Magnonic crystals: towards terahertz frequencies

When a spin system is driven out of equilibrium by a small perturbation, the most straightforward dynamics is the uniform precession, where all spins are equally canted and precess in phase. This uniform precession is referred to as ferromagnetic resonance (FMR) mode. In order to derive the equations for the FMR mode of a thin ferromagnetic film one may start with the Landau–Lifshitz–Gilbert (LLG) equation of motion

$$\begin{equation}\frac{1}{\gamma }\frac{\partial \mathbf{M}}{\partial t}=-\left[\mathbf{M}{\times}{\mu }_{0}{\mathbf{H}}_{\mathrm{e}\mathrm{ff}}\right]+\frac{\alpha }{\gamma M}\left(\mathbf{M}{\times}\frac{\partial \mathbf{M}}{\partial t}\right),\end{equation} \tag{ 1 }$$

where M represents the magnetisation, H_eff is the effective field, which consists of external as well as internal fields, γ is the gyro-magnetic ratio given by gμ_B/ℏ where g is the g-factor. α denotes the dimensionless damping parameter. Under some circumstances one can find analytical solutions describing the dynamics. Based on the FMR theory one can derive analytical expressions for the dispersion relation of the FMR mode (for details see [55]). Another approach is the one developed by Smit and Beljers [56], which is based on the consideration of the free energy of the system. Note that the FMR mode refers to the uniform precession of all spins and describes the magnons with the wavevector q = 0. Hence the dispersion relation in this case is usually referred to the dependency of the FMR frequency on the applied magnetic field. Using the first approach and after calculating the real and imaginary parts of the magnetic susceptibility, one can show that the imaginary part of the susceptibility, has the form of a Lorentzian function, which has its maximum at the FMR frequency.

Considering the coordinate system depicted in figure 2(a) for a thin film made of a single crystalline material ordered in a cubic structure [for example body-centered cubic (bcc) Fe] and considering only a uniaxial out-of-plane K_2⊥ and a cubic K₄ anisotropy the resonance frequency for the field applied along different directions will then be given by

$$\begin{equation}\mathbf{H}\parallel \left[001\right]:\frac{{\omega }_{\mathrm{F}\mathrm{M}\mathrm{R}}}{\gamma }={\mu }_{0}H+{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}+\frac{2{K}_{4}}{M}\end{equation} \tag{ 2 }$$

$$\begin{align}\hfill \mathbf{H}\parallel \left[100\right]:{\left(\frac{{\omega }_{\mathrm{F}\mathrm{M}\mathrm{R}}}{\gamma }\right)}^{2}& =\left({\mu }_{0}H+\frac{2{K}_{4}}{M}\right)\hfill \\ \hfill & \hfill \quad {\times}\left({\mu }_{0}H-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}+\frac{2{K}_{4}}{M}\right)\end{align} \tag{ 3 }$$

$$\begin{align}\hfill \mathbf{H}\;\left[110\right]:{\left(\frac{{\omega }_{\mathrm{F}\mathrm{M}\mathrm{R}}}{\gamma }\right)}^{2}& =\left({\mu }_{0}H-\frac{2{K}_{4}}{M}\right)\hfill \\ \hfill & \hfill \quad {\times}\left({\mu }_{0}H-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}+\frac{{K}_{4}}{M}\right).\end{align} \tag{ 4 }$$

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Here ${\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}=\frac{2{K}_{2\perp }}{M}-{\mu }_{0}M$ denotes the effective out-of-plane anisotropy field (for details see for example [57, 58]). One should note that equations (2)–(4) are only valid for the saturated condition, i.e., H is strong enough to align all magnetic moments parallel to its direction. In an FMR experiment, however, for specific anisotropy values and directions of the external magnetic field the resonance conditions can also be fulfilled when M is not parallel to H. This causes an additional signal at smaller fields with lower intensity (unsaturated mode), when the FMR spectra are recorded at a fixed frequency and the external magnetic field is swept. Such unsaturated resonance modes have been observed in several experiments on Fe films [58–60]. In order to find the dispersion relation (frequency versus field) for the unsaturated mode numerical calculations are required.

In figure 2(b) the dispersion relation of the FMR mode (the dependency of the FMR frequency on the applied magnetic field) calculated for an Fe film with μ₀M_eff = 2.14 T, the cubic anisotropy field of 2K₄/M = 55 mT and the g-factor of g = 2.09 is presented. The blue (red) curve represents the calculation for the applied magnetic field in the film plane along the [100]- ([110]) direction. The gray curves represent the calculations for intermediate directions between these two main symmetry axes. It is apparent that the dispersion relation even for the FMR mode is highly anisotropic in a magnetic film. This could also simply be observed by comparing equations (2)–(4). Another important consideration is that in the above mentioned formalism the confinement in the direction perpendicular to the film is neglected. This is a very good assumption for the limit of ultrathin films. Such an effect can simply be considered by adding an additional term into equation (2)

$$\begin{equation}\mathbf{H}\parallel \left[001\right]:\frac{{\omega }_{\mathrm{F}\mathrm{M}\mathrm{R}}}{\gamma }={\mu }_{0}H+{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}+\frac{2{K}_{4}}{M}+\frac{2A}{M}{q}_{z}^{2}.\end{equation} \tag{ 5 }$$

Here A represents the exchange stiffness constant and q_z denotes the magnon wavevector in the direction perpendicular to the film. Since along this direction the film possess a certain thickness, q_z takes only discrete values given by nπ/d, where n is an integer number. This implies that the thinner the film the larger the frequency associated with these modes. Such modes are traditionally called spin wave resonances and have been observed in FMR experiments performed on relatively thick films [61]. In the experiments only the odd modes could be observed. This was then explained by Kittel [62]. He showed that due to the boundary conditions, the modes with even numbers are not allowed to be excited (because the spins are assumed to be pinned at the boundaries of the sample, as a consequence of the surface magnetic anisotropy). Note that the uniform FMR mode is the one with n = 0. In equation (5) if one sets n to zero, one would obtain the same expression given by equation (2) i.e., the uniform FMR mode. The uniform FMR mode does not depend directly on the film thickness (it may indirectly depend on the thickness via thickness dependence of magnetic anisotropy and magnetisation).

Likewise one may introduce lateral confinements in the structure. In such a case it is important to consider the fact that the demagnetising factors need to be taken into account (in the above discussions we have assumed that the lateral dimensions are far larger than the film thickness). Furthermore, depending on the length-scales involved in the lateral confinement additional energy terms can come into play. For a simplified case with the field applied along the easy axis of the cubic anisotropy the dispersion relation will be given by

$$\begin{align}\hfill \;{\left(\frac{{\omega }_{\mathrm{F}\mathrm{M}\mathrm{R}}}{\gamma }\right)}^{2}& =\left[{\mu }_{0}H+\frac{2{K}_{4}}{M}+\left({N}_{x}-{N}_{z}\right){\mu }_{0}M\right]\hfill \\ \hfill & \quad {\times}\left[{\mu }_{0}H+\left({N}_{y}-{N}_{z}\right){\mu }_{0}M+\frac{2{K}_{4}}{M}\right],\hfill \end{align} \tag{ 6 }$$

where, N_x, N_y and N_z represent the demagnetising factors along the x, y and z directions, respectively (see the coordinate system depicted in figure 2(a)) [63].

Geometrically confined ferromagnetic objects prepared by patterning the ferromagnetic structures exhibit nonuniformities. Such nonuniformities might be due to different reasons e.g., the static magnetic configuration of the object, the dynamic demagnetising fields, or even a deliberate design of magnetic inhomogeneities. As a consequence the local FMR frequency and the local dynamic susceptibility at different parts of the object will be different. This has been demonstrated in reference [64], where a methodology to efficiently calculate the local FMR frequencies is developed.

