High-symmetry magnonic modes in antidot lattices magnetized

PHYSICAL REVIEW B 85, 104414 (2012)

High-symmetry magnonic modes in antidot lattices magnetized perpendicular to the lattice plane R. Bali,1 M. Kostylev,1,* D. Tripathy,2,† A. O. Adeyeye,2 and S. Samarin1 1

2

School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia Information Storage Materials Laboratory, Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 (Received 10 November 2011; published 22 March 2012)

Microwave magnetization dynamics on two-dimensional periodic arrays of nanoscale magnetic antidots with magnetic field applied perpendicular to the lattice plane have been studied using ferromagnetic resonance spectroscopy. Linear dependence of the resonant mode frequency on the applied field was observed experimentally. Theoretical calculations show that this linear dependence originates from the high symmetry imposed by applying the field perpendicular to the plane of the antidot lattice. The calculated mode profiles exhibit a fourfold symmetry in contrast to the twofold symmetry typical for the in-plane magnetization direction. From the calculated Bloch wave dispersion the group velocities along the [10] and [11] directions and close to the center of the first Brillouin zone are found to be the same, which demonstrates a very high degree of isotropy of magnonic modes for the center of this zone in this case. Perpendicular standing spin-wave modes due to microwave shielding were also observed on the antidot lattices. DOI: 10.1103/PhysRevB.85.104414

PACS number(s): 75.75.−c, 75.30.Ds, 75.78.−n, 78.67.Pt

I. INTRODUCTION

Artificial periodic lattices fabricated from magnetic thin films known as magnonic crystals (MCs) can act as media for propagating and standing spin-wave (SW) modes.1–5 Two types of two-dimensional (2D) MCs with fourfold symmetry patterned from a continuous film have attracted attention. The first type includes 2D-periodic arrays of magnetic islands (or nanodots).6–15 This material is discontinuous, and collective excitations are propagated via long range dipolar interactions. This type of coupling is inefficient due to flux lines having to traverse through air, and as a result the frequency bands for nanodot-based MCs are not wide.14,15 On the other hand, antidot nanostructures,16–39 which are 2D-periodic arrays of holes fabricated from a continuous film, maintain a continuity of the magnetic material. This enables dipole coupling through the magnetic material and avoiding flux traversal through low permittivity regions. The coupling takes the form of a SW travelling in the continuous space between the holes, allowing wider frequency bands and higher group velocities than for nanodots.25 For any given MC under an applied field He , the SW dispersion depends on the magnitude of He and its direction with respect to the main crystallographic axes of the MC. Application of He in the lattice plane induces a strong twofold contribution to the anisotropy of SW dispersion14 : collective SWs acquire considerable group velocities while propagating perpendicular to He but are practically immobile when their Bloch wave vector is directed along He . This effect implies that an MC effectively breaks into parallel wave guiding channels in the direction perpendicular to the field. For the antidot MC with fourfold symmetry there are two contributions to this effect of symmetry reduction. The first one is the underlying twofold symmetry of SW dispersion in in-plane (IP) magnetized continuous films. The second one is the static and dynamic demagnetizing fields emerging from the hole edges, which also possess twofold symmetry. As a result the canalized SW modes propagate along the edges of the rectangular antidot lattice and perpendicular to He .23–25 1098-0121/2012/85(10)/104414(12)

In recent years there has been a considerable interest in plane materials magnetized perpendicular to their plane (PP). This magnetization configuration is important for perpendicular magnetic recording. It is also advantageous for spin transfer torque nano-oscillator arrays.40–42 Saturating an MC with fourfold symmetry perpendicular to its plane should result in the symmetry of the ground sate of magnetization which is the closest one to the fourfold one. This should translate into the same type of symmetry of supported SW modes.43 Previously, the dynamic response of antidot arrays in the PP configuration was studied using cavity ferromagnetic resonance (FMR) and thus for limited frequencies.44,45 In contrast to the IP magnetization (see, e.g., Ref. 46) where the reduced symmetry results in a number of excited FMR modes with similar amplitudes, the FMR traces shown in this paper display just one prominent mode. As we demonstrate below, this is a signature of the higher degree of symmetry of dynamic excitations for the PP configuration with respect to the IP case. In the present work, by using broadband field modulated FMR, we measure the SW response for antidot structures for the center of the first Brillouin zone and for the static magnetic field applied perpendicular to the lattice plane. We characterize materials’ behavior and identify all resonant modes observed experimentally using the theory we develop. We also calculate the dispersion of Bloch SW modes across the whole width of the first Brillouin zone. The paper is organized as follows: Section II describes the sample fabrication and FMR experiments. Section III describes the experimental results. In Sec. IV we give details of the developed theory and apply it to the calculation of magnonic mode dispersion across the whole width of the first Brillouin zone for this artificial crystal. Section V compares the theory to the experimental results to identify the origin of the specific resonant modes. The conclusions are presented in Sec. VI. II. EXPERIMENT

Two 4 × 4 mm2 nanoscale antidot arrays with fourfold symmetry were fabricated on commercially available Si(001)

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PHYSICAL REVIEW B 85, 104414 (2012)

FIG. 1. (a) SEM image of the fabricated antidot array and (b) the geometry of the problem for the theoretical treatment.

