Collective spin excitations of helices and magnetic skyrmions: review and perspectives of magnonics in non-centrosymmetric magnets

We revisit spin-wave excitations in the field-polarized regime of a large bulk sample (denoted by FP in figure 2(a)). A large magnetic field H  >  H_c2 is assumed to be applied along the z-axis, $\vec{H}=H\hat{z}$. The magnetization is polarized so that ${{\hat{n}}_{\text{eq}}}=\hat{z}$. In this case, the local dreibein can be chosen to be independent of position, e.g. ${{\hat{e}}_{1}}=\hat{x}$ and ${{\hat{e}}_{2}}=\hat{y}$. The magnon Hamiltonian is then given by $\mathcal{H}={{\mathcal{H}}_{0}}+{{\mathcal{H}}_{\text{dip}}}$ where ${{\mathcal{H}}_{0}}$ derives from equation (1) and reads

$$\begin{eqnarray}&&{{\mathcal{H}}_{0}}=\mathcal{D}\left(-\mathbb{1}{{\vec{\nabla}}^{2}}-\text{i}2Q{{\tau}^{z}}\hat{z}\vec{\nabla}\right)+g{{\mu}_{B}}{{\mu}_{0}}H\mathbb{1}\end{eqnarray} \tag{ 9 }$$

where we used the stiffness $\mathcal{D}=2Ag{{\mu}_{B}}/{{M}_{s}}$ (table 2). In order to discuss the influence of the dipolar interaction, it is actually more convenient to switch from real to momentum space, ${{\mathcal{H}}_{0}}\left(-\text{i}\nabla \right)\to {{\mathcal{H}}_{0}}\left(\vec{k}\right)$. In momentum space, the Hamiltonian including the contribution from the dipolar interactions reads

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\mathcal{H}=\mathcal{D}\left(\mathbb{1}{{\vec{k}}^{2}}+2Q{{\tau}^{z}}\hat{z}\vec{k}\right)+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}\mathbb{1}\\ \,\,\,\,\,\,\,\,\,\,\,\,+\frac{g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}}{2{{k}^{2}}}\left(\begin{array}{c} {{k}_{+}}{{k}_{-}} & k_{-}^{2} \\ k_{+}^{2} & {{k}_{+}}{{k}_{-}} \end{array}\right)\end{array}\end{eqnarray} \tag{ 10 }$$

where ${{H}_{\text{int}}}=H-{{N}_{z}}{{M}_{s}}$ is the internal field, and we introduced ${{k}_{\pm}}={{k}_{x}}\pm \text{i}{{k}_{y}}$. The eigenfrequencies of the magnon spectrum then follow straightforwardly from solving the eigenvalue equation (6) and are given by  ±$\omega \left(\pm \vec{k}\right)$ with

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\hbar \omega \left(\vec{k}\right)=2\mathcal{D}Q{{k}_{z}}\\ \quad +\sqrt{\left(\mathcal{D}{{k}^{2}}+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}\right)\left(\mathcal{D}{{k}^{2}}+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}+\frac{g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}k_{\bot}^{2}}{{{k}^{2}}}\right)}\end{array}\end{eqnarray} \tag{ 11 }$$

where ${{\vec{k}}_{\bot}}=\left({{k}_{x}},{{k}_{y}},0\right)$ and $k=|\vec{k}|$. The second term with the square root corresponds to the Herring-Kittel formula for a ferromagnet [109]. The DMI gives rise to the additional first term and leads to a shift of the dispersion along k_z, i.e. the direction in momentum space singled out by the magnetic field.

In particular, for vanishing perpendicular momentum, ${{\vec{k}}_{\bot}}=0$, the dispersion reduces to a shifted parabola $\hbar \omega \left({{k}_{z}}\right)=\mathcal{D}\left({{k}_{z}}+Q\right){{}^{2}}+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}-\mathcal{D}{{Q}^{2}}$. As a consequence, the magnon dispersion $\omega \left({{k}_{z}}\right)$ is not symmetric with respect to k_z. This implies, for example, a finite group velocity ${{\partial}_{{{k}_{z}}}}\omega \left({{k}_{z}}\right)\to 2\mathcal{D}Q$ in the limit of small ${{k}_{z}}\to 0$ resulting in nonreciprocal spin wave propagation at small wavevectors. This group velocity is positive for a right-handed DMI as considered here. In case of a left-handed DMI, the momentum shift would be in the opposite direction leading to a negative group velocity. Nonreciprocal spin wave propagation was recently explored for Cu₂OSeO₃ [110]. The DMI-induced term in equation (11) also results in an asymmetry in the dynamic susceptibility with respect to k_z. Introducing the spin-flip contributions for the dynamic susceptibility of equation (8), ${{\chi}_{\pm \mp}}\left({{k}_{z}},\omega \right)=\hat{e}_{i}^{\mp}{{\chi}_{ij}}\left({{k}_{z}},\omega \right)\hat{e}_{j}^{\pm}$, we obtain for its imaginary parts at ${{k}_{\bot}}=0$

$$\begin{eqnarray}&&\chi _{\pm \mp}^{\prime \prime}\left({{k}_{z}},\omega \right)=\pm \frac{\pi g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}}{\hbar}\delta \left(\omega \mp \omega \left(\pm {{k}_{z}}\right)\right).\end{eqnarray} \tag{ 12 }$$

A magnon at a given momentum k_z and energy $\omega \left({{k}_{z}}\right)$ can be absorbed, but, different from a conventional ferromagnet, it cannot be emitted with the same energy as $\omega \left(-{{k}_{z}}\right)\ne \omega \left({{k}_{z}}\right)$. This asymmetry in absorption and emission of magnons has been very recently observed in inelastic neutron scattering on MnSi [111].

The energy dispersion $\omega \left(\vec{k}\right)$, equation (11), is mimimal for $\vec{k}=Q\hat{z}$ with an energy gap $g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}-\mathcal{D}{{Q}^{2}}$. This gap vanishes at the second critical field $H_{c2}^{\text{int}}$ where the transition occurs between the field-polarized (FP) and the conical phase, see figure 2(a). The stiffness $\mathcal{D}=g{{\mu}_{B}}{{\mu}_{0}}H_{c2}^{\text{int}}/{{Q}^{2}}$ is thus determined by the size of the pitch vector Q and the value of $H_{c2}^{\text{int}}$, which are both accessible from independent measurements [37, 44, 112].

We comment on the eigenmodes at zero wavevector attributed to uniform oscillations of the magnetization. In this limit, the Hamiltonian simplifies to

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}{{\mathcal{H}}_{\text{uniform}}}=g{{\mu}_{B}}{{\mu}_{0}}H\mathbb{1}+g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}\left(-{{N}_{z}}\mathbb{1}+\frac{{{N}_{x}}+{{N}_{y}}}{2}\mathbb{1}\right. \\ \left. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{{N}_{x}}-{{N}_{y}}}{2}{{\tau}^{x}}\right).\end{array}\end{eqnarray} \tag{ 13 }$$

For the field-polarized phase, the eigenfrequencies have the form of the well-known Kittel mode [4, 113]

$$\begin{eqnarray}&&\hbar \omega =g{{\mu}_{B}}{{\mu}_{0}}\sqrt{\left(H+\left({{N}_{x}}-{{N}_{z}}\right){{M}_{s}}\right)\left(H+\left({{N}_{y}}-{{N}_{z}}\right){{M}_{s}}\right)}.\end{eqnarray} \tag{ 14 }$$

The uniform mode within a sample of large size is thus not affected by the DMI.

When the magnetic field H is lowered below H_c2, the field-polarized state becomes unstable with respect to the formation of a conical spin helix. It is characterized by a helix axis that is aligned with the magnetic field. The conical helix is realized in the chiral magnets within the field range between H_c1 and H_c2. If the field is lowered below H_c1 the helix might reorient its axis from the field direction into a crystallographic high-symmetry direction, either $\langle 1\,1\,1\rangle$ or $\langle 1\,0\,0\rangle$ depending on the material [106]. This reorientation has its origin in the magnetocrystalline anisotropies F_aniso. If the field is applied along the favoured crystallographic direction then H_c1  =  0 in case of high-field cooling. In the following, we neglect these anisotropies and the associated complexities arising from the reorientation transition close to H_c1. We limit ourselves to the spin waves in the presence of a conical helix configuration. Here, we consider the situation of a bulk sample. Thin films will be discussed in section 4.6.1.

3.2.1. Spin-waves of the magnetic helix in bulk samples: finite wavevectors.

The conical helix configuration for a magnetic field applied along the z-axis is given by

$$\begin{eqnarray}&&{{\hat{n}}_{\text{helix}}}(z)=\sin \theta {{\text{e}}^{-\text{i}Qz{{\mathbf{L}}^{z}}}}\hat{x}+\cos \theta \hat{z}=\left(\begin{array}{c} \sin \theta \cos (Qz) \\ \sin \theta \sin (Qz) \\ \cos \theta \end{array}\right).\end{eqnarray} \tag{ 15 }$$

It consists of a homogeneous part pointing along the magnetic field direction and a helical part that rotates as a function of the z-coordinate within the plane perpendicular to the field. The helical part is favored by the DMI and can be generated with the help of the z-component of the spin-1 operator $\mathbf{\vec{L}}$ introduced in equation (2).

