Emergence of the Stoner-Wohlfarth astroid in thin films at dynamic regime

The Stoner-Wohlfarth (SW) model is the simplest model that describes adequately the magnetization reversal of nanoscale systems that are small enough to contain single magnetic domains. However for larger sizes where multi-domain effects are present, e.g., in thin films, this simple macrospin approximation fails and the experimental critical curve, referred as SW astroid, is far from its predictions. Here we show that this discrepancy could vanish also in extended system. We present a detailed angular-dependent study of magnetization reversal dynamics of a thin film with well-defined uniaxial magnetic anisotropy, performed over 9 decades of applied field sweep rate (dH/dt). The angular-dependent properties display a gradual transition from domain wall pinning and motion-like behaviour to a nucleative single-particle one, as dH/dt increases. Remarkably, in the high dynamic regime, where nucleation of reversed domains is the dominant mechanism of the magnetization reversal (nucleative regime), the magnetic properties including the astroid become closer to the ones predicted by SW model. The results also show why the SW model can successfully describe other extended systems that present nucleative regime, even in quasi-static conditions.

We described the experimental details of the angular-resolved dynamical magnetic study presented in the main body of the manuscript, performed in model thin films with well-defined uniaxial magnetic anisotropy. A detailed analysis of the data acquired in quasi-static conditions is also presented. The data are discussed in the framework of both domain wall pinning and macrospin Stoner-Wohlfarth reversal models.

EXPERIMENTAL
A schematic representation of the sample structure and the experimental configuration are shown in Fig. S1. Co thin films were fabricated by DC magnetron sputtering at room temperature (RT) on thermally oxidized Si substrates covered with a 2 nm thick Ta buffer layer to avoid oxidation from the substrate. The samples were grown in a 0.2 T in-plane external magnetic field to induce a well-defined uniaxial magnetic anisotropy (K U ) in the ferromagnetic (FM) layer, with the anisotropy direction parallel to the field direction. The samples were finally capped with a 2 nm thick Ta layer to prevent oxidation. Three samples with different thicknesses were prepared, t Co = 5, 10, and 20 nm, all of them presenting the aforementioned uniaxial magnetic anisotropy. Similar quasistatic and dynamic behaviors have been found for the three samples. Here we present a detailed analysis of the quasi-static properties of the thicker FM layer whereas the dynamical aspects are discussed in the manuscript. Quasi-static conditions in this context represents a field sweep rate in the order of a few mT/s. The frontier between what is called quasi-static and dynamic regimes will be more accurately stablished in the main body. Angular dependent, time-resolved, vectorial Kerr magnetometry measurements have been performed with our home-made vectorial magneto-optic Kerr effect (v-MOKE) setup at RT over 9 decades of applied magnetic field sweep rates (dH/dt). The basics of the experimental setup has been described elsewhere [1]. In this case, detectors and electromagnets have been adapted to suit the requirements of dynamical measurements over a wide range of dH/dt, from 10 −4 to 10 +4 T/s. In particular, the magnetic field response of the electromagnet is linear with the driving current up to 80 mT for frequencies ranging from 1 mHz to 150 kHz, and the rise-time of the detectors has been set to 20 ns. A cross-check that ensures that both electromagnet and detection systems are properly chosen comes from the experimental observation of non-hysteretic loops at the hard-axis direction in the whole dynamical range investigated (see bottom left graph of Fig.2 in the manuscript).
This setup allows simultaneous and quantitative acquisition of the two in-plane magnetization components during the hysteresis loop, i.e., the parallel (M || ) and transverse (M ⊥ ) magnetization components with respect to the applied magnetic field (see Fig. S1), for a given orientation, allowing full angular studies. It is able to perform automated angular-dependent dynamical magnetic measurements at RT, without readjusting the reflection plane or changing the Kerr geometry during the process. α H = 0 • corresponds to the field direction parallel to the magnetic anisotropy easy axis (e.a.). The dynamic measurements were carried out by using triangular and sinusoidal magnetic fields from quasi-static (mHz) to 5 kHz and from 10 Hz to 150 kHz, respectively. A sinusoidal signal has only the fundamental frequency, without oddharmonics, allowing to increase the frequency range without suffering shape distortions. Hysteresis curves were  obtained by averaging over several 10-1000 magnetization cycles to improve statistics. Two different set of experiments were performed. Detailed angular-dependent measurements over the whole angular range at representative dH/dt values (e.g., Fig.S1 and Fig.2), and detailed dynamic hysteresis measurements at representative α H angles (e.g., Fig.2 and Fig.3). In both cases, the results are cross-checked for completeness.

