Hysteresis-free switching between vortex and collinear magnetic states

We demonstrate a lossless switching between vortex and collinear magnetic states in circular FePd disks arranged in a square lattice. Above a bifurcation temperature T e ?> we show that thermal fluctuations are enough to facilitate flipping between the two distinctly different magnetic states. We find that the temperature dependence of the vortex annihilation and nucleation fields can be described by a simple power law relating them to the saturation magnetization.

the out-of-plane moments, and the chirality, the direction of the curling of the in-plane moments. The bistable nature of the vortex polarity, pointing either up or down, has been the subject of intense study [12][13][14][15] and a bistability between the magnetic vortex and the collinear state has also been observed. Upon thermal annealing of individual cobalt islands grown on a ruthenium surface, Ding et al found that for certain sizes of the islands the magnetic state could be converted from a vortex state to a collinear state and vice versa [16]. Furthermore, thermally active nano-patterned magnetic materials have recently gained a lot of interest [17][18][19][20][21], where elongated islands fluctuate between two collinear states, different only in the direction of the magnetization. Here we investigate the effects of thermal fluctuations in arrays of thin circular disks. Our results provide a new perspective on the role of thermal excitations in magnetic nano systems, illustrated by the presence of a bistable region above a bifurcation temperature ( ) T e . Above this temperature, the thermal fluctuations are sufficient for transitions between the vortex state and the collinear state, leading to a hysteresis-free switching.
The magnetic field response of vortices is characterized by two critical fields, the annihilation field (H an ) and the nucleation field (H n ). Upon reaching H an , the vortex is pushed out from the disk (i.e. annihilated), and the magnetization of the disk becomes close to collinear. A vortex reappears when the field is removed, albeit generally at a lower field H n . This hysteresis between the annihilation and the nucleation fields can be viewed as arising from an energy barrier between the two states. A vortex state has a very low stray field and the interaction between even densely packed disks in the vortex state is therefore negligible. When an external field is applied, the vortex cores shift from the center of the disks, resulting in a net magnetization and a stray field. The energy landscape of vortices forming a dense array can therefore be tuned by the distance between the disks [3,[22][23][24] as well as through altering the inter-disk coupling by changing the disk material. Understanding the influence of interactions and temperature on the magnetic states is essential for obtaining a complete and consistent picture of the different magnetic phases and their boundaries. However, the use of materials having high intrinsic Curie temperatures (T C ) has severely limited the previously accessible range in magnetization [16,[25][26][27]. Using materials with experimentally accessible T C , in which full magnetization is obtained at low temperatures and zero magnetization at and above the ordering temperature ( ⩾ T T C ), a maximum response is obtained. Here we utilize both the composition of the disks and the temperature to alter the energy landscape of the vortex/ collinear state and thereby control the region of bistability.
Two samples with different FePd compositions, Fe 13.5 Pd 86.5 and Fe 20 Pd 80 , were grown on × 30 30 mm 2 pre-patterned fused silica substrates prepared by surface conformal nano-imprint lithography [28]. The samples were co-sputtered using elemental Fe(99.95 at-%) and Pd(99.95 at-%) targets in an ultra-high vacuum magnetron sputtering system with a base pressure below 2 × − 10 7 Pa. Argon was used as the sputtering gas at a pressure of 0.8 Pa. The sample stage was rotated and the substrate kept at room temperature (RT) during growth. This resulted in highly uniform polycrystalline disks, see inset in figure 1(d). Each of the disks has a radius of R = 225 nm and a mean thickness of 10 nm. The lattice parameter of 513 nm gives an inter-disk distance d = 63 nm in the [10] direction and d = 212.5 nm in the [11] direction, well within the strong coupling regime of circular magnetic elements [22]. The saturation magnetization of 13.  [29] with T C ∼300 K and ∼500 K [30], respectively.
Imaging of the field dependence of the magnetic vortex state was obtained using photoemission electron microscopy (PEEM), conducted at the elliptically polarized microfocus soft x-ray undulator beamline UE49-PGM-a-SPEEM at the BESSY-II synchrotron facility (Helmholtz Zentrum Berlin), which was equipped with the Elmitec PEEM-III endstation. Imaging was performed using low-energy secondary electrons with a probing depth of around 5 nm and a spatial resolution of 30 nm (field of view 5 μm). The Fe L 3 resonance (705.6 eV) was used to obtain magnetic contrast, with images taken using both right-hand and left-hand circularly polarized light (RCP and LCP). The magnetic contrast is obtained as the asymmetry ratio of intensities for LCP and RCP polarizations. The technique is sensitive to magnetic moments which have a component in the beam direction. Several series of nucleation and saturation events were recorded.
As a field is applied, the vortex cores move from the center of the disks towards the edges where they annihilate, as seen in figures 1(a), (b) and (c). The shift of the vortex center is perpendicular to the applied magnetic field and depends on the vortex chirality with respect to the direction of the applied field. The disks exhibited no evident memory of chirality upon field cycling. No general preferable chirality was seen, although clusters of up to 14 disks, and lines with up to ten disks in a row, with the same chirality, were observed.