Plugging this Ansatz into the free energy functional and minimizing with respect to the cone angle θ one finds ${{M}_{s}}\cos \theta ={{\chi}_{\text{con}}}H=\chi _{\text{con}}^{\text{int}}{{H}_{\text{int}}}$ with the susceptibility $\chi _{\text{con}}^{-1}=\left(\chi _{\text{con}}^{\text{int}}\right){{}^{-1}}+{{N}_{z}}$. We assume here that the z-axis coincides with a principal axis of the sample characterized by the demagnetization factor N_z. The internal susceptibility is given by $\chi _{\text{con}}^{\text{int}}={{\mu}_{0}}M_{s}^{2}/\left(2A{{Q}^{2}}\right)$. On the mean-field level, the differential susceptibility $\partial M/\partial H={{\chi}_{\text{con}}}$ is thus expected to be constant within the conical phase. This is indeed observed approximately in chiral magnets except for relatively high temperatures close to T_c [114]. The experimentally determined value of the susceptibility $\chi _{\text{con}}^{\text{int}}$, see table 1, provides an important combination of parameters that will enter the spin-wave spectrum. Moreover, at the second critical field H_c2 the angle $\theta =0$ so that $\chi _{\text{con}}^{\text{int}}H_{c2}^{\text{int}}={{M}_{s}}$ which is consistent with the result of section 3.1.

The helix possesses the continuous screw symmetry. An arbitrary translation of the helix along the z-axis can be compensated by a rotation around the same axis leaving the helix invariant. Mathematically, this is reflected in the fact that $\left({{\mathbf{P}}_{z}}-Q{{\mathbf{L}}^{z}}\right){{\hat{n}}_{\text{helix}}}(z)=0$ where ${{\mathbf{P}}_{z}}=-i\mathbb{1}{{\partial}_{z}}$. An additional discrete symmetry arises in zero field where $\cos \theta =0$. In this case, the helix is also invariant with respect to a π rotation of real and spin space around the x-axis. These symmetries will be reflected in specific properties of the spin wave excitations as discussed below.

In the conical helix phase, the magnon Hamiltonian deriving from the exchange free energy density ${{\mathcal{F}}_{0}}$ of equation (1) reads [37, 115]

$$\begin{eqnarray}&&{{\mathcal{H}}_{0}}=\mathcal{D}[-\mathbb{1}{{\vec{\nabla}}^{2}}-\text{i}2{{\tau}^{z}}Q{{\hat{n}}_{\bot}}(z)\vec{\nabla}+\frac{{{Q}^{2}}{{\sin}^{2}}\theta}{2}\left(\mathbb{1}-{{\tau}^{x}}\right)].\end{eqnarray} \tag{ 16 }$$

The magnetic helix gives rise to a periodic potential, ${{\hat{n}}_{\bot}}(z)=\left(\sin \theta \cos (Qz),\sin \theta \sin (Qz),0\right)$, for the magnons along the z-direction. The periodic potential leads to Bragg scattering that opens gaps in the energy spectrum and results in magnon bands with different index n  =  0, 1, 2, 3,.... In accordance with Bloch's theorem, the spin wave spectrum ${{\omega}_{n}}\left(\vec{k}\right)$ is periodic ${{\omega}_{n}}\left(\vec{k}+m\vec{Q}\right)={{\omega}_{n}}\left(\vec{k}\right)$ for any $m\in Z$ with $\vec{Q}=Q\hat{z}$. In real space, relevant periodicities $2\pi /Q$ are on the order of 20 to 70 nm for the materials listed in table 1.

The wave equation is in general not diagonal in momentum space. The full stationary equation including the dipolar interactions reads

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} -g_{0}^{-1}\left(\omega,\vec{k}\right){{{\vec{\Psi}}}_{\omega}}\left(\vec{k}\right)+V\left(\vec{k}\right){{{\vec{\Psi}}}_{\omega}}\left(\vec{k}-\vec{Q}\right)+{{V}^{\ast}}\left(\vec{k}\right){{{\vec{\Psi}}}_{\omega}}\left(\vec{k}+\vec{Q}\right) \\ +\underset{\begin{array}{c} \alpha,\beta =-1,0,1, \\ \alpha \ne \beta \end{array}}{\sum}\mathcal{H}_{\text{dip}}^{\alpha \beta}\left(\vec{k}-\alpha \vec{Q}\right){{\Psi}_{\omega}}\left(\vec{k}-\left(\alpha -\beta \right)\vec{Q}\right)=0. \end{array}\end{eqnarray} \tag{ 17 }$$

The periodic potential of equation (16) leads to the off-diagonal potential $V\left(\vec{k}\right)=\mathcal{D}Q{{k}_{-}}\sin \theta {{\tau}^{z}}$ with ${{k}_{\pm}}={{k}_{x}}\pm i{{k}_{y}}$. The diagonal part is summarized in the Green function

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}-g_{0}^{-1}\left(\omega,\vec{k}\right)=-\hbar \omega {{\tau}^{z}}+\mathcal{D}\left[\mathbb{1}{{\vec{k}}^{2}}+\frac{{{Q}^{2}}{{\sin}^{2}}\theta}{2}\left(\mathbb{1}-{{\tau}^{x}}\right)\right]\\ \quad +\underset{\alpha =-1,0,1}{\sum}\mathcal{H}_{\text{dip}}^{\alpha \alpha}\left(\vec{k}-\alpha \vec{Q}\right).\end{array}\end{eqnarray} \tag{ 18 }$$

The contribution of the dipolar interaction (3) leads to the terms $\mathcal{H}_{\text{dip}}^{\alpha \beta}$ whose explicit form is given in appendix A. If the magnon only possesses a momentum along the z-direction, i.e. the direction of the helix, $\vec{k}={{k}_{z}}\hat{z}$, the problem simplifies considerably. In this case, magnon gaps do not exist in the spectrum because the Fourier transform of the periodic potential, $V\left(\vec{k}\right)$, vanishes, so that one is left with the empty lattice model for spin waves [21]. The vanishing of Bragg scattering in this limit is attributed to the continuous screw symmetry of the helix. The spectrum can then be obtained exactly and reads

$$\begin{eqnarray}&&\hbar \omega \left({{k}_{z}}\right)=\mathcal{D}|{{k}_{z}}|\sqrt{k_{z}^{2}+\left(1+\chi _{\text{con}}^{\text{int}}\right){{Q}^{2}}\left(1-{{\left(\frac{{{H}_{\text{int}}}}{H_{c2}^{\text{int}}}\right)}^{2}}\right)}\end{eqnarray} \tag{ 19 }$$

in the extended zone scheme where the term $\chi _{\text{con}}^{\text{int}}$ arises due to the dipolar interaction, and ${{H}_{\text{int}}}/H_{c2}^{\text{int}}=H/{{H}_{c2}}$.

In the other limit of large perpendicular momenta $|{{\vec{k}}_{\bot}}|\gg Q$ with ${{\vec{k}}_{\bot}}\hat{z}=0$, the dipolar interaction and the last term in equation (16) can be effectively neglected. The wave equation for the partial Fourier transform ${{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)$ then reduces to

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\hbar \omega {{\tau}^{z}}{{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)\approx \\ \quad \mathcal{D}\left[\mathbb{1}\left(\vec{k}_{\bot}^{2}-\partial _{z}^{2}\right)-\text{i}2{{\tau}^{z}}Q|{{\vec{k}}_{\bot}}|\sin \theta \cos \left(Qz-\alpha \right)\right]{{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)\end{array}\end{eqnarray} \tag{ 20 }$$

with $\vec{k}=|{{\vec{k}}_{\bot}}|\left(\cos \alpha,\sin \alpha,0\right)$. It describes a particle with quadratic dispersion in a periodic cosine potential. This wave equation can be identified with the Mathieu equation. Interestingly, the strength of the periodic potential can be tuned by the size of the perpendicular momentum ${{\vec{k}}_{\bot}}$ allowing to tune the band structure from the weak-binding to the tight-binding limit. For very large $|{{\vec{k}}_{\bot}}|\gg (n+1){{}^{2}}Q/\sin \theta$, the Bragg scattering is so strong that the band with index n is basically flat, i.e. non-dispersive, and the spin waves are localized within the z-direction. In this limit, the periodic cosine potential $\cos \left(Qz-\alpha \right)$ can be expanded around its minima. Solving the resulting wave equation one obtains [36, 37]

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\hbar {{\omega}_{n}}\left(\vec{k}\right)\approx \mathcal{D}\left[\vec{k}_{\bot}^{2}-2|{{\vec{k}}_{\bot}}|Q\sin \theta +2Q\sqrt{Q|{{{\vec{k}}}_{\bot}}|\sin \theta}\left(n+\frac{1}{2}\right)\right.\\ \left. \quad -\frac{{{Q}^{2}}}{8}\left({{n}^{2}}+n\right)\right]\end{array}\end{eqnarray} \tag{ 21 }$$

where $\sin \theta =\sqrt{1-\left(H/{{H}_{c2}}\right){{}^{2}}}$. The third term corresponds to the harmonic oscillator spectrum deriving from oscillations around the minima of the cosine potential. The last term is attributed to the anharmonicity of the potential. The spin wave spectra numerically solved for various values of ${{\vec{k}}_{\bot}}$ are shown in figure 6.

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At the Brillouin zone center close to zero wavevector the energy of the lowest band vanishes. This low-energy spin wave is a Goldstone mode and protected by translational symmetry. Both, at zero magnetic field and in the absence of magnetocrystalline anisotropies F_aniso that would break the rotational invariance of the theory, the dispersion of this mode is particularly soft [116, 117],

$$\begin{eqnarray}&&{{\omega}^{2}}\sim Bk_{z}^{2}+Kk_{\bot}^{4}.\end{eqnarray} \tag{ 22 }$$

It has the Landau-Peierls form characteristic for lamellar structures where B and K are the elastic constants of compression and splay [118], respectively. Such a mode would lead to a Landau-Peierls instability destroying long-range magnetic order of the conical helix. However, either a magnetic field or small magnetocrystalline anisotropies eventually lead to a $k_{\bot}^{2}$ contribution to the right-hand side of equation (22) that stabilizes the helimagnetic order [116].

The helimagnon spectrum has been measured in MnSi by inelastic neutron scattering [36, 37]. Kugler et al [37] resolved five helimagnon bands as a function of energy as shown in figure 7. Such measurements have been also performed on Cu₂OSeO₃ [1, 40] but here the band structure could not be resolved because the associated energy scale $\mathcal{D}{{Q}^{2}}$ is an order of magnitude smaller than in MnSi.

