The important static magnetic features of magnetic nanowires, to be used in the following discussion, are single-domain remanent state; well-defined easy axis (uniaxiality); controllable coercivity and anisotropy; sufficiently narrow SFD; thermally stable magnetization state; and tunable saturation magnetization (by the choice of the magnetic material and by porosity).

The single-domain state assumes that at least one dimension of the particle is smaller than the characteristic length for domain formation. The characteristic length has several definitions, all leading to the same order of magnitude: It can be defined as the exchange length ${\lambda }_{\mathrm{ex}}={\left[2A/M_{\mathrm{S}}^2\right]}^{1/2}$, the domain wall width ${\lambda }_{\mathrm{D}\mathrm{W}}={\left[A/K\right]}^{1/2}$, or the magnetostatic length ${\lambda }_{\mathrm{D}}=\left[2{\left(AK\right)}^{1/2}\right]/M_{\mathrm{s}}^2$, where A is the exchange interaction energy, K is the anisotropy energy, and M_s is the saturation magnetization at temperature T. For the existence of a thermally stable state at T, the size of the single-domain particle should be just around these limits; otherwise, the particle is superparamagnetic or multidomain. For traditional ferromagnetic metals (Fe, Co, Ni) the single-domain size is a few tens of nm, ideally fitting the nanowire technology.

For a single-domain particle the anisotropy has three contributions. For polycrystalline wires the magnetocrystalline anisotropy (H_A) is usually negligible unless there is a preferential growth induced ordering. There might be a magnetoeleastic contribution due to the nonzero magnetostriction of the wire material, which can be optimized by proper choice of the materials. Finally, the dominating contribution for nanowires is the shape anisotropy. For single-domain-diameter wire particles the easy axis is preferred to coincide with the wire axis, thus minimizing the stray fields. For a large enough aspect ratio a, that is, wire length/diameter, $a=\ell /d\gg 10$, the DMF along the wire can be taken as $N_z\cong 0$, whereas N_x = N_y = 0.5, as for an infinitely long ellipsoid of rotation. As a result, the nanowires are bistable, having only two magnetic states, “up” and “down” along the easy axis with a remanent magnetization M_r = M_s.

However, for nanowire systems, consisting of a large number of wires, an interesting situation takes place: The overall magnetic anisotropy of the array is determined not only by the shape anisotropy of the individual wires, which will induce a magnetic easy axis parallel to the wire axis, but also by the magnetostatic coupling among the wires, attempting to develop a magnetic easy axis perpendicular to the wire axis. Thus the uniaxial anisotropy is expected to be reduced for an interacting system of nanowires. Reducing the anisotropy field corresponds to a reduction in coercivity, measurable experimentally in hysteresis measurements.

The nonellipsoidal shape and the relatively large surface area of a small size particle produce a nonuniform internal field in nanoparticles. Close to saturation the magnetic moments at the wire ends are still canted to minimize the stray fields, and due to increased surface/volume ratio this canted structure persists to very high fields and contributes to switching field reduction by creating nucleation centers for easy reversal. For a solitary single domain particle the coercivity is equal to the switching field, H_c = H_sw < H_A. For real nanoassemblies, with a large number of particles, the mean value of the switching field, that is, the coercivity, is usually much lower than that arising from nonuniform magnetization rotation processes. The statistical distribution of individual switching fields (SFD) is quite broad, due to nucleation at defects. The mechanism, responsible for SFD, contributes to the observed FMR linewidth, that is, to dynamic properties. At the same time, the FMR linewidth characterizes the microwave losses in devices, based on nanowire arrays.

23.2.1.2 The role of shape anisotropy – demagnetizing tensor of nonellipsoidal magnetic elements

To discuss the role of interactions in large 2-D arrays of magnetic nanowires, first the knowledge of the demagnetizing tensor elements is needed. The equilibrium magnetic structure of the nanowire system is governed by the arrangement of the elements' magnetization according to the energy minimum, following the internal field. For zero magnetocrystalline and magnetoelastic anisotropy, the internal field H_i in an applied dc bias field H₀ is determined by the demagnetizing field, H_D = − NM created by the magnetized body, as:

${\mathit{H}}_{\mathrm{i}}={\mathit{H}}_0-\overleftrightarrow{N}\mathit{M}$

where $\overleftrightarrow{N}$ is the demagnetizing tensor. For magnetic bodies of a shape, different from the ellipsoid of rotation, the demagnetizing tensor elements can be calculated numerically (Pardavi-Horvath, 2009 and references therein). As shown earlier, in the first approximation the shape of the nanowires can be approximated by a long thin wire, ℓ  2r, and a corresponding approximation for the internal field can be used. However, to describe and understand the behavior of the magnetization of individual nanowires and that of an interacting array, more precise calculations are necessary. The magnetization is not uniform inside a nanowire (or any other magnetic nanoparticle) because the internal field is inhomogeneous due to shape effects. There is a noncollinear magnetic spin structure at the ends of the wires, even for wires of very large aspect ratio and very high magnetic field, where the material is assumed to be saturated.

The demagnetizing field dominates the internal field in individual nanoparticles and interacting nanoparticle arrays. The calculation of the demagnetizing tensor elements for nonellipsoidal bodies is the hardest task in any micromagnetic technique, and significant effort was devoted to this problem from the time when shape and size effects in design of machines and devices, based on magnetic materials, in industrial technology required the knowledge of the DMFs for nonellipsoidal shape magnetic materials (see Pardavi-Horvath, 2009 and references therein). By now the numerical methods for calculating the demagnetizing tensor elements for small particles are well established.

To study the magnetic structure and SW excitations in nanowire arrays, the analysis of magnetostatic coupling energy requires the evaluation of the elements of the demagnetizing tensor, or its diagonal terms, the DMFs. To obtain a qualitative estimate, a simplified approach is frequently used, where long wires were considered as long ellipsoids and interacting dipoles. However, in the dense arrays of nanowires this approach is insufficient, especially it is inconvenient for quantitative analysis of configurational phase transitions to be described in Section 23.2.1.4. In such a case the rigorous method of DMF calculation (Akhiezer et al., 1968), taking into account the specifics of nanoparticles' shape, should be performed.

The representation of a nanowire as an infinitely long cylinder is also inadequate when considering the configurational phase transition, as infinitely long cylinders do not interact via stray fields; the real aspect ratio of individual nanowire is large, but the finite length of the cylinders should be taken into account. Table 23.1 summarizes the calculated and experimentally obtained DMFs for single nanowires and interacting nanowires of actual geometry for a series of nanowire arrays. As shown in Table 23.1, row 5, the calculated values for a single nanowire are close to unity, but even this small difference is important for the analysis of phase equilibrium in the system. The DMF of the nanowire array can be calculated via summation of stray fields of all nanowires in array because of the long-range character of dipolar interaction (Table 23.1, row 7).

In general, there are two techniques of such summation: the real space approach, which was used particularly in Tartakovskaya et al. (2010) and the numerical technique of the Fourier transformation of dipolar tensor (Guslienko et al., 2000). Both methods have their advantages and disadvantages. For the array of cylindrical particles, the Fourier transformation simplifies the final calculation routine, but it is hard to apply to finite arrays and/or to disordered arrays. In both of these cases, the real space approach works without any restrictions. In the case of nanowires with rectangular cross section, according to the analytical results of Aharoni for DMF for rectangular prisms (Aharoni, 1998) the real space approach is the most useful for calculating of DMF in one-dimensional (1-D) and 2-D planar nanoparticle arrays, as it was demonstrated in Tartakovskaya (2010) and the same approach was applied to nanowire arrays in Tartakovskaya et al. (2010).

23.2.1.3 Static dipolar interaction effects in nanowire arrays

To understand the interactions and the changes in magnetic structure caused by the interacting wires in a 2-D array, a more precise calculation of the demagnetizing tensor of the individual wires and of the nanowire system is necessary.

