A semi-analytical approach to calculating the dynamic modes of magnetic vortices with Dzyaloshinskii-Moriya interactions

Interfacial Dzyaloshinskii-Moriya interactions (DMI) are important for the stabilization of Néel skyrmions Fert, Cros, and Sampaio (2013); Rohart and Thiaville (2013); Woo et al. (2016); Moreau-Luchaire et al. (2016), and have also been shown to influence domain wall formation and propagationRyu et al. (2013); Emori et al. (2013); DeJong and Livesey (2015). The DMI also lead to significant changes in the dispersion relations for surface spin waves in saturated magnetic thin films Moon et al. (2013); Kostylev (2014), where inclusion of a DMI energy term results in shifts in the frequencies of surface spin waves that propagate in opposite directions. These theoretically predicted frequency shifts have been verified experimentally by several groups, and the detection of this frequency difference is currently the best available method to obtain quantitative measurements of the DMI Stashkevich et al. (2015); Nembach et al. (2015); Di et al. (2015); Ma et al. (2017).

The DMI have intriguing potential for spintronics and magnonics applications, consequently it is important to develop a full understanding of how the DMI affect dynamics in patterned magnetic structures. The dynamic excitations in micro- and nano-sized structures are quantized based on the element size, hence the wavelengths of the spin excitations are short enough that they are likely to be affected by the DMI. In the presence of DMI, counterpropagating surface spin waves that have the same frequency will have different wavelengths and, as a direct consequence, the spin excitations in confined geometries can no longer be standing wave excitations Zingsem et al. (2019). Micromagnetic simulations have also shown that the DMI should lead to non-reciprocal modes that propagate around the edges of saturated structures Garcia-Sanchez et al. (2014), and to changes of the spin eigenmodes of nonuniform magnetization states Mruczkiewicz, Krawczyk, and Guslienko (2017), where the dynamic spectra for spin textures ranging from a magnetic skyrmion to a vortex were considered as the anisotropy and DMI were varied. Magnetic vortices, in particular, have often served as a model system for the study of spin textures. In the dynamic regime, vortices exhibit modes with gyrotropic Novosad et al. (2005), radial Demokritov and Demidov (2008); Vogt et al. (2011), and azimuthal Giovannini et al. (2004) motion patterns. The vortex radial modes, which have been described theoretically Guslienko et al. (2005); Zaspel et al. (2009) but without the inclusion of the DMI, involve spin waves that are quantized in the radial direction, also the direction that should be maximally affected by interfacial DMI. Hence this is a convenient system in which to examine the effects of the DMI on spin textures.

The paper is structured as follows: In section II, we introduce the Landau-Lifshitz based diagonalization (LLD) method to solve for the spin excitation frequencies and mode profiles for spin states with cylindrical symmetry and with DMI. In section III, the approach used for micromagnetic simulations is presented. In sections IV, we show that the LLD method and micromagnetic simulations yield frequencies and mode profiles that agree well for a magnetic vortex state. The results show that the DMI lead to important modifications of the spin excitations in patterned structures. The modifications of the vortex dynamics are illustrative of what should be expected for other spin configurations, and this approach can be easily extended to other systems with cylindrical symmetry, e.g., skyrmions. Finally, in section V we provide the conclusions.

We use a semi-analytical approach to study the effects of the DMI on the eigenfrequencies and eigenmodes of magnetic vortices confined in cylindrical nanodots of thickness $L$ and radius $R$. Radial-type modes with frequencies well above the vortex core gyrotropic frequency are considered (typically radial modes are in the GHz range, whereas the gyrotropic mode $\sim 100$ MHz), and we assume that the magnetization does not depend on the $z$-coordinate through the disk thickness. Further, the DMI is of the interfacial type with symmetry breaking along the $z$ direction as observed in heavy metal/ferromagnetic bilayer or multilayer thin films.

