Spectral fingerprinting: microstate readout via remanence ferromagnetic resonance in artificial spin ice

WM-S and WM-HDS samples, shown in figures 1(b) and (c) are square ASI with sublattices of wide (w₁) and thin (w₂) bars. Wider bars have lower coercivity (H_c1 < H_c2), allowing microstate control via the application of global field [19].

Mounting the width-modified samples along the crystallographic [11] axis and starting from a field saturated state type 2 (figure 1(f)) prepares a type 1 microstate (figure 1(g)) by applying a field such that only wide bars reverse. This is the system ground state (GS) [32–34] due to maximum dipolar flux closure, hence we term this field axis the 'GS' orientation. Rotating the samples and applying H_ext along $[1\bar{1}]$ direction (figures 1(a) and (b)) and reversing wide bars prepares the type 4 state (figure 1(h)), with 4 like polarity magnetic charges at each vertex. This results in highly unfavourable dipolar field interactions, termed the 'monopole state (MS)' [35–37]. We hence term this sample mounting the MS orientation. If there is any slight angular misalignment in the sample from the $[1\bar{1}]$ direction, the MS orientation also prepares type 3 states (figure 1(g)) with 3 like polarity and 1 opposite polarity vertex charges.

MOKE hysteresis loops for WM samples are shown in figure 2(a)–(d). Figures 2(a) and (b) show the GS orientation WM-S (a) and WM-HDS (b) loops. Figures 2(c) and (d) show the monopole orientation WM-S (c) and WM-HDS (d) loops. The shape of the hysteresis loop changes significantly between the two orientations. The GS loops show two sharp steps in magnetisation at fields H_c1 and H_c2 corresponding to wide and thin bar reversal respectively. The magnetisation plateau between H_c1 and H_c2 corresponds to the type 1 state. MS hysteresis loops show gradual magnetisation reversal due to the energetically unfavourable dipolar field landscapes of type 3 and type 4 states. Here a clear plateau is absent, with kinks in the magnetisation curve revealing locations of the type 4 state in the WM-S sample (21 mT) and type 3 state in the WM-HDS (20 mT). The dipolar interaction in the WM-HDS sample is too strong for a pure type 4 state to be observed.

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The remanence traces on the MOKE loops (coloured lines) leading from the major loop back to zero-field, showing the resultant magnetisation after the microstate preparation protocols. Figures 2(e)–(h) show remanence FMR spectra corresponding to microstates prepared by the remanence traces shown in figures 2(a)–(d). The spectral plots show two major absorption peaks at $\sim 7$ GHz and $\sim 8.6$ GHz corresponding to bulk centre-localized modes in wide (M₁) and thin (M₂) bars respectively. Spectra shown in figures 2(e)–(h) exhibit characteristic changes in the peak profiles of M₁ and M₂, both in terms of resonant frequency f₀ and differential amplitude $\frac{\partial P}{\partial H}$ depending on the microstate.

With H_ext along the [11] orientation, samples were saturated into the type 2− state by applying an initial −200 mT field. The preparation field H_prep was then stepped from 0–40 mT in 1 mT steps, measuring zero-field spectra after each field. Figures 3(a) and (b) show differential spectral heatmaps of the preparation field against FMR frequency. When wide bars reverse at H_c1, the sample switches from the type 2− to the type 1 state. Thin bars reverse at H_c2, switching from the type 1 to the type 2+ state. Due to the stable energetics of the type 1 GS, H_c2 is increased in this configuration relative to an isolated thin bar, broadening the type 1 field window.