So far we have limited our discussion to the FMR mode in the limit of q → 0, where all the magnetic moments precess in phase, resulting an infinite wavelength of the magnon. In the next step we aim to consider magnons with q ≠ 0. Depending on the wavelength, the magnetic interaction dominating the magnons energy will be different. For the long wavelength excitations the dominating magnetic interactions are dipolar interactions. Hence these magnon modes are usually referred to as dipolar magnons. In the case of short wavelength magnons (wavelengths comparable to the interatomic distances) the relevant magnetic interaction is the magnetic exchange interaction. Consequently these magnons are referred to as exchange magnons. Magnons in the intermediate regime have both characters and are referred to as dipolar-exchange magnons. We shall briefly discuss different types of magnons in the following sections.

The characteristics of dipolar or magnetostatic magnons have been introduced long ago [65]. Due to the anisotropic nature of the dipolar interaction the dispersion relation of this type of magnons depends strongly on the direction of the applied magnetic field and the shape of the sample. In the case of thin magnetic films one would expect three characteristic modes [55].

• The so-called magnetostatic surface wave (MSSW) refers to a magnon mode excited in a geometry in which q and H lie in the film plane but orthogonal to each other.
• The so-called magnetostatic backward volume wave (MSBVW) refers the case in which q and H are parallel and both lie in the film plane.
• Finally, the mode excited with q in the plane and H applied normal to the plane is called magnetostatic forward volume wave (MSFVW).

An illustration of the dispersion relation of all dipolar modes excited in a thin ferromagnetic film is presented in figure 3. The MSSW mode or the Damon–Eshbach (DE) mode shows a positive dispersion starting from the uniform FMR mode (Kittel mode) at q = 0. This mode is further characterised by the localisation of the magnons in the vicinity of the top and bottom surfaces of the film. In addition, this mode shows a nonreciprocal behaviour. The nonreciprocity of this mode implies that the magnons associated with this mode are differently localised at the top and bottom interfaces, depending on their propagation direction. This means that magnons with positive or negative sign of q travel either in the top or in the bottom layer. In other words the counter-propagating magnons propagate via the top or bottom part of the sample.

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The dispersion relation of the MSSW mode has the from of

$$\begin{align}\hfill {\left(\frac{{\omega }_{\mathrm{M}\mathrm{S}\mathrm{S}\mathrm{W}}}{\gamma }\right)}^{2}& =\left({\mu }_{0}H+\frac{2{K}_{4}}{M}\right)\left({\mu }_{0}H-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}+\frac{2{K}_{4}}{M}\right)\hfill \\ \hfill & \hfill \quad +{\left(\frac{{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}}{2}\right)}^{2}\cdot \left(1-{\mathrm{e}}^{-2{q}_{\parallel }d}\right).\end{align} \tag{ 7 }$$

The dispersion relation of the MSBVW mode is given by

$$\begin{align}\hfill {\left(\frac{{\omega }_{\mathrm{M}\mathrm{S}\mathrm{B}\mathrm{V}\mathrm{W}}}{\gamma }\right)}^{2}& =\left({\mu }_{0}H+\frac{2{K}_{4}}{M}\right)\hfill \\ \hfill & \hfill \quad {\times}\left[{\mu }_{0}H+\frac{2{K}_{4}}{M}-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}\left(\frac{1-{\mathrm{e}}^{-{q}_{\parallel }d}}{{q}_{\parallel }d}\right)\right].\end{align} \tag{ 8 }$$

In the equations above q_∥ represent the parallel momentum i.e., the component of the wavevector which lies the film plane. The MSBVW mode exhibits a negative group velocity. Another characteristic of this mode is that, in contrast to the MSSW mode, its amplitude is distributed throughout the volume part of the film.

The dispersion relation of the MSFVW mode is given by

$$\begin{align}\hfill {\left(\frac{{\omega }_{\mathrm{M}\mathrm{S}\mathrm{B}\mathrm{V}\mathrm{W}}}{\gamma }\right)}^{2}& =\left({\mu }_{0}H+\frac{2{K}_{4}}{M}\right)\hfill \\ \hfill & \quad {\times}\left[{\mu }_{0}H+\frac{2{K}_{4}}{M}-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}\left(1-\frac{1-{\mathrm{e}}^{-{q}_{\parallel }d}}{{q}_{\parallel }d}\right)\right].\hfill \end{align} \tag{ 9 }$$

Similar to MSSW, this mode also possesses a positive group velocity. Unlike MSSW the amplitude of this mode is distributed over the volume part of the film. The dispersion relation of this mode is isotropic in the film plane meaning that it does not depend on the direction of q_∥.

These magnons are governed by the magnetic exchange interaction, usually referred to as Heisenberg exchange. Since this interaction is rather short-range one has to use a microscopic picture in order to describe the exchange magnons. The Heisenberg Hamiltonian describing the interaction between magnetic moments siting on lattice site i and j reads as

$$\begin{equation}\mathcal{H}=-\sum _{i\ne j}{J}_{ij}{\boldsymbol{\hat{e}}}_{\boldsymbol{i}}\cdot {\boldsymbol{\hat{e}}}_{\boldsymbol{j}},\end{equation} \tag{ 10 }$$

where J_ij denotes the symmetric Heisenberg exchange constant and the unit vectors ê_i and ê_j represent the direction of the magnetic moments at the atomic sites i and j, respectively. For simplicity we set the amplitude of spin to unity S = 1. In this picture magnons are the low-energy excitations associated with small-amplitude precessions of moments around the main magnetisation direction. In the case of bulk single-element ferromagnets expanded in the three dimensional space one expects to observe a single magnon band [55]. However, in magnetic films composed of several atomic layers one can show that, in principle, m magnon modes are expected to be observed [66–68]. The dispersion relation of these modes can be calculated by starting from equation (10) and finding the solution of the following matrix equation

$$\begin{equation}\hslash \omega \left(\begin{matrix}\hfill {A}_{1}\hfill \\ .\hfill \\ .\hfill \\ .\hfill \\ \hfill {A}_{m}\hfill \end{matrix}\right)=\left(\begin{matrix}\hfill 2\sum _{j}^{m}{J}_{i=1,j}\left[{A}_{1}-{A}_{j}\mathrm{exp}\left(i\boldsymbol{q}\cdot \boldsymbol{R}\right)\right]\hfill \\ .\hfill \\ .\hfill \\ .\hfill \\ \hfill 2\sum _{j}^{m}{J}_{i=m,j}\left[{A}_{m}-{A}_{j}\mathrm{exp}\left(i\boldsymbol{q}\cdot \boldsymbol{R}\right)\right]\hfill \end{matrix}\right),\end{equation} \tag{ 11 }$$

where R = R_i − R_j with R_i and R_j being the position vector of sites i and j, respectively. A_j (j = 1 to m) represent the amplitude of the magnons. Equation (11) indicates that for a magnetic film composed of m atomic layers one should observe m different magnon modes. There will be two magnon modes, which are lower in frequency with respect to the others. Those modes are formed due to the lower coordination number of magnetic moments in the top and bottom layers of the film. At q = 0 these two modes are separated by ω(q = 2π/md) in frequency, where d represents the interatomic distance. We assume that the interatomic distances in all directions are the same. In case that the same values of exchange constants are considered at the top and bottom layers, these two modes will degenerate in frequency at high wavevectors, close to the zone boundary. It is apparent from equation (11) that this quantity depends on the exchange constants as well. For a given magnon mode A_j varies from one layer to another. Therefore this quantity might be considered as the contribution of each layer to that particular mode. For example the two lowest-frequency magnon modes have the largest amplitude of the eigenvectors in the top and bottom layers. This means that these modes are localised at these two layers. If these two layers are magnetically identical it is not possible to distinguish between them. However, if the layers are magnetically different (the values of the exchange parameters or the amplitude of the moments in the layers are different) the degeneracy of these two low frequency modes will break and one can associate a mode to each layer. This is of particular importance and will be discussed further in section 2.3.2.

