Coupled oscillations in noncollinear microscale rectangular magnets

PHYSICAL REVIEW B 82, 214422 共2010兲

Coupled oscillations in noncollinear microscale rectangular magnets S. Jain,1 M. Kostylev,2 and A. O. Adeyeye1,* 1Information

Storage Materials Laboratory, Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore 2School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia 共Received 8 April 2010; revised manuscript received 8 November 2010; published 21 December 2010兲

The eigenmode spectrums of dense arrays of single-layer rectangular elements periodically arranged in a two-dimensional 共2D兲 magnonic crystal with a complex unit cell have been characterized using broadband ferromagnetic resonance 共FMR兲 spectroscopy and magneto-optical Kerr effect measurements. The crystal’s unit cell consists of noncollinear orientations of constituting elongated rectangular elements, which makes it significantly different from a normal 2D magnonic crystal with similar elements. Microscopy and numerical simulations of the ground state reveal that the elements are statically dipole coupled and that the ground state is of collective nature. Two distinct collective FMR modes were observed in the frequency range from 1 to 20 GHz, the frequency positions of which depend on the angle of the applied field with the selected crystal axis. These modes were explained as predominantly localized in elements with different easy-axis orientations. Coherence of oscillations in the elements of the same type is ensured by mediators in the form of the elements of the other types. Angular FMR measurements confirmed uniaxial configurational anisotropy for the array with a not-so-well-resolved frequency gap between the two main modes. DOI: 10.1103/PhysRevB.82.214422

PACS number共s兲: 76.50.⫹g, 85.75.⫺d

I. INTRODUCTION

The static and dynamic properties of ferromagnetic nanostructures have been extensively investigated in recent years due to their potential in a wide range of applications including magnetic random access memory,1,2 high-frequency spintronic devices,3,4 bit patterned media,5 magnetic logic,6 and biological applications.7 Furthermore, there has been a growing interest in the fundamental understanding of spin-wave 共SW兲 propagation in magnonic crystals 共MCs兲 recently, which are coupled periodic magnetic nanostructures and are conceived as the magnetic microwave analog of photonic crystals. As in photonic crystals, periodicity of the medium is used to form a forbidden frequency gap in the spectrum, called the magnonic gap. This may find application in novel miniature microwave filters8 and generators.9 The wave excitation exploited in magnonic crystals is the microwave spin wave, whose wavelength can be smaller than the length of the optical waves. Thus, with spin waves it is possible to achieve microminiaturization which is hardly possible with optical waves. The band spectrum of MC consists of allowed states 共magnonic bands兲 and forbidden states 共magnonic gaps兲. The existence of magnonic gaps has been predicted in onedimensional 共1D兲,10,11 two-dimensional 共2D兲,12 and three-dimensional13 systems. Frequency band gaps have been observed experimentally in wirelike structures consisting of shallow grooves etched into an yttrium-iron-garnet films,14 one-dimensional array of homogenous Ni80Fe20 nanowires separated by an air gap,15,16 and synthetic nanostructures composed of periodic arrays of alternating Ni80Fe20 nanowires in direct contact with Co nanowires, also known as “bicomponent MC.”17 It has been clearly shown that the frequency band gaps can be tuned by the application of a magnetic field and also by changing the lateral dimensions of the nanowires18 and the air gaps between the 1098-0121/2010/82共21兲/214422共9兲