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As the sample enters a type 1 state the local dipolar field landscape shifts, blueshifting both modes in figures 3(d) and (e). M₁ blueshifts by 60 ± 7 MHz (WM-S) and 70 ± 5 MHz (WM-HDS) while M₂ blueshifts 46 ± 8 MHz (WM-S) and 111 ± 8 MHz (WM-HDS). Due to the smaller lattice parameter and larger dipolar field of WM-HDS, frequency shifts are enhanced relative to the WM-S sample. Similarly, this explains the relative difference in shift magnitude between M₁ and M₂. The dipolar field emanating from the wide bar is stronger due to its larger volume, so it induces a greater frequency shift on the thin bar, while the frequency shift in the M₁ mode is smaller.

Fitting the in-field FMR response with the Kittel equation to M₁ and M₂ modes, we extract the relative shift in the dipolar field from the f₀ shift when the microstate changes from type 2 to type 1. For the WM-S (WM-HDS) samples, this gives 1.9 ± 0.1 mT (4.6 ± 0.1 mT) for the thin bar and 2.5 ± 0.1 mT (2.9 ± 0.1 mT) for the wide bar. Point dipole simulations estimate type 2 to type 1 dipolar field shifts as 1.9 mT (4.6 mT) and 4.5 mT (7.2 mT) at the centre point of the thin and wide bars respectively for WM-S (WM-HDS) samples. The WM-S sample demonstrating efficacy of spectral fingerprinting at providing an absolute measurement of dipolar field textures. The discrepancy in the WM-HDS may arise from significant edge-curling, suggesting that the use of the field at the island centre instead of integrating over the whole mode area is not a good approximation for strongly-interacting samples.

The extracted amplitude of M₁ and M₂ are shown in figures 3(c) and (d) along with remanence magnetisation M_r for WM-S and WM-HDS samples. Remanence magnetisation was measured via MOKE using different H_prep and taking remanence curves, figures 2(a)–(d). H_c1 and H_c2 match with the sign change of remanence FMR measurements, allowing effective magnetisation measurement of each bar subset.

Rotating the sample along the MS orientation, samples are saturated in type 2− state at −200 mT and zero-field FMR spectra are again measured after applying preparation fields from 0–40 mT with 1 mT steps. Figures 4(a) and (b) shows spectral heatmaps of the preparation field against FMR frequency. Two modes are observed corresponding to wide (lower frequency) and thin bars (higher frequency). Type 4 vertices are energetically unfavourable hence transition in and out of this state occurs via the type 3 state (figure 4(c)). The field window for pure type 4 microstates is further reduced by the dipolar field from reversed wide bars lowering H_c2. Like the hysteresis loops shown in figures 2(c) and (d), the MS-orientation heatmaps show a continuous frequency shift in the M₁ and M₂ modes.

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As the population of type 3 and type 4 vertices grows the local dipolar field landscape changes, shifting the resonant frequency of both modes figures 4(c) and (d). Due to quenched disorder, type 3+ vertices populate the sample gradually. Type 3+ vertices increase M₁ mode splits as the reversed wide bars blueshift and the M₂ mode gradually redshifts. Once H_prep > H_c1 the population of type 4 vertices increases, resulting in greater blueshift in M₁ and redshift in M₂ until the maximum type 4 population is reached. For the WM-S (WM-HDS) samples total M₁ blueshift is 248 ± 4 MHz (348 ± 19 MHZ) while the M₂ redshifts 206 ± 15 MHz (249 ± 7 MHz). In the MS orientation the resonance frequency continuously shifts due to gradually changing the average local dipolar field from the ensemble of microstates throughout the reversal process. The mix of vertex population and quenched disorder leads to a more disordered transition into and out of type 4 and consequently creates a more varied dipolar field landscape.

Using Kittel fits, the relative MS orientation shift in dipolar field between type 2 and type 4 states is 8.6 ± 0.2 mT (10.6 ± 0.1 mT) and 10.5 ± 0.1 mT (15.0 ± 0.3 mT) in WM-S (WM-HDS) for thin and wide bars respectively. Point dipole modelling estimates bar centre-point shifts of 7.2 mT (11.6 mT) and 4.5 mT (7.3 mT) for WM-S (WM-HDS) in thin and wide bars respectively. The simulated dipolar field estimate underestimates the dipolar field shift in the wide bars again due to neglecting realistic magnetisation textures such as edge curling influence on the mode shift.

