Reconfigurable magnonic mode-hybridisation and spectral control in a bicomponent artificial spin ice
Jack C. Gartside, Alex Vanstone, Troy Dion, Kilian D. Stenning, Daan M. Arroo, Hide Kurebayashi, Will R. Branford
RReconfigurable magnonic mode-hybridisation andspectral control in a bicomponent artificial spin ice
Jack C. Gartside , Alex Vanstone , Troy Dion , Kilian D. Stenning , Daan M. Arroo ,Hide Kurebayashi , and Will R. Branford Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom These authors contributed equally to this work * Corresponding author e-mail: [email protected]
ABSTRACT
Strongly-interacting nanomagnetic arrays are finding increasing use as model host systems for reconfigurable magnonics. Thestrong inter-element coupling allows for stark spectral differences across a broad microstate space due to shifts in the dipolarfield landscape. While these systems have yielded impressive initial results, developing rapid, scaleable means to access abroad range of spectrally-distinct microstates is an open research problem.We present a scheme whereby square artificial spin ice is modified by widening a ‘staircase’ subset of bars relative to the restof the array, allowing preparation of any ordered vertex state via simple global-field protocols. Available microstates rangefrom the system ground-state to high-energy ‘monopole’ states, with rich and distinct microstate-specific magnon spectraobserved. Microstate-dependent mode-hybridisation and anticrossings are observed at both remanence and in-field withdynamic coupling strength tunable via microstate-selection. Experimental coupling strengths are found up to g / π = . GHz.Microstate control allows fine mode-frequency shifting, gap creation and closing, and active mode number selection.
Introduction
The field of magnonics aims to employ spin-waves to propagate and process information . Spin-waves offer a host ofattractive benefits as data carriers including low heat generation, power consumption and coherent coupling to photons ,phonons and other magnons . Functional magnonics has proliferated in recent years, with wide-ranging applications fromtransistors to multiplexers and logic gates . As the complexity of magnonic designs increases, so does the demand forversatile, reconfigurable host systems.Recently, a family of metamaterials termed reconfigurable magnonic crystals (RMC) has made strong progress inanswering this need. Typically comprising discrete nanopatterned magnetic elements closely-packed in arrays to promotestrong dipolar coupling, RMC support multiple microstates and exhibit distinct microstate-dependent magnonic dynamics andspectra with diverse functional benefits. A subset of RMC has emerged based on artificial spin ice (ASI) arrays wheregeometrical frustration gives rise to a vastly degenerate microstate space that features a long-range ordered ground state andhigh-energy ‘magnetic monopole’-like excited states . The potential to leverage these states for their magnonic properties isgreat and studies into fundamentals of state-spectra correspondence have set the scene for a new generation of ASI-basedRMC designs. An open problem in the field is developing reliable, versatile and rapid means for microstate access. While ASIpossesses a huge range of states, they remain largely unavailable for magnonic exploitation due to state preparation techniquesbeing overly simplistic (for example global-field protocols which may only prepare saturated or randomly demagnetised,unrepeatable states), overly slow (surface-probe microscope nanomagnetic writing techniques which may prepare anystate but on timescales unsuitable for technological integration) or difficult to realise with current nanofabrication techniques(for example, recently proposed multi-level stripline technique ). In the absence of such techniques, ASI systems have beenmodified to allow access to an enhanced microstate range using global fields, ‘magnetic charge ice’ which rotates a subsetof bars in square ASI to allow global-field preparation of three microstates (types 1-3 as seen in fig. 1) or bar subsetmodification via either material , shape-anisotropy or exchange-bias . The magnetic charge ice case is elegant, but theway in which bars are rotated leads to greater separations between neighbouring elements so that greater density is requiredto achieve an appreciable interelement coupling required for collective excitations. Moreover, the rotation means differentsublattices will in general experience different effective global excitation and bias fields.Here, we present a square ASI with a ‘staircase’-pattern subset of width-modified bars. Shown in figure 1, this enables a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n reparation of four distinct type 1-4 microstates (fig. 1 b-e), g-j) including the typically elusive ground-state (type 1) and‘monopole’ states (types 3 and 4). The four states display rich and distinct magnonic spectra with fine control over modefrequency shifting, gap opening and tuning, and number of active modes. Microstate-dependent mode-hybridisation andanticrossings are observed with coupling-strength and gap width tunable via state selection. Selective mode-hybridisation offersreconfigurable mode profile and index control in-field and crucially at remanence. Results and Discussion
Microstate access via width-modification