Due to the nonvanishing dipolar field of finite length nanowires there is always an in-plane component of this dipolar field from neighboring nanowires, coupling the individual 1-D wires into a 2-D system. For a saturated 2-D infinite array, the total dipole field acting on one wire is the sum of fields produced by all other wires and can be calculated micromagnetically or written in a simple approximation:

$H_{\mathrm{D}}=4.2M_{\mathrm{s}}\pi r^{2}L/D^3$

where r is the radius of the wire, L the length, and D is the separation of wires. When all the magnetic moments of the wires are aligned perpendicular to the wire by an in-plane field, the total field acting on one wire is the sum of the dipole fields and self-demagnetizing field of the wire:

$H_{\perp }=2.1M_{\mathrm{s}}\pi r^{2}L/D^3+2\pi M_{\mathrm{s}}$

where the first term is from the dipole fields, which is parallel to the applied field. The total effective anisotropy field is

$H_{\mathrm{k}}=2\pi M_{\mathrm{s}}-6.3M_{\mathrm{s}}\pi r^{2}L/D^3+H_{\mathrm{A}}$

As the wire length L increases, when L = L_c = 2D³/(6.3r²) then H_k decreases linearly to zero. It is assumed that H_A = 0 for polycrystalline materials. When L > L_c, H_k becomes negative, that is, there is a reorientation from easy axis to easy plane behavior, as discussed in Section 23.2.1.4. In this situation for strongly coupled wires the magnetic easy axis reorients from parallel to perpendicular to the wire axis (Pardavi-Horvath et al., 2008b; Tartakovskaya, 2010; Pardavi-Horvath, 2014).

Recognizing the importance of the long-range magnetostatic–dipolar interactions in the array anisotropy, that is, configurational anisotropy in the magnetic properties of 2-D nanoparticle arrays, there were several early experimental and numerical works, dealing with the topic. In Encinas-Oropesa et al. (2001) the dipolar interaction between wires was modeled by a mean field approach and measured on nanowire samples 50–250 nm diameter, with corresponding porosity P from 4% to 38%. Linear dependence of the effective field from shape anisotropy and dipolar coupling on porosity was predicted and observed as:

$H_{\mathrm{eff}}=2\pi M_{\mathrm{s}}\left(1-3P\right)$

The importance of magnetostatic interactions in dense nanowire arrays was demonstrated by Clime et al. (2006), by calculating the in-plane and out-of-plane interaction fields using a hybrid numerical-analytical method and showing that the high field behavior of such systems is dominated by the magnetostatic effect. Analytical calculations or numerical simulation is limited by computing power to relatively small size samples and first-order approximations. However, an analytical approach was published (Laroze et al., 2007) pointing out the fact that first-order approximations are valid only when the wires are far apart, as was shown in an early work for interaction in small particle systems (Pardavi-Horvath et al., 1996). The dipolar interactions were numerically modeled by micromagnetic simulations (NMAG package) on finite and infinite periodic systems. It was shown that the finite size models fail to capture some of the important details of the system (Zighem et al., 2011).

23.2.1.4 Configurational phase transitions in arrays of nanowires

There is an interesting question about the existence of a proximity effect in small-particle magnetic systems, similar to the percolation limit in conductivity. The case of electrical percolation is well known for metal–nonmetal composites. At this volume fraction a transition occurs from positive to negative values of the real part of the composite dielectric constant, the signature of a metal. The composite conductivity changes sharply at the point of the electrical percolation threshold. A deeper analysis of the dielectric properties of metal-insulator composite, in which the metal component is supposed to obey the Drude law, a complex dielectric function and the insulator component is represented by a dielectric constant, reveals the existence of an optical threshold, depending on photon energy. It was observed that the optical threshold is different from the electrical percolation threshold. For magnetic nanoparticles in an insulating matrix, similar electrical and optical percolation effects are expected. Although electrical percolation takes place when the first continuous path opens for conduction, magnetic percolation is rather a proximity effect, due to long-range dipolar interactions between particles. The transition from noninteracting uniaxial wires to a weakly interacting system of nanowires, coupled by long-range dipolar forces, is easy to understand and calculate, based on the approach described earlier. However, a strongly interacting dipolar system can't be described by the usual far-field dipole approximation, and near field effects should be included in the description of this state. At a high density of wires the interaction becomes very strong, comparable to the exchange coupling, leading to an in-plane thin film-like behavior, as observed and discussed in Pardavi-Horvath et al. (2008b) and Tartakovskaya (2010).

Transitions between ground states of saturated nanowire systems with different kinds of dominating anisotropies are called configurational reorientation phase transitions. Change of the easy axis in magnetic crystals with temperature or pressure are well known as microscopic spin reorientation phase transitions and are caused by competing anisotropies, that is, spin–orbit coupling effects. There is a physical difference between macroscopic CRTs of a large system of nanowires, which are described here, and the classic microscopic spin reorientation phase transition. The Landau theory of phase transitions was developed to describe the microscopic phase transitions; however, the formalism of the original Landau theory can be applied to describe the change of easy axis in interacting nanostructured systems.

Spin reorientation phase transitions in ferromagnets were described in Landau and Lifshitz (1960) in the next way. An expansion of the thermodynamic potential Φ by the order parameter η,

$\Phi =K_1{\eta }^2+K_2{\eta }^4$

can be interpreted in terms of magnetic media, if we assume $\eta =sin\theta$, where θ is the angle between the average magnetization vector and the easy anisotropy axis, K₁ and K₂ are the temperature-dependent magnetocrystalline anisotropy constants. Usually K₁  K₂, however, in the vicinity of the phase transition, where K₁ changes its sign at some temperature point, K₂ and its sign determine the order of the phase transition (first or the second one). If K₂ = 0, the next term of the expansion should be considered.

The formalism developed for macroscopic configurational reorientation phase transitions, as described later, is based on the formal analogy between the classic formalisms of phase transitions and reorientations in magnetic nanostructures. Returning now to the arrays of magnetic nanowires, one can utilize the same formula (23.6), but now the magnetic energy plays the role of thermodynamic potential, whereas the dominating contribution to the anisotropy parameters, especially in soft magnetic materials, are the dipolar, magnetostatic forces, leading to the shape anisotropy. The first anisotropy constant K₁ is proportional to the effective DMF of the wire in an interacting array. In the transition point the DMF approaches zero. FMR measurements, shown in Figure 23.2, unambiguously demonstrate that the transition from easy axis to easy plane behavior of dense nanowire array takes place due to the dipolar interactions between wires (Pardavi-Horvath et al., 2008b).

However, it is not a phase transition in a thermodynamic sense (a detailed discussion about difference of phase transitions in bulk magnets and in arrays of magnetic nanoparticles is in Tartakovskaya (2010)). Because of its dependence on geometrical configuration of nanowires in array, such a reorientation transitions in arrays of nanoparticles called a CRT.

Here the second-order anisotropy parameter, K₂, also has a dipolar origin; it can be calculated taking into account the nonuniformities of the magnetic structure, due to the nonuniform internal field near the top and bottom edges of nanowires. The sign of K₂ determines the order of the configurational phase transition in the same way, as the corresponding temperature-dependent parameter of magnetocrystalline anisotropy determines the order of the phase transition in the classic Landau's theory (Tartakovskaya, 2010; Tartakovskaya et al., 2001, 2010).

Similar phenomena can be investigated in 1-D and 2-D arrays of planar nanodots and nanostripes, based on the general approach to the calculation of DMFs, described in Section 23.2.1.2. For instance, in a 1-D array of ferromagnetic stripes the CRT between in-plane and out-of-plane ground states takes place due to the competition between shape anisotropy (which is reduced compared with a nonpatterned film) and out-of-plane magnetocrystalline anisotropy (Choe, 2008), or a combination of anisotropies of magnetocrystalline and magnetoelastic origin (Navas et al., 2010).

On the contrary, in 2-D arrays of ferromagnetic wires the CRT can arise in magnetically soft material as a result of the competition between anisotropies of the same dipolar origin: that is, intrawire and interwire shape anisotropies (Guslienko et al., 2000). The other difference of CRT in 2-D arrays of circular nanowires and 1-D array of stripes is that stripes can be treated as infinite in this context, whereas such CRT exists in 2-D arrays of circular nanowires of finite length. As it was shown in Tartakovskaya et al. (2010) by calculating the effective magnetostatic energy taking into account the nonuniform micromagnetic state at the end of the wires, the CRT in long ferromagnetic nanowires due to the competition between different kinds of dipolar interactions is a first-order magnetic phase transition in the Landau sense.

