1. Introduction

Potential applications in microwave devices motivate strong interest in high-frequency dynamics of arrays of magnetic nanowires [1,2]. In the case of relatively low angle precession, magnetic dynamics manifests itself through bulk magnetic excitations known as spin wave (SW) modes [3]. One of the attractive features of such structures stemming from their geometry, is the possibility to synthesize materials with a desired value of uniaxial magnetic anisotropy oriented along the wire axis. Breakthroughs in the technology of ferromagnetic metals electrodeposition (Co, Ni, permalloy) in arrays of self-assembled cylindrical nanopores created in alumina (Al₂O₃) films have accelerated the development of this concept [4–6]. Magnetic properties of ferromagnetic nanowires have recently been intensively studied both theoretically and experimentally (see [1] and references therein). Nanowires have been shown to exhibit magnetoresistance [7], giant magnetoresistance (GMR) [8–10], spin-transfer torque effects in “bi-layered” wires [11], non-reciprocity [5] and support coherent spin waves [12].

Previous experimental efforts relied mainly on the Ferromagnetic Resonance (FMR) measurements in the wide-band strip-line/VNA (Vector Network Analyzer) version [13–15]. The influence of inter-wire dipole interactions was investigated on dynamic properties of Ni nanowire arrays in a wide range of packing densities P (4% to 38%) and the wire diameters (50 to 250 nm). The dipolar interactions between the wires were modeled using a mean-field approach assuming an effective uniaxial anisotropy field oriented perpendicular to the wire axis and proportional to the membrane porosity.

Similarly, interesting effects in CoFeB ferromagnetic nanowire arrays, associated with two magnetization populations, one for nanowires with upward magnetization and one with downward magnetization have been addressed in [4]. It should be noted that until now all experimental studies have been performed on wires nanometric only in cross-section, their length typically being of the order of tens of microns (aspect ratio R typically over 100).

Brillouin light scattering (BLS) based on magneto-optical (MO) interaction of light with SW modes localized on nanowires, is an alternative technique to probe magnetic excitations in nanowires [16,17]. In contrast to earlier experimental results obtained on Co wires of practically infinite height (high aspect ratio R>100), the role of “vertical” SW resonances, along the axis of the nanorod of low aspect ratios (R<10), turned out to be very important as well as that of the near-field nature of MO interactions in such nanorod arrays.

In this paper we report the results of BLS study of the Co nanorod arrays with the aspect ratio R ≈6 (diameter 30 nm, height 175 nm, separation 70 nm). For the experimental data analysis, numerical methods have been used since purely analytical solutions, although effective and providing for more physical insight, are reliable only for R close to infinity, which is not our case. The results show the important role of spin wave modes along the nanorod axis in contrast to the previously studied infinite magnetic nanowires. Unusual optical properties featuring an inverted Stokes/anti-Stokes asymmetry of the Brillouin scattering spectra have been observed. The spectrum of spin wave modes in the nanorod array has been calculated and compared with the experiment. Magnetic nanorod arrays are important class of metamaterials that may have significant influence on magnetic and magneto-optical data storage as well as important for the development of active plasmonic metamaterials for optical signal control.

2. Sample preparation, experimental technique and modelling

Co nanorod arrays were electrodeposited in the anodized aluminium oxide (AAO) templates.Template fabrication involved the RF magnetron sputtering of a thin film multilayer comprising of Al on top of gold and tantalum buffer layers [18]. The tantalum oxide was necessary as an adhesion layer to avoid delamination on anodisation of the aluminium and the gold layer allows good electrodeposition into the pores. Anodisation and hence pore formation was carried out at constant voltage using a platinum counter electrode. For this work the aluminium was anodized at 30 V in 0.3 M sulphuric acid. The electrolyte was cooled below 275 K and the temperature monitored throughout the process. After a brief etch to remove the barrier layer at the bottom of the pores, Co rods were grown by potentiostatic electrodeposition from a 0.1 M CoSO₄ solution at a voltage of –1 V versus a saturated calomel electrode. The samples were grown for 60 seconds to produce rods of ~175 nm as estimated from electron microscope images. A typical transmission electron microscope (TEM) images of the nanorod array are shown in Fig. 1(a) . The sample under investigation consist of Co nanords of 30 nm diameter and 175 nm length with separation of 70 nm, corresponding to the filling factor of 15%.

