We investigate the influence of artificial defects (small holes) inserted into magnetic nanodisks on the vortex core dynamics. One and two holes (antidots) are considered. In general, the core falls into the hole but, in particular, we would like to remark an interesting phenomenon not yet observed, which is the vortex core switching induced by the vortex-hole interactions. It occurs for the case with only one hole and for very special conditions involving the hole size and position as well as the disk size. Any small deformation in the disk geometry such as the presence of a second antidot changes completely the vortex dynamics and the vortex core eventually falls into one of the defects. After trapped, the vortex center still oscillates with a very high frequency and small amplitude around the defect center.
It is well known that azimuthal spin wave modes of magnetic vortex state in permalloy nanodisks have circular symmetry. Intuitively, magnetic materials having magnetocrystalline anisotropy is not compatible with the circular symmetry of the azimuthal modes. In this article, however, we report cubic azimuthal modes in heterogeneous nanodisks consisting of a permalloy core and a Fe shell. The fourfold symmetry of azimuthal modes is due to the exchange, and magneto-static, interactions between the permalloy core and the Fe shell. In comparison to results of circular azimuthal mode, the vortex swi…
Magnetic vortex is topologically nontrivial and commonly found in ferromagnetic nanodisks. So far, three classes spin-wave eigenmodes, i.e., gyrotropic, azimuthal and radial modes, have been identified in ferromagnetic nanodisks. Here, using micromagnetic simulation and analytical calculation, we reveal twisted spin-wave modes in a thick permalloy (Ni0.8Fe0.2) nanodisk. The twisted spin-waves carry topological charges, which sign depends on the core polarity of the magnetic vortex in the nanodisk. By applying rotating magnetic fields at one end of the sample, we observe continuous generation o…
We reverse the magnetic vortex curling direction of ferromagnetic nanodisk by applying a circular Oersted field. The nanodisk is fabricated without breaking its symmetry. The Oersted field is induced by passing current through an atomic force microscope tip placed at the center of the disk. Micromagnetic simulation indicates that compared to the uniformly distributed current throughout the cross section of disk, the line current concentrated in the center can reverse the chirality more easily, which is in accordance with our experimental results.
Direct observation of vortex states in an antiferromagnetic layer have been recently reported [Wu, et al, Nature Phys. 7, 303 (2011)]. In contrast to their analogues in ferromagnetic systems, namely in nanomagnets, the vortex core of antiferromagnets are not expected (and have not been observed) to present gyrotropic or any other remarkable dynamics, even when external fields are applied. Using simulated annealing and spin dynamics techniques we have been able to describe a number of properties of such a vortex state. Besides of being in agreement with reported results, our results also indica…
A type of vortex spin texture with skyrmionic core and a series of circle spin stripes was obtained in thin nanodisk of the chiral magnets with Dzyaloshinskii-Moriya interaction and uniaxial anisotropy by means of micromagnetic approach and Monte Carlo simulation. The size of skyrmionic core can be modulated continuously by controlling the disk size. Moreover, in some certain values of the disk size, this vortex state may be spontaneous ground state even without the help of the external magnetic field and thermal fluctuation. In addition, the uniaxial anisotropy is able to stabilize this vorte…
The switching of a vortex core of a single disk in an array of a multilayer system is investigated by micromagnetic simulation. We found that the perpendicular uniaxial anisotropy (PUA) decreases the frequencies of the azimuthal mode in disks with magnetic vortex configuration. We obtained a phase diagram of magnetic field intensity vs. frequency of the azimuthal mode, as a function of the value of perpendicular uniaxial anisotropy. We demonstrated that rotating magnetic fields (CW and CCW) with frequency equal to azimuthal modes can be used to switch the vortex core of single disks in a disk …
We analyze the zero-energy sector of the trigonal zigzag nanodisk and corner based on the Dirac theory of graphene. The zero-energy states are shown to be indexed by the edge momentum and grouped according to the irreducible representation of the trigonal symmetry group $C_{3v}$. Wave functions are explicitly constructed as holomorphic or antiholomorphic functions around the K or K' point. Each zero-energy mode is a chiral edge mode. We find a texture of magnetic vortices. It is intriguing that a vortex with the winding number $2$ emerges in the state belonging to the $E$ (doublet) representat…
Vortex states in magnetic nanodisks are essentially affected by surface/interface induced Dzyaloshinskii-Moriya interactions. Within a micromagnetic approach we calculate the equilibrium sizes and shape of the vortices as functions of magnetic field, the material and geometrical parameters of nanodisks. It was found that the Dzyaloshinskii-Moriya coupling can considerably increase sizes of vortices with "right" chirality and suppress vortices with opposite chirality. This allows to form a bistable system of homochiral vortices as a basic element for storage applications.