Samples were designed by taking square ASI and increasing the width of one nanobar subset. Two samples were fabricated,square (S, fig. 1 a) and high-density square (HDS, fig. 1 b), using an electron beam lithography liftoff process and thermaldeposition of Ni Fe . Sample dimensions were selected such that the wider bar subset may be magnetically reversed via aglobal-field without thin-bars also reversing from the combination of global-field H ext and local dipolar field H loc , satisfying H c > H ext + H loc > H c with H c and H c the wide and thin-bar coercive fields respectively ( H c and H c visible in fig. 1MOKE loops). This enables preparation of the entire range of ordered, ‘pure’ (i.e. single vertex type across the array)microstates. Arrays comprising identical bars may only access a single pure microstate by saturating with global-field (type2). The width-modification employed here allows global-field access to four pure states with distinct local-field profiles andcorresponding magnonic spectral dynamics. Microstates are shown via magnetic force microscope (MFM) (fig. 1 g-j) andmagnetic charge schematics (fig. 1 k-n). The S sample comprises bars of 830 nm ×
230 nm (wide-bar), 145 nm (thin-bar) × ×
200 nm (wide-bar),125 nm (thin-bar) ×
20 nm, 100 nm gap.Bars are widened in alternating y -axis columns (axes defined in fig. 1 a) such that wide-bars may be reversed from asaturated background state (type 2, fig. 1 h,l) without reversing thin-bars. If the global-field H ext is oriented along the y -axis,reversing only wide-bars from a ˆ y -saturated state leaves the system in the antiferromagnetic type 1 state (fig. 1 g,k), whichforms the ASI ground state with and without width-modification . Here the microstate allows the maximum amount ofinter-bar dipolar flux-closure and lowest system energy. If H ext is oriented along the x -axis, reversing wide-bars results in thetype 4 state (fig. 1 j,n), termed a ‘monopole’ or ‘all-in, all-out’ state with four like-polarity magnetic charges at each vertex,highly-repulsive inter-bar dipolar field interactions and maximum system energy. If H ext has any angular misalignment from thewidth-modified columns, one of the ± ◦ wide-bars will experience a higher field along its easy-axis, resulting in that barreversing at lower H ext . The resulting state with just one of the ± ◦ wide-bars reversed is the type 3 state (fig. 1 i,m) with threelike-polarity and one opposite polarity magnetic charge per vertex. In experiment there is always some angular misalignmentand the array will transition between states 2 and 4 via state 3. The field window in which state 3 exists may be broadened bydeliberately increasing angular misalignment. The S array may access type 1-4 states, the HDS array may access states 1-3 butnot 4 as the increased H loc magnitudes arising from smaller inter-bar separation leads to spontaneous reversal of thin-bars froma thin-bar majority charge type 3 to a wide-bar majority type 3 when attempting state 4 access, i.e. H ext + H loc > H c − thin .We analyse mode frequencies following the Kittel equation f = µ γ π (cid:112) H ( H + M ) in the k=0 limit applicable to this work,with γ the gyromagnetic ratio and H = H ext + H loc . The local dipolar field landscape varies greatly between microstates, withresulting distinct microstate-dependent magnon spectra. Microstate-dependent magnonic spectra
Broadband FMR spectra were measured using a flip-chip method with samples excited by a coplanar waveguide. For S andHDS samples, spectra were taken with H ext in ˆ y (‘ground-state’ orientation, as in fig. 1 a) and ˆ x (‘monopole’ orientation).Samples were saturated at H ext = − ±
300 mT field range (fig. 2 a-e, relative H ext orientation inset) and 0 −
40 mT range (fig. 2 f-j) with corresponding simulated spectra (fig. 2 k-o).Spectra exhibit two main Kittel-like modes, the lower and higher frequency modes corresponding to bar-centre localisedmodes in the wide W and thin-bars T respectively. This correspondence is evidenced by frequency jumps and ∂ f ∂ H signinversions indicating bar reversal in the low- and high- f modes at H c and H c respectively, matching switching fieldsobserved via MOKE (fig. 1 c-f). Higher relative amplitude of the low- f mode matches the larger sample volume shareof the wide-bar, simulated spatial mode-power maps support the mode-bar correspondence. The wide-bar exhibits twosubharmonic modes. Harmonics are expected in the thin-bar and observed in simulation, but fall below the amplitude thresholdfor experimental detection. In addition to offering two well-defined frequency channels, the different width bar subsets allowclear identification of which subset has reversed or undergone microstate-dependent frequency shifting.For H ext >
200 mT, thin and wide-bar modes tend to the same frequency as shape-anisotropy is overcome and bar igure 1.
Schematic of width-modified square and high-density square reconfigurable magnonic crystals and their type 1-4microstates and hysteresis loops. y and x array axes are referred to as ‘ground-state’ and ‘monopole’ orientations throughoutthis work.a) Scanning electron micrograph of the square sample. Bars are 830 nm long, 230 nm (wide-bar) and 145 nm (thin-bar) wide,20 nm thick with 120 nm vertex gap (bar-end to vertex-centre).b) Scanning electron micrograph of the high-density square sample. Bars are 600 nm long, 200 nm (wide-bar) and 125 nm(thin-bar) wide, 20 nm thick with 100 nm vertex gap.c-f) MOKE hysteresis loops of S sample in ‘ground state’ (c) and ‘monopole’ (d) orientations, HDS sample in ‘ground state’ (e)and ‘monopole’ (f) orientations. Blue points show fully-saturating hysteresis loop, orange points show minor loops withmaximum positive field value chosen to prepare sample in type 1 (c,e), type 4 (d) and type 3 (f) states before sweeping back tonegative saturation.g-j) Magnetic force microscope images of type 1-4 microstates. Type 1 and 4 states have inset SEM images showing therelative orientation of H ext to the width-modified subsets required for state preparation. Type 2 and 3 states may be prepared ineither ± ◦ field orientation. Type 1 and 4 states are often termed ‘ground state’ and ‘monopole’ state in artificial spin ice.k-n) Magnetic charge dumbbell schematic of type 1-4 microstates. igure 2. Differential ferromagnetic resonance spectra of square (S) and high-density square (HDS) samples withcorresponding micromagnetic simulations. Peak amplitude occurs at boundary between white and black bands.Samples were saturated in 1000 mT negative field then swept in positive field direction, with relative field orientation indicatedin inset scanning electron micrographs. Measurements were performed at room temperature.Fields were swept from ±
