-tions ∇ B = 0 , ∇ E = 4 π P a e a n a , ∇ × E = 0 , ∇ × B = 4 πc X a e a n a v a + 4 π X a ∇ × M a , (13)where subindex ”a” describes the species of particles. TheMaxwell equations (13) presented here do not contain timederivatives of the electric and the magnetic fields, be-cause we have derived the QHD equations from the non-relativistic theory. III. DISPERSION EQUATION
We consider the collective eigen-waves in a system ofthe neutral spinning particles being in an external uniformmagnetic field. Hence, we have deal with the paramag-netic and diamagnetic dielectrics. There are the two fun-damental collective excitations in such physical systems,they are the sound waves and the spin waves. Followingthe QHD description of the three dimensional magnetizeddielectrics [42] one can show that there is one type ofthe spin waves in such systems. These waves have a con-stant eigen-frequency ω = 2 γB / ~ , which is the cyclotronfrequency, where B is an external magnetic field. Herewe have no dependency on wave vector, consequently thegroup velocity ∂ω/∂k of these waves equal to zero.We are interested in an interference of the spin-currentevolution on the dispersion properties of the medium. Tofind the dispersion dependence of eigen-waves in the de-scribed systems we consider small amplitude excitationsaround an equilibrium state of the medium. We considertwo different equilibrium states. In the first case we sup-pose that an equilibrium spin-current equals to zero, and inthe second case we consider a medium with an equilibriumspin-current under condition that an equilibrium velocityfield equals to zero.For getting of solution we consider hydrodynamic vari-ables as the sum of an equilibrium part and a small pertur-bation n = n + δn, E = 0 + E , B = B e z + δ B , v = 0 + δ v ,M α = M α + δM α , M α = χB α , J αβ = J αβ + δJ αβ , p αβ = pδ αβ , δp = mv F δn a , (14)where δ αβ is the Kronecker symbol, v F is the Fermi ve-locity, χ a = κ a /ν a is the ratio between the equilibriummagnetic susceptibility κ a and the magnetic permeability ν a = 1 + 4 πκ a . In the case κ a ≪ we have χ a ≃ κ a .Substituting relations (14) in the set of equations (3), (5),(8) and (13) and neglecting by the nonlinear terms, we ob-tain a system of linear homogeneous equations in the par-tial derivatives with constant coefficients. Passing to thefollowing representation for the small perturbations δfδf = f ( ω, k ) exp ( − ıωt + ıkx ) yields a homogeneous system of algebraic equations.Even when we consider an equilibrium spin-current thelinear set of QHD equations splits on four independent sets.One of them contains n , δv x , δM z , J zx , the second onecontains δM x , δM y , δJ xx , δJ yx , the third set includes δJ xy , δJ yy , and fourth includes δJ xz , δJ yz . Two last setshave the same solution, which is the cyclotron frequency ω = 2 γB / ~ . Hence, we have to consider solutions of thefirst and second sets. A. The first group of dispersion branches
The first set give the following dispersion equation ω + ω (cid:18) πn γ m − υ (cid:19) k + 4 πk M mn J zx ω − πn γ m υ k = 0 , (15)where υ = v F + ~ k m , (16)and v F = (3 π n ) / ~ m . (17)We can see that a non-zero equilibrium spin-current leadsto existence of the additional term in the dispersion equa-tion.In the absence of an equilibrium spin-current we havetwo solutions of the equation (15) ω = − πγ n k /m, (18)and ω = υ k , (19)where solution (19) is the well-known sound wave. Dis-persion of the sound wave consists of two parts (16). Thefirst of them is the usual linear term, which appears dueto the Fermi pressure. The second term is the contributionof the quantum Bohm potential giving dispersion of the deBroglie wave. Moreover the evolution of spin-current leadsto existence of the second solution (18). This solution canbe rewritten as ω = ± ı r πn m γk. (20)It shows faster damping and gives no wave behavior.The sound wave solution (19) and spin wave ω =2 γB / ~ appear in the simpler model without account ofthe spin-current evolution. In this case a system of neutralspinning particles can be described by the continuity, Eu-ler, and magnetic moment evolution equations, where wecan put J αβ = M α v β to