Current-induced exchange switching magnetic junctions with cubic anisotropy of the free layer
S. G. Chigarev, E. M. Epshtein, Yu. V. Gulyaev, P. E. Zilberman
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Current-induced exchange switching magneticjunctions with cubic anisotropy of the free layer
S. G. Chigarev, E. M. Epshtein ∗ , Yu. V. Gulyaev, P. E. ZilbermanV.A. Kotelnikov Institute of Radio Engineering and Electronicsof the Russian Academy of Sciences, 141190 Fryazino, Russia Abstract
The stability is analyzed of the equilibrium configurations of a mag-netic junction with a free layer that has cubic symmetry and two anisotropyaxes in the layer plane. Different variants of the switching between vari-ous configurations are considered. A possibility is shown of the substan-tial lowering of the threshold current density needed for the switching.Numerical simulation is made of the switching dynamics for various con-figurations.
Switching magnetic junctions by a spin-polarized current is one of themain spintronic effects. Besides the academic interest, this phenomenonmay be used for high-density information processing, since the character-istic scales are the exchange interaction and the spin diffusion lengths ofthe order of tens of nanometers.Nowadays interest has revived to magnetic junctions the layers ofwhich have cubic magnetic anisotropy; the well-studied iron may be anexample [1, 2, 3]. Thin Fe(001) films have two equivalent anisotropy axes,[100] and [010], in the layer plane. This allows switching the layer mag-netizations between different easy axes by means of magnetic field and/orspin-polarized current, which may be used in memory cells with more thantwo stable states.In the present work, we consider a magnetic junction with cubic anisotropyof the free layer placed between pinned and nonmagnetic ones. Switchingsuch systems by applied magnetic field has been studied in Refs. [2, 4].Here we investigate the switching by spin-polarized current at various rel-ative orientations of the pinned and free layer magnetization vectors. Thethickness of the free layer is assumed to be small compared to the spindiffusion length, so that the macrospin approximation is valid [5]. ∗ E-mail: [email protected] wo main mechanisms are known of the interaction between spin-polarized current and magnetic lattice, namely, spin transfer torque (STT) [6,7] and spin injection leading to appearing regions of nonequilibrium spinpolarization near the interfaces [8, 9] (the term “field-like torque” is usedsometimes in literature, because the mechanism action is equivalent toinfluence of some effective magnetic field in some cases). The first of themechanisms mentioned (STT) is related with appearing a negative damp-ing that prevails over the positive Gilbert damping above the thresholdcurrent density; this leads to instability of the original magnetic configu-ration. The other (“injection”) mechanism is based on increasing the sd exchange interaction energy between the nonequilibrium conduction elec-trons and the lattice under spin injection to the free layer of the magneticjunction, so that the original state becomes unstable, and a reorientationphase transition occurs. A unified theory of switching magnetic junctionsincluding actions of both mechanisms was presented in Refs. [10, 11]. Therelative contribution of the mechanisms indicated depends on the layerparameters and the applied magnetic field. Earlier, the conditions wereformulated [12, 13, 5] under which the injection mechanism plays the mainrole (see below for details). In this work, special attention is paid to thelatter.The main equations describing the magnetization of the thin free layerin the magnetic junction under spin-polarized current are presented inSec. 2. In Sec. 3 we analyze the stability of the stationary configurationsdepending on the current density through the magnetic junction, as wellas possible switching between these configurations. In Sec. 4 the current-voltage characteristic is found of the magnetic tunnel junction under for-ward and backward currents depending on the original configurations. InSec. 