 

Fig. 1 (a) TEM cross-section image of the Co nanorods. Please note that different nanorods are situated not in the same image plane and appear differently in the image. (b) The MOKE hysteresis loop taken in the polar magneto-optical configuration with the Co nanorod arrays.

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Magnetic properties of the Co nanorods were studied by sensing the magneto-optical Kerr effect (MOKE) at 633 nm wavelength using a photoelectric modulator based instrument. A typical hysteresis loop measured in a polar configuration with the applied magnetic field was out-of-plane of the sample and parallel to the nanorods is shown in Fig. 1(b). The saturation field is approximately 5 kOe. It has a characteristic square shape typical of a Stoner-Wohlfarth particle magnetized along its easy axis [19]. Non-vertical slopes of the magnetization curve, both left and right, can be attributed to the inter-rod dipolar interactions [4]. High remanence magnetization points to the fact that in the initial state of nanorods macrospins is parallel and practically no anti-parallel ordering takes place. It should be mentioned that the shape of the magnetization cycle is very close to that of the arrays of Ni nanorods of similar geometry (see [6], Fig. 4(a)).

Near-field magneto-optical analysis has been performed using finite element numerical method. Numerical magnetic simulations have been carried out using the Object Oriented Micro Magnetic Framework (OOMMF) package [20,21] for modeling dynamic behavior of an individual nanorod. This method was backed up by a simple analytic approach proposed in [14]. OOMMF, being a finite difference algorithm, is fast numerical technique but not reliable in the case of objects with curved surfaces. The accuracy of OOMMF simulations were cross-checked for selected modes using a more reliable home-made ad hoc code based on the fluctuation-dissipation theorem [22]. The discrepancy in the SW frequencies did not exceed 5% which is acceptable especially taking into account high absorption in cobalt and, hence, large width of BLS spectral lines.

The BLS measurements in the p-s polarization configuration were carried out in the Damon-Eshbach geometry: the bias magnetic $\overrightarrow{H}$ (from −13 kOe to + 13 kOe) field was applied parallel to the plane of the film and the direction of the in-plane wavevector probing magnetic excitation normal to $\overrightarrow{H}$. Light from a single-mode Ar⁺ laser of 350 mW power at wavelength λ = 514 nm was focused onto the sample and the frequency spectrum of the backscattered light was analyzed using a computer controlled Sandercock-type (2 ´ 3)-pass tandem Fabry-Pérot interferometer. The wavenumber K was changed by varying the angle of light incidence θ from 0° to 65°:

Cross-polarized incident and scattered beams were used in order to practically suppress the light scattered by phonons. The acquisition time in the BLS measurements was on the order of 1 hour which compares favorably with 8 hours reported in [17].

3. Results and discussion

The experimental studies of spin-waves in magnetic nanorods consisted of two series of measurements. Firstly, the BLS spectra were measured at the fixed angle of incidence (θ = 45°) for varying applied magnetic field from – 13 kOe to + 13 kOe with a step 1 kOe. In the second experiment, the dispersion of modes was measured at the fixed magnetic field by varying the angle of incidence.

Two representative BLS spectra are presented in Fig. 2 for the direct and reverted magnetic bias with the applied field H = − 11 kOe and + 11 kOe, respectively. The spectra consist of the central elastic scattering (Rayleigh) peak surrounded by the two dominating Stokes (down-shifted) and anti-Stokes (up-shifted) Brillouin scattering components with about 35 GHz (150 µeV) spectral shift and full width at the half-maximum (FWHM) of about 10 GHz (50 µeV). Similar to the case of magnetostatic modes in continuous magnetic films, the BLS spectra are characterized by a pronounced Stokes/Anti-Stokes (S/AS) asymmetry. As in the case of continuous films, this asymmetry increases with the angle of incidence. This points out to the fact that in both cases the polarization within a metal nanorod/film becomes more elliptical with the increase of the angle of incidence. However, contrary to the case of continuous films, in the measurements in positive fields, the intensity of the anti-Stokes components (up-shifted) exceeds the intensity of the Stokes (down-shifted) as can be clearly seen in Fig. 2(a). The asymmetry is reversed under the reversed bias, i.e., the conventional S/AS asymmetry is observed in negative saturating fields (Fig. 2(b)). In other words, the asymmetry of the observed S/AS pattern is inverted with respect to that occurring in continuous thin films.

 

Fig. 2 BLS spectra measured from the fully saturated samples at the angle of incidence θ = 45° (raw data): (a) H = + 11000 Oe, (b) H = − 11000 Oe. Red lines represent Lorentzian fits.

