Ferromagnetic resonance spectra of permalloy nano-ellipses as building blocks for complex magnonic lattices

Mesoscopic segments having an elongated (e.g., elliptical or barlike) cross section have been used to assemble interconnected lattices such as those used in artificial spin ice studies, which is a topic of considerable recent activity.^1–4 The magnetic response of the individual segments themselves is also of interest, and the case of permalloy ellipses has been studied via Brillouin scattering for magnetic fields along the in-plane principal axes^5–8 and via the time-resolved magneto-optic Kerr effect (TR-MOKE).⁹ We recently completed a study of the microwave response of threefold, 120°-symmetric, clusters of such ellipses¹⁰ and their angular dependence;¹¹ our future intention is to incorporate these vertices into honeycomb or Kagome spin-ice lattices. But prior to this, it is important to characterize the behavior of the isolated elements used to assemble these structures, so as to examine the extent to which the responses of the individual ellipses govern the overall response of the clusters and, ultimately, lattices assembled from them. In so doing, we can advance ferromagnetic resonance (FMR) as a tool that complements magnetic force microscopy and static magnetization studies, where the latter only probes the net magnetization of a structure.

Here, we report the angular dependence of in-plane FMR measurements using a vector network analyzer (VNA) on 15-nm thick permalloy nanoellipses with lateral dimensions of 500 × 200 nm² in the range of 2–10 GHz. Our discussion consists of three parts: (i) the observed angular dependence of the spectra of single nanoellipse arrays, (ii) a comparison of these data with micromagnetic calculations using the dynamical matrix method (DMM) where excellent agreement with the experiment is found, and (iii) a comparison of the single ellipse data with that of three- and fourfold nanoellipse clusters, showing that, semiquantitatively, the latter can be seen as a “superposition” of the single ellipse data.

Most of our devices consist of a square array of well separated (lattice constant = 1880 nm) permalloy nanoellipses patterned over the central strip of a coplanar waveguide (CPW). A limited number of samples were also fabricated with the ellipses between signal and ground lines. It has earlier been shown that having metallic contact with the underlying guide results in maximal coupling, thereby achieving high overall sensitivity.¹² In the CPWs, the central line has 20 μm width, and the two ground lines have 40 μm width. The spacing between the central line and each ground line is 8 μm. The devices are fabricated by the following process: First, the pattern for the CPWs was defined using a laser writer followed by electron beam evaporation of 5 nm of Ti and 120 nm of Au on an underlying intrinsic Si substrate having a 300 nm SiO₂ layer; the latter insures electrical isolation of the resulting guide elements. In the next step, a 15-nm thick permalloy (Py, Ni₈₀Fe₂₀) nanoellipse array was fabricated by electron beam lithography and electron beam evaporation using a lift-off process. The thicknesses of the deposited materials were monitored by a quartz crystal microbalance during the evaporation: the rates were ∼0.2 Å/s for Ti, ∼1.4 Å/s for Au, and ∼0.4 Å/s for Py. The pressure in the chamber during the metal depositions was ∼3 × 10^–7Torr.

The probe station in our instrumental setup cannot be rotated. Our goals in the present work included: (i) the angular dependence with respect to the angle between the ellipse axis and the static magnetic field direction, and (ii) studies of the polarization dependence, in which the microwave field is perpendicular to or parallel to the static magnetic field direction. To address these two behaviors, we fabricated two different types of CPWs: one set for which the static field was applied parallel to the guide axis and a second set in which it was perpendicular.

Figure 1 shows a schematic representation of the two geometries used for the polarization measurements. Figure 1(a) shows configuration 1 where the static field is parallel to the guide axis. For the case shown here, the Py ellipses were patterned with their long axes perpendicular to the guide axis. The angle between the magnetic field and the ellipse axis is designated as θ_H; here, θ_H = 0°.

Figure 1(b) shows configuration 2 where the static field is perpendicular to the guide axis. For the case shown here, the Py ellipses were patterned with their long axes parallel to the guide axis. The angle between the magnetic field and the ellipse axis is also designated as θ_H; here, θ_H = 90°. Although we designate the orientation of the ellipses as θ_H for the two configurations, they correspond to the same field direction; however, the direction of the dynamic field differs by 90° between the configurations.

