Interpreting FORC Diagrams Beyond the Preisach Model: An Experimental Permalloy Micro Array Investigation

First-order reversal curves (FORC) are a powerful tool that is increasingly used in material science and nano-magnetism research [1–4]. Besides its utility for fingerprinting magnetic systems, it can distinguish between different components in a complex magnetic system without the need for a laterally resolving magnetometer [5]. Furthermore, quantitative information about the intrinsic coercivities and interaction fields between magnetic particles can be extracted from FORC measurements. Unfortunately, the interpretation of a FORC diagram as a Preisach distribution is not straight forward as most real system violate the Mayergoyz criteria [6], i.e. congruency and wiping-out. In these cases, the interpretation of the peaks in a FORC diagram needs to rely on additional tools like magnetic microscopy [1, 2]. Thus, it would be desirable to extend the interpretability of FORC diagrams beyond the limits set by the Mayergoyz criteria [3, 4]. Here, we systematically design artificial multi-component systems that purposely violates the congruency criterion to gain insights into the interpretability of FORC diagrams of such systems. These systems are made up of 150 μm long Permalloy (Py, Ni80Fe20) stripes that are structured by photo lithography and arranged into micro arrays (cf. inset in Figure 1). The coercivity of these stripes is tuned by their width (10–50 μm), the interaction strength by their spacing (10–30 μm). Thus, we are able to specifically determine the coercive and interaction fields of these well-defined systems. Subsequently, FORCs are measured with a customized Durham Magneto Optics NanoMOKE3 magnetometer. This allows fast acquisition of high resolution FORC diagrams [5], thus, enabling us to systematically investigate a large number of samples. Additionally, 250.000 local hysteresis loops with a spatial resolution of 5 μm are measured per sample to gain an independent measure of the spatial distribution of coercive and interaction fields. Such a comparison between FORC diagram and the spatial coercivity distribution is shown in Figure 1. The two coercivities of the wide (1 Oe) and narrow (3 Oe) stripes are reflected in the FORC diagram. However, an additional peak-dip feature (marked as A in Figure 1) appears in the FORC diagram that cannot be individually attributed to a magnetic element in the system. Looking into the magnetization configuration during each minor loop, it becomes evident that this peak-dip feature is a consequence of the wide stripes (smaller coercivity) to reverse their magnetization at different applied fields when the initial magnetization is aligned parallel or anti-parallel to the narrow stripes (larger coercivity). This additional signal in the FORC diagram arises purely from the interaction when the system violates the Mayergoyz criteria. More precisely the interaction field, between narrow and wide stripes, which switch at different fields respectively. Following the evolution of this interaction peak while tuning the interaction strength of the system, a clear change in the peak size and position is observed. Figure 2 shows the integral of the interaction peak in dependence of the stripe spacing, which governs the interaction strength. As the interaction strength decreases, the peaks' integrated FORC density also decreases. Finally, the peak vanishes when the interaction strength is sufficiently decreased and the system fulfils the Mayergoyz criteria again (not shown in Figure 2). Thus, it can be deduced that here the integrated FORC density of the peak is also a quantitative measure for the interaction strength. We have used a systematic study of artificial micromagnet arrays that intentionally violate the Mayergoyz criteria, to investigate FORC diagrams beyond the Preisach interpretation of discrete hysterons. The combination of magnetic microscopy and fast FORC measurements to get to independent measures of coercive and switching field distributions allowed us to reliably interpret these FORC diagrams. Thereby, we identified an interaction peak that arises from the strong interaction of magnetic elements in the system. When such an interaction peak occurs, it is a mere manifestation of an interaction field. With this knowledge, interaction peaks in FORC diagrams of unknown system like NdFeB magnets can be interpreted. This opens up a pathway to fully interpreting FORC diagrams and elevating FORC beyond mere fingerprinting.