Abstract
To develop a full understanding of interactions in nanomagnet arrays is a persistent challenge, critically impacting their technological acceptance. This paper reports the experimental, numerical and analytical investigation of interactions in arrays of Co nanoellipses using the firstorder reversal curve (FORC) technique. A meanfield analysis has revealed the physical mechanisms giving rise to all of the observed features: a shift of the noninteracting FORCridge at the lowH_{C} end off the local coercivity H_{C} axis; a stretch of the FORCridge at the highH_{C} end without shifting it off the H_{C} axis; and a formation of a tilted edge connected to the ridge at the lowH_{C} end. Changing from flat to Gaussian coercivity distribution produces a negative feature, bends the ridge, and broadens the edge. Finally, nearest neighbor interactions segment the FORCridge. These results demonstrate that the FORC approach provides a comprehensive framework to qualitatively and quantitatively decode interactions in nanomagnet arrays.
Introduction
Nanomagnet arrays are basic building blocks^{1} for key technologies such as ultrahigh density magnetic recording media^{2,3,4}, magnetic random access memory (MRAM)^{5,6}, and logic devices^{7,8,9}. Interactions within the arrays critically affect the functionalities of the nanomagnets as well as enable new device concepts. For instance, dipolar interactions may trigger an analog memory effect in nanowire arrays^{10}, enable digital computation in magnetic quantumdot cellular automata systems^{7,8,11}, lead to frustrations in artificial “spin ice”^{12,13,14}, or adversely affect thermal stability and switching field distribution in magnetic recording media or MRAM elements^{5,15,16,17}. Probing and managing these interactions is often difficult because they are longranged, anisotropic, and configurationdependent^{17}.
The firstorder reversal curve (FORC) method^{18,19} has provided detailed characterization for a variety of magnetic^{20,21,22,23,24,25,26,27} and other hysteretic systems^{28,29}. However, a coherent framework to interpret the features of the FORC diagrams and extract quantitative information is still lacking despite decades of effort by numerous groups. In this work, using the FORC method we have quantitatively investigated tunable interactions in model systems of single domain nanomagnet arrays. With meanfield level simulations, supplemented with a cluster extension, we have reproduced all features and trends of the experimental FORC diagrams quantitatively and identified their physical origins. Our approach decodes interactions in nanomagnet arrays, even disordered arrays, and also presents a pathway to evaluate (de)stabilizing interactions in other hysteretic systems.
Results
Rectangular arrays of polycrystalline Co ellipses were fabricated with varying centertocenter spacing by magnetron sputtering, in conjunction with ebeam lithography and liftoff. Details are presented in Methods. The ellipses have major/minor axes of 220/110 nm, with a structure of Ta(1 nm)/Co(9 nm)/Ta(1 nm), forming 50 × 50 μm^{2} arrays. In arrays A1/2/3 the minoraxis spacings of 150/200/250 nm are less than the majoraxis spacing of 500 nm. Thus the mean dipolar interactions are demagnetizing, favoring antiparallel alignment. In arrays B1/2/3 the minoraxis spacing of 500 nm exceeds the majoraxis spacings of 250/300/350 nm. Therefore, the mean dipolar interactions are magnetizing, favoring parallel alignment. Scanning electron microscopy (SEM) and magnetic force microscopy (MFM) images, at remanence after DC demagnetization, of arrays A1 and B1 are shown in Figs. 1 and 2, respectively. MFM image contrast indicates the outofplane stray fields and confirms the ellipses' single domain state.
FORC measurements were performed to obtain magnetization M(H, H_{R}) under different reversal field H_{R} and applied field H. The FORCdistribution is then extracted^{18}, , where M_{S} is the saturation magnetization. In loose analogy to the Preisach model, the FORCdistribution in certain simple cases can be interpreted as the 2dimensional distribution of elemental hysteresis loops with unit magnetization called hysterons on the (H, H_{R}) plane, or on the corresponding (H_{C}, H_{B}) plane, defined by local coercivity H_{C} = (HH_{R})/2 and bias/interaction field H_{B} = (H+H_{R})/2.