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3.2.2. Dynamics of the spin helix at zero wavevector: uniform mode.

According to equation (15), the uniform magnetization is given by $\vec{M}=\hat{z}{{M}_{s}}\cos \theta =\hat{z}{{\chi}_{\text{con}}}H$. The excitation of the uniform mode results in a dynamic correction to the magnetization, $\delta \vec{M}(t)={\int}^{}\frac{\text{d}\omega}{2\pi}{{\text{e}}^{-\text{i}\omega t}}\delta \vec{M}\left(\omega \right)$ that, in first order in the magnon wavefunction, is determined by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \delta \vec{M}\left(\omega \right)=\frac{{{M}_{s}}}{V}{\int}^{}\text{d}\vec{r}\sqrt{\frac{g{{\mu}_{B}}}{{{M}_{s}}}}\frac{1}{2}\text{Re}\left\{\left({{{\hat{e}}}^{+}}(z),{{{\hat{e}}}^{-}}(z)\right)\vec{\Psi}\left(\vec{r},\omega \right)\right\} \end{array}\end{eqnarray} \tag{ 23 }$$

with the volume V. We can perform the real space integral using the representation ${{\hat{e}}_{1}}=\left(-\sin (Qz),\cos (Qz),0\right)$ and ${{\hat{e}}_{2}}=\left(-\cos \theta \cos (Qz),-\cos \theta \sin (Qz),\sin \theta \right)$ for the vectors ${{\hat{e}}^{\pm}}=\frac{1}{\sqrt{2}}\left({{\hat{e}}_{1}}\pm \text{i}{{\hat{e}}_{2}}\right)$.

Consider first the z-component

$$\begin{eqnarray}&&\delta {{\vec{M}}_{z}}\left(\omega \right)=\frac{\sqrt{g{{\mu}_{B}}{{M}_{s}}}}{2\sqrt{2}V}\sin \theta \,\text{Re}\left\{\text{i}(1,-1)\vec{\Psi}\left(0,\omega \right)\right\}\end{eqnarray} \tag{ 24 }$$

where $\vec{\Psi}\left(0,\omega \right)={\int}^{}\text{d}\vec{r}\,\vec{\Psi}\left(\vec{r},\omega \right)$. It is determined by the magnon wavefunction at zero momentum $\vec{\Psi}\left(0,\omega \right)$, which coincides with the Goldstone mode of the helix associated with the breaking of translational symmetry. Its eigenvector is therefore related to the derivative ${{\partial}_{z}}{{\hat{n}}_{\text{helix}}}(z)\propto {{\hat{e}}_{1}}$. Comparing this with equation (5) identifies the eigenmode ${{\vec{\Psi}}^{T}}\left(0,\omega \right)=\frac{1}{\sqrt{2}}(1,1)$ with eigenfrequency $\omega =0$. The scalar product $(1,-1)\vec{\Psi}\left(0,\omega \right)$ determining $\delta {{\vec{M}}_{z}}$ thus exactly vanishes. So we conclude that in linear order in the spin wave amplitude the mean magnetization does not vary in time along the z-direction, $\delta {{\vec{M}}_{z}}(t)=0$, within the linear spin-wave approximation considered here.

The uniform oscillation of the magnetization is therefore confined to the plane in spin space that is orthogonal to the magnetic field,

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\delta {{\vec{M}}_{x}}\left(\omega \right)=\frac{\sqrt{g{{\mu}_{B}}{{M}_{s}}}}{4\sqrt{2}V}\\ \quad \times \text{Re}\left\{i\left(\left(-1-\cos \theta,-1+\cos \theta \right),\left(1-\cos \theta,1+\cos \theta \right)\right)\left(\begin{array}{c} \vec{\Psi}\left(\vec{Q},\omega \right) \\ \vec{\Psi}\left(-\vec{Q},\omega \right) \end{array}\right)\right\}\end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\delta {{\vec{M}}_{y}}\left(\omega \right)=\frac{\sqrt{g{{\mu}_{B}}{{M}_{s}}}}{4\sqrt{2}V}\\ \quad \times \text{Re}\left\{\left(\left(1+\cos \theta,1-\cos \theta \right),\left(1-\cos \theta,1+\cos \theta \right)\right)\left(\begin{array}{c} \vec{\Psi}\left(\vec{Q},\omega \right) \\ \vec{\Psi}\left(-\vec{Q},\omega \right) \end{array}\right)\right\}\end{array}\end{eqnarray} \tag{ 26 }$$

where $\vec{\Psi}\left(\pm \vec{Q},\omega \right)={\int}^{}d\vec{r}{{e}^{\mp i\vec{Q}\vec{r}}}\vec{\Psi}\left(\vec{r},\omega \right)$. The oscillations within this (x,y) plane are determined by the two components $\vec{\Psi}\left(\pm \vec{Q},\omega \right)$ of the magnon wavefunction. It turns out that these two components are decoupled from the other modes and governed by the wave equation

$$\begin{eqnarray}\left(\begin{array}{c} -g_{0}^{-1}\left(\omega,\vec{Q}\right) & \mathcal{H}_{\text{dip}}^{1,-1}(0) \\ \mathcal{H}_{\text{dip}}^{-1,1}(0) & -g_{0}^{-1}\left(\omega,-\vec{Q}\right) \end{array}\right)\left(\begin{array}{c} \vec{\Psi}\left(\vec{Q},\omega \right) \\ \vec{\Psi}\left(-\vec{Q},\omega \right) \end{array}\right)=\mathbf{0}\end{eqnarray} \tag{ 27 }$$

where the Green function reads explicitly

$$\begin{eqnarray} &&\begin{array}{*{35}{l}} -g_{0}^{-1}\left(\omega,\pm \vec{Q}\right)=-\hbar \omega {{\tau}^{z}}+\mathcal{D}{{Q}^{2}}\left[\mathbb{1}+\frac{{{\sin}^{2}}\theta}{2}\left(\mathbb{1}-{{\tau}^{x}}\right)\right] \\ +\chi _{\text{con}}^{\text{int}}\left(\frac{{{\sin}^{2}}\theta}{2}\left(\mathbb{1}-{{\tau}^{x}}\right)+\frac{{{N}_{x}}+{{N}_{y}}}{4}\left(\left(\mathbb{1}+{{\cos}^{2}}\theta \right)1+{{\sin}^{2}}\theta {{\tau}^{x}}\right.\right.\\ \quad\left. \left. \left. \pm \cos \theta {{\tau}^{z}}\right)\right)\right]. \end{array}\end{eqnarray} \tag{ 28 }$$

The term in the last line is attributed to the dipolar interaction and, again, its strength is conveniently parametrized by $\chi _{\text{con}}^{\text{int}}$; note that $\mathcal{D}{{Q}^{2}}\chi _{\text{con}}^{\text{int}}=g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}$ (table 2). The off-diagonal parts read

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}\mathcal{H}_{\text{dip}}^{1,-1}(0)=\mathcal{D}{{Q}^{2}}\chi _{\text{con}}^{\text{int}}\frac{{{N}_{x}}-{{N}_{y}}}{4}\left(-{{\sin}^{2}}\theta \mathbb{1}-\text{i}2\cos \theta {{\tau}^{y}}\right. \\ \left. \quad -\left(1+{{\cos}^{2}}\theta \right){{\tau}^{x}}\right)\end{array}\end{eqnarray} \tag{ 29 }$$

and $\mathcal{H}_{\text{dip}}^{-1,1}(0)=\left(\mathcal{H}_{\text{dip}}^{1,-1}(0)\right){{}^{\dagger}}$. The solution of the $4\times 4$ matrix (27) determines the eigenfrequencies and eigenvectors of the uniform oscillations of the magnetization within the conical helix phase. There are two eigenmodes denoted by $\pm Q$ in the literature [34, 38]. The corresponding positive eigenfrequencies are given by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{\hbar {{\omega}_{\pm Q}}}{g{{\mu}_{B}}{{\mu}_{0}}{{H}_{c2}}}=\frac{1}{2(1+{{N}_{z}}\chi _{\text{con}}^{\text{int}})}((2+\chi _{\text{con}}^{\text{int}})(4+({{N}_{x}}+{{N}_{y}})\chi _{\text{con}}^{\text{int}}) \\ \ \ \ +{{h}^{2}}(-4+\chi _{\text{con}}^{\text{int}}(-4-{{N}_{y}}\chi _{\text{con}}^{\text{int}}+{{N}_{x}}(-1+2{{N}_{y}})\chi _{\text{con}}^{\text{int}})) \\ \ \ \ \pm \chi _{\text{con}}^{\text{int}}[N_{y}^{2}{{(2+\chi _{\text{con}}^{\text{int}}-{{h}^{2}}\chi _{\text{con}}^{\text{int}})}^{2}}\\ \ \ \ +N_{x}^{2}{{(2+(1+{{h}^{2}}(-1+2{{N}_{y}}))\chi _{\text{con}}^{\text{int}})}^{2}}-2{{N}_{x}}{{N}_{y}}({{(2+\chi _{\text{con}}^{\text{int}})}^{2}} \\ \ \ \ -2{{h}^{2}}(2+\chi _{\text{con}}^{\text{int}})(4+\chi _{\text{con}}^{\text{int}}+{{N}_{y}}\chi _{\text{con}}^{\text{int}})+{{h}^{4}}(8+\chi _{\text{con}}^{\text{int}}(8+\chi _{\text{con}}^{\text{int}} \\ \ \ \ +2{{N}_{y}}\chi _{\text{con}}^{\text{int}})))]{{}^{1/2}}){{}^{1/2}} \end{array}\end{eqnarray} \tag{ 30 }$$

where we expressed the energy $\mathcal{D}{{Q}^{2}}=g{{\mu}_{B}}{{\mu}_{0}}H_{c2}^{\text{int}}$ in terms of the second critical field ${{H}_{c2}}=H_{c2}^{\text{int}}\left(1+{{N}_{z}}\chi _{\text{con}}^{\text{int}}\right)$, and we abbreviated $h=\cos \theta =H/{{H}_{c2}}={{H}_{\text{int}}}/H_{c2}^{\text{int}}$. This result was first presented in [34] where the formula is given in terms of the susceptibility $\chi _{\text{con}}^{-1}=\left(\chi _{\text{con}}^{\text{int}}\right){{}^{-1}}+{{N}_{z}}$.

