We studied vortex nucleation/annihilation process and its temperature dependence in micromagnetic objects with lowered symmetry using micro-Hall magnetometry. Magnetization reversal curves were obtained for the Pacman-like nanodots placed directly on Hall probes. Lowered symmetry of the object leads to good control of its chirality. Vortex nucleation and annihilation fields strongly depend on the angle of the external in-plane magnetic field with respect on the nanodot symmetry. The micromagnetic simulations support the experimental results - the vortex nucleation fields are controlled by local magnetization configurations present in the object (C-, S-, and double S-states) for field just above vortex nucleation field. The experiments also confirm that the vortex nucleation proceeds via thermal activation over an energy barrier.
Magnetic nanoelements attract attention of researchers due to their interesting fundamental properties and also the potential in novel high-density magnetic memories. Controlled manipulation of magnetic domains and vortices in ferromagnet nanostructures have opened opportunities for novel fast, high-density, and low-power memories, including race track memories, magnetic random access memories, bit patterned media, and skyrmion memories.1–3 Magnetic properties of the nanomagnets are governed by magnetostatic and exchange energies, and are fundamentally influenced by their shape.
Significant experimental and theoretical work have been devoted within last years to understand magnetization reversal and vortex dynamics in submicron ferromagnetic disks.4–9 A disk is convenient high-symmetrical shape which simplifies both experimental and theoretical studies. However, vortex chirality and polarity is difficult to control in disks due to symmetry reasons.
Recently, a new prospective shape of a nanomagnet with broken symmetry was proposed (“Pac-man like”, PL10–13). It shows easy control of vortex chirality/polarity in the in-plane magnetic field for objects smaller than 100 nm.
Vortex nucleation was studied previously by micro-Hall technique in Ni8 and Co individual nanodots of cylindrical shape.6 Both papers present influence of temperature on the nucleation and annihilation fields with results not completely understood. The non-thermal dynamics of magnetic vortices in micron-size Py disks was reported. The measurement has been done on ensemble of Py dots by superconducting quantum interference device (SQUID). From the temperature dependence of the time relaxation constant, the athermal behavior was found and interpreted as a quantum depinning of vortex cores through the structural defects of the sample.7
Magnetization reversal can be observed experimentally on arrays of similar nanodots by techniques like magneto-optical Kerr effect14 or by SQUID.7 The state of individual nanomagnet can be explored non-invasively by scanning Hall probe microscopy or micro-Hall probe magnetometry.
In this letter, we present a temperature dependent study of the magnetization reversal in individual submicron Permalloy (Py) PL nanodots using micro-Hall magnetometry. The technique is used to gather quantitative information on stray magnetic field of individual ferromagnetic objects placed directly on micro-Hall probes.15–17 Changes of the stray field influence the magnetic-field flux through the active area of the Hall sensor. Therefore, the Hall voltage directly shows the magnetization changes of the PL nanodot including vortex nucleation and annihilation. We have found (similarly to the Ref. 6) that the vortex nucleation field increases with temperature rapidly for temperatures below ∼ 20 K, and only slightly for temperatures above ∼ 20 K. We also show that for the PL nanomagnet the vortex nucleation field depends significantly on the angle of the external field according its axis of symmetry.
It has to be stressed that lowered symmetry of the ferromagnet complicates the Hall voltage interpretation as compared with highly symmetric objects like disks. In asymmetric objects, the object shape and the vortex chirality and polarity can influence the Hall voltage signal significantly. Therefore, in this work we combine the Hall probe measurement with micromagnetic simulations to support our interpretation of the results obtained.