elements.19 One of the main challenges associated with the actual realization of magnonics based devices however, is the design and fabrication of high-quality MCs at the nanoscale level. On one hand ensuring a significant dipole coupling through the air gap requires gap widths below 100 nm,19 on the other hand it is also extremely challenging to fabricate periodic structures consisting of two ferromagnetic materials with sharp interfaces over a large area using multilevel electron-beam lithography 共EBL兲. Therefore, there is a need to explore other fabrication approaches that circumvent the inherent limitations of proximity effects with using conventional EBL. For a number of applications, large macroscopic arrays of microelements and nanoelements are of paramount importance. In particular, functionality of magnetic logic6,20 and of artificial spin ices21 is based specifically on noncollinear orientation of elongated magnetic microscopic elements and exploits a number of competing long-range static and dynamic dipole interactions on large periodical and quasiperiodical arrays of such elements. Alternatively, the interest in fabrication of large-area periodical structures is also driven by advances in lithography and other controlled nanofabrication techniques and the development of new characterization techniques such as nonresonant stripline ferromagnetic resonance 共FMR兲共Ref. 22兲 and polarized neutron reflectometry. In this paper, we fabricate and investigate a test large-area 2D bicomponent periodical MC-like microstructure. Its bicomponent functionality is provided by orientation in two perpendicular directions of elongated rectangular magnetic elements forming the structure’s unit cell. We investigate both the static and dynamic magnetic properties of this test structure. We also want to address an important issue of the strength and anisotropy of the dipole coupling of elements forming two-component noncollinear periodic arrays. In a very recent paper,23 it was found that on a 2D onecomponent array of magnetic nanosquares, only the elements along the direction perpendicular to the applied field are dy-

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©2010 The American Physical Society


JAIN, KOSTYLEV, AND ADEYEYE UNIT CELL Spacing between each unit cell and elements ~ 50 nm

(90°) Y

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FIG. 1. 共Color online兲 SEM micrograph of the C shape configuration along with the description of a unit cell and dimensions.

namically dipole coupled, such that parallel “waveguiding channels” are formed in this direction. This strong-coupling anisotropy was explained as due to strong static demagnetization in the direction of the applied field24,25 which impedes energy flow between the channels. In the present study we attempt answering the important question, is small or negligible dynamic dipole coupling along the direction of the applied field is a general rule for 2D arrays of elements, or one can get around this obstacle by a proper choice of geometry. The paper is organized as follows: in Sec. II, we shortly describe the choice of structure design and the fabrication method used in synthesizing the structures. In Sec. III, we present the results of detailed investigation of its ground state and show that the cell components with different easy-axis orientations are statically dipole coupled and undergo contrasting reversal processes when the external field is applied along specific orientations. Section IV contains the results of broadband FMR measurements. In this section, we demonstrate that dynamic dipole coupling is significant in the center of the first Brillouin zone 共⌫ point兲 for this MC. Subsequently, discussion of the obtained results is provided in Sec. V. We analyze the FMR mode structure for the sample and find that there are two dominant modes. They are characterized by localization of the resonant precession predominantly in one of the two types of the elements. Importantly, the elements of the other type being off-resonance act as conductors of dynamic dipole field which ensures dipole coupling across structure periods. Finally, we discuss the results of the angle-resolved FMR measurements. From these measurements it follows that the array exhibits a twofold configurational anisotropy and the gap in the spectrum at the ⌫ point becomes small and not well resolved for a magnetization direction close to the diagonal of the unit cell. II. EXPERIMENTAL DETAILS

The noncollinear array of rectangular elements was fabricated using deep ultraviolet lithography at 248 nm wavelength along with deposition and lift-off techniques. Each element was designed to be confined in three dimensions 共2 ␮m ⫻ 0.8 ␮m ⫻ 30 nm兲 with interelement spacing of ⬃50 nm. This extremely small spacing between the elements was chosen to ensure the maximum dipole coupling on the array and was maintained across the whole size of the array 共4 ⫻ 4 mm2兲. A 30-nm-thick polycrystalline Ni80Fe20 layer was subsequently deposited using electron beam deposition at a constant rate of 0.2 Å / s. Details of the fabrication technique are described elsewhere.26 The elements are arranged in a “C” cell configuration with each unit cell comprising of three rectangular elements. Shown in Fig. 1 are the

