Observation of anisotropic energy transfer in magnetically coupled magnetic vortex pair

magnetic vortex pair N. Hasegawa, S. Sugimoto, D. Kumar, S. Barman, A. Barman, K. Kondou, and Y. Otani Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India CEMS, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

The magnetic vortex has been drawing much attention as one of the fundamental structures confined in a nanometer scaled ferromagnetic disk. 1,2It is characterized by two degrees of freedom, the in-plane curling magnetization direction, "chirality" ðc ¼ 61), and the out of core magnetization, "polarity" ðp ¼ 61).The core of a magnetic vortex is known to behave as a quasiparticle in a harmonic potential. 3In the low energy excitation (or gyration) mode, the core continuously gyrates along an equipotential line.The resonant frequency for a single vortex can be determined by the potential shape which depends on the aspect ratio (thickness/ radius) of the magnetic disk 4 and not on polarity or chirality.
When two magnetic vortices are placed sufficiently close to each other, the dynamic dipolar interaction serves as a binding force.In the two-coupled vortices system, the resonant frequency splits into two branches and the split-width depends on the strength of the dipolar interaction. 5,6By arranging magnetic vortices in a two-dimensional array, it forms a so-called magnonic crystal, where the band structure can be tuned by polarity. 7Thus, its band structure has been intensively studied for several cases such as a one-dimensional chain, 8-10 a twodimensional array 7,11 and a carbon flake. 12In these systems, properties of the energy transfer and storage in each disk are also important for actual applications. 13,14The interesting properties of the energy transfer and storage within a group of magnetic vortices have recently been examined in a theoretical study; 15 however, no experimental observation has been reported.Here, we observed the specific gyrating motion due to anomalous energy storage, for a coupled vortices system by using the electrical detection method.
Samples were fabricated on a silicon substrate by means of electron beam lithography on polymethyl-methacrylate resist and a subsequent lift-off process after electron beam deposition.We fabricated two 30 nm thick disks 500 nm in radius of Permalloy (Py: Ni 80 Fe 20 ) aligned along the x axis with a 100 nm gap between their nearest edges.By attaching Cu electrodes in each Py disk, we can apply different ac currents I acL and I acR , where the subscripts "L" and "R" represent the left and right Py disks, respectively.The Cu electrodes were patterned at the same distance from the disk center and deposited by thermal evaporation.Before the Cu deposition, a careful Ar ion beam etching (600 V beam voltage) was carried out for 30 s in order to clean the Py surface.In Fig. 1(a), we show a scanning electron microscope (SEM) image of the two Py disks as well as a schematic image of the measurement circuit.Here, anti-parallel polarities were prepared by applying relatively large ac current 16 and confirmed it by measuring the splitting intensity of resonant frequencies as reported in Ref. 6.
][19][20] The resistance oscillation associates with the core gyration amplitude and the phase of the gyrating motion as follows.When the core position r ¼ (x, y) is shifted from the disk center r ¼ (x c , y c ), the resistance of the Py disk can be expressed as 19,20 where a is proportional to the sample resistance change due to AMR effect.When the ac current I ¼ I ac e ixt is applied to the disk, the core gyrates in the steady orbit.The time evolution of the core position is expressed as: y 0 þYe ixt Á , with ðX; YÞ¼ ðX 0 þiX 00 ; Y 0 þiY 00 Þ.Here, p is the polarity, ðx 0 ;y 0 Þ is the core position before the gyrating motion starts and X 0 (X 00 ) and Y 0 (Y 00 ) are the x and y component of the real (imaginary) part of the gyration amplitude.By substituting the above relation into Eq.