The temperature dependence of the magnetization was investigated using the magnetooptical Kerr effect in a longitudinal configuration [31]. The incident laser beam (λ = 635 nm) had a spot size of ∼1 mm 2 , averaging therefore the response of approximately four million disks. A magnetization loop at RT for the Fe 20 Pd 80 , with an external field applied in the [11] direction, is shown in figure 1(d). The results reveal a typical vortex behavior, with a sharp increase (decrease) of the magnetization at the field required for the vortex annihilation (nucleation). The individual magnetization loops for the Fe 13.5 Pd 86.5 sample are comparable to the results obtained for the Fe 20 Pd 80 sample on a reduced temperature scale. The magnetization as a function of temperature for the Fe 13.5 Pd 86.5 sample, at a field of 50 mT, is displayed in figure 2(a). The solid line is a fit assuming a modified Bloch behavior: The field response of the magnetization obtained at 222 K differs significantly from what is observed at 50 K, as seen in figure 2(b). The nucleation and the annihilation fields are now the same and the switching between vortex and collinear states is hysteresis-free. The field dependence of the magnetization still displays typical vortex features, with a sharp increase (decrease) at annihilation (nucleation). The bifurcation temperature T e at 212 K marks the onset of the collapse of the hysteresis, as illustrated in the phase diagram in figure 2(c). The temperature dependence of H an and H n for individual permalloy (Py) disks has previously been related to the decrease of M s with increased temperature [25]. Thermal excitations within the islands influence H an through oscillations of the vortex core [9]. Consequently, increased temperature will lead to larger oscillations, facilitating the annihilation process. For an array of disks the inter-disk interactions need to be included to obtain a more complete view of the creation-annihilation process.
In the absence of a full theoretical treatment of the effect of inter-disk coupling, we attempt the following phenomenological description of the relation between the critical fields and the magnetization.  (see  table 1) for the Fe 13.5 Pd 86.5 sample, resembling results obtained from individual disks of Py [25]. Increasing the Fe content, figure 3(b), does not change the interrelation for H an between the [10] and [11] directions, they are still indistinguishable. The absence of directional dependence of H an does not by necessity imply non-existing inter-disk coupling in the annihilation process. Inter-disk coupling is known to stabilize the collinear state, decreasing both H an and H n [22][23][24]. The observed stabilization can thus be viewed as a bias field, increasing with decreasing distance between the disks and decreasing with increased temperature as the saturation magnetization (M s ) decreases. The shift of H an is, however, intricate and involves the distribution of chirality of the vortex states since this will affect the stray field of each disk and that of its neighbors. The rigid vortex model [22] which has been successfully deployed earlier to describe various phenomena fails to explain this directional dependence. Supposedly the effective bias field, as the coupling can be described as, suppresses H an equally in all directions. Even though no directional dependence is seen even for the Fe-rich sample, the exponent b decreases, see table 1, due to the increased inter-disk coupling, as expected from the increase in M s .
The effect of the inter-disk coupling is more clearly seen for H n which increases with decreasing M s , the opposite to that of individual disks of Py [25]. Contrary to H an , there is also a clear directional dependence for H an arising from the different inter-disk distances in the [10] New J. Phys. 16 (2014) 053002 E Östman et al Figure 3. (a) Log-log plot of H an and H n versus normalized saturation magnetization in both the [10] and the [11] directions for the Fe 13.5 Pd 86.5 sample. Solid lines are fits to equation (1). H an is the same in both the [10] and the [11] directions whereas H n shows a distinct difference between the two. Panel (b) shows the corresponding data for Fe 20 Pd 80 . The difference for H n between the [10] direction and the [11] direction has increased due to a higher M s and consequently higher inter-disk coupling. The higher = T 462.5 C K shifts T e to higher values. The graph is cut off at 0.2 on the horizontal axis for clarity. Above 0.7 M s the hysteresis loops display a coercivity and the simple model, equation (1), is no longer applicable. and the [11] directions. The larger influence of inter-disk coupling in nucleation compared to annihilation can be understood considering the collinear state and the larger effective bias field strength relative to the applied field, since the nucleation occurs at much lower fields. Increasing the Fe content results in an increased M s and thus increased stray field, which in turn means an increased coupling leading to an enhanced splitting in the directional dependence of H n . Increasing the temperature decreases the directional difference as a result of the reduced interdisk coupling. The larger inter-disk coupling for the Fe-rich sample is also reflected in the b values where the difference for b in the [10] and the [11] directions is significantly increased. The energy landscape of the array above T e in the bistable region is schematically depicted in figure 4. In the absence of an external field (marked (1) in the figure), the energy of the vortex state is lower than the collinear state. The opposite is valid at (3), at which the collinear state has lower energy. The energy difference between the two states (ΔE) will therefore have to approach zero at some field strength, giving rise to a bistable condition. In the transition region marked as (2), ΔE is small and the transition between the two different states will be dictated by the activation barrier. Here we have to distinguish between two extreme cases: one where the transition energy barrier Δε ≫ k T B and one where Δε ≪ k T B . When both Δε → v c and Δε → c v (the energy barriers for transitions from vortex to collinear states and vice versa) are larger than k T B , the bi-stability will give rise to hysteresis, characteristic for the vortex annihilation and nucleation. However, when Δε → v c and Δε → c v are smaller than k T B , the switching will occur without hysteresis, as illustrated in figure 4 and seen in figure 2(c).
To conclude, a bistable region with hysteresis-free transition between collinear and vortex states is obtained when > T T e . Below T e , the switching between the vortex and the collinear states exhibit hysteresis and the transition is irreversible. Above T e , thermally induced bistability is obtained ( > > T T T c e ). In this bistable region, thermal fluctuations alone are enough for switching between the vortex and the collinear state. These findings enable the exploration of dynamics in a wider temperature range and provide a new insight into and pathways for the study of bistability in magnetic nano systems, such as artificial spin ice systems [21]. The hysteresis-free switching of vortices is not only of academic interest, it also has significant implications for applications of magnetic nano structures in spintronic devices, discrete recording media or magnetic field sensors [32,33].