In previous calculations it was carried out that in small PL objects (< 100 nm) is the vortex nucleation preceded by a C-state or an S-state formation.10 Our simulations showed that much more complicated picture with nucleation of several vortices appears for large objects (d > 500 nm). However, smaller PL nanodots (d < 350 nm) gave much more reproducible results, which were supported by presented experiments. Therefore, in this paper we concentrate on 310-nm object, for which simple magnetic state with one vortex appears, and still reasonable large Hall voltage signal is obtained.6
In the case of 310-nm PL object we have identified 3 different single-domain (SD) states for fields closely above the vortex nucleation: C-state, S-state and double S-state (or 2S-state). The state influences the vortex nucleation process, which is basically controlled by thermal activation over an energy barrier. The process is influenced by the object asymmetry, which can be represented by additional magnetic dipoles located within the object at its boundary imperfections. Such dipoles change their value and direction during the magnetization reversal measurements. The charge rearrangement should be sensitive also to thermal processes and represents small energy barriers for the vortex nucleation process.
Hall probes were fabricated from GaAs/AlGaAs heterostructure with a two-dimensional electron gas (2DEG) located 80 nm below the surface (electron density ∼7 x 1015 m−2, mobility ∼6 m2/Vs at 77 K). Technology of the probe: First, standard optical lithography and wet etching were used to define 15 μm Hall crosses and their leads. Then, electron beam lithography (EBL) and wet etching were applied to lower the linear Hall probe dimensions to ∼1 μm. Finally, Py PL nanomagnets of the thickness of 38 nm were formed. The process consisted of EBL process on thinned PMMA 950K resist (thickness 100 nm) with a dose of 110 μC/cm2 at 10 kV, and e-beam evaporation of Py layer at a pressure below 10−5 Pa, followed by lift-off. Fig. 1(a) shows the SEM image of the finished micro-Hall probes with PL nanomagnet (d = 310 nm) located on the probe. Only one PL nanodot was patterned per cross to evaluate single PL-dot properties.
Hysteresis loops were measured by applying an in-plane magnetic field and recording VH. The nanomagnet was fabricated asymmetrically with respect to the active zone of the Hall cross in order to improve the signal measured.6 The main reason for asymmetric placement of the nanomagnet is to obtain non-zero magnetic flux through the active zone of the Hall probe. In case of symmetric placement, the Hall signal should be zero due to dipolar nature of the magnetic field. In our experiments, best resolution was obtained in the Hall probe configuration (current flows from lead I+ to I-, Hall voltage is measured between leads V+ and V- (Fig. 1(a)).
The measurements were carried out at temperatures 4 – 100 K in the physical property measurement system (PPMS) using dc current (10 μA) across the Hall probe. Hall voltage was measured and amplified by the PPMS electronics, thereby voltage noise lower than ∼20 nV was achieved in the best case (integration time 10 s, T = 30 K).
In the magnetization reversal measurements, a homogeneous in-plane magnetic field Hext was applied in parallel with the active area of the Hall probe, thus not contributing to the measured signal. Hall voltage measured is proportional to the average magnetic flux through the active zone of the probe generated by the PL nanodot. Sweeping amplitude of the external field was fixed to 2 T, for which positive and negative branches of the hysteresis loop gave the same fields Hnuc (Han). This was not the case for much lower field amplitude (e.g. 200 mT), probably due to PL-boundary imperfections - only high external magnetic field brings the object into identical SD states for both field polarities.
We have analyzed the vortex nucleation/annihilation process for three different field directions, 90°, 120°, and 160°, according its “symmetry” axes (Fig. 1(b) shows that real nanomagnet is not ideally symmetric due to edge imperfections). The angle-selection was based on the outputs of micromagnetic simulations.
First, we have calculated the angular dependence of the single-domain (SD)-to-vortex (V) state transitions, Hnuc and Han, for the PL nanodot shown in the Fig. 1(b). Dynamic behavior of the vortex nucleation, propagation, and annihilation were evaluated by micromagnetic calculation using OOMMF software package.18 Parameters used in the simulation were: PL thickness t = 38 nm, diameter d = 310 nm, Py exchange constant A = 13 × 10−12 J/m, saturated magnetization Ms = 8.6 × 105 A/m, and damping parameter 0.5. The unit element size was fixed to 4 nm x 4 nm x 38 nm. The software solves Landau-Lifshitz-Gilbert equation and simulates experiment at 0 K.