close the set of equations. It hasbeen mostly used for systems of charged spinning particles[28], [34], [37].Presence of the equilibrium spin-current in the equa-tion (15) makes it rather complicate. So, we are goingto solve it numerically. It seems reasonable to chouse Ω ≡ ω/λ = p m/ ( πn ) ω/ (2 γk ) as dimensionless fre-quency, where we introduced λ = 4 πγ n m k . (21)In this case we get equation (15) in the form of Ω + (1 − α )Ω + β Ω − α = 0 , (22)where α = mυ / (4 πn γ ) is the parameter describ-ing contribution of the quantum Bohm potential, β = kM J zx / ( γ n ) is the parameter describing contributionof the equilibrium spin-current. On figures we presenta limit case: de-Broglie regime when α ≃ α ~ = ~ k / (16 πmn γ ) . The both parameters α and β dependon module of the wave vector k . Consequently equation(22) allows to get Ω( k ) dependence. Equation (22) givestwo solutions. One of them is stable solution, which disper-sion presented on Fig. (1). The second solution of equation(22) shows an instability, which exists due to the existenceof the equilibrium spin-current J zx . This solution is pre-sented on Fig. (2).Solutions (18) and (19) can be briefly written in termsof reduced frequency Ω and α , so we have Ω = − and Ω = α .Let us consider approximate solutions of equation (22)at nonzero equilibrium spin-current under assumption that β gives small contribution in dispersion of waves (19) and(20). Thus solutions of equation (22) emerge as ω = υk + ζ (23)and ω = ± ı r πn m γk + ζ , (24)where ζ i = βω i ω i (2( ω i /λ ) + 1 − α ) + λβ ≃ β ω i /λ ) + 1 − α ) . (25) FIG. 1. (Color online) The figure describes the dispersion depen-dence of the stable collective excitation described by equation(22). This figure is obtained in the de-Broglie regime. We seethat Ω approximately proportional to k . So this figure revealsquadratic dependence of the frequency ω of wave vector k .FIG. 2. (Color online) The figure describes the second solutionof equation (22) in the de-Broglie regime. This solution is un-stable.) The dispersion dependence and corresponding instabilityincrement of the collective excitation are presented. ω i is used for short representation of solutions (19) and(20). Including αλ = υk we come to the following dis-persion relation for the sound wave ω = υk + βλ υ k + λ ) ≡ U ∗ k, (26)where we introduced a modified sound velocity U ∗ , whichslightly depends on the wave vector k via the quantumBohm potential. U ∗ = υ + 2 πγ n ( β/k ) mυ + 4 πγ n . (27)Since β > and β ∼ k we see that U ∗ > υ .Contribution of the spin-current in unstable branch (24)in considering limit ω = ± ıλ − βλ υ k + λ )= ± ı r πn m γk − πγ n βmυ + 4 πγ n (28)appears to be real and negative. We see that real part ofsolution (28) is almost a linear function of the wave vector k , but it contains small additional dependence on the wavevector via υ containing contribution of the quantum Bohmpotential. This results differs from previously consideredcase depicted on Fig.2. Formula (28) is obtained in thelimit of small contribution of the equilibrium spin-current.When Fig.2 is obtained for a finite value of the equilibriumspin-current, so it reveals an interesting dependence of realand imaginary parts of the frequency on the wave vectordiscussed above. B. Wave dispersion for the second group of variables
The second set of equations containing evolution of δM x , δM y , δJ xx , δJ yx gives the following dispersionequation ω + 8 π Ω γ γ ~ ω kJ zx +4 π kω ( ω +Ω γ ) ωkn γ ~ + Ω γ γmJ zx m ~ ( ω − Ω γ ) − Ω γ (1 − πξ ) = 0 , (29)where Ω γ = γB ~ and ξ = M /B . For the ferromagneticsamples ξ is larger than one ξ ≫ , while ξ is rather small ξ ≪ for the para- and the dia-magnetics.In the absence of the equilibrium spin-current this equa-tion is simplified to ω + (cid:18) λ − γ (1 − πξ ) (cid:19) ω +Ω γ (cid:18) λ +Ω γ (1 − πξ ) (cid:19) = 0 . (30)Solving this equation we get ω = Ω γ (1 − πξ ) − λ ± r λ − λ Ω γ (1 − πξ ) + 16 π ξ Ω γ . (31)These are new solutions. Their occurrence is caused by theaccount of the spin-current evolution, so we called themspin-current waves.In the small k limit we come to ω = Ω γ (cid:18) πξ − πξ (cid:19) + λ (cid:18) − − πξ − + πξ (cid:19) . (32) In the paramagnetic limit formula (30) gives ω ξ = Ω γ − λ ± λ r λ − γ . (33)At large k at small magnetic field ( λ ≫ Ω γ ) only onesolution exists, which corresponds to the sign plus in frontof the square root in formula (31). This solution appears as ω λ, + = Ω γ s π ξ Ω γ λ − , (34)since Ω γ λ ≪ we have that this solution exists at ξ > λ π | Ω γ | ≫ . It corresponds to large magnetization M .Equation (29) can be rewritten as an equation of fifthdegree. One of its solution is ω = 0 . So we have equationof fourth degree ω + (cid:18) λ − γ (1 − πξ ) (cid:19) ω + ϑω + Ω γ (cid:18) Ω γ (1 − πξ ) + λ (cid:19) = 0 , (35)where ϑ = 16 πγ Ω γ J zx k/ ~ . (36)This equation (35) differs from (30) by one term only. Itis the third term ϑω . Thus we have changing of solutions(31) caused by the equilibrium spin-current ϑ ∼ J zx . So-lutions of equation (35) we calculated approximately underassumption that the equilibrium spin-current gives smallcontribution in dispersion dependence. Designating so-lutions of equation (30) presented by formula (31) as ̟ .Then solutions of equation (35) appear as ω = ̟ + ϑ̟ ̟ (2 ̟ + λ − γ (1 − πξ )) + ϑ ≃ ̟ + ϑ ̟ + λ − γ (1 − πξ )) . (37)Using explicit form of ̟ in the second term of formula(37) we come to the following formula ω = ̟ ± ϑ q λ − λ Ω γ (1 − πξ ) + 64 π ξ Ω γ . (38)Signs plus and minus in front of the second term of formula(38) correspond to signs in formula (31). C. Discussion
Formulas (18), (19), (31) and the cyclotron frequency ω = 2 γB / ~ appear as solutions of a dispersion equationobtained at account of the spin-current evolution in the ab-sence of the equilibrium spin-current.Solution (18) reveals two branches (20). One of themhas an increasing amplitude. Another one has a decreas-ing amplitude. Since we have no source of the energy inthe system we have no mechanism for the amplitude in-creasing. So, we conclude that the decreasing branch takesplace in considering case. This decreasing solution givesno contribution in the spectrum as it reveals monotonic de-creasing of the amplitude (non oscillating solution). Thuswe have got left the two unchanged solutions: the soundwave (19) and the spin wave ω = 2 γB / ~ , which can befound with no account of the spin-current evolution. Wehave also found solutions (31), which present the two spin-current waves reaching the wave spectrum of magnetizeddielectrics.At this step we can conclude that the account of thespin-current evolution, without changing of conditions sys-tem being at, we obtained additional information aboutprocesses happen in the system. In considering case wehave got the two additional wave branches (the spin-currentwaves) described by formula (31). Formula (32) revealsthat at small k and large enough external magnetic field B the cyclotron frequency Ω γ = 2 γB / ~ gives main contri-bution in the dispersion of the spin-current waves.Consideration of the spin-current as an independentphysical variable gives us possibility to consider some con-ditions, which can not be included in more simple model.One of these conditions is the existence of an equilib-rium spin-current with the zero equilibrium velocity field v = 0 . Presence of an equilibrium spin-current leads tochanges of solutions (18) and (19), and to complication ofdispersion equation (30) (degree of this equation increases,so we get equation (29)). However it gives no change inthe dispersion of spin waves with the cyclotron frequency ω = 2 γB / ~ .Presence of an equilibrium spin-current J zx leads tochange of the solutions (19) and (20). Real part of solu-tion (20) appears due to J zx . In the de-Brolie regime itis pictured by lowest curve on Fig. (2). It reveals as acurve with two linear areas, one at small k , and the sec-ond one at k ≥ · cm − . They connect smoothlyaround k = 3 · cm − . Thus, we can assume Ω = ν i k ,with different ν i for each area. Consequently, we have Reω = p πn m γν i k . Imaginary part of solution Imω is pictured by upper curve on Fig.(2). Its form is similarto the form of the real part of the frequency