To observe the angular dependence of the Py ellipses, multiple devices were prepared oriented at θ_H = 0°, 15°, 30°, 45°, 60°, 75°, and 90° relative to the guide axis. Scanning electron microscopy (SEM) images of devices in the first configuration are shown in Figs. 2(a)–2(g). Here, the microwave field is perpendicular to the static field [red arrow in Figs. 1(a) and 2]. To explore the polarization dependence, seven additional samples were prepared in configuration 2 where the CPW axis was perpendicular to the magnetic field, as shown in Fig. 1(b). For these experiments, the ellipses were also oriented at 0°, 15°, 30°, 45°, 60°, 75°, and 90°, with the magnetic field along the y axis [blue arrow in Figs. 1(b) and 2], parallel to the rf driving field.

To probe the dynamic magnetic response of the ellipses, we performed broadband FMR measurements of the transmission parameter S₂₁ at a nominal microwave power of 0 dBm using a vector network analyzer (VNA).¹³ The microwave absorption spectra were recorded using the following steps: First, the magnetic field was set at +3000 Oe, and the frequency swept between 2 and 10 GHz to establish a baseline that was subtracted from the data gathered at all other fields. This field is sufficient to erase any prior history and establish a well-defined starting point for measuring the subsequent responses. In the next step, frequency sweeps were then carried out between 2 and 10 GHz for discrete magnetic fields ranging between +900 and –900 Oe in steps of 10 Oe.

The theoretical calculations were performed using the dynamical matrix method (DMM).^14,15 The DMM is based on the solution of the Hamilton equations for the precessional motion through the diagonalization of a dynamical matrix, which contains the second derivatives of the energy density with respect to the angular coordinates; all the contributions to the energy of the system are computed as interactions among the magnetic moments of the elemental (discretization) cells: the dipolar interaction (through the computation of the specific demagnetizing tensor corresponding to the geometry of the sample), the exchange interaction (among nearest neighbor elemental cells), and the Zeeman interaction (between the applied field and the magnetic moment of each elemental cell).

Advantages of the method are that a single calculation provides all the independent modes allowed by the mesh through which the sample is discretized (i.e., equal to the number of micromagnetic cells), independently of their symmetry and strength. Conversely, full simulations obtain results that depend on characteristics of the excitation (e.g., the polarization and profile) and, hence, require separate runs for each case, together with extended simulation times to detect weakly coupled modes with the same precision.^16,17 Moreover, if periodic boundary conditions are included, the calculation time is the same as for a single particle; conversely, in full micromagnetic simulations, it is often necessary to include several primitive cells to accurately simulate the full periodic system, with a dramatic increase of the computation times. The computation times strongly depend on the system size (i.e., the number of active elemental cells) and the computer used. In our case, with an 8-CPU Intel® Xeon® E5462, the calculations typically run for approximately 300 s (3150 elemental cells).

The following parameters of permalloy were used: saturation magnetization, M_s= 650 kA/m; exchange stiffness parameter, A = 1.00 × 10⁻¹¹ J/m; gyromagnetic ratio, γ = 185 rad GHz/T; and anisotropy coefficient, K = 0 (no anisotropy).

Figure 3 shows false color images of the FMR spectra for the case where the dc magnetic field is aligned along the x axis, which is parallel to the CPW axis. The red solid lines show the results of the theoretical calculations and will be described in detail below. We again note that θ_H is the angle between the applied static magnetic field and the major axis of the ellipses, as shown in Fig. 3(a). At θ_H = 0°, the magnetization initially remains parallel to the field as it is reduced and the FMR frequency falls. When the field passes through zero,¹⁸ a metastable regime is entered in which the field and magnetization are oppositely aligned. At approximately −240 Oe in Fig. 3(b), an instability is encountered, where the magnetization abruptly switches by 180°, so as to again align parallel to the external field, but now in the –x direction. When this realignment occurs, the FMR frequency abruptly increases to about 7 GHz; note this value is essentially identical to the value it had at +240 Oe.