( 1), one can obtain the oscillating resistance.Therefore, the applied ac current is rectified by the oscillating resistance and a dc voltage V dc normalized by I ac is given by 19,20 V dc =I ac ¼ Àpaðx 0 À x c ÞX 00 þ aðy 0 À y c ÞY 0 : ( From the analysis based on the Thiele's equation, it is known that X 00 shows the anti-Lorentzian, while Y 0 shows Lorentzian 19,21,22 spectrum shape.Eq. ( 2) implies that the normalized dc voltage V dc /I ac is proportional to the shift in the core position from the disk center, (x 0 -x c ) and (y 0 -y c ). Figure 1(b) shows the normalized dc voltage spectra measured with I ac ¼ 2:3 mA for the left disk (black dots) and the right disk (red dots).The normalized dc voltage spectra show almost the same peak structures, which ensure that the disks are not so different.We can also find that the spectra show Lorentzianlike shape, which implies that the value of ).In that case, the ratio of gyration amplitude in each vortex core can be approximated by using only the y component, i.e., Y 0 R =Y 0 L ¼ ðV dcR =I acR Þ=ðV dcL =I acL Þ.In order to estimate the gyration amplitude in each disk, we injected two ac currents with different amplitudes and same frequency f, i.e., I L ¼ I acL sin ð2pf Þ for the left disk and I R ¼ I acR sin ð2pf þ D Iac Þ for the right disk, where D Iac is the phase difference between the two ac currents.The D Iac can be monitored by using an oscilloscope.We applied a large current to the left disk ðI acL ¼ 2:3 mAÞ to excite and detect the gyrating motion and another current to the right disk ðI acR ¼ 0:23 mAÞ which is small enough not to affect the induced gyrating motion but large enough for detection.
Figure 2(a) shows the normalized dc voltages as a function of the phase difference D Iac at 230 MHz, which is the middle frequency of the double resonant peaks at 215 and 245 MHz.At the left disk (black dots), the normalized dc voltage shows almost a constant value of 0.30 mX,, which is proportional to gyration amplitude in left disk.This implies that the collective dynamics is not perturbed by I acR .At the right disk (red dots), the D Iac dependence shows the sinusoidal curve originating from the phase difference between the core oscillation and the I acR .When the core and I acR oscillate in-phase, the V dc /I ac takes the maximum value of 0.69 mX, which is proportional to gyration amplitude in the right disk.Surprisingly, at this frequency, the gyration amplitude in right disk is 2.3 times larger than that of left disk, i.e., Y 0 R =Y 0 L ¼ 2:3.From this result, we can estimate the ratio of the stored energy c in each disk by using the for- In this frequency, the value of c in right (left) disk can be estimated to be 0.85 (0.15) of stored energy in coupled vortices system.
Figure 2(b) shows the obtained values of c as a function of the input frequency.For both resonant frequencies of 215 and 245 MHz, the values are about 0.5.It means that for both cores the gyration amplitude is almost same, and the same amount of energy was stored in both disks.Interestingly on the other hand, in the off-resonant frequency range from 215 to 245 MHz, the c in the right disk is larger than that of the left disk.The maximum difference in c between left and right disks appears at around the middle frequency between two resonant peaks.This anisotropic energy storage results in the amplification effect is discussed in Ref. 15.
In order to understand the physical mechanism of the frequency dependence of c, we performed a micromagnetic simulation 23 by solving numerically the Landau-Lifshitz-Gilbert equation with spin transfer torque terms.The material parameters for Py in our simulations are: saturation magnetization M S ¼ 0.93 T, the exchange stiffness constant A ¼ 1.05 Â 10 11 J/m, the spin polarization P ¼ 0.4, and the damping coefficient a ¼ 0.01.For the simulation, the disk is divided into rectangular prism like cells of 5 Â 5 Â 30 nm 3 and the time step of 0.25 ps.For simplicity, non-adiabatic torque term has been ignored.The dimensions of the disks are the same as the actual sample setup, and only the I L was applied to the left disk.
Figure 3(a) shows the simulation result for frequency dependence of c calculated from gyration amplitudes at 150 ns after the beginning of the current injection.The frequency is also swept between the resonant frequencies (215 MHz and 245 MHz).In general, our micromagnetic simulation reproduces the experimental result in a good approximation.Note that even if the ac current was applied to the right disk, almost the same result was obtained, i.e., the gyration amplitude in the left disk is larger than that in the right disk.
