Fine structure of excitation spectra of magnetic vortex dot clusters

Patterned magnetic films in the form of regular arrays of small flat particles (dots) with different shapes are considered as artificial magnetic nanostructures, which are actively investigated nowadays. The isolated ferromagnetic dots reveal discrete spin excitation spectra due to geometrical confinement.¹⁾ The interdot coupling (pure magnetostatic, if the ferromagnetic dots are put on a dielectric substrate) leads to collective spin excitation spectra of the dot lattices (magnonic crystals) or dot clusters. The standard Bloch approach along with the periodic Born–von Karman boundary conditions²⁾ can be applied to describe the eigenfunctions and eigenfrequencies of magnonic crystals within momentum representation introducing a quasi-continuous Bloch wave vector.³⁾ However, the excitation spectra of small clusters consisting of N magnetostatically coupled dots (typically, N ≤ 9) can be considered only in the coordinate representation.^4,5) It was found⁶⁾ that these spectra are analogous to the lattice vibration spectra of N-atomic molecules bound with dipolar interactions.

The lateral arrays of soft magnetic dots with submicron and micron in-plane sizes attract extra attention owing to the presence of a vortex (topological soliton) magnetization configuration in their ground states.⁷⁾ The vortices are characterized by in-plane curling magnetization in the main dot area and a small vortex core with the perpendicular magnetization bearing a topological charge. The vortex core excited by an external magnetic field or a spin-polarized electric current can oscillate in the 100 MHz–1 GHz range (gyrotropic modes) providing a small frequency linewidth and a relatively high microwave power output.⁸⁾ The vortex oscillators coupled to each other are now under intensive investigation as promising candidates for spin-torque nano-oscillators emitting microwaves or transferring the microwave signals in magnonic crystals.⁹⁾ One of the unresolved problems is the synchronization of several coupled nano-oscillators to increase essentially the generated microwave power. Recently, clusters of the coupled vortex state dots have been proposed as efficient bipolar junction transistors.¹⁰⁾ The need of close packing of such magnetic dots to increase interdot magnetostatic interactions leads to the necessity of considering the collective vortex dynamics in dot pairs and clusters. The simplest case of vortex gyrotropic excitation in a dot pair was considered theoretically in Refs. 4 and 5.

The effect of the interdot interaction on the ferromagnetic resonance spectra of closely packed rectangular lattices of ferromagnetic dots was explored for the first time by Kakazei et al.¹¹⁾ A series of experimental and theoretical papers on the coupled magnetic vortex dynamics in lateral nanostructures have been published recently. Two-dimensional (2D) arrays, chains, and clusters of the circular and square (rectangular) dots were investigated theoretically¹²⁾ and by different experimental techniques, such as broadband ferromagnetic resonance,^13–16) Brillouin light scattering,¹⁷⁾ and time-resolved X-ray imaging.^18–20) In this letter, we present an analytical approach to the problem of excitation spectra of coupled magnetic vortex dots if the number of dots is more than 2 and they form a cluster. The collective eigenfrequencies are explicitly calculated for triangular (N = 3) and square (N = 4) dot clusters, and their dependence on interdot separation is represented via analytic expressions for magnetostatic coupling integrals.

For the consideration of magnetization dynamics in coupled vortex dot arrays, we use a theoretical approach based on the Landau–Lifshitz (LL) equation of magnetization (M) motion, $\mathbf{\dot{M}} = - \gamma \mathbf{M} \times \mathbf{H}_{\text{eff}}$, where $\mathbf{H}_{\text{eff}} = - \delta w/\delta \mathbf{M}$, w being the magnetic energy density. It is convenient to rewrite the LL equation of motion of dot magnetizations, $\mathbf{M}_{n}(\boldsymbol{{\rho,t}}) = \mathbf{M}_{n}(\boldsymbol{{\rho,}}\mathbf{X}_{n}(t))$, in the form of Thiele equations of motion²¹⁾ for the vortex core positions $\mathbf{X}_{n} = (X_{n},Y_{n})$ in the n-th dot via the complex variables $s_{n} = s_{xn} + is_{yn}$, $\mathbf{s}_{n} = \mathbf{X}_{n}/R$:

