Spin-polarized current effect on antiferromagnet magnetization in a ferromagnet - antiferromagnet nanojunction: Theory and simulation
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Spin-polarized current effect on antiferromagnetmagnetization in a ferromagnet–antiferromagnetnanojunction: Theory and simulation
E. M. Epshtein ∗ , Yu. V. Gulyaev, P. E. ZilbermanV. A. Kotelnikov Institute of Radio Engineering and Electronicsof the Russian Academy of Sciences, Fryazino, 141190, Russia Abstract
Spin-polarized current effect is studied on the static and dynamic mag-netization of the antiferromagnet in a ferromagnet–antiferromagnet nano-junction. The macrospin approximation is generalized to antiferromag-nets. Canted antiferromagnetic configuration and resulting magnetic mo-ment are induced by an external magnetic field. The resonance frequencyand damping are calculated, as well as the threshold current density cor-responding to instability appearance. A possibility is shown of generatinglow-damping magnetization oscillations in terahertz range. The fluctu-ation effect is discussed on the canted antiferromagnetic configuration.Numerical simulation is carried out of the magnetization dynamics of theantiferromagnetic layer in the nanojunction with spin-polarized current.Outside the instability range, the simulation results coincide completelywith analytical calculations using linear approximation. In the instabil-ity range, undamped oscillations occur of the longitudinal and transversemagnetization components.
The discovery of the spin transfer torque effect in ferromagnetic junctionsunder spin-polarized current [1, 2] has stimulated a number of works inwhich such effects were observed as switching the junction magnetic con-figuration [3], spin wave generation [4], current-driven motion of magneticdomain walls [5], modification of ferromagnetic resonance [6], etc. It iswell known that the spin torque transfer from spin-polarized electronsto lattice leads to appearance of a negative damping. At some currentdensity, this negative damping overcomes the positive (Gilbert) damping ∗ E-mail: [email protected] ith occurring instability of the original magnetic configuration. The cor-responding current density is high enough, of the order of 10 A/cm .This, naturally, stimulates attempts to lower this threshold. Various wayswere proposed, such as using magnetic semiconductors [7], in which thethreshold current density can be lower down to 10 –10 A/cm becauseof their low saturation magnetization. However, using of such materialsrequires, as a rule, low temperatures because of low Curie temperature.Besides, the ferromagnetic resonance frequency is rather low in this case.In connection with these difficulties, the other approaches were pro-posed, based on high spin injection [8] or joint action of external magneticfield and spin-polarized current [9, 10]. It seems promising, also, usingmagnetic junction of ferromagnet–antiferromagnet type, in which the fer-romagnet (FM) acts as an injector of spin-polarized electrons. The anti-ferromagnetic (AFM) layer, in which the magnetic sublattices are cantedby external magnetic field, may have very low magnetization that pro-motes low threshold [11]. The AFM resonance frequency may be bothlow and high reaching 10 s − , i.e. terahertz (THz) range. However,investigation and application of THz resonances is prevented because oftheir large damping. Such a damping in ferromagnetic junctions can besuppressed, as mentioned above, by means of spin-polarized current. Thequestion arises about possibility of such a suppression in FM/AFM junc-tions. Note, that this problem has been paid attention of a number ofauthors [12]–[21].Another interesting feature of the FM/AFM junctions with spin-polarizedcurrent is the possibility of canting the AFM structure by spin-polarizedcurrent without magnetic field. Let us consider a FM/AFM junction (Fig. 1) with current flowing perpen-dicular to layers, along x axis. An ultrathin spacer layer is placed betweenthe FM and AFM layers to prevent direct exchange coupling between thelayers. An external magnetic field is parallel to the FM magnetizationand lies in the layer plane yz . The simplest AFM model is used with twoequivalent sublattices.The AFM energy (per unit area), with uniform and nonuniform ex-change, anisotropy, external magnetic field, and the sd exchange inter-action of the conduction electrons with the magnetic lattice taking intoaccount, takes the form [22] W = Z L AFM dx ( Λ( M · M ) + 12 α ((cid:18) ∂ M ∂x (cid:19) + (cid:18) ∂ M ∂x (cid:19) ) + α ′ (cid:18) ∂ M ∂x · ∂ M ∂x (cid:19) − β (cid:8) ( M · n ) + ( M · n ) (cid:9) − β ′ ( M · n )( M · n ) − (( M + M ) · H ) − α sd (( M + M ) · m ) ) , (1) M AFM NMM F H − + n xyz0 L AFM j/e
