Magnetic Field Controlled Transition in Spin-Wave Dynamics in Kagome Artificial Spin Ice Structure
Avinash Kumar Chaurasiya, Amrit Kumar Mondal, Jack C. Gartside, Kilian D Stenning, Alex Vanstone, Saswati Barman, William R. Branford, Anjan Barman
11 Magnetic Field Controlled Transition in Spin-Wave Dynamics in Kagome Artificial Spin Ice Structure
Avinash Kumar Chaurasiya, Amrit Kumar Mondal, Jack C. Gartside, Kilian D Stenning, Alex Vanstone, Saswati Barman, William R. Branford, and Anjan Barman Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block – JD, Sector-III, Salt Lake, Kolkata 700 106, India Blackett Laboratory, Department of Physics, Imperial College London, SW7 2AZ, United Kingdom Institute of Engineering and Management, Sector-V, Salt Lake, Kolkata 700 091, India London Centre for Nanotechnology, Imperial College London, SW7 2AZ, United Kingdom * Corresponding author’s email address: [email protected]
Abstract
Artificial spin ice systems have seen burgeoning interest due to their intriguing physics and potential applications in reprogrammable memory, logic and magnonics. In-depth comparisons of distinct artificial spin systems are crucial to advancing the field and vital work has been done on characteristic behaviours of artificial spin ices arranged on different geometric lattices. Integration of artificial spin ice with functional magnonics is a relatively recent research direction, with a host of promising early results. As the field progresses, studies examining the effects of lattice geometry on the magnonic response are increasingly significant. While studies have investigated the effects of different lattice tilings such as square and kagome (honeycomb), little comparison exists between systems comprising continuously-connected nanostructures, where spin-waves propagate through the system via exchange interaction, and systems with nanobars disconnected at vertices where spin-waves are transferred via stray dipolar-field. Here, we perform a Brillouin light scattering study of the magnonic response in two kagome artificial spin ices, a continuously-connected system and a disconnected system with vertex gaps. We observe distinctly different high-frequency dynamics and characteristic magnetization reversal regimes between the systems, with key distinctions in system microstate during reversal, internal field profiles and spin-wave mode quantization numbers. These observations are pertinent for the fundamental understanding of artificial spin systems and the design and engineering of such systems for functional magnonic applications.
I. Introduction
Artificial spin ice (ASI) systems are engineered magnetic metamaterials designed by nanopatterning strongly interacting single domain nanoislands in a geometrically frustrated array [1]. Frustration in a physical system emerges from inability to simultaneously minimizing all interactions [2]. In 2006, Wang et al. carried out pioneering experiments on lithographically-defined arrays of interacting nanomagnets and explored interesting physics analogous to spin ice materials (Pyrochlores, Dy Ti O etc) [3]. This triggered research efforts to investigate ASI for exotic fundamental physics as well as potential applications such as reprogrammable memory, logic and more recently reconfigurable magnonics [4-7]. Arising from its parallels with water ice systems, studies on ASIs have revealed the occurrence of classical spin liquid states [8], Coulomb phases [9], monopole-like excitation [10] and spin fragmentation [11]. Apart from conventional ASI based on square and kagome lattice arrangement, there has been a considerable progress in other possible geometrically frustrated arrays such as shakti lattice [12], tetris lattice [13] and coupled islands [14]. In the context of experimental probing of the spin dynamics, ferromagnetic resonance (FMR) technique has been mostly employed which is based on the global excitation and probing of a large area ASI subject to an external sweep field [13-23]. Because of the large array of nano islands in such studies, it is quite unclear to attribute that the observed changes in FMR frequencies relies upon the number of nanobars that have been reversed or whether the precise spin configuration of the array plays a crucial role. The main motivation behind this work is to understand the spin dynamics in a limited array size which gives the better way of understanding the switching behaviour