substrates using deep ultraviolet (DUV) lithography at 248 nm exposing wavelength. The Si(001) substrates were first coated with a 60-nm-thick antireflective layer followed by 480 nm of positive DUV photoresist. A Nikon lithographic scanner with KrF excimer laser radiation was used for exposure, leading to the formation of resist dots. This was followed by e-beam deposition of Ni80 Fe20 films of thickness L = 40 nm and 80 nm at room temperature. The magnetic film was then removed from the unexposed areas by ultrasonic assisted liftoff in OK73 resist thinner. Lift-off completion was determined by the color contrast of the patterned film and confirmed by inspection under a scanning electron microscope (SEM). Details of the fabrication process are described in Ref. 47. An SEM image of one of the samples is shown in Fig. 1(a). For both samples, the hole diameter was 165 nm, and the distance between the centers of the nearest holes (“pitch”) was a = 415 nm. Continuous films prepared in the same deposition run were studied as reference samples. The measurements have been performed by using the Field Modulated Broadband FMR setup. A microstrip antenna was used to apply a microwave magnetic field to the antidot array. The microwave signal was applied to the microstrip from a high-stability c.w. microwave generator at selected frequencies between 2 and 18 GHz, with frequency resolution better

than 3 kHz. The 0.3-mm-wide microstrip was parallel and in close contact to the sample plane, and the assembly was placed between the poles of an electromagnet. The field was applied perpendicular to the sample plane and swept between 0 and 17 kOe for each selected frequency. A small coil was placed in the vicinity of the sample to apply a modulating field of ±4 Oe at a fixed frequency of 220 Hz. The signal applied to the coil was also used as the reference for the Stanford SR850 lock-in amplifier. The microwave magnetic field of the microstrip antenna excited SWs in the antidot array, which was seen as a static-field dependence of absorption of microwave power by the microstrip line. To observe the power absorption, the microwave signal from the output of the microstrip was rectified using a microwave diode. Throughout the experiment, the rectified voltage was kept at ∼150 mV, which ensured the maximum of sensitivity for the setup as well as linear regimes for operation of the microwave diode and of excitation of the magnetization dynamics in the sample. The rectified signal was fed to the lock-in amplifier, and the output voltage of the lock-in was registered as the proxy to the microwave power absorbed due to excitation of the standing SWs in the sample. The output signal observed using this field modulation technique represents the first derivative (differential absorption) of the original Lorentzian absorption curve with respect to the parameter varied in the experiment (in this case the applied field was varied). Zero-crossings of the applied-field dependence of the differential absorption yield the resonant modes. In addition, the spectra were also measured in a Varian-4 microwave cavity, which has a resonant frequency of 9.529 GHz. Measurements in the cavity are restricted to its resonant frequency, however, asymmetry effects from the microstrip due to exposure of only one side of the sample to the microwave field are avoided. Hysteresis loops of the antidot arrays were measured at 300 K with the field applied perpendicular to the film plane R ) magnetometer. using a SQUID (Quantum Design MPMS This was to confirm that the resonant modes measured in FMR were measured mostly above the saturating field for the samples and also in order to relate peculiarities observed in the FMR data to the particular points in the hysteresis curve. III. RESULTS

Figure 2 shows typical FMR absorption spectra for antidots and the reference continuous films. Figure 2(a) shows the spectrum for the 40 nm continuous film compared with the antidot array of the same thickness, measured at 16 GHz. [Figure 2(d) is a blow-up view of the lower-field part of Fig. 2(a).] The spectrum for the continuous film is characterized by a strong signal of the fundamental thickness mode (FF) at 15.680 kOe and by the presence of just one higher-order standing SW peak (FH1) at 13.480 kOe. In comparison, the fundamental mode for the antidot array (MF) is shifted to lower field by 0.9 kOe. The antidot spectrum also shows at least four higherorder small-amplitude modes at 13.760, 12.740, 12.360, and 12.040 kOe. The mode at 12.360 kOe (H1) has the largest amplitude from these small-amplitude modes. Figure 2(b) shows similar data for the 80-nm continuous film and the antidot array. [Figure 2(e) is a blown-up view of

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FIG. 2. (Color online) (a) and (b) Microstrip broadband FMR absorption traces for 16 GHz. (a) Nanostructure and film thicknesses are 40 nm. (b) Thicknesses are 80 nm. Thick solid lines: experiment, antidots; thin solid lines: experiment, reference continuous films. Thin dashed lines: theory for antidots. The theoretical curves are shifted down in the field by 380 Oe in (a) and by 650 Oe in (b). (c) Experimental absorption traces for the 80-nm-thick antidot nanostructure. Thin solid line: microstrip FMR at 9.5 GHz; thick solid line: cavity FMR at 9.529 GHz. (d) and (e) Blown-up views of the lower-field parts of (a) and (b), respectively.

the lower-field part of Fig. 2(b).] The spectrum of the reference continuous film shows two pronounced peaks at 15.953 kOe (FF) and at 15.026 kOe (FH1), with the peak at 15.953 kOe identified as the fundamental thickness mode. There is also a small-amplitude peak FH2 seen at 13.964 kOe. Similar to the reference film the fundamental (MF) peak for the antidot array at 14.442 kOe is closely followed by a high-amplitude peak H1 at 13.950 kOe. Furthermore, the antidot array with

80-nm thickness shows well-resolved higher-order lowamplitude peaks at 13.280, 12.600, and 11.820 kOe. An important observation in Fig. 2(b) is the second largeamplitude peak FH1 for the continuous reference film. Similar behavior was recently observed in Ref. 48, where it was explained as efficient excitation of a higher-order thickness mode due to asymmetry of the total microwave driving field. In the microstrip FMR this asymmetry is caused by microwave