The DMI between two atomic spins $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$ is given by

$\displaystyle\mathcal{H}_{DMI}=\mathbf{D}_{ij}\mathbf{\cdot}(\mathbf{S}_{i}\times\mathbf{S}_{j}),$

(2)

where $\mathbf{D}_{ij}$ is the Dzyaloshinskii-Moriya vector, which is perpendicular to both the asymmetry direction and the vector $\mathbf{r}_{ij}$ between the spins $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$. This atomistic model translated to a continuum energy density in cylindrical coordinates with symmetry breaking along the $\mathbf{\hat{z}}$ direction, reads

$$E_{DM}=-D\left[(\hat{\phi}\times\hat{z})\cdot\left(\mathbf{m}\times\frac{1}{\rho}\frac{\partial\mathbf{m}}{\partial\phi}\right)\right.+\\ \left.(\hat{\rho}\times\hat{z})\cdot\left(\mathbf{m}\times\frac{\mathbf{m}}{\partial\rho}\right)\right].$$

(3)

If the magnetization is only a function of $\rho$, the DM energy density reduces to

$\displaystyle E_{DM}=-D\left[m_{\rho}\frac{\partial m_{z}}{\partial\rho}-m_{z}\left(\frac{\partial m_{\rho}}{\partial\rho}+\frac{m_{\rho}}{\rho}\right)\right],$

(4)

with the associated effective field

$\displaystyle\mathbf{h}_{DM}=\frac{2D}{\mu_{o}M_{s}^{2}}\left[\frac{\partial m_{z}}{\partial\rho}\hat{\rho}-\left(\frac{\partial m_{\rho}}{\partial\rho}+\frac{m_{\rho}}{\rho}\right)\hat{z}\right].$

(5)

Since the vortex magnetization has radial symmetry, an assumption that is supported by micromagnetic simulations for $|D|\geq 0$ mJ/m², the two-dimensional spin distribution is effectively reduced to a one-dimensional system parameterized by the radial coordinate $\rho$. Under this simplification, we set the magnetization to a random state and find the static configuration $\mathbf{m}_{0}(\rho)$ by minimizing the disk energy, Eq. (1), using the nonlinear conjugate gradient method with the modified Polak-Ribiere-Polyak-method with guarantee conjugacy Fischbacher et al. (2017). The results, shown in Fig. 1, indicate that the static magnetization of the vortex-state with DMI is tilted from the azimuthal direction by an angle $\psi=\tan^{-1}{\left(-m_{\rho}/m_{\phi}\right)}$ that is well described by a Cauchy function for most values of DMI and dot aspect ratios, $\psi(\rho)=\frac{a}{1+\left(\frac{\rho}{\sigma R}\right)^{2}}$, where $a$ and $\sigma$ are fitting parameters. The tilt $\psi$ goes to zero as $D$ approaches zero, and it also disappears for a ring, which occurs because a vortex in a ring lacks an out-of-plane core and there is consequently no DMI energy advantage to a tilt. A small out-of-plane canting of the spins also develops at the disk edge due to the DMI, as Fig. 2 shows.

To describe small oscillations of the magnetization about $\mathbf{m}_{0}(\rho)$, we write the Landau-Lifshitz (LL) equation in a new orthogonal system $(s,\xi,z)$, where the magnetization is rotated around the $z$ axis at each $\rho$ position by the tilt angle $\psi{(\rho)}$ using the rotation matrix

$\displaystyle\mathcal{R}(\psi)=\begin{pmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{pmatrix}$

(6)

such that the reduced magnetization and reduced total effective field in the rotated frame are

	$\displaystyle\mathbf{\widetilde{m}}=\mathcal{R}(\psi)\mathbf{m},$		(7)
	$\displaystyle\mathbf{\widetilde{h}}=\mathcal{R}(\psi)\mathbf{h},$		(8)

respectively, with

$\displaystyle\mathbf{m}=\begin{pmatrix}m_{\rho}\\ m_{\phi}\\ m_{z}\end{pmatrix},\mathbf{\widetilde{m}}=\begin{pmatrix}\widetilde{m}_{s}\\ \widetilde{m}_{\xi}\\ m_{z}\end{pmatrix},\mathbf{h}=\begin{pmatrix}h_{\rho}\\ h_{\phi}\\ h_{z}\end{pmatrix},\textrm{and}~{}\mathbf{\widetilde{h}}=\begin{pmatrix}\widetilde{h}_{s}\\ \widetilde{h}_{\xi}\\ h_{z}\end{pmatrix}.$

(9)

The LL equation in the rotated frame with no damping reads

$\displaystyle\frac{\partial\widetilde{\mathbf{m}}}{\partial t}=-|\gamma|M_{s}\mathbf{\widetilde{m}\times\widetilde{h}},$

(10)

where $\gamma$ is the gyromagnetic ratio, and the reduced total effective field in the rotated frame $\mathbf{\widetilde{h}}$,