As an example we show in figure 4 the exchange-dominated magnons calculated based on a Heisenberg model for an Fe film composed of six atomic layers. The calculations are performed for a simplified case, where only the nearest and next nearest neighbours are taken into account. The assumption here is that the nearest neighbour exchange constant is J_n = 8.9 meV, the next nearest neighbour exchange constant is J_2n = 0.6J_n and the higher order terms are zero (J_3n = J_4n = ... = 0). The calculations are performed for the first and second Brillouin zone (BZ). The characteristics of the exchange magnons are as follows. (i) The modes with low quantum number show a semiparabolic dispersion for the limit of small q_∥. This means that the group velocity of such magnons, given by ${\mathbf{v}}_{\boldsymbol{g}\parallel }=\left(\partial \omega /\partial {q}_{\parallel }\right){\hat{\boldsymbol{q}}}_{\parallel }$, increases linearly with q_∥ up to the zone boundary. This behaviour can be different for modes with a large quantum number. The fact that the film possess a translational symmetry in the plane is reflected in the fact that the dispersion relation is perfectly reproduced while going from the first to the second BZ. (ii) The frequency of these magnons exceeds quickly the terahertz regime.

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Conventionally, the lowest frequency magnon mode, which satisfies the Goldstone criteria, is the so-called 'acoustic mode' (curve shown in blue colour in figure 4). The higher energy magnon modes are the so-called 'optical modes', which possess a finite energy at q_∥ = 0, in analogy to the phonon modes in solids.

Note that the assumptions made above break in the limit of q_∥ → 0. This is due to the fact that at this limit the other interactions become important and must be considered in the description of the magnon dispersion relation. This is why even the uniform FMR mode possesses a finite frequency (see figure 3).

It is important to notice that although the magnon modes shown in figure 4 are the result of quantum confinement in the direction perpendicular to the film, they exhibit a finite momentum in the film plane (finite q_∥). This means that these magnons can propagate through the system with a certain group and phase velocity.

In a similar manner one can also calculate the eigenstates based on equation (11). The eigenstates would provide the profile of different magnon modes. An easier way to treat the mode profiles would be to just analyse the amplitude of the eigenstates. This would provide all the necessary information regarding the localisation of different modes. As discussed above the first two modes indicated by the blue and red colours are the modes mainly localised in the top and bottom surfaces. Since the spins sitting in these two layers possess less neighbours than those sitting in the volume part of the film, these two modes are lower in frequency, compared to the others. In the calculations shown in figure 4 all the layers are treated equally and hence the top and bottom layers are identical. This means that the amplitudes of the eigenvectors for magnons describing these two modes, given by equation (11), will be equal in the top and bottom layers. In the case of real systems (ferromagnetic film grown on a nonmagnetic substrate) the presence of the substrate breaks the degeneracy of these two modes and leads to the nonequivalent localisation of these modes in the top and bottom layer. Another consequence of the identical exchange parameters in these two layers is the degeneracy of the modes near and at the zone boundary ($\overline{\mathrm{M}}$-point).

The magnon momentum in the direction perpendicular to the layers is confined and takes only discrete values q_⊥ = nπ/m, where m is the number of atomic layers and n is the mode number, being 0, 1, ..., m − 1 [see the right side of figure 4]. Note that here it is assumed that the spins are all exchange coupled. This, in turn, means that the spins are unpinned at the two interfaces. In a thin magnetic film with a finite thickness the magnons with quantised q_⊥ = q_z and zero parallel momentum, q_∥ = 0, can be excited also in the gigahertz regime, when the film thickness is several tens of nanometers. These magnon modes are the spin wave resonances as described earlier in section 2.1.1. Their dispersion relation is given by equation (5). It has been demonstrated that due to the boundary conditions only the odd resonance modes can be observed in experiments (see for example [61] for details).

The intermediate regime between the dipolar and exchange magnons is the so-called dipolar-exchange regime. Obviously, both the dipolar and exchange terms need to be considered in order to describe this regime. It has been shown that in the presence of the exchange field additional boundary conditions need to be considered. These boundary conditions are usually referred to as exchange boundary conditions, which imply that at the film boundaries the precession of moments depends on the surface anisotropy present in the system. Kalinikos and Slavin have derived a practical expression for the dispersion relation of dipole-exchange magnons in a thin ferromagnetic film subject to arbitrary exchange boundary conditions, when the magnetic field is applied in the film plane [69]

$$\begin{align}\hfill {\left(\frac{\omega }{\gamma }\right)}^{2}& =\left({\mu }_{0}H+\frac{2{K}_{4}}{M}+\frac{2A}{M}{q}_{n}^{2}\right)\hfill \\ \hfill & \quad {\times}\left({\mu }_{0}H+\frac{2{K}_{4}}{M}+\frac{2A}{M}{q}_{n}^{2}-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}{T}_{nn}\right).\hfill \end{align} \tag{ 12 }$$

The wavevector q can be decomposed to the parallel and perpendicular components. Since for a thin film the perpendicular component is quantised one can therefore write ${q}_{n}^{2}={q}_{\parallel }^{2}+{\left(n\pi /d\right)}^{2}$. Here again q_∥ denotes the in-plane component of the magnon wavevector. T_nn describes the matrix element of the dipole-dipole interaction and in a general form is given by

$$\begin{equation}{T}_{nn}=1-{P}_{nn}{\mathrm{cos}}^{2}\;{\varphi }_{q}-{\mu }_{0}{M}_{\mathrm{e}\mathrm{ff}}\frac{{P}_{nn}\left(1-{P}_{nn}\right){\mathrm{sin}}^{2}\;{\varphi }_{q}}{\left({\mu }_{0}H+\frac{2{K}_{4}}{M}+\frac{2A}{M}{q}_{n}^{2}\right)}.\end{equation} \tag{ 13 }$$

Here φ_q denotes the angle between q and M. Under the assumption that the surface moments are fully unpinned and if the magnetisation is fully uniform across the thickness, P_nn can be simplified to

$$\begin{equation}{P}_{nn}=\frac{{q}^{2}}{{q}_{n}^{2}}-\frac{{q}^{4}}{{q}_{n}^{4}}\frac{2}{qd}\left[1-{\left(-1\right)}^{n}{\mathrm{e}}^{-qd}\right]\frac{1}{1+{\delta }_{0n}}.\end{equation} \tag{ 14 }$$

A graphical visualisation of the dipolar-exchange magnons in an Fe film is presented in figure 5. It is apparent that confinement effects can drastically change the dispersion relation of these magnons. The most simplest picture is that the wavevector will no longer be a continuous quantity and will only choose discrete values. In the next section we will discuss how the confinement effects change the magnon dispersion.

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The magnon dispersion relation in MCs with periodic modulations in the magnetic properties can be understood in the same way as the formation of the electronic bands in a solid as a result of the periodic arrangement of atoms. The magnon dispersion relation is therefore commonly called the magnonic band structure. Such a band structure can be efficiently calculated using the so-called plane wave method, which is analogous to the Bloch description for electronic states in solids [72, 86–91]. In the following we just briefly introduce this approach. Another approach is based on the Walker's equation [92]. We do not, however, discuss it in the present topical review.