of the ASI nanostructures as opposed to the widely used FMR technique. In addition, the optical technique such as conventional Brillouin light scattering (BLS), which is a powerful tool for probing spin dynamics, has never been employed for the investigation of spin waves in kagome ASI till date. However, microfocused BLS has recently been employed to probe the topological defects modified magnon mode patterns [24]. Here we have performed a comprehensive study of the spin-wave (SW) dynamics and evolution under magnetic bias field in connected and disconnected kagome ASI (c-ASI and d-ASI hereafter) systems made of Ni Fe (permalloy, Py hereafter) in magnetically saturated and disordered states via BLS and micromagnetic simulation. II. Experimental and Simulation Methods
Arrays of kagome ASI of area 90 × 90 µm were fabricated using electron-beam lithography (EBL) lift-off process. Single layer polymethyl methacrylate (PMMA) resist (950K) resist was used on Si/SiO substrate. After writing and development of the resist pattern using MIBK developer, 25-nm-thick Py film was deposited by thermal evaporation at a base pressure of 9.0 × 10 -7 Torr. The samples were then capped with 3.7-nm of Al O to protect it from oxidation. For the c-ASI sample, the nanobars have length ( l ) and width ( w ) 300 nm and 80 nm respectively. The d-ASI has l and w matching the c-ASI but a 50nm vertex gap defined as the distance between nanobar-end and vertex centre) is introduced. Preliminary characterization was performed using scanning electron microscopy (SEM) to inspect the ASI geometry and nanofabrication quality. Magnetic force microscopy (MFM) was performed at the remanent state, confirming single-domain nanobars and system microstates obeying the ice rules. Magnetic hysteresis loops were measured using magneto-optical Kerr effect (MOKE). BLS measurements were performed in Damon-Eshbach (DE) geometry using a Sandercock-type six-pass tandem Fabry-Pérot interferometer [25]. Conventional 180° backscattered geometry was used to investigate the field evolution of spin waves. In the light scattering process, total momentum is conserved in the plane of the thin film. As a result, the Stokes (anti-Stokes) peaks in BLS spectra correspond to creation (annihilation) of magnons with momentum 𝑘 = sin 𝜃 , where 𝜆 is the wavelength of the incident laser beam (532 nm in our case), and 𝜃 refers to the angle of incidence of laser. To get well defined BLS spectra for the larger incidence angles, the spectra were obtained after counting photons for several hours. Free spectral range (FSR) of 30 GHz and a 2 multi-channel analyser were used during the BLS measurement. Frequency resolution is determined by estimating FSR/2 (≈0.03 GHz) for the Stokes and anti-Stokes peaks of the BLS spectra. A variable bias magnetic field ( H ) was applied during the measurements. Further details can be found in reference [26]. To support the experimental results, micromagnetic simulations were performed using Object-Oriented Micromagnetic Framework (OOMMF) software [27]. The SEM images of the arrays were mimicked and converted into discretized mesh. It is important to mention here that using a real SEM images allow us to introduce the slight discrepancies in the nanobars dimensions resulting due to the nanofabrication imperfections often termed as quenched disorders. A cell size of 5×5×25 nm dimension was employed, with lateral dimensions below the Py exchange length. Typical material parameters of Py were used in simulation, namely: gyromagnetic ratio, γ = 2.211 ×10 A -1 s -1 , saturation magnetisation, M S =
800 kA/m, and the exchange stiffness constant, A = 1.3 × 10 -11 J/m and Gilbert damping, =
III. Results and Discussion Figure 1.
SEM images of (a) connected (c-ASI) and (b) disconnected (d-ASI) kagome ASI. Experimental MFM images taken at remanence for (c) c-ASI and (d) d-ASI. Simulated MFM images at remanence for (e) c-ASI and (f) d-ASI. (g) Schematic of the Brillouin light scattering measurement geometry.
Figures 1(a) and (b) represent the scanning electron micrographs (SEM) images for connected (c-ASI) and disconnected (d-ASI) kagome ASI samples. The variation in the measured dimensions are: ±3% in length, ±2% in width and ± 1.5% in the inter-bar distance for d-ASI, while the heights of the nanobars are found to vary by ±3% measured via atomic force microscope (AFM) [29]. The SEM images confirm good quality of the samples.