Good agreement between measured and simulated FORCdistributions was obtained for all studied arrays [e.g., Figs. 1(c) and (d) for array A2]. All demagnetizing FORCs A1/2/3 exhibited a ridge with the highH_{C} end on the H_{C} axis and the lowH_{C} end shifted in the +H_{B} direction^{30}. Increasing interactions in A3→A2→A1 increased the lowH_{C} shift and the length of the ridge. In addition, an edge emerged from the lowH_{C} end towards negative H_{B}, highlighted by the arrows, forming a “wishbone” or boomerang structure^{24}. A negative feature is observed at negative H_{B} values near the highH_{C} end. Similarly, FORC distributions for the magnetizing arrays also exhibit a ridge with the highH_{C} end on the H_{C} axis, but with the lowH_{C} end shifted towards −H_{B} [e.g., Figs. 2(c) and (d) for array B2]. Increasing interactions again increase the lowH_{C} shift, but reduce the length of the ridge^{30}. A negative feature below the ridge is more prominent than that in the demagnetizing case.
Noninteracting case
A FORC gives a nonzero contribution to ρ(H, H_{R}) only if dM/dH along the FORC depends on H_{R}. We first show that in the noninteracting case ρ coincides with the coercivity distribution D(H_{K}), spread along the H_{C} axis. Indeed, particle P_{i} with coercivity H_{K}^{i} downflips at H_{dn}^{i} = −H_{K}^{i} and upflips at H_{up}^{i} = H_{K}^{i}. Therefore, on a FORC starting at H_{R} > −H_{K}^{i}, P_{i} starts out upflipped and remains so, not contributing to dM/dH nor ρ.
In contrast, on FORC(H, H_{R} = −H_{K}^{i}), P_{i} is the last to downflip along the major loop, and has the highest coercivity among the downflipped particles. Therefore, P_{i} is the last to upflip as H increases past H_{K}^{i} on the same FORC, causing a dM/dH > 0 jump. Since this dM/dH jump is unmatched by the neighboring FORC(H, H_{R} > −H_{K}^{i}), dM/dH exhibits a dependence on H_{R}, making d(dM/dH)/dH_{R} nonzero. dM/dH increases as H_{R}_{ }decreases, making a positive contribution to ρ at (H = H_{K }^{i}, H_{R} = −H_{K}^{i}).
For all subsequent FORCs at H_{R} < −H_{K}^{i}, P_{i} starts out downflipped but still upflips at H = H_{K}^{i}. The dM/dH jumps on these FORCs are matched since they occur at the same field on each FORC(H, H_{R} < −H_{K}^{i}). Thus dM/dH is independent of H_{R}, and doesn't contribute to ρ. This reasoning highlights that only dM/dH jumps on individual FORCs that are unmatched by neighboring FORCs contribute to ρ. Each particle P_{i} contributes to ρ only once, at (H = H_{K}^{i}, H_{R} = −H_{K}^{i}) or equivalently at (H_{C} = H_{K}^{i}, H_{B} = 0). The contributions of all particles gives rise to a ridge along the H_{C} (H_{R} = −H) axis, which reflects D(H_{K}).
Interacting case
Next, we introduce interactions between nanomagnets on the meanfield level by including an interaction field H_{int} = αM(H), where α<0 for demagnetizing systems and α>0 for magnetizing ones^{21}. Fig. 3(a) shows a sequence of FORCs for a demagnetizing system with a rectangular D(H_{K}), and a zoomin view of the boxed region (right panel). The three FORC segmentpairs (1)/(2)/(3) show that the last dM/dH jump on each FORC(H, H_{R} = −H_{K}^{i}αM(H_{R}))  caused by the last upflipping particle P(H_{K}^{i})  is unmatched by the neighboring FORC(H, H_{R} > −H_{K}^{i}αM(H_{R})).