Let us consider a magnetic thin film with a thickness much smaller than the lateral extension. Now if one introduces lateral variation of magnetic properties in one or two dimensions an MC may be realised. This general idea is schematically illustrated in figure 7. Note that for simplicity we only consider MCs of ultrathin films. One may also think of periodicity in the third direction. The physics of those systems would be somewhat similar. We do not treat those structures in the present topical review.

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The periodic variation of magnetic properties may be realised either by modulation of the intrinsic magnetic properties (without modulation of the structure) or by modulation of the structure. According to equations (3)–(9) and (12) the magnetic properties, which may be periodically modulated are magnetisation, local internal fields and magnetic anisotropy. Such modulations may be realised by several means e.g., variation in the thickness [93–96], materials composition [49, 50, 97–103], chemical properties [104, 105] or local strain, temperature, magnetic states, spin texture and magnetic field [51–54]. The structure might be single or bi-component [89, 106–113]. The basic idea behind is essentially the same.

Considering a periodic modulation of the saturation magnetisation and solving the LLG equation [equation (1)] one can calculate the magnonic band structure. In such a case the H_eff includes the external applied field H, the anisotropy fields H_ani, the exchange field H_ex and finally the magnetostatic field H_ms caused by magneostatic interactions. If one assumes that the magnons having a long wavelength have a uniform profile in the z direction, perpendicular to the periodic structure, one can simplify the picture for a 2D array of magnetic elements. Now if one assumes the form of $\boldsymbol{M}\left(\boldsymbol{r},t\right)=M\left(\boldsymbol{r}\right)\hat{\boldsymbol{i}}+{m}_{y}\left(\boldsymbol{r},t\right)\hat{\boldsymbol{j}}+{m}_{z}\left(\boldsymbol{r},t\right)\hat{\boldsymbol{k}}$ and $\boldsymbol{H}\left(\boldsymbol{r},t\right)=H\left(\boldsymbol{r}\right)\hat{\boldsymbol{i}}+{h}_{y}\left(\boldsymbol{r},t\right)\hat{\boldsymbol{j}}+{h}_{z}\left(\boldsymbol{r},t\right)\hat{\boldsymbol{k}}$) for the dynamical magnetisation and the magnetic field, respectively, one can simplify the solution of the LLG equation to the following coupled equations

$$\begin{align}\hfill i{\Omega}{m}_{y}-{m}_{z}+\frac{M}{H}{h}_{z}& =0\hfill \\ \hfill i{\Omega}{m}_{z}+{m}_{y}-\frac{M}{H}{h}_{y}& =0.\hfill \end{align} \tag{ 16 }$$

Here Ω represents the normalised frequency and is given by Ω = ω/γμ₀H. Considering the limit of the uniform mode, one can write the component of the dynamical magnetisation as

$$\begin{equation}{m}_{y}\left(\boldsymbol{r}\right)={m}_{y}{\mathrm{e}}^{\mathrm{i}\left({q}_{x}x+{q}_{y}y\right)}\quad {m}_{z}\left(\boldsymbol{r}\right)={m}_{z}{\mathrm{e}}^{\mathrm{i}\left({q}_{x}x+{q}_{y}y\right)}.\end{equation} \tag{ 17 }$$

In order to find the solution now one has to apply the boundary conditions

$$\begin{align}\hfill {h}_{y}& =-{m}_{y}\frac{qd}{2}{\mathrm{e}}^{\mathrm{i}\left({q}_{x}x+{q}_{y}y\right)}{\mathrm{sin}}^{2}\varphi \hfill \\ \hfill {h}_{z}& =\left(-{m}_{z}+{m}_{z}\frac{qd}{2}\right){\mathrm{e}}^{\mathrm{i}\left({q}_{x}x+{q}_{y}y\right)}.\hfill \end{align} \tag{ 18 }$$

Inserting equations (17) and (18) into equation (16), one can obtain an equation for the dispersion relation.

Now one should consider the periodic modulation of M in the structure. This can be done in the same way as it is done for electrons in a solid. The magnetisation can be written in terms of its Fourier component.

$$\begin{equation}M\left(\boldsymbol{r}\right)=\sum _{\boldsymbol{G}}M\left(\boldsymbol{G}\right){\mathrm{e}}^{\mathrm{i}\boldsymbol{G}\cdot \boldsymbol{r}},\end{equation} \tag{ 19 }$$

where $\boldsymbol{G}={\left[{G}_{n},\;{G}_{m}\right]}^{\mathrm{T}}={\left[\frac{2\pi n}{a},\frac{2\pi m}{b}\right]}^{\mathrm{T}}$ represents a two-dimensional vector of the reciprocal lattice with periodicity along the two main directions x and y, given by the lattice constants a and b, respectively. The Fourier component M(G) is the key quantity defining the properties of the MC, as it can define the geometry (and the symmetry) of the system. Note that in this simple formalism modulations in other material constants are not taken into consideration.

The solutions of equation (16) in general form are Bloch waves

$$\begin{equation}\boldsymbol{m}\left(\boldsymbol{r}\right)=\sum _{\boldsymbol{G}}{\boldsymbol{m}}_{\boldsymbol{q}}\left(\boldsymbol{G}\right){\mathrm{e}}^{\mathrm{i}\left(\boldsymbol{q}+\boldsymbol{G}\right)\cdot \boldsymbol{r}}.\end{equation} \tag{ 20 }$$

Inserting equation (20) in equation (16) the dynamic magnetic field components can be calculated. Equations (19), (20) and (16) provide a system of equations in the case of finite number of G, which can be summarised in the form of

$$\begin{equation}\overline{\overline{\mathcal{M}}}{\boldsymbol{m}}_{\boldsymbol{q}}^{j}=i{{\Omega}}_{j}{\boldsymbol{m}}_{\boldsymbol{q}}^{j},\end{equation} \tag{ 21 }$$

where, ${\boldsymbol{m}}_{\boldsymbol{q}}^{j}={\left[{m}_{y,\boldsymbol{q}}^{j}\left({\boldsymbol{G}}_{1}\right),\dots ,{m}_{y,\boldsymbol{q}}^{j}\left({\boldsymbol{G}}_{N}\right),{m}_{z,\boldsymbol{q}}^{j}\left({\boldsymbol{G}}_{1}\right),\dots ,{m}_{z,\boldsymbol{q}}^{j}\left({\boldsymbol{G}}_{N}\right)\right]}^{T}$. The eigenvalues Ω_i = ω_i/γμ₀H would provide the dispersion relation of the modes associated with the respective periodic structure. In addition to this the eigenvectors ${\boldsymbol{m}}_{\boldsymbol{q}}^{j}$ would provide the mode profile. The calculation of the eigenvectors is of particular importance when one aims to see how different magnon modes are formed and where these modes are mainly localised. The norm of the amplitude of the eigenvectors would results in the degree of localisation of each mode in different parts of the structure. The 2N × 2N-matrix $\overline{\overline{\boldsymbol{\mathcal{M}}}}$ has a block-diagonal form

$$\begin{equation}\overline{\overline{\boldsymbol{\mathcal{M}}}}=\left(\begin{pmatrix}\hfill {\overline{\boldsymbol{\mathcal{M}}}}_{yy}\hfill & \hfill {\overline{\boldsymbol{\mathcal{M}}}}_{yz}\hfill \\ \hfill {\overline{\boldsymbol{\mathcal{M}}}}_{yz}\hfill & \hfill {\overline{\boldsymbol{\mathcal{M}}}}_{zz}\hfill \end{pmatrix}\right),\end{equation} \tag{ 22 }$$