Figures 1(c) and (d) show the remanent state MFM images for the samples c-ASI and d-ASI, respectively. The MFM images reveal that the magnetization orientation for both c-ASI and d-ASI obey the spin ice rule, i.e. “two-in one-out” or “one-in two-out” with no observed vertices exhibiting three like-polarity magnetic charges. MFM images simulated at remanence using LLG simulator for both c-ASI and d-ASI (
Fig. 1(e) and Fig. 1(f) ) show excellent agreement with the experimental MFM images.
Fig. 1(g) shows the schematic of the BLS measurement geometry.
Magnetization Reversal of Kagome ASI Figure 2 represents the magnetic hysteresis loops measured by longitudinal MOKE for both samples c-ASI and d-ASI. The difference in structure of c-ASI and d-ASI arrays leads to
Figure 2. (a) MOKE hysteresis loops showing the magnetization reversal for connected (c-ASI) and disconnected (d-ASI) kagome ASI samples. (b) Simulated hysteresis loop for both the samples. a lower coercivity for the c-ASI structure as once domain walls are introduced to the array by the bias field, they may propagate throughout all nanobars in the system through the connected vertices, whereas the d-ASI sample requires a separate domain wall nucleation event for every nanobar and corresponding higher coercivity. The arrays exhibit coercive fields of 370 Oe for c-ASI and 450 Oe for d-ASI, remanence of 70% for c-ASI and 82% for d-ASI, and a two-step reversal for c-ASI as opposed to a single step reversal for d-ASI. This two-step feature indicates switching of two different parts of the sample at different magnetic fields.
Figure 2(b) shows the simulated hysteresis loops which are in qualitative agreement with the experimental loops. We will further investigate how the magnetization reversal behaviours affect the high frequency SW dynamics of these samples.
Spin-Wave Dynamics of Kagome ASI Figures 3(a) and 3(c) show the representative BLS spectra for c-ASI and d-ASI taken at k ≈ 0 in the DE geometry at different H values. The corresponding simulated SW spectra are shown in Figs. 3(b) and
In experiment, H was varied between +1.4 kOe ≤ H ≤ -1.4 kOe, tracing the upper branch of the magnetic hysteresis, and SW frequencies recorded from BLS spectra. For both c-ASI and d-ASI, we observe three dominant modes whose frequencies and BLS intensities vary significantly and non-monotonically with H . In addition, a stark mode frequency difference is observed between c-ASI and d-ASI due to their differing inter-nanobar coupling mechanism as also evidenced in the MOKE hysteresis loops. Figures 3(e) and are the experimental BLS spectra taken at H = -415 Oe and -550 Oe for c-ASI and d-ASI respectively, halfway through magnetisation reversal as measured via MOKE. It is clear from this figure that two additional higher frequency SW modes (M4 and M5) appear for both samples. These modes grow at the expense of modes M2 and M3 as the magnitude of negative H increases further. The origin of these additional modes further discussed later in this paper. Figure 3.
Representative BLS spectra measured at wave vector k ≈ 0 for different magnetic fields from (a) c-ASI and (c) d-ASI. Each spectrum corresponds to a different magnetic field value as indicated next to the spectrum. The solid grey curves represent Lorentzian multipeak fits. The SW modes M1, M2, and M3 are marked in ascending peak frequency. Simulated SW spectra are shown for c-ASI (b) and d-ASI (d). BLS spectra taken halfway through magnetisation reversal are shown for c-ASI (e) and d-ASI (f) with additional modes present relative to the saturated state. Next, we have investigated the detailed magnetic field dispersion and nature of the observed SW modes. The magnetic field dispersion of SW frequency is plotted in Figs. 4(a) and for c-ASI and d-ASI respectively. The experimental BLS spectra taken at varying H values are presented as yellow (high spin wave intensity) to blue (low intensity) heatmaps with numerical simulation results from OOMMF superimposed as filled symbols. The SW modes exhibit distinct and non-monotonic dispersion across the field range investigated. Figure 4.