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FIG. 3. (Color online) (a) Ferromagnetic resonance frequency vs applied field for the 40-nm nanostructure. (c) A section of the out-of-plane hysteresis loop for the same range of applied fields for the 40-nm-thick sample. (b) and (d) The same, but for 80-nm-thick sample. (c) and (d) Insets: Respective complete hysteresis loops. In (a) and (b) dots are experiment; lines: theory. Solid lines are the calculation for the main mode family using Eq. (9). The theoretical curves are shifted down in the field by 380 Oe in (a) and by 650 Oe in (b). The dashed lines in (a) and (b) represent linear fits to the experimental data with the slope given by the gyromagnetic coefficient extracted from the measurements on the respective reference films. The lines in (c) and (d) are given as guides for the eye.

shielding of the microwave field of the microstrip by eddy currents induced in the sample.48–50 One may suppose that a similar effect may also exist for nanostructured materials. As shown in Refs. 48 and 50, the asymmetry of the driving field due to the microstrip geometry can be removed in a cavity measurement. This allows identifying thickness-non-uniform resonance modes by comparing microstrip and cavity data. To check this idea and to find out if similar thickness-non-uniform resonance modes can be seen for the nanostructures, we performed a test measurement with a microwave cavity. The result of characterization of the 80-nm-thick nanostructure with the cavity (9.529 GHz) is shown in Fig. 2(c), where it is compared with the microstrip data taken at 9.5 GHz. It is seen that in the trace taken with the cavity, the amplitude of the second-highest peak (located at 11.800 kOe in the cavity data) is significantly smaller than that of the second-largest peak (H1, located at 11.720 kOe) seen in the trace taken with the microstrip. The resonant modes for the observed frequency and applied field range taken using the microstrip transducer are plotted in Fig. 3. Figures 3(a) and 3(b) demonstrate the broadband FMR data for the antidot arrays and continuous films, and Figs. 3(c) and 3(d) show sections of the respective hysteresis loops, which correspond to the same range of the applied fields as for

the FMR data. The overall behavior of the hysteresis curves is typical for magnetization of magnetic material along a hard magnetization direction [see insets to Figs. 3(c) and 3(d)]. The main observation from Fig. 3 is that the spectra are multimodal and that the frequency vs field dependencies for both nanostructure thicknesses is linear in the range where the nanostructures are well saturated magnetically (to the right from the vertical dashed lines in Fig. 3). They start to deviate from the straight lines when the samples start to be released from saturation (to the left from the vertical dashed lines). IV. THEORY

In this section we present the theory we developed and discuss those results of our numerical calculations, which have no direct relevance to the discussion of the experiment described in the previous section. The calculation results pertinent to the experimental data are discussed in the next section. The derivation of our theory is given in the Appendix. The system of equations (A9) we obtain allows one to numerically calculate dispersion of the collective SWs on the antidot arrays with fourfold symmetry for the PP magnetization configuration. The system is obtained in a way such that it is suitable

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for interpretation of both present applied-field resolved FMR data and possible future Brillouin light scattering (BLS)5 experiments in which eigenfrequency of thermal excitations is measured as a function of the transferred wave vector for a given value of the applied field. By shifting the respective terms from the left-hand side to the right-hand side, the system of equations (A9) can be easily reformulated as an eigenvalue problem in which either the resonant frequency ω (suitable for BLS) or the resonant field H0 (suitable for FMR) plays the role of an eigenvalue. The eigenvalue-eigenvector problem (A9) allows numerical solutions, provided the infinite series on the right-hand side of the system of equations are truncated by some finite order. R We used MathCAD software to program the solution of the eigenvalue-eigenvector problem. Truncation of the series in Eqs. (A4) and (A9) after the tenth harmonic of the periodic distribution of dynamic magnetization (|n|,|j |,|n |,|j  |  10) results in convergence of all physically important eigenvalues and eigenvectors. In contrast to standard micromagnetic software such as OOMMF,51 NMAG,52 or similar, this customized numerical code is quite quick: it takes half a minute to calculate the whole set of eigenvectors and eigenvalues for a given set of input parameters (e.g., the whole set of resonance fields for a given value of the driving frequency). One may expect that PP magnetization of low-aspect ratio nano-patterned materials results in significantly more uniform distribution of directions of the vector of static magnetization and in a more uniform internal magnetic field. The precise degree of uniformity of these parameters determines the accuracy of our theory, since the theory relies on the often used approximation53 of the uniform static magnetization Eq. (A2). For the IP magnetized materials this approximation often fails to provide sufficient accuracy in order to explain experimental data, and inclusion of the non-uniform magnetization ground state in the theory is necessary (see, e.g., Ref. 54). Therefore before proceeding to the numerical calculation using the developed theory, the accuracy of the approximation of the uniform ground state of magnetization was checked with a direct NMAG simulation of the magnetic ground state for the sample. In the simulation made for the middle of the experimentally measured field range (15 kOe), the static magnetization vector is seen to be perfectly aligned along the normal to the array plane everywhere except the very edges of the holes, where the z-component of magnetization 4π Ms is smaller by just 0.2% (20 G) than the saturation magnetization value for our samples (about 10700 G). Therefore one may expect that the error in the calculation of the resonance field resulting from this approximation should not exceed 20 Oe, which should be enough to adequately explain our experimental FMR data in which the accuracy of setting the applied field is of the same order of magnitude (40 Oe). Two types of numerical calculations were performed. The first one is related to the FMR experiment in Sec. III. To fit the experimental data, calculations using Eq. (A9) are performed for the center of the first Brilloiun zone (kB = 0). The obtained eigenvalues and eigenvectors are analyzed in two ways. First, we find the eigenvalues that correspond to the eigenvectors having the largest value of the element |m0,0 |. This element is the zeroth spatial Fourier harmonic of the areal profile of dynamic magnetization, which gives the spatially uniform