$\displaystyle\mathbf{\widetilde{h}}=\mathcal{R}(\psi)(\mathbf{h_{ex}}+\mathbf{h_{d}}+\mathbf{h_{DM}}),$

(11)

includes the exchange $\mathbf{h_{ex}}$, demagnetization $\mathbf{h_{d}}$, and DMI $\mathbf{h_{DM}}$ fields. The magnetization and the effective field can each be separated into static and dynamic parts

	$\displaystyle\mathbf{\widetilde{m}}=\mathbf{\widetilde{m}_{0}}(\rho)+\bm{\widetilde{m}_{\sim}}(\rho,t),$
	$\displaystyle\mathbf{\widetilde{h}}=\mathbf{\widetilde{h}_{0}}(\rho)+\mathbf{\widetilde{h}_{\sim}}(\rho,t),$

The demagnetizing field contribution is $\mathbf{h}_{d}(\mathbf{r})=\widehat{G}[\mathbf{m}(\mathbf{r})]$ where $\widehat{G}$ is a tensorial non-local integral operator (the tensorial magnetostatic Green’s function $G_{\alpha\beta}(\mathbf{r,r^{\prime}})=-(\nabla_{\mathbf{r}})_{\alpha}(\nabla_{\mathbf{r^{\prime}}})_{\beta}(4\pi|\mathbf{r}-\mathbf{r^{\prime}}|)^{-1}$) expressed in cylindrical coordinates with $\mathbf{r}=(\rho,\phi,z)$ as $\widehat{G}[\mathbf{m}(\mathbf{r})]=\int\widehat{G}[\mathbf{r,r^{\prime}}]\mathbf{m(\mathbf{r^{\prime}})}d^{3}\mathbf{r^{\prime}}$. Due to the radial symmetry and constant magnetization $\mathbf{m(r)}$ across the disk thickness, the Green’s functions can be averaged over $\phi\phi^{\prime}$ and $zz^{\prime}$

$\displaystyle g_{\alpha\beta}(\rho,\rho^{\prime})=\frac{1}{2\pi L}\int^{L}_{0}dz\int^{2\pi}_{0}d\phi\int^{L}_{0}dz^{\prime}\int^{2\pi}_{0}d\phi^{\prime}G_{\alpha\beta}(\mathbf{r,r^{\prime}}).$

(15)

The only terms that are nonzero are

$\displaystyle g_{\rho\rho}(\rho,\rho^{\prime})=-\frac{1}{\rho^{\prime}}\delta(\rho-\rho^{\prime})+\frac{(2-\gamma^{2})K_{ell}\left[\gamma^{2}\right]-2E_{ell}\left[\gamma^{2}\right]}{L\pi\gamma\sqrt{\rho\rho^{\prime}}}-\frac{(2-\gamma_{L}^{2})K_{ell}\left[\gamma_{L}^{2}\right]-2E_{ell}\left[\gamma_{L}^{2}\right]}{L\pi\gamma_{L}\sqrt{\rho\rho^{\prime}}}$

(16)

$\displaystyle g_{zz}(\rho,\rho^{\prime})=-\frac{2}{\pi L}\left\{\frac{1}{\rho^{\prime}}K_{ell}\left[\frac{\rho^{2}}{{\rho^{\prime}}^{2}}\right]\left(1-\Theta\left[\rho-\rho^{\prime}\right]\right)+\frac{1}{\rho}K_{ell}\left[\frac{{\rho^{\prime}}^{2}}{\rho^{2}}\right]\Theta\left[\rho-\rho^{\prime}\right]-\frac{1}{\gamma_{2}}K_{ell}\left[\frac{\gamma_{1}^{2}}{\gamma_{2}^{2}}\right]\right\}$

(17)

where $\Theta[\rho-\rho^{\prime}]$ is the Heaviside step function, $K_{ell},E_{ell}$ are elliptic integrals, and

	$\displaystyle\gamma=\sqrt{\frac{4\rho\rho^{\prime}}{(\rho+\rho^{\prime})^{2}}},\ \ \gamma_{L}=\sqrt{\frac{4\rho\rho^{\prime}}{L^{2}+(\rho+\rho^{\prime})^{2}}},$		(18)
	$\displaystyle\gamma_{1,2}=\frac{1}{2}\left[\sqrt{(\rho+\rho^{\prime})^{2}+L^{2}}\mp\sqrt{(\rho-\rho^{\prime})^{2}+L^{2}}\right].$		(19)

It is important to mention that Eqs. (16) and (17) can also be used for the dynamic calculations since we are in the magnetostatic regime, i.e., $\omega<<ck$, where $k$ is the wave vector.

