Spin wave spectral probing of degenerate microstates in building-block of square artificial spin ice

Square artificial spin ice (S-ASI) is the 2-dimensional realization of spin ice behavior observed in 3-dimensional oxide pyrochlore compounds such as Dy₂Ti₂O₇, Ho₂Ti₂O₇, etc.¹ Such designer nanostructured materials offer possibilities to investigate variety of fascinating phenomena such as physics of frustration,² emergent magnetic monopoles,³ collective magnetization dynamics of interacting spins,⁴ and phase transition,⁵ etc. Recently, there is a growing interest in the spin wave behavior in S-ASI due to multiple macroscopically degenerate microstates at the ground state. These microstates offer a route to manipulate spin wave excitation which make these structures a promising candidate for reconfigurable magnonic crystals.^6,7 Moreover, possibilities of locally changing the magnetic behavior of individual nanostructures, such as by the stray field at the magnetic tip of magnetic force microscopes, allows tuning of magnonic band structure of such systems. Indeed, recent studies indicate that ASI can be conceived as analogs of magnonic crystals with tunable band structures offering useful functionalities for applications such as in logic devices.^8–10 Therefore, these structures received a great deal of consideration due to their potential in reprogrammable magnonic crystals,^6,7 computation, data storage,¹¹ Spin logic gate,^8–10 microwave filter,¹² etc.

In particular, experimental and theoretical studies have been performed to understand the magnetization dynamics in S-ASI and it is found that spin wave modes or magnons excited in the systems are affected by the defects,¹³ thickness of the nanoislands,¹⁴ topological excitations (emergent magnetic monopoles),¹⁵ etc. Recent studies in this direction show the possibility of realizing spin wave channels,¹⁶ manipulation of spin wave response,^17,18 influence of vertex region¹⁹ and reversal order at the vertices,²⁰ etc.^21,22 However, the question of how the underlying magnetic state or the dipolar coupling among the nanostructures affects the spin wave behavior in such ASI systems is still far from being properly understood. In this work, we report the results of investigation on how spin wave modes are effected due to the magnetization reversals of single-domain nanostructures obtained by performing micromagnetic simulations. We also investigated the influence of different magnetic microstates evolved due to reversals on the spin wave properties of the system. Our results show that the overall spin wave behavior of individual vertices of S-ASI with closed edges can be understood from the studies of a square shaped ring of dipolar coupled nanomagnets. Considering such rings as building blocks of individual S-ASI vertices, we find that parallelly placed islands after the magnetization switching show unique signature of low frequency spin wave modes localized at the switched islands. Hence, our results can be used to probe the evolution of microstates in dipolar coupled ring type structures.

For the micromagnetic simulations, we used finite difference discretization based GPU-accelerated open source software, MuMax3.²³ Here, Landau-Lifshitz equation (Eq. (1)) is employed to calculate the evolution of reduced magnetization $m⃗(r⃗,t)$⁠.

with γ_LL the gyromagnetic ratio (rad/Ts), α the dimensionless damping parameter, and $B⃗eff$ the effective field (T). Here, $B⃗eff=B⃗ext+B⃗demag+B⃗exch+B⃗anis+⋯$ where $B⃗ext$⁠: externally applied field, $B⃗demag$⁠: magnetostatic field, $B⃗exch$⁠: Heisenberg exchange field, and $B⃗anis$⁠: anisotropy field.

For the micromagnetic simulations, ring type structures consisting of nanoislands of Ni₈₀Fe₂₀ of elliptical cross-section with dimensions 300 nm × 100 nm × 25 nm and lattice constant (edge-to-edge distance) 150 nm (see Fig. 1) are discretized in cuboidal cells of 5 nm length which is less than the exchange length (λ_exch ≈ 5.3 nm) of permalloy.²⁴ Experimentally reported value of saturation magnetization M_sat = 8.6 × 10⁵ A/m, exchange stiffness constant A_ex = 1.3 × 10⁻¹¹ J/m, and damping coefficient α = 0.5 for Ni₈₀Fe₂₀ are used throughout the study.²⁵ For dynamic simulations, the damping coefficient was changed from 0.5 to 0.008, for prolonged precession of weak modes.²⁶ 