5 the junction switching dynamics is simulated numerically. Let us consider a magnetic junction consisting of a pinned ferromagneticlayer 1, a free ferromagnetic layer 2, and a nonmagnetic layer 3 whichcloses the electric circuit. There is a thin spacer between the layers 1 and2 that prevents the direct exchange interaction between the magnetic lat-tices of the layers. The electric current flows perpendicular to the layers(CPP mode). The free layer 2 has cubic symmetry with three mutually or-thogonal symmetry axes and, correspondingly, magnetic anisotropy axes,one of which, [001], is perpendicular to the layer plane, while two other,[100] and [010], lie in the plane. The anisotropy energy of that layer (perarea unit) is [14] U a = 12 MH a L n(cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) o , (1)where L is the thickness of layer 2, M is the saturation magnetization ofthat layer, ˆ M = M / | M | is the unit vector along the magnetization, n , n , n are the unit vectors along [100], [010] and [001] axes, respectively, H a is the effective anisotropy field. he free layer thickness L is assumed to be small compared to thespin diffusion length l and the inhomogeneity scale of the magnetic latticein that layer (such a scale, the measure of the “spatial inertia” of thelattice, is the domain wall thickness δ ). Under such conditions, layer 2manifests itself as united whole (“macrospin”) in respect of its magneticbehavior. This leads to the modification of the Landau–Lifshitz–Gilbertequation for the layer magnetization with the disappearance of the spatialderivative term and the introduction of a new term describing the currenteffects. With the cubic anisotropy taken into account, the equation takesthe form (cf. [5]) d ˆ M dt − κ ˆ M × d ˆ M dt ! + γ (cid:16) ˆ M × H (cid:17) + γ (cid:16) ˆ M × H d (cid:17) − γH a (cid:26)(cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) (cid:26)(cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17)(cid:27) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) (cid:26)(cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17)(cid:27) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M · n (cid:17) (cid:26)(cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17) + (cid:16) ˆ M · n (cid:17) (cid:16) ˆ M × n (cid:17)(cid:27)(cid:27) = − aL n p ( ˆ M ) (cid:16) ˆ M × ˆ M (cid:17) + κ ( ˆ M ) (cid:16) ˆ M × (cid:16) ˆ M × ˆ M (cid:17)(cid:17)o . (2)Here ˆ M is the unit vector along the magnetization of layer 1, H is theapplied magnetic field, H d is the demagnetization field, κ is the Gilbertdamping factor, γ is the gyromagnetic ratio, a = γH a δ is the latticemagnetization diffusion constant, p ( ˆ M ) = µ B γατ Q ea jλ Z Z (cid:20) Z Z + Z Z λ − (cid:16) ˆ M · ˆ M (cid:17) + 2 bλ (cid:18) λ + Z Z (cid:19) (cid:16) ˆ M · ˆ M (cid:17)(cid:21) × (cid:20) Z Z + Z Z λ + (cid:16) ˆ M · ˆ M (cid:17) (cid:21) − , (3) k ( ˆ M ) = µ B Q eaM j (cid:18) Z Z + Z Z λ (cid:19) (cid:20) Z Z + Z Z λ + (cid:16) ˆ M · ˆ M (cid:17) (cid:21) − , (4)where e is the electron charge, µ B is the Bohr magneton, α is the sd exchange interaction constant, τ is the spin relaxation time, λ = L/l ≪ Q is the conduction spin polarization, Z i = ρ i l i − Q i ( i = 1 , ,
3) (5)is the layer spin resistance [13], ρ is the electric resistivity; the quanti-ties without index refer to the free layer 2. The b = ( α M τ ) / ( αMτ )parameter describes influence of the pinned layer 1. he parameters p and k in the right-hand side of Eq. (2) describe in-jection and STT mechanisms of the spin-polarized current effect on themagnetic lattice under current flowing in the “forward” direction, corre-sponding to the electron drift in the 1 → → (cid:16) ˆ M · ˆ M (cid:17) → (cid:16) ˆ M · ˆ M (cid:17) − , j → −| j | corresponds to the backward direc-tion (3 → → Z Z , Z Z ≪ λ ≪ sd exchange interaction energy betweenconduction electrons and magnetic lattice.Let us consider a configuration with x axis along the