Quantitative Decoding of Interactions in Tunable Nanomagnet Arrays Using First Order Reversal Curves

Interacting Case

• Measured and simulated FORC distributions agreed; stronger interactions lengthened the ridge in demagnetizing arrays but shortened it in magnetizing arrays.The low-HC ridge end shifted toward +HB in demagnetizing arrays and toward -HB in magnetizing arrays as interactions increased.Source: Results · pp. 3–4
• Demagnetizing arrays developed a low-HC edge toward negative HB and a negative feature near the high-HC end; the below-ridge negative feature was more prominent in magnetizing arrays.The demagnetizing edge forms a wishbone or boomerang with the ridge.Source: Results · pp. 3–4

Non-interacting Case

• Without interactions, the FORC distribution coincides with the coercivity distribution D(HK), forming a ridge along the HC axis at HB = 0.Each particle contributes once, when an unmatched dM/dH jump occurs on a FORC relative to its neighboring FORCs.Source: Results · p. 4

Interacting Case

• In the mean-field demagnetizing model, interactions shift the low-HC ridge end toward +HB by -αMS without shifting its HC coordinate.The common interaction field preserves the coercivity-governed flip order; the unmatched jump at the lowest coercivity defines this ridge end.Source: Results · p. 5
• The high-HC ridge end remains on the HC axis but stretches along it by -αMS; the H-axis projection retains the intrinsic coercivity distribution without interaction-induced distortion.The high-HC endpoint is defined after the rest of the system is negatively saturated.Source: Results · p. 6
• Interactions misalign low-HC up-flip jumps across neighboring FORCs, producing an edge whose tilt and asymmetry directly measure the width of D(HK).The edge arises from first up-flip field shifts that make the dM/dH jumps unmatched.Source: Results · pp. 6–7

• For a flat coercivity distribution, interactions align jumps from different particles on neighboring FORCs, leaving the region between the edge and ridge at zero and reproducing most measured features.The matching occurs because aligned jumps have the same values on neighboring FORCs.Source: Results · p. 7
• With Gaussian coercivity, partial matching of decreasing-magnitude jumps generates a negative feature; the ridge bends and the edge broadens.The negative region tracks the decreasing slope of the Gaussian distribution.Source: Results · p. 8

Nearest-neighbor correlations

• Nearest-neighbor interactions segment the FORC ridge as particles flip in varying neighbor configurations, shifting ridge pieces by different interaction fields.For sufficiently narrow coercivity distributions, reversal first forms a checkerboard pattern; subsequent flips occur in intermediate configurations.Source: Results · pp. 8–9

• Demagnetizing interactions shift the low-H_C ridge toward +H_B, stretch the high-H_C ridge along H_C without shifting it off-axis, and form a tilted low-H_C edge.The shifts and stretch are by -αM_S.Source: Results · p. 9

Magnetizing interactions

• Magnetizing interactions shift the low-H_C ridge toward -H_B and compress its high-H_C end without moving it off the H_C axis, while unmatched jumps form a negative edge.With Gaussian coercivity, the negative edge moves toward the positive ridge and the ridge curves; nearest-neighbor terms trigger avalanche reversal that collapses the ridge to a single value.Source: Results · p. 9

Quantifying Interaction Fields

• Experimental FORC field shifts agree remarkably well with calculated interaction fields, demonstrating quantitative predictive power and a way to extract numerical interaction-field values.The authors identify this as particularly important for disordered arrays, where calculating interactions is difficult.Source: Results · p. 10

• Unmatched jumps account for all experimentally observed FORC features, including shifted ridge-and-edge structure and negative features.Source: Discussion · p. 10
• Ridge-and-edge tilting, shifting, and stretching provide quantitative information about the system and demonstrate FORC's predictive power.Source: Discussion · p. 10
• The unmatched-jump construction treats (de)magnetizing interactions as (de)stabilizing cases, yielding an approach for evaluating any hysteretic system with FORC.Source: Discussion · p. 10