Extracted M₁ and M₂ amplitudes are shown in figures 4(e) and (f) along M_r for WM-S and WM-HDS samples. The MS orientation remanence magnetisation curve does not show a distinct magnetisation plateau due to the similar coercive fields of the thin and wide bars in this direction. In FMR measurements the distinct resonance frequency of the thin and wide bars allows each subset to be probed individually, in contrast to conventional magnetometry which can only access the bulk magnetisation. The FMR differential amplitude of M₁ and M₂ correspond well with the remanence magnetisation measurement.

We now apply spectral fingerprinting to the S-S sample, while width-modified samples follow a well-defined microstate trajectory during reversal, the S-S sample evolves through disordered microstates with mode frequencies determined solely by local dipolar field texture and quenched disorder.

Figure 5(a) shows the MOKE hysteresis loop with highlighted remanence magnetisation traces at a range of H_prep and corresponding to the prepared microstates identified via MFM images (figures 5(c)–(e)) and differential FMR spectra (figure 5(b)). The sample was saturated along −x before measuring MFM and FMR. Spectra exhibit two dominant modes corresponding to unreversed ${M}_{1}^{-}$, (−x magnetized, 20 mT trace) and reversed ${M}_{1}^{+}$ (+x magnetized, 30 mT trace) bars, with partially reversed microstates a combination of both (i.e. 24.4 mT trace). Figure 5(f) shows remanence spectral heatmaps with H_prep = 20–30 mT, 0.2 mT steps. We observe 2 additional lower intensity modes in addition to the dominant bulk centre-localized mode M₁, corresponding to bulk edge-localized E₁ and the edge E₂ modes with simulated spatial powermaps for each mode. All modes exhibit characteristic sign change (figure 5(g)) of $\frac{\partial f}{\partial H}$ around H_c associated with magnetisation reversal.

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At remanence, isolated reversed and unreversed bars will have the same resonant frequency. However, in a lattice the surrounding bars and lattice imperfections lead to resonant frequency shifts throughout reversal. Within the coercive field distribution both ${M}_{1}^{-}$ and ${M}_{1}^{+}$ contribute and are identified via careful fitting of two opposite sign differential Lorentzian curves. The extracted f₀ figure 5(f) shows a characteristic asymmetric frequency shift around H_c from the combination of local dipolar field texture and quenched disorder.

The distribution of coercive fields over the sample follows a distribution of bar dimensions due to imperfections in the nanofabrication (quenched disorder) which varies the demagnetisation factors and hence f₀. This can be interpreted as a distribution of widths with on average wider bars having lower H_c and lower f₀. These bars will typically reverse at lower field and upon reversal will be surrounded by unreversed neighbouring bars, experiencing local dipolar field oriented opposite their magnetisation. This reduces H_eff and hence redshifts f₀. The total frequency shift for reversed bars is a sum of the redshift due to quenched disorder and local dipolar field ${\Delta}{f}_{0}^{+}=-{\Delta}{f}_{0}^{\text{QD}}-{\Delta}{f}_{0}^{\mathrm{d}\mathrm{i}\mathrm{p}}$. The corollary of these arguments is that bars reversing at higher fields typically have higher shape anisotropy and hence higher f₀. The dipolar field frequency shift will still result in an f₀ redshift as the symmetry of the argument remains the same and so the total frequency shift for unreversed bars is ${\Delta}{f}_{0}^{-}={\Delta}{f}_{0}^{\text{QD}}-{\Delta}{f}_{0}^{\mathrm{d}\mathrm{i}\mathrm{p}}$. This asymmetry in the contribution of the dipolar field to frequency shift results in ${M}_{1}^{-}$ showing a smaller frequency shift than ${M}_{1}^{+}$ throughout reversal. In figure 5(g) simulated curves generated via point dipole simulation are superimposed for the reversed (red) and unreversed (blue) bar populations, including quenched disorder of 4% and converting H_eff at each bar to f₀ via results from MuMax3 simulation. Close correspondence is observed between measured and simulated behaviour, confirming the attribution of the initial low field increase in f₀ due to quenched disorder and the later higher field sharp decrease in f₀ to the local dipolar field texture. Crucially, this allows absolute measurements of both. Deconvoluting these effects in strongly interacting magnetic nanoarrays is historically extremely challenging, and the demonstration here is a key strength of the spectral fingerprinting method.