A quite different reorientation transition results if an external magnetic field is applied to the nanowire sample, perpendicular to the wire axis. In this case the competition between the Zeeman interaction and shape anisotropy leads to a transition of the second order (Tartakovskaya, 2005), with the usual square root dependence of the order parameter near the phase transition point, $\theta \propto {\left(2-2H_0/2\pi M_{\mathrm{s}}\right)}^{1/2}$, where the order parameter θ is the angle of the magnetization with respect to the wire axis. This has a special influence on the SW spectra. The field dependence of frequencies of the dipolar-exchange modes in a nanowire arrays is quite different in the cases, when the external field is applied parallel or perpendicular to the wire. As it was observed in several BLS and FMR experiments (Encinas-Oropesa et al., 2001; Wang et al., 2002; Nguyen et al., 2006; Stashkevich et al., 2009), the SW dispersion behavior is more or less monotonic in parallel external field, whereas in perpendicular field it has a spike, when the value of this field approaches the shape anisotropy field of the infinite nanowire, $H_0=2\pi M_{\mathrm{s}}$, as shown in Figure 23.3. This well-defined minimum of the dispersion relation corresponds to the softening of modes in the vicinity of the second-order reorientation phase transition. In addition, a special splitting of SW mode arises, due to the violation of the circular symmetry (Tartakovskaya, 2005).

23.3.1.1 Dipolar-exchange SW modes of individual cylindrical nanowires

When fresh FMR results stimulated the development of a new theory on spin dynamics in nanowires, a definite theoretical basis already existed. The main ideas of SW formalism in magnetic films and multilayers and established calculations of magnetostatic modes in ferromagnetic cylinders were naturally applied to nanowires. Taking into account the dominant interactions, all the variety of magnetic films and multilayers are divided into three classes (Camley, 1999): thick, thin, and ultrathin. For thick films (more than a few hundreds nanometers), the theory implements a continuum model where the dipolar field dominates, whereas exchange interactions could be neglected. It is the same domain of small dimensions, where the theory of magnetostatic SW in particles with cylindrical shape was developed (Joseph and Schlonann, 1961). In ultrathin films (less than 100 atomic layers) a discrete model is more appropriate. The so-called thin films with thickness of a few tens of nanometers lay between these two limiting cases: the continuum model is still valid; however, both dipolar and exchange interactions should be taken into account (Kalinikos and Slavin, 1986). It is evident, that magnetic nanowires with radii from 10 to 100 nm belong to this last dimension range. Such nanowires were grown for a long time. Electrodeposited Ni wires in polymer templates were probed by the FMR technique Figure 23.4 (Ebels et al., 2001; Encinas-Oropesa et al., 2001). A special double-peak of absorption derivative spectra was observed for nanowires with R = 40 nm, whereas for nanowires with much larger or smaller radii this additional peak vanished (or it was much less pronounced). This strange behavior was not observed previously and could not been explained in the frame of magnetostatic theory.

Here, the theoretical approach to calculate dipolar-exchange SW in nanowires, given in Arias and Mills (2001), will be reviewed, as it provides a good understanding of the possible generalization of the theory, as well, as unavoidable restrictions of it. Following Arias and Mills (2001), we consider an infinitely long uniform cylindrical nanowire embedded in the external magnetic field of a strength H₀, directed along the nanowire. In such a case the average magnetization is parallel to the nanowire (to axis z), whereas the linearized SW mode has two perpendicular components, m_x and m_y. For the following analysis we should note an important consequence of the infinite length and uniformity of the nanowire: It gives the possibility to write the spatial dependence of SW components and of the corresponding magnetic potential Φ_M as

$\begin{array}{l}{m}_{\alpha }\left(x,y,z\right)=m_{\alpha }\left(x,y\right)exp\left(\mathrm{i}kz\right)\\ {\Phi }_{\mathrm{M}}\left(x,y,z\right)={\Phi }_{\mathrm{M}}\left(x,y\right)exp\left(\mathrm{i}kz\right)\end{array}$

Here k is a wave vector along the nanowire. In finite cylinders, where the dependencies like Eq. (23.8) are not valid, the theory of dipolar-exchange SW is usually based on use of a basic set of functions followed by the implementation of the Ritz method of finding the approximate solutions, like in Kakazei et al. (2004).

The SW components obey the Landau–Lifshitz (L–L) equations,

$\begin{array}{l}\mathrm{i}\Omega m_x=\left(H_0-D{\nabla }^2\right)m_y+M_{\mathrm{s}}\partial {\Phi }_{\mathrm{M}}/\partial y\\ -\mathrm{i}\Omega m_y=\left(H_0-D{\nabla }^2\right)m_x+M_{\mathrm{s}}\partial {\Phi }_{\mathrm{M}}/\partial x\end{array}$

where Ω is the frequency of SW, the parameters of the magnetic material of the wire are the saturation magnetization M_S and an exchange stiffness D. The magnetic potential Φ_M inside the wire obeys the magnetostatic condition

${\nabla }^2{\Phi }_{\mathrm{M}}-4\pi \left(\frac{\partial m_x}{\partial x}+\frac{\partial m_y}{\partial y}\right)=0$

whereas outside the magnetic material Φ_M satisfies the Laplace's equation. The set of differential equations (23.8) and (23.9) can be reduced to one equation of higher order for the magnetic potential:

$\left[\left(D{\nabla }^2-H_0\right)\left(D{\nabla }^2-H_0-4\pi M_{\mathrm{s}}\right)\right]{\nabla }^2{\Phi }_{\mathrm{M}}+4\pi M_{\mathrm{s}}\left(D{\nabla }^2-H_0\right)\frac{{\partial }^2{\Phi }_{\mathrm{M}}}{\partial z^2}=0$

Such a symmetrical form of the equation for the magnetic potential is obtained due to the unperturbed circular symmetry of the given geometry of the nanowire. As we will see later, this equation cannot lead to the exact solution in the case when this symmetry is violated.

Using Eq. (23.8) a solution of (23.4) can be written in cylindrical coordinates as

${\Phi }_{\mathrm{M}}\left(x,y,z\right)=J_{\mathrm{m}}\left(K\rho \right)exp\left(\mathrm{i}m\varphi +\mathrm{i}kz\right)$

where J_m(Kρ) are Bessel function of the first kind.

Inserting Eq. (23.12) into Eq. (23.11), we find a bicubic equation by the radial wave vector K,

$D^2{\left(K^2+k^2\right)}^3+D\left(2H_0+4\pi M_{\mathrm{s}}\right){\left(K^2+k^2\right)}^2+\left(H_0\left(H_0+4\pi M_{\mathrm{s}}\right)-4\pi M_{\mathrm{s}}D{k}^2\right)\left(K^2+k^2\right)-4\pi M_{\mathrm{s}}{H}_0{k}^2=0$

The magnetic potential inside the wire can be represented as a linear combination of three linearly independent solutions (which correspond to the roots of Eq. (23.13)) of the form Eq. (23.12), whereas outside there exists one independent solution of Laplace's equation in the form of the modified Bessel function. The usual way to find a dispersion relation for the SW is to apply the appropriate boundary conditions, the number of which should be the same as the number of solutions, that is, in our case it is four. Two of these boundary conditions stem from general magnetostatics: they are defined by the continuity of the radial component of the induction and the tangential component of the dipolar field. The other two boundary conditions should define the magnetization behavior on the side surface of the cylinder.

In the article described earlier, Arias and Mills considered the most general case of mixed boundary conditions (medium pinning), not specifying their origin. They proved that the boundary conditions lead to the special effects in nanowires, in particular to the hybridization of the uniform FMR mode with the exchange modes with small wave vector, what strongly distinguishes the SW dynamics in nanowires and thin films. In such a case some exchange modes can lie below the FMR mode in wires (Figure 23.5), whereas in films they always lie above. The situation, when the frequencies of FMR mode and one of exchange modes are close, can be realized in wires for specific values of radii. In such a way this theory explains the satellite of the FMR peak, observed in the experiments (Figure 23.6).

The agreement between the double-peaked FMR signal and the explanation of this shape by dipolar-exchange SW theory highlighted the validity of the theoretical approach, but could not be considered a comprehensive approval of the theory. In early 2000, Ni nanowire samples were fabricated by electrochemical deposition of ferromagnetic material (Ni) into hexagonally arranged alumina membrane, shown in Figure 23.7 (Nielsch et al., 2000). These new, improved samples were investigated by BLS technique. These experiments quantitatively proved Arias's and Mills's theory (Wang et al., 2002). The dependence of the measured SW frequencies in the wires on the applied field and wire radius demonstrated the predicted quantization of SW in the confined nanowires' geometry and unambiguously showed the validity of dipolar-exchange approach, whereas previous theories, where exchange was ignored, turned out to be insufficient in this case.