300 mT, with full sweeps (a-d), 0-40 mT sweeps around the coercive fields (e-h) and micromagneticMuMax3 simulations of the coercive field region (i-l) presented. Sample geometry and H ext orientation are shown inset.Switching fields are labelled by vertical dashed lines, a and b subscripts refer to separate ±
45 and ∓
45 subset reversal whereapplicable, type 3- and 3+ refer to thin-bar majority and wide-bar majority type 3 states respectively. From left to right, verticalcolumns of spectra relate to samples: S (‘ground-state’ orientation), S (‘monopole’ orientation), HDS (‘ground-state’orientation), HDS (‘monopole’ orientation). agnetisation rotates from an Ising-like state to lie parallel to H ext . At these H ext , the bar demagnetising fields become negligbleand the Kittel equation is dominated by H ext .At lower fields around H c and H c , rich and distinct spectra are observed between samples and orientations. Figure 2 f)shows S sample spectra in ‘ground-state’ orientation. At H ext = M x component of all bars oriented against positive H ext and both modes exhibiting negative ∂ f ∂ H . At H c =
16 mT thewide-bars reverse, its mode jumping 6.2-7.8 GHz and displaying positive ∂ f ∂ H . The thin-bar mode is blueshifted 0.1 GHz at H c due to the change in local dipolar field landscape as the system enters a type 1 microstate (fig. 1 g,k). For H c < H ext < H c thesystem is in a type 1 state, the two modes exhibiting opposite ∂ f ∂ H and crossing at H ext = mT . Opposing frequency gradientsand presence of a mode crossing in this field range afford sensitive mode and gap tunability via H ext . At H c =
29 mT thethin-bars reverse, preparing a type 2 state aligned with H ext and redshifting the wide-bar mode 0.1 GHz via the shift in dipolarfield landscape. Above H c both modes exhibit positive ∂ f ∂ H .Rotating H ext ◦ accesses the ‘monopole’ orientation. Fig. 2 b) shows that at high-field saturated states, ‘monopole’ and‘ground-state’ orientations behave similarly. Around the coercive fields, fig. 2 g) shows stark spectral differences betweenthe orientations. While the ‘ground-state’ orientation transitions directly between a type 2 and type 1 state via simultaneousreversal of both wide-bars, the highly-unfavourable type 4 dipolar field landscape separates the wide-bar switching into twodistinct reversal events occurring at different fields. At H ext =
13 mT the ± ◦ subset of wide-bars better aligned to H ext reverses, placing the system in a ‘thin-bar majority’ type 3 state (fig. 1 i,m) where both thin-bars and a single wide-bar sharelike-polarity charges. This splits the low frequency mode as half the wide-bars ( ± ◦ ) reverse while the rest ( ∓ ◦ ) remainaligned against H ext . The reversed wide-bar mode jumps from 6.0-7.7 GHz and exhibits positive ∂ f ∂ H . The unswitched wide-barmode is blueshifted 0.3 GHz and the thin-bar mode redshifted 0.1 GHz by the type 3 dipolar field landscape. This reducesthe gap between unswitched wide and thin-bar modes by 0.4 GHz without modifying the magnetisation state of either bar,demonstrating the degree of spectral control available via microstate engineering. Remaining wide-bars reverse at 20 mT,placing the system in a type 4 microstate (2 e) and unifying the wide-bars in a single 7.5 GHz mode, redshifted 0.1 GHz relativeto the already-reversed wide-bar mode. Thin and wide-bar modes now occupy the same frequency, obscuring the expectedthin mode blueshift in the experimental data. The mode gap width may be modified by varying the relative bar widths at thefabrication stage, and the overlap here is a consequence of the specific dimensions employed. At H c =
23 mT the thin-barreverses, placing the system in a type 2 state aligned with H ext and restoring the mode gap.To demonstrate the spectral control available via array design choices, the HDS sample was fabricated. By reducingbar-length and inter-bar separation, stronger dipolar interactions between bars and greater variation in the dipolar field landscapeis achieved, resulting in larger spectral shifts when transitioning state. Fig. 2 h) shows the HDS ‘ground-state’ orientation,qualitatively matching that of the S sample but with frequency shifts of 0.2 GHz (twice that observed in the S sample) and abroadened type 1 field window due to the increased stability. Fig. 2 i) shows the HDS ‘monopole’ orientation, again qualitativelymatching that of the S sample (with enhanced 0.3 GHz frequency shifts) up to 23 mT where the system transitions from athin-bar majority type 3 state to a wide-bar majority type 3 rather than type 4. A wide-bar majority type 3 persists between23-30 mT. At 30 mT the remaining ± ◦ thin-bar reverses, causing a 0.3 GHz redshift in the thin-bar mode and transitioning toa type 2 state. Negative field evolution of microstate-dependent magnonic spectra
So far spectra have been measured while positively sweeping H ext after negative saturation. This allows study of the system asit evolves through a range of microstates, but each microstate is stable in a limited field window. Alternatively, states may beprepared via negatively saturating then applying a microstate-specific positive field (i.e. 22 mT for the S sample type 1 state, fig.2 f), then recording spectra while negatively sweeping H ext until saturation. This allows mode dynamics to be studied for eachmicrostate over its entire stable field range, revealing additional spectral details not accessed in fig. 2 e-h) 0-40 mT sweeps.Figure 3 shows negatively-swept spectra for the S (FMR panels a-d), simulations e-h) and HDS samples (FMR j-m),simulations n-q) for all microstates and orientations alongside mode extractions (S sample i), HDS r) allowing state comparison.Figure 3 a) shows the S sample ‘ground-state’ orientation prepared in a type 1 state at H ext =