The development of future spintronic applications requires a thorough and fundamental understanding of the magnetisation dynamics. Of particular interest are magnetic nanodisks, in which the vortex state emerges as a stable spin configuration. Here, we focus on how the vortex core polarisation can be reversed periodically by an oscillating magnetic field, applied perpendicularly to the disk's surface. By means of micromagnetic simulations, we demonstrate the presence of several subharmonic switching modes, i.e., the commensurate ratio between the switching frequency of the core and the driving…
Interacting magnetic nanoobjects constitute one key component for many proposed spintronic devices, from microwave nano-oscillators to magnetic memory elements. For this reason, the mechanism of this interaction and its dependence with distance $d$ between these nanoobjects has been the subject of several recent studies. In the present work, the problem of the interaction between magnetic nanodisks with magnetic vortex structures is treated both analytically and through micromagnetic simulation. The coupling of two magnetic nanodisks is obtained, in the case of nanodisks with different gyrotro…
We report the experimental and theoretical characterization of the angular-dependent spin dynamics in arrays of ferromagnetic nanodisks arranged on a honeycomb lattice. The magnetic field and microwave frequency dependence, measured by broadband ferromagnetic resonance, reveal a rich spectrum of modes that is strongly affected by the microstate of the network. Based on symmetry arguments with respect to the external field, we show that certain parts of the ferromagnetic network contribute to the detected signal. A comparison of the experimental data with micromagnetic simulations reveals that …
We give a theoretical study on the magnetic properties of monolayer nanodisks with both Heisenberg exchange and Dzyaloshinsky-Moriya (DM) interactions. In particular, we survey the magnetic effects caused by anisotropy, external magnetic field, and disk size when DM interaction is present by means of a new quantum simulation method based on mean-field theory. This computational approach finds that uniaxial anisotropy and transverse magnetic field enhances the net magnetization as well as increases the transition temperature of the vortex phase while preserving the chiralities of the nanodisks.…
Strongly-interacting nanomagnetic systems are pivotal across next-generation technologies including reconfigurable magnonics and neuromorphic computation. Controlling magnetisation state and local coupling between neighbouring nanoelements allows vast reconfigurable functionality and a host of associated functionalities. However, existing designs typically suffer from an inability to tailor inter-element coupling post-fabrication and nanoelements restricted to a pair of Ising-like magnetisation states. Here, we propose a new class of reconfigurable magnonic crystal incorporating nanodisks as t…
A target skyrmion is a flux-closed spin texture that has two-fold degeneracy and is promising as a binary state in next generation universal memories. Although its formation in nanopatterned chiral magnets has been predicted, its observation has remained challenging. Here, we use off-axis electron holography to record images of target skyrmions in a 160-nm-diameter nanodisk of the chiral magnet FeGe. We compare experimental measurements with numerical simulations, demonstrate switching between two stable degenerate target skyrmion ground states that have opposite polarities and rotation senses…
Using micromagnetic simulation, we show that in-plane uniaxial magnetic anisotropy (IPUA) can be used to obtain FAN-OUT, AND and OR gates in an array of coupled disks with magnetic vortex configuration. First, we studied the influence of the direction of application of the IPUA on the energy transfer time (τ) between two identical coupled nanodisks. We found that when the direction of the IPUA is along the x axis the magnetic interaction increases, allowing shorter values of τ, while the IPUA along the y direction has the opposite effect. The magnetic interactions between the nanodisks along x…
