Resonant properties of dipole skyrmions in amorphous Fe/Gd multilayers

The dynamic response of dipole skyrmions in Fe/Gd multilayer films is investigated by ferromagnetic resonance measurements and compared to micromagnetic simulations. We detail thickness- and temperature-dependent studies of the observed modes as well as the effects of magnetic field history on the resonant spectra. Correlation between the modes and the magnetic phase maps constructed from real-space imaging and scattering patterns allows us to conclude that the resonant modes arise from local topological features such as dipole skyrmions but do not depend on the collective response of a close-packed lattice of these chiral textures. Using micromagnetic modeling, we are able to quantitatively reproduce our experimental observations which suggests the existence of localized spin-wave modes that are dependent on the helicity of the dipole skyrmion. We identify four localized spin-wave excitations for the skyrmions that are excited under either in-plane or out-of-plane rf fields. Lastly we show that dipole skyrmions and nonchiral bubble domains exhibit qualitatively different localized spin-wave modes.

Figure 1

(a) Schematic shows the cross-section view of a dipole skyrmion where the domain wall has a helical structure across the skyrmion depth with opposite Néel caps at the top and bottom surface and a chiral Bloch-line center. (b) Schematic of the experimental setup with the film on a coplanar waveguide. (c) The normalized absorption spectra, measured at 160 K, for a [Fe(0.36 nm)/Gd(0.4 nm)] $\times 80$ multilayer. The spectra are obtained while sweeping a perpendicular magnetic field, from $H_z=-10\mathrm{kOe}$ to $+10\mathrm{kOe}$, at a fixed frequency $h_{\mathrm{rf}}$ pump field. Resonances at/above the saturation field exhibit a smaller linewidth than resonances elicited by nonhomogeneous magnetic textures. The dashed line in (c) indicates the magnetic saturation field: resonances appearing above this field are known as Kittel peaks, and resonant peaks appearing below magnetic saturation are denoted domain resonances.

Figure 2

Resonance and magnetic properties of a [Fe(0.36 nm)/Gd(0.4 nm)] $\times 80$ multilayer. (a) The field dependence of the resonance resulting from the precession of the homogenous magnetization, e.g., Kittel mode, is shown from 240 K to 80 K. (b) Fitting results from (a) and the linewidth of the Kittel peaks, such as those in Fig. 1; we determine the temperature dependence of the effective magnetization and effective damping. (c) The extracted effective and intrinsic uniaxial anisotropy calculated from results in (b) and Fig. S2 of the Supplemental Material [28].

Figure 3

Temperature dependence of the resonant modes of a [Fe(0.36 nm)/Gd(0.40 nm)] $\times 80$ multilayer. The frequency-field $f\text{−}H$ dispersions illustrate the resonant modes that exist at temperatures ranging from 240 K down to 80 K. These $f\text{−}H$ dispersions were constructed from measurements in which the perpendicular magnetic field is swept from $H_z=-10\mathrm{kOe}$ to $+10\mathrm{kOe}$ at each fixed frequency, and detail the resonances observed from zero field to 8000 Oe. At each temperature, several regions are identified that correspond to specific domain morphologies: (I) stripes, (II) coexisting stripes and skyrmions, (III) skyrmion lattice, (IV) disordered skyrmions, and (V) Kittel region as described in Ref. [18]. These regions are delimited with dashed lines in the $f\text{−}H$ dispersions. The saturation field $H_{\mathrm{sat}}$ is obtained from magnetic hysteresis measurements performed in the same geometry.

Figure 4

Film thickness dependence of the resonant modes at room temperature. (a), (b) Images of the domain morphology field evolution of a [Fe(0.34 nm)/Gd(0.41 nm)] $\times 80$ and [Fe(0.34 nm)/Gd(0.41 nm)] $\times 120$ under positive perpendicular magnetic fields. The images are obtained with transmission x-ray microscopy along the Fe $L_3$ absorption edge (708 eV), where white features depict magnetization $(-m_z)$ and gray regions exhibit magnetization $(+m_z)$. (c), (d) The $f\text{−}H$ dispersions for the latter Fe/Gd films show the field-varying resonant modes observed at room temperature. From images such as (a), (b) we correlate the modes to specific domain morphologies: (I) stripes, (II) coexisting stripes and skyrmions, (III) skyrmion lattice, (IV) disordered skyrmions, and (V) Kittel region. These regions are delimited with dashed lines in the $f\text{−}H$ dispersions.