22 mT. At 22 mT thin and wide-barmodes occupy a single frequency at 8 GHz. As field is negatively swept, modes exhibit opposite gradient due to opposing wideand thin bar magnetisation, reaching a maximum mode-frequency gap of 3 GHz at -16 mT, after which the wide-bars reverse.This prepares a type 2 state, with a wide-bar frequency jump and thin-bar redshift as observed in the 0-40 mT positive sweeps.The broadly tunable 0-3 GHz gap and wide field-stability window of type 1 state are desirable for functional magnonic systemswhere mode-frequency gap control is crucial.Figure 3 b) shows the ‘ground-state’ orientation S sample prepared in a type 2 state at 10 mT. Sweeping H ext negatively,both modes exhibit a constant gradient and frequency gap of 2 GHz. Figure 3 c) shows a monopole-orientation type 3 spectra.A 21 mT preparation field reverses half the wide-bars, preparing a thin-bar majority type 3 state. The preparation here isdistinct from earlier discussion of type 3 states in the HDS sample, where a well-defined ± ◦ subset reverses due to closer igure 3. Differential ferromagnetic resonance spectra taken while negatively sweeping H ext after microstate preparation inpositive field. Microstates were prepared by -1000 mT saturation then applying the positive field required to reverse the desiredbars, hence differing positive field limits for different microstate spectra.a-d) Experimental spectra for S sample microstates taken in ‘ground-state’ (a,b) and ‘monopole’ (c,d) orientations.e-h) Simulated S sample microstate spectra for ‘ground-state’ (e,f) and ‘monopole’ (g,h) orientations.i) Mode peak-extractions for all S sample microstate-spectra.j-l) Experimental spectra for HDS sample microstates taken in ‘ground-state’ (j,k) and ‘monopole’ (l) orientations. ‘Monopole’orientation signal-to-noise is lower due to array-waveguide alignment issues. Modes are still well-resolved and correspond wellwith simulation.m-o) Simulated HDS sample microstate spectra for ‘ground-state’ (m,n) and ‘monopole’ (o) orientations.p) Mode peak-extractions for all HDS sample microstate-spectra. lignment to H ext . Here, the Gaussian spread of H c throughout the system due to nanofabrication imperfections (termedquenched disorder ) is leveraged for state-preparation. By selecting a 21 mT field at the centre of the H c distribution,half the wide-bars are reversed and on average the system placed in a type 3 state, with a random distribution of ± ◦ wide-barsreversed. While sweeping field back from 21 mT the thin-bar exhibits negative ∂ f ∂ H . The wide-bar mode is split into reversedand unreversed modes exhibiting opposite ∂ f ∂ H sign. The two modes should cross at 5 mT if no deviation from Kittel-likebehaviour is observed. However, the modes are bent away from each other around 5 mT with an anticrossing frequency gapremaining between them. The gap is observed in both experimental and simulated (fig. 3 g) spectra with 0.27 GHz width, and acorresponding 0.30 GHz gap in the HDS type 3 spectra (experimental and simulated in fig. 3 l and p respectively). Whereaspreviously discussed mode-frequency shifting occurs due to magnetostatic inter-bar interactions, i.e. the microstate-dependentdipolar field landscapes giving different H loc values for the Kittel equation, mode anticrossings are an effect of dynamicmode-hybridisation . High-resolution anticrossing spectra are shown in figure 4 with accompanying discussionbelow. In addition to the anticrossing the type 3 state offers a high-degree of spectral control, with 3 active modes and tunablemode-gaps.Figure 3 d) shows the type 4 state, prepared at 22 mT. Qualitatively the spectra resembles that of the type 1 state but modesexhibit enhanced separation due to different local dipolar field landscape and are redshifted relative to type 1. This is bestvisualised through peak extractions shown in fig. 3 i). 0.4 and 0.2 GHz mode gaps at 22 mT are observed for type 4 and type 1states respectively, along with a 0.35 GHz blueshift of the type 1 wide mode relative to type 4 demonstrating the fine controlavailable. At -12 mT one ± ◦ subset of wide-bars reverses, preparing a type 3 state with accompanying mode shifts andwide-bar mode-splitting. The remaining wide subset reverses at -16 mT, preparing a type 2 state.The HDS sample exhibits qualitatively similar behaviour to the S sample with increased magnitude microstate-dependentfrequency shifts due to stronger inter-element interaction. Figure 3 j) shows the ‘ground-state’ orientation HDS sample preparedin a type 1 state at 30 mT. Here a 0-3.6 GHz mode gap is observed over a -12 - 30 mT field range. Figure 3 k) shows a type 2state prepared at 16 mT, exhibiting a constant 2 GHz gap across the field sweep. Figure 3 l) shows the system in a thin-barmajority type 3 state at 21 mT. The reversed and unreversed wide-bar modes exhibit opposite gradient with a 7.3 GHz crossingat 6 mT. Microstate-dependent mode hybridisation and anticrossings
In the type 3 spectra (fig. 3 c),g),l) and p), where reversed and unreversed wide-bar modes approach a single frequency theydo not overlap. The modes instead bend away from a Kittel-like form as they approach, leaving an anticrossing gap which has been predicted to occur in ASI due to a microstate-dependent band structure . Figure 4 shows the anticrossingregion in type 3 microstates. Samples were prepared in type 3 states in ground-state and monopole orientations, spectrameasured while negatively-sweeping field. Mode-bending and anticrossings are exhibited in all experimental spectra (fig.4 a-d) and corresponding simulations (fig. 4 f-h). Experimental spectra show anticrossing gaps of ∆ S , GS = .