$$\begin{equation} iG_{zn}\dot{s}_{n} = \frac{2}{R^{2}}\frac{\partial U}{\partial \bar{s}_{n}}, \end{equation} \tag{ 1 }$$

where $\partial /\partial s = (\partial /\partial s_{x} - i\partial /\partial s_{y})/2$, $\partial /\partial \bar{s} = (\partial /\partial s_{x} + i\partial /\partial s_{y})/2$, $G_{zn} = - p_{n}|G|$, and $|G| = 2\pi M_{\text{s}}L/\gamma$ are the z-projection and absolute value of the gyrovector of the n-th dot in the array, L is the dot thickness, R is the dot radius, $U = \sum _{n}\kappa _{n}s_{n}\bar{s}_{n}/2 + U_{\text{int}}$ is the total magnetic energy of the vortex dot array, $p_{n} = \pm 1$ is the vortex core polarization, and κ_n is the stiffness coefficient²²⁾ of the n-th dot. The dots are assumed to be sufficiently thin (L < 50 nm). Therefore, the dependence of the dot magnetization on the thickness coordinate (z) can be neglected. We are interested in finding eigenfrequencies of the coupled vortex state dots in the positions described by the in-plane vector $\vec{\Re }_{n}$; therefore, the damping term is not included in Eq. (1).

Equation (1) using the interaction energy calculated in Ref. 12,

$$\begin{align} U_{\text{int}} &= 16\pi M_{\text{s}}^{2}V\beta \sum_{nn'l}P_{l}\frac{R^{l + 1}}{\Re_{nn'}^{l + 1}} C_{n}C_{n'}\\ &\quad\times\left[\frac{(l + 1)}{\Re_{nn'}^{2}}(\mathbf{s}_{n}\cdot \vec{\Re}_{nn'})(\mathbf{s}_{n'} \cdot \vec{\Re}_{nn'}) - l\mathbf{s}_{n}\cdot \mathbf{s}_{n'}\right], \end{align} \tag{ 2 }$$

where $V = \pi R^{2}L$ is the dot volume, $\beta = L/R$, P_l are the coefficients of multipole decomposition, $\vec{\Re }_{nn'} = \vec{\Re }_{n} - \vec{\Re }_{{n'}}$, and $C_{n} = \pm 1$ are the vortex chiralities,⁷⁾ can be applied to consider the coupled vortex dynamics in small dot clusters consisting of cylindrical dots $(N = 2,3,4, \ldots )$. Equation (2) is obtained from the multipole decomposition of the dot pair interaction energy. The dot multipole moments $Q_{\nu }^{m},Q_{{\nu '}}^{{m'}}$ ($|m| \leq \nu$, the index $\nu = 1,2, \ldots$) are not zero only for odd $\nu ,\nu '$; therefore, the index $l = \nu + \nu ' = 2,4, \ldots$ is even.¹²⁾ The anisotropic interdot interaction (2) breaks the circular dot symmetry, and the eigenmodes correspond to elliptical vortex core trajectories.⁵⁾

The system of the coupled Thiele equations of motion (1) after conducting the Fourier transform $s_{n}(t) = s_{n}\exp (i\omega t)$ is reduced to pairs of conjugated equations (κ_n = κ for identical dots),

$$\begin{align} (p_{n}\omega - \omega_{0})s_{n} &= \frac{2}{|G|R^{2}}\frac{\partial U_{\text{int}}}{\partial\bar{s}_{n}},\\ (p_{n}\omega - \omega_{0})\bar{s}_{n} &= \frac{2}{|G|R^{2}}\frac{\partial U_{\text{int}}}{\partial s_{n}},\quad \end{align} \tag{ 3 }$$

where $\omega_{0} = (1/R^{2})(\kappa/|G|)$.

