The effect of material defects on resonant spin wave modes in a nanomagnet

We have theoretically studied how resonant spin wave modes in an elliptical nanomagnet are affected by fabrication defects, such as small local thickness variations. Our results indicate that defects of this nature, which can easily result from the fabrication process, or are sometimes deliberately introduced during the fabrication process, will significantly alter the frequencies, magnetic field dependence of the frequencies, and the power and phase profiles of the resonant spin wave modes. They can also spawn new resonant modes and quench existing ones. All this has important ramifications for multi-device circuits based on spin waves, such as phase locked oscillators for neuromorphic computing, where the device-to-device variability caused by defects can be inhibitory.

lithographically delineated windows opened in e-beam resist. The resists are patterned with electron beam lithography and the nanomagnets are produced by lift-off. The lateral dimensions of these nanomagnets are on the order of 100-300 nm and their thickness is on the order of 16 nm. Note that in Fig. 1, the nanomagnets have "rims", i.e. the thickness is much larger along the periphery than at the center. This type of defect is an aftermath of the lift-off process and is fairly common.
Other types of defects will involve "voids" (missing material) or "pimples" (excess material) at certain locations on the nanomagnet's plane. They are usually caused during metal evaporation. The types of defects that we have studied in this work are shown schematically in Fig. 2. Note that defect type C4 approximates the structure shown in the right panel of Fig. 1, while the other types are representative of the defects shown in the left panel.
Excitation of spin waves in the defect-free and defective nanomagnets. Spin waves are generated in a magnet whenever its magnetization is perturbed by an external agent. They can be excited in a nanomagnet Figure 1. Atomic force micrographs of arrays of Co nanomagnets deposited on a substrate using electron beam lithography, electron beam evaporation of Co on to the patterned substrate, followed by lift-off. The nanomagnets have various defects such as thickness variation along the plane (classified as defects of type C2 and C3), a raised region in the center (classified as defect C5) and cratering or larger thickness along the periphery (classified as defect C4). Reproduced from ref. 30 with permission of the American Physical Society © American Physical Society.

Figure 2.
Classification of defects in an elliptical cobalt nanomagnet of major axis dimension 186 nm, minor axis dimension 180 nm and thickness 16 nm: C0 (defect-free), C1 (hole in the center with diameter 5 nm and depth 12 nm), C2 (thickness variation where one half of the nanomagnet is 3 nm thicker, C3 (thickness variation where one half is 4 nm thicker), C4 (the periphery forms an annulus of width 20 nm and height 10 nm, C5 (thickness variation where a central circular region of diameter 5 nm is 12 nm thicker), and C6 (a through hole of 3 nm diameter at the nanomagnet's center; this nanomagnet dimension is slightly larger with major axis of 190 nm and minor axis of 186 nm). This figure is not drawn to scale. www.nature.com/scientificreports www.nature.com/scientificreports/ in a variety of ways. One common approach is to apply a bias magnetic field in the plane of the nanomagnet and then induce precession of the magnetization around this field with an ultrashort laser pulse. This is easily achieved in a time-resolved magneto-optical Kerr effect (TR-MOKE) and ferromagnetic resonance set-up. The precession spawns confined spin waves in the nanomagnet. In order to study them in the presence of defects, we simulate the following scenario: We consider cobalt nanomagnets in the form of elliptical disks whose major axis dimension is 186 nm, minor axis is 180 nm and thickness is 16 nm. A bias magnetic field is applied along the minor axis. Then an out-of-plane magnetic field pulse of amplitude 30 Oe, rise time 10 ps, and duration 100 ps is applied perpendicular to the nanomagnet's plane to simulate the effect of the laser pulse. This out-of-plane field sets the precession of the magnetization about the bias field in motion.
We track the time evolution of the nanomagnet's magnetization by using the micromagnetic simulator MuMax3 23 which allows us to determine the out-of-plane micromagnetic component M z (x, y, z, t) at every coordinate point within the nanomagnet at every instant of time. The nanomagnet is discretized into cells of dimension 2 × 2 × 2 nm 3 . The cell size in all directions is kept well below the exchange length of cobalt to consider both dipolar and exchange interactions in the magnetization dynamics of nanoscale magnets as well as to accurately reproduce the shapes of the nanomagnets under study. The time step used is 1 ps. The magnetic parameters used for the simulation are: saturation magnetization M s = 1100 emu/cm 3 , gyromagnetic ratio γ = 17.6 MHz/Oe and exchange stiffness constant A ex = 3.0 × 10 −6 erg/cm. These parameters correspond to cobalt nanomagnets. We spatially average M x y z t ( , , , ) z over space to find the out-of-plane magnetization component M t ( ) z as a function of time.
We start the simulation by preparing the magnetic ground state upon applying the bias magnetic field (H) along the minor axis of the elliptical nanomagnet at time t = 0. The initial magnetization is assumed to have been directed along the major axis which is the easy axis. We wait until the micromagnetic distributions reach steady state and the spatially averaged magnetization points in the direction of the applied bias field H along the minor axis. Next, we apply the out-of-plane magnetic field pulse and study the time evolution of the out-of-plane magnetization M t ( ) z (associated with precession of the magnetization around the bias magnetic field) for 4 ns (4000 time steps). Figure 3 shows M t ( ) z versus t for the seven different (defect-free and defective) nanomagnets at three different bias magnetic fields of strengths, H = 650 Oe, 760 Oe and 1000 Oe. We perform a fast Fourier transform (FFT) of each of these "oscillations" to extract the dominant frequencies (frequency peaks) in the oscillation. These are the frequencies of the resonant spin wave modes in the nanomagnet. The frequency resolution in the generated FFT depends upon the total simulation time. Since the simulation time is 4 ns, the frequency resolution is 0.25 GHz. www.nature.com/scientificreports www.nature.com/scientificreports/ Figure 4 plots the Fourier spectra for the seven nanomagnets (nanomagnets with seven different types of defects illustrated in Fig. 2) at three different bias fields. The peaks in these spectra correspond to the frequencies of the resonant spin wave modes in the seven nanomagnets.
There are three interesting features to note in Fig. 4. First, the spectral peaks, which are the frequencies of the resonant spin wave modes, are different in the seven different nanomagnets at the same bias magnetic field. This shows that defects affect the frequencies of the resonant spin wave modes and that has important implications for spin wave microwave generators employing resonant spin waves. Second, the bias field dependence of the resonant mode frequencies (peaks in the spectrum) are sensitive to defects. Thus, the "tunability" of the oscillation frequency of microwave oscillators with a magnetic field is affected by the presence of defects. Third, defects spawn some new resonant modes that are absent in the defect-free nanomagnet. Conversely, defects can also quench resonant modes that are present in the defect-free nanomagnets.    www.nature.com/scientificreports www.nature.com/scientificreports/ lie in the film plane 27 . III) Forward Volume (FV) mode -In the so-called magnetostatic forward volume mode (MSFVM) geometry, the magnetization is perpendicular to the film 28 .  www.nature.com/scientificreports www.nature.com/scientificreports/ In our system of nanomagnets, modes that are similar to the first two types of modes can exist since we apply the bias magnetic field in the plane of the sample. Nothing like the third type of mode can exist. Since the modes in the nanomagnets are not propagating modes, they are not exactly classifiable as DE or BV modes. The generated spin waves get reflected from the boundaries of the nanomagnets and form standing spin-wave modes similar to resonant cavity modes. Hence, we call them resonant modes. Because of their confined nature, instead of assigning wave vector to the modes, we count the number of nodal planes and assign a mode quantization number to the observed modes. The quantization numbers are defined according to whether the quantization axis is along the magnetic field direction (n, in BV geometry) or perpendicular to the field direction (m', in DE  www.nature.com/scientificreports www.nature.com/scientificreports/ geometry). In some cases, the modes are quantized along the azimuthal axis. We name those as azimuthal modes with a corresponding quantization number (m).