Reω , but ithas larger value. Limit of small contribution of the equilib-rium spin-current allows to obtain some analytical solution(28). In this approximation the equilibrium spin-current J zx gives rise to appearance of the negative real part of the frequency. It gives no changes in the imaginary part of thesolution.Let us discuss the sound wave. In the absence of anequilibrium spin-current it is described by formula (19).In the presence of an equilibrium spin-current we numeri-cally consider a limit case: the de-Broglie regime. In thede-Broglie regime we find that the equilibrium spin-currentdoes not change form of the dispersion curve (formula (19)gives us ω = ~ k m in the de-Broglie regime). Thus we seethat Ω linearly depends on the wave vector k , and ω ∼ k .The approximation of small equilibrium spin-current letsto trace contribution of the equilibrium spin-current on thesound wave analytically. In the de-Broglie regime an addi-tion to spectrum of the free quantum particles as △ ω S = 8 πmn γ β ~ k + 16 πmn γ ∼ kk + χ , where χ does not depend on the wave vector k . One cansee that the frequency shift is positive and decreases at theincreasing of the wave vector. In the Fermi regime we findthat the sound velocity increases on a constant value de-fined by formula (27).Let us discuss now contribution of the equilibrium spin-current in the dispersion of the spin-current waves. For-mula (38) shows that the spin-current wave having plus(minus) in front of the square root in formula (31) gets pos-itive (negative) contribution of the equilibrium spin-currentin the dispersion dependence. So we find an increasing (adecreasing) of the frequency. Including the fact that ϑ ∼ k and λ ∼ k we have the following dependence of the lastterm in formula (38) (a frequency shift caused by the equi-librium spin-current) on the wave vector appears as △ ω J = 8 √ πm Ω γ J zx ~ √ n k p k − ˜ ak + ˜ b , where ˜ a = 2 m (1 − πξ )Ω γ / ( n γ ) and ˜ b =4 m ξ Ω γ / ( n γ ) are positive constants. At k → weobtain △ ω J → . △ ω J increases with increasing of thewave vector. However △ ω J reaches its maximum at anintermediate wave vector k = √ mξ Ω γ γ √ n . This maximumvalue of the frequency shift is △ ω J ( k ) = 8 √ πm Ω γ J zx ~ √ n q p ˜ b − ˜ a . At large k the frequency shift △ ω J decreases as /k . IV. CONCLUSION
To get influence of the spin-current on dynamics of mag-netized dielectrics we have derived equation of the spin-current evolution as a part of the set of the QHD equations.In the result we have set of four equations: the continuityequation (particle number evolution equation), the Eulerequation (momentum balance equation), the Bloch equa-tion (magnetic moment balance equation), and equation ofthe spin-current evolution. These equations are used in theself-consistent field approximation.With no account of the spin-current evolution equationwe find two wave solutions: the sound wave and one spinwave solution. Including the spin-current evolution equa-tion leads to account of new solutions. We found three newwave solutions. Two of them have frequencies near the cy-clotron frequency. These solutions make spectrum of spinwaves richer. It appears as a splitting of one spin wavebranch on three branches. The third solution has nega-tive square of the frequency and reveal monotonic dampingof perturbation amplitude. These solutions emerge whenthe equilibrium spin-current equals to zero. Account of anequilibrium spin-current does not give new solution, but achange of wave dispersion was obtained.
APPENDIX A: GENERAL FORM OF THE FORCEFIELD AND THE SELF-CONSISTENT FIELDAPPROXIMATION
In the Euler equation (5) the force field F α = enE α + ec ε αβγ nv β B γ + M β ∇ α B β (39)is presented in the self-consistent field approximation.Here we are going to present general form of this forcefield and explain how we got formula (39) from the gen-eral formula.Force field consists of two parts F α = F αext + F αint . (40)The first of them is the force of the particle interaction withan external field F αext = enE αext + ec ε αβγ nv β B γext + M β ∇ α B βext , (41)and the second part is the inter-particle interactions F αint = − e Z ( ∇ α G ( r − r ′ )) n ( r , r ′ , t ) d r ′ + Z ( ∇ α G βγ ( r − r ′ )) M βγ ( r , r ′ , t ) d r ′ , (42)where n ( r , r ′ , t ) = Z X p,n = p δ ( r − r p ) δ ( r − r ′ n )Ψ ∗ ( R, t )Ψ(