For generic sample shapes eigenfrequencies of modes  +Q, −Q for H  <  H_c2 as well as the uniform Kittel mode of equation (14) for H  >  H_c2 in the FP phase are shown in figure 8. For ${{N}_{x}}={{N}_{y}}\ne 0$ the modes  +Q and  −Q are found to be degenerate at zero field H  =  0 and to split when H approaches H_c2 (figures 8(a) and (b)). In figure 8(c) the modes  +Q and  −Q exhibit different frequencies for all H when ${{N}_{x}}\ne {{N}_{y}}$. In the limit of a thin platelet, ${{N}_{x}}={{N}_{y}}=0$ and N_z  =  1, the eigenfrequencies are degenerate for all fields H  <  H_c2, see figure 8(d). In this limit they are given by Kataoka's formula [89]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \hbar {{\omega}_{\pm Q}}{{|}_{\text{disc}}}=g{{\mu}_{B}}{{\mu}_{0}}H_{c2}^{\text{int}}\sqrt{1+\left(1+\chi _{\text{con}}^{\text{int}}\right)\left(1-{{h}^{2}}\right)} \end{array}\end{eqnarray} \tag{ 31 }$$

with $h={{H}_{\text{int}}}/H_{c2}^{\text{int}}$. This formula is consistent with the bulk spectrum of equation (19) at k_z  =  Q. The branch in the FP phase is found either above or below the paramagnetic limit (dashed lines) depending on whether the shape anisotropy field adds to or subtracts from the applied field, respectively [4].

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The eigenvectors of equation (27) determine the helicity, i.e. the sense of precessional motion, the polarization and weights of the resonances. The weights as a function of field are shown in figure 9. The $\pm Q$ modes have a well-defined helicity provided that they are non-degenerate. The uniform magnetization oscillates counterclockwise for the  +Q mode and clockwise for the  −Q mode. As a result, the weight of the  +Q mode continuously connects to the Kittel mode at the critical field, H_c2, because they possess the same helicity, i.e. for both modes the uniform magnetization precesses counterclockwise. In contrast, the weight of the  −Q mode vanishes as H_c2 is approached. Note that in general the helicity of the dynamical precession is determined by the sign of the gyromagnetic ratio. Individual magnetic moments attributed to electrons always precess counterclockwise around a static field and so does the uniform Kittel mode in the field-polarized phase. This is independent of the chirality of the crystal, and, in particular, it remains unaffected by the reversal of the DMI, i.e. by considering a left-handed instead of a right-handed material.

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If the modes are degenerate as for ${{N}_{x}}={{N}_{y}}=0$, one can always choose a basis of, for example, two circularly polarized degenerate eigenmodes. However, in the generic case of two non-degenerate modes, each of them possesses a distinct polarization and ellipticity. This leads to an interesting dependence of their weights on the orientation ${{\vec{H}}_{\text{AC}}}$ of a linearly polarized ac field as illustrated in figure 9. Both modes are in general elliptically polarized where the axes are defined by the sample shape. We introduce the ellipticity defined by

$$\begin{eqnarray}&&\varepsilon =\text{sign}\left\{\delta {{M}_{x}}-\delta {{M}_{y}}\right\}\frac{\sqrt{|\delta M_{x}^{2}-\delta M_{y}^{2}|}}{\text{max}\left\{\delta {{M}_{x}},\delta {{M}_{y}}\right\}}\end{eqnarray} \tag{ 32 }$$

where $\delta {{M}_{x,y}}$ are the positive amplitudes of the oscillations within the (x, y) plane. We introduce a sign so that $\varepsilon >0$ if the major axis is aligned with the x-axis and $\varepsilon <0$ if it is aligned with the y-axis. The ellipticity for the $\pm Q$ modes is shown in figure 10 for a specific value of N_z as a function of N_x. Interestingly, we find that at zero field $\vec{H}=0$ the two modes are generically linearly polarized, i.e. $\varepsilon =\pm 1$.

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The linear polarization is related to the π-rotation symmetry of the helix that is present at zero field, see the discussion below equation (15). This symmetry is represented by the matrix operator

$$\begin{eqnarray}\left(\begin{array}{c} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{array}\right)\end{eqnarray} \tag{ 33 }$$

that commutes at $\vec{H}=0$ with the Hamiltonian, i.e. with the $4\times 4$ matrix of the wave equation (27). As a result, the two eigenmodes $\pm Q$ are also eigenvectors of the matrix (33) with two distinct eigenvalues. Moreover, the amplitudes $\delta {{M}_{x}}$ and $\delta {{M}_{y}}$ are determined by the projection of the wavefunction $\left({{\vec{\Psi}}^{T}}\left(\vec{Q},\omega \right),{{\vec{\Psi}}^{T}}\left(-\vec{Q},\omega \right)\right)$, see equation (25), onto different degenerate subspaces of the π-symmetry operator (33). At zero field, either of the two amplitudes thus vanishes leading to a linear polarization. In less mathematical terms, the π-rotation symmetry ensures that there are always pairs of spins within the helix whose local precessions conspire such that in total only a linear polarization remains. This situation seems to be similar to easy-plane antiferromagnets where the precession of magnetic moments on the two sublattices also combine to yield a linearly polarized uniform mode at antiferromagnetic resonance [4, 119]. In contrast to antiferromagnets, the linear polarization for chiral magnets is however an interplay between DMI and sample shape. Because the relevant energy scale of DMI is orders of magnitude smaller compared to antiferromagnetic exchange interaction, the linearly polarized modes of Cu₂OSeO₃ reside at small frequencies [4].

Figure 10 also indicates that the ellipticity of the  +Q mode changes sign implying that it changes the major axis of its elliptical polarization as a function of field. Generally, this is expected to occur at the magnetic field

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{H_{+}^{\text{circ}}}{{{H}_{c2}}}= \\ \frac{2+\chi _{\text{con}}^{\text{int}}}{\sqrt{\left(2+\chi _{\text{con}}^{\text{int}}\right)\left(2+\left(2-{{N}_{z}}\right)\chi _{\text{con}}^{\text{int}}\right)+\sqrt{\left(1-{{N}_{z}}\right)\chi _{\text{con}}^{\text{int}}{{\left(2+\chi _{\text{con}}^{\text{int}}\right)}^{2}}\left(4+\left(1-{{N}_{z}}\right)\chi _{\text{con}}^{\text{int}}\right)}}}. \end{array}\end{eqnarray} \tag{ 34 }$$

For Cu₂OSeO₃ with $\chi _{\text{con}}^{\text{int}}=1.76$ and a sample with N_z  =  0.53 this yields $H_{+}^{\text{circ}}/{{H}_{c2}}\approx 0.76$. At this specific field, the  +Q mode is circularly polarized irrespective of the values of the demagnetization factors N_x and N_y.

In the early experiments by Date et al [35] both $\pm Q$ excitation modes were detected on MnSi. Onose et al [38] reported the two modes for Cu₂OSeO₃ (figure 4). Finally, Schwarze et al [34] performed microwave resonance experiments on MnSi, Cu₂OSeO₃ and Fe_0.8Co_0.2Si, and found quantitative agreement with the theory reviewed above. The linear polarization of the $\pm Q$ modes in zero field was explored and experimentally confirmed by Stasinopoulos et al [120].

The Dzyaloshinskii-Moriya interaction not only favours helices but also topological solitons. Within the field-polarized state, a single soliton is generated by the map

$$\begin{eqnarray}&&{{\hat{n}}_{\text{Sk}}}\left(\vec{r}\right)={{\text{e}}^{\text{i}{{{\vec{f}}}_{\text{Sk}}}\left(\vec{r}\right)\mathbf{\vec{L}}}}\hat{z}=\left(\begin{array}{c} -\sin \chi \sin \theta \left(\rho \right) \\ \cos \chi \sin \theta \left(\rho \right) \\ \cos \theta \left(\rho \right) \end{array}\right)\end{eqnarray} \tag{ 35 }$$

where $\mathbf{\vec{L}}$ is the spin-1 operator of equation (2) and ${{\vec{f}}_{\text{Sk}}}\left(\vec{r}\right)=\theta \left(\rho \right)\hat{\rho}$ with $\vec{r}=\left(\rho \cos \chi,\rho \sin \chi,z\right)$ in cylindrical coordinates and $\hat{\rho}=\left(\cos \chi,\sin \chi,0\right)$. Note that ${{\vec{f}}_{\text{Sk}}}$ is a conservative vector field and possesses a potential according to ${{\vec{f}}_{\text{Sk}}}\left(\vec{r}\right)=\vec{\nabla}{{\Theta}_{\text{Sk}}}\left(\rho \right)$ with $\theta \left(\rho \right)={{\partial}_{\rho}}{{\Theta}_{\text{Sk}}}\left(\rho \right)$. The function $\theta \left(\rho \right)$ obeys the asymptotics $\theta (0)=\pi$ and $\theta \left(\rho \right)\to 0$ as $\rho \to \infty$. This soliton is translational invariant along the field-direction, i.e. the z-axis, and it varies within the plane perpendicular to the field. In a three dimensional bulk magnet, it defines in fact a string. The soliton carries a finite topological skyrmion number so that it is referred to simply as a skyrmion (figure 2(d)). The topological skyrmion density within the (x,y) plane is defined by ${{\rho}_{\text{top}}}=\frac{1}{4\pi}\hat{n}\left({{\partial}_{x}}\hat{n}\times {{\partial}_{y}}\hat{n}\right)$. Integrating the topological density of the texture (35) yields an integer, ${\int}^{}\text{d}x\text{d}y{{\rho}_{\text{top}}}=-1$. Minimizing the exchange energy of equation (1) with the Ansatz (35), i.e. neglecting the dipolar interactions one obtains an ordinary differential equation that determines the function $\theta \left(\theta \right)$, which was first discussed by Bogdanov and Hubert [121]. Importantly, one finds that $\theta \left(\rho \right)$ decays exponentially so that the skyrmion is confined to an area that scales approximately as $1/{{\vec{H}}^{2}}$ where $\vec{H}$ is the magnetic field [122].

