Dynamics of skyrmionic states in confined helimagnetic nanostructures

In confined helimagnetic nanostructures, skyrmionic states in the form of incomplete and isolated skyrmion states can emerge as the ground state in absence of both external magnetic field and magnetocrystalline anisotropy. In this work, we study the dynamic properties (resonance frequencies and corresponding eigenmodes) of skyrmionic states in thin film FeGe disk samples. We employ two different methods in finite-element based micromagnetic simulation: eigenvalue and ringdown method. The eigenvalue method allows us to identify all resonance frequencies and corresponding eigenmodes that can exist in the simulated system. However, using a particular experimentally feasible excitation can excite only a limited set of eigenmodes. Because of that, we perform ringdown simulations that resemble the experimental setup using both in-plane and out-of-plane excitations. In addition, we report the nonlinear dependence of resonance frequencies on the external magnetic bias field and disk sample diameter and discuss the possible reversal mode of skyrmionic states. We compare the power spectral densities of incomplete skyrmion and isolated skyrmion states and observe several key differences that can contribute to the experimental identification of the state present in the sample. We measure the FeGe Gilbert damping, and using its value we determine what eigenmodes can be expected to be observed in experiments. Finally, we show that neglecting the demagnetization energy contribution or ignoring the magnetization variation in the out-of-film direction—although not changing the eigenmode's magnetization dynamics significantly—changes their resonance frequencies substantially. Apart from contributing to the understanding of skyrmionic states physics, this systematic work can be used as a guide for the experimental identification of skyrmionic states in confined helimagnetic nanostructures.

Figure 1

(a) A thin film FeGe disk sample with $10\text{nm}$ thickness and diameter $d$. An external magnetic bias field $\mathbf{H}$ is applied uniformly and perpendicular to the sample (in the positive $z$ direction). (b) A cardinal sine wave excitation magnetic field $\mathbf{h}\left(t\right)$, used in the ringdown method, is applied for $0.5\text{ns}$ in either in-plane $\left(\widehat{\mathbf{x}}\right)$ or out-of-plane $\left(\widehat{\mathbf{z}}\right)$ direction. (c) The Fourier transform of excitation field $h\left(t\right)$ shows that all eigenmodes (allowed by the used excitation direction) with frequencies lower than $f_{\text{c}}=100\text{GHz}$ are excited approximately equally.

Figure 2

The Power spectral densities (PSDs) of an incomplete skyrmion (iSk) ground state at zero external magnetic field in a $80\text{nm}$ diameter FeGe disk sample with $10\text{nm}$ thickness. (a) Spatially averaged and (b) spatially resolved PSDs for an in-plane excitation, together with overlaid resonance frequencies computed using the eigenvalue method. The resonant frequencies obtained using the eigenvalue method are marked with a triangle symbol $\left(▵\right)$ if they can be activated using a particular excitation and with a circle symbol $(\circ )$ otherwise. (c) Spatially averaged and (d) spatially resolved PSDs for an out-of-plane excitation. (e) Schematic representations of magnetization dynamics associated with the identified eigenmodes. Schematically, we represent the skyrmionic state core with a circle symbol, together with a directed loop if it gyrates around its equilibrium position. Contour rings represented using dashed lines revolve/breathe out of phase with respect to the those marked with solid lines. The magnetization dynamics animations of all identified eigenmodes are provided in video 1 in Ref. [55].

Figure 3

Power spectral density (PSD) maps showing the dependence of incomplete skyrmion (iSk) state resonant frequencies on the external magnetic bias field changed between $-0.5\text{T}$ and $1.2\text{T}$ in steps of $10\text{mT}$ for $d=80\text{nm}$ when the system is excited using (a) in-plane and (b) out-of-plane excitation. The dependence of resonance frequencies on the disk sample diameter varied between $40\text{nm}$ and $180\text{nm}$ in steps of $2\text{nm}$ at zero external magnetic field for (c) in-plane and (d) out-of-plane excitation. We show two plots for every PSD map: one for the complete studied frequency range $\left(0–50\text{GHz}\right)$ and another plot in order to better resolve the low-frequency $\left(0–10\text{GHz}\right)$ part of the PSD map.

Figure 4

The power spectral densities (PSDs) of an isolated skyrmion (Sk) ground state in a $150\text{-nm}$ diameter FeGe disk sample with $10\text{nm}$ thickness at zero external magnetic bias field. (a) Spatially averaged and (b) spatially resolved PSDs for an in-plane excitation, together with overlaid resonance frequencies computed using the eigenvalue method. The resonant frequencies obtained using the eigenvalue method are marked with a triangle symbol $\left(▵\right)$ if they can be activated using a particular excitation and with a circle symbol $(\circ )$ otherwise. (c) Spatially averaged and (d) spatially resolved PSDs computed when the Sk state is perturbed from its equilibrium with an out-of-plane excitation. (e) Schematic representations of magnetization dynamics associated with the identified eigenmodes. Schematically, we represent the skyrmionic state core with a circle symbol, together with a directed loop if it gyrates around its equilibrium position. Contour rings represented using dashed lines revolve/breathe out of phase with respect to the those marked with solid lines. The magnetization dynamics animations of all identified eigenmodes are provided in videos 2 and 3 in Ref. [55].

Figure 5

Power spectral density (PSD) maps showing the dependence of isolated skyrmion (Sk) state resonant frequencies on the external magnetic bias field changed between $-0.5$ and $1.2\text{T}$ in steps of $10\text{mT}$ for $d=150\text{nm}$ when the system is excited using (a) in-plane and (b) out-of-plane excitation. The dependence of resonance frequencies on the disk sample diameter varied between 40 and $180\text{nm}$ in steps of $2\text{nm}$ at zero external magnetic field for (c) in-plane and (d) out-of-plane excitation. We show two plots for every PSD map: one for the complete studied frequency range $\left(0–50\text{GHz}\right)$ and another plot in order to better resolve the low-frequency $\left(0–10\text{GHz}\right)$ part of the PSD map.

Figure 6

Comparisons of power spectral densities (PSDs) of ground incomplete skyrmion (iSk) state (solid red line) and metastable isolated skyrmion (Sk) state (dashed blue line) in a $100\text{-nm}$ disk sample with $10\text{-nm}$ thickness at different values of external magnetic field $H$, computed for an in-plane (left column) and an out-of-plane (right column) excitation.

Figure 7

The linewidth $\mathrm{\Delta }H$ (half width at half maximum) measurement points at different resonance frequencies $f$ for a FeGe thin film, together with a first degree polynomial fit from which the Gilbert damping was extracted.

Figure 8

The spatially resolved power spectral densities (PSDs) of an incomplete skyrmion state in a $80\text{-nm}$ diameter FeGe disk sample with $10\text{-nm}$ thickness at zero external magnetic bias field for (a) in-plane and (b) out-of-plane excitation directions. The isolated skyrmion state in a $150\text{nm}$ diameter thin film disk with $10\text{-nm}$ thickness at $H=0$ when the system is excited with (c) in-plane and (d) out-of-plane excitation. The PSDs are computed using the experimentally measured value of FeGe Gilbert damping $\alpha =0.28$.

Figure 9

The comparison of power spectral densities computed using three-dimensional and two-dimensional models in absence of demagnetization energy contribution with the PSD obtained using a full simulation model for an isolated skyrmion state in the case of (a) in-plane and (b) out-of-plane excitations. Simulated sample is a $150\text{-nm}$ diameter disk with $10\text{-nm}$ thickness at zero external magnetic field.