Dynamics of coupled vortices in a pair of ferromagnetic disks
Satoshi Sugimoto, Yasuhiro Fukuma, Shinya Kasai, Takashi Kimura, Anjan Barman, YoshiChika Otani
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Dynamics of coupled vortices in a pair of ferromagnetic disks
Satoshi Sugimoto, Yasuhiro Fukuma, Shinya Kasai, Takashi Kimura, ∗ Anjan Barman, and Y. Otani
1, 2, † Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan Advanced Science Institute, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan National Institute for Material Sciences, Sengen, Tsukuba 1-2-1, Japan Department of Material Sciences, S. N. Bose National Centre for Basic Science,Block JD, Sector III, Salt Lake, Kolkata 700 098, India (Dated: August 19, 2018)We here experimentally demonstrate that gyration modes of coupled vortices can be resonantlyexcited primarily by the ac current in a pair of ferromagnetic disks with variable separation. Thesole gyration mode clearly splits into higher and lower frequency modes via dipolar interaction,where the main mode splitting is due to a chirality sensitive phase difference in gyrations of thecoupled vortices, whereas the magnitude of the splitting is determined by their polarity configuration.These experimental results show that the coupled pair of vortices behaves similar to a diatomicmolecule with bonding and anti-bonding states, implying a possibility for designing the magnonicband structure in a chain or an array of magnetic vortex oscillators.
Magnetic vortex structure [1, 2] is one of the fundamen-tal spin structures observed in submicron-sized ferromag-netic elements. It is well characterized by two degrees offreedom, one is ’chirality’ ( c = ± p = ± ∼ Ni ; Py) disks with variable edge to edgeseparation d from 75 nm to 250 nm and electrical leadsare fabricated on thermally oxidized Si (100) by means ofelectron beam lithography combined with electron beamevaporation techniques. The polarities of vortices wereconfirmed by means of magnetic force microscopy priorto all the electrical measurements. Hereafter, the num-ber 1 or 2 respectively represents the vortex confined inthe left or that in the right disk shown in the SEM imageof Fig. 1. One of the paired vortices is excited by a radiofrequency ac current I ac [14–16, 18, 19] and the result-ing dc voltage V dc through a bias tee is synchronouslydetected by the same electrical contact probes [15, 20].In order to set the configuration of polarities p p = 1or p p = − p of the excited vortex is switched byapplying high I ac of about 20 mA (3 . × A/m ) atthe resonant frequency [21].Figure 1(b) shows the measured V dc /I ac as a functionof the frequency of I ac for the Py disk pair with d =75 nm. The reference spectrum for single disk shown bygreen symbols exhibits a sole dip at 352 MHz, which cor-responds to the resonant frequency of the vortex core gy-ration. When I ac is applied to one of the two neighboringPy disks, clear mode splitting takes place as can be seen FIG. 1: (color on line). (a) Schematic diagram of the measure-ment circuit and an SEM image of the sample. Two Copperelectrodes are attached to one of the disks in the Permalloydisk pair and the enveloped core is excited by a radio fre-quency current. Dynamics of the cores can be detected as dcvoltages through the spin-torque diode effect utilized by a re-sistance oscillation associated with the core gyration. A lock-in technique is adopted at room temperature. (b) Frequencydependence of the normalized dc voltage V dc /I ac measuredfor an isolated disk (green triangles) and for the paired diskswith different polarities; black squares for p p = 1, and redcircles for p p = −
1. The ac current amplitudes I ac used forthe measurements are I ac = 3 . I ac = 6 . d = 75 nm. Solid curves in each spectrum rep-resents the best fit to the data points using Eq. (1), therebythe dipolar coupling is evaluated. in the black and red spectra in the figure. Important tonote is that the gyration mode of the single vortex with I ac = 7 . p p = − p p = 1.To gain insights into the dynamics of magnetostaticallycoupled vortices, micromagnetic simulations based on theLandau-Lifshitz-Gilbert equation [23] were performed onpair of Py disks with identical physical dimensions. Typ-ical material parameters for Py are used : the saturationmagnetization M s = 1 T, the exchange stiffness constant A = 1 . × − J/m, the spin polarization P = 0.4 andthe damping coefficient α = 0 .