part of this distribution. Since the FMR is a response to a spatially uniform microwave magnetic field and is given by the overlap integral of the driving field with the mode profile, this analysis identifies the modes which will show the largest FMR responses and thus represents the mode selection rule for the FMR experiment. More information may be obtained by calculating magnetization amplitudes driven by the uniform microwave magnetic field. Proxies to these amplitudes are given by the expression as follows: m(He ) =

∞ 

m0,0 mty0,0 /(He − Hej + iH ),

(1)

j =−∞

where H is the resonance linewidth, Hej are resonant fields of the modes which are obtained as the matrix eigenvalues, and mty0,0 is the element of the respective left-hand eigenvector of Eq. (A9). [It can be calculated as the respective element of the matrix which is inverse to the matrix which has eigenvectors of Eq. (A9) as its columns.] The field-modulated FMR response R(He ) represents differential absorption and scales as the first derivative of Eq. (1) with respect to He . The latter can be calculated analytically:  ∞   t 2 R(He ) = −Im my0,0 my0,0 /(He −Hej +iH ) . (2) j =−∞

Examples of calculation using Eq. (2) are shown in Figs. 2(a)–2(d). These results will be discussed in the next section. The other type of calculation we perform is the one of Bloch wave dispersion across the whole width of the first Brillouin zone for a number of principal directions of the zone. The results are shown in Fig. 4. Figure 4(a) demonstrates the dispersion along the diagonal of the unit cell [the axis [11] in Fig. 1(b)]. A number of wide magnonic frequency bands are seen in which propagation of collective SWs is allowed that are separated by relatively narrow prohibited zones (magnonic gaps). The fact that the frequency bands are wide is explained by the important peculiarity of the antidot geometry: the magnetic material is continuous in each direction in the sense that any two points inside the magnetic material can be connected by a continuous path lying completely inside the magnetic material. This conclusion is well supported by the experiment in Ref. 55, which showed a significant increase in the width of a magnonic band upon filling in the air gaps between Permalloy stripes with a different highpermeability magnetic material. Figure 4(b) demonstrates the blown-up picture of the two lowest magnonic frequency bands for two principal directions on the crystal lattice [see Fig. 1(b)]: along the edge of the square [01] and along its diagonal [11]. The magnonic gap is larger for the Bloch waves travelling along [01] direction. Figure 5 shows the modal profiles of dynamic magnetization for some particular collective modes from Fig. 4(a) for kB = 0. Figure 6 shows the widths of the first magnonic band and the first magnonic gap as a function of the hole diameter for a given value of the pitch (415 nm). As expected, the band becomes narrower and the gap wider with an increase in hole diameter.

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FIG. 4. (Color online) Calculated Bloch wave dispersion of collective modes of the main family for the 40-nm-thick sample. The static magnetic field is 15 kOe and is applied perpendicular to the film surface. (a) Dispersion for a number of lowest-order modes for Bloch wave propagation along [11] crystallographic axis [see Fig. 1(b) for the definition of the crystallographic directions]. MF and M1 denote branches for the respective modes seen in the FMR experiment. (Mode M2 is located at 25 GHz+ and therefore is not seen in this graph.) (b) Two lowest-frequency modes separately for propagation in [11] (dashed line) and [10] (solid line) directions. Note different periodicity of dispersion for the two directions, √ which originates from a larger width of the first Brillouin zone (2 2π/a) along [11] direction than along [10] axis (2π/a).

Interestingly, for any diameter, the band is larger for √ [11] direction, although when the hole diameter is larger than 2a any straight line along [11] direction will cross the holes. The latter implies that SW propagation in this direction for large R is due to wave scattering from the hole edges and to dipole coupling across the holes. Presumably this should be a less efficient process than propagation along a straight line which always lies inside the magnetic material. The [01] direction is the one along which the material remains always connected by straight chains of magnetic bridges for any values of R. Consequently, smaller gaps and larger bands are expected in this direction; however, this is contradicted by the simulation. This suggests that the physics is more complicated than just this simple argument that considers material continuity and assumption that scattering from the hole edges resulting in

FIG. 5. (Color online) Calculated modal profiles for modes MF, M1, M2, and M3 from Figs. 2 and 3. These profiles have been calculated for the 40 nm-thick antidot lattice. Profiles for the 80 nm-thick antidot lattice are qualitatively the same. The lowermost panel shows the cross-sections of the amplitude distribution along the lines shown in MF modal profile (uppermost panel): solid line shows the cross-section along the centers of the holes, and dotted line is the cross-section running through the middle distances between the holes. The amplitudes corresponding to the color scheme for the MF also apply to all modal profiles.