The demagnetization field in cylindrical coordinates can now be reduced to

$\displaystyle\mathbf{h_{d}}(\rho)=\hat{\Gamma}_{d}[\mathbf{m(\mathbf{\rho^{\prime}})}]$

(20)

where $\hat{\Gamma}_{d}$ is the magnetostatic tensorial non-local integral operator, which operates on $\mathbf{m}(\mathbf{\rho^{\prime}})$ as follows

$\displaystyle\hat{\Gamma}_{d}[\mathbf{m(\mathbf{\rho^{\prime}})}]=\int\hat{g}(\rho,\rho^{\prime})\mathbf{m(\mathbf{\rho^{\prime}})}\rho^{\prime}d\rho^{\prime},$

(21)

with

$\displaystyle\hat{\Gamma}_{d}=\begin{bmatrix}\hat{A}_{\rho\rho}&0&0\\ 0&0&0\\ 0&0&\hat{A}_{zz},\end{bmatrix}$

(22)

	$\displaystyle\hat{A}_{\rho\rho}=\hat{g}_{\rho\rho}\ \ \mathrm{diag}(\rho^{\prime})\ \ \Delta\rho^{\prime},$		(23)
	$\displaystyle\hat{A}_{zz}=\hat{g}_{zz}\ \ \mathrm{diag}(\rho^{\prime})\ \ \Delta\rho^{\prime},$		(24)

where the non-local integral operators $\hat{g}_{\rho\rho}$ and $\hat{g}_{zz}$ are written in $n\times n$ discretized matrix form with $\rho$ entries as columns, and $\rho^{\prime}$ entries as rows following Eqs. (16-17), $\mathrm{diag}(\rho^{\prime})$ is an $n\times n$ diagonal matrix, and $\Delta\rho^{\prime}$ is the cell size. Note that the integration in Eq.(21) is accomplished by the matrix multiplication between the non-local integral operator $\hat{\Gamma}_{d}$ with the column vector $\mathbf{m(\mathbf{\rho^{\prime}})}$ (for more details see the Appendix: Definitions of matrix operators).

The DMI Eq. (5) and exchange effective fields can be expressed in terms of differential operators as

$\displaystyle\mathbf{h_{DM}}(\rho)=\hat{\Gamma}_{DM}\mathbf{m(\mathbf{\rho})},\ \ \mathbf{h_{ex}}(\rho)=\hat{\Gamma}_{ex}\mathbf{m(\mathbf{\rho})},$

(25)

with

$\displaystyle\hat{\Gamma}_{DM}=\begin{pmatrix}\hat{D}_{\rho\rho}^{BC}&0&\hat{D}_{\rho z}\\ 0&0&0\\ \hat{D}_{z\rho}&0&\hat{D}_{zz}^{BC}\end{pmatrix},\hat{\Gamma}_{ex}=\begin{pmatrix}\hat{E}_{\rho\rho}&0&\hat{E}_{\rho z}^{BC}\\ 0&\hat{E}_{\phi\phi}&0\\ \hat{E}_{z\rho}^{BC}&0&\hat{E}_{zz}\end{pmatrix}.$

(26)

The operators with the superscript $BC$ contain only terms that contribute to the boundary conditions and in the absence of DMI these terms vanish. For spin textures with radial symmetry the extended form of the above operators are

	$$\displaystyle\hat{D}_{\rho z}=\frac{2D}{\mu_{0}M_{s}^{2}}\frac{d}{d\rho}$$		(27)
	$$\displaystyle\hat{D}_{z\rho}=-\frac{2D}{\mu_{0}M_{s}^{2}}\left(\frac{d}{d\rho}+\frac{1}{\rho}\right)$$		(28)
	$$\displaystyle\hat{E}_{\rho\rho}=\hat{E}_{\phi\phi}=\frac{2A_{ex}}{\mu_{0}M_{s}^{2}}\left(\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}-\frac{1}{\rho^{2}}\right)$$		(29)
	$$\displaystyle\hat{E}_{zz}=\frac{2A_{ex}}{\mu_{0}M_{s}^{2}}\left(\frac{d^{2}}{d\rho^{2}}+\frac{1}{\rho}\frac{d}{d\rho}\right)$$		(30)

subject to the boundary conditions

$\displaystyle\frac{d\mathbf{m}}{dn}=\frac{D}{2A_{ex}}(\hat{z}\times\hat{n})\times\mathbf{m}.$

(31)

This form guarantees that the edge magnetization rotates in a plane containing the edge surface normal Rohart and Thiaville (2013).