In order to understand the influence of evolved microstates on the spin wave behavior, we first investigated the magnetization reversal behavior of the four different structures, involving single ring to four rings. By minimizing the total energy using Runge-Kutta (RK45) method at an external field of every 2 mT within the range of ±300 mT applied along x-axis, switching behavior and corresponding microstates are obtained. Fig. 2 shows the hysteresis loops for the four structures where two switchings for all the structures are identified as S1 and S2. For the studies of spin wave dynamics, an additional square time-dependent pulse of magnitude (B_max) of 3 mT with rise time (t_rise), duration (t_dur), and fall time (t_fall) of 20 ps each, was applied along z-direction. After applying the transient pulse field, magnetization dynamics was observed up to 4 ns at every 10 ps time gap. Spin wave (sw) spectra are then calculated by performing fast Fourier transformation (FFT) on reduced magnetization $m⃗(r,t)$⁠. Corresponding power and phase profiles of the identified sw modes in the calculated spectra are mapped by performing FFT of time evolution of magnetic moment at each unit cell of the discretized geometry.

In S-ASI and other ring type structures, arrangement of magnetization configuration in constituting nanoislands decide the microstates present in these systems. Switching the magnetization direction in any constituting nanoisland transform their previous microstates. To investigate the effect of these emerging microstates on their respective spin wave dynamics, applied external field values are taken according to the hysteresis loop. The results shown in Fig. 2(a-d) indicate the microstate’s evolution for structures displayed in Fig. 1. For single square ring, nanoisland’s magnetization switchings lead to the evolution of microstates from horseshoe (124 mT) to micro-vortex (-136 mT) to horseshoe (-144 mT) [Fig. 2(a) insets]. These switchings in the nanoislands appeared as steps in the hysteresis curve of the respective structure while the slope of the curve denotes the magnetization behavior in the vertical nanoislands for which easy-axis ⊥ B_ext. Observed evolution of microstates can be understood in terms of the total energy $E$ of the magnetic system which depends on relative orientation of magnetization (⁠$Ehead-to-tail$ < $Ehead-to-head$⁠) direction in neighbouring islands.²⁴ Since, square ring and other extended ring structures are first saturated along the x-axis, therefore, after removing the field all the horizontal islands remain aligned along +ve x direction (see Fig. 1) due to high shape anisotropy and thus exhibit horseshoe microstate at remanent. Similarly, other ring-type structures follow microstate path which ensure maximum head to tail configuration to retain it’s minimum energy state. When the number of rings increases beyond 2, onion microstate appeared in the dynamics. Square rings exhibiting onion state at remanent follow onion → horseshoe → onion microstate’s path. Also, results show that at remanence both triple and quadruple ring conserve one of the degenerate type II states at the vertex.²⁴ 

To identify the spin wave modes corresponding to individual islands in selective microstates, we have first examined the spin wave modes excited in a single (individual) nanoisland in saturation. Fig. 2(e) shows the spin wave spectra in horizontal and vertical nanoisland with associated mode power profiles above respective spectral peaks. For the horizontal nanoisland, bulk mode at 18.28 GHz and edge mode at 13.2 GHz were observed as reported in the literature²⁷ which indicate the reliability of the simulation performed for other structures. In vertical island, edge mode and bulk mode appear at 4.56 GHz and 8.37 GHz, respectively [see Fig. 2(e)]. An additional high-frequency mode of reduced amplitude appears at 10.15 GHz which carries the nature of higher order end mode, possibly due to the localization of exchange dominated spin wave modes in the region of inhomogeneous internal field. For vertical island, easy axis makes 90° angle to the applied field (B_eff) which results in the increase of the demagnetization field which consequently reduces the effective field in the nanoisland and thereby decreasing the mode frequency (ω_L = γ_LLB_eff). Thus, we observe lower-frequency modes for vertical islands and relatively high-frequency modes for horizontal island at high bias field [see Fig. 2(e)].

We have then systematically investigated the sw modes of all the ring type structures for four different magnetic configurations (see Fig. 3) which include saturated state ( + ve x-axis), remanent state, microstates formed after first and second switching (S1 and S2) as observed from the hysteresis loop [Fig. 2(a-d)]. We observed that all the ring type structures including S-ASI in saturation show a strong correlation in spin wave spectra which is evident from the results shown in Fig. 3(a). In saturation, five modes found to appear for all structures with three strong modes corresponding to outer-end mode (OE1, 4.57 GHz), bulk mode (B, 8.38 GHz) confined in vertical islands, and bulk mode (B, 18.27 GHz) confined in horizontal islands [see Fig. 3(a) and 4(a)]. Corresponding power profiles of excited modes are similar for all the structures therefore power profile of only single square ring is shown in Fig. 4(a). In this case, effect of magneto-static interaction to affect the magnetic moment distribution in neighbouring islands appears to be insignificant. It can be clearly seen from the sw spectral behavior as it shows the signature of all the five sw-modes localized in individual horizontal and vertical islands at saturation [see Fig. 2 and 3(a)].