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FIG. 6. (Color online) Frequency widths of the first magnonic band (thick lines) and the first magnonic gap (thin lines) for the 40-nm-thick nanostructure. Solid lines: Bloch wave propagates along [10] axis; dashed line: wave propagates along [11] axis. Applied field is 15 kOe.

formation of a plane Bloch wave in [11] direction is more efficient than expected. V. DISCUSSION

In this section we discuss the results of the experimental FMR investigations and of the respective aspects of the theory. A. Resonant modes in the 40-nm-thick antidot lattice

Let us start with the 40-nm-thick antidot array [Fig. 3(a)]. In this section we will identify the resonances observed for it by comparing the experimental data with our theoretical calculation. A success in identification of all experimentally observed resonances will imply that the theory we suggest is valid. The linear slope of the ω(He )-dependence for the 40-nmthick reference continuous film allows the extraction of the gyromagnetic coefficient γ . The resonance frequency for the fundamental mode (uniform precession) of FMR with the static field applied perpendicular to the film plane is given by the Kittel formula56 ω = γ (He − 4π Ms ).

(3)

In contrast to the IP case the ω(He )-dependence is linear and is given by the gyromagnetic coefficient. This is a very useful property of the PP configuration, since the value for γ /(2π ) can be obtained independently from extracting 4π Ms . From the least-square fit of the experimental FMR data for the reference 40-nm-thick film with a straight line we obtain 2.9113 MHz/Oe, which corresponds to the value of g-factor of 2.082. In the next step from the point of intersection of the obtained straight line with the ω axis we find the value of the saturation magnetization for the sample 4π Ms = 10184G. We used these magnetic parameters as input for the numerical calculations with the model (A9) developed in Sec. III. The calculated ω(He )-dependences for the sample well above the saturating field (He > 12 kOe) represent straight

lines, which agree well with the experiment. The lowestorder resonance modes possess the same slope of 2.9113 MHz/Oe as in the experiment. It is also the same as for the reference continuous film. Thus, theory confirms the experimental finding that the slope should be the same as for the continuous film and should be given by the gyromagnetic coefficient. The calculation predicts resonant fields larger than in the experiment by 380 Oe. When all the calculated fields are shifted by this value, a good agreement with the experiment for three higher-field theoretical modes (MF, M1, and M2) is obtained (Fig. 3) for the entire frequency range. The same difference in predicted and observed peak positions for three modes simultaneously suggests that the theory is valid. This difference may be an indication of some additional contribution to the resonant fields which was not originally considered in the theory, for instance, the normal uniaxial anisotropy induced in the magnetic material by the process of nano-patterning.57 Note that two modes seen in the experiment [H1 and H2 in Fig. 2(a)] do not have their counterparts in the theory. One of these modes (H1) has the second-largest intensity. A higher-order mode FH1 [Figs. 2(a) and 2(d)] is seen in the absorption spectrum for the reference film. Based on its field position, this extra mode may be identified as a higher-order exchange perpendicular standing spin-wave (PSSW) across the thickness of the reference film.58 Such modes can be seen in the microstrip-based broadband FMR due to microwave shielding effect48 and also due to partial surface spin pinning in both microstrip and cavity FMR. When we shift the absorption trace for the reference film toward lower fields such that the fundamental modes for the antidot structure and the film overlap [Fig. 2(a)], the mode FH1 for the reference film appears to be located close to the additional modes H1 and H2 for the nanostructure. Based on this observation, the modes H1 and H2 of the nanostructure are attributed to the PSSW family, which originates from the mode FH1 of the reference film. Further analysis of the thickness-non-uniform dynamics possibly originating from surface spin pinning is beyond the scope of this paper. For completeness of the study, in Fig. 3(a) we just fit ω(He )-dependencies for H1 and H2 with a straight line with slope equal to the value of γ extracted from the measurement on the reference film. Agreement between theory and experiment suggests that MF, M1, and M2 may belong to the main mode family. Identification of the individual modes from this family can be done based on the calculated modal profiles for the theoretical modes whose resonant fields are in agreement with the experiment. We find that the modes MF, M1, and M2 correspond to the simplest distributions of dynamic magnetization across the area of the unit cell (Fig. 5). The mode MF is identified as fundamental, i.e., the mode having the most uniform distribution of dynamic magnetization across the area of the unit cell. Successful identification of all the modes seen in the simplest case of the thinner film proves validity of the theory we developed. Now we proceed to discussion of the most general properties of the FMR dynamics on antidots and also to discussing the more complicated case of the 80-nm-thick nanostructure based on this theory.

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B. Origin of the linear ω(He )-dependence

The general property we want to discuss is the origin of the linear slope of ω(He )-dependence seen in the experiment. As seen from Fig. 3(a), the theory also shows the same linear slope for most of the field range except for fields close to the saturating field for the nanostructure (about 11 kOe). Note that both ω and H0 enter the left-hand side of Eq. (A9). In other words they are located on the diagonal blocks (n = n , j = j  ) of the block matrix (A9). Equation (A9) shows that if the diagonal matrix on the left-hand side and the more dense (band) matrix on the right-hand side had the same eigenvectors, the dependence of the frequency on the field would be linear. The eigenvectors of the left-hand side matrix are mx,n,j = ±imy,n,j .