current, yz planeparallel to the layer planes H = { , H sin ψ, H cos ψ } , n = { , , } , n = { , , } , n = { , , } , H d = − πM { ˆ M x , , } , ˆ M = { , , } .In spherical coordinates with the polar axis along [100] axis, ˆ M = { sin θ cos φ, sin θ sin φ, cos θ } , the dimensionless (in MH a L units) mag-netic energy under forward current takes the form U ( θ, φ ) MH a L = − HH a cos( θ − ψ ) + 12 sin θ cos θ + 12 sin θ sin φ cos φ + 2 πMH a sin θ cos φ − jj cos θ + b ( Z /Z λ ) cos θ Z /Z λ ) + ( Z /Z λ ) cos θ , (6)where j = eH a Lµ B ατ Q .The corresponding formula for backward current is obtained with sub-stitution cos θ → (cos θ ) − , j → −| j | in the last term of Eq. (6) describingthe current effect: U ( θ, φ ) MH a L = − HH a cos( θ − ψ ) + 12 sin θ cos θ + 12 sin θ sin φ cos φ + 2 πMH a sin θ cos φ + jj cos θ + b ( Z /Z λ )( Z /Z λ ) + [1 + ( Z /Z λ )] cos θ . (7)
45 90 135 180 225 270-1.5-1.0-0.50.00.51.01.5 , degU/MH a L j/j = 0 j/j = 0.1 j/j = 0.272 j/j = 0.5 j/j = 1 j/j = 1.44 Figure 1: The magnetic junction energy as a function of the angle between themagnetization vectors of the pinned and free layers with various values of the(dimensionless) forward current density.
The stationary states of the system in study correspond to the extremaof the U ( θ, φ ) function; the minima of the function correspond to thestable equilibrium states. Because of the positive definiteness of the termwith the azimuthal angle φ , it is sufficient to consider only the energydependence on the polar angle θ at fixed value φ = 90 ◦ during the minimafinding (this corresponds to in-plane position of the magnetization vector).There are three minima in absence of magnetic field ( H = 0) andcurrent ( j = 0): θ = 0 ◦ , θ = 90 ◦ , and θ = 180 ◦ , corresponding to theparallel, perpendicular and antiparallel relative orientations of the pinnedand free layers.Let us present the stability analysis results for these stationary statesin presence of the current under intense injection conditions Z Z , Z Z ≪ λ ≪ U energyas a function of the θ angle (Figs. 1 and 2) at various values of the currentdensity in forward and backward directions.1) The parallel configuration ( θ = 0 ) in absence of magnetic field( H = 0) is stable under forward and backward currents.2) Under the forward current increasing from zero, the energy mini-
45 90 135 180 225 270-2-10123 , degU/MH a L j/j = 0 j/j = 0.1 j/j = 0.272 j/j = 0.5 j/j = 1 j/j = 1.44 Figure 2: The magnetic junction energy as a function of the angle between themagnetization vectors of the pinned and free layers with various values of the(dimensionless) backward current density. mum corresponding to the original perpendicular configuration ( θ = 90 ◦ )shifts in the direction of the parallel configuration (i.e., θ angle decreases).This takes place up to the current density value j = j p / ≈ . j ,when the deviation from the original position reaches arcsin (cid:0) / √ (cid:1) ≈ ◦ (i.e., θ ≈ ◦ ). At this value, the energy minimum disappears (it changesto an inflection point), and the system switches abruptly to a parallelconfiguration and remains in that configuration under further variationsof the current density, so that the 90 ◦ → ◦ switching is irreversible.Under backward current increasing from zero, the minimum corre-sponding to the original perpendicular configuration ( θ = 90 ◦ ) shifts inthe direction of the antiparallel configuration, and the corresponding θ angle increases up to some value θ ( j ) that tends to θ ( ∞ ) = arccos − s Z /Z λ Z /Z λ ) ! (8)at the high current limit. Under returning to zero current, the perpendic-ular orientation restores, so that the switching by the backward currentis of “temporary” character.3) The antiparallel configuration ( θ = 180 ◦ ) in a magnetic field parallelto the magnetization of the pinned layer ( ψ = 0 ◦ ) becomes unstable andswitched to parallel one under high enough forward current. The threshold urrent density is j th = j [1 + ( Z /Z λ ) + ( Z /Z λ )] Z /Z λ ) − ( Z /Z λ ) (cid:18) − HH a (cid:19) ≈ j (cid:18) − HH a (cid:19) . (9)We see from Eq. (9) the mentioned possibility of the switching thresh-old lowering by means of an applied magnetic field close to (but lowerthan) the anisotropy field. Such an assistance of the magnetic field doesnot break the local character of the switching, because the magnetic fieldlower than the anisotropy field cannot do switching alone (without a cur-rent).The parallel configuration appeared after switching is stable againstfurther variations of the current, so that the switching by the forwardcurrent is irreversible.Under the backward current, the antiparallel configuration becomesunstable at the same (in magnitude) current density | j | = j th , however,in this case switching takes place to a nonequilibrium stationary state θ = θ ( j th ) or to the symmetrical state θ = 360 ◦ − θ ( j th ). With returningto zero current, the system does not return to antiparallel configuration,but comes to one of two perpendicular configurations θ = 90 ◦ or θ = 270 ◦ .Thus the following variants are possible of the switching between sta-tionary states: 1) switching an antiparallel configuration to a parallel oneby turning up the forward current of j > j th density and subsequent turn-ing off; 2) switching an antiparallel configuration to a perpendicular oneby turning up the backward current of the same density and subsequentturning off; 3) switching a perpendicular configuration to a parallel one byturning up and subsequent turning off the forward current of substantiallylower density j > . j . In experiments, the current-driven switching magnetic junction manifestsitself, in the first place, as a change of the junction resistance. The re-sistance depends substantially on the relative orientation of the layersforming the junction; this is the cause of the well-known tunnel magne-toresistance effect.The conductance of a magnetic tunnel junction with θ angle betweenthe magnetization vectors of the layers takes the form [17] G ( θ ) = G P cos θ G AP sin θ , (10)where G P , G AP are the junction conductances at parallel ( θ = 0 ◦ ) andantiparallel ( θ = 180 ◦ ) relative orientation of the layers, respectively.It is convenient to describe the change of the junction resistance withthe following ratio: R ( θ ) − R P R P = ρ (1 − cos θ )2 + ρ (1 + cos θ ) , (11) j/j [R(j) - R P ]/R P Figure 3: The magnetic tunnel junction resistance as a function of the (di-mensionless) current density. The dots on the ordinate show the stationaryconfigurations without current: parallel (square), perpendicular (circle), andantiparallel (skew cross). The arrows with corresponding tails show the resis-tance changes under the current change for different initial configurations.8
0 100 150 200 250 30000.511.522.533.5 ), radian Figure 4: Switching dynamics of the antiparallel configuration to parallel oneby the forward current. where R ( θ ) = 1 /G ( θ ), R P = 1 /G P ; ρ = [ R (180 ◦ ) − R P ] /R P is the tunnelmagnetoresistance defined by usual way [18].To find the resistance dependence on the current direction and density R ( j ), it is necessary to substitute θ ( j ) dependence to Eq. (11). With theforegoing analysis taking into account, the results are obtained shown inFig. 3. A possibility is seen of the switching between different stationarystates corresponding to different electric resistances. Together with the investigation of the stationary states and the switchingbetween them, the switching dynamics is of great interest, because itdetermines the speed of response of the devices based on the magneticjunctions.The time-dependent vector equation (2) describing the dynamics withusing polar coordinates ( θ, φ ) takes the form of a set of equations dθdT = sin θ κ {− κA ( θ, φ ) + B ( θ, φ ) } , (12) dφdT = 11 + κ { A ( θ, φ ) + κB ( θ, φ ) } , (13)