where ${\mathcal{M}}_{yy}^{ij}={\mathcal{M}}_{zz}^{ij}=0$ and the terms of the off-diagonal sub-matrices are given by

$$\begin{align}\hfill {\mathcal{M}}_{yz}^{ij}& ={\delta }_{ij}-\frac{1}{H}\left(-1+\frac{\vert \boldsymbol{q}+{\boldsymbol{G}}^{j}\vert d}{2}\right)M\left({\boldsymbol{G}}^{i}-{\boldsymbol{G}}^{j}\right)\hfill \\ \hfill & \quad +\frac{{H}_{\mathrm{d}\mathrm{e}\mathrm{m}}\left(\left({\boldsymbol{G}}^{i}-{\boldsymbol{G}}^{j}\right)\right)}{H}\hfill \\ \hfill {\mathcal{M}}_{zy}^{ij}& =-{\delta }_{ij}-\frac{1}{H}\frac{\vert \boldsymbol{q}+{\boldsymbol{G}}^{j}\vert \vert d}{2}{\mathrm{sin}}^{2}\;{\varphi }_{{\boldsymbol{G}}^{j}}M\left({\boldsymbol{G}}^{i}-{\boldsymbol{G}}^{j}\right)\hfill \\ \hfill & \quad -\frac{{H}_{\mathrm{d}\mathrm{e}\mathrm{m}}\left(\left({\boldsymbol{G}}^{i}-{\boldsymbol{G}}^{j}\right)\right)}{H}.\hfill \end{align} \tag{ 23 }$$

In the above equation the locally varying static demagnetising field H_dem has been taken into account by including a Bloch type of function of this field in the form of

$$\begin{equation}{H}_{\mathrm{d}\mathrm{e}\mathrm{m}}\left(\boldsymbol{r}\right)=\sum _{\boldsymbol{G}}{H}_{\mathrm{d}\mathrm{e}\mathrm{m}}\left(\boldsymbol{G}\right){\mathrm{e}}^{\mathrm{i}\boldsymbol{G}\cdot \boldsymbol{r}}.\end{equation} \tag{ 24 }$$

The exact form of H_dem(r) can be obtained either by analytical expressions of the demagnetising factor or more practically by micromagnetic simulations. The presence of H_dem(r) can, in principle, change the boundary conditions and hence has a direct consequence on the magnonic band structure of MCs. In order to obtain the dispersion relation one needs to accurately calculate the periodic potential of the system. For that one needs to consider many reciprocal lattice vectors. This may violate the assumption that the magnon wavelength is much larger than the thickness of the sample. This means that while comparing the results of such calculations to those of real systems, measured experimentally, the lowest order branches should show a good quantitative agreement.

Note that for simplicity in the above formalism modulations in the exchange field has not been taken into account. Such effects can be included by adding the following expression into the effective field in the LLG equation.

$$\begin{equation}{H}_{\mathrm{e}\mathrm{x}}\left(\boldsymbol{r},t\right)=\left(\nabla \cdot {l}_{\mathrm{e}\mathrm{x}}^{2}\left(\boldsymbol{r}\right)\nabla \right)\boldsymbol{m}\left(\boldsymbol{r},t\right),\end{equation} \tag{ 25 }$$

where the exchange length l_ex(r) is given by ${l}_{\mathrm{e}\mathrm{x}}\left(\boldsymbol{r}\right)=\sqrt{\frac{2A\left(\boldsymbol{r}\right)}{{\mu }_{0}{M}^{2}\left(\boldsymbol{r}\right)}}$. Consequently one needs to define the Fourier component of the exchange length as

$$\begin{equation}{l}_{\mathrm{e}\mathrm{x}}^{2}\left(\boldsymbol{r}\right)=\sum _{\boldsymbol{G}}{l}_{\mathrm{e}\mathrm{x}}^{2}\left(\boldsymbol{G}\right){\mathrm{e}}^{\mathrm{i}\boldsymbol{G}\cdot \boldsymbol{r}}.\end{equation} \tag{ 26 }$$

In order to better understand the effect of the lateral confinement and the periodicity of the structure on the magnonic band structure let us first consider the so-called empty lattice model (ELM). Based on the Bloch theorem in solids the presence of the translational symmetry in a crystal introduces the periodicity of the dispersion relation. The period of this periodicity is given by G. For an illustration this effect is calculated for the MSBVW mode excited in a permalloy film with a thickness of d = 20 nm. The lattice parameter of the artificial square lattice was a = 314 nm. The results are shown in the right side of figure 8. For a clear representation we use the so-called extended zone scheme. It is apparent from figure 8 that the single mode of the extended thin film, as expected from equations (8) and (12), is now changed to several magnon branches, as a result of back-folding. The single-band system is now developed into a multi-magnon band system. The magnonic band structure exhibits several degenerate magnon states at the zone centre. In order to shed light on the origin of these states one may analyse the magnon eigenvectors. The eigenvectors provide the mode profiles of different magnon modes. The calculated mode profile of different crossing points at the zone centre are shown in the left side of figure 8, indicating that the degenerate magnon states at this high symmetry point are the result of the fact that these modes represent magnons which differ only in their phase by G. The q_x value of these magnons is quantised and is given 2π/a, 4π/a, 6π/a, .... It is important to mention that the folding of the dispersion band within the ELM approximation is rather fictional. This is opposed to the case of real MCs.

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Based on the same arguments one can show that for the DE geometry in which the applied magnetic field and the wavevector are orthogonal to each other one would expect the appearance of additional magnon modes as a result of periodicity in the structure.

Using the formalism discussed above one may calculate the magnon band structure for an MC composed of ferromagnetic nano-elements embedded in a matrix made out of a different ferromagnetic material. A comparison between such results with those of ELM would provide useful information regarding the effect of the periodic potential on the magnonic band structure. An example of such analysis is provided in reference [89]. The magnon band structure for a system composed of Co nanodots embedded in a permalloy matrix as calculated by Krawczyk et al is shown in figure 9. The band structure calculations have been performed for a sample with a thickness of d = 20 nm, a lattice constant of a = b = 600 μm and a dot diameter 2R = 310 nm. Note that for this system the demagnetising factors can be calculated analytically [114, 115]. A magnetic field of μ₀H = 20 mT, applied along one of the main symmetry axis in the plane has been assumed, as depicted in figure 9.

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The magnonic band structure exhibits several bands, even in the low-frequency part of the spectrum. However, the first order mode seems to appear at very low frequencies, even below the FMR frequency of the system. This has its origin in the fact that internal fields along this particular direction are small. The effect is very similar to the one discussed for the magnetic stripes in section 2.2 and in figure 6. Owing to the presence of boundaries the internal field is not homogeneous, unlike the case of a continuous film. The field profile leads to the formation of periodic potentials, which confine the magnons. Since the magnons representing these modes propagate through the part of the structure with internal fields of lower magnitude, the frequency of these modes becomes lower, compared to a continuous film.

Another essential observation is that the degeneracy of the modes at the zone centre is lifted. This is due to the fact that the mode profile associated with each of these points has a different shape across the structure (these modes are differently distributed). Consequently, it is important to see where the maximum amplitude of these modes is located (on the dots or on the spaces between the dots). In other words nonuniform pattern of demagnetising fields across the structure leads to the fact that the modes do not degenerate at the zone centre. The splitting of the modes depend on the frequency. As a rule of thumb the higher the frequency of the modes the smaller the splitting.

Another important phenomenon which originates from the interaction of different magnon modes in an MC is the appearance of the so-called avoided-crossing regions in the magnon band structure. These regions which are shown by circles in figure 9 indicate the presence of interaction between different magnon modes. Since avoided-crossing takes place only for modes having the same symmetry, one should not expect it for all modes. This is a general phenomenon observed for quasi-particles in periodic potentials. Hence an analysis of the symmetry of the modes would account for the identification of the regions with avoided-crossing [116].