The magnon scattering off such a single skyrmion will be discussed in section 4.4. Here, we focus on the spin wave excitation of a skyrmion crystal. When it is energetically advantageous to condense skyrmions, they proliferate and form a skyrmion lattice (figure 2(e)). Such a magnetic state is observed in bulk crystals of chiral magnets at intermediate fields and close to the critical temperature (figure 2(a)). A skyrmion crystal state can be generated with the help of the map

$$\begin{eqnarray}&&{{\hat{n}}_{\text{SkX}}}\left(\vec{r}\right)={{\text{e}}^{\text{i}{{{\vec{f}}}_{\text{SkX}}}\left(\vec{r}\right)\mathbf{\vec{L}}}}\hat{z}\end{eqnarray} \tag{ 36 }$$

with $\mathbf{\vec{L}}$ of equation (2). The potential of the conservative vector function ${{\vec{f}}_{\text{SkX}}}\left(\vec{r}\right)=\vec{\nabla}{{\Theta}_{\text{SkX}}}\left({{\vec{r}}_{\bot}}\right)$ again only varies within the plane perpendicular to the magnetic field, ${{\vec{r}}_{\bot}}=(x,y,0)$. It possesses the periodicity of a two-dimensional hexagonal Bravais lattice L, i.e. ${{\Theta}_{\text{SkX}}}\left({{\vec{r}}_{\bot}}\right)={{\Theta}_{\text{SkX}}}\left({{\vec{r}}_{\bot}}+{{\vec{R}}_{i}}\right)$ for any vector ${{\vec{R}}_{i}}\in L$. If the lattice constant a is large compared to the skyrmion radius, this potential can be approximated to be the sum of individual skyrmion potentials ${{\Theta}_{\text{SkX}}}\left({{\vec{r}}_{\bot}}\right)\approx \sum\nolimits_{{{{\vec{R}}}_{i}}\in L}{{\Theta}_{\text{Sk}}}\left({{\vec{r}}_{\bot}}-{{\vec{R}}_{i}}\right)$. However, it turns out that the skyrmions are closely packed, and the lattice constant a is in fact comparable to the skyrmion diameter so that ${{\Theta}_{\text{Sk}}}$ is expected to differ from the single-skyrmion solution. In any case, minimization of the free energy (1) with such a type of Ansatz does not yield a smaller energy than the competing conical state of equation (15). The phase diagram of figure 2 can therefore not be explained on the mean-field level. It was argued in [50] that fluctuations around the mean-field potential stabilizes the SkL, which was subsequently confirmed by Monte-Carlo simulations [123].

Once the SkL is stabilized the magnon spectrum can be derived on the mean-field level taking into account both contributions, the exchange and the dipolar energies of equations (1) and (3), respectively. However, for practical purposes it is more convenient to relax the constraint of $\hat{n}$ being a unit vector and to work instead within the framework of a linear sigma model. In the following, we present numerical results for the spin wave spectrum of skyrmion crystals in bulk chiral magnets. For computational details we refer the reader to [34] and [124].

3.5.1. Spin waves of the skyrmion crystal in bulk samples: finite wavevectors.

The periodicity of the two-dimensional hexagonal skyrmion crystal gives rise to a magnon band structure with a two-dimensional Brillouin zone in the (x,y) plane perpendicular to the applied magnetic field (inset in figure 11(a)). Figure 11(a) shows the magnon dispersion for vanishing out-of-plane momentum k_z  =  0 along particular symmetry directions within the first Brillouin zone for the parameter $\chi _{\text{con}}^{\text{int}}=0.34$ corresponding to MnSi. Figure 11(b) shows the spectrum for Cu₂OSeO₃ with $\chi _{\text{con}}^{\text{int}}=1.76$ yielding a compressed band structure as compared to panel (a). For better comparison, the first 14 bands in (a) and (b) are shown as solid lines and the remaining bands are displayed as dashed lines. Figure 11(c) shows the complete magnon spectrum of MnSi for wavevectors in the plane of the SkL.

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Similarly to the conical helix, the skyrmion crystal breaks translational invariance so that the spectrum possesses a Goldstone mode. This mode arises at the $\Gamma$ point where the lowest band touches zero energy. Its excitation energy vanishes quadratically with momentum, $\omega \sim {{k}^{2}}$, which has been attributed to the topological nature of the skyrmions in [115, 125].

The non-trivial topology has further consequences. As will be explained in detail in section 4.4, each skyrmion acts like an orbital magnetic field with quantized flux resulting in skew scattering. It is therefore expected that a magnon in a skyrmion crystal experiences this magnetic flux per magnetic unit cell. As a consequence, magnons should occupy Landau levels that are reflected in non-trivial Chern numbers of the band structure [126]. The Chern numbers of the magnon bands for $\chi _{\text{con}}^{\text{int}}=0.34$ are shown on the right-hand side of figure 11(c). We find that for the lowest 14 bands, for which we were able to compute these numbers reliably, most of the bands have a Chern number 1 with a few exceptions, in particular, at low energies where the Chern number is zero. For the parameters of MnSi, the magnon bands are sufficiently well separated allowing for an unambiguous computation. For the parameters of Cu₂OSeO₃ some of the bands come very close rendering the computation of Chern numbers difficult. Up to such ambiguities, the Chern numbers listed in figure 11(c) are consistent with the work of Roldán-Molina et al [127]. The band structures of figure 11 will be further discussed in section 4.

3.5.2. Spin waves of the skyrmion crystal in bulk samples: uniform mode.

Due to the back-folding induced by the periodicity of the skyrmion crystal, there is a multitude of modes present at the $\Gamma$ point, i.e. at zero wavevector. However, similar to the conical helix, only a few of them are magnetically active and excited by an homogeneously oscillating magnetic field. For the skyrmion crystal, there are three uniform magnetic modes, breathing, CCW and CW mode, first identified theoretically by Mochizuki [85].

The breathing mode possesses a macroscopic dipole moment oscillating out-of-plane, i.e. it is linearly polarized and excited with a longitudinal ac magnetic field (figure 12). The size of each skyrmion in the crystal performs an oscillating motion as illustrated in figure 13. Its resonance frequency decreases with magnetic field similar to the $\pm Q$ modes of the adjacent conical phase. In case of phase coexistence within the sample, the breathing mode can be distinguished from the former by its polarization which is orthogonal to the ones of the $\pm Q$ modes.

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The uniform magnetization of the CCW and CW modes oscillates counterclockwise and clockwise, respectively, in the plane of the skyrmion crystal. They are excited with an in-plane ac field. These modes are in general elliptically polarized along different axes that are determined by the sample shape. Their ellipticity however varies only slightly within the limited magnetic field range where the skyrmion crystal is found [124]. Generally, an ac field linearly polarized within the plane excites both modes but the CCW mode always has a larger weight (figure 12). Whereas the resonance frequency of the CCW increases with field, the CW frequency only depends weakly on the magnetic field consistent with experimental observations [34, 38].

The chiral magnets with bulk DMI provide strikingly new characteristics in view of magnonic crystals and manipulation of spin waves in solids. The DMI-induced spin helix introduces a periodic potential (equation (16)) for magnons that leads to backfolding of spin-wave dispersion relations and Brillouin zone boundaries at half the pitch vector Q (figure 6). For Cu₂OSeO₃ the intrinsic pitch length that defines the periodicity of the helix amounts to about 60 nm. Thereby magnonic crystal equivalent band structures are formed in a bottom-up strategy avoiding any top-down nanopatterning that typically introduces roughness and inhomogeneous broadening. In addition, in the helical phase, the band structure changes qualitatively as a function of momentum ${{\vec{k}}_{\bot}}$ transverse to the helix axis, see figure 6. For large $|{{\vec{k}}_{\bot}}|\gg Q$, the lowest bands become flat implying that the magnons are localized along the helix axis and only propagate transverse to it, i.e. the spin waves experience a channeling effect. Channeling of spin waves is well known for two-dimensional magnonic crystals made from antidot lattices [128, 129]. In their case the inhomogeneous demagnetization field induced by in-plane magnetic fields creates narrow channels that are perpendicular with respect to the applied field direction. In the helical or conical phase, spin waves are localized within planes perpendicular to the pitch vector $\vec{Q}$. The orientation of the latter can be manipulated with an applied magnetic field allowing to control the channeling. Future research might explore the multitude of tuneable parallel channels in view of spin wave multiplexer on the nanoscale.

Within the phase of the skyrmion lattice, the magnetic Brillouin zone becomes two-dimensional and also contains dispersive magnon bands. The dispersion is particularly pronounced for the magnetically active bands associated with the breathing and CCW modes at their respective $\Gamma$ point, see figure 11. Their group velocities $v={{\partial}_{k}}{{\omega}_{{\vec{k}}}}$ along certain lines within the two-dimensional Brillouin zone are shown in figure 14. The velocities are on the order of ${{v}_{c2}}=\hbar \omega _{c2}^{\text{int}}/\left(\hbar Q\right)$ which corresponds to ${{v}_{c2}}\approx 300$ m s⁻¹ for MnSi and ${{v}_{c2}}\approx 140$ m s⁻¹ for Cu₂OSeO₃. The band structures of figure 11 feature well separated bands and pronounced forbidden frequency gaps. To obtain complete band gaps for top-down fabricated magnonic crystals nanoscale lattice constants would be needed that introduce Bragg scattering for exchange-dominated spin waves. Such small lattice constants are however technologically challenging.