(4)

If the eigenvectors of the matrix on the right-hand side were the same, this would explain the linear behavior. However, a simple analysis of the right-hand part of Eq. (4) demonstrates that the eigenvectors of that side should have a more complicated form than (4). On the other hand, the theory clearly demonstrates that the linear behavior is asymptotic: one sees noticeable deviation of the theoretical curves from straight lines at smaller fields in Fig. 3. Hence, the eigenvectors on the left-hand side are allowed to obey (4) approximately or asymptotically for the fields >12 kOe. To find the appropriate approximation, we may rely on the physical basics of the FMR method. Expression (4) implies that for all spatial harmonics of the dynamic magnetization, precession should be perfectly circular as it is for the continuous films. From the derivation of the Kittel formula for the PP configuration, it is seen that the reason for the linear character of ω(He )-dependence for continuous films is the circular precession of magnetization. This highest symmetry of precession originates from the highest magnetic symmetry which is achieved when a continuous isotropic film is magnetized perpendicular to its plane.59 Thus, the reason for the linear behavior of ω(He ) for the antidot nanostructure is a similar level of degree of magnetic isotropy. The previous analysis implies that the degree of ellipticity of magnetization precession ε = |mx /my | can serve as an indicator of anisotropy of SWs on the antidot lattice. The ellipticity of precession for the fundamental mode is obtained from its profile (Fig. 5) for the center of the first Brillouin zone. The precession is perfectly circular for point C of intersection of straight lines along the directions [11] and [−1, −1], which connect the centers of neighboring holes [Fig. 1(b)]. This suggests the complete isotropy for this point. The ellipticity for the middle of the distance between the centers of the neighboring holes along [01] and [10] axes [points A and B in Fig. 1(b)] does not vanish, however, the calculation shows that ε(A) = 1/ε(B) . This implies that the deformation of the precession cone is the same for these two points, but the long axes of ellipses are oriented in perpendicular directions to each other in these two cases. This is consistent with the direction of the dynamic demagnetizing field the hole edges induce. Such symmetry suggests that the ellipticity of precession may be circular on average. To check this idea we calculate the ratio of values of mx and my averaged over the unit cell area. These average values are given by the elements

of the eigenvector of Eq. (A9), which correspond to the zeroth harmonic of the Fourier transform of the dynamic magnetization m0,0 . We calculate |mx,0,0 /my,0,0 | and find that this ratio is perfectly 1 for the fundamental mode and for all the other modes from Fig. 5. Thus, the precession is circular on average, and the valid approximation for the eigenvalues on the right-hand side of Eq. (A9) is given by the condition of the circular precession of magnetization on average. The FMR is by definition a dynamic response averaged over a microscopic area of the sample; therefore, the local fourfold symmetry inside the unit cell does not significantly influence the results. From the previous analysis it can be concluded that the reason for the linear dependence of the resonance frequency on the applied field is the circular precession of magnetization on average across the area of the unit cell. C. Resonant modes in the 80-nm-thick antidot lattice

A key observation for this sample is the presence of two intense absorption peaks [Fig. 2(b)]. Fitting the experimental data with the theory using the material parameters extracted from the measurement on the 80-nm-thick reference film and the value of induced normal uniaxial anisotropy of 650 Oe results in a good overlap for four modes [Figs. 3(b) and 3(e)], which are identified in this way as modes MF, M1–M3. Here one has to note lesser accuracy of the small-aspect ratio approximation used to derive Eq. (A9) in the case of the thicker nanostructure [see discussion before Eq. (A4) in the Appendix]. The overlap of the experimental and theoretical curves for three modes simultaneously suggests that the accuracy of this approximation remains sufficient for the sizes of this sample. Similar to the 40-nm-thick case, two modes remain unexplained by the theory. Among them is the largest intensity mode [denoted as H1 in Fig. 2(b)]. Our cavity FMR data [Sec. III and Fig. 2(c)] provide strong experimental evidence that the additional modes (H1 and H2) also originate from the higher-order PSSW of the reference film. But in contrast to the thin 40-nm sample, where either surface pinning or microwave shielding48 may be responsible for the smallamplitude response of these modes,60 the very large amplitude for H1 in the microstrip experiment has no explanation other than the microwave shielding effect. As found in Sec. III, the amplitude of H1 is much smaller in the cavity, and H2 completely vanishes. Therefore, based on the analogy between the continuous films and the nanostructures and using the previous theoretical49 and experimental48,50 results for continuous films, we identify H1and H2 as originating from one (FH1) or two different (FH1 and FH2) higher-order PSSWs of the continuous film. If we assume that both H1 and H2 originate from FH1 of the continuous film, it is possible that H1 is the fundamental mode of the PSSW family and H2 is the analog of the mode M1 from the main mode family. This was checked with a simple calculation in which we substitute the function F (q) in Pxx (q) and Pyy (q) with P11 (q) from61 [Eq. (19) in that paper]. In this way we effectively include the dispersion of the first antisymmetric higher-order PSSW mode (FH1) of the continuous film in the calculation [Eq. (26) in Ref. 62].