The final and the most important observation is the appearance of several band splittings in the band structure (shown by shaded area in figure 9). The origin of these splittings which can clearly be observed for the DE mode lies in the destructive interference of Bragg reflected magnons at the boundary of BZ [43, 117, 118]. The most intuitive picture of the Bragg mechanism is as follows. Consider a wave travelling through a one dimensional periodic medium. If the Bragg conditions are not fulfilled the wave can easily propagate through. However, if the Bragg conditions are fulfilled the wave will be reflected backwards. In such a case one would expect a constructive or distractive interference, depending on the phase relation between the waves. The interference leads to the formation of standing waves and the presence of band splittings in the dispersion relation at the high symmetry points. Such an effect is expected for magnons as well. The magnitude of such a band splitting depends on the materials parameter, the applied magnetic field and the geometry of the sample [119]. For some cases an analytical expression for the amplitude of the band splitting can be derived (see for example [114]).

The band splitting is the first step towards designing a system with a bandgap through the whole BZ. In order to open such a bandgap one should reduce the anisotropy of the DE and MSBVW modes. If the frequency of the DE mode at the boundary of the first BZ is smaller than the frequencies of MSBVW, one would expect a bandgap opening. Such conditions are never fulfilled in a square lattice. In references [120–124] the changes in the geometrical structure have been examined. However, no full magnonic gap could be observed for the studied structures. As a side remark, it is rather straightforward to imagine that since in the case of MSFVW the dispersion relation is isotropic one would be able to open full magnonic bandgaps [125].

The above discussion was limited to the dipolar and dipolar-exchange magnons. Exploiting the pure exchange magnons would open new possibilities to take advantage of high group/phase velocity as well as high frequency of this type of magnons. Moreover, since these magnons are governed by the short range exchange interaction, they would allow one to design MCs on much smaller length-scales.

As discussed in section 2.1.3 the magnon band structure of the exchange-dominated magnons can be calculated based on equation (11). Since the magnetic exchange interaction is rather short range in a single-crystalline sample the periodicity in the magnetic energy is given by the atomic arrangement i.e, the periodicity of the lattice. Assuming that the magnetic exchange interaction is isotropic (this is the case for most of the cubic crystals) the dispersion relation is also isotropic along the equivalent symmetry directions. For an infinitely extended sample in three dimensional space the Heisenberg model predicts only a single magnon band. This is due to the fact that all the magnon bands of such a system degenerate in energy. If the idea is to break the degeneracy of these bands, one may introduce additional periodicity on the atomic scales, for instance by replacing every second magnetic atom with another type. Under the assumption that the Heisenberg exchange is not affected, this leads to the fact that the spins on different sublattices will have different magnitudes and hence the degeneracy of the magnon bands will be lifted. In other words the magnetic unit cell becomes bigger. The larger the number of atoms in the unit cell the larger the number of magnon bands. In the next step if the idea is to open bandgaps in the magnonic band structure one needs to satisfy the conditions that the maximum frequency of the lower band is always lower than the minimum frequency of the upper band. This condition may be satisfied by taking different strategies (see the discussion below).

In the case of ferrimagnetic materials, in which the magnitude of the spin sitting on different sublattices is different, it has been often observed that the system exhibits several branches for exchange magnons. However, the magnitude of magnetic moments and the exchange parameters describing the interaction of spins are somehow interconnected. Since both the magnitude of moments and the values of exchange parameters are intrinsic properties of the material, tuning them in a desired way is not an easy task. Moreover, ferrimagnets often exhibit a rather complex structure, which adds to the complication of the problem.

Another approach as has recently been proposed is using the synthetic layered structures made of atomically smooth ultrathin ferromagnetic layers [126]. In order to illustrate the idea let us first discuss how the magnonic band structure develops in a layered ferromagnet. For simplicity we consider a ferromagnet ordered in a face-centered cubic (fcc) structure. The choice of fcc structure is based on the fact that due to its symmetry the nearest neighbours exist in both the individual atomic plane as well as the neighbouring atomic planes. This is not the case for a bcc lattice and hence one would expect additional effects due to the change in the number of nearest neighbours. The calculated magnon dispersion relation for an atomic layer of such a structure, based on equation (10) and considering only the nearest neighbour interactions, is shown in figure 10(a). The system shows a single magnon band with the maximum frequencies at the zone boundary. Now if one adds an additional layer on top of this layer, due to the fact that magnons with a phase relation of π are also allowed to form in such a bilayer structure, one will observe an additional magnon band, as shown in figure 10(b). In the case of a film composed of three atomic layers one should observe three magnon modes [see figure 10(c)]. The first two modes exhibit a similar dispersion relation and are equally distributed in the top and bottom atomic layer. The third mode is the result of strong coupling of the middle layer to these two layers. The opened partial bandgap along the $\overline{{\Gamma}}$–$\overline{\mathrm{X}}$ direction is a result of the complete 'magnetic unit cell'. The magnitude of the bandgap is determined by the minimum of the n = 2 magnon band and the maximum of the n = 1 magnon band. The minimum of the n = 2 magnon band is either at the $\overline{\mathrm{X}}$-point or at the $\overline{{\Gamma}}$-point. The maximum of the n = 1 magnon is at the $\overline{\mathrm{X}}$-point. The bandgap is therefore given by ${{\Delta}}_{\bar{{\Gamma}}-\bar{\mathrm{X}}}^{\mathrm{O}\mathrm{M}\mathrm{B}\mathrm{G}}=\left(1/2\pi \right)\left[{\omega }_{n=2}\left(\overline{{\Gamma}}\;\mathrm{o}\mathrm{r}\;\overline{\mathrm{X}}\right)-{\omega }_{n=1}\left(\overline{\mathrm{X}}\right)\right]=8JS/h$. Since this gap in opened between the optical magnons it may be called optical magnon bandgap (OMBG).

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Under the assumption that the exchange parameter J is isotropic, there will be no bandgap along the other high symmetry direction i.e., $\overline{{\Gamma}}$–$\overline{\mathrm{M}}$. One can, however, show that in the case of an anisotropic exchange interaction one can open an OMBG through the whole BZ. In order to illustrate this, in figure 10(d) we show the results of the calculations of the magnonic band structure for a trilayer structure, assuming that the interlayer exchange constant J_⊥ is larger than the intralayer one J_∥. Interlayer exchange parameter J_⊥ refers to the coupling strength of the Heisenberg type exchange interaction describing the coupling of spins sitting in different (neighbouring) layers and the intralayer exchange parameter J_∥ refers to the coupling strength between spins sitting in the same atomic layer. In this case, in addition to the partial OMBG along the $\overline{{\Gamma}}$–$\overline{\mathrm{X}}$ direction, a full OMBG opens as well. The partial OMBG is given by ${{\Delta}}_{\bar{{\Gamma}}-\bar{\mathrm{X}}}^{\mathrm{O}\mathrm{M}\mathrm{B}\mathrm{G}}=\left(1/2\pi \right)\left[{\omega }_{n=2}\left(\overline{\mathrm{X}}\right)-{\omega }_{n=1}\left(\overline{\mathrm{X}}\right)\right]=8{J}_{\perp }S/h$ and the full OMBG is given by ${{\Delta}}_{\bar{\mathrm{X}}-\bar{{\Gamma}}-\bar{\mathrm{M}}}^{\mathrm{O}\mathrm{M}\mathrm{B}\mathrm{G}}=\left(1/2\pi \right)\left[{\omega }_{n=2}\left(\overline{\mathrm{X}}\right)-{\omega }_{n=1}\left(\overline{\mathrm{M}}\right)\right]=8S\left[{J}_{\perp }-{J}_{\parallel }\right]/h$. This means that in addition to the fact that the bandgap lies in the terahertz frequency regime, its value is also several terahertz. The larger the anisotropy of the exchange the larger the bandgap. For the data shown in figure 10(d) we assume J_∥ = J and J_⊥ = 1.5J (therefore ${{\Delta}}_{\bar{\mathrm{X}}-\bar{{\Gamma}}-\bar{\mathrm{M}}}^{\mathrm{O}\mathrm{M}\mathrm{B}\mathrm{G}}=4JS/h$).