Semi-analytical solutions were obtained by extending the approach presented in Ref. [Guslienko et al., 2005] for the case without DMI. In the absence of DMI, the linearized LL equation Eq.(12) can be reduced to a single integro-differential equation and then solved numerically as an eigenvalue problem. The effective torque exerted by the DMI, however, yields two uncoupled integro-differential equations, hence the problem is consequently more complicated than the dipolar-only case. With DMI, the problem can be set up as an eigenvalue problem of the form

$\displaystyle\frac{i\omega}{|\gamma|M_{s}}\bm{m}_{\sim,n}=\hat{\Gamma}(\rho,\rho^{\prime})\bm{m}_{\sim,n}$

(32)

with eigenfrequencies obtained as follows

$\displaystyle\omega_{n}=-i\gamma M_{s}\lambda_{n},$

(33)

where $\bf{m}_{\sim,n}(\rho)$ is the $n^{th}$ eigenmode with eigenfrequency $\omega_{n}$, and $\lambda_{n}$ are the eigenvalues of the operator $\hat{\Gamma}$, which contains the magnetostatic nonlocal integral operator, and the exchange and DMI differential operators. The extended form of Eq. (32) reads

$\displaystyle\frac{i\omega}{|\gamma|M_{s}}\begin{pmatrix}\widetilde{m}_{\rho}\\ m_{z}\end{pmatrix}=\begin{pmatrix}\widetilde{m}_{0}\hat{\Gamma}_{z\rho}\cos\psi&\widetilde{m}_{0}\hat{\Gamma}_{zz}-\tilde{h}_{o,\xi}\\ \widetilde{h}_{o,\xi}-\widetilde{m}_{0}\left(\hat{\Gamma}_{\rho\rho}\cos^{2}\psi+\hat{\Gamma}_{\phi\phi}\sin^{2}\psi\right)&-\widetilde{m}_{0}\hat{\Gamma}_{\rho z}\cos\psi\end{pmatrix}\begin{pmatrix}\widetilde{m}_{\rho}\\ m_{z}\end{pmatrix},$

(34)

with

	$\displaystyle\hat{\Gamma}_{\rho\rho}=\hat{A}_{\rho\rho}+\hat{E}_{\rho\rho}+\hat{D}^{BC}_{\rho\rho}$		(35)
	$\displaystyle\hat{\Gamma}_{zz}=\hat{A}_{zz}+\hat{E}_{zz}+\hat{D}^{BC}_{zz}$		(36)
	$\displaystyle\hat{\Gamma}_{\phi\phi}=\hat{E}_{\phi\phi}$		(37)
	$\displaystyle\hat{\Gamma}_{\rho z}=\hat{D}_{\rho z}+\hat{E}_{\rho z}^{BC}$		(38)
	$\displaystyle\hat{\Gamma}_{z\rho}=\hat{D}_{z\rho}+\hat{E}_{z\rho}^{BC}.$		(39)

Expressions for the operator matrices can be found in the appendix.

Micromagnetic simulations were also performed using MuMax3 Vansteenkiste et al. (2014) and compared with the results from the matrix method. The simulations were conducted by first relaxing a particular spin structure in zero magnetic field to obtain the ground state. Next, a small out-of-plane perturbation field of approximately 20 mT was applied, chosen such that the perturbation of the spins from the zero-field equilibrium state is a few percent. The spins were next allowed to precess after removing the perturbation field using a damping parameter of $\alpha=0.01$, and Fourier transforms of the $z$-component of the magnetization versus time were performed to obtain the spectra. Simulations with a sinusoidal driving field were conducted at selected resonance frequencies and the modes were constructed from the spin distributions saved out over two periods after a stead-state response to the driving field was reached, typically after approximately 50 oscillations. Simulations were conducted for both disks and rings in the vortex state, where the vortex state in the ring lacks the central out-of-plane core. Structures with $R$ and $L$ ranging from 100 to 350 nm and 1 to 10 nm, respectively, were considered. Parameters typical for Permalloy were used for the magnetic layer: $M_{s}=8\times 10^{5}$ A/m, exchange of $A=1\times 10^{-11}$ J/m², anisotropy was neglected, and the interfacial DMI was varied from 0 to 1.5 mJ/m², which is within the range of values that have been reported for heavy metal/ferromagnetic thin film bilayers such as Permalloy/Pt Stashkevich et al. (2015); Nembach et al. (2015).