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Agreement in the absolute values for the resonant fields of this theory with the experiment is quite fair: at 16 GHz for H1 the theory predicts 14.3 kOe, and 14.0 kOe is observed experimentally. For H2 the predicted and the observed values are 13.7 kOe and 13.4 kOe, respectively. This is despite the approximate character of this approach and the strong dependence of the overall downshift of the resonant fields for the PSSW modes on the assumed value for the exchange constant. However, the difference between the resonance fields for H1 and H2 mainly depends on the difference in the strengths of dynamic dipole field they produce and thus should not be very sensitive to the value of the exchange constant. For this difference we find good agreement with experiment: about 600 Oe at 16 GHz both measured and calculated. Furthermore, similar to the experiment, the calculated H1 is located at a smaller field than MF, and both H1 and H2 are located above the resonant field for M1. Also similar to the experiment, the theoretical H1 has considerably larger absorption amplitude than H2. In this way, and also based on its calculated modal profile, H1 is identified as the fundamental mode for PSSW family. VI. CONCLUSION

In this work we studied experimentally and theoretically the microwave magnetic dynamics on 2D periodic arrays of nanoscale magnetic antidots in the special case of array magnetization perpendicular to its plane. A linear dependence of the mode eigenfrequencies on the applied field for the center of the first Brilloiun zone is observed. We show that this behavior originates from the highest degree of magnetic symmetry for this nanopattern for the perpendicular direction of magnetization, which results in circular magnetization precession on average. The microstrip-based broadband FMR technique allowed the identification of two classifications of modes of the antidots arrays: the main mode family originating from the uniform precession in the continuous films and the PSSW mode family originating from one of the higher-order exchange standing SW modes across the thickness of the continuous films. The PSSWs are usually not seen in the FMR experiment at all or show negligible amplitudes. We demonstrate that the microwave shielding effect recently observed for continuous films in the microstrip-based FMR allows observation of an extremely strong response of the PSSWs for nanostructures provided this particular experimental technique is used. The calculated mode profiles exhibit a fourfold symmetry in contrast to the twofold symmetry typical for the IP magnetization direction. Furthermore, the calculated Bloch wave dispersion shows the same slope and thus the same group velocity for the two main directions on the crystal lattice [10] and [11] close to the center of the first Brillouin zone. The two dispersions practically overlap for most of the width of the first Brillouin zone for [10] direction. The differences in dispersions for these two directions is mainly due to the larger size of the first Brillouin zone along [11] axis with respect to [10] axis. This calculation evidences a very high degree of isotropy of collective SWs for the magnetization of the antidot array perpendicularly to its surface. The higher magnetic symmetry of the perpendicular-

to-plane magnetization configuration with respect to the IP magnetization may represent an important advantage for some applications exploiting travelling SWs on the antidot lattices.

ACKNOWLEDGMENTS

Financial support from Australian Research Council is acknowledged. We would like to thank Dr. N. Singh for template fabrication.

APPENDIX: THEORY OF COLLECTIVE SW ON AN ANTIDOT LATTICE MAGNETIZED PERPENDICULAR TO ITS PLANE

To derive the final equations we proceed in the same way as in Ref. 14 but use a different nanopattern and the different direction of magnetization as “input parameters” for our derivation. Referring to the reference frame defined in Fig. 1(b), we start with the linearized Landau-Lifschitz equation: −iωm(r) = −γ [M(r) × (hex (r) + hd (r)) + m(r) × H0 (r)], (A1) where m(r) is the dynamic magnetization, r is the radiusvector, hex (r) and hd (r) are the dynamic exchange and dipole fields, respectively, H0 (r) = H0 (r)ey is the static internal field inside the magnetic material, M(r) is the static magnetization, and γ is the gyromagnetic coefficient. Following the approach in Refs. 53 and 63, we neglect inhomogeneity of the static magnetization inside the elements. Thus  M(r) =

M0 ez , outside the holes (|r|>R) . 0, in the holes (|r|  R)

(A2)

The effective exchange field is defined as follows: hex (r) = α∇ 2 m(r),

(A3)

where α = A/(2π M02 ), and A is the exchange constant for the material. We also neglect the inhomogeneity of distribution of dynamic magnetization and of its dipole and exchange field across the dot thickness, i.e., along the axis z. For elements with a small aspect ratio L/R  1, this approximation is often appropriate, as shown in many previous studies.61,63–65 In this way we are able to do calculations for the modes which belong to the main mode family only; however, as we show in Sec. V, a simple modification of the theory provides meaningful results for the higher-order thickness mode families. As the system is periodical in both directions, we use the Bloch-Floquet theorem to write down a solution for m(ρ) = m(x, y) in the form of plane Bloch waves: m(ρ) = ˜ ˜ + a) is a periodic func˜ = m(ρ m(ρ) exp(ikB ρ), where m(ρ) tion, a is the lattice vector for the 2D artificial crystal, and kB is the Bloch vector of the collective mode. We then expand all periodic quantities into Fourier-series and obtain the following

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relations: m(x,y) =

∞ ∞  

mj,l j =−∞ l=−∞ ∞ ∞  

hex (x,y) = −α

exp(ikxj x + ikyl y) exp(ikBx x + ikBy y),

Kj,l2 mj,l exp(ikxj x + ikyl y) exp(ikBx x + ikBy y)

(A4)

j =−∞ l=−∞ ∞ ∞  

hd (x,y) = 4π

Pˆ (Kj,l )mj,l exp(ikxj x + ikyl y) exp(ikBx x + ikBy y).