corresponding scanning electron micrograph 共SEM兲 and a schematic of the unit-cell arrangement. In this geometry, the vertical components form columns and horizontal components are placed near both ends of each vertical component. The goal is to investigate the strength of the static and dynamic dipole coupling between the vertical and horizontal elements. An important problem in each FMR study is unambiguous identification of the observed modes. By placing two horizontal components in a unit cell 共C configuration兲 we want to facilitate the mode identification: the amount of magnetic material contained in the horizontal elements is twice larger than in the vertical ones, which may affect amplitudes of specific modes and thus, “tag” specific resonances. This also provides inversion symmetry for the structure which affects the structure ground state in a specific way, as we will show below. The in-plane magnetization curves were recorded using a magneto-optics Kerr effect 共MOKE兲 setup with a laser spot size of about 50 ␮m in the longitudinal geometry at room temperature. Domain configurations at remanence were investigated using magnetic force microscopy 共MFM兲 imaging in the phase detection mode with commercial CoCr coated Si cantilever tips magnetized along the tip axis. The dynamic properties 共resonance frequencies兲 were measured in the 1–20 GHz range using a broadband vector network analyzer 共VNA, Agilent 8363C兲 ferromagnetic resonance technique27 with a microstrip microwave transducer. To obtain the highfrequency response, the sample was positioned on top of the transducer, with the magnetic elements facing the board. The external magnetic field was applied parallel to the microstrip line which will also be considered as 0° with respect to the sample orientation 共Fig. 1兲. This geometry ensures a strong coupling between the transducer and the magnetic elements. Understanding of the reversal mechanism in the elements was facilitated by micromagnetic modeling of the ground state and the microwave dynamics, which was performed using LLG micromagnetic software.28 Periodic boundary conditions were utilized to incorporate the effect of magnetostatic interactions between the neighboring cells. Standard parameters were used to characterize the properties of Ni80Fe20 共cell size= 10 nm⫻ 10 nm, exchange constant A saturation moment M s = 860 = 13⫻ 10−12 J m−1, ⫻ 103 A m−1, anisotropy K1 = 0兲. To simulate the dynamic behavior, a 2 ns long pulse of magnetic field oriented inplane and perpendicular to the static field was applied. Temporal Fourier analysis of the recorded precessional motion of magnetization upon application of the pulse yielded frequencies of the structure’s eigenmodes and also the modes’ 2D profiles. III. CHARACTERIZATION OF THE MAGNETIC GROUND STATE

Shown in Figs. 2共a兲–2共c兲 are the representative experimental hysteresis loops for ␪ = 0°, 45° and 90°, respectively. The magnetization reversal behavior is observed to be extremely sensitive to the orientation of applied field due to the induced configurational anisotropy in the system. The reversal process is characterized by the sharp switching fields in

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FIG. 2. 共Color online兲 The M-H loops obtained using MOKE magnetometry are shown for 共a兲 ␪ = 0°, 共b兲 ␪ = 45°, and 共c兲 ␪ = 90°. The remanent MFM images obtained after saturating the patterns at +3.9 kOe and bringing it back to zero are shown in 共d兲 and 共e兲 for ␪ = 0° and 90°, respectively. The corresponding LLG simulated spin states are shown as insets.

magnetization for both ␪ = 0° and 45°. However, for ␪ = 90°, a boxlike behavior is observed near zero field. To gain a better understanding of the magnetization states, MFM imaging was performed at remanence, after saturating the structures at +4 kOe and subsequently decreasing the field to zero. Shown in Figs. 2共d兲 and 2共e兲 are the MFM images taken at remanence for field applied along ␪ = 0° and 90°, respectively. It is clearly evident that both the horizontal and vertical magnetic elements attain significantly different domain configurations when the external field is applied in two orthogonal directions due to shape induced magnetic anisotropy. For ␪ = 0°, all the elements with their easy axes aligned along the field direction adopted a single domain state configuration, whereas, all the elements with their easy axes aligned perpendicular to the applied field direction adopted a vortex configuration due to the strong demagnetization field. The inset in Fig. 2共d兲 is the corresponding simulated spin state of a single unit cell. There is a good agreement between the MFM image and the simulated magnetic state. As evidenced from the simulated spin state, the effect of magnetostatic coupling due to the vertical element on the horizontal element can be distinctly observed in the form of shifted edge domain positions. The bright contrast 共of red and yellow online兲 at the edges of the two horizontal elements in a unit cell are toward the inner side. When they are in close proximity 共i.e., when two unit cells are placed together兲, the side charges repel each other, as they are of same type and hence, move away to the neighboring corner. The unit cell of C shape is chosen in order to attain different remanent domain configurations in both the horizontal and vertical mag-