This simplified fundamental idea can be realised by growing atomically flat ultrathin synthetic ferromagnetic layers. The value of exchange parameters and their anisotropy can experimentally be tuned by using the epitaxial growth [127, 128] and also the choice of the materials involved (section 3.2).

Another important quantity which one may derive from the simple Heisenberg model, discussed above, is the amplitude of different magnon modes in each layer. In order to illustrate this we discuss an fcc film composed of three atomic layers. For sake of completeness we consider the general case of anisotropic exchange J_∥ ≠ J_⊥. The dispersion relation along the $\overline{{\Gamma}}-\overline{\mathrm{X}}$ ([110] direction of the fcc(100) surface) can be derived starting from equation (11) as following

$$\begin{align}\hfill {\omega }_{n=0}& =\frac{4{J}_{\parallel }}{\hslash }\left[1-\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)\right]\hfill \\ \hfill & \quad +\frac{4{J}_{\perp }}{\hslash }\left[3-\sqrt{5+4\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)}\right],\hfill \end{align} \tag{ 27 }$$

$$\begin{equation}{\omega }_{n=1}=\frac{4{J}_{\parallel }}{\hslash }\left[1-\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)\right]+\frac{8{J}_{\perp }}{\hslash },\end{equation} \tag{ 28 }$$

$$\begin{align}\hfill {\omega }_{n=2}& =\frac{4{J}_{\parallel }}{\hslash }\left[1-\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)\right]\hfill \\ \hfill & \quad +\frac{4{J}_{\perp }}{\hslash }\left[3+\sqrt{5+4\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)}\right].\hfill \end{align} \tag{ 29 }$$

Here ω_n=0, ω_n=1, and ω_n=2 represent the frequencies of the magnon modes with the quantum number of n = 0, 1, and 2, respectively. The magnon amplitude A_j in each atomic layer can be calculated again using equation (11). In the present case one obtains the amplitude of the three magnon modes to be

n = 0:

$$\begin{equation}{A}_{1}={A}_{3}=A,\quad {A}_{2}=\frac{\sqrt{5+4\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)}-1}{2\mathrm{cos}\left(\frac{a}{2\sqrt{2}}{q}_{\parallel }\right)}A\end{equation} \tag{ 30 }$$

n = 1:

$$\begin{equation}{A}_{1}=-{A}_{3}=A,\quad {A}_{2}=0\end{equation} \tag{ 31 }$$

n = 2:

$$\begin{equation}{A}_{1}={A}_{3}=A,\quad {A}_{2}=-\frac{\sqrt{5+4\mathrm{cos}\left(\frac{a}{\sqrt{2}}{q}_{\parallel }\right)}+1}{2\mathrm{cos}\left(\frac{a}{2\sqrt{2}}{q}_{\parallel }\right)}A.\end{equation} \tag{ 32 }$$

A₁ and A₃ refer to the amplitude of magnons in the bottom and top atomic layers, and A₂ refers to the amplitude of the magnons in the middle layer. These quantities provide information on the localisation of different magnon modes in different layers. According to equations (30)–(32), for the first two magnon modes of such a system (modes with n = 0 and n = 1), the largest amplitude appears at the bottom and top atomic layers. An illustration is provided in figure 11, where the magnon amplitude in different layers is plotted for all thee magnon modes. The amplitude is plotted for three representative points of the surface BZ e.g., $\overline{{\Gamma}}$-point, middle of $\overline{{\Gamma}}$–$\overline{\mathrm{X}}$ and $\overline{\mathrm{X}}$–pint. For a better representation we plot the normalised values i.e., ${A}_{j}/{\sum }_{j=1}^{3}{A}_{j}$. The magnon amplitude for the mode with n = 0 near the $\overline{{\Gamma}}$-point is identical in all three layers. This indicates the uniform precession of all layers and their equal contribution to this magnon mode. The magnon's amplitude in the middle layer decreases while moving from the zone centre towards the zone boundary. At the $\overline{\mathrm{X}}$-point only the magnon amplitude in the bottom and top layer remains. This means that at this point the magnons are localised entirely in these two layers. For the n = 1 magnon mode, the zero amplitude at the middle layer indicates the standing wave character of this magnon with one node in the central atomic layer. The magnon mode with n = 2 has the largest amplitude in the middle layer (the amplitude is larger than the one at the bottom and top layers). The amplitude of this mode becomes the largest at the zone boundary. Thus, this magnon mode is mainly localised in the middle layer. Note that in the present consideration the top and bottom layers are identical. The pattern of the localisation will change when the top and bottom layers are no longer identical.

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The discussion provided above is for atomically thin ferromagnetic layers. A model based on Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation has been developed in references [129–131] for periodic and quasi-periodic crystals. The model assumes thin films of different materials which are exchanged coupled and, in a similar way, predicts existence of several bands and also partial bandgaps in such structures.

Dzyaloshinskii–Moriya interaction (DMI) is an antisymmetric magnetic interaction originating from spin–orbit coupling in the absence of inversion symmetry. This interaction has the form of ${\mathcal{H}}_{\mathrm{D}\mathrm{M}\mathrm{I}}=-{\sum }_{i\ne j}{\boldsymbol{D}}_{\boldsymbol{i}\boldsymbol{j}}\cdot {\boldsymbol{\hat{e}}}_{\boldsymbol{i}}{\times}{\boldsymbol{\hat{e}}}_{\boldsymbol{j}}$, where D_ij is the so-called microscopic DMI vector. It has been shown that DMI modifies the properties of magnons in chiral magnets [247]. The most prominent effects of DMI on the magnon dispersion relation is that it leads to an asymmetry in the dispersion relation [259] and the magnon lifetime [136]. These effects result in a nonreciprocity in the propagation characteristics of counter propagating magnons. Recently, it has been proposed that a periodic modulation of this interaction leads to very interesting effects on the magnonic band structure of gigahertz magnons [50]. Analytical calculations combined with micromagnetic simulations have revealed the interesting features of the magnon band structure in these systems e.g., indirect magnonic bandgaps. Furthermore, several interesting effects such as dispersionless bands have been observed in the simulations. As the presence of the DMI leads to the nonreciprocity of magnons, this leads to unusual temporal behaviour of standing spin waves, exhibiting nonzero phase velocities (for a review on the topic see chapter 5 of reference [19]).

As pointed out in section 2.3.1 periodic spatial modulation of any magnetic parameter may be used to design MCs. Such modulation may be done on the domain pattern of a thin film [260, 261] or periodic magnetic vortices [262, 263]. Likewise this may be realised by introducing ordered topological defects in the system. Based on numerical simulations Ma et al have demonstrated a novel MC based on a linear array of skyrmions [54, 264–266]. Skyrmions are nanoscale topological spin textures emerging in some chiral magnets. A magnetic stripe, acting as a waveguide, with 1D arrangement of skyrmions has been considered. It has been shown that the skyrmion-induced spatial variation of the magnetisation along the waveguide results in a drastic modification of the magnon dispersion relation, which leads to the appearance of allowed and forbidden bands in its band structure.