Now, for the microstates available in remanence, each contributing island has magnetization along it’s easy axis. Therefore, both horizontal and vertical islands behave similarly under uniform transient field excitation which results in only two modes [Fig. 3(b)]. Here, lower frequency mode ( ≈ 6.09 GHz) corresponds to the outer end mode and higher frequency mode ( ≈ 10.41 GHz) corresponds to bulk mode. In remanence, the bulk mode appears with a shoulder peak or shows bifurcation in S-ASI. We have analyzed this behavior as a function of island’s separation (lattice constant: 300 nm to 1000 nm) and our results suggest that either bifurcation or shoulder peak is not present (not shown). Hence, this behavior is due to magnetostatic interaction among the islands. However, power profile corresponding to lower-frequency outer end mode shows power variation at the corners which is found to carry the information of the net magnetic charge at the junction. According to dumb-bell model, each single domain nanoisland can be considered as a dipole of magnetic charge ±Q_M separated by a certain distance (length of the nanoisland). For single square ring, perpendicularly placed islands can have the macrospin configuration of head to head/tail to tail with net magnetic charge (ΣQ) of ±2Q_M and/or head to tail/tail to head with ΣQ = 0. Accordingly, net magnetic charges of magnitude ≈ ±2Q_M appear near lower horizontal nanoisland at remanence [Fig. 1(a)]. Remaining three nanoislands have closed magnetic flux line distribution with zero magnetic charge at the junction. Close observation of the power-profile shown in Fig. 5(a) denotes that power of the outer end modes is mostly concentrated in the corners with nonzero net magnetic charge. Line profiles of power through the center of the horizontal islands shown in Fig. 5(b) show that power associated with OE mode drops to 97% in the region of zero net magnetic charge as compared to charged region. However, the power distribution profile exactly matches in all the corners [Fig. 5(c)].

Finally, for the case of S1 and S2 state, sw power spectra [Fig. 3(c,d)] show extensive variation in spectral nature as well as the corresponding mode profiles [Fig. 4(b-e)]. A weak mode localized in unswitched horizontal island appears at 4.5 GHz. As the frequencies of this mode and the OE mode (5.07 GHz) in vertical islands lie close to each other, therefore, we observe hybridization of the modes. However, we observe OE modes are oriented at certain angle with respect to the easy axis. Our analyses suggest that this orientation is directed along the net magnetization direction due to lower effective field region at the edges. In square single ring, high-frequency sw mode (∼ 14.72 GHz) is localized in lower island which is the switched island (observed from micromagnetic simulation, see Fig. 2(a) inset). Thus, high-frequency bulk mode of the horizontal islands show marked power variation with respect to unswitched islands [Fig. 4(b-f)]. This strong power variation is consistent in all the four ring type structures. Therefore, this mode can be considered as a switching identifier mode. The observed variation in sw modes’ frequencies for all the states (Fig. 3) is due to variation in applied external field to access desirable microstates. Also, power spectra shown in Fig. 3 clearly indicate that all the sw modes are excited at the approximately same frequencies for all the four cases. Corresponding power profiles shown in Fig. 4 for the excited sw modes are consistent in all the structures. Thus, our results suggest that studies of the sw behavior of single square ring may provide information on the overall sw mode of S-ASI. Further detailed investigation at higher temperatures, different shapes of nanoislands etc. will be required to ascertain this observation.

In summary, we have studied the spin wave response in four different magnetic configurations for square single ring, double ring, triple ring, and quadruple ring (S-ASI) structure. We have reported the presence of ubiquitous high-frequency switching identifier mode and mapping of the local magnetic charges in terms of power profiles associated with excited outer end modes in remanence. Hence, our results suggest the possibility of employing spin wave modes to probe the switching and evolved microstates in S-ASI and ring type of structures.

We acknowledge High Performance Computation (HPC) facility of IIT Delhi for enabling us to carry out the micromagnetic simulations. N.A. is thankful to the Council of Scientific Industrial Research (CSIR), Government of India for research fellowship. P.D. acknowledges financial support from IIT Delhi through Grant No. MI01687G.

The data that support the findings of this study are available from the corresponding author upon reasonable request.