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Typically, exchange modes, due to the fact that their polarization differs considerably from that of the dipole excitations, produce a fairly symmetrical S/AS pattern [6]. Since in our case the Stokes/anti-Stokes asymmetry is fairly pronounced, one can conclude that the contribution of the fundamental dipolar magnetostatic mode is predominant.

The Brillouin scattering spectra observed above reflect the spin wave mode spectrum of the magnetic nanorod arrays. In order to determine the spin wave eigenmodes of the magnetic nanorods and their dynamic properties, the simulations were carried out in three stages. The main goal of the simulations was to obtain the eigenfrequencies of the spin wave modes and their dependence on the magnetic field. First of all, the static magnetic properties of the nanorods were evaluated. The saturating magnetic field, applied normally to the axis of the cylinder, was changed step-by-step from zero to a maximum value of 13 kOe, with a step of 1 kOe.

To simulate the dynamic response of the nanorod, one of the minimum energy states, calculated in the previous step, was taken as the initial condition. The magnetization was excited by a short, strong, spatially uniform magnetic field pulse with the Gaussian time dependence. The field direction in the pulse was perpendicular to the static bias field (see the insert in Fig. 3 ). The pulse amplitude was 100 Oe, large enough to excite eigen spin-wave modes, but much less than the bias field. Its full width at half-maximum was chosen equal to 20 ps, so that spin-wave dynamics up to 60 GHz could be modeled. As a result, we obtained the time evolution of the magnetization vector in each point of the nanorod. Finally, following the procedure proposed in [21], we applied discrete Fourier transformation to obtain local power spectrum. To have a better overall view of the magnetization behavior, the power spectra were averaged over all points of the nanorod. Please note that since the applied magnetic pulse was spatially uniform, only the even modes were efficiently excited.

 

Fig. 3 Profiles of dynamic magnetization localized on a nanorod: (a) fundamental (Kittel mode n = 0) and (b) first excited symmetric mode (n = 2) with the saturating magnetic field H = 10 kOe applied normally to the nanorod axis. Insert in (a) shows the geometry of numerical simulations of magnetic properties. (c) Dependence of the theoretical and experimental values of resonant frequencies on the magnetic field applied normally to the axis of a nanorod: (red triangles, green squares) experimental BLS data for positive and negative magnetic field, respectively; (blue crosses) numerical modeling (the accuracy is within the data marker dimensions).

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To simulate magnetic properties of the array of Co nanorods in the geometry used in the experiment, an “ad hoc” semi-analytical approach was used. First, we numerically analyzed the SW modes existing in an isolated cylindrical Co nanorod. To avoid unrealistically long computation time, we have begun with a nanorod of twice smaller dimensions: a diameter d = 15 nm and a height h = 87.5 nm, retaining the value of the aspect ratio (R≈ 6). Since the lowest fundamental mode (also known as the magnetostatic Kittel mode) is a magnetic dipolar one, its frequency depends only on R and not on the absolute size of the nanorod. As in the case of thin continuous ferromagnetic layers, the frequency splitting between the higher modes is due to the exchange interactions whose contribution to the eigenfrequency grows as a square of the film thickness. This holds in our case of a thin nanocomposite film [6]. This consideration was taken into account to rescale the frequencies of the next higher mode which is determined by the exchange interaction. To take into account the dipolar interactions between nanorods, we made use of the approach based on the effective field of magnetic anisotropy proposed in [14]. In this approach, the shape magnetic anisotropy is regarded as a result of a competition of the anisotropy of a single rod, which is equal to 2πM_s for an infinite cylinder, and that of a quasi-continuous planar film engendered though inter-wire magnetic dipolar interactions. It was supposed that the latter contribution is simply proportional to the packing density P, being expressed as -6πM_sP. Since in our case the cylinders are finite, we extracted the true value of the anisotropy of an individual rod from our OOMMF based numerical simulations.

In the following simulations, the magnetic parameters for Co media were used from [23], the saturating magnetization was M_s = 1.4x10⁶ A/m, the damping coefficient was α = 0.01, the exchange constant was A = 16x10⁻¹² J/m, and the magnetic anisotropy was neglected. To simulate the dynamic behavior of the cylinder, we took all experimental values of the bias field ranging from 1000 to 13000 Oe with the step of 1000 Oe.