The numerically obtained magnonic band structure by Ma et al [54] in the 1D periodic skyrmion lattice formed in a waveguide is presented in figure 18(a) in the extended zone scheme. The band structure exhibits a periodic character of the three BZs. Back folding of the bands into the first BZ would results in bandgaps with the magnitude of 6, 4, and 2.5 GHz. It has been also illustrated that magnons in such a system exhibit nonreciprocity due to their chiral nature. The nonreciprocal behaviour of these magnons is demonstrated in figures 18(b) and (c), where the magnon amplitudes at the two edges of the waveguide are presented as the colour scale. On the edges of the waveguide the counter-propagating magnons are localised. This means that the magnons travelling to one side of the waveguide are localised at one edge and those travelling to the opposite direction are localised in the other edge of the waveguide. The chiral character of these magnons originates from DMI.

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Such a skyrmion-based MC is dynamically reconfigured by changing the diameter of the skyrmions e.g., by applying a current, and bias magnetic or electric field [54]. The same may be realised by changing the periodicity of the lattice using external stimuli.

In a 2D lattice of ferromagnetic as well as aniferromagnetic skyrmions it has been shown that one would expect to observe a very interesting magnonic band structure containing allowed magnon bands, bandgaps as well as chiral edge states, which are topologically protected [267, 268], similar to those in topological magnon insulators [269].

The idea may be extended to 3D skyrmion crystals. In bulk chiral magnets the skyrmion lattices are usually stabilised by temperature and application of a magnetic field. It has been shown that the dynamics of such systems do not obey the dynamics of a simple ferromagnetic sample given by equations (2)–(4). The FMR response of such systems has been investigated in detail [270]. It has been observed that in the skyrmion phase there exist three magnon modes describing clock wise motion, counter clock wise motion and breathing of the skyrmions. A generalised description of the resonance frequency has been reported for materials hosting skyrmion crystals through the whole phase diagram [270]. Furthermore, it has been illustrated that such MCs would allow for nonreciprocal microwave components and grating couplers controlled by external stimuli (for review see for example reference [271]). This is a promising research direction that requires further experimental efforts for realisation of the effects predicted by the theory and simulation.

It has recently been demonstrated that magnons excited in a thin ferromagnetic film can be coupled to a flux lattice of a thin superconducting film, if these two are brought in a close proximity. Experiments on Py/Nb bilayers have shown that the magnonic band structure of such a system contains allowed states as well as forbidden gaps, which can be tuned by a magnetic field. In addition the Doppler shifts in the frequency spectra of magnons scattered on a flux lattice has been observed when the lattice was moved by sending an electrical current through the superconducting layer [53].

One of the main challenges hindering the application of MCs in practical devices is the lifetime of magnons [102, 283–288]. While in the case of ferromagnetic insulators the magnon lifetime is reasonably long, in most of the metallic ferromagnets magnons suffer from damping. In order to overcome this problem two alternatives have been suggested. First, efforts have been devoted to grow high quality thin films of magnetic insulators (mainly YIG films) with low damping [289] and use them further for nano-patterning and device fabrications [290]. Although fabrication of nano-elements based on ferromagnetic insulators is another challenge, the long magnon lifetime and several decade of research on these materials have made them still as the prime candidate for magnonic devices. Second, efforts have been devoted to suppress the magnon damping in metallic ferromagnets either by tuning the damping parameter through the modification of the material's property or by extrinsic effect such as magnon amplifications and exploiting the antidamping phenomena [11, 291–294]. Third, it has been suggested that by increasing the magnon group velocity one may enhance the mean-free-path of magnons and thereby overcome effect of damping [295].

The magnon damping is still a challenge in the case of terahertz magnons. The damping of terahertz magnons in itinerant ferromagnets is a result of their decay into Stoner excitations. Generally the probability of decay of a magnon to a Stoner pair can depend nonmonotonously on the frequency and wavevector. This behaviour in ultrathin ferromagnets and low dimensional magnetic structures can be very much different from that in the bulk ferromagnets. A way to gain a detailed knowledge on the Landau damping of terahertz magnons in itinerant ferromagnets is to calculate the Landau map. The concept has been introduced and discussed in references [144, 145, 147, 296]. It has been shown that by calculating the Landau map one can assess the contribution of different electronic states of the system to the Landau damping of magnons. Generally, the map provides direct information about the probability and the number of Stoner transitions, which contribute to the damping of a magnon with a given wavevector and frequency, over BZ. Places in BZ where the Stoner transitions happen with a large intensity are the hot spots. In the systems for which the number and the intensity of the Landau hot spots are larger the Landau damping is stronger. Hence, an efficient way to suppress the damping would be to modify the electronic band structure of the sample such that the intensity and the number of such hot spots are drastically reduced in the system. This idea has been realised for example in the case of an ultrathin Fe film alloyed with Pd [147].

On the other hand one should notice that the terahertz magnons possess a rather high group and phase velocity [see for example figure 13(c)]. Hence, although the damping parameter of terahertz excitations may seem to be high compared to the Gilbert damping parameter of the FMR mode, one may try to take advantage of the high group and phase velocity of terahertz magnons. The mean-free-path of a magnon wave packet is given by λ = v_g ⋅ τ, where τ denotes the lifetime. In addition, the wavelength of terahertz magnons is rather short (it is on the order of interatomic distances, being a few Å). This fact enables scaling of the devices down to the atomic scales.

Magnons with a long wavelength (mainly dipolar and dipolar-exchange magnons) can be very efficiently excited and detected using microwave and optical techniques, methods based on electrical transports or combinations of these. Techniques like FMR, both in conventional and broad-band scheme provide excellent opportunities to excite and detect magnons near the zone centre. BLS and time-resolved optical methods provide frequency, wavevector, time, space and phase resolved measurement of long wavelength magnons. The recently developed techniques such as spin-torque- or spin-Hall-nano-oscillators have contributed very largely to the realisation of all electric magnon excitation and detection in magnonic devices (see for example [297, 298]). This has pushed the development in the field one big step further with the perspective of integrating the magnonic circuits with the current state-of-the-art spintronic technologies [299, 300]. However, in the case of terahertz magnons with short wavelengths one still needs to develop techniques for excitation and detection of these excitations. Spin-polarised high-resolution electron energy-loss spectroscopy (traditionally called spin-polarised electron energy-loss spectroscopy) has been used for fundamental investigation of such magnons in low-dimensional magnetic structures [248, 301–309]. The technique provides the magnonic band structure in a momentum-resolved manner with a high momentum resolution an extremely high sensitivity. Several interesting effects have already been observed using this technique [68, 126–128, 136, 142, 143, 147, 245–247, 251, 253, 254, 259, 296, 308–311]. Alternatives excitation schemes, which may enable the practical use of terahertz magnons in logic devices are as follows. Terahertz magnons can be excited by e.g., terahertz radiations (see for example see [312–314]), using nonlinear effects, spin torque nano-oscillators [11, 315] or very recently demonstrated vortex core excitation method [316]. The latter method allows coherent excitation of short wavelength excitations down to 80 nm wavelength (but still in gigahertz regime). The mechanism of magnon excitations in this case has been explained in terms of the mechanism initially introduced by Schlömann [84, 85, 317, 318]. In addition is has been demonstrated that terahertz magnons can be very efficiently excited in tunnelling junctions [319] and by electron tunnelling via an STM tip (see for example [240] and references therein). In these ways one may be able to excite terahertz magnons for application in logic elements. However, the first important step towards any future device application is the proof of concepts. Realisation of the concepts for practical applications is another challenge, which demands further efforts.