0 100 150 200 250 30000.20.40.60.811.21.41.6 ), radian Figure 5: Switching dynamics of the perpendicular configuration to parallel oneby the forward current. where A ( θ, φ ) = h cos ψ + h a cos θ cos 2 θ + cos θ cos φ + P ( θ ) , (14) B ( θ, φ ) = cos φ sin φ − K ( θ ) , (15) h = H πM , h a = H a πM , T = 4 πγMt. The P ( θ ) and K ( θ ) functions describing the current effect take theform P ( θ ) = jj λh a Z Z (cid:20) Z Z + Z Z λ − cos θ + 2 bλ (cid:18) λ + Z Z (cid:19) cos θ (cid:21) × (cid:18) Z Z + Z Z λ + cos θ (cid:19) − , (16) K ( θ ) = jj h a ατ γM (cid:18) Z Z + Z Z λ (cid:19) (cid:18) Z Z + Z Z λ + cos θ (cid:19) − (17)for the forward current, and P ( θ ) = jj λh a Z Z (cid:20) − (cid:18) Z Z + Z Z λ (cid:19) cos θ − bλ (cid:18) λ + Z Z (cid:19) cos θ (cid:21) × (cid:20) (cid:18) Z Z + Z Z λ (cid:19) cos θ (cid:21) − , (18)
00 200 300 400 500 60000.511.522.533.5 ), radian Figure 6: Switching dynamics of the antiparallel configuration to perpendicularone by the backward current. The step shows the current turning-off time.11
00 200 300 400 500 60000.511.522.533.5 , radian 4 Mt/(1+ ) Figure 7: The perpendicular configuration evolution under turning on and turn-ing off a rectangular pulse of the backward current. K ( θ ) = − jj h a ατ γM (cid:18) Z Z + Z Z λ (cid:19) cos θ (cid:20) (cid:18) Z Z + Z Z λ (cid:19) cos θ (cid:21) − (19)for the backward current.Since h a ≪ h a cos φ were omitted in derivation of Eqs. (12)–(15).In Ref. [16] an analytical solution was found of Eqs. (12), (13) at K = 0 , h ≪ , h a ≪ κ = 0)and strong damping ( κ ≫ κ = 0 . , h a = 0 . , h =0 , λ = 0 . , Z /Z = Z /Z = 0 . , ατ γM = 60 (cf. [5]). The thermalnoises initiating deviation of the free layer magnetization from the originalunstable equilibrium state were imitated with giving a small initial devia-tion from such a state by an angle of 0.01 radians in the layer plane wherethe demagnetization field does not prevent fluctuation-induced deviations,so that minimal fluctuation energy is needed. The time dependences ofthe magnetization deviation from [100] axis θ ( T ) under forward current of j = 2 j density for antiparallel and perpendicular original configuration, espectively, are shown in Figs. 4 and 5, the similar ones for the backwardcurrent are shown in Figs. 6 and 7. The dimensionless time is laid off asabscissa with t = (1 + κ ) / (4 πγM ) as the time unit; at M = 900 G onenanosecond corresponds to 200 scale divisions of the abscissa. The stepsin Figs. 6 and 7 show the current turning on and turning off times.The numerical solution (simulation) results consist completely withforegoing analysis based on the angular dependence of the magnetic en-ergy. It is seen that direct switching occurs of the antiparallel and perpen-dicular configurations to parallel one under the forward current. Underthe backward current, the switching occurs to an intermediate nonequilib-rium stationary state, from which a transition (or return, when the initialconfiguration is perpendicular) occurs to the perpendicular state. As wasto be expected, the switching is accompanied with damped oscillation dueto precession of the magnetization vector.With given parameter values, the characteristic switching times areof the order of nanoseconds, while the oscillation period is of the orderof fractions of nanosecond. With increasing the magnetization and thedamping constant, the speed of response rises (the latter up to some limits,because the switching process becomes aperiodic and slows with increasingdamping at too strong damping ( κ ≫ The analysis shows a possibility of increasing the number of the switch-able states by using magnetic junctions with cubic-anisotropy layers. Thefact is of interest that the switching of the perpendicular configurationto the parallel one requires current density lower by several times, thanthe switching of the antiparallel configuration. The fruitfulness should benoted of the combination of the energy approach to determining stationarystates with numerical simulation of the switching processes.
Acknowledgments
The authors are grateful to Yu. G. Kusraev, N. A. Maksimov and G. M.Mikhailov for useful discussions.The work was supported by the Russian Foundation for Basic Re-search, Grant No. 08-07-00290.
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