We investigated the spin dynamics of a vortex state in a core-shell magnetic nanodisk driven by an oscillating field applied perpendicular to the disk plane by means of micromagnetic simulations. The nanodisk comprises a Py (Fe0.2Ni0.8) core of 100 nm in radius, surrounded by a 50 nm thick Fe shell. Fourier transform analyses show that the Py core and the Fe shell dominate spin-wave oscillation at the fundamental and higher order radial modes, respectively. For oscillating driving field tuned to the fundamental eigenfrequency, the Py/Fe interface effectively confines spin-wave excitation in th…
The magnetic field-driven evolution of chiral spin textures in thin helimagnet nanodisk with varied size are investigated by means of Monte Carlo simulation. It is demonstrated that the complex spin texture may simply be regarded as the superposition of the edged state with in plane spin orientation perpendicular or parallel to the edge and the bulk state with the features similar to two-dimensional chiral magnetic films. With the increase of the external field, the proportion of the parallel spins of the edge state increases, and the spin textures finally transfers into edged magnetic vortex.…
We employ a self-consistent simulation approach based on quantum physics here to study the magnetism of antiferromagnetic skyrmions formed on manolayer nanodisk planes. We find that if the disk is small and the Dzyaloshinsky-Moriya (DM) interaction is weak, a single magnetic vortex may be formed on the disk plane. In such a case, when uniaxial anisotropy normal to the disk plane is further considered, the magnetic configuration remains unchanged, but the magnetization is enhanced in that direction, and reduced in other two perpendicular orientations. Very similarly, a weak external magnetic fi…
We propose the use of the gyrotropic motion of vortex cores in nanomagnets to produce a magnetic echo, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g., nanodisks, is magnetized with an in-plane (xy) field, and after a time τa field pulse inverts the core magnetization; the echo is a peak in M_{xy} at t=2τ. Its relaxation times depend on the inhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant α. Its feasibility is demonstrated using micromagnetic simulation. To illustrate an application of the echoes, we have determin…
Quantization effects of the nonlinear magnon-vortex interaction in ferromagnetic nanodisks are studied. We show that the circular geometry twists the spin-wave fields with spiral phase dislocations carrying quantized orbital angular momentum (OAM). Meanwhile, the confluence and splitting scattering of twisted magnons off the gyrating vortex core (VC) generates a frequency comb consisting of discrete and equally spaced spectral lines, dubbed as twisted magnon frequency comb (tMFC). It is found that the mode spacing of the tMFC is equal to the gyration frequency of the VC and the OAM quantum num…
The ability to modulate light at high speeds is of paramount importance for telecommunications, information processing, and medical imaging technologies. This has stimulated intense efforts to master optoelectronic switching at visible and near-infrared frequencies, although coping with current computer speeds in integrated architectures still remains a major challenge. As a partial success, midinfrared light modulation has been recently achieved through gating patterned graphene. Here we show that atomically thin noble metal nanoislands can extend optical modulation to the visible and near-in…
A model for describing structural pointlike defects in nanoscaled ferromagnetic materials is presented. Its details are explicitly developed whenever interacting with a vortex-like state comprised in a thin nanodisk. Among others, our model yields results for the vortex equilibrium position under the influence of several defects along with an external magnetic field in good qualitative agreement with experiments. We also discuss how such defects may affect the vortex motion, like its gyrotropic oscillation and dynamical polarization reversal.
Under the action of an alternating perpendicular magnetic field the polarity of the vortex state nanodisk can be efficiently switched. We predict the regular and chaotic dynamics of the vortex polarity and propose simple analytical description in terms of a reduced vortex core model. Conditions for the controllable polarity switching are analyzed.