netic elements. If another horizontal magnet was placed such that it becomes an “E” shape unit cell, then there is a strong possibility of forming a vortex configuration in all the horizontal elements 共due to strong repelling of charges and energy minimization兲.29 Interestingly, when the magnetic field is applied along the orthogonal direction 共␪ = 90°兲, the magnetization states change accordingly 共i.e., those elements with the easy axes aligned along the field direction adopted a single domain state configuration while those elements with the easy axes aligned perpendicular to the field undergo transition to single or double core vortex states兲 as shown in the MFM image of Fig. 2共e兲. The simulated magnetic state at remanence is shown as an inset, which is in good agreement with the MFM image. Moreover, the formation of double core vortex state may account for the boxlike behavior in the hysteresis curve 关Fig. 2共c兲兴 which has been observed previously.30 IV. FERROMAGNETIC RESONANCE STUDIES

In order to characterize the two distinct behaviors in the system due to the induced configurational anisotropy, FMR spectroscopy was employed. In terms of the collective spin wave dispersion on 2D periodic structures,23 FMR probes the center of the first Brillouin zone, i.e., the ⌫ point. Shown in Figs. 3共a兲–3共d兲 are the FMR absorption curves as a function of the magnitude of the applied field 共in the range 3–2 kOe兲 for ␪ = 0° and 90° along with the simulated frequency spectrum and the mode profiles obtained using LLG software. The field was initially applied at the highest magnitude of

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JAIN, KOSTYLEV, AND ADEYEYE 3.2 kOe

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FIG. 3. 共Color online兲 Frequency spectrum obtained experimentally at 共a兲 ␪ = 0° and 共b兲 90° for field varied from 3 to 2 kOe. The corresponding simulated frequency response as a function of the external magnetic field is shown in 共c兲 and 共d兲. The simulated mode profiles for the two resonant peaks are also shown as insets. The mode profile cross sections of the lines indicated in 共c兲 are shown in 共e兲 for line 1, 共f兲 for line 2, and 共g兲 for line 3.

+3.9 kOe to ensure homogenous saturation magnetization state, and was thereafter decreased to the desired field values. From the superposition of the frequency sweeps at constant field values, we observed the presence of a number of eigenmodes which vary with the magnitude of the applied field. Two of them were distinct by their large amplitude. In the following, we will concentrate on the behavior of these two prominent modes. There is one significant observation that can be made from the spectra in Fig. 3: the amplitude of the higher resonance frequency mode is larger than that of the lower frequency mode for ␪ = 0° and vice versa for ␪ = 90°. For simplicity, we can name the lower frequency mode for ␪ = 0° as RF1 and higher one as RF2. We have systematically varied the strength of the applied field from +3.9 kOe to 0 and then to −3.9 kOe, and plotted the corresponding experimentally extracted peak positions of the two resonance frequencies for ␪ = 0° as a function of the applied magnetic field in Fig. 4共a兲. Figures 5 and 6 show the results of in-plane angleresolved FMR measurements. Figure 5 demonstrates variation in frequency of RF1 and RF2 for a given value of the applied field. It is clearly observed that the array is charac-

terized by a twofold configurational anisotropy. Correspondingly, the areas of mode crossing are specifically addressed in Fig. 6. To obtain this figure a different method of FMR detection was used in order to increase resolution.22,31 This technique is similar to one used in conventional cavity FMR: we have employed field modulation and lock-in amplification 共FMFMR兲 and carried out field sweeps instead of frequency sweeps.31 This method measures the derivative of the resonance line with respect to the applied field 关solid blue lines in the panels of Figs. 6共a兲–6共e兲兴 which improves resolution. The goal was detection of an eventual gap in the spectrum in the areas of crossings of the lines in Fig. 5. From the raw traces 关shown in Figs. 6共a兲–6共e兲兴 and the comparison of frequency peaks for various modes 关shown in Fig. 6共f兲 by blue lines兴, we observed that near the point of crossing 共40°兲, the mode structure is complicated and the gap is not well resolved. The presence of gap can be effectively speculated by directly observing these lines. The resonance line itself 共red lines兲 obtained as an antiderivative of the blue line, indicates just broadening at the point of cross section, similar to the VNA data in Fig. 5.