Spin wave mode frequencies of (a) c-ASI and (b) d-ASI as a function of applied bias magnetic field along the horizontal bar axis. The experimental bias field dependent SW frequencies are shown as heat maps with yellow region denoting high spin wave intensity. Simulated SW mode frequencies are shown by filled symbols for. The different SW branches are denoted by M1, M2 and M3. The experimental data taken halfway through magnetization reversal (corresponding to the SW spectra shown in Fig. 3(e) and 3(f))) are denoted by star symbols for both c-ASI and d-ASI. In
Fig. 4(a) , the intensity of the lowest frequency mode M1 is significantly lower than the other two modes. The frequency splitting between M2 and M3 decreases with the reduction of field from +1.4 kOe reaching a minimum separation of ~2 GHz between H = -200 Oe to -400 Oe, beyond which it increases again. The frequencies of all modes experience a minimum at H = -400 Oe, which is close to the mid-point of reversal as shown by the black arrow in Fig. 2(a) . For H < -400 Oe, the mode frequencies experience a sharp upwards frequency transition with an increase of ~3 GHz observed in M3. The relative intensities of M2 and M3 are also exchanged, with M2 the dominant mode for H < -400 Oe. As the magnitude of the negative field further increases the intensities of the modes tend towards equal. Figure 4(b) shows the field dependence SW frequencies for d-ASI where again three distinct modes are observed. Unlike for c-ASI, the BLS intensity of Mode M2 is highest for d-ASI at positive fields. Here, the lowest frequency mode M1 appears very close to M2. Here also, the frequency splitting between M2 and M3 decreases with decreasing field from +1.4 kOe and they merge together between H = -200 Oe and -400 Oe. Moreover, a crucial distinction between the d-ASI and c-ASI response exists most likely due to the exchange interaction at the vertices of the nanobars. After magnetisation reversal is complete at -600 Oe modes M2 and M3 split again with the frequency gap between them increasing with negative field amplitude. The distinct and characteristic spin wave field dispersion in connected and disconnected kagome ASI demonstrate a high degree of tunability, inviting integration with future spintronic and magnonic devices for microwave filtering applications where the forbidden and allowed frequency regimes may be finely manipulated via bias field. Numerical Simulations
To aid interpretation of the experimental findings, the field-dependent static spin configurations and spatial power and phase profiles have been simulated using OOMMF with additional spin-wave analysis performed using in-house developed code [30].
Figure 5 shows the static spin configuration (a,f,m), x component of demagnetizing field (b,g,n) and SW mode profiles (remaining panels) for c-ASI at three magnetic bias fields corresponding to just below (-400 Oe), at (-550 Oe) and above (-800 Oe) magnetisation reversal and the corresponding transition in mode frequency.