Fig. 3(b) shows that with interactions the unmatched dM/dH jumps still generate the ridge, but at shifted H_{B} values. Importantly, on the meanfield level all particles experience the same interaction field and thus the order of flips continues to be governed by the order of the coercivities: along the major loop the particles downflip in the order of their coercivities, lowest (highest) coercivity particle P(H_{K}^{min}) first [P(H_{K}^{max}) last]. Starting at the lowH_{C} end, the lowest coercivity particle P(H_{K}^{min}) downflips at H_{dn}^{min} = −H_{K}^{min}αM_{S}, as no other particles have flipped yet: M(H_{dn}^{min}) = M_{S}. Increasing H along FORC(H, H_{R} = H_{dn}^{min}), P(H_{K}^{min}) upflips at H_{up}^{min} = H_{K}^{min}αM_{S}, causing a positive jump dM/dH > 0 as shown by the lower FORC(H, H_{R} = H_{dn}^{min})segment of pair (1). This jump, caused by P(H_{K}^{min}), is absent on the upper FORC(H, H_{R} > H_{dn}^{min})segment and is thus unmatched, contributing to ρ at (H = H_{K}^{min}αM_{S}, H_{R} = −H_{K}^{min}αM_{S}), or similarly at (H_{C} = H_{K}^{min}, H_{B} = −αM_{S}), defining the lowH_{C} end. These flipping fields are shifted from their noninteracting values, as shown by the arrow set (1) in Fig. 3(b). Since P(H_{K}^{min}) defines the lowH_{C} end of the FORCridge, one concludes that interactions shift the lowH_{C} end of the FORCridge to the +H_{B} direction by −αM_{S} (recall α < 0), but leave its H_{C} coordinate unshifted at H_{C} = H_{K}^{min}.
FORCsegment pairs (2)/(3) in Fig. 3(a) illustrate the upflip of higher coercivity particles P(H_{K}^{i}). P(H_{K}^{i})s create unmatched dM/dH jumps on FORCs where they were the last to downflip at H_{dn} = −H_{K}^{i}αM(H_{R}) [vertical arrows in Fig. 3(b)] and also are the last to upflip at H_{up} = H_{K}^{i} αM_{S} [horizontal arrows in Fig. 3(b)], since M(H_{up}^{i}) = M_{S} when P(H_{K}^{i}) upflips.
The highH_{C} end of the FORCridge is defined by P(H_{K}^{max}) that is the last to downflip when the rest of the system is already negatively saturated (M = −M_{S}), thus H_{dn}^{max} = −H_{K}^{max}(−αM_{S}). P(H_{K}^{max}) upflips along the FORC(H, H_{R} = H_{dn}^{max}) only after the rest of the system is positively saturated: H_{up}^{max} = H_{K}^{max}αM_{S}. Accordingly, the unmatched dM/dH jumps caused by P(H_{K}^{max}) contributes to ρ only at (H = H_{K}^{max}αM_{S}, H_{R} = −H_{K}^{max}+αM_{S}), or similarly at (H_{C} = H_{K}^{max}αM_{S}_{, }H_{B} = 0), defining the highH_{C} end. As observed before, the highH_{C} end of the FORCridge remains on the H_{C} axis^{24}, but stretched along the H_{C} axis by −αM_{S} (α<0).
Note that interactions shift the FORC ridge feature in H uniformly [Fig. 3(b)], i.e., the resultant projection of FORC distribution onto the Haxis is only displaced from its intrinsic values, but not distorted. This Hprojection therefore mirrors the noninteracting case, where H^{Up} = H_{C} = H, reflecting the intrinsic coercivity distribution, simply displaced by −αM_{S}. Thus the intrinsic coercivity distribution can be  without distortion from interactions – directly identified from the FORC distribution.