The behavior at θ_H = 15°, 30°, and 45° shown in Figs. 3(c)–3(e) is quite similar to that at 0°; however, the region of metastability is reduced. At θ_H = 60° in Fig. 3(f), there is a field range where no FMR signal is observed ranging from about 0 to −225 Oe. This occurs because the static magnetization in this region has rotated such that it largely lies parallel to the H_f of the CPW, in which case the torque in the Landau-Lifshitz equation is greatly diminished.

Figure 3(h) shows the data θ_H = 90° that display some additional features. At high fields, the magnetization is aligned with the short axis; this is the hard direction and corresponds to what is termed a leaf state.^10,19 We find that the 90° saturation field occurs at 640 Oe: at this field value, the magnetization parallel to the applied field is larger than 95% with respect to the saturation value, while the perpendicular one is lower than 10%.

As the field is reduced from +900 Oe, the FMR frequency falls since the magnetization initially stays aligned with the field. This holds down to a point where this alignment is no longer stable, at which point the magnetization gradually rotates in the easy axis, to one side or the other; this occurs at approximately +450 Oe where a “cusp” occurs, after which the FMR frequency rises. As the field is reduced to zero, the magnetization rotates so as to lie parallel to the dynamic H_f, where the torque arising from the H_f vanishes; the signal then vanishes accordingly [Fig. 3(h) from about +200 Oe to −200 Oe]. A black arrow at −380 Oe indicates a discontinuous behavior that approximately coincides with a feature in the calculations, to be discussed next [see Fig. 3(h)], that is assigned to a soft mode.^8,20

The red solid lines in Fig. 3 show the results of the theoretical calculations for the nanoellipses with different angles corresponding to the experimental data, when the applied dc magnetic field lies along the CPW axis, i.e., the x axis. The calculations were performed using the DMM as described above, and the single nanoellipse was discretized with 5 × 5 × 15 nm³ square-based elemental cells. Overall, the experimental data shown are in good agreement with the theoretical calculation.

Figure 4 shows the data for our second configuration in which the CPWs have been rotated by 90° and where the dc magnetic field is aligned along the y axis that is perpendicular to the guide axis. We note again that θ_H is the angle between the applied static magnetic field and the major axis of the nanoellipse, as shown in Fig. 4(a). However, the intensities differ since the static and dynamic fields are now aligned. As an example, the signal in the missing field range of Figs. 3(f)–3(h) is now particularly intense in Figs. 4(f)–4(h).

In Fig. 4(h), in particular, the static magnetic field is along the minor axis of the ellipses. However, the magnetization favors the long axis at low fields. The dynamic field, which is here parallel to the static field, is then perpendicular to the magnetization yielding maximal torque, and hence, the signal is large. The field also lies along the minor axis in Fig. 3(h), and again the magnetization is perpendicular to the static field. However, here the dynamic field is parallel to the magnetization and the torque vanishes, consistent with the disappearance of the signal at low static fields. In Fig. 4(b), the field lies along the major axis of the ellipses; however, the dynamic field is again in the same direction leading to a vanishing torque. In Fig. 3(b), the magnetization and field are again parallel (or antiparallel) to the static field, but the dynamic field is perpendicular and we have a large signal.

The DMM yields the full angular range of the modes. Experimentally, however, we see that, depending on the orientation between the dynamic in-plane field and the magnetization, one can highlight different regions of the curves. This suggests that if the dynamic field were to be perpendicular to the plane (i.e., out-of-plane), then the measurement would reproduce the full angular dependence. To test this hypothesis, we fabricated a sample with the ellipses patterned in the gap between the signal and the ground lines [see Fig. 5(a)]. Here, the field is largely perpendicular to the plane of the ellipses although due to the limited space their total number is smaller; this, together with the lack of direct metallic contact with the wave guide (known empirically to enhance coupling, see above), results in overall weaker signals.