The problem is reduced to the calculation of the interdot interaction energy U_int accounting for the geometry of the dot clusters. We consider the clusters with the numbers of identical dots, i.e., N = 3 [the dots are placed in the vertexes of the equilateral triangle in the xOy plane (Fig. 1)] and N = 4 (the dots are placed in the vertexes of a square).

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The vortex chiralities do not change the fine structure of the excitation spectra keeping the values of the eigenfrequencies; they change only the symmetry of the eigenmodes belonging to each eigenfrequency.²³⁾ The set of vortex core polarizations for the dot array, {p_n}, is of principal importance for calculations of the eigenfrequencies. Therefore, the excitation spectrum is determined by the choice of a particular {p_n} set. In the cluster of N dots, the number of eigenfrequencies is N; however, the number of nonequivalent sets of core polarizations is essentially smaller than 2^N. The core polarizations can be easily controlled by applying and releasing a perpendicular bias magnetic field. No simple means exist to control the chirality of the circular dots. Fortunately, in the case of linear vortex dynamics, the vortex chiralities can be excluded from the equations of motion using the relation $C_{n}\mathbf{s}_{n} = \mathbf{\hat{z}} \times \boldsymbol{{\mu}}_{n}/\xi$,⁷⁾ where $\boldsymbol{{\mu}}_{n} = \langle \mathbf{M}_{n} \rangle _{V}/M_{\text{s}}$ is the n-th dot volume average magnetization, and the model-dependent parameter ξ is on the order of 1 (ξ = 2/3 for the two-vortex model of the displaced vortex⁷⁾). The interdot interacting energy

$$\begin{align} U_{\text{int}}(\boldsymbol{{\mu}}) &= 16\pi M_{\text{s}}^{2}V\frac{\beta}{\xi^{2}}\sum_{nn'l}P_{l}\frac{R^{l + 1}}{\Re_{nn'}^{l + 1}} \\ &\quad\times\left[\boldsymbol{{\mu}}_{n} \cdot \boldsymbol{{\mu}}_{n'} - \frac{(l + 1)}{\Re_{nn'}^{2}}(\boldsymbol{{\mu}}_{n}\cdot \vec{\Re}_{nn'})(\boldsymbol{{\mu}}_{n'}\cdot \vec{\Re}_{nn'})\right] \end{align} \tag{ 2′ }$$

allows writing the equations of motion of the dot average magnetization (μ_n) in the form similar to the LL equation ($\mathbf{\dot{M}}_{n}/\gamma = \mathbf{M}_{n} \times \delta w/\delta \mathbf{M}_{n}$) as

$$\begin{align} \dot{\boldsymbol{{\mu}}}_{n} &= \hat{\mathbf{z}} \times \frac{\partial U}{\partial \boldsymbol{{\mu}}_{n}}\frac{1}{|G|R^{2}},\\ U &= \int dVw[\boldsymbol{{\rho,}}\boldsymbol{{\mu}}_{1},\ldots,\boldsymbol{{\mu}}_{N}] = \frac{1}{2\xi^{2}}\sum_{n}\kappa_{n}\boldsymbol{{\mu}}_{n}^{2} + U_{\text{int}}(\boldsymbol{{\mu}}). \end{align} \tag{ 4 }$$

The eigenmodes of the linear Eq. (4) in terms of μ_n do not depend on the vortex chiralities. If one only wants to recalculate the eigenmodes to elementary motions of the vortex core coordinates (S_n) in each dot, then one needs to specify the set of chiralities (C_n). In many practical cases, the μ_n representation is sufficient because it allows us to determine which eigenmodes can be excited by the in-plane magnetic field of a given direction (the total mode dynamical magnetization should not be zero in that direction) and what their relative intensities are.