Figs. 1(c,d) and 3(b,c) show that besides the ridge, ρ exhibit an edge as well with interactions^{27}. As discussed earlier, in the absence of interactions, the dM/dH jumps along a FORC(H, H_{R} = −H_{K}^{i}) are matched by the jumps on the subsequent FORC(H, H_{R} < −H_{K}^{i})s, not contributing to ρ. The arrows of Fig. 3(a) show that the interactions destroy this matching specifically at the lowH_{C} end by shifting the first upflip field H_{up}^{min} of each FORC, caused by P(H_{K}^{min}), by −αM(H_{R}). These shifts make the dM/dH jumps misaligned, see FORCsegmentpairs (4) and (5) (above and below), thus contributing to ρ at (H = H_{K}^{min}αM(H_{R}), H_{R}). These unmatched jumps give rise to the edge in Fig. 3(c). The endpoints of the edge are (H = H_{K}^{min}αM_{S}, H_{R} = −H_{K}^{min}αM_{S}) and (H = H_{K}^{min}+αM_{S}, H_{R} = −H_{K}^{max}+αM_{S}), or alternatively (H_{C} = H_{K}^{min}, H_{B} = −αM_{S}) and (H_{C} = (H_{K}^{min}+H_{K}^{max})/2, H_{B} = αM_{S}+(H_{K}^{min}H_{K}^{max})/2). Accordingly, the tilt and asymmetry of the edge provide a direct measure of the width of coercivity distribution D(H_{K}) = H_{K}^{max}H_{K}^{min}. In the extreme case of nearly identical nanomagnets with D(H_{K}) = H_{K}^{max}H_{K}^{min}≈0, the edge is vertical in the H_{C} –H_{B} plane, at H_{C} = H_{K} and H_{B} within ± αM_{S}, as observed experimentally in Ni nanowire arrays^{10}.
In short, on the meanfield level the FORCdistribution of a system with demagnetizing interactions exhibits an edge and a ridge, shifted by the unmatched first and last dM/dH jumps along each FORC. The FORCdistribution vanishes between them for the considered flat coercivity distribution, because jumps between the first and the last jumps along each FORC are matched by jumps on the neighboring FORCs. Here, the matched jumps are not caused by the same particles, as in the noninteracting system, but rather by different particles whose upflipping fields were shifted into alignment by the interactions. Still, the jumps are matched because the values of the aligned jumps are the same on neighboring FORCs for a flat distribution D(H_{K}). Visibly, the flat coercivity distribution on the meanfield already reproduces most features of the measured FORCdistribution.
To improve our model we introduce a more realistic Gaussian D(H_{K}) to elucidate the origin of the negative features, which represent a clear distinction between FORC and a literal Preisach interpretation. A Gaussian breaks the matching of jumps as now shown by examining the set of particles {P(H_{K}^{Cent})} around the center of the coercivity distribution. A P(H_{K}^{Cent}) particle is the last to downflip at H_{R} = −H_{K}^{Cent}, where M(H_{R}) = 0, and the last to upflip on the FORC(H, H_{R} = −H_{K}^{Cent}), contributing to ρ at (H = H_{K}^{Cent}αM_{S}, H_{R} = −H_{K}^{Cent}). On FORC(H, H_{R} < −H_{K}^{Cent})s P(H_{K}^{Cent}) is no longer the last to upflip. On subsequent FORCs, upflip jumps from different particles get shifted into alignment with this jump. For the flat distribution, the number of particles shifted into alignment is steady, making the match complete and thus zero contribution to ρ. However, for the Gaussian distribution, the particles shifted into alignment come from the decreasing slope of the Gaussian, leading to dM/dH jumps with a decreasing magnitude, providing only a partialmatch and generating a negative contribution to ρ. Fig. 3(d) shows the FORCdistribution for a Gaussian model that indeed develops a negative region specifically tracking the decreasing slope of the Gaussian, highlighted by the dashed line. Analogous arguments show that a Gaussian D(H_{K}) also bends the ridge and broadens the edge.
Nearestneighbor correlations
The last unaccounted feature of the measured ρ is the segmenting of the FORCridge into separate lowH_{C} and highH_{C} ends, with different amplitudes. To explain this we include the nearestneighbor interaction fields H_{nn}^{i}(conf) as the first terms of a systematic clusterexpansion.