Figure 5(a) shows that the spins in the horizontal nanobars of c-ASI continue to align along the +x direction even at H = -400 Oe, while the spins in the diagonal nanobars rotate in the clockwise (anticlockwise) direction in the upper (lower) half of the hexagon forming a forward onion-like state. At H = -550 Oe, the spins in certain nanobars start to reverse their orientation in the direction of H forming vortex-like configuration in some of the hexagons in the lower half of the array (shown by ticks (√) Fig. 5(f) ). However, in some of the hexagons it forms neither a vortex nor an onion state as shown by asterisks (*) in
Fig. 5(f) . When H is further increased to -800 Oe, full reversal of spins in all the nanobars occurs ( Fig. 5(m) ) forming reverse onion-like states in all hexagons. The corresponding spatial maps of the x-component of demagnetizing field ( H d ) are presented in the Fig. 5 (b, g, n) . As expected, the direction of H d lies in the direction opposite to the applied field H = -400 Oe. At H = -550 Oe, the direction of H d points towards H only for the reversed nanobars at that field. As we move further to H = -800 Oe, the direction of H d lies completely along H . Figure 5:
Simulated static spin configuration (a,f,m) and x-component of the demagnetizing field (b,g,n) for c-ASI. The vortex like configuration in some of the hexagons are marked with tick (√) symbol. The star (*) marks refer to the hexagons which forms neither vortex nor an onion state. The flow of the magnetization is shown by curved arrows. Simulated SW power profiles at H = -400 Oe (c-e), -550 Oe (h-l) and -800 Oe (o-q) for c-ASI. (r) The schematic of the measurement and simulation geometry is also shown at right bottom corner. The phase profiles are shown in the inset at the left corner of each power profile. The color maps are given at the right top corner. Figures show the power profiles of M1, M2 and M3 for c-ASI at H = -400 Oe. The corresponding phase profiles are shown in the insets at the left corner of each image. We define different quantization numbers m and n for SW modes in backward volume (BV) geometry or in Damon-Eshbach (DE) geometry respectively. The mode M1 ( Fig. 5(c) ) is the edge mode (EM) at the vertex junction of the nanobars. The power is concentrated where the two diagonal bars join the horizontal bar. The diagonal nanobars show DE like behaviour with m = 3. Mode M2 shows a BV character in the horizontal nanobar with n = 9 and DE character in the diagonal nanobars with m = 9. M3 also shows similar character with n = 11 and m = 7. At H = -550 Oe, i.e. in the transition regime, the number of SW modes have increased to five. Here M1 becomes a BV-like mode with n = 3 in the horizontal nanobars, and DE-like mode with m = 5. M2 has n = 9, m = 7 with higher power than for H = -400 Oe. M3 has n = 11, m = 7, similar to M3 for H = -400 Oe. M4 has n = 11, m = 9. M5 has mixed BV-DE character in the individual nanobars and it is difficult to resolve the mode numbers from the phase profile. As we further increase H to -800 Oe where a complete magnetization reversal occurred to a reverse onion mode, the mode characters are very different from those at H = -400 Oe. Here, M1 has n = 2, m = 3, while M2 has n = 2, m = 5. Although the phase contrast in the horizontal nanobars on both sides of the nodal plane is not high (far less than π) and the power is also small, it can still be considered as a BV-like mode with n = 2. M3 has n = 9 and m = 9, similar to M2 at H = -400 Oe. This drastic variation in the mode quantization number with the magnetic field leads to the observed abrupt change in the frequencies of the SW modes. Next, we investigate the static spin configuration, x-component of the demagnetizing field and SW mode profiles for d-ASI at three different applied bias fields, H = -400, -600 and -800 Oe as shown in Fig. 6 . In d-ASI, static spin configuration follows a similar trend as c-ASI. However, only one vortex state (√) and three asterisk states (*) are formed in this case (
Fig. 6(f) ). In addition, the direction of H d in the horizontal nanobars is opposite to that observed in c-ASI. The partial reversal occurred at H = -600 Oe causes local reversal in H d as shown in Fig. 6(g) ). The striking difference in the frequency of the various SW modes between c-ASI and d-ASI are related with this change in the spin configuration and H d . Figure 6 : Simulated static spin configuration (a,f,m) and x-component of the demagnetizing field (b,g,n) for d-ASI. The vortex like configuration in some of the hexagons are marked with tick (√) symbol. The asterisk (*) marks refer to the hexagons which forms neither vortex nor an onion state. The flow of the magnization is shown by curved arrows. Simulated power maps for various precessional modes at H = -400 Oe (a-c), -600 Oe (d-h) and -800 Oe (i-k) for d-ASI. The schematic of the geometry is also shown at right bottom corner. Phase profiles are shown inside the rectangular box at the left corner of each images. The color maps are given at the right top corner. (r) SEM image of the d-ASI sample is shown at the bottom right corner. The simulated power maps are shown together with the corresponding phase profiles in the inset of each image. Here, at H = -400 