A magnetic vortex has attracted significant attention since it is a topologically stable magnetic structure in a soft magnetic nanodisk. Many studies have been devoted to understanding the nature of magnetic vortex in isolated systems. Here we show a new aspect of a magnetic vortex the dynamics of which strongly affects the magnetic structures of environment. We exploit a nanodot of an exchange-coupled bilayer with a soft magnetic Ni81Fe19 (permalloy; Py) having a magnetic vortex and a perpendicularly magnetized L10-FePt exhibiting a large switching field (Hsw). The vortex dynamics with azimut…
We report on the switching of the magnetic vortex core in a Pac-man disk using a magnetic field pulse, investigated via micromagnetic simulations. The minimum core switching field is reduced by 72 % compared to that of a circular disk with the same diameter and thickness. However, the core switches irregularly with respect to both the field pulse amplitude and duration. This irregularity is induced by magnetization oscillations which arise due to excitation of the spin waves when the core annihilates. We show that the core switching can be controlled with the assist magnetic field and by chang…
We study vortex domain wall dynamics in wide Permalloy strips driven by applied magnetic fields and spin-polarized electric currents. As recently reported [V. Estévez and L. Laurson, Phys. Rev. B, 93, 064403 (2016)], for sufficiently wide strips and above a threshold field, periodic dynamics of the vortex core are localized in the vicinity of one of the strip edges, and the velocity drop typically observed for narrow strips is replaced by a high-velocity plateau. Here, we analyze this behavior in more detail by means of micromagnetic simulations. We show that the high-velocity plateau originat…
We report manipulation of the magnetic states of elliptical cobalt magnetostrictive nanomagnets (of nominal dimensions ~ 340 nm x 270 nm x 12 nm) delineated on bulk 128° Y-cut lithium niobate with Surface Acoustic Waves (SAWs) launched from interdigitated electrodes. Isolated nanomagnets that are initially magnetized to a single domain state with magnetization pointing along the major axis of the ellipse are driven into a vortex state by surface acoustic waves that modulate the stress anisotropy of these nanomagnets. The nanomagnets remain in the vortex state until they are reset by a strong m…
Influence of the spin-transfer torque on the vortex state magnetic nanodisk is studied numerically via Slonczewski-Berger mechanism. The existence of a critical current is determined for the case of same-directed electrical current, its spin polarization and polarity of the vortex. The critical current separates two regimes: (i) deformed but static vortex state and (ii) essentially dynamic state under which the spatio-temporal periodic structures can appear. The structure is a stable vortex-antivortex lattice. Symmetry of the lattice depends on the applied current value and for high currents (…
Magnetic vortices are characterized by the senses of in-plane magnetization chirality and by the polarity of the vortex core. The electrical control of vortex polarity and chirality is highly demanded not only for fundamental understanding on spin dynamics in nano-disks under different circumstances, but also for technological applications, such as magnetic non-volatile memories and spin torque oscillators for neuromorphic computing. Here we report a novel approach that enables one to electrically control both the vortex chirality and polarity with low energy consumption. Thorough micromagneti…
The thermal stability in nanostructured magnetic systems is an important issue for applications in information storage. From a theoretical and simulation perspective, an accurate prediction of thermally-activated transitions is a challenging problem because desired retention times are on the order of 10 years, while the characteristic time scale for precessional magnetization dynamics is of the order of nanoseconds. Here, we present a theoretical study of the thermal stability of magnetic elements in the form of perpendicularly-magnetized ferromagnetic disks using the forward flux sampling met…
The interplay between magnetic vortex polarity, chirality and the curvature of the underlying surface results in a dependence of the vortex polarity switching efficiency on the vortex chirality. The switching is studied numerically by applying a short Gauss pulse of the external magnetic field to a spherical cap within its cut plane. The minimum field intensity required for the switching essentially depends on the vortex chirality and it does not depend on the initial vortex polarity. This effect decreases with the curvature radius increasing and it vanishes in the planar limit.