Let us consider first the cluster of N = 3 dots placed in the vertexes of the equilateral triangle. Applying the equations of motion (3) with the interdot interaction energy given by Eq. (2), we can calculate the eigenfrequencies analytically. We need to find the eigenvalues and the eigenvectors of the dynamical matrix corresponding to the system of equations of motion (3), which can be written in the standard form $\hat{\Gamma }\mathbf{S} = \omega \mathbf{S}$. The nondiagonal components of the matrix $\hat{\Gamma }$ are determined by the interdot magnetostatic coupling, and the vector $\mathbf{S} = \text{col}(s_{1},\bar{s}_{1},s_{2},\bar{s}_{2}, \ldots ,s_{N},\bar{s}_{N})$ is composed of dot variables $s_{n},\bar{s}_{n}$. The matrix elements of $\hat{\Gamma }$ and the frequency ω are normalized to the gyrotropic frequency of the isolated dot, ω₀, which is $\omega _{0} = (20/9)\gamma M_{\text{s}}\beta$ for the dot aspect ratio β = L/R ≪ 1. The secular equation for the eigenfrequencies, $R(\omega ) = \det (\hat{\Gamma } - \omega \hat{I}) = 0$, is cubic with respect to ω²:

$$\begin{equation} R(\omega) = \omega^{6} + a_{1}\omega^{4} + a_{2}\omega^{2} + a_{3} = 0, \end{equation} \tag{ 5 }$$

where the coefficients are: $a_{1} = - 3 - 2(\nu _{1}^{2} - \nu _{2}^{2})p\sigma$, $a_{2} = 3 + (\nu _{1}^{2} - \nu _{2}^{2})^{2}\sigma ^{2} - 6(\nu _{1}^{2} + \nu _{2}^{2}) + 2p\sigma (2\nu _{1}^{3} + \nu _{1}^{2} + \nu _{1}\nu _{2}^{2} - \nu _{2}^{2})$, $a_{3} = - 4\nu _{1}^{6} - 12\nu _{1}^{5} + 4\nu _{1}^{3} + 6\nu _{1}^{2} + 3\nu _{1}^{4}(4\nu _{2}^{2} - 3) + (\nu _{2}^{2} - 1)^{2}(4\nu _{2}^{2} - 1) + 3\nu _{1}\nu _{2}^{2}(5\nu _{1}\nu _{2}^{2} + 2(\nu _{1}^{2} + \nu _{2}^{2}) - 6\nu _{1} - 2)$, $\nu _{j} = (\beta \omega _{M}/\omega _{0})I_{j}$, $\omega _{M} = \gamma 4\pi M_{\text{s}}$, $p = p_{1}p_{2}p_{3}$, and $\sigma = p_{1} + p_{2} + p_{3}$.

The interdot coupling integrals [$I_{j}(d)$] are ($d = D/R$)

$$\begin{equation} I_{1}(d) = \sum_{l}\frac{(l - 1)}{d^{l + 1}}P_{l},\quad I_{2}(d) = \sum_{l}\frac{(l + 1)}{d^{l + 1}}P_{l}, \end{equation} \tag{ 6 }$$

where $l = 2,4,\ldots$ and we considered that, for the dot cluster, all interdot center-to-center distances are equal, namely, $|\vec{\Re }_{nn'}| = D$.

The expressions for the eigenfrequencies are complicated. Therefore, we present these eigenfrequencies in the limit of weak interdot coupling, $\nu _{j} \ll 1$ (the coupling energy is typically lower than 10% of the in-dot magnetic energy). The eigenfrequencies are functions of two combinations of vortex core polarizations, p and σ. Thus, we have two different cases of vortex motion: all polarizations are parallel (p = +1 or p = − 1, σ = ±3), all cores move counterclockwise (σ = +3) or clockwise (σ = −3), and two polarizations are parallel, one is antiparallel to them ($\sigma = \pm 1$, $p = \mp 1$). The parameters p and σ are related by the equation $2\sigma p = \sigma ^{2} - 3$. There are two sets of three eigenfrequencies of the dot cluster, and these sets depend on σ². The values ±σ correspond to the same eigenfrequencies; therefore, we can choose σ > 0 and distinguish the eigenfrequency sets using the parameter p = ±1. In the case of p = +1, the vortex eigenfrequencies are

$$\begin{equation} \omega_{1}^{+} = \omega_{0}(1 - 2\nu_{1}),\quad \omega_{2,3}^{+} = \omega_{0}(1 + \nu_{1}). \end{equation} \tag{ 7 }$$

