Spin dynamics in permalloy nano-ellipses for honeycomb and square lattices

The exploration of spin dynamics in artificial spin ice (ASI) systems has recently gained much attention due to their potential applications in magnonics-based devices and data storage concepts.^1–10 Strong dipolar interaction between neighboring elements in ASI results in frustration and a highly degenerate energy landscape. The dynamic behavior of these systems is linked to their magnetic states.¹¹ In particular, the exact configuration of the nano-ellipses making up the ASI lattice affects the dipole interactions at the vertices and thus determines the spin dynamics.^12,13 It was previously shown that a localized mode exists in the vertex region of a kagome ASI comprising contacting nano-ellipses clusters, while this mode was absent in the non-contacting system. However, the influence of the ASI lattice structure on this localized mode has not yet been studied.^12,13

Here, we present a study of the ferromagnetic resonance (FMR) spectra of non-contacting or contacting permalloy nano-ellipses and compare the effect of honeycomb and square lattice structures on the dynamics. The lattice structures were patterned using electron-beam lithography and deposited using electron-beam evaporation, followed by a lift-off process. The ASI structures were patterned directly on the top of the signal line of a coplanar waveguide, which was previously demonstrated to provide sufficiently strong coupling of the magnetic system to the driving microwave magnetic field.¹¹ The dynamic response of the nanostructured arrays is studied using a broadband ferromagnetic resonance technique using a vector-network analyzer. The experimentally obtained spectra are compared to micromagnetic simulations using mumax³.¹⁰ The micromagnetic simulations show an excellent agreement with the experimentally detected spectra. Our results show that a careful design and analysis of the static and dynamic response in combination with detailed simulations enable the realization of tailored magnonic device concepts. Furthermore, our findings demonstrate that the complex dynamics of artificial spin-ice structures can be understood from the fundamental properties of their individual building blocks.

The devices for VNA-FMR (vector network analyzer-ferromagnetic resonance) measurements consist of permalloy (Ni₈₀Fe₂₀) nanostructured arrays patterned on coplanar waveguides (CPW) as shown in Fig. 1. The 50-Ohm matched CPW has a central signal line and two ground lines that are designed to achieve maximal coupling and sensitivity in broadband FMR measurements. The CPW was defined by photolithography followed by lift-off of a 100 nm Au film on a 5 nm Ti film deposited by e-beam evaporation. The permalloy nano-ellipses arrays are defined in honeycomb or square lattices, which were directly patterned on top of the signal line of the CPW using e-beam lithography followed by deposition of a 15-nm permalloy film using e-beam evaporation and lift off. Each ellipse is 500 nm long and 200 nm wide as shown in the scanning electron microscopy (SEM) images in Fig. 2. A double poly-methyl methacrylate (PMMA) layer was used for the e-beam lithography process to ensure a reliable lift-off after permalloy deposition. Deposition rates, monitored by a quartz crystal microbalance, were ∼0.2 Å/sec for Ti, ∼1.4 Å/sec for Au, and ∼0.4 Å/sec for permalloy. The base pressure during all evaporations was ∼3 x 10⁻⁷ Torr.

Figure 2 shows SEM images of the permalloy nano-ellipse lattice structures. In order to study the effect of dipolar interaction in the contacting region of the nano-ellipse vertex, we prepared non-contacting and contacting lattices. Figures 2(a) and (b) show non-contacting and contacting honeycomb lattices, respectively; non-contacting square and contacting square lattice structures are shown in Figs. 2(c) and (d), respectively. One of the three ellipses making up a vertex in the honeycomb lattice is aligned parallel to the x-axis, while one of the two ellipses making up a vertex in the square lattice is parallel to the x-axis.

The experimental results are compared to micromagnetic simulations performed using MuMax³.¹⁰ The ASI is simulated using a single vertex and periodic boundary conditions, with the same geometry and dimensions as the nanofabricated sample. The number of cells, and hence the cell size, is different for different structures. For the honeycomb lattice where the x and y size of the unit cell is different, different numbers of cells are defined to keep the x and y cell sizes almost the same.¹⁴ However, all cell sizes are defined to be smaller than exchange length of permalloy 5 nm (∼4.1 to 4.8 nm) for all structures. First, the equilibrium state of the magnetization is found and then the magnetization dynamics is simulated by applying a sinc field pulse with a cutoff frequency of 50 GHz. The average magnetization components are saved for a total time of 20 ns following the excitation. The eigenmodes of the system are found by performing a fast Fourier transformation on the time traces. We obtain the field-dependent intensity spectra by performing the FFT on the average z-component of the magnetization in the field range of +1200 to −1200 Oe with 40 Oe steps.

Broadband FMR measurements using a VNA were performed to study the dynamic spin responses of the permalloy nano-ellipses lattices. The VNA is connected via pico-probes to the CPW for recording the microwave absorption spectra. The S₂₁ transmission parameter is measured at a nominal microwave power of 0 dBm. The following VNA measurement procedure was used: firstly, the external magnetic field H is set to +3000 Oe to saturate the magnetization of permalloy nano-ellipses to the direction of the external magnetic field, and the frequency is swept between 2 and 12 GHz. This establishes a baseline from which data at other fields are subtracted. Then, the frequency is swept over the same range (2 to 12 GHz) at fixed magnetic fields between +1200 Oe to −1200 Oe in 10 Oe intervals. All measurements are performed with an in-plane external magnetic field directed along the signal line of the CPW and at room temperature.

