Quenched Slonczewski-Windmill in Spin-Torque Vortex-Oscillators
Volker Sluka, Attila Kákay, Alina M. Deac, Daniel E. Bürgler, Riccardo Hertel, Claus M. Schneider
QQuenched Slonczewski-Windmill in Spin-Torque Vortex-Oscillators
V. Sluka, ∗ A. K´akay, A. M. Deac, D. E. B¨urgler, R. Hertel, and C. M. Schneider Peter Gr¨unberg Institute, Electronic Properties (PGI-6) and J¨ulich-Aachen Research Alliance,Fundamentals of Future Information Technology (JARA-FIT),Forschungszentrum J¨ulich GmbH, D-52425 J¨ulich, Germany Institute of Ion Beam Physics and Materials Research,Helmholtz-Zentrum Dresden-Rossendorf e. V., D-01314 Dresden, Germany Institut de Physique et Chimie des Mat´eriaux de Strasbourg,Universit´e de Strasbourg, CNRS UMR 7504, F-67034 Strasbourg Cedex 2, France (Dated: November 10, 2018)We present a combined analytical and numerical study on double-vortex spin-torque nano-oscillators and describe a mechanism that suppresses the windmill modes. The magnetization dy-namics is dominated by the gyrotropic precession of the vortex in one of the ferromagnetic layers. Inthe other layer the vortex gyration is strongly damped. The dominating layer for the magnetizationdynamics is determined by the current polarity. Measurements on Fe/Ag/Fe nano-pillars supportthese findings. The results open up a new perspective for building high quality-factor spin-torqueoscillators operating at selectable, well-separated frequency bands.
The advent of spintronics lead to the development ofexciting new concepts for nano-scale devices using thespin-degree of freedom of the electron besides its charge-property [1]. Much effort has been spent on spin-torquenano-oscillators (STNOs) [2–7], which typically consistof two single domain ferromagnetic layers separated bya metallic spacer or a tunnel barrier, one with its mag-netization fixed (polarizing layer), the other one suscep-tible to torques (free layer). An electric current travers-ing the system perpendicular to the layers becomes spin-polarized and exerts torques on the magnetizations [8–10], leading to magnetization dynamics of the free layer.These excitations are typically in the range of a fewgigahertz and can be detected by measuring the timevariation of the magnetoresistance (MR). The pinning ofthe polarizing layer can, for example, be achieved by ex-change coupling to an antiferromagnet [3] or by extendingits thickness and lateral dimension [2]. In the absence ofpinning both ferromagnetic layers will be excited and inthe case of increasingly symmetric STNOs, this resultsin a dynamic equilibrium state called the Slonczewski-windmill [8, 11]. In this state both layers’ magnetizationsrotate in the same direction with a constant relative an-gle, resulting in a vanishing MR time-dependence.Here we investigate STNOs containing two stacked mag-netic vortices, i.e., a system consisting of two ferromag-netic disks, each in a vortex state and separated by ametallic, nonmagnetic spacer. Employing analytical andnumerical methods, we study the coupled spin torque-driven motion of the magnetizations in the two disks,which are not pinned by any of the above mentionedmechanisms. We find that in the double vortex system,Slonczewski-windmill modes are quenched by an intrigu-ing mechanism. Our results show that that the currentpolarity determines which disk is excited and thereby se-lects the STNO-frequency band. This property is shownto arise from a spin torque-mediated vortex-vortex inter- action. Thus, it is an entirely different principle than thespin accumulation based mechanism suggested by Tsoiet al. [12]. We analyze in detail the underlying torquesand the resulting forces. The force exerted by one vor-tex onto the other can be split into two contributions;one part arising from the polarizer vortex in-plane mag-netization acting on the free vortex core (disk-core part)and another one which is due to the core-disk interaction.We compute the dependence of these terms on the lateralcore-core distance. These results provide insight into thefascinating dynamics of coupled magnetic vortices. Thetheoretical findings are supported by our experimentaldata obtained from double-vortex Fe/Ag/Fe STNOs.The motion of the magnetic vortex in each of the disksis governed by the Thiele equation [13] which we writehere for the vortex in the top disk, G × d X d t − d W d X − D d X d t + ¯ hjP e F = 0 . (1) G = − πκ ( µ M s L/γ ) ˆe z is the gyro vector, where L and κ are the disk thickness and the vortex core po-larity, respectively. X is the core position with re-spect to the disk center, W refers to the effective mag-netostatic potential in which the core is moving, and D = ( αµ M s /γ ) Lπ ln( R/r ) characterizes the dampingof the vortex motion. The parameters R and r are theradii of the disk and the vortex core, and the indices 1and 2 correspond to the top and bottom disks, respec-tively. The spin-transfer torque-induced force acting onthe vortex generated by a vortex-polarizer can be decom-posed into two contributions F = F d + F c . F d arisesfrom the in-plane magnetization of