Figure 1: Scheme of the ferromagnet (FM)–antiferromagnet (AFM) junction;NM being a nonmagnetic layer. The main vector directions are shown. where M , M are the sublattice magnetization vectors, Λ is the uniformexchange constant, α, α ′ are the intra- and inter-sublattice nonuniformexchange constants, respectively, β, β ′ are the corresponding anisotropyconstants, n is the unit vector along the anisotropy axis, H is the ex-ternal magnetic field, m is the conduction electron magnetization, α sd isthe dimensionless sd exchange interaction constant. We do not includedemagnetization term because its contribution is small compared to theuniform exchange. The integral is taken over the AFM layer thickness L AF M . We are interested in the spin-polarized current effect on the AFMlayer, so we consider a case of perfect FM injector with pinned latticemagnetization and without disturbance of the electron spin equilibrium,that allows to not include the FM layer energy in Eq. (1).Two mechanisms are known of the spin-polarized current effect onthe magnetic lattice, namely, spin transfer torque (STT) [1, 2] and analternative mechanism [23, 24] due to the spin injection and appearanceof nonequilibrium population of the spin subbands in the collector layer(this is the AFM layer, in our case). In the case of antiparallel relativeorientation of the injector and collector magnetization vectors, such a statebecomes energetically unfavorable at high enough current density, so thatthe antiparallel configuration switches to parallel one (such a process inFM junction is considered in detail in review [25]). The latter mechanismis described with the sd exchange term in Eq. (1). As to the formermechanism, it is of dissipative character (it leads to negative damping),so that it is taken into account by the boundary conditions (see below),not the Hamiltonian.The equations of the sublattice motion with damping taking into ac-count take the form ∂ M i ∂t − κM (cid:20) M i × ∂ M i ∂t (cid:21) + γ h M i × H ( i ) eff i = 0 ( i = 1 , , (2) here M is the sublattice magnetization, κ is the damping constant, H ( i ) eff = − δWδ M i ( i = 1 ,
2) (3)are the effective fields acting on the corresponding sublattices.From Eqs. (1)–(3) the equations are obtained for the total magnetiza-tion M = M + M and antiferromagnetism vector L = M − M : ∂ M ∂t − κM (cid:26)(cid:20) M × ∂ M ∂t (cid:21) + (cid:20) L × ∂ L ∂t (cid:21)(cid:27) + γ [ M × H ] + γ [ M × H sd ]+ 12 γ ( β + β ′ )( M · n )[ M × n ] + 12 γ ( β − β ′ )( L · n )[ L × n ]+ 12 γ ( α + α ′ ) (cid:20) M × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) L × ∂ L ∂x (cid:21) = 0 , (4) ∂ L ∂t − κM (cid:26)(cid:20) L × ∂ M ∂t (cid:21) + (cid:20) M × ∂ L ∂t (cid:21)(cid:27) + γ [ L × H ] + γ [ L × H sd ] − γ Λ [ L × M ]+ 12 γ ( β + β ′ )( M · n )[ L × n ] + 12 γ ( β − β ′ )( L · n )[ M × n ]+ 12 γ ( α + α ′ ) (cid:20) L × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) M × ∂ L ∂x (cid:21) = 0 , (5)where H sd ( x ) = δδ M ( x ) Z L AFM (cid:0) M ( x ′ ) · m ( x ′ ) (cid:1) dx ′ (6)is the effective field due to sd exchange interaction. This field determinesthe spin injection contribution to the interaction of the conduction elec-trons with the antiferromagnet lattice.To find H sd ( x ) field, the conduction electron magnetization m ( x ) isto be calculated. The details of such calculations are presented in ourpreceding papers [26, 9]. Here we adduce the result for the case, wherethe antiferromagnet layer thickness L AF M is small compared to the spindiffusion length l with the current flow direction corresponding to theelectron flux from FM to AFM: m = ( m + ∆ m ) ˆ M , ∆ m = µ B τ QjeL AF M (cid:16) ˆ M (0) · ˆ M F (cid:17) , (7)where m is the equilibrium (in absence of current) electron magnetization,∆ m is the nonequilibrium increment due to current, ˆ M = M / | M | is theunit vector along the AFM magnetization, ˆ M F is the similar vector forFM, µ B is the Bohr magneton, e is the electron charge, τ is the electronspin relaxation time, j is the current density.It should have in mind in varying the integral (6), that the electronmagnetization m depends on the vector M orientation relative to the FMmagnetization vector M F . From Eqs. (6) and (7) we have [9] H sd = α sd m ˆ M + α sd µ B τ QjeL AF M (cid:16) ˆ M (0) · ˆ M F (cid:17) ˆ M + α sd µ B τ Qje ˆ M F δ ( x − . (8) y substitution (8) into (4) and (5), we obtain ∂ M ∂t − κM (cid:26)(cid:20) M × ∂ M ∂t (cid:21) + (cid:20) L × ∂ L ∂t (cid:21)(cid:27) + γ [ M × H ] + γα sd µ B τ Qje h M × ˆ M F i δ ( x − γ ( β + β ′ )( M · n )[ M × n ] + 12 γ ( β − β ′ )( L · n )[ L × n ]+ 12 γ ( α + α ′ ) (cid:20) M × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) L × ∂ L ∂x (cid:21) = 0 , (9) ∂ L ∂t − κM (cid:26)(cid:20) L × ∂ M ∂t (cid:21) + (cid:20) M × ∂ L ∂t (cid:21)(cid:27) + γ [ L × H ] + γα sd µ B τ Qje h L × ˆ M F i δ ( x − − γ (cid:18) Λ − α sd mM − α sd µ B τ QjeL AF M M (cid:16) ˆ M (0) · ˆ M F (cid:17)(cid:19) [ L × M ]+ 12 γ ( β + β ′ )( M · n )[ L × n ] + 12 γ ( β − β ′ )( L · n )[ M × n ]+ 12 γ ( α + α ′ ) (cid:20) L × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) M × ∂ L ∂x (cid:21) = 0 . (10) The equations of motion (9) and (10) contain derivative over the spacecoordinate x . Therefore, boundary conditions at the AFM layer surfaces x = 0 and x = L AF M are need to find solutions. The way of derivationwas described in Ref. [9] in detail. The conditions depend on the electronspin polarization and are determined by the continuity requirement of thespin currents at the interfaces.The terms with the space derivative in Eq. (9) may be written in theform of a divergency:12 γ ( α + α ′ ) (cid:20) M × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) L × ∂ L ∂x (cid:21) = ∂∂x (cid:26) γ ( α + α ′ ) (cid:20) M × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) L × ∂ L ∂x (cid:21)(cid:27) ≡ ∂ J M ∂x . (11)The J M vector is the lattice magnetization flux density.Let us integrate Eq. (9) over x within narrow interval 0 < x < ε withsubsequent passing to ε → +0 limit. Then only the mentioned terms withthe space derivative and the singular term with delta function will con-tribute to the integral. As a result, we obtain an effective magnetizationflux density with sd exchange contribution at the AFM boundary x = +0taking into account: J eff (+0) = J M (+0) + γα sd µ B τ Qje h M (+0) × ˆ M F i . (12) he magnetization flux density coming from the FM injector is J ( −
0) = µ B Qe j ˆ M F . (13)The component J k = (cid:16) J ( − · ˆ M (+0) (cid:17) ˆ M (+0) remains with the elec-trons, while the rest, J ⊥ = J ( − − J k = µ B Qe j n ˆ M F − ˆ M (+0) (cid:16) ˆ M F · ˆ M (+0) (cid:17)o = − µ B QeM j h M (+0) × h M (+0) × ˆ M F ii , (14)is transferred to the AFM lattice owing to conservation of the magnetiza-tion fluxes [1, 2].By equating the magnetization fluxes (12) and (14), we obtain J M = − µ B QeM j h M × h M × ˆ M F ii − γα sd µ B τ Qe j h M × ˆ M F i , (15)all the M vectors being taken at x = +0.Since the AFM layer thickness is small compared to the spin diffusionlength and the exchange length, we may use the macrospin approximationwhich was described in detail in Ref. [9]. In this approximation, themagnetization changes slowly within the layer thickness. This allows towrite ∂ J M ∂x ≈ J M ( L AF M ) − J M (+0) L AF M = − J M (+0) L AF M , (16)because the magnetization flux is equal to zero at the interface betweenAFM and the nonmagnetic layer closing the electric circuit, J M ( L AF M ) =0. This allows to exclude the terms with space derivative from Eq. (9).In the rest terms, M ( x, t ) and L ( x, t ) quantities are replaced with theirvalues at x = 0. Then Eq. (9) takes a more simple form: d M dt − κM (cid:26)(cid:20) M × d M dt (cid:21) + (cid:20) L × d L dt (cid:21)(cid:27) + γ [ M × H ]+ 12 γ ( β + β ′ )( M · n )[ M × n ] + 12 γ ( β − β ′ )( L · n )[ L × n ]+ K h M × h M × ˆ M F ii + P h M × ˆ M F i = 0 , (17)where K = µ B QeL
AF M M j, P = γα sd µ B τ QeL AF M j. (18)The term with delta function does not present here, since it is taken intoaccount in the boundary conditions.Now we are to use again the macrospin approximation to exclude thespace derivatives from Eq. (10), too.Owing to known relationships [22] between M and L vectors, namely, M + L = 4 M and ( M · L ) = 0, we have the following conditions: (cid:18) M · ∂ M ∂t (cid:19) + (cid:18) L · ∂ L ∂t (cid:19) = 0 , (cid:18) L · ∂ M ∂t (cid:19) + (cid:18) M · ∂ L ∂t (cid:19) = 0 . (19) y substituting Eqs. (10) and (17) in (19) we find that conditions (19)are fulfilled if the terms in (10)12 γ ( α + α ′ ) (cid:20) L × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) M × ∂ L ∂x (cid:21) ≡ X (20)satisfy the following equations:( X · M ) + K (cid:16) L · h M × h M × ˆ M F ii(cid:17) + P (cid:16) L · h M × ˆ M F i(cid:17) = 0 , ( X · L ) = 0 . (21)Let us decompose the considered X vector on three mutually orthog-onal vectors: X = a M + b L + cγ [ L × M ] . (22)The substitution (22) in (21) gives a = K (cid:16) L · ˆ M F (cid:17) − P (cid:16) [ L × M ] · ˆ M F (cid:17) , b = 0. As to c coefficient, it is a current-induced correction to the co-efficient of γ [ L × M ] term in Eq. (10), i. e., a correction to the uniformexchange constant Λ. Let us estimate the correction. Multiplying (22)scalarly by [ L × M ] with (20) taking into account gives c = 1 M L (cid:18) [ L × M ] · (cid:26) γ ( α + α ′ ) (cid:20) L × ∂ M ∂x (cid:21) + 12 γ ( α − α ′ ) (cid:20) M × ∂ L ∂x (cid:21)(cid:27)(cid:19) = 12 (cid:26) ( α + α ′ ) 1 M (cid:18) M · ∂ M ∂x (cid:19) − ( α − α ′ ) 1 L (cid:18) L · ∂ L ∂x (cid:19)(cid:27) . (23)It is seen that c ∼ α/L AF M , while Λ ∼ α/a , where a is the latticeconstant [22]. Since L AF M ≫ a , the mentioned correction to Λ may beneglected.As a result, Eq. (10) takes the form d L dt − κM (cid:26)(cid:20) L × d M dt (cid:21) + (cid:20) M × d L dt (cid:21)(cid:27) + γ [ L × H ] − γ Λ [ L × M ]+ 12 γ ( β + β ′ )( M · n )[ L × n ] + 12 γ ( β − β ′ )( L · n )[ M × n ]+ K h L × h M × ˆ M F ii + P h L × ˆ M F i = 0 . (24)Here, Λ constant contains also the equilibrium contribution of the con-duction electrons − α sd m/M .Equations (17) and (24) are the result of applying the macrospin con-cept to AFM. It is shown that such an approximation may be justifiedformally for AFM layer. Earlier, it was justified for FM layers [1, 2] andgeneralized [9] with spin injection taking into account. The macrospinapproach corresponds well to experimental conditions and simplifies cal-culations substantially. The terms with K coefficient in Eqs. (17), (24)describe effect of STT mechanism, while the terms with P coefficient takethe spin injection effect into account. The magnetization wave spectrum anddamping
We assume that the easy anisotropy axis lies in the plane of AFM layerand is directed along y axis, the FM magnetization vector is parallel tothe positive direction of z axis, the external magnetic field is parallel to z axis too (see Fig. 1).We are interesting in behavior of small fluctuations around the steadystate M = { , , M z } , L = { , L y , } , i. e. the small quantities M x , M y , f M z = M z − M z , L x , e L y = L y − L y , L z .Let us project Eqs. (17), (24) to the coordinate axes and take theterms up to the first order. The zero order terms are present only in theprojection of Eq. (24) to x axis. They give M z = H z + Pγ Λ + 12 ( β − β ′ ) ≈ H z + Pγ Λ ,L y = ± q M − M z ≈ ± M . (25)Note that the spin-polarized current takes part in creating magneticmoment together with the external magnetic field due to the spin injectioninduced interaction of the electron spins with the lattice [23, 24], which P parameter in Eq. (25) corresponds to. Such an interaction leads to appear-ance of an effective magnetic field parallel to the injector magnetization.As a result, a canted antiferromagnet configuration may be created with-out magnetic field. However, such a configuration corresponds to parallelorientation of FM and AFM layers, M k M F . As is shown below, the insta-bility does not occur with this orientation, so that an external magneticfield is to be applied to reach instability.With Eq. (25) taking into account, the equations for the first orderquantities take the form dM x dt − κM (cid:26) − M z dM y dt + L y dL z dt (cid:27) + ( γH z + P ) M y − γ ( β + β ′ ) M z M y − γ ( β − β ′ ) L y L z + KM z M x = 0 , (26) dM y dt − κM M z dM x dt − ( γH z + P ) M x + KM z M y = 0 , (27) d f M z dt + 12 κM L y dL x dt + 12 γ ( β − β ′ ) L y L x = 0 , (28) dL x dt − κM ( L y d f M z dt − M z d e L y dt ) − γH z L y M z f M z = 0 , (29) d e L y dt − κM M z dL x dt − γ ( β − β ′ ) M z L x = 0 , (30) L z dt + 12 κM L y dM x dt + ( γH z + P ) L y M z M x + KM z L z = 0 . (31)The set of equations (26)–(31) splits up to two mutually independentsets with respect to ( M x , M y , L z ) and ( L x , e L y , f M z ). They describe twoindependent spectral modes, one of them corresponds to precession of theAFM magnetization vector around the magnetic field, while another to pe-riodic changes of the vector length along the magnetic field. We begin withthe spectrum and damping of the first mode. We consider monochromaticoscillation with ω angular frequency and put M x , M y , L z ∼ exp( − iωt ).Then we obtain from Eqs. (26), (27), (31) (cid:0) − iω + KM z (cid:1) M x + (cid:26) γH z + P − γ ( β + β ′ ) M z − iκωM M z (cid:27) M y − (cid:26) γ ( β − β ′ ) − iκωM (cid:27) L y L z = 0 , (32) (cid:0) − iω + KM z (cid:1) M y − (cid:26) γH z + P − iκωM M z (cid:27) M x = 0 , (33) (cid:0) − iω + KM z (cid:1) L z + (cid:26) γ Λ + 12 γ ( β − β ′ ) − iκωM (cid:27) L y M x = 0 . (34)Note that aforementioned additivity (in the algebraic sense, the signtaking into account) of the external magnetic field and the injection-driveneffective field takes place not only in the steady magnetization (25), butalso in the oscillations of the magnetization and antiferromagnetism vec-tors, so that both fields appear in Eqs. (32), (33) “on an equal footing”.Usually, Λ ≫ β, β ′ . With these inequalities and stationary solu-tion (25) taking into account we find the dispersion relation for the mag-netization oscillation (1 + κ ) ω + 2 iνω − ω = 0 , (35)where ω = q γ H A H E + ( KM z ) + ( γH z + P ) , (36) ν = κγH E + KM z , (37) H E = Λ M is the exchange field, H A = ( β − β ′ ) M is the anisotropyfield. Formulae (36) and (37) (without current terms KM z and P ) coin-cide with known ones [22, 27]. At H E ∼ –10 Oe, H A ∼ Oe wehave oscillations in THz range, ω ∼ s − . In absence of current thedamping is rather high: at κ ∼ − νω = κ r H E H A ∼ . (38) et us consider the contribution of spin-polarized current to the fre-quency and damping of AFM resonance. At first we consider STT mech-anism effect [1, 2]. According to (18) and (25), KM z = µ B Q Λ eL AF M H z j. (39)At H z <
0, that corresponds to direction of the magnetic field (and,therefore, the AFM magnetization) opposite to the FM magnetization,this quantity is negative. The total attenuation becomes negative also(an instability occurs), if j > eκγM | H z | L AF M µ B Q ≡ j . (40)At κ ∼ − , γM ∼ s − , | H z | ∼ Oe, L AF M ∼ − cm, Q ∼ j ∼ A/cm . At j near to j weakly damping THzoscillation can be obtained. At j > j , instability occurs which may leadto either self-sustained oscillations, or a dynamic stationary state. Thelatter disappears with the current turning off. To answer the questionabout future of the instability it is necessary to go out the scope of thelinear approximation. We have simulated numerically the behavior ofthe AFM magnetization behind the linear approximation (see section 8below).The spin-polarized current contributes also to the oscillation frequency.At the mentioned parameter values, we have | KM z | ∼ s − that iscomparable with the frequency in absence of the current. This allowstuning the frequency by the current or excite parametric resonance bymeans of the current modulation. Now let us discuss the injection mechanism effect [23, 24]. As mentionedbefore, the role of the mechanism is reduced to addition of an effectivefield
P/γ to the external magnetic field. At reasonable parameter values,that field is much less than the exchange field H E , so that it does notinfluence directly the eigenfrequency (36). Nevertheless, that field canmodify substantially the contribution of the STT mechanism, becauseEq. (39) with (25) taking into account now takes the form KM z = µ B Q Λ eL AF M ( H z + P/γ ) j. (41)Such a modification leads to substantial consequences. At H z < P <γ | H z | the instability threshold (40) is lowered, since | H z | − P/γ differenceappears now instead of | H z | . If, however, P > γ | H z | then the AFMmagnetization steady state M z = H z + P/γ
Λ (42)becomes positive that corresponds to the parallel (stable) relative orienta-tion of the FM and AFM layers. In this case, the turning on current leads o switching the antiparallel configuration (stated beforehand by meansof an external magnetic field) to parallel one. With turning off current,the antiparallel configuration restores.Since the mentioned injection-driven field depends on the current (see (18)),the instability condition (40) is modified and takes the form j η < j < j η , (43)where η = α sd κγM τ , j being defined with Eq. (40). In absence ofthe injection mechanism, this condition reduces to (40). Under risingrole of this mechanism we have lowering the instability threshold, on theone hand, and the instability range narrowing, on the other hand. At j > j /η the antiparallel configuration switches to parallel one. Therelative contribution of the injection mechanism is determined with η parameter. At typical values, α sd ∼ , κ ∼ − , γM ∼ s − , τ ∼ − s, this parameter is of the order of unity, so that the injectioneffect may lower noticeably the instability threshold.Now let us return to the set of equations (26)–(31) and consider thesecond mode describing with Eqs. (28)–(30). The current influences thismode by changing steady magnetization M z due to the injection effectivefield effect (see (26)), while the STT mechanism does not influence thismode. A calculation similar to previous one gives