Figure 3 shows false-color-coded images of the experimental FMR results. The main resonances in the spectra shown in Fig. 3 are the fundamental modes of each nano-ellipse. The frequency difference between different modes at the same magnetic field depends on the angle between the major axes of the nano-ellipse and the applied external magnetic field together with the demagnetizing fields.¹³ 

The experimental FMR spectra of the non-contacting honeycomb lattice structures, as shown in Fig. 3(a), show that two dominant modes are clearly detected. The mode at the highest frequency shows slightly smaller intensities in the FMR spectra. This mode is due to nano-ellipses whose major axis is parallel to the applied magnetic field. Note the lower frequency mode has larger intensity. This can be understood by the fact that two nano-ellipses in the honeycomb unit cluster, those oriented at ±60° with respect to the applied magnetic field, contribute to the signal. The experimental FMR results shown in Fig. 3(a) are in good agreement with the micromagnetic simulation shown in Fig. 4(a).

Figure 3(b) shows the experimental FMR spectra for the contacting honeycomb lattice. The main difference in the FMR spectra between contacting and non-contacting honeycomb lattices is the existence of additional modes which occur at the lowest and highest frequencies. We previously demonstrated that the lowest frequency mode in the FMR spectra arises from a localized mode in the vertex region, Fig. 3(b). This localized mode is the so-called vertex center mode (VCM).¹³ In stark contrast, this mode is absent in the non-contacting lattice, as shown in Fig. 3(a). Both experimental and simulated FMR spectra for the contacting honeycomb lattice show a much stronger VCM signal at the lowest frequency compared to the spectra found for the non-contacting honeycomb lattice. The VCM exists in a larger area when the three nano-ellipses merge at the inner ends. This is different from localized end mode in a narrow area at the inner ends of the non-contacting honeycomb unit cluster. When the three nano-ellipses merge at the inner ends, the VCM resides over an extended area, unlike the end mode localized in narrow area at the inner ends of the non-contacting honeycomb unit cluster. This is due to exchange interaction which increases the coherence of the spin oscillation to a wider area at the vertex center. Multiple branches above 10 GHz at higher magnetic fields, that are barely visible in the simulation in Fig. 4 (b), correspond to higher order backward modes.

In the experiments, the square lattice structures, both non-contacting and contacting, show two primary curves and reverse their magnetization between −150 and −200 Oe, as shown in Figs. 3 (c) and (d). The mode at the highest frequency arises from the nano-ellipses with their major axis aligned parallel to the applied magnetic field. Overall, the dynamic behaviors for both non-contacting and contacting square lattice shown in Figs. 3 (c) and (d) are very similar. However, different behaviors are found in the low-field regime between −500 Oe and +500 Oe due to contacting inner ends of the nano-ellipses in the square lattice. It was previously shown that these elements, whose major axes are perpendicular to the direction of the applied magnetic field, exhibit a characteristic W-shape curve in the spectra at low frequency.¹⁵ However, the expected behavior in the range −500 Oe and +500 Oe was not clearly observed in the experiments, see Fig. 3(c), likely due to the microwave absorption intensity from the primary ellipse being much stronger in this magnetic field range. When the inner ends of the nano-ellipses merge in the square lattice structure, i.e., in the contacting square lattice structure, the middle parts of the W-shape behavior between −500 Oe and + 500 Oe are clearly detected, as shown in Fig. 3 (d). This corresponds to a stronger low frequency signal in the simulation, as shown in Fig. 4 (d). Since the region with near-zero magnetization is reduced, as can be seen in Fig. 5(c), the relative intensity of the spectrum from the nano-ellipses aligned perpendicular to the applied magnetic field was increased.

The zero-field static magnetization alignment obtained from micromagnetic simulations after the system has been saturated is shown in Fig. 5. The color code represents the vertical magnetization component which is the magnetization component perpendicular to the applied magnetic field direction. In the honeycomb lattice structures, flux lines pass through the vertices from right to left and the net magnetization in the vertical direction is zero as shown in Figs. 5(a) and (b). In the square lattice structures, a non-zero magnetization alignment in the vertical direction when H = 0 is shown in Figs. 5(c) and (d). Remarkably, magnetizations at the both ends of the nano-ellipses with non-zero angles between their major axes and the applied magnetic fields have changed close to the zero magnetization at the merging vertices in the two contacting nano-ellipses lattice structures in Figs. 5(b) and (d).

We have performed experimental and theoretical studies of artificial spin ice systems consisting of non-contacting honeycomb and square lattice structures and compared the results with their contacting counterparts. The broadband ferromagnetic resonance measurements of the permalloy arrays of nano-ellipses are compared to micromagnetic simulations using mumax³. Overall, the experimental results show a good agreement with the simulations. The FMR spectra from the non-contacting and contacting lattices that we studied are important to understand the spin dynamics of the artificial spin ice systems.

Work at Korea University of Technology and Education was supported by the new professor research program of KOREATECH in 2021 and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1G1A1092937). Northwestern participation involved support from the U.S. Department of Energy through grant DE-SC0014424. Research at the University of Delaware including micromagnetic simulations supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0020308.

The authors have no conflicts to disclose.

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