the polarizer and actson the core of the free vortex. The second term F c iscaused by the core of the polarizer-vortex and acts onthe in-plane magnetization of the free vortex. Both forcecontributions depend on the lateral core-core distance l .Following Ref. [14], we obtain the expressions a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec FIG. 1. Spin-torque magnitudes exerted by disk 2 on thevortex in disk 1 and their dependence on the lateral core-coreseparation. The torque arising from the polarizer core, F c isnegligible at large distances, explaining the results obtainedby micromagnetic simulations and presented in Ref. [15]. F d = (cid:90) A d x [ m ,x sin( ϕ ) − m ,y cos( ϕ )] ∇ θ + sin( ϕ ) cos( ϕ ) [ m ,x cos( ϕ ) + m ,y sin( ϕ )] ∇ ϕ = C C κ F d ˆe , (2)and F c = − (cid:90) A d x m ,z sin ( θ ) ∇ ϕ = κ F c ˆe , (3)where we introduce the vectors ˆe = − ˆe = ˆe z × ( X − X ) (cid:112) ( X − X ) + ( Y − Y ) . (4) C i = ± m ,i refers to the i thcomponent of the bottom vortex unit magnetization vec-tor. The top layer magnetization is written in spheri-cal coordinates ( ϕ, θ ), where ϕ and θ are the azimuthaland polar angles, respectively. In the calculation we as-sume rigid vortices with thickness-independent magneti-zation. The cylinder axis is chosen as the z -axis. Usingthe ansatz of Feldtkeller and Thomas [16] for the out-of-plane core magnetization ( | m i,z | = exp ( − a r ) with a ≈ ln 2 / (25 nm ) to mimic the experimentally obtainedvortex core size in Fe [17]), we obtain the forces and theirdependence on the lateral core-core distance l as shownin Fig. 1. For large l , the contribution of F c to thetotal force becomes negligible. This explains the simu-lation results reported in Ref. [15], where the influenceof the polarizer-vortex core on the dynamics was foundto be small. From Eq. (3) we see that this is causedby the reduction of the function |∇ ϕ | with increasingdistance from the top vortex core. In contrast to theasymptotic decrease of F c to zero, the magnitude of F d approaches a finite value for large l . For small distances l < r ≈
10 nm, however, we observe that both torquesfall to zero. This can be attributed to the gain in symme-try with decreasing core-core distance. The small torqueintroduced by F c is neglected in our investigation on thedynamics of the system with the vortices coupled by theelectric current. This is justified by the fact that we areinterested in the general behavior of the solutions. Thedecrease of F d at small l must however be included. Thecoupled Thiele equations read, with ˜ j := ¯ hjP/ (4 e ) G × d X d t − d W d X − D d X d t + ˜ jC C κ F d ˆe = 0 (5) G × d X d t − d W d X − D d X d t − ˜ jC C κ F d ˆe = 0 . (6)The sign of ˜ j is positive for electron flow from the topto the bottom layer. For a quantitative analysis of thesolutions, we use parabolic approximations to the effec-tive magnetostatic potentials. We let W ( X ) / | G | =6 .
28 ns − X /
2, resulting in a top vortex eigenfrequencyof f = 1 . W / | G | = (5 / W / | G | ( f = 1 . j/ | G | = (5 /
23) ns − corre-sponding to about 1 . × A / m (for P = 1), whichis within the range of experiments. The spin-transfertorque-induced force F d is assumed to increase linearlyfrom l = 0 to l = 10 nm, from whereon it is set to theconstant value F d ∞ = 23 nm.The solutions are obtained numerically using Maple’srkf45 implementation [18]. The results can be summa-rized as follows: For positive currents and equal vortic-ities, the top vortex gyrates around the disk center ona trajectory of about 50 nm in radius, regardless of thecore polarity. The sense of rotation is determined bythe core polarization (counterclockwise for positive andclockwise for negative core polarity). The gyration fre-quency is 1 . . . . . . FIG. 2. Trajectory radius r and frequency f of the bottomvortex as functions of the phase φ with respect to the topvortex that gyrates at the same frequency for positive (red)and negative (green) core polarity. The blue lines mark theeigenfrequency f of the free running bottom vortex for pos-itive (b) and negative (c) core polarity. The shaded regionscorrespond to equivalent solutions but negative radius. vortices depends on the relative core alignment. For pos-itive currents and parallel cores, the bottom core gyratesapproximately 90 ◦ ahead of its top counterpart, while forantiparallel cores a 90 ◦ lag is observed. From Eqs. (5)and (6) it is clear that the solutions for opposite vortic-ities are identical to those obtained for equal vorticitieswith a negative current polarity.For large enough | j | , the obtained characteristics of thedynamics are the generalization of the criterion foundin Ref. [15]. In the model used by those authors, thepolarizer was assumed to be a fixed, rigid vortex, andonly magnetization dynamics in the other, free disk wasallowed. In our case, both disks can be polarizing orfree layer. For a given combination of vorticities C C and applied current polarity, the system responds with adamped and a dominant gyration, the former defining thepolarizing