the former dispersionrelation (35), but now ω = 2 γ H E H A γH z γH z + P , (44) ν = κγH E γH z γH z + P . (45)At H z < P > | H z | , that corresponds to current density j > j /η , thetotal attenuation becomes negative, while the frequency becomes imagi-nary, that means switching the antiparallel configuration to parallel one. Let us consider briefly the situation where AFM has easy-plane anisotropy.We take the AFM layer yz plane as the easy plane and x axis as the (hard)anisotropy axis. The magnetic field, as before, is directed along z axis.Without repeating calculations, similar to previous ones, we presentthe results. A formal difference appears only in Eq. (36) for the eigenfre-quency ω of the first of the modes considered above. We have for thatfrequency ω = q ( γH z + P ) + ( KM z ) . (46)The damping has the former form (37), so that the instability thresholdis determined with former formula (43).In absence of the current ( K = 0 , P = 0) with not too small dampingcoefficient κ , the frequency appears to be much less than damping, sothat the corresponding oscillations are not observed. The current effectincreases the frequency, on the one hand, and decreases the damping (at H z < Fluctuation effect
It follows from Eq. (43) that the threshold current density is proportionalto the external magnetic field strength | H z | and decreases with the fieldlowering. A question arises about permissible lowest limit of the total field | H z | + P/γ . In accordance with Eq. (25), such a limit may be the fieldwhich create magnetization | M z | comparable with its equilibrium valuedue to thermal fluctuations. Let us estimate this magnetization and thecorresponding field.The AFM energy change in V volume under canting the sublatticemagnetization vectors with θ < ◦ angle between them is∆ E = Λ M (1 − cos θ ) V = 12 Λ V M z , (47)the anisotropy energy being neglected compared to the exchange energy.The equilibrium value of the squared magnetization is calculated usingthe Gibbs distribution: h M z i = ∞ R −∞ M z exp (cid:18) − Λ V M z kT (cid:19) dM z ∞ R −∞ exp (cid:18) − Λ V M z kT (cid:19) dM z = kT Λ V (48)(strictly speaking, the magnetization may be changed within ( − M , M )interval, however, Λ V M ≫ kT , so that the integration limits may betaken infinity).To observe the effects described above, the magnetization M z whichappears under joint action of the external field and the current (see (25))should exceed in magnitude the equilibrium magnetization h M z i / . Atthe current density j = j / (1 + η ) corresponding to the instability thresh-old, this condition is fulfilled at magnetic field | H z | > r Λ kTV (1 + η ) ≡ H min . (49)At Λ ∼ , η ∼ L AF M ∼ − cm and lateral sizes of the switchedelement 10 × µ m we have V ∼ − cm and H min ≈
30 Oe at roomtemperature. This limit can be decreased under larger element size.It should be mentioned also about other mechanisms of AFM canting.The most known and studied one is the relativistic Dzyaloshinskii–Moriaeffect (see, e.g. [22, 28]). Besides, possible mechanisms have been discusseddue to competition between sd exchange and direct exchange interactionof the magnetic ions in the lattice [29]. At the same time, there are noindications, to our knowledge, about measurements of canting in conduc-tive AFM. So, present theory is related to conductive AFM, in which thelattice canting is determined with external magnetic field. With the purpose of simulating, it is convenient to modify slightly Eqs. (17)and (24), namely: i) to use sublattice magnetizations M , M instead of , L , ii) to describe damping by the Landau–Lifshitz representation withdouble vector product, and iii) to introduce the following dimensionlessvariables: ˆ M i = M i /M ( i = 1 , , h = H /M , T = γM t κ ,K = µ B QjeL
AF M γM , P = η K , η = α sd γM τ. With these variables, the set of equations (17), (24) take the form d ˆ M i dT = − h ˆ M i × h ( i ) eff i − κ h ˆ M i × h ˆ M i × h ( i ) eff ii , i = 1 , , (50) h (1) eff = h − Λ ˆ M + β (cid:16) ˆ M · n (cid:17) n + β ′ (cid:16) ˆ M · n (cid:17) n + P ˆ M F + K h(cid:16) ˆ M + ˆ M (cid:17) × ˆ M F i(cid:16) ˆ M + ˆ M (cid:17) ≡ F (cid:16) ˆ M , ˆ M (cid:17) , (51) h (2) eff = F (cid:16) ˆ M , ˆ M (cid:17) . (52)It is seen from Eqs. (50)– (52) that the motions of two sublattices are cou-pled each other. The sources of such a coupling are the uniform exchange,intersublattice anisotropy (coefficient β ′ ), as well the effective field due tothe spin torque (the last term in Eq. (51)).We assume the following orders of magnitude of the used parame-ters: κ ∼ − –10 − , Λ ∼ –10 , β, β ′ ∼ − ( β = β ′ ), M ∼ G, γM ∼ s − . Under such conditions, h = 1 corresponds to mag-netic field H ∼ Oe, and T = 1 corresponds to time t ∼ − s.With α sd ∼ , τ ∼ − s, Q ∼ L AF M ∼ − cm we have η ∼ ,and K = 1 value corresponds to current density j ∼ A/cm .Equations (50)– (52) written in coordinates represent six ordinary dif-ferential equations of the first order in the Cauchy form for ˆ M x , ˆ M y ,ˆ M z , ˆ M x , ˆ M y , ˆ M z . The equations are not mutually independent be-cause of the normalization conditions (cid:12)(cid:12)(cid:12) ˆ M (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ M (cid:12)(cid:12)(cid:12) = 1 . (53)Nevertheless, all the six equations are used in our simulation, while thementioned normalization conditions serve for checking correctness of thecalculations.The simulation was carried out by means of Simulink program inMATLAB system with using Differential Equation Editor (DEE). Right-hand sides of the equations resolved with respect to derivatives wereentered into the DEE block. The parameters Λ , β, β ′ , κ, K , P weregiven as input signals, while three projections of the magnetization vec-tor ˆ M x = ˆ M x + ˆ M x , ˆ M y = ˆ M y + ˆ M y , ˆ M z = ˆ M z + ˆ M z wereoutput to oscilloscope blocks. Besides, (cid:12)(cid:12)(cid:12) ˆ M (cid:12)(cid:12)(cid:12) = (cid:16) ˆ M x + ˆ M y + ˆ M z (cid:17) / nd (cid:12)(cid:12)(cid:12) ˆ M (cid:12)(cid:12)(cid:12) = (cid:16) ˆ M x + ˆ M y + ˆ M z (cid:17) / values were output to digital dis-plays; these values must be equal to 1 (or, at least, be close to 1) undercorrect calculation.In the present work, we assume that the magnetic field and current areturned on at the initial time instant T = 0 and hold constant. However,the procedure used allows to consider arbitrary time dependence of thesequantities, specifically, to vary turning on and turning off instants.We began simulation with “verifying” results of the linear theory (ac-tually, this was a test for the model adequacy). As above, we assumethat the AFM layer lies in yz plane, the current flows along x axis, theeasy anisotropy axis coincides with y axis ( n = { , , } ), the current ispolarized along the positive direction of the z axis ( ˆ M F = { , , } ), themagnetic field is collinear with z axis ( h = { , , h z } ). In such a configu-ration, the collinear relative orientation of the FM and AFM layer magne-tizations is stationary (although, possibly, unstable), so that a small initialdeviation from such orientation was given to imitate thermal fluctuations.The initial value of the ˆ M z component was chosen to be equal to theequilibrium value ˆ M z = h z / Λ in the given magnetic field without current.Thus, at Λ = 10 , h z = − M z = ˆ M z = − × − (so that ˆ M z = − × − ), ˆ M x = ˆ M x = − × − , ˆ M y = − ˆ M y =(1 − ˆ M x − ˆ M z ) / are used taking the normalization conditions (53) intoaccount.With the dimensionless variables, the main results of the linear theorytake the following form.The stationary magnetization in z direction isˆ M z ≡ ˆ M z + ˆ M z = h z + η K Λ + ( β − β ′ ) ≈ h z + η K Λ . (54)Under deviation from the stationary value, two modes appear with di-mensionless (referred to γM ) frequencies ω , and damping ν , definedwith the following formulae: ω = 2( β − β ′ )Λ + ( h z + η K ) + (cid:18) Λ K h z + η K (cid:19) , (55) ν = κ Λ + Λ K h z + η K , (56) ω = 2( β − β ′ )Λ h z h z + η K , (57) ν = κ Λ h z h z + η K . (58)The instability of the antiparallel relative orientation of the FM andAFM layers at h z < κ | h z | κη < K < | h z | η . (59)To start, the case of absence of the current ( K = 0) has been con-sidered. Perfect agreement has been observed with Eqs. (55)–(58). Inparticular, the magnetization oscillation frequency drops abruptly when
100 200 300 400 500 600−3.8−3.6−3.4−3.2−3−2.8−2.6 x 10 −4 t, ps M z / M Figure 2: Oscillations of the longitudinal magnetization in the instability rangeat K = 7 × − . we put β = β ′ , and the oscillations disappear completely, if we take, more-over, h = 0. At β = β ′ the observed frequency consists with Eq. (55).In absence of the magnetic field, the frequencies of the two modes coin-cide, so that the oscillation takes the form of a simple sinusoid. Underrising magnetic field, the frequencies become different, and beats appearbecause of interaction between the modes. The simulation results consist,also, with Eq. (54) for the stationary magnetization in presence of themagnetic field. Turning on the magnetic field at T = 0 instant leads to anaperiodic transient process which decays completely by T = 0 .