In a theoretical fitting of the experimental data (Wang et al., 2002), the approximate formalism based on general dipolar exchange theory was used. It employed the concept of small or large pinning of magnetic modes on the surfaces of nanowires. In the absence of the external field, it gives a possibility to write the expression for the frequency of the bulk standing modes (that is, when $k\ll K$) in the simple analytical form.

$\Omega ={\left[D{\left(\frac{a_{\mathrm{m}}}{R}\right)}^2\left[D{\left(\frac{a_{\mathrm{m}}}{R}\right)}^2+4\pi M_{\mathrm{s}}\right]\right]}^{1/2}$

where the parameters a_m correspond to the antinodes or nodes of the SW functions on the surfaces and, in such a case, the ratios a_m/R play the role of radial wave vectors of SW. A good agreement between the experimental data and the calculations by formula (23.14) was achieved under the assumption of low pinning on the surfaces, as shown in Figure 23.8.

It is appropriate to mention here about the discussion of pinning in confined systems. A great deal of attention was paid to the question of dipolar boundary conditions in ferromagnetic nanostructures. It was shown that in thin nanoelements with rectangular cross section (Guslienko and Slavin, 2005), as well as in planar disks (Ivanov and Zaspel, 2002), the inhomogeneity of dipolar field near the sharp angles on side leads to the strong pinning of SW. However, in infinite cylindrical wires such strong dipolar inhomogeneities can be avoided, so only the surface anisotropy, which is not large on smooth surfaces, can be the source of the pinning. It explains why the low pinning conditions used in Wang et al. (2002) for fitting the data in circular nanowires is a physically reasonable choice.

Summarizing, the quantized dipolar-exchange SW modes in long isolated cylindrical nanowires were investigated in early 2000s, and remarkable agreement between theoretical results and experimental data (both of FMR and BLS techniques) was achieved. From the short overview given here, it is evident that it was possible to find exact analytical solutions just due to the unperturbed circular symmetry of the problem. Future extension of the initial investigations should answer the questions: How did the dipolar interaction between nanowires influence the dynamical behavior of the magnetization in both cases of arrays of saturated or nonsaturated wires? What happens if the cross section of the nanowire is not completely circular? How can the violation of the circular symmetry by the external field with nonzero perpendicular component change the result?

The attempts to generalize the theory by taking into account all these physical conditions simultaneously demonstrated that it is a complicated task (Nguyen and Cottam, 2004, 2005a,b, 2007). Further, we clarify these questions in more details.

23.3.1.2 Collective SW modes in arrays of interacting nanowires

The influence of dipolar interaction between nanowires on the SW modes should be much more pronounced than between thin magnetic layers in multilayers. Actually, the static dipolar interaction between parallel, saturated infinite nanowires is equal to zero in the same way as interaction between ideal films in a multilayer. However, there is a remarkable difference in the interaction between SW modes in thin films in a multilayer and the nanowires in an array. The dipolar field generated by SW in a film decreases with a distance of its surface exponentially, whereas the dipolar field from the uniform mode of ferromagnetic cylinder falls of inversely with the square of the distance (Arias and Mills, 2003). For instance, a clear influence of dipolar interaction of the lowest mode was observed in FMR experiments with samples of Ni nanowire arrays with different porosity (Encinas-Oropesa et al., 2001).

The theory of collective SW in arrays of ferromagnetic nanowires, based on a multiple scattering approach, was developed by Arias and Mills (2003). The formalism demonstrated the advantages of real space approach, which is applicable to any ordered or even disordered nanowire array. An explicit expression for SW frequencies was found for collective SW of a nanowire pair. Numerical calculations of SW dispersion in a linear array of ferromagnetic cylinders show hybridization between collective modes and the standing wave modes of individual cylinders. Results of this theory were experimentally proved by BLS in hexagonal arrays of permalloy (Liu et al., 2005) and FeCo (Wang et al., 2006) nanowires. An excellent agreement between the theory and experiment was achieved. It was shown, that in 2-D arrays with usual magnetic and geometrical parameters only the lowest frequency SW mode can be influenced by the interwire interaction. The reduction of the frequency of the collective SW, in comparing with the frequency of the lowest mode of an individual wire, was demonstrated by the investigation of BLS data in arrays with various interwire spacing as shown in Figure 23.9.

A comprehensive study, both experimentally and by micromagnetic simulation, of a densely packed array of permalloy nanowires with wire diameter much greater than the exchange length demonstrated new features of SW dynamics in nanowires (Dmytriiev et al., 2012). Particularly, the tunneling of end modes was observed, which is more efficient in arrays with smaller distance between nanowires, due to the common action of static and dynamic parts of the dipolar field in an array.

For the present, we considered individual nanowire or array of the nanowires magnetized in the same direction. But in a real case, the external field might not be strong enough to saturate the sample. The shape anisotropy of long individual nanowire forces the average magnetization to lie along the nanowire. The individual wires are bistable, having two possible ground states, up and down. As was mentioned, static dipolar interaction between parallel circular nanowires with homogeneous magnetization and infinite length is impossible. But nanowires never happen to be infinitely long, and, so they interact via stray fields arising due to poles on their end surfaces. If we consider only two wires, the energetically favorable is opposite alignment, as it allows flux closure through the wires (Figure 23.10).

However, if the interacting nanowires are arranged on a hexagonal lattice, they cannot establish such an antiferromagnetic order where the closest neighbors to every nanowire are oppositely aligned (Figure 23.10). In such a case, it is convenient to consider a nonsaturated array of axially magnetized interacting nanowires as an aggregate of two oppositely magnetized wire populations, without specifying the magnetic order of the whole array (the same concerns disordered arrays of parallel wires). The calculation of the effective permeability tensor on the basis of a Maxwell–Garnett homogenization procedure (Boucher and Ménard, 2008) was done in this representation. Further, the calculation results were compared and found to agree well with the data of broadband microstrip line measurements of interacting amorphous CoFeB nanowires (Boucher et al., 2009).

The investigation of unsaturated arrays has a strong motivation: to develop microwave devices, which can be operated in low magnetic fields. As a result, attention was paid to FMR studies of magnetic excitations of such materials. As it was observed, in both dilute (Encinas et al., 2007) and dense (Kou et al., 2009) arrays, the FMR signal strongly depends on the remanent state of the sample. It was shown, that the Kittel formula, which is usually applied to FMR data in thin films, is insufficient in the present case. Nanowires magnetized in two different directions have different resonance frequencies, so two FMR peaks arise due to the presence of two (up and down) populations of wires. Double FMR resonance was observed both in parallel (Carignan et al., 2009a,b) and perpendicular (Boucher et al., 2011) external static fields. It was proven that the origin of double-peak shape in such a case is just the existence of two unsaturated populations: Above the saturation one peak vanished, and only one FMR peak was observed. An influence of magnetocrystalline anisotropy on microwave properties was probed by magnetometry and BLS in cobalt, permalloy, and nickel arrays of unsaturated nanowires (Cherif et al., 2011).

In De La Torre Medina et al. (2010), the FMR absorption properties of two bistable nanowire arrays (30 nm diameter Co₅₅Fe₄₅ and 40 nm diameter Ni₈₃Fe₁₇) were investigated. The in-field measurements over the full hysteresis cycle demonstrated, that the two SW branches that correspond to the double FMR peak have different dispersion relations, dependently on the magnetic configuration (Figure 23.11), whereas the intensity of every FMR peak is proportional to the fraction of the correspondent nanowire population. The FMR- FORCs (first-order reversal curves diagrams) technique allowed measuring of the dipolar interaction field and its dependence on the magnetic configuration of the bistable array of nanowires, and the results were explained on the basis of analytical mean-field theory (De La Torre Medina et al., 2010).