To identify the modes appearing in the power spectrum of a single nanorod, we calculated their profiles applying an external alternating microwave magnetic field at the resonance frequencies (Fig. 3). In our case of a nanosized cylinder with the value of the aspect ratio R, the results are quite predictable: the modes with the number n < R, in our case the first five modes, are vertical, i.e. resonances are along the nanorod axis. This means that radial and azimuthal distributions can be supposed to be uniform. The latter consideration allowed simplification of numerical simulations by using relatively rough meshing, thus leaving the computation time within realistic limits. At the same time, the chosen meshing is dense enough in the direction along which dynamic magnetization is expected, to be highly inhomogeneous (the axis of the nanorod, z). Figures 3(a) and 3(b) present the distributions of the dynamic magnetization in the zx plane (along the nanorods) for the fundamental (n = 0) and n = 2 modes with the external magnetic field H = 10000 Oe applied normally to the axis of the nanorods. The xy cross-section (normal to the nanorods) of the magnetization distribution is practically uniform with no azimuthal dependence. Such analysis was made for all solutions with different magnetic bias field values. As in the case of Ni nanorods of the same length with the aspect ratios of 2.5 and 9.0, the noticeable “dipolar pinning” has been observed which especially pronounced for the fundamental mode [6]. In other words our numerical simulations have demonstrated that although no pinning of spins on the top and the bottom of a nanocylinder takes place, the distribution of the dynamic magnetization in the fundamental mode is appreciably inhomogeneous. As has been shown in [24], this effect of purely dipolar nature, this why it was termed “dipolar pinning”.

The experimental data on the magnetic field dependence of the SW eigen frequencies are presented in Fig. 3(c) together with numerical estimations. To reliably assess the values of the frequency shifts from the raw data of the BLS measurements (Fig. 2), we used conventional Lorentzian fitting. In Fig. 2 these fits are shown as red lines. As can be seen, in the case of high saturating fields (H = ± 11 kOe), they ensure reliable fitting of the raw data. It is less reliable for the low bias fields when the elastic Rayleigh peak becomes too close to the BLS spectral lines. However, it should be noted that magnetic dynamic behavior of arrays of non-saturated nanorods magnetically interacting via dipolar coupling as well as the BLS scattering cross sections by SW modes localized on the nanorods require a more detailed theoretical analysis; BLS in non-saturated arrays of ferromagnetic nanorods, both theory and experiment, will be presented elsewhere.

The theoretical dependence in Fig. 3(c) was plotted in the following way. First, we used the data obtained in OOMMF numerical simulations taking into account the shape anisotropy of an individual nanorod. The latter shifts the minimum of the F_r(H) from the value H_min = 2πM_s = H_a = 8.8 kOe corresponding to the anisotropy field in an infinitely long cylinder to approximately H_a (R)≈7.5 kOe, the value, corresponding to a finite aspect ratio R = 6. To take into account the dipolar interaction between the nanorods in the array, we made use the formula proposed in [14]

The angular resolved spectra have not shown any visible SW mode dispersion in the metamaterial under investigation. This can be due to a low filling factor of the metamaterial. The density of the nanorods is high enough to result in the strong dipolar interaction between nanorods but not enough to enable formation of the delocalized spin waves.

At the same time, the strong dependence of the S/AS asymmetry of the Brilloin spectra on the angle of incidence has been revealed. Interestingly, the asymmetry is inverted with respect to the conventional case of a continuous metal film. Similar atypical S/AS pattern was observed in the arrays of Ni nanorods of approximately the same shape and size [6]. This type of asymmetry has been predicted if the film is transparent enough so that the peculiarity of the optical response of magnetic metamaterial determines this BLS feature. Interestingly, the S/AS symmetry was not inverted in the case of the Ni nanorods with the same filling factor P and the same length, but with thinner nanorods (R = 9), while in the arrays of thicker nanorods (R = 2.5) we did observe an inversion of the S/AS pattern. The latter suggests that the variation of the nanorod diameter, even on nanoscale level, could appreciably affect optical parameters of the magnetic metamaterial. In the case of Co nanorods, plasmonic effects may also be important [25].

We have performed, using finite element numerical modeling, a detailed analysis of the electromagnetic field distribution in the optical wave within a nanorod which interacts with its SW modes. The magnetic metamaterial was assumed to be a square lattice array of Co nanorods with the parameters corresponding to the experiment. Optical constants of Co were taken from [26]. Full vectorial electromagnetic modelling was used to calculate the evolution in time, during a full period, of the electric field vector E_i components of the incident p-polarized optical wave inside the metamaterial for the light wavelength used in the experiment (514 nm).