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It has been already demonstrated that the SkL in Cu₂OSeO₃ can be manipulated with an electric field due to magnetoelectric coupling [79, 84, 130]. Electric-field controlled magnetic crystals based on SkLs should thus be possible. Technologically electric field control is most advantageous as electric fields can be confined very precisely and allow for a low power consumption. At the same time, the SkL might be exploited as either a resonant or non-resonant magnonic grating coupler if integrated to a coplanar waveguide [11, 131]. Relevant wavelengths for emitted spin waves are on the order of the inverse of reciprocal lattice vectors G [11, 131]. In case of an SkL we assume for the smallest non-zero reciprocal lattice vector G  =  Q leading to spin-wave wavelengths $\lambda =2\pi /G=2\pi /Q$ on the order of 20 to 70 nm for the materials listed in table 1. Applying an electric field would rotate the orientation of the hexagonal magnetic grating around the field direction and allow to emit short-wavelength spin waves in different and tuneable directions transverse to $\vec{H}$. The continuous tunability of propagation directions has not yet been foreseen with conventional ferromagnets. By tilting the magnetic field $\vec{H}$ stabilizing the SkL the grating coupler can be further adjusted and becomes versatile for even more spatial directions.

Applied electric or thermal currents have been shown to couple efficiently to a SkL. At very small electric current densities a SkL is depinned and shifted in space via spin-transfer torque (STT) [132, 133]. A magnon current, e.g. generated by a thermal gradient is also able to move skyrmions [134–138]. It has been also experimentally demonstrated that a thermal current might induce a continuous rotation of the SkL [139, 140]. Interestingly, micromagnetic simulations have suggested that a linearly polarized microwave field applied to a chiral magnet can displace both an individual skyrmion and a SkL in the presence of a symmetry-breaking magnetic field component. The maximum velocity is found for a frequency of the microwave that coincides with the resonance frequency of the breathing mode of the skyrmions [141]. These effects allow for the field- or current-controlled positioning of skyrmion-based magnonic crystals. In a moving SkL Doppler-shift like frequency variations of spin waves might be anticipated [142].

It is interesting to compare the magnon band structures found for hexagonal skyrmion lattices with the ones that were discussed for hexagonal lattices formed by bubble domains several decades ago. The dynamics of individual bubble domains and corresponding lattices was reviewed in [31]. We follow [31] and summarize the main aspects here. Bubble domains exist in epitaxial layers of ferrimagnetic garnets that are usually several micrometer thick and possess a perpendicular magnetic anisotropy but no or only a small DMI. Due to dipolar interaction, a periodic lattice of bubble domains is formed when a specific magnetic field is applied perpendicular to the film plane. If the distance between periodically arranged bubble domains is small, a magnonic crystal like band structure results from mainly magnetic dipole interactions. Allowed minibands can be classified in terms of two acoustic branches and one optical branch covering a frequency band of a few tens of megahertz for a 5 micrometer thick garnet. For the acoustic branches relatively large group velocities on the order of 100 m s⁻¹ were expected in case of small edge-to-edge separations of bubble domains compared to their diameter. Already in 1979, the authors of [31] speculated that corresponding spin waves possessed potential as information signal carriers. Interestingly, bubble domain lattices allowed for reconfigurable magnon band structures [143]. However it was noted that the experimental realization of a lattice with identically hard bubbles appeared to be difficult [31]. In chiral magnets reviewed here, the pitch vector introduced by DMI enables the formation of regular periodic lattices with identical skyrmions.

The resonance frequencies of the uniform $\pm Q$ modes of the conical helix strongly depend on the shape of the sample as illustrated in figure 8. For the breathing, CW and CCW modes of the skyrmion lattice the variation of eigenfrequencies with demagnetization factors is less pronounced [34]. Still the order of magnon bands in the skyrmion lattice varies with the strength of the effective dipolar interaction as shown for MnSi and Cu₂OSeO₃ in figure 11.

The interplay of bulk DMI and shape anisotropy leads to a linear polarization of the magnetization dynamics in the spin-helix state at small magnetic field, see section 3.4. Relevant eigenfrequencies are in the few GHz frequency regime for which transmission lines and coplanar waveguides are impedance matched and guide linearly-polarized microwaves efficiently on microchips. In conventional ferro- and ferrimagnets such linearly polarized spin-precessional motion is not accomplished. For them the spin-precessional motion is elliptically or circularly polarized. Helical modes in chiral magnets with bulk DMI are linearly polarized at small fields and thus ideally couple to the linear polarization provided by the excitation fields of the conventional microwave technology [120].

The Dzyaloshinskii-Moriya interaction imposes also boundary conditions on the surface spins that result in a surface twist. In the FP phase, this leads to exchange spin waves that are localized to the surfaces of the sample [144, 145]. They will be further discussed below in section 4.6.1. In the SkL phase, surface spin-wave modes are also expected for a different reason. We have discussed in figure 11 that some of the magnon bands possess a non-trivial Chern number resulting from the non-trivial topology of the magnetic texture. According to the bulk-boundary correspondence, this implies the presence of topologically protected magnon edge states. Such magnon edge states have been previously discussed in a different context for the pyrochlore Lu₂V₂O₇ [146]. Moreover, the magnon bands with a finite Chern number should contribute to a magnon Hall effect, i.e. spin accumulation transverse to the magnon flow [136, 137, 139].

The three uniform excitation modes of the skyrmion lattice have been observed also for the Néel-type skyrmion lattice in the rhombohedral GaV₄S₈ [45, 46]. GaV₄S₈ hosts cycloidal spin structures and is characterized by a uniaxial anisotropy. Here, in the skyrmion lattice phase the breathing mode was experimentally found to possess a smaller resonance frequency than the CCW mode, i.e. the hierarchy of resonances differed from the one of the cubic chiral magnets in figure 13. This has also been theoretically confirmed by numerical simulations in [147]. Moreover, the eigenfrequencies and set of modes in this material class are expected to vary distinctly if an applied magnetic field is misaligned with respect to the easy axis. The misaligned field leads to a distortion of the skyrmion lattice with the concomitant change in resonance frequencies. In the cycloidal phase of GaV₄S₈, two prominent modes were found that exhibited a large frequency splitting of several GHz. Here, the isotropic exchange and exchange anistropy were argued to provide the energy scale attributed to the frequency splitting [45]. For the splitting of the two modes  +Q and  −Q in the helical modes of chiral magnets, the weaker dipolar interaction was relevant.

The magnetoelectric coupling in Cu₂OSeO₃ imply that magnons exhibit both magnetic and electric activities which undergo interference effects and lead to directional dichroism. The relevant electric susceptibilities of the rotational and breathing modes in the SkL were reviewed in [84]. Linearly polarized microwaves guided through an SkL experience directional dichroism in both configurations for which the wavevector ${{\vec{k}}_{\text{em}}}$ of the electromagnetic wave is either perpendicular or parallel to the magnetic field $\vec{H}$ stabilizing the SkL. Counterpropagating electromagnetic waves encounter different absorption leading to nonreciprocal microwave properties (directional dichroism). For the case that ${{\vec{k}}_{\text{em}}}$ is collinear with $\vec{H}$ the directional dichroism is called magnetochiral effect [84]. This effect has been found not only in the SkL of bulk Cu₂OSeO₃ but also in the conical and field-polarized phases. This kind of microwave nonreciprocity can thus be exploited over a broad frequency regime. For a detailed discussion we refer the reader to [87, 88]. To harvest such characteristics in microwave technology bulk materials with DMI are needed that are insulators and form skyrmions at room temperature. While bulk metals with DMI and ultrathin metallic layers incorporating interfacial DMI provide magnetic skyrmions at room temperature [63, 73], to our knowledge, bulk insulators with room-temperature skyrmions have not yet been reported. Corresponding materials still need to be identified.

Finally, we remark that a direct electric field control of spin wave eigenfrequencies was also demonstrated in case of the cycloid spin structure of the room-temperature magnetoelectric material BiFeO₃ [148]. Here, the spin-wave frequency (>600 GHz) was tuned electrically by over 30%, in a non-volatile way and with virtually no power dissipation.

For skyrmion-based spin-electronics an individual skyrmion formed in a ferromagnetic matrix is expected to be technologically relevant [55]. In insulators, spin waves can then be used to move the skyrmion, e.g. in a nanotrack [134, 136, 137, 138, 139, 149, 150]. For the functionalization of this magnon-skyrmion momentum-transfer, the understanding of the spin wave scattering off a skyrmion is essential.

When spin waves approach the skyrmion they scatter off it, and they can even form bound states. This magnon-skyrmion scattering problem was considered in [136, 137, 151] on the level of the exchange energy of equation (1). Here, we review the results of [137] where the magnon Hamiltonian was derived analytically and solved numerically for a finite in-plane magnon momentum ${{\vec{k}}_{\bot}}=\left({{k}_{x}},{{k}_{y}},0\right)$. The resulting spectrum (without uniaxial anisotropy) is shown in figure 15. Apart from scattering states above the magnon gap, there exist a few in-gap states corresponding to magnon-skyrmion bound modes. Above the critical field H_c2, there is only a single breathing mode for which the skyrmion radius oscillates in time. An additional quadrupolar and sextupolar mode appears only for smaller fields where the field-polarized state in a three-dimensional bulk material is already unstable. In the presence of a finite magnetic anisotropy, K, another mode with gyrotropic character might materialize below the magnon gap [152].