From the OOMMF data we have calculated also z-component of the PL stray magnetic field 10 nm below the PL nanomagnet, and also at the distance of the active area of the Hall sensor (i.e. 80 nm). Magnetic stray fields are calculated as gradients of scalar potential:
where the potential Φm is determined by the magnetization in a ferromagnetic object. Numerical evaluation of the stray field of the object calls for spatial discretisation of the system. This is done by splitting the object into rectangular cells as is done in the OOMMF simulation package. Under assumption of uniform magnetization inside these cells, we calculate the values of stray fields by methods proposed by either Newell et al.19 or Abert et al.20
The OOMMF calculations have shown three basic different configurations of local magnetizations in the PL nanomagnet. They are depicted in the Fig. 2, line 1, for external field value just above the vortex nucleation field Hnuc- for field angle α = 90° (left column, C-state), for α = 120° (middle column, S-state), and for α = 160° (right column, 2S-state). Red line follows the direction of local magnetization lines. Lines 2 and 3 depict the map of relative values of the z-component of the PL stray field in the active plane of the Hall probe before (line 2), and after (line 3) the vortex nucleation, blue color is for the negative and red color for the positive z-component of the magnetic field, respectively. The Hall voltage is proportional to the integral magnetic flux through its active zone, i.e. through the the region above the dashed lines shown in the Figs. 2, second line. Overall flux through the active area of the probe is much larger in the case of its asymmetrical position as compared to its central position.
Figs. 2, line 4, show the same fields components like in line 3, but at the distance of 10 nm only from the PL object. Stray field at the PL’s opening is strong enough at the distance of 10 or 80 nm and it defines clearly vortex chirality (clockwise, CW, for left column, and counter-clockwise, CCW, for middle and right columns). On the other hand, vortex polarity can be much better recognized at the distance of 10 nm (Figs. 2, line 4) as compared to the distance of 80 nm (Figs. 2, line 3).
Vortex state nucleated from C-state differs from the one from S-state in chirality, and vortex states created from S-state and 2S-state differ in polarity. The chirality contribution to the Hall signal is about 20%, meanwhile vortex polarity contribution is about 2% only for the distance 80 nm. The polarity signal is on the noise level in our experiment.
Now we discuss the influence of the PL shape, its location on the HP, and vortex chirality/polarity on the magnetization reversal Hall signal. Figs. 3(a) and 3(b) show magnetization reversal curves for field angle 90° at 30 K. The signal obtained from HP differs significantly from the signal from disk.6 In both objects abrupt changes in the stray field correspond to significant changes of the magnetization pattern (vortex nucleation and annihilation). The changes are directly connected with the step change of the system energy or its redistribution between exchange and magnetostatic energies. On the other hand, smooth changes of the signal can be attributed to the smooth transformation of the magnetization field (shift of the vortex, C-state or S-state modification, etc.) in the ferromagnet.
As compared to signal from a disk,6 Hall signal from the PL nanomagnet depends significantly on vortex chirality (Fig. 3(a) and 3(b)), and can be easily controlled by the time sequence of the applied field. Fig. 3(c) depicts the time sequence used to set desired chirality of the object. Chirality setting in other shapes was also shown recently by other authors.21,22
Now we explain the hysteresis loops shown in the Fig. 3(a) and 3(b) according the time sequence of the applied external magnetic field. Let us start CW chirality at zero field (point 1 in the Fig. 3(c)). Then, positive field up to +2 T is applied (with detailed measurement only up to +10 mT, shown in the figure 3(a)), and the vortex core is expelled from PL object at Han = 88 mT at the right side of the object (see Fig. 3(d)). Then is the field lowered, vortex nucleates at Hnuc = 22.5 mT and the chirality is set to CCW.10 Negative field then expels the vortex core again to the right side due to the CCW chirality, and vortex annihilates at Han = -88 mT. Then is the field increased, vortex nucleates again at Hnuc = -22.5 mT with CW chirality.10 Explained sequence (points 1-2-3-4-5 in the Fig 3(c)) represents basic cyclic loop, for which is the vortex core expelled only to the right boundary of the object (Fig. 3(d)). Fig. 3(a) shows described magnetization reversal loop obtained for 100% runs with described magnetic-field time sequences, which proves that the vortex chirality is perfectly controlled in our experiments. The physics behind such behavior is explained in Ref. 11.