23.3.1.3 SWs in ferromagnetic nanowires with noncircular cross section

Interesting attempts to apply Yablonovitch's ideas related to photonic crystals to magnetic superlattices were made in the context of 2-D arrays of long nanocylinders (Vasseur et al., 1996; Tiwari and Stroud, 2010), but for the present time planar arrays of nanoparticles are considered the most suitable material for magnonics application (Kruglyak et al., 2010). A large domain of SW investigations in planar arrays of dots, antidots, and nanostripes is beyond the scope of the present review (see, for instance, Wang et al., 2009; Kumar et al., 2014 and references therein). The significant progress in fabrication and investigation of thin magnetic nanowires with rectangular cross section (which also are called nanostripes or strips) was the driving force to attempt the theoretical investigation of nanowires with noncircular cross sections. A calculation of SW dispersion of linear chain of parallel cylindrical nanowires (Arias and Mills, 2003) was an attempt to approach this physical situation. For a quantitative fitting of the experimental data the more exact description of the shape of the wires was a challenging task. However, to develop the exact theory of dipolar-exchange modes in nanostripes, based on the example of circular nanowires, seemed unfeasible. The method of calculations of SW frequencies and wave vectors was presented for nanowires in the magnetostatic limit, with exchange interaction neglected. The key to the solution was to apply the extinction theorem. The special attractiveness of this approach is that it does not use a basic set of functions, and as a result, it can be applied to wires of arbitrary cross section. In the first article on the subject (Arias and Mills, 2004), the application of the general results to the wires with square, rectangular, and ellipsoidal cross section was demonstrated. In the next article (Arias and Mills, 2005), the generalization of the approach to the case of large width/height ratio was presented, which allowed to apply the calculations to long, thin nanostripes. It is interesting that this problem was solved at the same time in another way: The rectangular cross section was approximated by an ellipsoidal, then the exact solution for dipolar-exchange SW was found by the use of the Mathieu basic set of functions (Gubbiotti et al., 2004).

23.3.1.4 Magnetic structure and dynamics of multilayered nanowires and magnetic nanotubes

To make devices with controlled properties, recently nanomaterials of new geometries were proposed, like multilayered nanowires (MNWs) and nanotubes. The attractive common feature of MNW and nanotubes is the presence of extra dimensions, which makes their properties more easily tunable, compared with nanowires.

Fabrication of the first MNW was stimulated by the interest in giant magnetoresistance, observed in multilayers, and was reported almost simultaneously by several research groups (Piraux et al., 1994; Blondel et al., 1994; Liu et al., 1995). These novel materials, consisting of alternating layers of Co and Cu, were made by deposition into a disordered array of polimer pores. Later a variety of materials was used for MNW fabrication (Saidin et al., 2013; Yang et al., 2014; Wang et al., 2010). Remarkable progress was achieved in electrochemical synthesis of large-area highly ordered MNW by using ordered templates like nanoporous anodic alumina. (See in Part I of this book.) Due to this technique the magnetic segments in MNW became ordered not only along the wires, but they are arranged into a planar ordered matrix as well, that is, it is an ordered crystal in all three directions.

Together with extensive investigation as a new magnetoresistive material (Dubois et al., 1997; Nasirpouri et al., 2007; Pullini et al., 2007), MNW were recognized as a perfect model structure that opens new possibilities for investigations of interactions between nanoparticles. Depending on varying parameters, like the aspect ratio of magnetic segments and its relation to the exchange length, the distance between the wires and the height of nonmagnetic components, MNW can combine properties of planar magnetic nanodisks or elongated magnetic nanorods; moreover, it can be investigated how the properties of arrays of such nanoparticles, due to the dipolar interaction between them, are related to the geometries of long wires or multilayered thin films. Three different micromagnetic states, in-plane flower, out-of-plane flower, and vortex states, were recognized in Ni/Cu MNW by analysis of hysteresis loops, depending on the different aspect ratios of magnetic components (Chen et al., 2003). The corresponding phase diagram, which shows what micromagnetic state can be realized in the system with various diameter/exchange length and aspect ratios, qualitatively agreed with the phase diagrams of 2-D arrays of elongated cylinders (Ross et al., 2002) and planar disks (Scholz et al., 2003).

Configurational phase transition and reversal modes in CoNi/Cu MNWs were investigated by Tang et al. (2007). It was shown, that by varying the thickness of magnetic segment while keeping the thickness of Cu constant, the easy axis can be turned from the direction perpendicular to nanowire to parallel to nanowire, whereas the magnetization reversal will be of different types: from coherent rotation in disk-shaped magnetic segments, to a combination of coherent rotation and a curling mode in a rod-shaped ones. Targeted investigations of dipolar interactions between magnetic segments in CoCu/Cu MN by FMR and magnetometry were presented in De La Torre Medina (2010). It can be observed from the FMR spectra (Figure 23.12) that the resonance field of the multilayered sample is higher than in the sample of pure Co nonlayered sample. A comparison of experimental results and model calculations allowed to develop a comprehensive anisotropy diagram (Figure 23.13) that describes what is the direction of the easy axis (parallel or perpendicular to the wire axis), depending on the aspect ratio of magnetic segments, exchange length, and thickness of Cu. This anisotropy was proved to be only of dipolar origin.

Special attention was paid recently to the magnetostrictive iron–gallium alloys (FeGa), named galfenol (for a review on galfenol alloy, see Atulasimha and Flatau, 2001), because of their anticipated use in actuators, transducers, and magnetic sensors. The first fabrication of highly ordered galfenol nanowires, deposited into nanoporous anodic alumina membranes, was reported by McGary and Stadler (2005). Then, a comprehensive study of galfenol-based magnetostrictive MNW was undertaken (Reddy et al., 2011). MNW samples with different diameters of wires, aspect ratio of galfenol segments, and different thickness of nonmagnetic Cu were grown to investigate magnetization reversal properties (Figure 23.14).

Magnetocrystalline anisotropy was excluded due to the randomization of crystalline easy axes of the polycrystalline segments, so the magnetic properties of such MNW can be treated as the result of the interplay between intrasegment exchange, dipolar interaction, and Zeeman energy. Comprehensive study of hysteretic properties and magnetization reversal mechanisms in FeGa/Cu MNW and comparison to individual MNWs with measurements on interacting MNW in arrays (Park et al., 2013; Reddy et al., 2012) conclusively shows the importance of dipolar interactions both between segments in nanowire and between nanowires in the array. Depending on the size of the magnetic segments, multilayer nanowires can present very interesting vortex structure from rigid vortex in disks, to specific vortices of elongated cylinders (Park et al., 2010), it makes the theoretical investigation of dipolar interaction between vortices convenient (Altbir et al., 2007).

Investigation of SW dynamics in cylindrical MNW was performed only theoretically. SW spectra including damping were obtained by solving the Landau–Lifshitz–Gilbert equation in the “effective-medium” approximation. The results could be used in constructing magnonic crystals (Kruglyak et al., 2005).

A special case of the nanowire geometry is the case of nanotubes. Recently the interest to the theoretical and experimental investigation of ferromagnetic nanotubes is intensified. (See details of fabrication in Chapter 24.) There are several advantages of the tubular shape that make the potential application of magnetic nanotubes even wider than that of the homogeneous nanowires. For example, changing the wall thickness gives an additional varying geometrical parameter, allows tuning of the magnetic properties more effectively. It was shown, that dipolar fields in arrays of Ni nanotubes can be controlled by varying the thickness of the tube wall. It was shown also that the dipolar interaction between nanotubes is weaker than between solid nanowires, which observation gives the possibility to decrease the undesirable effect of dipolar interaction in dense arrays (Velázquez-Galvan et al., 2014). Basic magnetic properties can be manipulated in a nanotube due to its large surface fraction. The temperature dependence of the magnetization also can be changed. It was shown (Sharif et al., 2013) that reduced dimensionality and enhanced surface effects lead to a deviation from Bloch law, $M=\left(1-B{T}^{\lambda }\right)$, where the exponent λ decreases considerably compared with both nanowire and bulk. Manipulation of domain wall motion is quite important for applications (see Part II of this book). It is known, that Walker breakdown exists in bulk materials, which is an unavoidable restriction on domain wall velocity, V. However, it was shown (Yan et al., 2011) by numerically solving the Landau–Lifshitz–Gilbert equation that the Walker breakdown does not exist in nanotubes: although above the critical velocity of the vortex domain wall the slope of the curve V(H) suddenly decreases, whereas after this change V(H) continues to increase (Figure 23.15). It was shown that this striking feature of domain wall motion in nanotubes is connected with magnon emission by the domain wall, which process is similar to the Cherenkov emission of photons. Dispersion of SW, associated with the domain wall, was calculated numerically (Figure 23.16), and the result on dispersion is in agreement with analytical calculations (Gonzalez et al., 2010).