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FIG. 4. 共Color online兲共a兲 Experimentally extracted resonant frequency versus magnetic field for two resonant frequencies at ␪ = 0°, along with the response of the continuous film. Solid line is the theoretically obtained frequency response from Kittel’s equation. The two half-filled square dots indicate the resonance frequency obtained from simulations at 3.3 kOe. 共b兲 shows the response for RF2, with solid dots denoting experimental data and the solid line corresponds to the fit of the experimental data. V. DISCUSSION

quency response of the continuous film is very distinct from the array response: it matches the low-field response with RF1 and high-field response with RF2. To identify the modes RF1 and RF2, we conducted numerical simulations of magnetization dynamics using the LLG software with the material parameters given above. The simulated mode frequencies for external field of 3.3 kOe are plotted as half-filled square dots in Fig. 4共a兲. From the figure it is evident that the calculated frequencies are in good agreement with the experimental data. Based on this good agreement, we may use the simulated mode profiles 关Fig. 3共c兲兴 to identity the modes seen in the experiment. In this way, RF2 is identified as the mode predominantly localized in the horizontal elements, and RF1 as predominantly localized in the vertical elements. This identification also explains the modes’ absorption amplitudes: the amount of magnetic material contained in all horizontal elements is twice larger than that contained in all vertical ones. This suggests that the mode localized in the horizontal elements should absorb more microwave power. From Fig. 3共a兲, it is obvious that RF2 amplitude is larger than RF1 which is consistent with this idea. The same applies to the simulated amplitudes in Fig. 3共c兲.

The peak positions of RF1 and RF2 follow linear frequency dependence in the saturated state, a behavior qualitatively consistent with the dynamics expected from Kittel’s equation,32 f = ␥兵关Hx + 4␲共Nz − Nx兲M x兴 ⫻ 关Hx + 4␲共Ny − Nx兲M x兴其1/2 , 共1兲 where ␥ is the gyrometric ratio, Nx, Ny, Nz are the demagnetizing factors 共Nx , +Ny , +Nz = 1兲 of the element under study and M x which is substituted by M s, is the saturation magnetization of Ni80Fe20. “x” is assumed to be the direction of the applied field. Magnetic parameters for the structure were extracted from FMR data obtained on a continuous film of identical thickness deposited under similar conditions are shown in Fig. 4共a兲. For a continuous film, Eq. 共1兲 is reduced to f = ␥关共Hy + M y兲Hy兴1/2 since Nx = Ny = 0 and Nz = 1. A least-squares fit to the experimental data 关the solid line in Fig. 4共a兲兴 yielded a ␥ value of 2.93 GHz/kOe and confirmed the standard value for saturation magnetization for our material: M s = 860 ⫻ 103 A m−1 共4␲ M s = 10.75 kG兲. We also note that the fre-

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FIG. 5. 共Color online兲 The variation in two resonance modes 共RF1 and RF2兲 as a function of the angle between the rectangular elements and the external magnetic field at two different saturation fields of 共a兲 800 Oe and 共b兲 3.5 kOe are shown. 214422-5


JAIN, KOSTYLEV, AND ADEYEYE Anti‐derivative

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Angle (θ) FIG. 6. 共Color online兲 Experimentally obtained results for rotational measurements. 共a兲–共e兲 show the absorption curves measured at the proximity of the mode hybridization point. Thin 共blue兲 curves: first derivative of the resonance line with respect to applied field which are the raw data obtained in this measurement. Thick 共red兲 solid lines: calculated anti-derivative of the raw data. 共f兲 resonant fields vs the angle ␪ extracted from the raw data. The lines for the peaks marked by 共color兲 dashed lines in 共a兲–共e兲 are shown in 共f兲共by respective colors online兲. To increase the resolution, a field modulation and lock-in amplification in combination with field sweeps at a constant frequency 7 GHz was used for these measurements. 共Not that a higher field peak in a field sweep corresponds to a lower frequency peak in a frequency sweep.兲