Decreasing the field from positive saturation, the weakest coercivity particles downflip first. For demagnetizing interactions, H_{nn}^{i}(conf) of these downflipped particles stabilize their nearestneighbors in their up state. Therefore, for a sufficiently narrow D(H_{K}), the magnetization decreases towards zero by developing a checkerboard pattern [Fig. 1(b)]. The checkerboard naturally forms defects where the sequence of increasing coercivities selects thirdnearest neighbor particles to downflip. Still, the dominant reversal mechanism for nearly half of the particles is the checkerboard formation: downflipping with allup neighbors. Accordingly, the (nearly) half of the FORC ridge with H_{K}^{i} < H_{K}^{Cent} gets shifted along the +H_{B} axis by H_{nn}^{i}(up), where H_{nn}^{i}(up) is the interaction field for the allneighborsup configuration. Once the checkerboard pattern is formed, the rest of the particles flip with neighbors in various intermediate configurations. Therefore, the H_{K}^{i} > H_{K}^{Cent} half of the ridge is broken into several pieces, shifted by varying H_{nn}^{i}(conf) fields. Consequently, the nearest neighbor interactions manifest as segmenting of the ridge^{30}.
To reiterate, the demagnetizing interactions (α < 0) (1) shift the noninteracting FORCridge at the lowH_{C} end to the +H_{B} direction by −αM_{S}; (2) stretch the noninteracting FORCridge at the highH_{C} end along H_{C} by −αM_{S} without shifting it off the H_{C} axis; and (3) form a tilted edge connected to the ridge at the lowH_{C} end. Changing from flat to Gaussian D(H_{K}) distribution (4) produces a negative feature, bends the ridge, and broadens the edge. Finally, (5) nearest neighbor interactions segment the FORCridge.
Magnetizing interactions
Adapting the above arguments for magnetizing interactions (α > 0): (1) the lowH_{C} end is shifted in the −H_{B} direction, and (2) the highH_{C} end is compressed without shifting it off the H_{C} axis [Fig. 3(f)]. (3) Regarding the edge, the first upflip along each branch is shifted by interactions in the opposite direction as the demagnetizing case [Fig. 3(e) right panel]. Therefore, the first dM/dH jumps are unmatched, decreasing in magnitude with more negative H_{R}, thus negatively incrementing the FORC, forming a negative edge [Fig. 3(g)]. Changing from flat to Gaussian D(H_{K}) distribution (4) the negative edge gets pressed towards the positive ridge, and the FORCridge becomes curved [Fig. 3(h)]. The inclusion of nearest neighbor terms leads to (5) an avalanche reversal, collapsing the FORCridge to a singlevalue^{31}.
Quantifying Interaction Fields
Finally, we demonstrate the quantitative predicting power of the above considerations. The lowest H_{K}^{i}, which is shifted in H_{B} by αM_{S}, is extracted from the FORC ridge by selecting an H_{C}(threshold) such that 10% of the particles have H_{K}^{i} < H_{C}(threshold), and averaging ρ over the H_{C} = 0→H_{C}(threshold) range. The averaged (dM/dH_{B})' are shown in insets of Figs. 4(a) and 4(b). The H_{B} shift is determined by linearly extrapolating (dM/dH_{B})' at the high H_{B} end to zero. The interaction field is calculated by a finite element method (using the NIST OOMMF code) for the nearest and next nearest neighbors, and treating the remainder of the array as point dipoles. The experimental H_{B} shifts and the calculated interaction fields agree remarkably well (Fig. 4), confirming the validity of the meanfield description of the FORCdistribution and its quantitative predictive power, making the FORC technique a tool to extract numerical values of interaction fields. This is particularly important for disordered arrays where calculations of interactions are not easily achievable.