Figure 5

Calculated domain morphologies and magnetic susceptibilities. (a) The field-dependent domain morphology at $H_z=0\mathrm{Oe}$, 1200 Oe, 1600 Oe, 2500 Oe, and 2900 Oe showing the evolution of disordered stripe domains to ordered skyrmions to disordered skyrmions ($m_z$ at $z=40\mathrm{nm}$; topside view). The in-plane magnetization along the center of the slab reveals the helicity of these textures ($m_x$ at $z=0\mathrm{nm}$; topside view). The magnetization distribution ($m_x,m_z$) of these equilibrium states is based on the red/blue color bar. (b) The numerically computed susceptibility that results from whole slab at several fixed $H_z$ fields is detailed. (c) Fitting the resonance spectra with $n$-Lorenztian peaks allows us to construct frequency-field $f\text{−}H$ dispersions. The marker size corresponds to the normalized intensity of the susceptibility at each fixed $H_z$ field and the marker color distinguishes the resonance spectra from either exciting the equilibrium states with an rf field $h_{\mathrm{rf}}$ along the $x$ axis or the $z$ axis. Using the equilibrium states, such as those in (a), we correlate resonant modes to specific domain textures: (I–II) stripes and coexisting stripes and skyrmions, (III) skyrmion lattice, (IV) disordered skyrmions, and (V) Kittel region. These regions are delimited with dashed lines in the $f\text{−}H$ dispersions.

Figure 6

(a) The skyrmion localized spin-wave dynamics observed from perturbing the skyrmion phase, at $H_z=2000\mathrm{Oe}$, along both $x$ axis and $z$ axis geometries are detailed for both skyrmion helicities. Using contour plots, the motion of the skyrmion is traced along three different depths positions ($z=-40\mathrm{nm}$, 0 nm, 40 nm) for each resonant mode. Different contour lines depict position of the skyrmion with reference to the (b) sinusoidal pulse field and arrows are used as guide for the eye to describe the motion of the spin wave.

Figure 7

(a) The perpendicular field dependence of equilibrium states with a constant $H_x$ field show the evolution of aligned stripes to cylindrical domains (topside; $z=40\mathrm{nm},m_z$). The cross section across the center of the slab ($z=0\mathrm{nm}$) shows the magnetic texture has a mixture of Bloch-line ordering (topside; $m_x$). (b) The computed susceptibility spectra for magnetization states like those in (a) result in four-mode branches below magnetic saturation and two-mode branches above magnetic saturation. The magnetization states observed include (I) stripes, (II) coexisting stripes and cylindrical domains, (III) ordered lattice of cylindrical domains, (IV) disordered cylindrical domains, and (V) Kittel region. (c) The spin-wave dynamics associated with a bubble domain when probed under an in-plane sinusoidal pulse field.

Figure 8

The topside view of the Néel caps and Bloch-line configuration for dipole skyrmions with (a) clockwise and (b) counterclockwise helicity, and a (c) achiral bubble are detailed at different depths ($z=40\mathrm{nm}$, 20 nm, 0 nm, $-20\mathrm{nm}$, $-39\mathrm{nm}$). The left column shows a simple schematic of the magnetic spin configuration with emphasis on the arrangement of the in-plane magnetic spins (arrows) and the right column shows the domain state obtained from micromagnetic simulations where the white arrows represent the in-plane magnetization and the perpendicular magnetization is represented in terms of the color bar.

Figure 9

(a)–(c) Schematic of a simplified purely radial achiral bubble domain and its corresponding dynamics under an in-plane microwave field. (a) The in-plane magnetization of the achiral bubble is emphasized with arrows at three depth positions ($z=40\mathrm{nm}$, 0 nm, $-40\mathrm{nm}$). The 3D structure of the bubble domain only shows the in-plane magnetic spins along the direction of the perturbation field, and the topside view shows parallel and orthogonal in-plane magnetic spins. (b), (c) The redistribution of the magnetic spins is detailed at the maxima (b) and minima (c) of the sinusoidal perturbation field.