j =−∞ l=−∞ 2 In these expressions, kxj = 2πj/a, kyl = 2π l/a, j,l = · · · − 1,0,1 . . .; Kj,l = (kxj + kBx )2 + (kyl + kBy )2 ; Pˆ (|q|) is the dy2 namic dipole-field tensor with components Pyy (|q|) = −F (|q|) sin (ϕ(q)), Pxx (|q|) = −F (|q|) cos2 (ϕ(q)), sin(ϕ(q)) = qy /|q|; a is the size of the unit cell for the artificial crystal in x and y directions; and q is some “dummy” wave vector. The function F (q) is defined as follows: F (q) = 1 − [1 − exp(−|qL|)]/|qL|61 (recall that L is the nanostructure size along the z-direction, i.e., the sample thickness). Substituting Eqs. (A3) and (A4) into Eq. (A1), results in a system of coupled equations as follows:

i(ω/γ )mn,m,κB =

∞ 

[Mn−n ,m−m ×(−α|kn ,m + kB |2 mn ,m ,κB +Pˆ (|kn ,m +kB |)mn ,m ,κB ) + H0 n−n ,m−m ×mn ,m ,κB ], (A5)

n ,m =−∞

where Mj,l is the Fourier-series transform of Eq. (A2). The Fourier-series transform H0 j,l of the internal static field and H0 is related to Mj,l by the out-of-plane component of the same dipole-field function Pˆ : Pzz (|q|) = F (|q|) − 1, H0 j,l = He − 4π Pzz (Qj,l )Mj,l , (A6) √ 2 2 = kxj + kyl . (Note that in the approach of the thickness-averaged field from Ref. 53 the

where He is the applied field and Qj,l IP components of H0 j,l vanish.) Now we need a Fourier-space representation for our geometry. We present the 2D Fourier-series transform of Eq. (A2) as a difference of two Fourier transforms Ff and Fh . Ff is the transform of the geometry of a continuous film, and Fh is the one of the hole in it. The hole is modeled as a circle with the center at the origin of the frame of reference and with the radius equal to the hole radius R [Fig. 1(b)]. The transform Ff is simply a product of two Delta functions δn0 δm0 , and the transform Fh is given by a p double integral Mn,m over the area of the circle. The total expression reads:   p (A7) Mn,m = δj 0 δl0 − Mn,m Ms , where Ms is saturation magnetization for the magnetic material and  R p Mn,m =2 dy sin(kxn R 2 − y 2 ) exp(ikym y)/(kxm w 2 ).

(A8)

0

The Fourier transform of the static demagnetizing field is obtained by substituting (A7) into (A6) and by making use of the fact that the Fourier transform of the static demagnetizing field for the continuous film is given by 4π Pzz (q = 0)Ms = 4π Ms . Finally, with (A6)–(A8) and in projections on two IP coordinate axes, Eq. (A5) reads: ∞ ∞    (1)  He mx,n,j + i(ω/γ )my,n,j = A(1) An,j,n ,j  my,n ,j  + Bn,j,n ,j  mx,n ,j  m + B m + n,j x,n,j n,j y,n,j n =−∞ j =−∞

He my,n,j + i(ω/γ )mx,n,j =

A(2) n,j mx,n,j

∞ 

+ Bn,j my,n,j +

∞   (2)  An,j,n ,j  mx,n ,j  + Bn,j,n ,j  my,n ,j 

(A9)

n =−∞ j =−∞

where

and

  2 2 A(1) n,j = Ms 1 − αKn,j − P (|Kn,j |) cos (ϕn,j ) ,   2 2 A(2) n,j = Ms 1 − αKn,j − P (|Kn,j |) sin (ϕn,j ) , Bn,j = −Ms P (|Kn,j |) sin(ϕn,j ) cos(ϕn,j )

 p 2 2 A(1) n,j,n ,j  = Ms Mn−n ,j −j  P (Qn−n ,j −j  ) − 1 + αKn ,j  + P (|Kn ,j  |) cos (ϕn ,j  ) ,

 p 2 2 A(2) n,j,n ,j  = Ms Mn−n ,j −j  P (Qn−n ,j −j  ) − 1 + αKn ,j  + P (|Kn ,j  |) sin (ϕn ,j  ) p

Bn,j,n ,j  = Ms Mn−n ,j −j  P (Qn−n ,j −j  ) sin(ϕn ,j  ) cos(ϕn ,j  ), cos(ϕn,j ) = (kx,n + kBx )/|Kn,j | (or sin(ϕn,j ) = (ky,n + kBy )/|Kn,j |). 104414-10

(A10)

(A11)

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59

For this magnetization direction both dynamic components of magnetization lie in the plane of the continuous isotropic film and both produce no dipole magnetic field. As a result the precession is circular. 60 For the 80-nm-thick nanostructure H1 is also observed at 9.5 GHz, which is the typical frequency for the cavity FMR experiment. However H1 and H2 are not visible for the 40-nm-thick antidot structure at 9.5 GHz [Fig. 3(a)]. Therefore we are unable to check the origin of these peaks with the cavity method. 61 B. A. Kalinikos, IEE Proc. H 127, 4 (1980). 62 B. A. Kalinikos, Sov. Phys. J. 24, 718 (1981). 63 C. Bayer, J. Jorzick, S. O. Demokritov, A. N. Slavin, K. Y. Guslienko, D. V. Berkov, N. L. Gorn, M. P. Kostylev, and B. Hillebrands, Spin Dynamics in Confined Structures III, Topics in Applied Physics, Vol. 101 (Springer, Berlin, Heidelberg, 2006), p. 57. 64 K. Yu. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B 66, 132402 (2002). 65 M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T. Ono, and R. L. Stamps, Phys. Rev. B 76, 054422 (2007).

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