The magnetization singularity on the axis, which exists in wires, is “cut out” in a nanotube. It allows to provide a comprehensive and rigorous theoretical investigation of technologically important properties of nanotubes. The magnetic configurations and reversal in thick nanotubes, with opposite chirality of end domains (Chen et al., 2011), and thin ferromagnetic nanotubes, with the same chirality of end domains (Chen et al., 2010) were investigated recently by both by micromagnetic simulation and analytical approach. A theoretical procedure was proposed to control the propagation and chirality switching of a vortex domain wall in nanotubes by magnetic field pulses (Otalóra et al., 2012).

23.4.1.1 EMW interactions with nanowire structures

For successful application of EM phenomena, the understanding of the details of the EMW propagation in these devices is required. As early as in 1887, Lord Raleigh discussed wave propagation in periodic lattices. He developed the mathematical formalism to describe the wave–material interaction. From there many fruitful applications followed; one of them was the formalism of Bloch waves for the description of electrons in a crystal. In a review by Elachi on waves in active and passive periodic structures, the theory of propagation in unbounded and bounded periodic media, and a wide variety of applications were discussed (Elachi, 1976). In the 1960s the main emphasis was on the exact solution of EMW equations in periodic, sinusoidal, and laminated media. With the arrival of periodic nanostructures and potential applications, it was necessary to develop theories of the interaction of these nanostructures with EMW and describe the propagation and diffraction effects in geometrically confined (bounded) structures. It should be kept in mind that EMW treatments include optical frequencies (light propagation) and X-rays.

With the advent of periodic magnetic nanostructures the task became even more complex. The characteristic size of the elements, 2r, of these 1-D, 2-D, and 3-D periodic structures is much below the wavelength λ of the EM wave in the material, 2r < λ. Still, the EM field over any of the elements is rapidly changing, making it necessary to include propagation effects. Moreover, the periodicity of the nanoelement lattice Λ (wires, layers, spherical particles) is not commensurate with the rf field periodicity, Λ ≠ nλ, as illustrated in Figure 23.19.

To describe the EMW propagation in magnetic nanostructures, it is necessary to solve Maxwell's equations together with the L–L equation rigorously and with proper electrodynamic boundary conditions, including the possible nonlinear response of the system. At the same time, to model very long, parallel nanowires is a relatively simple task among the periodic magnetic nanostructures. Usually there is an easy axis of magnetization, coinciding with the wire long axis and with the direction of the applied bias dc magnetic field.

An often-neglected aspect of the high-frequency behavior of small particles is that for the nanowire array geometry there is a large microwave magnetic field generated outside the wire of radius r, by the precession of the magnetization. For a SW with finite wave vector k, this field falls off as ~ exp(2kr)/r^1/2 far from the wire. For long wavelength modes, this field has a very long range. In the limit of zero wave vector, the field created by the precession of the magnetization falls off inversely with the square of the distance from the center of the wire. The existence of this large field outside the nanowire is a substantial change with respect to the thin film case. For the thin film, for the uniform precession mode, excited in an FMR experiment, the macroscopic magnetic field created by precession of the magnetization is completely confined to the film. For a SW whose wave vector k is parallel to the film surfaces, there is a macroscopic field outside with the spatial variation ~ exp(2kz), if the film surfaces are parallel to the xy plane, but the prefactor that controls the strength of this field scales as 4πM_s(kd), where d is the film thickness. This field is thus very weak in the thin film limit, or whenever the wavelength of the SW is large compared to the film thickness.

On the other side, the existence of large rf demagnetizing fields outside the nanowire has interesting implications. They contribute to the interactions between the nanowires, so one is led to explore the collective excitations of the nanowire array. By changing the ratio of the interwire separation to the range of interaction, SWs of variable wavelength k could be excited. In the case of nanowire arrays, the nature of their collective excitations due to dynamic demagnetizing effects is a most interesting topic, and it was discussed in Section 23.3.1. This dynamic contribution to the interactions at high frequencies will affect the “proximity effects,” as described earlier. The increased interaction strength is expected to reduce the threshold wire separation for the onset of magnetic percolation. This effect has a very strong shape dependence, as the dynamic demagnetizing fields are very sensitive to the shape.

Several approaches were developed to treat the EMW interaction with nanowire structures; however, all those methods have to face some numerical complications: the devices, based on magnetic nanostructures are of finite size, roughly scaled with half of the wavelength, that is, the size can be several mm, too big for a usual micromagnetic simulation, whereas the nm size elements require fine discretization. Another significant problem is to deal with the statistical distribution of the huge number of particles in any device-size geometry. The question of how to take into account the statistical distribution of particle size, shape, separation in calculating the interactions in a nanowire (or other nanostructured magnetic system) is mostly open. Likewise, the temperature dependence is usually ignored; however, these nanoparticle devices have to operate in a feasible wide temperature range, where the stability of such small magnetic particles against thermal fluctuation is uncertain.

Most of the theoretical modeling of numerical simulation of FMR and spin dynamics in an applied rf field, is based on the L–L formalism, including damping, and usually neglecting exchange interactions. The “magnetic” approach either deals with the system as an assembly of isolated spins, or the magnetic particles are treated as macrospins on a regular array. A review on the simulations of magnetodynamics in nanometric systems was published by Schmool (Schmool, 2009). Micromagnetic calculations were performed by several authors. It is based on solving the L–L equation with damping included. The most time-consuming part of the calculations is to find the demagnetizing tensor elements, necessary to obtain the equilibrium magnetic structures. The nanostructure chapter of Schmool (2009) is mostly about nanodots, but the methodology is adaptable for other geometries, like to wires. Rivkin et al. (2007) applied translational symmetry principles to analyze an infinite array of magnetic dots and got good agreement with experiments.

Another way to model composite magnetic materials is using an effective media approach. Calculating the permeability tensor and the permittivity, the EMW propagation in a nanocomposite of magnetic particles embedded in a dielectric matrix can be analyzed by using the Maxwell–Garnett formula. The task is to calculate microscopic parameters and from there to determine the macroscopic behavior of the nanostructure. Hillion analyzed the propagation of a harmonic plane wave in a medium consisting of magnetic particles in a periodic dielectric matrix and showed the resulting TE and TM modes (Hillion, 2010). Boucher and Ménard (2008) applied a similar, general model to obtain the effective permeability and permittivity tensors of ferromagnetic nanowire arrays, based on a Maxwell–Garnett homogenization procedure. It incorporates the effects of the geometric parameters of the array, the shape of the wires, and their intrinsic dielectric and magnetic properties. Effective external tensors, which include the dynamic dipolar dielectric and magnetic interwire interactions, are then derived to provide a link between the effective susceptibilities of the array and those measured in microwave cavity experiments. The authors Wang et al. (2013) and Liberal et al. (2011) applied a numerical homogenization technique to create an effective medium and extract the dispersive and nonreciprocal permeability tensor of a magnetic nanowire system. Using the resulting parameters, they analyzed the performance of microwave devices (isolators, phase shifters) by full wave simulations.

It is possible to solve the problem rigorously by applying a mathematical technique, developed for MMIC microwave devices, based on the decomposition into autonomous blocks with virtual Flouqet channels (FAB; Golovanov, 2006). A rigorous electrodynamic model was built (Makeeva et al., 2009c), based on the solution of the nonlinear 3-D propagation–diffraction boundary problem, to describe the interaction of EMW with anisotropic 3-D nanostructures. The model was applied to solve the size and shape effects in nanowire arrays up to photonic frequencies. The model solves the nonlinear Maxwell's equations (23.15) and (23.16) with electrodynamic boundary conditions, together with the L–L equation, including the exchange term (23.17) and (23.18):

$\mathrm{curl}\overrightarrow{H}\left(t\right)={\varepsilon }_0\varepsilon \frac{\partial \overrightarrow{E}\left(t\right)}{\partial t}+\sigma \overrightarrow{E}\left(t\right)$

$\mathrm{curl}\overrightarrow{E}\left(t\right)=-\frac{\partial \overrightarrow{B}\left(t\right)}{\partial t}$

$\frac{\partial \overrightarrow{M}\left(t\right)}{\partial t}=-\gamma \overrightarrow{M}\left(t\right)\times \left(\overrightarrow{H}\left(t\right)+{\overrightarrow{H}}_q\left(t\right)\right)+{\omega }_r\left({\chi }_0\overrightarrow{H}\left(t\right)-\overrightarrow{M}\left(t\right)\right)$

${\overrightarrow{H}}_{\mathrm{q}}\left(t\right)=q\mathrm{\Delta }\overrightarrow{M}\left(t\right)$

where E and H are the EM fields, M is the magnetization, H_q is the effective exchange field, σ is the electrical conductivity, γ is the gyromagnetic ratio, ω_r = αγH₀ is the relaxation frequency, χ₀ is the susceptibility, q = 2A/μ₀M₀ is the exchange constant, and H₀ and M₀ are the bias magnetic field and the saturation magnetization.