and the latter the free disk. The current polar-ity determines which disk is dominantly excited. There-fore, the generalized jCC -criterion reads: For jC C > jC C < jC C and replace thedenominator in Eq. (4) by d := (cid:112) X + Y , yielding thefollowing relations between the radius r of the bottomvortex trajectory, its phase φ (relative to the top vortex)and the common frequency f of the two oscillators: FIG. 3. X -components of the top (red) and bottom (blue)core coordinates versus time for the case of symmetric disks.The cores are aligned parallel in (a) and antiparallel in (b).In the parallel case, a windmill-mode only appears in a tran-sient time interval, but is hindered afterwards. The oscillationdecays due to the low core-core separation and the related de-crease of the spin-transfer torque-induced force. r ( φ ) = ˜ jC C F d ∞ πD cos φ + κ ˜ jC C F d ∞ π | G | sin φf + ˜ jC C F d ∞ πD d (7) f ( φ ) = f − ˜ jC C F d ∞ π | G | r ( φ ) sin φ (8)These relations are displayed in Fig. 2 and reproducethe behavior observed in the numerical solutions: Forboth cases of positive and negative bottom vortex corepolarity, the bottom vortex can adapt to the ( a priori arbitrary) frequency of the top vortex by adjustingthe phase. Positive (negative) frequency correspondsto counterclockwise (clockwise) gyration. As displayedin Fig. 2(a), this phase shifting comes with a strongreduction of the orbit radius- or, in other words, aquenching of the windmill-modes. The dashed lines inthe shaded regimes correspond to solutions of negativeradius. Since a reverse of the sign of the radius is equiv-alent to a phase shift of π , these negative- r solutionsare identical to the trajectories represented by the solidlines. By means of a phase adaption and reduction of theradius, the vortex can use a fraction of the spin-transfertorque-induced force to assist or counteract the forcedue to its magnetostatic potential. The resulting radialforce component can differ strongly from the purelymagnetostatic force. It may even lead to an inversionof the relation between the sense of gyration and thecore polarity. A special case is the configuration, forwhich a frequency adaption is not necessary, i.e., ifthe two disks are identical and the cores parallel [Fig.3a)]. In this case, the vortices start rotating in phase,but as they reach the limit cycle and the core-core FIG. 4. Resistance versus field for (a) I = −
21 and (b) I = +21 mA sample current. I < distance drops below 10 nm, the mutual spin-transfertorque-induced force decreases leading to a decay ofthe oscillation amplitudes. For antiparallel alignment,the windmill-modes are quenched by the mechanism offrequency and phase adaption [Fig. 3b].For an experimental confirmation of the frequency andphase adaption mechanism and the related quenchingof the windmill modes we study the current-inducedmagnetization dynamics of an Fe/Ag/Fe nanopillar witha Fe layer thickness ratio of 5/3. According to our model,we expect to observe excitations for both current polari-ties, but with different frequencies yielding a frequencyratio of approximately 5/3. Cylindrical nanopillarsare patterned using e-beam lithography and Ar ionmilling from molecular beam epitaxy-grown GaAs(001)/Fe(1)/Ag(150)/Fe(25)/Ag(6)/Fe(15)/Au(25) stacks(layer thicknesses in nm). The pillar diameter is 210 nm.The milling was stopped after reaching the 150 nm thickAg buffer layer. Thus, the oscillator consists of twoferromagnetic disks of equal diameter and comparablethickness stacked on top of each other. Figures 4(a)and (b) display the field dependence of the nanopillarresistance for I = ∓
21 mA ( ∓ . × A/cm ), re-spectively. The external magnetic field was applied inthe sample plane. The magnetoresistance profiles arecharacteristic for this sample type [19] and reflect twomagnetization states: The first one comprises a vortexin one disk, while the other nanomagnet remains ina quasi-homogeneous state. These configurations arecharacterized by a nearly linear field dependence of theresistance caused by a continuous lateral displacementof the vortex with changing field. The second state isobserved from low field magnitudes up to about 200 mT and is characterized by low resistance values near thelevel in magnetic saturation. Here, each disk containsa vortex with the vorticity given by the circumferentialOersted field. This results in locally parallel alignmentof the two disks’ magnetizations explaining the observedlow resistance. For both current polarities in Figs. 4(a)and (b) we detected magnetization dynamics in thosefield intervals, in which the double-vortex state occurs.The excitation frequencies are shown in Fig. 4(c)along with the corresponding electron flow directions ofthe externally applied currents I . All frequencies arebelow 2 GHz, which is typical for vortex gyration inFe/Ag/Fe nanopillars [19–21], but the frequencies areclearly different and well separated for the two currentpolarities. At low external fields their ratio is about 1 . (cid:39)
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