22 GHz, ∆ S , monopole = .
27 GHz, ∆ HDS , GS = .
22 GHz, ∆ HDS , monopole = . H loc values along the H ext -axis due to broken microstate symmetry and resultinglycrossings occur at different fields, ground-state at -1 mT (-1 mT), ‘monopole’ at 3 mT (5 mT) in S (HDS) samples. Theantiferromagnetic macrospin ordering in type 3 states is crucial for mode-hybridisation. The difficulty in preparing suchantiferromagnetic states is a key barrier to observing dynamic coupling effects such as anticrossings, and a key strength of themicrostate-access protocol presented here.The microstate control demonstrated allows tailoring of spectra such that modes may also cross with no resolvableanticrossing. In type 1 states (fig. 1 g,k), crossings are observed between thin and wide-bar modes with no observable gapor deviation from Kittel-like behaviour. Simulated spectra of the crossing point (fig. 4 g) show no anticrossing gap andspatial power maps (fig. 4 h) show a single-node bulk mode throughout the type 1 field range. While the type 1 state exhibitsantiferromagnetic order between the thin and wide bars, it occurs at weaker effective interaction than type 3 states as thewide-thin bar vertex separation and dipolar-coupling are reduced relative to the type 3 wide-wide bar case. The lack of aresolvable type 1 anticrossing is testament to the sensitivity of dynamic coupling phenomena to interaction strength. igure 4. Mode-hybridisation and anticrossings.a-d) Negative-swept field dependent experimental FMR spectra of type 3 states for S and HDS samples in ground-state and‘monopole’ orientations. Red scatter points are peak extractions of the upper mode branch, blue points extractions of the lowerbranch. Mode frequency gaps or anticrossings and mode bending in the field range around the crossing point are observed in allsamples and orientations. Monopole-orientation crossing points (panels b and d) are offset in positive H ext due to the net H loc atthe vertex, ground state orientation crossings (a and c) occur at H ext = H loc .e) Simulated spectra of monopole-orientation HDS sample. Anticrossing gap of 0.32 GHz is observed at 4.3 mT.f) Simulated spatial mode-power maps for monopole-orientation HDS sample. Maps I-VI relate to corresponding pointslabelled on spectra in panel e). High-frequency, single-node mode branch is seen in I-III. Low-frequency, multi-node branch isIV-VI. g) Simulated type 1 spectra of ground-state orientation HDS sample. Crossing occurs at 24 mT with no observable gap.f) Simulated spatial mode-power maps for ground-state orientation type 1 HDS sample. Maps I-V relate to correspondingpoints labelled on spectra in panel e). Mode-hybridisation is not observed, with matching profile pairs I and V, IV and III. onclusions We have demonstrated that introducing a width-modified sublattice to ASI permits rapid, scalable and reconfigurable controlover rich and diverse spectral features. This approach offers an attractive addition to the host of spectral and microstatecontrol methodologies, requiring only widely-available global-field and nanofabrication protocols. The state-dependent spectraobserved suggest microwave-assisted state preparation as a promising direction for integrated read-write functionality.The magnitude and diversity of microstate-dependent mode and gap control exhibited invite a host of functional applicationsincluding tunable microwave filters and enable further study of how spin-wave characteristics and band structure of nanomagneticsystems may be employed in magnonic logic and neuromorphic devices . In particular, the observation of previously elusivemicrostate-dependent mode-hybridisation and anticrossings suggests a magnonic device which may reconfigurably transmitor reflect spin-waves depending on its state. In this regard we emphasise that anticrossing behaviour depends only on themicrostate and does not require width-modification except as a means for microstate access. As state-preparation techniquesdevelop, we expect mode-hybridisation to become observable and exploitable in other artificial spin systems. Author contributions
JCG, AV, TD and WRB conceived the work.JCG and KDS fabricated the samples.AV performed experimental MOKE and FMR measurements and analysis.AV, JCG and KDS performed MFM measurements.TD wrote code for simulation of the magnon spectra and performed micromagnetic simulations. Noticed avoided crossings insimulation prompting further investigation into experimental data.DMA wrote code for simulation of the magnon spectra.JCG drafted the manuscript, with contributions from all authors in editing and revision stages.
Acknowledgements
This work was supported by the Leverhulme Trust (RPG-2017-257) to WRB.TD and AV were supported by the EPSRC Centre for Doctoral Training in Advanced Characterisation of Materials (Grant No.EP/L015277/1).Simulations were performed on the Imperial College London Research Computing Service .The authors would like to thank Professor Lesley F. Cohen of Imperial College London for enlightening discussion andcomments, and David Mack for excellent laboratory management. Competing interests
The authors declare no competing interests.
Data availability statement
The datasets generated during and/or analysed during the current study are available from the corresponding author onreasonable request.