These panels show the profile cross sections along lines 1 and 3 关as labeled in Fig. 3共c兲兴, respectively. These curves display the imaginary part of the complex precession amplitude. Recall, that the profiles were obtained as inverse Fourier transforms of time resolved data, which resulted in a noticeable digital noise in the figure. From the cross section 1 of Fig. 3共e兲, it is clearly observed that the amplitudes of oscillation in both types of elements are comparable. This evidences a strong dynamic dipole coupling between the differently aligned elements. This result suggests that contrary to the highly symmetric case of the square dot array from Ref. 23, it is possible to efficiently couple parallel horizontal waveguiding channels which are formed by the double chains of horizontal elements interleaved by the vertical elements in this geometry. This may have potential applications in microwave directional couplers. However, only simulation of traveling collective modes and measurements with Brillouin light scattering across the whole width of the first Brillouin zone for this magnonic crystal can deliver a definitive answer to the question about the anisotropy of dipole coupling on the array. These measurements and simulations are beyond the scope of this paper. In the present work, we only want to point out that simulations performed on a small number of elements, like in this study or carried out in Ref. 33, may only suggest a more isotropic coupling if the element shape deviates from a square. For the completeness of the study it is crucial to obtain some quantitative description of the dipole coupling. It is important because in previous studies of dipole-coupled stripes 共see, e.g. Ref. 11兲, it was found that often the theory overestimates the strength of dynamic dipole coupling on the real MCs. It is difficult to come up with a theory which quantifies each of the number of couplings between different neighbors for different resonance modes. Therefore, we concentrate on just one mode 共RF2兲 and observe the interactions involving the horizontal elements. To quantify the experimental coupling strength we compare the experimental data with the theory of uncoupled horizontal elements. The horizontal elements can be approximated to be rectangular prisms with finite dimensions of a, b, and c. For an isolated single element the demagnetizing factor in the direction of applied field 共which is along c in this case兲 is given as,34

␲Nx =

From the 2D profiles shown as insets in Fig. 3共c兲, we first draw a qualitative-level conclusion: the oscillations in the vertical and horizontal elements are indeed dynamically coupled. For each peak in Fig. 3共c兲, we observe noticeable amplitude of precession in both types of elements. A more quantitative result can be found from Figs. 3共e兲 and 3共g兲. 214422-6

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共a2 + b2兲3/2 + 共b2 + c2兲3/2 + 共c2 + a2兲3/2 , 3abc