Figure 5(h) shows a comparison of MOKE remanence magnetisation measurement (black curve, left y-axis) with M₁ FMR amplitude. As in figures 3 and 4(f)–(g), an extremely close correspondence is observed—illustrating the effectiveness of spectral fingerprinting at elucidating not just fine microstate details, but also the system magnetisation.

Here we compare three M = 0 microstates; prepared by applying H_prep = H_c after negative saturation (figure 6(a)), AC field demagnetisation (figure 6(b)), and as grown (figure 6(c)). Macroscopic magnetisation measurements are unable to distinguish between these M = 0 states, but their remanence FMR spectra are distinct due to different local dipolar field textures in each microstate. The microstate in figure 6(a) is dominated by small domains of type 2 vertices magnetized in random directions, the microstate in figure 6(b) is a roughly equal mixture of type 1 and randomly oriented type 2 domains and the as grown sample is close to perfectly periodic type 1 order. Figures 6(d)–(f) shows the remanence spectra for the three microstates. Using the MFM images, the local field at the centre of each bar was estimated using dipolar simulations. The resonance frequency was estimated from the local dipolar field and Kittel fits of the S-S sample with the field. Summing together differential Lorentzian peaks for each macrospin gives the dipolar sim. curves shown in figures 6(d)–(f). This allows for the full microstate from the MFM images to be included in the spectral response. The H_c state spectra shows a low amplitude mode indicating a wide distribution of dipolar field landscapes with equal populations of negatively and positively magnetized bars, reflected in the low amplitude of the simulated curve.

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The demagnetized state shows a shift in the resonance frequency because of the high population of type 1 states but only a single magnetisation direction. The progression to the demagnetized state can be seen in figures 6(g) and (h) as remanence FMR spectra were measured throughout AC demagnetisation from 28 mT − 20 mT in 0.1 mT steps. The sample starts in a saturated type 2 state and as the preparation field is reduced, bars with higher-than-average coercive field lock into the last magnetisation state where the field was large enough to reverse. As the H_prep spectra in the positive field direction changes phase while in the negative field direction it remains the same and the net FMR power does not reach zero (figure 6(h)). This is evidence of residual magnetisation M_demag due to the imperfection of demagnetisation routines in reaching the GS [24, 38]. The different populations of type 1 and type 2 vertices results in the difference between the M = 0 spectra.

In the as grown state the vertices of square ASI will tend towards type 1 (figure 6(c)) as the lowest energy configuration. The spectrum (figure 6(f)) shows a superposition of ${M}_{1}^{+}$ and ${M}_{1}^{-}$ with a slight shift in frequency between the two modes, however, the antiferromagnetic ordering of the GS results in an identical dipolar field on both the positively and negatively magnetized bars. In zero-field this will result in a cancellation of the positively and negatively magnetized modes. However, incorporating a slight magnetic field of −0.5 mT, possibly due to trapped flux, into the simulation reproduces the splitting of the ${M}_{1}^{+}$ and ${M}_{1}^{-}$ modes. Despite the small sample area of the MFM measurements the simulated and experimental spectra for the M = 0 microstates agree very well. The ability to distinguish between equal magnetisation microstates using a fast and low power technique is essential for readout in reservoir computation.