Oe, M1 corresponds to n = 5, m =5, M2 has n = 3, m = 7, and M3 shows uniform mode (UM), m = 9. At H = -600 Oe, M1 has n = 5, m = 5, M2 has n = 7, m = 5, M3 has n = 3, m = 7, M4 has n = 6, m = 9, M5 has n = 1, m = 9. Hence, M2 and M4 are two new modes generated at transition, while three other modes retain their character with some modulation of power and phase distribution. This key observation is also evident in the experimental BLS spectra taken in the transition regime where the partial reversal of nanobars occurred for both c-ASI and d-ASI as shown in Fig. 3(e,f) . At H = -800 Oe, the mode characters have become drastically different. Here M1 corresponds to n = 4, m = 5, M2 has n = 7, m = 9 and M3 has n = 1, m = 11. Again, this drastic change in mode character has led to the observed transition in frequency which show a significantly different behaviour from H = -400 Oe for d-ASI, while for c-ASI, only one mode gets modified and the other two retain their identity. The characteristic features of these SW spectra as observed from BLS experiment matches qualitatively well with that obtained using simulation. The additional modes appear most likely due to the local spin configuration during magnetization reversal. Moreover, the local modification of the internal field due the variation of H d leads to the significant change in the SW frequencies associated with the reversed nanobars between c-ASI and d-ASI. In addition, to validate the simulation, a comparative study of the simulation with/without 2D-PBC performed on a unit cell and large array of ASI has been given in the Supplementary Materials [29]. The Simulated SW spectra and mode profiles (Fig. S2 & S3) at a positive saturation field (at H = 1400 Oe) infers the existence of three SW modes in all the cases and the peak frequency do not vary much [29]. IV. Conclusions
In conclusion, we have experimentally and numerically studied the magnetization dynamics of connected (c-ASI) and disconnected (d-ASI) kagome artificial spin ice nanostructures made of Py nanobars using Brillouin light scattering and micromagnetic simulation. The MFM images reveal that the magnetic microstates of kagome ASI obey the spin ice rule and a good agreement is found with the simulated images using micromagnetic simulation. The magnetic hysteresis loop measured by MOKE exhibits two-step reversal process for c-ASI, but a single step reversal for d-ASI. Bias field dependent SW spectra measured by BLS reveal distinct features both in SW frequency ( f ) as well as BLS intensity for c-ASI and d-ASI. Both c-ASI and d-ASI exhibit frequency minimum, occurring at negative bias magnetic fields, when ramped down from a positive saturation field. A sharp jump in the mode frequencies was observed beyond the minima. This jump is linked to the switching field observed from the MOKE loops and the corresponding mixed state of nanobars aligned parallel and anti-parallel to the magnetic bias field. This jump is associated with a stark change in the SW mode profiles and mode quantization numbers, demonstrating strong control of SW modes via a small variation in magnetic field. Simulated internal field profiles yield further insights into the variation of SW modes in this system. Investigation and control of SW modes of these kagome artificial spin-ice nanostructures will be important for fundamental understanding and applications in devices requiring frequency selective SW propagation such as microwave filters. Acknowledgements
AB gratefully acknowledges the financial assistance from Department of Science and Technology (DST), Government of India under grant no. SR/NM/NS-09/2011 and S. N. Bose National Centre for Basic Sciences under project no. SNB/AB/18-19/211. AKC gratefully acknowledges DST, Government of India for INSPIRE fellowship (IF150922) and British Council UK for providing Newton Bhabha fellowship under Newton Bhabha Fund PhD Placement Program. AKM acknowledges S. N. Bose National Centre for Basic Sciences, India for senior research fellowship. SB acknowledges Science and Engineering Research Board (SERB), India for funding (grant no. CRG/2018/002080). WRB gratefully acknowledges financial support by the Leverhulme Trust (RPG-2017-257). AV was supported by the EPSRC Centre for Doctoral Training in Advanced Characterisation of Materials (Grant No. EP/L015277/1). References [1] L. J. Heyderman, and R. L. Stamps,
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Magnetic Field Controlled Transition in Spin-Wave Dynamics in Kagome Artificial Spin Ice Structure
Avinash Kumar Chaurasiya, Amrit Kumar Mondal, Jack C. Gartside, Kilian D Stenning, Alex Vanstone, Saswati Barman, William R. Branford, and Anjan Barman Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block – JD, Sector-III, Salt Lake, Kolkata 700 106, India Blackett Laboratory, Department of Physics, Imperial College London, SW7 2AZ, United Kingdom Institute of Engineering and Management, Sector-V, Salt Lake, Kolkata 700 091, India London Centre for Nanotechnology, Imperial College London, SW7 2AZ, United Kingdom * Corresponding author’s email address: [email protected]
I. Atomic force microscopy : The large area AFM images from samples S1 and S2 are presented in the
Fig. S1 (a) and (b) , respectively.