Supplementary Information
Simulation details
Simulations were performed using MuMax3. To maintain field sweep history, ground state files are generated in a separatescript and used as inputs for dynamic simulations. S sample dimensions are; wide: 800 by 230 by 20 nm, narrow: 800 by130 by 20 nm and lattice parameter: 1120 nm (gap = 160 nm). HDS sample dimensions are; wide: 600 by 200 by 20 nmand narrow: 600 by 130 by 20 nm and lattice parameter: 800 nm (gap = 100 mn). Perturbation of dimensions from SEMimages were introduced to more accurately reproduce both static and dynamic magnetisation behaviour. Material parametersfor NiFe used are; saturation magnetisation, M sat = 750 kA/m, exchange stiffness. A ex = 13 pJ and damping, α = 0.001 Allsimulations are discretized with lateral dimensions, c x , y = 5 nm and normal direction, c z = 10 nm and periodic boundaryconditions applied to generate lattice from unit cell. A broadband field excitation sinc pulse function is applied along z-directionwith cutoff frequency = 20 GHz, amplitude = 0.5 mT. Simulation is run for 25 ns saving magnetisation every 25 ps. Staticrelaxed magnetisation at t = 0 is subtracted from all subsequent files to retain only dynamic components, which are then subjectto a FFT along the time axis to generate a frequency spectra. Power spectra across the field range are collated and plotted as acolour contour plot with resolution; ∆ f = 40 MHz and ∆ µ H = 1 mT. Spatial power maps are generated by integrating over arange determined by the full width half maximum of peak fits and plotting each cell as a pixel whose colour corresponds to itspower. Each colour plot is normalised to the cell with highest power. High-resolution simulations performed for figure 4 have ower damping, α = . ∆ f = 10MHz and ∆ µ H = 0.2 mT. H ext is offset from the array ˆ x , ˆ y -axes by 1 ◦ to better match experiment. Experimental methods
Samples were fabricated via electron-beam lithography liftoff method on a Raith eLine system with PMMA resist. Ni Fe (permalloy) was thermally evaporated and capped with Al O . A ‘staircase’ subset of bars was increased in width to reduce itscoercive field relative to the thin subset, allowing independent subset reversal via global field.Ferromagnetic resonance spectra were measured using a NanOsc Instruments cryoFMR in a Quantum Design PhysicalProperties Measurement System. Broadband FMR measurements were carried out on large area samples ( ∼ × ) mounted flip-chip style on a coplanar waveguide. The waveguide was connected to a microwave generator, coupling RFmagnetic fields to the sample. The output from waveguide was rectified using an RF-diode detector. Measurements were donein fixed in-plane field while the RF frequency was swept in 20 MHz steps. The DC field was then modulated at 490 Hz with a0.48 mT RMS field and the diode voltage response measured via lock-in. The experimental spectra show the derivative outputof the microwave signal as a function of field and frequency. The normalised differential spectra are displayed as false-colourimages with symmetric log colour scale.Magnetic force micrographs were produced on a Dimension 3100 using commercially available normal-moment MFM tips.MOKE measurements were performed on a Durham Magneto-Optics NanoMOKE system. The laser spot is approximately20 µ m diameter. The longitudinal Kerr signal was normalised and the linear background subtracted from the saturatedmagnetisation. The applied field is a quasistatic sinusoidal field cycling at 11 Hz and the measured Kerr signal is averaged over300 field loops to improve signal to noise. Positions of the MOKE magnetization plateaux
Bar reversal occurs by a domain wall nucleation process at a field determined by the aspect ratios of the bars. The wide barsubset can be characterized by a mean coercive field H c , and a magnetization M wide and the thin bar subset by a mean coercivefield H c and a magnetization M thin . Because the other bar dimensions are identical the ratio of the volumes and magnetisationsis the same as the ratio of the bar widths. For a sample with all bars identical, the type 1 and type 4 state should have zero netmagnetization. For the width modified sample the different volumes of reversed and unreversed bars would be expected (forperfect Ising spins and nominal bar widths) to give M / M type = M wide − M thin M wide + M thin = t wide − t thin t wide + t thin = 0.226 for the S sample and 0.231 forthe HDS sample (for both type 1 and type 4). The relative magnetization of the ground-state minor loop to the saturated majorloop at zero field is approx. 0.3 in the S sample and 0.2 in the HDS sample. For the type 4 state it is approx. 0.5 and 0.6 forS and HDS respectively. In a type 3 that is formed by both wide bars switching and triggering one thin bar to also reverse,then M / M S would be t wide t wide + t thin = 0.613 for S and 0.615 for HDS. Imperfections in the nanofabrication (quenched disorder)give the sublattice switching fields a Gaussian distribution about the mean, with a standard deviations σ wide around H c and σ thin around H c . If bars were all sufficiently spaced to be not interacting then desired states could be accessed by applying H ext = H c + H c / H c − H c >> σ wide + σ thin . However we are in the strongly-interacting regime and so each barexperiences an effective field H e f f = H app + H loc . The reversal of wide bars will change H loc experienced by the thin bars. If ∆ H loc increases H e f f then this makes it more difficult to realise the ordered state, as do the cases where we are preparing a type4 state. Where we are writing the type 1 state from the saturated type 2 state, ∆ H loc decreases H e f f and so the interactionsincrease the operating window where the ordered state may be prepared. The difference can be seen in the MOKE hysteresisloops in fig. 1 c,e) and 1 d,f). In fig. 1 c,e) we have very clear plateaux in the major (blue) hysteresis loops and can very easilyand reproducibly send minor loops to the type 1 microstate and back to saturated. Note that the data is the average of thousandsof individual loops and so the sharp switching and flat plateuax show there is no significant stochasticity in this major hysteresisloop and we go through the same microstates at the same fields in each measurement. Similarly in the minor (orange) loops werepeatedly go the the same expected plateau magnetization. For the same sample, with the same extrinsic disorder and sigmas,in the monopole geometry, switching the wide bars causes the dipolar field of all neighbouring bars to help the reversal of thethin bars and so the two Gaussian distributions start to overlap. We know from MFM we can access large areas of pure type 4with the correct protocol, but the hysteresis loop shows very broad reversal with no clear plateaux. It is not clear from our datawhether the broadening we see is from averaging similar broad loops or different loops with sharper individual features. Thedisorder could be spatial within the measurement spot, temporal with loop cycle number or both. Certainly there is a significantstochastic contribution in the measurement. References Kruglyak, V., Demokritov, S. & Grundler, D. Magnonics.