Figure 5 shows the data for the samples prepared (the patterning of the 90° sample failed); here the applied magnetic field is parallel to the axis of the CPW (the y axis). Note that the signal amplitude for the 60° data shown in Fig. 5(f) is relatively uniform as the field is swept and the magnetization rotates in the sample plane. This is to be contrasted with the 60° data shown in Fig. 3(f), where the field is also parallel to the guide axes, but now there is no “missing region” in the vicinity of −150 Oe. A similar contrast is seen at 75° although the overall amplitude is much weaker with this sample.

The motivation for the present study is to identify whether individual building blocks, such as ellipses or bars, could be useful for the understanding of the overall dynamic response of networks assembled from them. As examples, Figs. 6(e) and 6(f) show SEM images of 120° symmetric clusters formed from three nanoellipses having the same dimensions as those studied here and where the distance of the closest approach is ∼60 nm. The orientation in Fig. 6(f) is rotated by 90° counterclockwise from Fig. 6(e).

The FMR spectra for the 120° (threefold) cluster with the magnetic field parallel to the long axis of an ellipse is shown in Fig. 6(a). On the other hand, Fig. 6(c) shows the sum of the spectra of the single ellipses oriented at 0° [taken from Fig. 3(b)] and 60° [from Fig. 3(f)]. Note that the spectrum shown in Fig. 6(a) closely corresponds to a superposition of Figs. 3(b) and 3(f) with the branch involving two symmetry-equivalent ellipses having an enhanced intensity.

Figure 6(b) shows the spectrum for the 120° cluster with the field perpendicular to the long axis of an ellipse. On the other hand, Fig. 6(d) shows a sum of the spectra shown in Figs. 3(h) and 3(d). Again, we see that the spectrum of the cluster closely corresponds to a superposition of that for the individual nanoellipses with the intensities higher for the branches involving two symmetry-equivalent ellipses.

Figures 7(f) and 7(g) show a 90° symmetric cluster formed from four nanoellipses with the same dimensions for the magnetic field at 0° and 45°, respectively. Figure 7(a) shows the spectrum of the 90° (fourfold) symmetric nanoellipse clusters with the field parallel to a principal axis, while Fig. 7(c) shows the sum of the spectra shown in Figs. 3(b) and 3(h). Again, the superposition “principle” approximately holds. In Fig. 7(a), three branches are observed. Branch 1 corresponds to Fig. 3(b); however, the branch is shifted about 1 GHz to lower frequency. Branch 2 corresponds to Fig. 3(h); however, in low negative fields (indicated by a black arrow), the mode increases with a decreasing magnetic field. Branch 3 in the lowest frequency regime is not observed in the sum of spectra of the single nanoellipses in Fig. 7(c). Finally, Fig. 7(b) shows the spectrum of the fourfold clusters with the field rotated by 45°, while Fig. 7(d) shows the 45° spectrum taken from Fig. 3(e); note the close correspondence and that, by symmetry, there is now only a single branch, although the spectrum is shifted to a lower frequency by about 1 GHz.

We have performed a broadband study of the angular dependence of the ferromagnetic resonance (FMR) spectra of permalloy nanoellipses having dimensions of 500 × 200 × 15 nm³. By orienting the wave guide axis parallel and perpendicular to the dc magnetic field, the dependence of the coupling on the microwave polarization is exhibited. It is demonstrated that a detailed understanding of the behavior of these individual ellipses qualitatively explains the response of clusters of ellipses although with some quantitative shifts that occur due to the interaction between ellipses within the cluster. We propose that this superposition principle can be applied to more complex structures involving lattices formed from such clusters, such as artificial spin ice structures.

Work at Northwestern, including experimental design and FMR measurements, was supported under NSF Grant No. DMR 1507058. Work at Delaware, including data analysis and manuscript preparation, was supported under NSF Grant No. 1833000. Device fabrication was carried out at Argonne and supported by the U.S. Department of Energy (DOE), Office of Science, Materials Science and Engineering Division. Lithography was carried out at the Center for Nanoscale Materials, an Office of Science user facility, which is supported by DOE, Office of Science, Basic Energy Science under Contract No. DE-AC02-06CH11357.