01. The disk is dividedinto rectangular prisms of 5 × ×
50 nm for the simu-lation. A uniform I ac of 1 . × A/m is applied only FIG. 2: (color on line). (a) Simulated time evolutions of vor-tex cores at resonance frequencies for ( p p , c c ) = (1,1)under ac currents (365 MHz and 390 MHz). Blue solid linesshow the motions of the current-excited core in the left diskand red lines correspond to those of the indirectly excitedcore in the right disk. (b) Dispersion relations of amplitude ofsteady gyrations. Values of δy max show the radii of steady gy-rations (50 nanoseconds after beginning of the current flow).Black symbols correspond to the parallel polarities p p = 1and red ones to anti-parallel polarities p p = − c c = −
1. Results of same chiralities c c = 1are plotted by open symbols. Simulation results for a singlevortex are also presented by green symbols for comparison. to the left disk.After several nanoseconds from the start of excitation,the core gyration settles in an almost circular orbit andits amplitude is strongly enhanced at the resonance fre-quency [14, 15]. This induces the core gyration in theneighboring disk, which also settles in the steady cir-cular orbit, and the collective gyration of the two vor-tices becomes fully synchronized and the eigenfrequen-cies of these modes appear in the spectra as character-istic resonance frequencies. Figure 2(a) represents thetime evolution of the core deviations δy at lower andhigher resonance frequencies with respect to the singlevortex for parallel polarities p p = 1 and the same chi-ralities c c = 1. At the lower frequency (365 MHz)both left and right cores rotate almost in-phase, whereasat the high frequency (390 MHz) the phase of the rightcore is retarded by approximately half a period. This isvery much in analogy with covalent bonding in diatomicmolecules or other forms of coupled oscillators. In thecase of in-phase excitation, magnetic charges at side sur-faces form magnetic dipoles resulting in attractive forcebetween two cores, which corresponds to a bonding or-bital. On the other hand, the side charges of the disksrepel each other for the out-of-phase excitation, havingan analogy with an anti-bonding orbital.Figure 2(b) shows the maximum deviation of the corefrom the center ( δy max ) of the steady orbital for the cur-rent excited vortex in the left disk as a function of theac frequency for the sample with d = 75 nm. The singlevortex has a clear resonance peak at around 380 MHzand exhibits a small discrepancy with the experimentalresult (Fig. 1(b)) caused by self reduced magnetization[21]. For the coupled vortices, two clear resonance peaksare observed on both higher and lower frequencies rel-ative to the sole peak for the single vortex. The mag-nitude of mode splitting for p p = − p p = 1, as experimentally observed in Fig. 1(b). At both low and high frequencies, the in-phase andout-of-phase modes are degenerated with respect to thechirality. Figure 3 summarizes all four resonance modesfrom A to D characterized by rotational directions andthe phase difference in gyrations. The lower frequencymodes (A and C) stabilize with the help of attractive in-teraction due to the magnetic charges appearing alongthe disk circumferences. On the other hand, the higherfrequency modes (B and D) stabilize with the help of re-pulsive interaction due to the opposite magnetic charges.Therefore the appearance of either in-phase or out-of-phase gyration in Fig. 2 (a) depends only on the polarityconfiguration p p . FIG. 3: (color on line). Schematic diagrams of four differentresonance modes. Each mode is identified with the sign ofpolarities p p , chirarities c c and the phase difference ofthe gyrations. In both experimental and simulated results, the split-ting amplitude tends to be enhanced with the decrease inthe normalized separation distance d n = d/r , as shownin Fig. 4. To check the effect of the current induced Oer-sted field on the excitation of the gyration of the right Py FIG. 4: (color on line). Eigenfrequencies of the coupled gy-ration modes as a function of the separation distances. Thesimulation and experimental results are shown in (a) and (b),respectively. The separation distance is given by a dimen-sionless value d n = d/R . (c) Estimated values of couplingstrength η x from both experiment and simulation with differ-ent separation distances. A d − curve from the rigid vortexmodel is also plotted as a dotted line. disk, the