A 3-D periodic array of nanowires is assumed, as shown in Figure 23.20, with a, b, c periodicity along axes x, y, z. The 3-D magnetic nanocomposite is divided by using the decomposition of the elementary cell of array onto FABs (2r—diameter, l—length of nanowire). The cells are in the form of rectangular parallelepipeds with virtual Floquet channels (FAB). A monochromatic homogeneous plane EMW (TEM-wave, fields E = Ex₀, H = Hy₀; wave vector k; frequency ω) is incident on the input cross-section S₁ of a 2-D periodic array of metallic magnetic nanowires, embedded in a nonmagnetic, dielectric matrix, having relative permittivity ɛ_r = 5 and relative magnetic permeability μ_r = 1. A bias magnetic field H₀ = H₀y₀ is applied normal to the propagation direction z (Figure 23.20a), along the axis of nanowires (Figure 23.20b). For the calculations the model of the elementary cell of the 2-D periodic array is considered, where each cell contains one ferromagnetic nanowire (Figure 23.20c) as an autonomous block with Floquet channels (FAB). The values of parameters used in the calculations for magnetic nanowires are 2r = 50 nm, l = 500 nm of Fe (4πM₀ = 21,580 kG, exchange constant A = 2.2 × 10 ⁻⁹ Oe cm², Gilbert damping α = 0.0023; conductivity σ = 1.03 × 10⁵ S m^− 1). The periodicity of the array is a = b = 256 nm, c = 550 nm.

23.4.1.2 Shape and size effects in the EMW propagation in nanoarrays

High permeability, low loss magnetic materials are required to replace the low magnetic moment ferrites in mm-wave devices. Metallic ferromagnetic nanoparticles in an insulating matrix, are good candidates for the new generation of microwave materials. The microwave properties of such a composite can be controlled by the shape, size, and periodicity of the novel nanocomposite microwave material, as it was shown by (Pardavi-Horvath et al., 2009). Using the computational algorithm for calculating the FAB conductivity matrix Y (Golovanov and Makeeva, 2009), the complex wave number Γ₀ of the fundamental mode of the clockwise and counterclockwise polarized modes and quasi-extraordinary EMWs in 3-D arrays of magnetic nanowires were determined from the characteristic equation (Makeeva et al., 2009b). The main results of the calculation of the real and imaginary parts of complex wave number Γ₀ of the fundamental mode of quasi-extraordinary EMW (k = ky₀), depending on bias magnetic field H₀ = H₀y, at f = 26 GHz for the parallel orientation of the bias magnetic field H₀ to the axis of magnetic nanowires, are shown in Figure 23.21.

For the case, presented in Figure 23.21, the separation of the magnetic nanowires of diameter 2r = 25 nm is large (a = b = 256 nm, c = 550 nm), and as for lower density arrays, and the model of noninteracting nanowires is applicable. The wires behave as thin long cylinders in a bias magnetic field H₀. Respectively, for longitudinal orientation of the magnetic field H₀ the DMFs are N_x = N_y = 2π, N_z = 0, and in this case the eigenfrequency of the FMR of the array is (Gurevich and Melkov, 1996)

${\omega }_0=\gamma \left(H_0+2\pi M_0\right)$

Upon reducing the periodicity of the magnetic nanowires to a < 340 nm, the system becomes strongly coupled, and the exchange interaction plays the dominant role. For the diameter of the nanowires 2r = 25 nm at separation a = b = 67 nm the array behaves as an effective quasi-continuum, that is a thin magnetic film with DMFs N_x = 0, N_y = 4π, N_z = 0, and the eigenfrequency of the FMR for high-density arrays becomes

${\left({\omega }_0/\gamma \right)}^2=H_0\left(H_0+4\pi M_0\right)$

The increase of the resonance field with decreasing separation as given in Eq. (23.20) is illustrated in Figure 23.22. The calculated bias magnetic field H₀ dependence of the imaginary part of the effective permeability Im μ of a 3-D Co₈₀Ni₂₀ nanowire array (4πM₀ = 15,356 kG, α = 0.005, σ = 1.0 × 10⁷ S m^− 1, A = 1.5 × 10 ⁻⁹ Oe cm²) is shown in Figure 23.22, for variable periodicity of the arrays: H₀ = H₀y₀ at f = 26 GHz.

There is an interesting and important consequence of the strong interaction in nanoparticle arrays. By a similar model, the full nonlinear propagation effects were studied on a regular 3-D array of Fe nanospheres, assuming that a wave packet is propagating in the nanostructure and in the arrays the propagating EMW is a superposition of inhomogeneous plane waves. The dependence of the propagation constants Γ₀ of the EMWs on the lattice period of nanoparticles on a 3-D periodic array of r = 150 nm Fe nanospheres was calculated for the fundamental mode of clockwise and counterclockwise polarized EMWs, and for the ordinary and extraordinary EMWs, propagating in the array at f = 30 GHz, and shown in Figure 23.23 (Makeeva et al., 2009c).

Figure 23.23 demonstrates, that due to the change of the character of the propagation of EMWs with decreasing separation r/a, the propagation constants change significantly on the interval 0.1 < r/a < 0.35. The propagation constants of counterclockwise polarized and extraordinary modes become imaginary for r/a > 0.25 and, consequently, these waves are not propagating waves. As it is well known, EMW do not propagate in metals – that is, the nanoarray behaves like a metal, reaching the percolation length for conduction without direct metallic contact between the particles! Upon further reducing the separation of nanospheres to r/a > 0.25 (transition to the range of exchange length), the exchange interaction in the system of strongly coupled magnetic nanoparticles plays a dominant role, and the magnetic nanoarray starts to behave like a quasi-bulk continuum (Pardavi-Horvath et al., 2008a). As for the case of ferromagnetic metals, from the four normal modes there are two counterclockwise polarized and ordinary modes with real propagation constants, having a large phase velocity, propagating in the gyromagnetic nanostructured media as the separation r/a of magnetic nanoparticles 0.25 < r/a < 0.35 approaches the exchange length. There is an interaction range where extremely low loss nanocomposites can be designed. Using the values of the computed propagation constants and the dispersion relations, the complex diagonal μ₁ and off-diagonal μ₂ components of the effective permeability tensor, and the effective permittivity ɛ of the periodic magnetic nanosystems, depending on r/a, were calculated (Pardavi-Horvath et al., 2009). For weakly interacting nanospheres the magnetic losses are about two orders smaller, whereas the nonreciprocity is doubled as compared to YIG. Upon reducing the separation of nanospheres to r/a = 0.35 the imaginary part of the permeability increases due to an additional loss mechanism by the excitation of SWs in the system.

23.4.1.3 EMW scattering in nanowires at THz frequencies

In the GHz frequency range, the EM wavelength is still much larger than the size of the particles, and any treatment has to take care of propagation and magnetostatic effects, as before. What happens when the EMW frequency is getting comparable to the characteristic nm size of a particle on a 2-D or 3-D nanoarray? Indeed, this frequency range is in the THz, where many intriguing new phenomena are taking place, and potential magnetophotonic band gap effects are anticipated, and applications can be expected to be developed. In the THz frequency range the penetration (skin) depth δ₁ of EMW in conducting (metallic) media can be close to the nanowire diameter δ₁ ≤ 2r, leading to interesting resonance and SW effects.