Hysteresis and magnetization reversal
Representative in-plane resolved hysteresis loops acquired in quasi-static conditions at different α H angles are shown in Fig. S2. The angles have been selected to show the rich variety of hysteresis and magnetization reversal processes that a single film with well-defined uniax-ial magnetic anisotropy presents. Two different representations, standard M −H (top graphs) and polar M ⊥ −M (bottom graphs) curves, have been used in order to identify relevant magnetic properties. In the middle, the sample orientation has been depicted for each case with the corresponding relative directions. Remarkably, a simple inspection of the in-plane resolved hysteresis loops provides direct information about the magnetization easy-axis (e.a.) and hard-axis (h.a.) directions, critical fields, domain wall angles and magnetization reversal processes.
At a first glance, vectorial (in-plane) resolved hysteresis loops display different magnetization reversal pathways in both magnetization components, which strongly depend on α H , highlighting the importance of the simultaneous determination of both components provided by our setup. The different pathways are more evident for the M ⊥ (H) loop. In general, irreversible (sharp) transitions and/or fully reversible (smoother) transitions are observed in both M || (H) and M ⊥ (H) loops, corre-sponding to nucleation of magnetic domains followed by domain wall propagation and rotation processes respectively. The relative weight of these two reversal mechanisms depends on α H . The directional dependence originates from the symmetry breaking introduced by the magnetic anisotropy of the film.
At α H = 0 • the parallel component presents a perfect The quantitative information obtained from our v-MOKE setup allows the visualization of the in-plane trajectory of the magnetization vector during reversal (see bottom graphs of Fig. S2). In this polar plot representation, the data lying on the circle of radius unity, depicted with a dashed line, represent rotation processes. Every time the data is off this circle, magnetic domains are present. The specific mechanisms of the magnetization reversal are easily to detect with this plots. In all the cases, except for e.a., as the field is decreased from the maximum field, the magnetization vector rotates reversibly along the circle of radius unity. The rotation continues beyond zero field with the opposite field sense until an irreversible process is initiated, as indicated by the departure of the magnetization vector from that circle. Notice that both departure and return points are closed to e.a. direction, which accounts for irreversible process due to nucleation of magnetic domains oriented along the e.a. directions and subsequent domain wall propagation. The return point is found c.a. 180 • away from departure point, depending on α H . Note that the magnetization vector is far from being saturated along the field direction even for the largest field used, except for α H = 0 • . Therefore, the magnetization reversal proceeds by reversible magnetization processes, starting at saturation magnetization, all the way up to the closest e.a. direction, and from there it continues with a subsequent irreversible transition, which is related to nucleation of antiparallel magnetic domains and subsequent propagation of 180 • domain walls. In addition, the magnetic domains are not oriented necessarily parallel to the field direction but rather to the e.a. direction. In order to have quantitative analysis, relevant magnetic properties associated with the reversible and irreversible processes are discussed in the following.
Relevant magnetic parameters, such as remanences and critical fields can be readily obtained as a function of α H from the hysteresis loops, as indicates in Fig. S3. In both cases, the parameters are obtained after averaging the corresponding values of the forward and backward field branches at zero field and during reversal transitions for the remanences and the critical fields, respectively. In this sense, the two in-plane remanence values reads: . Notice that M R, is always positive, whereas M R,⊥ can be either positive (e.g., Fig. S3(a)) or negative (e.g., Fig. S3(b)), so there must be an angle where it vanishes. That angle is used to identify the characteristic magnetization easy-axis and hard-axis directions with great accuracy (see Fig.4a). In turn, the critical fields reads: ). The coercive field H C refers to the magnetic field required to reduce the magnetization to zero, i.e., M (H C ) = 0, whereas the switching field H S is the magnetic field where the irreversible transition takes place. In analogy, H S can be derived from the magnetization curve of the transverse component, i.e., M ⊥ (H S ) = 0, or more accurately from its derivative curve (shadow area in Fig. S3) looking for the field where its maximum is located (indicated with vertical dashed lines in Fig. S3). Notice that there is an angular range around the e.a. where both coercivity and switching field coincide (e.g., Fig. S3(a)), and in particular for α H < ±70 • ). Within (out of) this angular range, coercivity and switching field have the same (different) meaning, as discussed below.

Remanence and magnetic symmetry
The simplest information that can be extracted from the angular evolution, is the anisotropy directions, determined by the magnetic symmetry of the system. This can be done easily by plotting the normalized remanence values of the two in-plane magnetization components as a function of α H (see Fig. S4). Both magnetization components display a pronounced oscillation with periodicity of 180 • . The parallel component follows a | cos 2α H | law dependence, the perpendicular component changes the sign when a characteristic e.a. or h.a. direction is crossed, and both components are complementary, i.e., M 2 S = (M 2 R, + M 2 R,⊥ ). The polar plots of M R, and M ⊥,R display characteristic "two-lobes" shapes rotated 90 • (see Fig. S4(b)). This originates from the two-fold magnetic symmetry induced by the field applied during growth. All these features confirm a well-defined uniaxial magnetic anisotropy behavior of the film, where the anisotropy axis was field-induced as already explained.