共2兲

where a = 400 nm, b = 15 nm, and c = 1000 nm. For ␪ = 0°, the demagnetizing factors are calculated to be: along the applied field c : Nx⬘ = 0.0643733, along the in-plane size a : N⬘y = 0.16604 and along the out-of-plane size b : Nz⬘ = 0.7695867. Next we fit the experimental data for RF2 共␪ = 0°兲 using the same Kittel’s equation. The straight line in Fig. 4共b兲 is the fitted response for RF2. The extracted values of demagnetizing factors for this mode are Nxef f = 0.036299, Nyef f = 0.127149, and Nzef f = 0.836552. Clearly, Nxef f 共coupled elements兲 ⬍ Nx⬘ 共single element兲, Nyef f 共coupled el共single element兲 and Nzef f ements兲 ⬍N⬘y 共coupled elements兲 ⬎ Nz⬘ 共single element兲. All the extracted components are between N for a single element and a continuous film 关N = 共0 , 0 , 1兲兴, and for a continuous film the coupling is the largest possible. This quantifies the dynamic dipole coupling of the horizontally oriented elements on the array as very efficient, as it reduces Nx almost by a factor of 2 and Ny by a quarter. Note that the demagnetization factors we have extracted above are effective ones and characterize the strength of collective dipole field for a particular mode of the whole array 关see e.g., Eq. 共10兲 in Ref. 11 and discussion below this equation for the definition of the effective dynamic demagnetizing factors for collective modes兴. These factors include all possible dipole interactions on the array, including coupling between structure periods. In contrast to the 1D case,11 our 2D structure also includes contribution from the collective static demagnetizing field. Once the essentially collective character of dynamics in our experiment has been confirmed, in different ways, we proceed with the analysis of the collective modes. From Fig. 3共e兲, it can be seen that the simulated precession amplitude has the same sign for both vertical and horizontal elements. Furthermore, an identical profile can be found along the respective cross section of the lower horizontal element 共not shown兲. This suggests that RF2 represents the acoustic oscillation for the array, since precession in both types of elements is in phase 共actually there is a phase difference of 45° between the elements, however this is much smaller than 180°, therefore this mode is certainly acoustic兲. The detailed structure of the acoustic mode can be explained based on the mode structure for uncoupled elements. Cross sections 1 关Fig. 3共e兲兴 and 2 关Fig. 3共f兲兴 show that the acoustic mode is a combination of the fundamental modes for both 共uncoupled兲 horizontal elements and the third longitudinal mode for the uncoupled vertical element. There is also a considerable admixture of the fundamental mode for the uncoupled vertical element, since the mean value of the amplitude 共the net magnetic moment兲 for the cross section 2 关Fig. 3共f兲兴, does not vanish, but is finite and positive. It was previously theoretically shown35 and confirmed experimentally17 that dipole coupling of elements is considerably increased with magnetic “bridges” or “mediators” in which magnetization motion is forced to act as conductors of dynamic dipole field which link areas of resonant precession. In the above-mentioned cases, the gaps between resonating

Ni80Fe20 stripes were filled in with a different magnetic material 共Co兲. The increase in the dipole coupling was seen as considerable increase in the frequency width for the lowest magnonic band in Ref. 17 with respect to Ni80Fe20 stripes separated by low microwave magnetic permittivity spacers 共air gaps兲.15 From Fig. 3共f兲, it is observed that the vertical elements play the role of such bridges. As stated above, the distribution of dynamic magnetization for RF2 in the vertical elements does not represent a pure mode rather than a superposition of eigenmodes for the uncoupled stripes in such combination that the phase of precession is the same on both sides of the gaps which separate the horizontal and the vertical elements. This implies that the motion of magnetization for the vertical elements is forced and these elements are bridges for dipole coupling of the horizontal elements. A similar RF1 mode profile for cross section 3 is shown in Fig. 3共g兲. This mode is a combination of the fundamental mode for an uncoupled vertical element and of the seventh longitudinal mode for an uncoupled horizontal element 共seven half wavelengths of a standing wave across the horizontal element are seen in the figure兲. Again, there is also a small mixture of the fundamental mode of an uncoupled horizontal element, such that the net magnetic moment for the horizontal element does not vanish but is finite and negative. This negative value of the net magnetic moment for the horizontal element and a positive one for the vertical element suggests that this mode represents the optical oscillation for the array. Significantly smaller amplitude of dynamic magnetization in the horizontal elements of Fig. 3共g兲, than the amplitude of the forced motion in the vertical elements seen in Fig. 3共f兲, implies that the oscillations in the parallel columns of vertical elements are less coupled via the horizontal bridges than the oscillations in the parallel rows of the horizontal elements. Furthermore, note that the FMR response is proportional to the net dynamic magnetic moment of the structure. Therefore, responses of optical type are generally characterized by smaller or even vanishing amplitude with respect to the acoustic mode. If one compares the peak amplitudes in Figs. 3共c兲 and 3共d兲关and also in the experimental traces of Figs. 3共a兲 and 3共b兲兴, in all cases one finds that the amplitude of the smaller peak is smaller than that of the larger one by more than 2 times. The latter figure 共two times兲 is expected from the consideration of the amount of material contained in the vertical elements and in assumption of negligible coupling. Any smaller value may suggest that the magnitude of the net magnetic moment is affected by phase coherence across the whole array in the form of the antiphase response of the elements in which the precession is forced. The phase coherence is obviously ensured by dynamic dipole coupling between the horizontal and the vertical elements. This provides another experimental evidence for its strength. From Fig. 3共b兲, we observed that the peak amplitudes swap when the sample is rotated by 90°: the lower-frequency peak now has smaller amplitude. Based on our finding above that the mode which is more localized in the horizontally oriented elements should be characterized by the largest amplitude, we infer that the lower-frequency mode in Fig. 3共b兲 is also a response with dominating contribution from the horizontally oriented elements. This is confirmed by our numerical simulations 关Fig. 3共d兲兴.