It is easy to calculate the coupling integrals [I_j(d)] defined by Eq. (6). The parameters (P_l) of the integrals in the multipole decomposition (6) are model-dependent. For the two-vortex model of the displaced vortex core, we obtain $P_{2} = 1/36$, $P_{4} = 1/80$, $P_{6} = 113/(48 \times 70)$, ..., see Ref. 12. The term with l = 2 corresponds to dipole–dipole, that with l = 4 corresponds to dipole–octupole, and that with l = 6 corresponds to octupole–octupole and dipole–triacontedipole interdot magnetostatic coupling of the dot multipole moments. The l = 2 term is the main term in the multipole decomposition (6) at large and moderate d > 2.5; however, for $d \to 2$, the high-order multipole terms are important.

In the case of p = −1, the vortex eigenfrequencies are

$$\begin{align} \frac{\omega_{2,3}^{-}}{\omega_{0}} &= 1 + \frac{2\sqrt{3}}{3}\sqrt{\nu_{1}^{2} + 2\nu_{2}^{2}} \cos \left(\alpha \mp \frac{\pi}{3}\right),\quad\\ \frac{\omega_{1}^{-}}{\omega_{0}} &= 1 - \frac{2\sqrt{3}}{3}\sqrt{\nu_{1}^{2} + 2\nu_{2}^{2}}\cos (\alpha), \end{align} \tag{ 8 }$$

where the angle α is determined by the expression $\cos (3\alpha ) = (3\sqrt{3} /2)\nu _{1}\nu _{2}^{2}/(\nu _{1}^{2} + 2\nu _{2}^{2})^{3/2}$.

To calculate the eigenfrequencies of the cluster consisting of N = 4 dots, we introduced 4 coupling integrals, corresponding to the interaction of the x- and y-components of the dynamical magnetizations (μ_n) of the dots located on the square side and square diagonal, $\nu _{j} = (\beta \omega _{M}/\omega _{0})I_{j}(d)$ and $\nu _{j}^{d} = (\beta \omega _{M}/\omega _{0})I_{j}(\sqrt{2} d)$, respectively. The secular equation $R(\omega ) = 0$ is complicated. However, for physically interesting cases of aligned vortex core polarizations (all p_n = +1 or −1) and checkerboard ordering of the polarizations (p₁ = p₃ = +1, p₂ = p₄ = −1, the cluster ground state), it reduces to bilinear and biquadratic equations and has a simple solution. By using the smallness of the coupling parameters $\nu _{j},\nu _{j}^{d} \ll 1$, the eigenfrequencies can be written for p_n = +1 as

$$\begin{equation} \frac{\omega_{1,4}^{+}}{\omega_{0}} = 1 - \nu_{1}^{d}\mp 2\nu_{1},\quad \frac{\omega_{2,3}^{+}}{\omega_{0}} = 1 + \nu_{1}^{d}, \end{equation} \tag{ 9 }$$

and for p₁ = p₃ = +1 and p₂ = p₄ = −1 as

$$\begin{equation} \frac{\omega_{1,4}^{-}}{\omega_{0}} = 1 + \nu_{1}^{d}\mp 2\nu_{2},\quad \frac{\omega_{2,3}^{-}}{\omega_{0}} = 1 - \nu_{1}^{d}. \end{equation} \tag{ 10 }$$