Discussion
In this work, systems of interacting nanomagnets were examined experimentally, numerically, and analytically, using the FORC technique. A meanfield analysis based on the concept of unmatched jumps accounted for all experimentally observed features of the FORC diagram, including its shifted ridgeandedge structure and negative features. The tilting, shifting, and stretching of these structures were identified as tools to extract quantitative information about the system, demonstrating the predictive power of the FORC technique. Construction of the FORC distribution through unmatched jumps, and recognizing the (de)magnetizing interactions as a particular case of (de)stabilizing interactions, presents an approach which can be used to evaluate any hysteretic system with the FORC technique.
Methods
Arrays of Co ellipses were fabricated by DC magnetron sputtering, in a vacuum chamber with a base pressure of 1 × 10^{−8} Torr and Ar sputtering pressure of 2 × 10^{−3} Torr, on Si (100) substrates, in conjunction with electron beam lithography and liftoff techniques. Magnetic hysteresis loops were measured at room temperature using the magnetooptical Kerr effect (MOKE) magnetometer with a 632 nm HeNe laser having a 30 μm spotsize, capturing the reversal behavior of ~5,000 ellipses^{32}. The magnetic field was applied parallel to the major axis of the ellipses. Each measurement was averaged over ~10^{3} cycles at a rate of 11 Hz. The arrays were coated with a 60 nm ZnS layer to improve the signaltonoise ratio^{33}. FORC measurements were performed as follows^{22}: from positive saturation the magnetic field is swept to a reversal field H_{R}, where the magnetization M(H, H_{R}) is measured under increasing applied field H back to saturation, tracing out a FORC. The process is repeated for decreasing reversal field H_{R}^{30}.
Ellipses were modeled as dipoles oriented parallel to their major axes. The interdipole spacing and magnetic moment per dipole in the 100 × 100 array were representative of the experimental system. Each dipole i was assigned an intrinsic coercivity H_{K}^{i} with a distribution experimentally determined from the sample having the weakest interactions, A3. The H_{int}^{i} dipolar interaction fields at dipole i were calculated on the meanfield level as αM(H), where α was calibrated such that αM_{S} equals the analytically calculated H_{int} at saturation. This meanfield formulation was extended by the first term of a cluster expansion, representing the nearest neighbor dipole interaction H_{nn}^{i} explicitly: H_{int}^{i} = αM(H)+H_{nn}^{i}. At each field step (ΔH = 1Oe) the total field H_{tot}^{i} = H+H_{int}^{i} was compared to H_{K}^{i}, downflipping occurred when H+H_{int}^{i} < −H_{K}^{i} and upflipping occurred when H+H_{int}^{i} > H_{K}^{i}, until all dipoles became stable.
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Acknowledgements
This work was supported by NSF (ECCS0925626, DMR1008791, ECCS1232275) and BaCaTec (A4 [20122]). Work at UCM and IMDEA was supported by the Spanish MINECO grant FIS200806249 and CAM grant S2009/MAT1726.
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Affiliations
Dept. of Physics, University of California, Davis, California, 95616, USA
 Dustin A. Gilbert
 , Gergely T. Zimanyi
 , Randy K. Dumas
 , Nasim Eibagi
 & Kai Liu
Dept. of Earth & Environmental Sciences, LudwigMaximiliansUniversität München, Germany
 Michael Winklhofer
Dept. Fisica Materiales, Universidad Complutense, 28040 Madrid, Spain
 Alicia Gomez
 & J. L. Vicent
IMDEANanociencia, Cantoblanco 28049, Madrid, Spain
 J. L. Vicent
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Contributions
D.A.G. obtained the experimental and simulation results, and wrote the first draft of the paper. G.T.Z. and M.W. participated in the simulation design. R.K.D., A.G., N.E., J.L.V. and K.L. participated in the experimental design, fabrication and characterization. K.L. and G.T.Z. designed and coordinated the whole project. All authors contributed to analysis, discussion and revision of the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Gergely T. Zimanyi or Kai Liu.
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Journal of Materials Science (2018)

4.
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