The static equilibrium state and the dynamic modes were calculated using the 1D LLD approach and micromagnetic simulations. The tilt angles $\psi$ in Fig. 1, obtained using the 1D expressions, were used as input to calculate the the kernel $\hat{\Gamma}(\rho,\rho^{\prime})$ in Eqs. (32) and (34). The static profiles obtained from the LLD approach and the micromagnetic simulations agree well, as shown in Fig. 1b. The DMI leads to an in-plane tilt of the magnetization that increases with increasing $D$. The DMI also leads to changes in $m_{0,z}$, as shown in Fig. 2. For a given $L$ and $R$, the core size increases with increasing $|D|$, where the size depends on the magnitude but not the sign of $D$, and the spins near the edges tilt out-of-plane in the direction opposite to the core magnetization. Both of these effects are most pronounced for small $L$; for $L=10$, 5, and 1 nm, the core sizes at $D=2$ mJ/m² are 28%, 45%, and 100% larger than the core at $D=0$, respectively. Note that these calculations assume a constant $D$ to to provide a straightforward means to compare the effects of DMI with that of $L$, which mainly changes the demagnetization energy, but since the effect is interfacial in nature, the DMI should be very small for $L=10$ nm in a real sample. The tilting of the edges is also observed in rings and is a general feature of patterned magnetic structures with DMI.

The dynamic modes of the magnetic vortex are quantized along the radial direction. Figure 3 shows a comparison of the eigenvectors obtained from the LLD method and the simulations for the first three modes and the two agree well. These modes are also shown as cross-sectional plots and full mode maps of the out-of-plane motion $m_{\sim,z}$ in Figs. 4 and 5, respectively. For $D=0$ (left panels), the modes are standing wave excitations with well defined nodes and antinodes, whereas for $D=1.5$ mJ/m² (right panels), the modes have similar wavelengths but are propagating rather than standing waves. Nodes are still present but they are much less pronounced than they are for the $D=0$ case. Similar effects have been noticed in confined 1-D systems with DMI Zingsem et al. (2019). As shown in Fig. 5, the spin wave excitations appear to move from the center, outwards. The edge amplitudes are also much larger when DMI is present. Cross sections and full mode maps of the in-plane dynamic magnetization $\widetilde{m}_{\sim,s}$ show similar behavior (see appendix).

The direction of the propagation depends on the circulation of the vortex $c$ and the sign of $D$, but it is independent of the polarity $p$ of the vortex. The propagation direction is inwards for $cD<1$, and outward for $cD>1$, where $c=1$ ($-1$) corresponds to a counterclockwise (clockwise) circulation. These same effects are observed in ring-shaped structures in a vortex state that lack the complication of the central out-of-plane core and the resulting DMI-induced tilt angle $\psi$, hence the propagating nature of the modes is due to the DMI and is not a consequence of the static tilt. In fact, the sign of the static tilt depends on the sign of $D$ and the polarity $p$ of the vortex core but is independent of $c$. An inward (outward) static tilt is observed for $Dp>1$ ($Dp<1$) where $p=1$ corresponds to a polarity in the $+\hat{z}$ direction. The resonance frequencies are the same for any $D$, $p$, or $c$.