To expel the vortex core to the opening of the PL object (PL left side, Fig. 3(d)), we have to return from point 5 (zero field, Fig. 3(c)) to negative fields, point 6. The vortex with CW chirality then annihilates at Han = -56 mT, and again nucleates at Hnuc = -22.5 mT with CW chirality when the field is again increased. Such half-loop (points 5-6-7) is depicted in the left part of the Fig. 3(b). Similar operation can be realized for the positive part of the loop (points 9-10-11, Fig. 3(b), right part), for which we operate with CCW chirality, and vortex annihilates at 56 mT, and nucleates at 22.5 mT.
Fig. 4 shows the temperature dependence of the vortex nucleation field with field angle as a parameter. The aim is to find the role of the magnetic state in the SD-V transitions. Therefore we do not deal more with the vortex annihilation – magnetic state before vortex annihilation is for all 3 angles similar (vortex state). Each point of the magnetization reversals (except low temperature part of the curve for angle 120°) is a mean value of at least 8 measurements from its both, positive and negative branches. Typically these values are distributed for each shown points (all temperatures, field angles) within the interval ± 0.5 mT, and the typical standard deviation is ∼ 0.15 mT. Similarly to Ref. 6, the curves show two basic slopes – larger one for temperatures lower than 20 K, and small one above 20 K. We assume that the large slopes are caused by presence of many shallow minima of the total energy functional caused by edge corrugations as well as by granular structure of the Permalloy. At low temperatures, all small barriers presents obstacle for magnetization dynamics. Thus at low temperatures the systems has to overcome many barriers which contribute significantly to the final nucleation time, or at fixed rampage of the external field it is manifested as decrease of the nucleation field. At higher temperatures the small barriers does not represent obstacle due to the thermal energy of individual magnetic moments, and only the last barrier represents obstacle for nucleation. Overcoming the last barrier the system lowers its energy significantly, and vortex state is created.
Described mechanism needs further detailed analysis and modeling, which is over the scope of this paper and will be published elsewhere.
Special attention has to be paid to field angle 120° at low temperatures, for which nucleation field shows sort of “digital” noise for both, negative and positive branches. The curve resembles two metastable states in which the system can be trapped.
The appearance of two states (or the “digital” noise) can be explained by two chiralities of the nanomagnet (CW or CCW). To confirm this assumption, we have analyzed the height of Hall-voltage abrupt changes connected with the vortex nucleation. We have found that the heights of the voltage jumps depend on the vortex nucleation field. If we select temperature 5 K, we have for positive branch of the loop and vortex nucleation fields Hnuc = 24 mT and Hnuc = 32 mT (Fig. 4), voltage jumps 4.5±0.1 μV and 3.8±0.4 μV, respectively. For negative branch corresponding voltage jumps read 3.9±0.2 μV and 2.9±0.25 μV. Based on our simulation, so high relative differences in the Hall signal (∼25%) can be explained easily only by different chirality of the final states obtained (see also Fig. 2, line 3). It means that the ‘digital’ noise described for field angle 120° at low temperatures is a direct consequence of two chiralities generated in the nanomagnet.
In conclusion, we studied vortex nucleation/annihilation process and its temperature dependence in Pacman-like nanoobjects using micro-Hall probe magnetometry. Chirality of the object can be easily controlled due to its lowered symmetry. We show experimentally that vortex nucleation field strongly depends on the angle of the external in-plane magnetic field. The experiments also confirm that the vortex nucleation proceeds via thermal activation over an energy barrier.
This work has been supported by Slovak Grant Agency APVV, project APVV-0088-12 and project APVV-0036-11, and by the Research & Development Operational Program funded by the ERDF, “CENTE 1”, ITMS code 26240120011 (0.5).