R, t ) dR (43)is the two-particle concentration, and M αβ ( r , r ′ , t ) = Z X p,n = p δ ( r − r p ) δ ( r − r ′ n ) ×× µ B Ψ ∗ ( R, t ) σ αp σ βn Ψ( R, t ) dR (44)is the two-particle magnetization. It has been shown that a two-particle function f ( r , r ′ , t ) (see formulas (43) and (44)) appears as a sum of two terms f ( r , r ′ , t ) = f ( r , t ) f ( r ′ , t ) + g ( r , r ′ , t ) . The first ofthe two terms is the product of corresponding one-particlefunctions. It corresponds to the self-consistent field ap-proach suitable for the long-range interaction. The secondterm gives contribution of quantum correlations, particu-larly the exchange correlation.Thus, in the self-consistent field approximation wehave that the two-particle concentration represents asthe product of the concentrations in points r and r ′ : n ( r , r ′ , t ) = n ( r , t ) n ( r ′ , t ) ; and the two-particle mag-netization gives us the product of the one-particle magneti-zation M αβ ( r , r ′ , t ) = M α ( r , t ) M β ( r ′ , t ) . Putting theserepresentations in formula (42) we can introduce electricfield caused by the electric charges and the magnetic fieldcaused by magnetic moments. These fields emerge as E α = − e ∇ α Z G ( r − r ′ ) n ( r ′ , t ) d r ′ , (45)and B α = Z G αβ ( r − r ′ ) M β ( r ′ , t ) d r ′ . (46)They satisfy the Maxwell equations (13). Using these fieldswe come to the force field (39). ∗ [email protected] † [email protected][1] S. Takahashi, S. Maekawa, J. Magnetism and Magnetic Ma-terials , 1423 (2004).[2] I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. , 323(2004).[3] J. Sinova and I. Zutic, Nature Materials , 368 (2012).[4] K. Uchida, H. Adachi, T. An, H. Nakayama, M. Toda, B.Hillebrands, S. Maekawa, and E. Saitoh, J. Appl. Phys. ,053903 (2012).[5] F. Pezzoli, F. Bottegoni, D. Trivedi, F. Ciccacci, A. Gior-gioni, P. Li, S. Cecchi, E. Grilli, Y. Song, M. Guzzi, H. Dery,and G. Isella, Phys. Rev. Lett. , 156603 (2012).[6] Z. Rashidian, F. Kheirandish, Int. J. Theor. Phys. , 1989(2012).[7] R. T. Hammond, Appl. Phys. Lett. , 121112 (2012).[8] J. Lee, R. Oszwaldowski, C. Gothgen, and I. Zutic, Phys.Rev. B , 045314 (2012).[9] Y. Li, M. B. A. Jalil, and S. Ghee Tan, J. Appl. Phys. ,093717 (2012).[10] Rong Zhang, Long Bai, Chen-Long Duan, Physica B ,2372 (2012).[11] M. J. Ma, M. B. A. Jalil, and Z. B. Siu, J. Appl. Phys. ,07C326 (2012).[12] N. T. Bagraev, O. N. Guimbitskaya, L. E. Klyachkin, A.A. Kudryavtsev, A. M. Malyarenko, V. V. Romanov, A. I.Ryskin, I. A. Shelykh, A. S. Shcheulin, Physica C , 470(2010).[13] J. Flipse, F. L. Bakker, A. Slachter, F. K. Dejene and B. J.van Wees, Nature Nanotechnology , 166 (2012). [14] D. Pesin and A. H. MacDonald, Nature Materials , 409(2012).[15] I. J. Vera-Marun, V. Ranjan and B. J. van Wees, NaturePhysics , 313 (2012).[16] Wei Han, K. M. Mc Creary, K. Pi , W. H. Wang, Yan Li,H. Wen, J. R. Chen, R. K. Kawakami, J. Magnetism andMagnetic Materials , 369 (2012).[17] R. Jansen, Nature Materials , 400 (2012).[18] Z. An, F. Q. Liu, Y. Lin, C. Liu, Sci. Rep. , 388 (2012).[19] Q. F. Sun, X. C. Xie, Phys. Rev. B , 245305 (2005).[20] Q. F. Sun, X. C. Xie, J. Wang, Phys. Rev. B , 035327(2008).[21] P. Jin, Y. Li, F. Zhang, J. Phys. A , 7115 (2006).[22] J. Shi, P. Zhang, D. Xiao, Q. Niu, Phys. Rev. Lett. ,076604 (2006).[23] Farkhad G. Aliev, Juan F. Sierra, Ahmad A. Awad,Gleb N. Kakazei, Dong-Soo Han, Sang-Koog Kim, VitaliMetlushko, Bojan Ilic, and Konstantin Y. Guslienko, Phys.Rev. B , 174433 (2009).[24] A. A. Awad, K. Y. Guslienko, J. F. Sierra, G. N. Kakazei,V. Metlushko, and F. G. Aliev, Appl. Phys. Lett. , 012503(2010).[25] G. N. Kakazei, P. E. Wigen, K. Yu. Guslienko, V. Novosad,A. N. Slavin, V. O. Golub, N. A. Lesnik, and Y. Otani, Appl.Phys. Lett. , 443 (2004).[26] Ki-Suk Lee, Konstantin Y. Guslienko, Jun-Young Lee, andSang-Koog Kim, Phys. Rev. B , 174410 (2007).[27] K. Y. Guslienko, R. H. Heredero, O. Chubykalo-Fesenko ,Phys. Rev. B , 014402 (2010).[28] P. A. Andreev, L.S. Kuz’menkov, Moscow UniversityPhysics Bulletin , N.5, 271 (2007).[29] L. S. Kuz’menkov and S. G. Maksimov, Teor. i Mat. Fiz.,
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