The DMI magnitude has a significant effect on the resonant frequencies and mode amplitudes. Fig. 6 shows spectra obtained from micromagnetic simulations for selected $R$ and $D$. The spectra for $D=0$ show a strong mode that is the lowest-order radial mode, and a weaker mode that is the third order mode. A small peak that is due to the second mode is barely visible between the first and third modes. The spectra for $D>0$, in contrast, show strong peaks for each of the first, second, and third order modes. Without DMI the even modes are only weakly excited because they have a net moment near zero and the odd modes are favored, whereas with DMI, the odd and even modes have comparable amplitudes. A similar effect was observed in Ref. Zingsem et al., 2019 for simulations of saturated elements. The DMI leads to a reduction in symmetry that allows all possible modes to couple to a uniform driving microwave field. Fig. 7 shows that the frequency increases as a function of $L/R$ for $D=1.5$ mJ/m² (panel b) in a similar manner as what is observed for $D=0$ (panel a). The LLD results for $D=0$ are the same as those calculated using Ref. [Guslienko et al., 2005] if the exchange contribution is neglected. Here the exchange is included, which slightly raises the calculated resonance frequencies. Fig. 8 shows that the frequencies of all of the observed modes decrease as a function of increasing $D$ for $R=250$ nm and $L=5$ nm, and similar trends are observed for other structure dimensions (additional plots are included in the appendix). In all cases the LLD and micromagnetic simulation results agree well.

The spectra for the vortices with the DMI show shifts in the frequency of the respective modes that are similar to what is observed for spin waves with comparable wavelengths in extended films with one important difference: the frequency shifts in the confined structures are always to lower frequencies as compared to what is observed at $D=0$, and this corresponds to a dominant mode direction. For a thin film, surface waves with a given wavelength that propagate in opposite directions have different frequencies, and surface spin waves at a particular frequency that travel in opposing directions have different wavelengths. In a confined structure, the wavelength quantization imposed by the structure size and/or the spin texture creates a situation where it should be possible to excite just one of the inward or outward propagating modes of a particular order at a slightly smaller frequency than the $D=0$ resonance frequency, and the other at a slightly higher frequency. The outward-propagating mode is shown in Fig. 5; as mentioned previously, the direction can be changed by either reversing the sign of $D$ or the vortex chirality. The oppositely directed, higher frequency mode is, however, suppressed in the simulations and, according to the LLD calculations this is because this mode is not an eigenmode of the system. This aspect of how the DMI affects spin dynamics in confined geometry could be useful for enhancing non-reciprocity for magnonics applications.

In Fig. 7, the resonance frequencies shown from the LLD calculations are the real part of $\omega$, as calculated from Eq. (34). For $D=0$, the eigenvalues are real, but for $|D|>0$ the eigenvalues are complex. For $D=0$, small imaginary parts of order $~{}1\times 10^{-14}$ are obtained, which is a numerical artifact; the imaginary parts of the eigenfrequencies of the first mode are as large as $\sim 1$ percent for $|D|>0$, which may be indicative of a DMI contribution to the damping. In the micromagnetic simulations the linewidths of the first mode are slightly narrower in Fig. 6 for larger $D$. The linewidths $\Delta f$ are 0.47 and 0.40 GHz for $R=250$ nm with $D=0$ and 1.5 mJ/m², respectively, which correspond to fractional linewidths of $\Delta f/f_{1}$ of 8.81% and 8.75%, respectively. Theoretical calculationsMoon et al. (2013) predict that the DMI should lead to a change in not just the frequency but also the linewidth for extended thin films, so it is not surprising that the DMI also lead to linewidth changes for modes in confined structures.

In conclusion, the DMI lead to important modifications of the static spin state and the dynamic excitations of magnetic vortices. These modifications provide insight into the types of effects that should be observed for other patterned magnetic structures. These changes are captured well by the LLD method, which provides a rapid means to gain insight into the behaviour not just for vortex-based systems, but also for other spin textures with cylindrical symmetry. For the vortex, the inclusion of the DMI induces a static in-plane tilt of the spins that increases as a function of $D$ and that disappears if the core is absent, i.e., in a magnetic ring. In the presence of DMI the radial-type vortex modes are still quantized, however, the mode frequencies are shifted, the modes are propagating rather than standing modes, and the even and odd modes are both excited effectively by a spatially uniform mode due to the symmetry breaking provided by the DMI. There are also differences of at the edges, specifically an out-of-plane static tilt at the edges is observed, and the edges show higher dynamic amplitudes in the presence of the DMI. The changes in the mode frequencies induced by the DMI are similar to what is predicted for extended films, however, unlike the case of an extended film, only the downward-shifted modes are eigenmodes of the dynamic equations, which suggests that the combination of DMI and confined geometries could lead to new strategies for non-reciprocal spin wave excitation, or, in the case of vortices, the outward-only mode could serve as a point-like source for spin waves if coupled to an extended film.

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