The results are illustrated in Fig. 4 . In the right panel is presented characteristic time evolution of all three electric components of the incident p- polarized optical wave, during a full period. The left inset illustrates the configuration of the points 1-7 for which calculations have been made. These curves in the right panel correspond to the angle of incidence θ = 45° for the point 3. It is clearly seen that the two dominating field components have a significant phase shift within the metamaterial, resulting in an elliptical polarization. It is the elliptical polarization of the electromagnetic wave in the metamaterial that is essential for producing the S/AS asymmetry. As expected, the field in the middle of the nanorod is attenuated due to a small skin depth of Co which is on the order of 30 nm. The near-field nature of electromagnetic diffraction by nanorods results in the situation that amplitudes in symmetric points on opposite sides of the rods are practically equal (points 3 and 7). The polarization is almost linear in all the points along the center line (points 2, 4, and 6). In contrast, there is a non-negligible ellipticity in the polarization of E_i on the surface of the rod (1, 3, and 5), being especially pronounced in point 3. Interestingly, even in the case of normal incidence, there are zones where the ellipticity is nonzero in particular in points 3 and 7 that are situated symmetrically with respect to the center-line of the rod. However, the sign of the phase mismatch in point 3 is reversed with respect to that in point 7.

 

Fig. 4 Numerical simulations of the distribution of the electric field components within a nanorod in the metamaterial. (Left) Plot of the norm of the total electric field in the metamaterial simulated at the angle of incidence of 45° and p-polarised incident light at 514 nm wavelength. White arrows indicate the instantaneous direction of the electric field in points 1—7 of the nanorod. The wavevector of the incident light and the direction of the incident electric field are also shown. z-axis is along the nanorod axis, y-axis is normal to the plane of incidence. The metamaterial parameters correspond to the experimental ones. (Right) Temporal evolution (during a full period) of all the components of the electric field vector E_(i) of the incident p-polarized optical wave inside a nanorod in point 3.

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This polarization behavior explains the presence of S/AS asymmetry in the Brillouin scattering spectra. There are two characteristic zones within each nanorod where the light polarization ellipticity is strongly asymmetric at the angle of incidence used in the experiment: these are symmetric points 3 and 7. Besides, the direction of the rotation of the E_i vector in these points is opposite, which means that the zones engender two S/AS patterns with opposite asymmetry. In the case of normal incidence, for obvious reasons, the distribution of the E_i field is identical in both zones which means that the two contributions to the optical response are equal but opposite in phase, and their superposition finally produces an entirely symmetric S/AS pattern. With the increase in the angle of incidence the contribution of the zone 3 becomes predominant. In order to numerically estimate the plausibility of this hypothesis, we calculated the ratio I_AS/I_S using procedure outlined in [6], taking into account the field polarization within the nanorods. To this end, the actual values of light polarization were extracted from numerical simulations, while the ellipticity of the magnetic mode was taken equal to that of the Kittel mode. This model predicts the increase in the S-AS asymmetry with the angle increase. Moreover, for the specific experimental values of 15° and 45°angles of incidence, I_AS/I_S = 1.6 and I_AS/I_S = 2.4, which correspond well to the experimental values (1.3 and 2.2, respectively).

4. Conclusion

The BLS study of SW modes in Co nanorod metamaterials with low aspect ratio nanorods has revealed new magnetic and magneto-optical properties compared to continuous magnetic films or infinite magnetic nanowires. Magnetic behavior of such metamaterials is characterized by softening of dipole-exchange modes at a characteristic value of the saturating magnetic field. The latter corresponds to a pronounced minimum in the dependence of the eigen-mode frequencies on the applied out-of-plane saturating field. Eigenmode frequencies of dipole-exchange modes are shifted from those corresponding to an individual infinite wire as the result of two effects: finite aspect ratio and dipolar inter-rod interaction. These effects can be adequately modeled by a combined numerical–analytical approach, providing for a good agreement between the theory and the experiment. More specifically, the numerical part, based on the standard OOMMF program package, was aimed at modeling of the dynamic behavior of an individual nanorod of finite aspect ratio, while a simple analytical approach has allowed to take into account dipolar interrod coupling. In optical properties, we have observed an inversion of the Stokes/anti-Stokes asymmetry with respect to the case of continuous films and nanocomposite films of similar geometry, which emphasizes the sensitivity of optical properties of this type of metamaterial to variations of their geometry on a nanoscale level (tens of nanometres).