J. Phys. D: Appl. Phys. , 264001 (2010). Lenk, B., Ulrichs, H., Garbs, F. & Münzenberg, M. The building blocks of magnonics.
Phys. Reports , 107–136 (2011). . Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics.
Nat. Phys. , 453–461 (2015). Tabuchi, Y. et al.
Hybridizing ferromagnetic magnons and microwave photons in the quantum limit.
Phys. Rev. Lett. ,083603 (2014). Zhang, X., Zou, C.-L., Jiang, L. & Tang, H. X. Cavity magnomechanics.
Sci. advances , e1501286 (2016). Kalinikos, B. & Slavin, A. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchangeboundary conditions.
J. Phys. C: Solid State Phys. , 7013 (1986). Liensberger, L. et al.
Exchange-enhanced ultrastrong magnon-magnon coupling in a compensated ferrimagnet.
Phys. Rev.Lett. , 117204 (2019). Shiota, Y., Taniguchi, T., Ishibashi, M., Moriyama, T. & Ono, T. Tunable magnon-magnon coupling mediated by dynamicdipolar interaction in synthetic antiferromagnets.
Phys. Rev. Lett. , 017203 (2020). Sud, A. et al.
Tunable magnon-magnon coupling in synthetic antiferromagnets.
Phys. Rev. B , 100403 (2020).
Chumak, A. V., Serga, A. A. & Hillebrands, B. Magnon transistor for all-magnon data processing.
Nat. communications ,1–8 (2014). Vogt, K. et al.
Realization of a spin-wave multiplexer.
Nat. communications , 1–5 (2014). Stenning, K. D. et al.
Magnonic bending, phase shifting and interferometry in a 2d reconfigurable nanodisk crystal.
ACSnano (2020).
Grundler, D. Reconfigurable magnonics heats up.
Nat. Phys. , 438 (2015). Topp, J., Heitmann, D., Kostylev, M. P. & Grundler, D. Making a reconfigurable artificial crystal by ordering bistablemagnetic nanowires.
Phys. review letters , 207205 (2010).
Krawczyk, M. & Grundler, D. Review and prospects of magnonic crystals and devices with reprogrammable band structure.
J. Physics: Condens. Matter , 123202 (2014). Haldar, A., Kumar, D. & Adeyeye, A. O. A reconfigurable waveguide for energy-efficient transmission and localmanipulation of information in a nanomagnetic device.
Nat. nanotechnology , 437 (2016). Barman, A., Mondal, S., Sahoo, S. & De, A. Magnetization dynamics of nanoscale magnetic materials: A perspective.
J.Appl. Phys. , 170901 (2020).
Wang, . R. et al.
Artificial ‘spin ice’in a geometrically frustrated lattice of nanoscale ferromagnetic islands.
Nature ,303–306 (2006).
Lendinez, S. & Jungfleisch, M. Magnetization dynamics in artificial spin ice.
J. Physics: Condens. Matter , 013001(2019). Gliga, S., Iacocca, E. & Heinonen, O. G. Dynamics of reconfigurable artificial spin ice: Toward magnonic functionalmaterials.
APL Mater. , 040911 (2020). Skjærvø, S. H., Marrows, C. H., Stamps, R. L. & Heyderman, L. J. Advances in artificial spin ice.
Nat. Rev. Phys. , 13–28(2020). Talapatra, A., Singh, N. & Adeyeye, A. Magnetic tunability of permalloy artificial spin ice structures.
Phys. Rev. Appl. ,014034 (2020). Morgan, J. P., Stein, A., Langridge, S. & Marrows, C. H. Thermal ground-state ordering and elementary excitations inartificial magnetic square ice.
Nat. Phys. , 75–79 (2011). Gartside, J. C. et al.
Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing.
Nat. nanotechnology , 53 (2018). Ladak, S., Read, D., Perkins, G., Cohen, L. & Branford, W. Direct observation of magnetic monopole defects in an artificialspin-ice system.
Nat. Phys. , 359–363 (2010). Mengotti, E. et al.
Real-space observation of emergent magnetic monopoles and associated dirac strings in artificialkagome spin ice.