simulation was performed by replacing the leftPy disk with an electrode where the current flows withthe identical condition. The excited gyration amplitudedue to the Oersted field is much smaller than that forthe coupled vortices, implying that the magnetic dipolarinteraction is the dominant factor for the indirect excita-tion of the right Py disk.We here discuss the experimental results analyticallyon the basis of Thiele’s equation [24, 25], which describesthe gyrations of vortex cores r i = ( X i , Y i ) with i = 1 , V dc = I ac C/ X ] cos δ + Re[ Y ] sin δ ) , (1)where C is a constant and δ is the phase difference be-tween core and resistance oscillations in the left disk. Themodel [5] assumes the magnetostatic interaction energyterm in Thiele’s equation as U int = c c /R ( η x X X − η y Y Y ) + O ( | r/R | ) . (2)According to the rigid vortex model [24], the values of η x and η y are decided only by the shape of ferromagneticelement and the separation distance, independent fromthe excitation amplitude. As a simple solution forthese problems, η x and η y should be treated as phe-nomenological fitting parameters including the influenceof volume charges under the assumption of completelinear response with the fixed parabolic potential U = 1 / k x X i + 1 / k y Y i ( k x , k y ≡ const. ) and theinteraction energy U int . The best fitted curves to theexperimental data using the above equation are shownin Fig. 1(b) by solid curves. The values of k x and k y are decided by a result of single vortex with I ac = 3.8mA as ( k x , k y ) = (4 . × − J/m , 6 . × − J/m ).These results agree well with the previous work [20],implying adequacy of Thiele’s equation for the presentanalyses. The obtained values from experiments with 75nm separation are ( η x , η y ) = (1 . × − J, 4 . × − J) and those from simulations performed under identicalconditions are ( η x , η y ) = (1 . × − J, 8 . × − J).While the value of η x agrees well each other, a largediscrepancy of η y is observed. This is possibly due toanisotropy along y -direction in the experiment due tothe presence of attached Cu electrodes, which causesdeviation of stray field from the simulation, resulting incloser resonance frequencies for coupled vortices with p p = ± η x is shown in Fig. 4 (c). Itis clear that the coupling strength increases@with@thedecrease in@the@separation@distance, whichcan@be@clearly@reproduced by the two-dimensionalmicromagnetic simulation. It should be noted that the d dependence of η x does not follow the d − dependenceexpected from a rigid vortex model [5, 6]. A strongmagnetostatic coupling modifies the trajectories of thegyrations and the magnetization configuration near theedge region thus causes a deviation from the model,where a circular magnetization around the core isassumed.In summary, we have experimentally demonstrated theresonant excitation of coupled gyration modes in pairedvortices by means of local excitation by an ac currentpassing through one of the disks in the pair. Excitedcoupled modes are identified by rotational directions anda phase difference as four different eigen-modes. The unique property in this system give us a guiding prin-ciple for designing the magnonic crystal in further ex-panded systems such as one-dimensional chains and two-dimensional arrays and a candidate for novel tunable os-cillators using vortices [6, 9, 27].We would like to thank Y. Nakatani for fruitful discus-sions. This work is supported in part by a Giant-in-Aidfor Scientific Research in Priority Area ”Creation andControl of Spin Current” (Grant No. 19048013) fromthe Ministry of Education, Culture, Sports, Science andTechnology of Japan. ∗ Present address: Inamori Frontier Research Center,Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan † Electronic address: [email protected][1] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono,Science , 930 (2000).[2] A. Wachowiak et al. , Science , 577 (2002).[3] V. Novosad et al. , Phys. Rev. B , 024455 (2005).[4] S.-B. Choe et al. , Science , 420 (2004).[5] J. Shibata, K. Shigeto and Y. Otani, Phys. Rev. B ,224404 (2003).[6] J. Shibata and Y. Otani, Phys. Rev. B , 012404 (2004).[7] Y.A. Galkin, B.A. Ivanov and C.E. Zaspel, Phys. Rev. B , 144419 (2006).[8] K.S. Buchanan et al. , Nature Phys. , 172 (2005).[9] A. Ruotolo et al. , Nature Nanotech. , 528 (2009).[10] A. Vogel, A. Drews, T. Kamionka, M. Bolte andG. Meier, Phys. Rev. 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