All the eigenfrequencies ($\omega _{n}^{ \pm }$) increase with the number n and satisfy the equation $\sum _{n}\omega _{n}^{ \pm }/N = \omega _{0}$. The frequency splitting between the gyrotropic eigenfrequencies is determined by the parameter ν_j. In the limit of thin dots, β ≪ 1, $\nu _{j}(d) = (9\pi /5)I_{j}(d)$ depends only on the interdot distance d, i.e., the frequency splitting in thin dots in relative units depends only on d, whereas in absolute units (MHz), the splitting is also proportional to β. We explicitly obtain $\nu _{2} \approx 3\nu _{1} \approx (3\pi /20)/d^{3}$ within the dipole–dipole approximation. The calculated eigenfrequencies of the dot clusters with different sets of {p_n} are shown in Figs. 2 and 3. The degenerated gyrotropic frequency ω₀ splits into several eigenfrequencies (fine structure of the spectra) owing to the interdot interaction, and the splitting increases with decreasing lateral distance d between the dot centers. The frequency splitting is essentially larger in the case of unaligned vortex core polarization of dots forming a cluster. For the N = 3 cluster (p = −1), the lowest $\omega _{1}^{ - }$ and highest $\omega _{3}^{ - }$ eigenfrequencies determine the total frequency splitting. For the N = 4 dot cluster, the spectrum consists of two double-degenerated frequencies ($\omega _{2,3}^{ + }$) and two frequency doublets ($\omega _{1,4}^{ \pm }$). There is a degenerated frequency ($\omega _{2,3}^{ + }$) for 3- and 4-dot clusters, which splits only in the limit of touching dots, $d \to 2$, when nonlinear coupling terms become important in the eigenfrequency decompositions (7) and (9). The eigenfrequency $\omega _{1}^{ + }$ is essentially lower than $\omega _{2,3}^{ + }$ and decreases with decreasing d for N = 3 and 4. The eigenfrequency $\omega _{2}^{ - }$ (N = 3) and the double-degenerated frequency $\omega _{2,3}^{ - }$ (N = 4) depend very weakly on the interdot separation. Owing to the relation $\nu _{2} \approx 3\nu _{1}$, the overall frequency splitting of the initially degenerated frequency ω₀ of the 4-dot cluster is essentially larger for $p_{1} = p_{3} = + 1$ and $p_{2} = p_{4} = - 1$. The frequency spectra of the N = 3 and 4 clusters are similar in the case of antiparallel vortex core polarization.

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In the recent experiment by Wang et al.,¹⁹⁾ the excitation frequencies of the triangle permalloy (NiFe alloy) vortex dot clusters were measured by time-resolved photoemission electron microscopy (PEEM) imaging. The dot thickness L was 30 nm, the radius R was 2 µm (β = 0.015, ω₀/2π = 71 MHz), and the dot center-to-center distance D was 4.5–5.0 µm (d = 2.25–2.50). The relatively large bandwidths of the splitted eigenfrequencies, 0.16ω₀ for d = 2.25 and 0.09ω₀ for d = 2.50, indicate that the vortex core polarizations in the dot clusters are antiparallel (p = −1). The results of the experiments are compared with those of the calculations using Eqs. (5) and (6) in Fig. 2. The agreement is good, especially considering that the accuracy in the frequency measurements would be about 10% if only two periods of dot magnetization oscillations after excitation by an in-plane field pulse were detected. This initial stage of vortex motion in each dot was fitted using a sinusoidal function with only one dominating frequency, which does not generally match any of the dot cluster eigenfrequencies.¹⁹⁾ The dominating frequencies are shown in Fig. 2 by symbols. The dominating frequencies are close to the frequency of the maximum of the function of intensity, $|\sum _{n}\boldsymbol{{\mu}}_{n}(\omega )|^{2}$, versus the frequency ω for a given excitation field direction. The eigenmode with the lowest eigenfrequency $\omega _{1}^{ - }$ [see Eq. (8)] having the maximal average magnetization along the field is excited by the field pulse directed parallel to the triangle base. On the other hand, the eigenmode with the eigenfrequency $\omega _{2}^{ - }$ is excited by the field pulse directed perpendicularly to the triangle base, i.e., the red (green) squares in Fig. 2 are just three different estimations of the eigenfrequency $\omega _{1}^{ - }$ ($\omega _{2}^{ - }$) in the PEEM experiment.¹⁹⁾