4, followingwhich the magnetization remains constant value determined by Eq. (54).At Λ = 10 , κ = 10 − , η = 10 , h z = − K = 5 × − and disappear at K =1 × − . In our numerical experiments, increasing K parameter fromzero to the indicated threshold leads, in accordance with Eq. (56), todecrease of the damping of the magnetization vector precession about z axis because of the negative damping caused by the STT mechanism.Incidentally, the absolute value of the (negative) ˆ M z component decreasesfrom 10 − to 10 − because of influence of the injection mechanism, thatconsists, also, with Eq. (54). Thus, the simulation results agree completelywith the theory in the range below the instability threshold.Of course, the instability range ( K > × − ), which is not described
100 200 300 400 500 600 700−2−1.5−1−0.500.511.52 x 10 −4 t, ps M x / M Figure 3: Beats of the transverse magnetization in the instability range at K =8 × − . by the linear theory, is of more interest. At K = 5 . × − undamped os-cillations are observed. At K = 7 × − the precession oscillations of thetransverse components ˆ M x , ˆ M y become almost sinusoidal with ∆ T ≈ . ∼ × s − at the chosenparameter values). The longitudinal component ˆ M z oscillates periodicallywith a negative stationary background, however, the oscillation form is farfrom sinusoidal one (Fig. 2). At K = 8 × − the oscillations of ˆ M x , ˆ M y components take on a form of beats (Fig. 3), while they again becomesinusoidal at K = 9 × − . At K = 1 × − (this is the right bound-ary of the instability range in the linear theory) all the three componentsoscillate around zero values. Further, at K = 1 . × − the oscillationshold yet, but at K = 1 . × − they disappear almost completely, andthey are absent at K = 1 . × − .The subsequent increasing of the current leads only to rising magne-tization in the positive direction due to the injection mechanism. At K =2Λ /η , the longitudinal component ˆ M z reaches the maximal possible value ˆ M z =2 (the sublattices are flipped to a parallel position), then the (dimension-less) angular frequency is equal to 2Λ (this corresponds to ∼ s − ).Such a situation corresponds to rather high current density, it is estimatedas 2 × A/cm at the above-mentioned parameter values. Conclusions
The obtained results show a principal possibility of controlling frequencyand damping of AFM resonance in FM/AFM junctions by means of spin-polarized current. Under low AFM magnetization induced by an externalmagnetic field perpendicular to the antiferromagnetism vector, the thresh-old current density corresponding to occurring instability is less substan-tially than in the FM–FM case. Near the threshold, the AFM resonancefrequency increases, while damping decreases, that opens a possibility ofgenerating oscillations in THz range.Numerical simulation allows to trace behavior of the FM/AFM junc-tion in the whole current density range. The instability range predictedby the linearized theory is broadened only slightly because of nonlineareffects. In the instability range undamped oscillation of sinusoidal or morecomplicated form including beats.Under magnetic fields low compared to the exchange field, the inducedmagnetization is small in comparison with the sublattice magnetization, sothat the stationary oscillation amplitude beyond the instability thresholdis low, too.Thus, the following features may be expected in comparison with thesimilar effects in FM/FM junctions. First, the instability threshold is tobe lower because of the lower magnetization. This is a favorable fact facili-tating observations. Second, the oscillation intensity beyond the thresholdalso lowers as a square of the magnetization. This may make the effectdifficult to observe. Nevertheless, studying the current-driven nonlinearoscillations in FM/AFM structure is of principal interest, because the cur-rent induced instability can occur at relatively low current density, ∼ A/cm .The simulation results reveal an interesting possibility of a spin-fliptransition without magnetic field under the action of a high-density spin-polarized current only. Such a current overcomes the exchange forces andaligns the sublattice moments in parallel. Under such conditions, applyinga low alternating magnetic field can excite precession of the magnetizationvector at the AFM resonance frequencies, which may be as high as 3 × s − or more. Such a THz resonator might be useful to detect and measuresignals in THz range. Acknowledgments
The authors are grateful to Prof. G. M. Mikhailov for useful discussions.The work was supported by the Russian Foundation for Basic Re-search, Grant No. 10-02-00030-a.
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