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Only the breathing mode gives rise to an oscillating average magnetic dipole moment that can be excited with an ac magnetic field longitudinal to the dc field, i.e. along the z-axis. In the insulator Cu₂OSeO₃ the quadrupolar mode is accompanied with an oscillating electric dipole moment and could therefore be electrically excited (electromagnon) [137]. The breathing mode for Bloch-like skyrmions is also associated with an oscillating toroidal moment so that it can be excited also with a combination of in-plane magnetic and electric fields. In a magnetic layer, the conical state is suppressed and the field-polarized state remains stable below H_c2. The vertical dashed-dotted line in figure 15 then indicates a global instability where the energy of skyrmions becomes negative triggering the formation of a skyrmion crystal. Below this field, the skyrmion in the metastable field-polarized phase becomes elliptically unstable [121] when the resonance frequency of the quadrupolar mode vanishes.

The non-trivial topology of the skyrmion texture has a striking impact on the properties of the scattering states. Figure 16(a) shows the scattering cross section for magnons propagating in-plane with an energy that is large compared to the magnon gap: it is characterized by oscillations and strongly asymmetric with respect to forward scattering $\chi =0$, i.e. there is skew scattering. This is also illustrated in panel (b) that shows a monochromatic magnon wave travelling from the left to the right hand side and scattering off a skyrmion. Both aspects can be transparently discussed by considering the magnon-skyrmion scattering problem in the high-energy limit [153]. In this limit, the magnon wave function ψ of equation (5) is simply governed by the Schrödinger equation

$$\begin{eqnarray}&&\text{i}\hbar {{\partial}_{t}}\psi =\left[\frac{{{\left(-\text{i}\hbar {{{\vec{\nabla}}}_{\bot}}-\vec{A}\left(\vec{r}\right)\right)}^{2}}}{2m}+{{\varepsilon}_{\text{gap}}}\right]\psi \end{eqnarray} \tag{ 37 }$$

where, on the level of equation (1), the magnon gap ${{\varepsilon}_{\text{gap}}}$ is proportional to the magnetic field, i.e. ${{\varepsilon}_{\text{gap}}}=g{{\mu}_{B}}{{\mu}_{0}}H$. The high-energy magnon propagating within the plane perpendicular to the field, ${{\vec{\nabla}}_{\bot}}=\left({{\partial}_{x}},{{\partial}_{y}},0\right)$, only scatters off a vector potential $\vec{A}=\left({{A}_{x}},{{A}_{y}},0\right)$. The associated flux density ${{\mathcal{B}}_{z}}={{\partial}_{x}}{{A}_{y}}-{{\partial}_{y}}{{A}_{x}}$ is finite so that the magnon scatters off an emergent orbital magnetic field which is localized to the skyrmion area. This emergent orbital magnetic field arises from the non-trivial topology of the textured magnetization. When a high-energy magnon traverses the skyrmion texture, it locally adjusts its quantization axis to the spatially varying magnetization. Mathematically, the geometric constraint imposed by the varying quantization axis leads to a spin connection which enters the Hamiltonian as a vector potential. Its total flux

$$\begin{eqnarray}&&{\int}^{}\text{d}x\text{d}y\,{{\mathcal{B}}_{z}}=4\pi \hbar \end{eqnarray} \tag{ 38 }$$

is quantized and related to the topological skyrmion charge of the soliton.

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The magnon experiences an emergent Lorentz force due to ${{\mathcal{B}}_{z}}$ that deflects its trajectory leading to the skew scattering. A finite density of skyrmions are thus expected to give rise to a topological magnon Hall effect [136]. For a given skyrmion configuration this Hall effect and the resulting spin accumulation might be used to read out a propagating spin wave signal. The oscillations in the scattering cross section of figure 16(a) arise from a phenomenon known as rainbow scattering. The emergent field ${{\mathcal{B}}_{z}}={{\mathcal{B}}_{z}}\left(\rho \right)$ only depends on the distance ρ to the skyrmion center. As a consequence, classical magnon trajectories with impact parameters $\pm b$, that pass the skyrmion at the same distance but either on its right- or left-hand side, will see the same flux and experience the same deflection angle. These trajectories interfere and lead to oscillations in the scattering cross section. The emergent orbital field ${{\mathcal{B}}_{z}}$ for magnon excitations is characteristic for a topologically non-trivial magnetic texture. If the texture itself is moving with a finite velocity, an emergent electric field also arises according to Faraday's law [154]. This emergent electrodynamics is one of the exciting aspects of skyrmion textures, see [48] for a review.

An individual skyrmion might be stabilized dynamically in a ferromagnetic thin film by using a current-biased nanocontact providing STT locally [155]. Such skyrmions could serve as scattering centers as discussed above. On the other hand, it was predicted that, in the presence of DMI, the breathing mode of the individual skyrmion can serve as an efficient and tuneable microwave signal generator [155].

The eigenmodes of an individual skyrmion within confined geometries like a disc were discussed in a series of theoretical works [156–159]. In [160] Mruczkiewicz et al investigated the spin-wave band structure arising from a periodic chain of nanodisks containing individual skyrmions. In [161], Ma et al studied theoretically a long stripe in which a one-dimensional lattice of periodically ordered skyrmions were introduced dynamically via nanocontacts. In both cases, interfacial DMI and ultrathin films were considered. The skyrmions induced magnonic band structures with a dispersive character and collective properties [160, 161]. The stripe-based magnonic crystal has the advantage that both dipolarly and exchange-coupled modes can be controlled via the one-dimensional skyrmion lattice, while in chains of separated nanodisks only the dipolar coupling promotes propagating spin waves. Consequently, the work of Ma et al addressed spin waves propagating along the channel with wavelengths down to about 50 nm, thereby entering the exchange dominated regime with spin-wave eigenfrequencies of several ten GHz [161]. In [160] in contrast, the allowed minibands with dispersive spin waves were near the original eigenfrequencies of the gyrotropic and breathing modes below 1 GHz and around 13 GHz, respectively. Group velocities were found to be small and only a few 10 m s⁻¹. These values reflect the small coupling between the skyrmion-containing nanodisks. In [161], velocities are found to be an order of magnitude larger, offering a better signal transmission in a magnonic device.

Helimagnets incorporating bulk DMI have already been prepared via thin-film deposition techniques on different substrates. Typically, growth- and strain-induced anisotropies are found to modify the magnetic phase diagram with respect to the bulk material. In addition thin films often consist of domains in which the crystallites exhibit a different chirality. Preliminary measurements on the spin-wave damping in thin films of the helimagnet FeGe deposited by two different groups have provided damping parameters α that differ by almost two orders of magnitude [43, 157]. Here further research is needed to optimize chiral magnets in thin-film technology and exploit the full potential of bulk DMI. Propagating spin waves have been explored in the field-polarized phase of a thin platelet of Cu₂OSeO₃ that was extracted from a large single crystal using focused ion beam etching. When an in-plane field was parallel to the spin-wave wave vector a nonreciprocal spin-wave dispersion relation was extracted from the experimental spectra [110]. Cu₂OSeO₃ thus offers nonreciprocity of both transmitted electromagnetic waves and spin waves. A bulk sample of noncentrosymmetric LiFe₅O₈ (space group P4₁32) with DMI was reported to exhibit a nonreciprocal spin-wave dispersion relation as well [162].

Before we discuss interesting properties that theory predicts for thin films with bulk DMI we comment on the role of thin films that host skyrmions due to interfacial DMI. Spin dynamics in metallic magnetic multilayers exhibiting relevant interfacial DMI has been explored both theoretically and experimentally. Theoretically it has been shown that such layers form narrow domain walls that channel or scatter spin waves [163, 164]. In the saturated magnetic state the interfacial DMI has been predicted to induce nonreciprocal spin-wave dispersion relations [165] that can be tailored via different multilayer compositions. Inelastic scattering of both electrons and light has been applied to such materials systems and showed spin-wave spectra exhibiting different eigenfrequencies for opposing wave-vector directions consistent with theoretical considerations [166–168]. Interfacial DMI thus enriches a magnonics-related thin-film technology beyond skyrmions [169]. For instance, nanomagnonic waveguides with unidirectional spin-wave propagation become possible [170]. However, metallic magnetic layers with DMI are expected to exhibit a broad linewidth [171], i.e. an increased damping of spin waves. We note that the microscopic mechanism behind interfacial DMI lies in spin–orbit coupling. In metallic ferromagnets, spin–orbit coupling leads to spin-wave damping due to precession-induced intra- and interband excitations of conduction electrons and subsequent electron scattering [172]. At the same time, metallic multilayers with non-collinear spin structures might experience enhanced spin-wave damping via spin pumping [81, 173, 174]. We suppose damping parameters α of metallic thin films hosting magnetic skyrmions to be four to five orders of magnitude worse compared to YIG which is the prototypical magnetic material in microwave technologies exploiting spin-precessional motion. When aiming at low spin-wave damping we suggest to focus on thin-film materials where an insulating magnet carries the spin waves. In insulating magnets, excitations of conduction electrons are not relevant and spin-wave damping is potentially small.

4.6.1. Spin waves in thin films: field-polarized and magnetic helix state.

We now turn to the theoretical discussion of the spin-wave spectrum in a film of a cubic chiral magnet described by equations (1) and (3). The film is assumed to possess a thickness d that is comparable to an intrinsic length scale of the system, in particular, the pitch length $2\pi /Q$ (table 1). In the following we further assume an infinitely large film whose normal is oriented along the z-axis. For the thin film, one might take into account an additional magnetic anisotropy in the free energy functional ${{F}_{\text{aniso}}}={\int}^{}\text{d}\vec{r}{{\mathcal{F}}_{\text{aniso}}}$ with

$$\begin{eqnarray}&&{{\mathcal{F}}_{\text{aniso}}}=K\hat{n}_{z}^{2}\end{eqnarray} \tag{ 39 }$$

and anisotropy constant K. It corresponds to an easy-plane anisotropy for K  >  0 and an easy-axis anisotropy for K  <  0.