In the study by Vogel et al.,¹⁴⁾ the vortex frequencies were measured for different sizes of dot arrays using the broadband ferromagnetic resonance and varying the interdot distance d from 2.15 to 4.12.¹⁴⁾ The resonance linewidth is larger than the frequency splitting, and the fine structure of the spectra is not resolved. Therefore, it is reasonable to introduce a frequency spread due to the interdot coupling as the difference in resonance linewidth between the interacting and isolated dots (β = 0.1, ω₀/2π = 378 MHz). For 2 × 2 circular dot arrays (N = 4 clusters), the measured frequency spread was approximately 20 MHz for d = 2.32. This spread is around the frequency of 372 MHz corresponding to the resonance signal intensity maximum. The results of the ferromagnetic resonance experiments¹⁴⁾ are compared with those of calculations using Eqs. (9) and (10) for the N = 4 cluster in Fig. 3. We calculated the maximum frequency spreads of 41 MHz (d/R = 2.15) and 30 MHz (d/R = 2.32) for the same core polarization, and 102 MHz (d/R = 2.15) and 77 MHz (d/R = 2.32) for the chessboard ordering of the vortex core polarizations. Considering that the vortex core polarization set {p_n} was not especially prepared/controlled, the agreement with the experiment¹⁴⁾ is quite good. Most probably, the core polarizations of the square dot clusters measured in Ref. 14 are aligned (all p_n = +1 or −1).

Very recently, Haenze et al.²⁰⁾ have measured coupled vortex modes for the N = 9 dot cluster by scanning transmission X-ray microscopy. For such a large cluster, there are approximately 100 nonequivalent configurations of the vortex core polarizations ({p_n}), and nine, in general, different eigenfrequencies. However, it was shown directly that a stripe pattern consisting of columns (rows) of alternative core polarizations was established in the cluster that essentially simplifies the description of the dynamics. For larger dot clusters (N > 9) of square/rectangular shape, the coordinate approach is very complicated, and we can use the momentum representation¹²⁾ with the discrete wave vector k, where its components k_x and k_y are chosen within the first Brillouin zone $(0,\pi /D)$. The eigenfrequencies are then determined using Eq. (23) of Ref. 12 as $\omega _{m} = \omega (\mathbf{k}_{m})$. The choice of the discrete values of k_m and the corresponding standing waves of the dot magnetization $\boldsymbol{{\mu}}_{n}$ oscillations as a function of the set of {p_n} is beyond the scope of this study and will be considered elsewhere.

In summary, the low-frequency gyrotropic dynamics in clusters of 3 or 4 laterally placed magnetic vortex state cylindrical dots were considered. The eigenfrequencies of collective vortex oscillations were calculated analytically and compared with the results of recent broadband ferromagnetic resonance and X-ray imaging experiments. The fine structures of the spectra of N = 3 and 4 dot clusters are similar. The splitting of the degenerated gyrotropic frequency of isolated dots was determined by interdot coupling integrals, which are expressed explicitly in the form of multipole decomposition. The fine structure of the dot spectra can be tuned by changing the vortex core polarizations by an external magnetic field. The spread of the collective mode frequencies is approximately 3 times larger for the antiparallel vortex core polarization than for the aligned case. The accuracy of the current experimental techniques is insufficient to clearly detect the fine structure of the spectra of the cluster-forming coupled vortex dots (the frequency splitting is about 10–30 MHz). This can be a research subject in the near future.

Acknowledgments