• Co nanoellipse arrays were fabricated by DC magnetron sputtering and measured at room temperature with MOKE over roughly 5,000 ellipses.Source: Methods · pp. 10–11
• FORCs were traced by sweeping from positive saturation to a reversal field, increasing the field back to saturation, and repeating at decreasing reversal fields.Source: Methods · p. 11
• The 100×100 dipole model aligned ellipses with their major axes, used experimental-system spacing and moments, and assigned coercivities from weakly interacting sample A3.Source: Methods · p. 11
• Dipolar fields were modeled as αM(H), calibrated at saturation, then extended with a nearest-neighbor cluster-expansion term.Source: Methods · p. 11
• In simulations, each dipole's interaction field was compared with its intrinsic coercivity for down- and up-flipping until all dipoles were stable.Source: Methods · p. 11

Experiments-Demagnetizing arrays

• Demagnetizing-array FORC families become more sheared as interactions strengthen.The reported progression is A3 to A2 to A1.Source: §12 · p. 20, Figure S1 · p. 21
• In demagnetizing arrays, the ridge shifts toward +HB at low HC but stays on HB=0 at high HC, where it lengthens with stronger interactions.The low-HC shift also increases with interaction strength.Source: §12 · p. 20, Figure S1 · p. 21
• Demagnetizing arrays develop an extended -HB edge and a negative feature; nearest-neighbor interactions segment both the ridge and edge.The edge attaches to the ridge at its low-HC end and extends with stronger interactions.Source: §12 · p. 21

Experiments-Magnetizing arrays

• Magnetizing-array FORC families become less sheared and more square as interactions strengthen, opposite to the demagnetizing-array trend.The reported progression is B3 to B2 to B1.Source: §12 · p. 21, Figure S2 · p. 22
• In magnetizing arrays, the ridge shifts toward -HB at low HC but remains on HB=0 at high HC, while its HC extent decreases with stronger interactions.The ridge becomes compressed as interaction strength increases.Source: §12 · pp. 21–22, Figure S2 · p. 22
• Magnetizing arrays develop a negative edge below the ridge, and simulations reproduce the experimental trends.Source: §12 · p. 22, Figure S2 · p. 22

Magnetizing interactions

• Magnetizing mean-field interactions shift the low-HC ridge end in the -HB direction, compress its high-HC end, and create a negative edge.For positive α, the low-end shift is -αMS and the high-end stretch of -αMS acts as a compression.Source: §12 · p. 23

Nearest Neighbor Interactions – demagnetizing model

• The demagnetizing nearest-neighbor model yields three primary and three secondary FORC peaks, with P1 and P2 strongest because their pathways pass through the lowest-energy antiferromagnetic state.P3 is next strongest and is associated with flips from the checkerboard state to saturation; P4–P6 are weaker because their pathways involve higher-energy frustrated configurations.Source: §12 · pp. 24–25
• In demagnetizing arrays, dipole reversals nucleate local checkerboard ordering that can organize into larger patterns, a tendency mean-field theory misses.The checkerboard is an energetically favorable intermediate state through which most reversal pathways pass.Source: §12 · pp. 25–26
• Combining mean-field and nearest-neighbor interactions reproduces the demagnetizing ridge-and-edge structure and six-peak pattern, explaining the observed FORC features.Nearest-neighbor terms segment the ridge and edge; the combined model captures both that structure and the six peaks.Source: §12 · pp. 24, 26

Nearest Neighbor Interactions – magnetizing model

• Magnetizing arrays lack the lower-energy checkerboard intermediate, so reversal proceeds directly between saturated states without strong segmentation.The nearest-neighbor-only model accordingly produces a single localized FORC structure.Source: §12 · pp. 26–27
• Magnetizing nearest-neighbor interactions trigger dipole-chain avalanches that produce a localized FORC peak, broadened somewhat by wide coercivity distributions.Each dipole flip destabilizes its neighbors; propagation stops at a high-coercivity particle, where HK>2Hn.n.+H.Source: §12 · p. 27

12 Wang, R. F. et al. Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale

• In the combined mean-field and nearest-neighbor-only model, the single avalanche peak overlaps the mean-field ridge, preventing segmentation and producing a well-formed ridge maximum at the peak.The combined model's FORC does not exhibit segmentation.Source: §12 · p. 27, Figure S3 · p. 28