The situation in thin films is generally complicated due to the boundary conditions of equation (4) that the magnetization must obey at the surfaces of the film. It is still tractable for a magnetic field applied along the film normal, i.e. along the z-axis, $\vec{H}=H\hat{z}$. In this case the magnetization is fully polarized at large fields and ${{\hat{n}}_{\text{eq}}}=\hat{z}$. As the film is infinite and translationally invariant, we can perform a partial Fourier transform and consider the magnon wavefunction ${{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)$ as a function of in-plane momentum ${{\vec{k}}_{\bot}}=\left({{k}_{x}},{{k}_{y}},0\right)$, frequency ω, and the spatial z coordinate. The stationary wave equation can be cast into the form

$$\begin{eqnarray}&&\hbar \omega {{\tau}^{z}}{{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)={{\mathcal{H}}_{\text{loc}}}{{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)+{\int}_{0}^{d}\text{d}{{z}^{\prime}}\mathcal{K}\left({{\vec{k}}_{\bot}},z-{{z}^{\prime}}\right){{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}\left({{z}^{\prime}}\right)\end{eqnarray} \tag{ 40 }$$

for a film located within the interval $z\in \left[0,d\right]$. The local part of the Hamiltonian reads

$$\begin{eqnarray}&&{{\mathcal{H}}_{\text{loc}}}=\mathcal{D}\left(-\mathbb{1}\partial _{z}^{2}-\text{i}2Q{{\tau}^{z}}{{\partial}_{z}}\right)+\left(\mathcal{D}k_{\bot}^{2}+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{\text{int}}}-\frac{g{{\mu}_{B}}2K}{{{M}_{s}}}\right)\mathbb{1}\end{eqnarray} \tag{ 41 }$$

with the internal field ${{H}_{\text{int}}}=H-{{M}_{s}}$ corresponding to a demagnetization factor N_z  =  1. The nonlocal part derives from the dipolar interaction and its kernel is given by

$$\begin{eqnarray}\begin{array}{*{20}{l}}\mathcal{K}\left({{\vec{k}}_{\bot}},z\right)=\frac{g{{\mu}_{B}}{{\mu}_{0}}{{M}_{s}}}{4|{{{\vec{k}}}_{\bot}}|}{{\text{e}}^{-|{{{\vec{k}}}_{\bot}}||z|}}\left(\begin{array}{c} {{k}_{+}}{{k}_{-}} & k_{-}^{2} \\ k_{+}^{2} & {{k}_{+}}{{k}_{-}} \end{array}\right)\end{array}\end{eqnarray} \tag{ 42 }$$

in the same notation as in equation (10). Importantly, on the two surfaces of the film, z  =  0 and z  =  d, the wavefunction must obey the boundary conditions that follow from equation (4). In the absence of surface pinning, they are given by

$$\begin{eqnarray}&&\left. \left(-\text{i}{{\partial}_{z}}+Q{{\tau}^{z}}\right){{\vec{\Psi}}_{{{{\vec{k}}}_{\bot}},\omega}}(z)\right|_{\text{surface}}=0.\end{eqnarray} \tag{ 43 }$$

For zero in-plane momentum ${{\vec{k}}_{\bot}}=0$ the non-local part vanishes, $\mathcal{K}\left(\mathbf{0},z\right)=0$, and the eigenvalue problem becomes local. The eigenfunctions obeying the above boundary conditions are then readily obtained

$$\begin{eqnarray}&&{{\vec{\Psi}}_{\sigma,p}}(z)=\frac{1}{{{N}_{p}}}\cos \left(\frac{\pi p}{d}z\right){{\text{e}}^{-\text{i}Q{{\tau}^{z}}z}}\left(\begin{array}{c} 1+\sigma \\ 1-\sigma \end{array}\right)\end{eqnarray} \tag{ 44 }$$

with $\sigma =\pm 1$ and a normalization constant N_p. These eigenfunctions describe the so-called perpendicular standing spin-wave (PSSW) modes of the film [175] in the presence of DMI specified by the discrete quantum number p  =  0, 1, 2, 3,.... The corresponding eigenfrequencies are ${{\omega}_{\pm,p}}=\pm {{\omega}_{p}}$ with

$$\begin{eqnarray}&&\hbar {{\omega}_{p}}=\mathcal{D}{{\left(\frac{\pi p}{d}\right)}^{2}}+g{{\mu}_{B}}{{\mu}_{0}}\left(H-{{H}_{c2,z}}\right).\end{eqnarray} \tag{ 45 }$$

We introduced the critical field ${{H}_{c2,z}}={{M}_{s}}+\frac{2K}{{{\mu}_{0}}{{M}_{s}}}+\frac{\mathcal{D}{{Q}^{2}}}{g{{\mu}_{B}}{{\mu}_{0}}}$ that is assumed to be positive.

The spectrum given by equation (45) differs from that of a field-polarized ferromagnetic film. The mode p  =  0 does not correspond to the uniform mode but carries a finite wavevector. For decreasing H, its gap vanishes at the critical field H_c2,z triggering the formation of the helical state. The boundary conditions of equation (43) result in the additional oscillating factor ${{\text{e}}^{-\text{i}Qz}}$ (for $\sigma =1$). It was pointed out in [176] that the eigenfunction given by equation (44) corresponds to the superposition of two non-reciprocal spin waves with wavevectors $\frac{\pi p}{d}+Q$ and $-\frac{\pi p}{d}+Q$. Importantly, the dynamic correction to the uniform magnetization is proportional to the integral ${\int}_{0}^{d}\text{d}z{{\vec{\Psi}}_{+,p}}(z)$ that is finite for all quantum numbers p due to the additional phase factor ${{\text{e}}^{-\text{i}Qz}}$. As a result, all PSSW modes couple to a uniform oscillating magnetic field [176]. This is in contrast to a ferromagnetic film for which only the Kittel mode is excited by a uniform oscillating field if spins were free at the surfaces.

In the presence of a finite in-plane momentum ${{\vec{k}}_{\bot}}$ the nonlocal part $\mathcal{K}$ of the wave equation (40) must be considered. This is usually done either approximately or numerically, and, for ferromagnets, it gives rise to the magnetostatic forward volume mode (MSFVM). Recent microwave absorption experiments on films of FeGe for an applied out-of-plane field have found two resonances at high fields that were interpreted as PSSW modes [43].

For a large magnetic field within the plane of the film, say, along the x-axis, the magnetization is not completely polarized and ${{\hat{n}}_{\text{eq}}}\ne \hat{x}$. It is twisted close to the film surfaces due to the boundary conditions given by equation (4), and this boundary twist was experimentally investigated, e.g. in a film of MnSi [108] as well as in nanowires and lamellae of FeGe [177, 178]. The influence of this twist on the magnon spectrum was addressed theoretically in [144, 145] neglecting however dipolar interactions. It was found that the twist acts as an attractive potential for the exchange spin waves resulting in modes localized to the two surfaces. The mechanism for localization is thus different from surface- and edge confined magnetostatic modes already explored in magnonics, i.e. Damon-Eshbach modes [179] and modes in spin-wave wells [180], respectively. Intriguingly, a recent theoretical work predicts that in the presence of a surface twist and corresponding edge magnons a specific field protocol can be implemented to create a chain of skyrmions at the edge of a field-polarized two-dimensional chiral magnet [145]. Omitting the surface twist, [165] theoretically discussed the full dipole-exchange spin-wave spectrum. A numerical micromagnetic simulation of the magnon spectrum in this configuration was recently presented in [181] together with a comparison to microwave absorption experiments on FeGe films. Measurements in this geometry have been also performed on Cu₂OSeO₃ [110].

We now turn to a discussion of spin-wave properties in the spin helix state. The treatment for a general orientation of the magnetic field is again complicated by the boundary conditions of equation (4). Only for a magnetic field aligned along the film normal (z-axis), the conical helix of equation (15) automatically fulfils the boundary conditions of equation (4). We restrict our discussion to the case of vanishing in-plane magnon momentum ${{\vec{k}}_{\bot}}=0$. Following similar steps as in the context of equation (45), the eigenfrequencies of the PSSW modes are found to read

$$\begin{eqnarray}&&\hbar {{\omega}_{p}}=\sqrt{\mathcal{D}}\frac{\pi p}{d}\sqrt{\mathcal{D}{{\left(\frac{\pi p}{d}\right)}^{2}}+g{{\mu}_{B}}{{\mu}_{0}}{{H}_{c2,z}}\left(1-{{\left(\frac{H}{{{H}_{c2,z}}}\right)}^{2}}\right)}\end{eqnarray} \tag{ 46 }$$

with p  =  0, 1, 2,... and the critical field ${{H}_{c2,z}}={{M}_{s}}+$$\frac{2K}{{{\mu}_{0}}{{M}_{s}}}+\frac{\mathcal{D}{{Q}^{2}}}{g{{\mu}_{B}}{{\mu}_{0}}}>0$. This formula is valid for $H\leqslant {{H}_{c2,z}}$. At the critical field H_c2,z the eigenfrequencies of equation (46) directly connect to equation (45) for each p smoothly.

For a magnetic field within the plane of the film, different regimes might be realized. For strong easy-plane anisotropy⁵, the helix axis might remain aligned along the film normal. Similar to monoaxial chiral magnets [182], the in-plane field then distorts the helix to a chiral soliton lattice, which has been recently observed in films of FeGe [183]. The resonances of the chiral soliton lattice have been studied theoretically in [184]. In the other limit of easy-axis anisotropy, the helix axis rotates into the film plane, and the boundary conditions (4) result in a deformation of the helix close to the surfaces. Even more complicated situations arise for weak or canted magnetic fields.