Mutual influence between macrospin reversal order and spin-wave dynamics in isolated artificial spin-ice vertices

We theoretically and experimentally investigate magnetization reversal and associated spin-wave dynamics of isolated threefold vertices that constitute a Kagome lattice. The three permalloy macrospins making up the vertex have an elliptical cross section and a uniform thickness. We study the dc magnetization curve and the frequency versus field curves (dispersions) of those spin-wave modes that produce the largest response. We also investigate each macrospin reversal from a dynamic perspective, by performing micromagnetic simulations of the reversal processes, and revealing their relationships to the soft-mode profile calculated at the equilibrium state immediately before reversal. The theoretical results are compared with the measured magnetization curves and ferromagnetic resonance spectra. The agreement achieved suggests that a much deeper understanding of magnetization reversal and accompanying hysteresis can be achieved by combining theoretical calculations with static and dynamic magnetization experiments.

Figure 1

Hysteresis of the magnetization component $M_H$ parallel to the applied field H for the symmetric (solid line) and asymmetric (dashed line) samples normalized to the saturation magnetization $M_S$. The geometry of the three-lobed vertex structures are shown in insets (a) and (b).

Figure 2

Equilibrium magnetization configurations at $B={\mu }_{0}H=50\mathrm{mT}$, referred to as an “Ising saturated state,” for the symmetric (a) and asymmetric (b) vertices. Internal effective field maps (component parallel to the applied field) at (c) $B=-28.3\mathrm{mT}$, (d) $B=-41.5\mathrm{mT}$, (e) $B=-29\mathrm{mT}$, (f) $B=-34\mathrm{mT}$, and (g) $B=-75.8\mathrm{mT}$. Note that deep blue ($-1$, in arbitrary units) refers to minima. H, in the figure, is directed to the left.

Figure 3

Frequency of the largest power spin-wave modes as a function of applied field (decreasing from 100 mT), and (insets on the right) corresponding profiles (real, out-of-plane component of the dynamic magnetization). Bold red line labeled (a) is the first fundamental mode, going soft at $B_{c1}=-28.3\mathrm{mT}$; (b) is a backward-volume-like mode, (c) is a second fundamental mode (though highly hybridized), and (d) is the soft mode at $B_{c2}=-41.5\mathrm{mT}$. Profiles (a), (b), and (c) were calculated at 100 mT, and profile (d) (which was magnified $300\%$) was calculated at $-41.5\mathrm{mT}$.

Figure 4

Frequency vs applied field curves for the main modes discussed in the text. Modes (a)–(c) are the fundamental modes of each macrospin, while (d)–(f) are the soft modes at the corresponding transition fields: $B_{c1}=-29\mathrm{mT}$, $B_{c2}=-34\mathrm{mT}$, $B_{c3}=-75.8\mathrm{mT}$. The insets (a) to (f) are real $z$ components of the dynamic magnetization of the spin-wave modes discussed in the text. Modes (a), (b), and (c) were calculated at $H=50\mathrm{mT}$, while mode (d) was calculated at $-29\mathrm{mT}$, mode (e) was calculated at $-34\mathrm{mT}$, and mode (f) was calculated at $-75.8\mathrm{mT}$. In insets (d)–(f), the amplitude has been magnified three times with respect to (a)–(c).

Figure 5

Illustration of the triggering mechanism by which the soft mode initiates instabilities in the symmetric (a,b) and asymmetric (c,d,e) vertices. (a) $H=-28.3\mathrm{mT}$; (b) $H=-41.5\mathrm{mT}$; (c) $H=-29\mathrm{mT}$; (d) $H=-34\mathrm{mT}$; (e) $H=-75.8\mathrm{mT}$. The first column shows the equilibrium magnetization (red/blue color scale is the $y$ component) at applied field $H$; the second column shows the imaginary $y$ component of the soft mode at $H$ (rainbow color scale is the phase amplitude); the third column shows a snapshot of the magnetization at an early stage of the simulated reversal process. In the oommf simulations of reversal, the field variation $\epsilon$ was set to 1 mT for all cases, while the time variation $\partial t$ was set arbitrarily close to the starting point, until some evidence of the soft-mode profile was apparent. The fourth column shows the equilibrium $(t\to \infty )$ magnetization at the applied field $H+\epsilon$.

Figure 6

SEM images of the two sample types with a spacing of $1.88\mu \mathrm{m}$ prepared: (a) is a symmetric sample for which all three lobes have the same aspect ratio, and (b) is an asymmetric sample where the width of one lobe is reduced relative to the other two lobes.

Figure 7

Experimental dc magnetization loops at 5 K, either for the symmetric (black, open triangles) or asymmetric (red, full squares) vertices. The arrow and dashed circle indicate a small magnetization plateau region, corresponding to the predicted region between $B_{c1}$ and $B_{c2}$ of Fig. 1 (dashed line), between the first and second macrospin reversal.

Figure 8

Experimental FMR spectra for (I) symmetric (thickness 20 nm), (II) symmetric (thickness 15 nm), and (III) asymmetric vertices. The arrow shows the direction in which the magnetic field was swept. Note the close correspondence with the calculated results of Figs. 3 and 4 (where the same labels indicate the corresponding measured curves).