The normal skin effect is present, if the mean free path of conductivity electrons is smaller than the penetration depth δ₁ of the EM field in a metal. For a ferromagnetic metal nanowire, the skin depth is given by

${\delta }_1=\delta /{\left(\left|{\mu }_{\mathrm{eff}}\right|+{\mu }_{\mathrm{eff}}^{″}\right)}^{1/2}=\frac{1}{k_0}\sqrt{\frac{\omega }{2\pi \sigma }}/{\left(\left|{\mu }_{\mathrm{eff}}\right|+{\mu }_{\mathrm{eff}}^{″}\right)}^{1/2}$

where δ is the skin depth in a metal, ω is the frequency, σ is the electrical conductivity of the metal, ${\mu }_{\mathrm{eff}}={\mu }_{\mathrm{eff}}^{\prime }-\mathrm{i}{\mu }_{\mathrm{eff}}^{″}$ is the effective scalar permeability (Gurevich and Melkov, 1996) determined by the components of the high-frequency dynamic permeability tensor $\hat{\mu }$ (Akhiezer et al., 1968). The effective scalar permeability μ_eff depends on the dynamic permeability $\hat{\mu }$, and it is a function of frequency ω, the orientation of the bias magnetic field H₀, and the rf magnetic field H of the exciting EM wave. The skin depth in metallic ferromagnetic wires δ₁ (23.21) depends not only on the conductivity, but also on the magnetization, and at THz frequencies it can become the order of the size 2r ~ 10 nm (nanowire diameter) due to the large values of permeability (${\mu }_{\mathrm{eff}}={\mu }_{\mathrm{eff}}^{\prime }-\mathrm{i}{\mu }_{\mathrm{eff}}^{″}$).

If the nanowire diameter is larger than the penetration depth of the EM field 2r > δ₁ there is an inhomogeneous distribution of the magnetization and the rf magnetic field across the nanowire. This results in the modification of the spectrum of excited collective exchange SW-modes in such ferromagnetic nanowire arrays at THz frequencies.

To get an insight into the THz behavior of a 2-D nanowire array, the diffraction of EMW on arrays of ferromagnetic metallic (iron) nanowires was modeled at THz frequencies at EM accuracy by solving the 3-D diffraction problem for Maxwell's equations with electrodynamic boundary conditions, complemented by the L–L equation, including the exchange term. Using the computational algorithm based on the decomposition approach by autonomous blocks with Floquet channels (FABs), as in Golovanov (2006; Golovanov and Makeeva, 2009) the scattering parameters of the S matrix of 2-D magnetic nanowire arrays, depending on the bias magnetic field H₀, were calculated for nanowire diameters 10 < 2r < 60 nm at a frequency of 30 THz (Makeeva et al., 2009a). The geometry of the wire array and the parameters are the same as in Figure 23.20.

The transmission |T₂₁| and reflection |R₁₁| coefficients were calculated at a bias field of 75 Oe for 2-D periodic arrays of nanowires. The scattering parameters depend very strongly on the nanowire diameter in the 10 ≤ 2r ≤ 60 nm wire diameter range. There is an optimal geometry where there is a transmission maximum |T₂₁|_max at 2r = 27 nm, coinciding with the minimum of the reflection |R₁₁|_min. Upon increasing the wire diameter, the reflection is increasing to |R₁₁| = 1, while the transmission is gradually decreasing.

When in the array of metallic nanowires the diameter of the nanowires 2r is larger than the skin-depth δ in a metal at a frequency f = 30 THz, the EMWs are damped with the attenuation coefficient k″ ~ 1/δ (Gurevich and Melkov, 1996).

$k^{″}=k_0\sqrt{\frac{2\pi \sigma }{\omega }}{\left(\left|{\mu }_{\mathrm{eff}}\right|+{\mu }_{\mathrm{eff}}^{″}\right)}^{1/2}=\sqrt{{\mu }_{\mathrm{effR}}}/\delta =1/{\delta }_1$

where k₀ is the wave number of EMWs in free space, ${\mu }_{\mathrm{eff}}={\mu }_{\mathrm{eff}}^{\prime }-\mathrm{i}{\mu }_{\mathrm{eff}}^{″}$ is the effective scalar permeability: ${\mu }_{\mathrm{effR}}=\left|{\mu }_{\mathrm{eff}}\right|+{\mu }_{\mathit{eff}}^{″}$, and δ₁ is the skin-depth in the ferromagnetic metallic medium.

According to Eq. (23.22) the minimum of the absorption of EMWs is determined by the maximum of the skin-depth δ₁ = δ(μ_effR)^−1/2. This maximum is realized when the effective permeability μ_effR has a minimum (i.e., the real part of the effective permeability ${\mu }_{\mathrm{eff}}^{\prime }=0$ and the imaginary part of the effective permeability ${\mu }_{\mathrm{eff}}^{″}\to 0$), that is, in the antiresonance point. The frequency of the ferromagnetic antiresonance (FMAR) in an unbounded ferromagnetic metallic medium is

${\omega }_{\mathrm{a}\mathrm{r}}={\omega }_{\mathrm{H}}+{\omega }_{\mathrm{M}}$

where ω_ar is the frequency of antiresonance; ω_H = γH₀ and ω_M = 4πM_s.

Applying the condition for antiresonance in Eq. (23.23) to a finite size ferromagnetic ellipsoid, the resonance condition becomes

${\omega }_{\mathrm{a}\mathrm{r}}={\omega }_0+{\omega }_{\mathrm{M}}$

where ω₀ is the eigenfrequency of the FMR of the magnetized ellipsoid, determined by the internal magnetic field H_0int

${\mathit{H}}_{0\mathrm{i}\mathrm{n}\mathrm{t}}={\mathit{H}}_0-\overleftrightarrow{N}{\mathit{M}}_0$

where $\overleftrightarrow{N}$ is the demagnetizing tensor (introduced in Section 23.2.1.2).

Upon increasing the bias magnetic field H₀ several types of exchange SW modes are excited in the ferromagnetic metal (iron) nanowire array at f = 30 THz. These are radial surface SW modes with complex wave numbers, having hyperbolic distributions of the rf magnetization, satisfying the boundary conditions on the surface of the nanowire, and depending on H₀. When the standing SW resonance conditions in magnetic nanoarrays are satisfied, the frequencies of the radial antiresonance modes become (Gurevich and Melkov, 1996):

${{\omega }_{\mathrm{ares}}}_n={{\omega }_0}_n+{\omega }_{\mathrm{M}}$

where ω_0n are the eigenfrequencies of the standing SW resonances of exchange SW modes of order n in a thin ferromagnetic nanowire, determined by the internal magnetic field H_0int. The maxima of the transmission coefficients |T₂₁| are located in the antiresonance points as given by Eq. (23.26).

When the skin-depth δ₁ = δ(μ_effR)^− 1/2 near these antiresonance points increases, the transmitted EMWs through the magnetic nanoarray are also increasing. The effective permeability depends on the interactions in the array through the separation and the diameter of the nanowires, thus the transmission can be influenced by array geometry. The maxima of |T₂₁| can be tuned by the bias magnetic field H₀ for different wire diameter/periodicity ratios. Upon increasing the wire diameter the maximum of the transmission coefficient is moving to lower fields, and the transmission at the maximum is decreasing.

The rf magnetization profiles of the exchange SW modes modes are numerically simulated taking into account the skin depth of the EMW in a ferromagnetic metal (Fe) at THz frequencies (Pardavi-Horvath et al., 2011). The EMW having a wave vector k = kz₀ and fields E = Ex₀, H = Hy₀ is incident on a 2-D array of ferromagnetic metallic nanowires, biased by a magnetic field H₀ ⊥ k. The wire axis is y₀, the wire diameter 10 ≤ 2r ≤ 25 nm, length ℓ = 300 nm. If the nanowire diameter is somewhat larger than the penetration depth of the EM field, there is an inhomogeneous distribution of the magnetization and the magnetic field close to the nanowire surface. This results in the modification of the spectrum of excited collective exchange SW modes in such ferromagnetic arrays at THz frequencies. The rf magnetization profiles of these modes are numerically simulated for iron nanowires and shown in Figure 23.24 (Pardavi-Horvath et al., 2011). Figure 23.24a shows the bias field dependence of the transmission spectra for different nanowire diameters. The extrema correspond to specific SW mode excitations. The transmission maxima are the FMAR points (Makeeva et al., 2009a). The minima of transmission curves correspond to the minima of the skin depth, that is, when the effective permeability μ_effR^″ has a maximum, that is, to the FMR. At standing wave resonances μ_effR^″ and thus the power absorption increases and δ collapses.

Upon increasing the bias field H₀, the second-order SW modes n = 2 and then n = 1 (k = 0) modes are excited by the spatially uniform TEM wave. However, these modes are influenced by the skin effect. These are radial surface SW modes with complex wave numbers, having a hyberbolic distribution of the rf magnetic field, as shown by curves 1–4 in Figure 23.24b. The distribution of the rf magnetization component M_φ for the second order (n = 2) and then n = 1 SW modes, depending on the coordinate r across the nanowire are shown at the field points 1–4 marked in the (a) part of the figure for the 10 nm wire array.