Figure S1 : AFM images taken at remanence for S1 (a) and S2 (b). The corresponding scale bars are shown next to each image. (c) and (d) are the corresponding line profiles used to estimate heights of the nanobars (regions marked with dotted line in the AFM images). II. A comparison of spin wave spectra simulated for d-ASI with/without applying 2D-periodic boundary conditions (2D-PBC):
Figure S2 represents the simulated SW spectra at H = +1400 Oe. This is a comparison of the SW spectra simulated under various conditions. It is important to mention that for unit cell, the cell size was chosen to be 2 nm × 2 nm × 25 nm and the dynamic simulation was run for 4 ns at the step of 10 ps with 2D-PBC. In the case of large array (3.4 µm × 2.7 µm), the cell size was chosen to be 2.5 nm × 2.5 nm × 25 nm and the dynamic simulation was run for both 4 ns and 20 ns at the step of 10 ps with and without applying 2D-PBC. The material parameters used for the simulations are: gyromagnetic ratio, γ = 2.211 ×10 A -1 s -1 , saturation magnetisation, M S =
800 kA/m, and the exchange stiffness constant, A = 1.3 × 10 -11 J/m and Gilbert damping, = 0.008. It is evident that the primarily three SW modes are observed in all the cases and the peak frequency do not vary much which validates the presented simulation in the main text of the article. Figure S2:
Simulated FFT power spectra for d-ASI in various unit cell with 2D-PBC (black line), large array with (without) PBC (red (blue) line), and large array simulated for 20ns (magenta) duration. III. A comparison of SW mode profiles simulated with/without 2D-PBC:
To investigate the nature of SW modes, we have simulated the power and phase maps of various SW modes observed in d-ASI at H = +1400 Oe. Here we present the comparison of the simulation results obtained under various protocol. Figure S3 : Simulated power maps for various precessional modes in d-ASI at H = +1400 Oe for (a) unit cell with 2D-PBC (b) large array with 2D-PBC (c) large array without 2D-PBC and (d) large array without 2D-PBC for dynamics captured for 20 ns. Phase profiles are shown inside the rectangular box at the left corner of each images. In Fig. S3 (a) , the power and phase profiles of all three modes simulated for a unit cell with 2D-PBC is shown. The characters of the modes are edge mode (EM), BV-like mode (quantization number, n ) in the horizontal nanobars and DE-like modes (quantization number, m) in the tilted nanobars. Here M1 corresponds to n = 3 and m = 5, M2 has n = 5, m = 3 and M3 has n = 1. Subsequently, we have also simulated the mode profiles by considering a large array with 2D-PBC (without 2D-PBC) as shown in Fig. S3 (b) and
S3 (c) respectively. Here the dynamic simulation was run for 4 ns duration. Shown in
Fig. S3 (d), we also tested the validity of the presented simulation by running the dynamics for 20 ns for a large array without 2D-PBC. It is evident that the characteristics of SW mode profiles does not vary much whether 2D-PBC is applied or not.we also tested the validity of the presented simulation by running the dynamics for 20 ns for a large array without 2D-PBC. It is evident that the characteristics of SW mode profiles does not vary much whether 2D-PBC is applied or not.