The magnetic interaction energy between two vortex cores due to dipole interaction is analytically approximated as 5 When cores gyrate in the steady orbital, the long-time average of U int can be written as where c, g, and R are the chirality, the coefficient of the dipole interaction, and radius of magnetic disk, respectively.Thus, the amplitude of U int is determined by the phase difference of vortex core gyration D core .Figure 3(b) shows calculated values of D core by Landau-Lifshitz-Gilbert equation.We found that the D core of x and y components changed smoothly from p to 2p and from 0 to p, respectively.When the input frequencies match the resonant frequencies of 215 MHz and 245 MHz, the values of U int , respectively, take minimum and maximum value, 9 so that the torque s dip due to U int arises along the azimuthal direction which is antiparallel to the gyro torque s gyro at 215 MHz and parallel at 245 MHz as shown in schematics in Fig. 3(c).As a result, the resonant frequency in the coupled vortex core system changes from the resonant frequency (230 MHz) in an isolated magnetic disk. 6n the other hand, in the off-resonant region from 215 MHz to 245 MHz, the direction of s dip changes to the radial direction, meaning that the effective damping constant of vortex core can be modulated by the s dip .Fig. 3(c) shows the frequency dependence of the radial component of the s dip obtained by assuming that gyration amplitudes of disks are the same.The positive (negative) sign of s dip corresponds to parallel (anti-parallel) direction to the damping torque s damp .Thus, the effective damping constant for the left (right) core increases (decreases) at the off-resonant frequency.In such case, it is easy to store the magnetic energy in the right disk rather than the left disk, such as represented in Fig. 3(a).By using this mechanism, we can control the gyrating motion and amount of stored energy in each disk in magnonic crystals.
Note that the above discussion is also applicable to the case of parallel core polarities.In this case, there is also a frequency at which the relative phase difference of cores is p/2 (Àp/2), 6 i.e., the damping constant is modified by the dipole interaction.][7] Therefore, the damping constant modulation, namely, the anomalous energy storage should be suppressed.In order to confirm this trend, we have measured the same sequence for parallel core polarities and estimated c L ¼ 0.38 and c R ¼ 0.62 at 230 MHz.From micromagnetic simulation, we obtained c L ¼ 0.42 and c R ¼ 0.58 at 230 MHz.Thus, we found that the anomalous energy storage is strongly suppressed compared to that of the anti-parallel core polarities, which agrees with the calculated result. 15n summary, we have investigated the energy storage in coupled magnetic vortices by using the electrical detection method of core gyration amplitude.The specific gyrating motion due to anomalous energy storage is observed at off-resonant frequencies.Our micromagnetic simulations qualitatively reproduce the experimental results and explain that the behavior arises from the modulation of effective damping constant.These findings about the specific energy storage in coupled vortices may be important for magnetic vortex based signal processing and logic operations.

FIG. 1 .
FIG. 1.(a) Schematic diagram of the measurement circuit and an SEM image of the sample.(b) Normalized dc voltage spectrum measured at the left disk (black dots) and the right disk (red dots), independently.

FIG. 2 .
FIG. 2. (a) Normalized dc voltage as a function of the D Iac at 230 MHz for the left (black dots) and right disk (red dots).Solid lines show the fitting with constant value (black line) and sinusoidal curve (red line).(b) Frequency dependence of the c for both disks.

FIG. 3 .
FIG. 3. Simulated c obtained from the amplitude of steady gyration amplitudes at 150 ns after beginning of the current injection applied only into the left disk.(b) Phase difference between the left and right core gyration for x (black) and y (red).(c) Frequency dependence of the dipole torque working as the damping torque.Bottom schematics show the gyrating cores with each torque.The purple, blue, and green arrows show the direction of the gyro torque, damping torque, and dipole torque, respectively.
This work was supported by Grant-in-Aid for Scientific Research on Innovative Area, "Nano Spin Conversion Science" (Grant No. 26103002) and the RIKEN Junior Research Associate Program.