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This behavior can be explained by a swap of the underlying type of the standing wave formed across the area of components and variation in the static demagnetization conditions upon rotation. For ␪ = 0°, the larger peak is characterized by a Damon-Eshbach 共DE兲共Ref. 36兲 type standingwave oscillation in the horizontal elements. 共As these elements are magnetized along the easy axis we will call this configuration “easy-axis DE.”兲 The smaller peak is characterized by the backward volume wave-type standing-wave 共BVMSW兲 oscillation in the vertical elements. It is the fundamental property36 that DE wave has frequency larger than BVMSW. In our case the difference in frequencies is magnified by magnetization of the vertical elements along the hard axis 共hard-axis BVMSW兲. For ␪ = 90°, the horizontal elements are characterized by the hard-axis BVMSW standing wave configuration and the vertical ones by the easy-axis DE one. This implies that the lower-frequency peak should now have larger amplitude. This prediction is in agreement with Figs. 3共b兲 and 3共d兲. From the analysis above, it can be predicted that the frequency of the mode localized in the horizontal elements will gradually shift down with increase in ␪, and the mode localized in the vertical elements will gradually shift upwards in frequency. This is confirmed by our angle resolved measurements which show a uniaxial anisotropy for the array 共Fig. 5兲. For a system with dipole-coupled elements, repulsion of the “dispersion branches” and formation of a prohibited zone 共gap兲 in angle-resolved measurements is expected, when the two branches come close to each other. This is similar to the mode repulsion seen in the field-resolved data on dipolecoupled stripes.37 The absence of a well-resolved gap for our structure 共Fig. 6兲 suggests that the experimental dipole coupling between the vertical and the horizontal elements weakens for angles around 40°, due to the complicated character of the underlying spin wave dispersion in continuous films for the respective direction of magnetization.36

VI. CONCLUSION

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In this work, we have investigated the ground state and dynamic spectra of a periodic array consisting of noncollinearly arranged magnetic elements. We found strong static and dynamic dipole coupling of components having easy axes oriented in different directions. From FMR mode analysis we inferred that in contrast to previous studies of 2D arrays with higher symmetry, our geometry is likely to circumvent the drawback of strong anisotropy of dipole coupling caused by the static demagnetization. This may have potential application in miniature microwave directional couplers. The strength of the dipole coupling between the horizontal and vertical elements has been determined in the form of effective demagnetizing factors. The simulated FMR mode profiles demonstrate the predominant localization of oscillations in horizontal elements for the high-amplitude mode and in vertical elements for the lower-amplitude mode. For the acoustic mode the phase coherence across the whole array is efficiently mediated by the forced magnetization motion in the horizontal stripes. Mediation by the forced magnetization motion for the optic mode is less efficient which is seen as smaller amplitude of precession in the horizontal elements. Angle-resolved FMR measurements demonstrated a twofold configurational anisotropy for the array. The relative frequency positions for the modes depend on the direction of the magnetization. Upon rotating the applied field by 90°, the relative positions of the modes swap.

ACKNOWLEDGMENTS

This work was supported by National Research Foundation, Singapore under Grant No. NRF-G-CRP 2007-05, and the Australian Research Council. We would like to thank Dr. N. Singh for template fabrication.

6 A.

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