Nat. Phys. , 68–74 (2011). Gliga, S., Kákay, A., Hertel, R. & Heinonen, O. G. Spectral analysis of topological defects in an artificial spin-ice lattice.
Phys. review letters , 117205 (2013).
Zhou, X., Chua, G.-L., Singh, N. & Adeyeye, A. O. Large area artificial spin ice and anti-spin ice ni80fe20 structures:static and dynamic behavior.
Adv. Funct. Mater. , 1437–1444 (2016). Arroo, D. M., Gartside, J. C. & Branford, W. R. Sculpting the spin-wave response of artificial spin ice via microstateselection.
Phys. Rev. B , 214425 (2019).
Dion, T. et al.
Tunable magnetization dynamics in artificial spin ice via shape anisotropy modification.
Phys. Rev. B ,054433 (2019).
Iacocca, E., Gliga, S., Stamps, R. L. & Heinonen, O. Reconfigurable wave band structure of an artificial square ice.
Phys.Rev. B , 134420 (2016). Bang, W. et al.
Influence of the vertex region on spin dynamics in artificial kagome spin ice.
Phys. Rev. Appl. , 014079(2020). Sklenar, J., Bhat, V., DeLong, L. & Ketterson, J. Broadband ferromagnetic resonance studies on an artificial square spin-iceisland array.
J. Appl. Phys. , 17B530 (2013).
Gartside, J., Burn, D., Cohen, L. & Branford, W. A novel method for the injection and manipulation of magnetic chargestates in nanostructures.
Sci. reports , 32864 (2016). Wang, Y.-L. et al.
Rewritable artificial magnetic charge ice.
Science , 962–966 (2016).
Lehmann, J., Donnelly, C., Derlet, P. M., Heyderman, L. J. & Fiebig, M. Poling of an artificial magneto-toroidal crystal.
Nat. nanotechnology , 141–144 (2019). Gartside, J. C. et al.
Current-controlled nanomagnetic writing for reconfigurable magnonic crystals.
Commun. Phys. , 1–8(2020). Wang, Y.-L. et al.
Switchable geometric frustration in an artificial-spin-ice–superconductor heterosystem.
Nat. nanotech-nology , 560–565 (2018). Iacocca, E., Gliga, S. & Heinonen, O. G. Tailoring spin-wave channels in a reconfigurable artificial spin ice.
Phys. Rev.Appl. , 044047 (2020). Lendinez, S., Kaffash, M. T. & Jungfleisch, M. B. Emergent spin dynamics enabled by lattice interactions in a bicomponentartificial spin ice. arXiv preprint arXiv:2010.03008 (2020).
Parakkat, V. M., Macauley, G. M., Stamps, R. L. & Krishnan, K. M. Configurable artificial spin ice with site-specific localmagnetic fields.
Phys. Rev. Lett. , 017203 (2021).
Nisoli, C. et al.
Ground state lost but degeneracy found: The effective thermodynamics of artificial spin ice.
Phys. reviewletters , 217203 (2007). Möller, G. & Moessner, R. Magnetic multipole analysis of kagome and artificial spin-ice dipolar arrays.
Phys. Rev. B ,140409 (2009). Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice.
Nature , 42–45 (2008).
Farhan, A. et al.
Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice.
Sci. advances ,eaav6380 (2019). Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, coulomb phase and magnetic monopoles in artificialsquare ice.
Nature , 410–413 (2016).
Kittel, C. On the theory of ferromagnetic resonance absorption.
Phys. review , 155 (1948). Gurevich, A. G. & Melkov, G. A.
Magnetization oscillations and waves (CRC press, 1996).
Libál, A., Reichhardt, C. O. & Reichhardt, C. Creating artificial ice states using vortices in nanostructured superconductors.
Phys. review letters , 237004 (2009).
Budrikis, Z., Politi, P. & Stamps, R. L. A network model for field and quenched disorder effects in artificial spin ice.
NewJ. Phys. , 045008 (2012). Tacchi, S. et al.
Forbidden band gaps in the spin-wave spectrum of a two-dimensional bicomponent magnonic crystal.
Phys. review letters , 137202 (2012).
MacNeill, D. et al.
Gigahertz frequency antiferromagnetic resonance and strong magnon-magnon coupling in the layeredcrystal crcl 3.
Phys. review letters , 047204 (2019).
Yuan, H., Zheng, S., Ficek, Z., He, Q. & Yung, M.-H. Enhancement of magnon-magnon entanglement inside a cavity.
Phys. Rev. B , 014419 (2020).
Thirion, C., Wernsdorfer, W. & Mailly, D. Switching of magnetization by nonlinear resonance studied in single nanoparti-cles.
Nat. materials , 524–527 (2003). Podbielski, J., Heitmann, D. & Grundler, D. Microwave-assisted switching of microscopic rings: Correlation betweennonlinear spin dynamics and critical microwave fields.
Phys. review letters , 207202 (2007). Nembach, H. et al.
Microwave assisted switching in a ni 81 fe 19 ellipsoid.
Appl. Phys. Lett. , 062503 (2007). Bhat, V. et al.
Magnon modes of microstates and microwave-induced avalanche in kagome artificial spin ice withtopological defects.
Phys. Rev. Lett. , 117208 (2020).
Chumak, A., Serga, A. & Hillebrands, B. Magnonic crystals for data processing.
J. Phys. D: Appl. Phys. , 244001(2017). Grollier, J. et al.
Neuromorphic spintronics.
Nat. Electron.