Irreversible transitions and relevant fields
The more relevant transition fields are the coercivity (µ 0 H C ) and the switching field (µ 0 H S ), which are related to the field required to compensate the magnetization along the field direction and to the effective anisotropy of the system, respectively. Coercivity describes the stability of the remanent state and gives rise to the classification of magnets into hard magnetic materials (permanent magnets), semihard materials (storage media) and soft magnetic materials (core applications). The switching field determines the field needed to control the stable magnetization direction in any magnetic system, crucial in both permanent magnet and spintronic applications. Both can easily be determined from the M − H curves by looking for the applied field values at which M R, and M R,⊥ crosses zero, respectively. Notice that the coercive field does not always indicates the field value at which the irreversible transition takes place, while the switching field does. The sharp irreversible transitions are associated with the nucleation field and results in an abrupt change in the M − H loop, i.e., in a large peak in the corresponding numerical derivative curves (see Fig. S3). Notice that the irreversible hysteresis jumps found in both components, and their corresponding derivative curves, are not necessarily of the same height. Close to the e.a. direction the M (H) loops present significantly larger irreversible jumps, and higher maximum values of the corresponding derivative curves, in comparison with the ones found in the M ⊥ (H) loops, whereas approaching the h.a. direction the difference exchanges, as the right graphs of Fig. S3 shows. In this case, the angular evolution of switching field is more precise to determine from the M ⊥ (H) loops. Fig. S5 displays the angular dependence of the relevant fields extracted from M − H loops. In general, the twofold symmetry of the magnetic properties is also found in the experimental data of the angular dependence of both µ 0 H C and µ 0 H S , as revealed by the 180 • periodicity presented in both cases ( Fig. S5(a)). The coercive field presents a roughly constant value around the e.a. direction, i.e., µ 0 H C (0 • ) = 4.5 mT, increasing slightly until α H ≈ ±75 • . The switching field presents an identical angular evolution in this angular range. Above this angle, µ 0 H C starts to decrease, vanishing at the h.a. direction, i.e., µ 0 H S (90 • ) = 0 mT. In contrast, µ 0 H S presents a larger rising evolution as the angle approaches to the h.a. direction. The switching field at the h.a. direction is the effective anisotropy field, i.e., µ 0 H S (90 • ) = µ 0 H K = 17 mT. Notice that there is a factor of nearly four between the coercive field at the e.a. and the anisotropy field.
In order to understand the angular evolution of the relevant fields, we have compared the experimental results in quasi-static conditions with the expected ones from two, in principle, opposite models, which assumes rotation and single domain switching (SW model [2]) and rotation and nucleation of magnetic domains and subsequent propagation of domain walls pinned at de- fects (pinning-model [3]). There are two angular ranges where each model reproduces the experimental data. The SW model assumes a single particle behavior, i.e., reversal by switching (irreversible process) and/or rotation (reversible) of the whole magnetization, without taking into account any other irreversible magnetization reversal process like nucleation and propagation of magnetic domains. The latter is, in fact, energetically much more favorable in extended systems, in which defects of any kind-structural or morphological-play the main role to activate irreversible magnetic domain nucleation processes at magnetic fields that are significantly lower than expected by the SW model. In particular, even in epitaxial thin films, low coordination sites such as sample edges and topographic roughness are unavoidable. This defects will act as pinning centers for the created magnetic domain walls, and the reversal subsequently continues via domain wall propagation, as the field pressure allows the wall to unpin (overcoming its local energy barrier), just to reach other (higher energy barrier) pinning centers, and so on until all the reversal is completed. In this case, a simple model of pinned 180 • magnetic domain walls (pinning model) predicts a 1/| cos α H | law for the angular dependence of the reversal field, which has been already observed in both perpendicular [4] and inplane [5] anisotropy systems.
In a wide angular region around the e.a. direction, i.e., |α H | < 70 • , both fields are similar and follows a 1/| cos α H | law (discontinuous line in Fig. S5(b)), accordingly to the domain pinning model prediction. Thus, nucleation and subsequent propagation of pinned magnetic domains is the relevant process during the irreversible transitions. Close to the h.a. direction, i.e., |α H | > 70 • , the pinning model cannot reproduce the experimental data, and the predicted evolution tends to separate more and more from experimental values. On the other hand, SW model, which does not reproduce well the experimental results below 70 • becomes closer and closer as h.a. is reached, reproducing quite satisfactory the results at the exact h.a. For instance, the coercive field (switching field) decreases (increases) to zero (up to the anisotropy field) as approaching the h.a. direction, as predicted by the SW model (solid lines in Fig. S5). This indicates that close to the h.a. direction the magnetization reversal is mainly governed by rotation processes. Note that the SW model only reproduces their angular evolution around the h.a. directions, where reversible processes are the relevant mechanism during reversal (clear area in Fig. S5). It fails around the e.a. directions, where irreversible (nucleation and propagation of magnetic domains) processes dominate, as described above.
In summary, the quasi-static magnetic properties of a well-defined uniaxial magnetic anisotropy system have been disentangled by means of a detailed angular dependence vectorial-resolved magnetization reversal study. In general, both reversible and irreversible transitions ta-kes place during reversal, and their relevance depends strongly on α H . The reversible transitions correspond to magnetization rotation processes whereas the irreversible ones are related to nucleation of magnetic domains aligned with respect to the anisotropy axis and a subsequent domain wall propagation. The data are discussed in the framework of both domain wall pinning and macrospin SW reversal models for different angular ranges. In particular, the pinning and SW models reproduce the experimental data close to the e.a. and h.a., respectively. * Electronic address: julio.camarero@uam.es.