Results and Discussion
From Figs 5-11, we see that certain types of defects (C1 and C5) are relatively innocuous and affect the resonant spin wave modes slightly. They do not spawn new modes or quench existing ones. The changes they introduce in the power and phase profiles are also moderate. These types of defects are very tolerable.
Defect C6 is similar to C1, but unlike C1, this is a through-hole which makes it more invasive (the size is also slightly larger). The through-hole spawns a new mode at 650 Oe bias magnetic field and quenches an existing  www.nature.com/scientificreports www.nature.com/scientificreports/ mode at 760 Oe field. It does not alter the power profiles significantly, but affects the phase profiles much more. These types of defects are moderately tolerable, except in applications that require phase sensitivity.
Defects C2 and C3 are associated with thickness variation across one-half of the nanomagnet's surface. These types of defect are found to be extremely invasive and spawn new resonant modes at all magnetic fields. They also alter the power and phase profiles of the resonant modes quite significantly. Thickness variation across a significant fraction of a nanomagnet's surface (an extended defect) is therefore more serious than having localized defects such as a "hole" (C1, C6) or a "hillock" (C5). These types of defects are found to be the most harmful among the ones studied.
Defect C4 is important since it is commonplace in nanomagnets fabricated by electron-beam evaporation of a ferromagnetic metal into a lithographically delineated window in an e-beam resist. Curiously, it is not as invasive as C2 and C3. Like C6, it changes the power profiles slightly, but affects the phase profiles much more. The frequency of the edge mode decrease significantly. It also quenches a quantized mode that appears in the defect-free nanomagnet at the intermediate field of 760 Oe. This type of defect is, again, moderately tolerable, except in applications that hinge on phase sensitivity.
Based on these observations, it appears that maintaining thickness uniformity across a significant fraction of the nanomagnet's surface would be critical in applications that require reproducibility of resonant spin wave power and phase profiles. Expectedly, extended defects have a more serious effect on the resonant spin wave power and phase profiles than localized defects.

Conclusion
Our study has shown that several features of resonant spin wave modes in a nanomagnet (frequencies, magnetic field dependence of the frequencies, number and nature of resonant modes, power and phase profiles) are affected by the presence of defects associated with localized or extended thickness variations in the plane of the nanomagnets. This has serious consequences for many applications that rely on spin wave modes. In the past, it was found that in magnetostrictive nanomagnets, strain-induced magnetization reversal (switching) probability is dramatically affected by the presence of defects 29,30 . Defects are also known to have a serious deleterious effect on the stochastic behavior of low energy barrier nanomagnets that have been proposed for use in stochastic computing 31 . Here, we have found that defects have a dramatic effect on spin wave modes as well. For example, C1 and C6 are slightly different defects and yet the spin wave modes are vastly different in them. This indicates that spin waves are very sensitive to defects and that will cause significant device-to-device variability since the defect morphology will be different in different devices. In single (or few) device applications like magnonic holography 32 or a magnonic gate 2 or spin wave interferometer 33 or modulator 34 , this will not matter much since the number of devices involved is one or few, but in large-scale spin wave "circuits" where numerous devices have to behave in nominally identical manner for overall circuit functionality, the device-to-device variability caused by defects could be debilitating. Spin wave circuits that have little tolerance for variations of spin wave frequencies, or their power distributions, or their phase profiles -e.g. phase locked nano-oscillators for neuromorphic computing 5,22 -are especially vulnerable. Designing these systems for targeted applications in the presence of random defects will be extremely challenging.