Magnetic vortex cores exhibit a gyrotropic motion, and may reach a critical velocity, at which point they invert their z-component of the magnetization. We performed micromagnetic simulations to describe this vortex core polarity reversal in magnetic nanodisks presenting a perpendicular anisotropy. We found that the critical velocity decreases with increasing perpendicular anisotropy, therefore departing from a universal criterion, that relates this velocity only to the exchange stiffness of the material. This leads to a critical velocity inversely proportional to the vortex core radius. We ha…
We consider a nanodisk possessing two coupled materials with different ferromagnetic exchange constant. The common border line of the two media passes at the disk center dividing the system exactly in two similar half-disks. The vortex core motion crossing the interface is investigated with a simple description based on a two-dimensional model which mimics a very thin real material with such a line defect. The main result of this study is that, depending on the magnetic coupling which connects the media, the vortex core can be dramatically and repeatedly flipped from up to down and vice versa …
A simple mechanism of controllable switching of magnetic vortex chirality is proposed. We consider curvilinear magnetic nanoshells of spherical geometry whose ground state is a vortex magnetization distribution. Chirality of this magnetic vortex can be switched in controllable way by applying a Gaussian pulse of spatially uniform magnetic field along the symmetry axis of the shell. The chirality switching process is explored in detail numerically for various parameters of magnetic pulse: the corresponding switching diagram is build. The role of the curvature is ascertained by studying the swit…
We theoretically investigate the radial-spin-wave induced magnetic vortex switching in the presence of Dzyaloshinskii-Moriya interaction (DMI). From micromagnetic simulations, we observe a circular-to-radial vortex phase transition by increasing the DMI strength. The radial spin-wave excitation spectrum for each magnetization configuration is analyzed, showing that the frequency of spin-wave mode with a given radial node number monotonically increases (decreases) with the DMI parameter of the radial (circular) vortex. Interestingly, we find that the DMI can significantly facilitate the polarit…
The interaction of a magnetic vortex with a rotating magnetic field causes the nucleation of a vortex--antivortex pair leading to a vortex polarity switching. The key point of this process is the creation of a dip, which can be interpreted as a nonlinear resonance in the system of certain magnon modes with nonlinear coupling. The usually observed single-dip structure is a particular case of a multidip structure. The dynamics of the structure with $n$ dips is described as the dynamics of nonlinearly coupled modes with azimuthal numbers $m=0,\pm n,\pm 2n$. The multidip structure with arbitrary n…
Because of its stability, the polarity of a magnetic vortex core (VC) is a candidate for binary data storage. Switching can be accomplished, e.g, by driving the VC above a critical velocity $v_c$. Here, we report on controlled and repeated switching of VC polarity by significantly reducing $v_c$ locally. We excite vortex dynamics in thin Permalloy disks with a magnetic field pulse, and map the two-dimensional VC trajectory using time-resolved Kerr microscopy. In pristine samples, we observe normal gyrotropic motion of the VC. After laser-induced generation of defects, however, we observe repea…
Spin vortices in magnetic nanopillars are used as GHz oscillators, with frequency however essentially fixed in fabrication. We demonstrate a model system of a two-vortex nanopillar, in which the resonance frequency can be changed by an order of magnitude, without using high dc magnetic fields. The effect is due to switching between the two stable states of the vortex pair, which we show can be done with low-amplitude fields of sub-ns duration. We detail the relevant vortex-core dynamics and explain how field anharmonicity and phase control can be used to enhance the performance.
Vortex core polarity switching in NiFe disks has been evidenced using an all-electrical rectification scheme. Both simulation and experiments yield a consistent loss of the rectified signal when driving the core at high powers near its gyrotropic resonant frequency. The frequency range over which the loss occurs grows and shifts with increasing signal power, consistent with non-linear core dynamics and periodic switching of the core polarity induced by the core attaining its critical velocity. We demonstrate that core polarity switching can be impeded by displacing the core towards the disk's …
The influence of the in-plane uniaxial anisotropy (IPUA) in the mutual energy transfer time ($τ$) between two identical coupled nanodisks was studied. Using an analytical dipolar model we obtained the interactions between the disks along x and y directions (the coupling integrals) as a function of the uniaxial anisotropy constant (K$_σ$) and the distance. We find that the IPUA increases the interaction between the disks allowing shorter energy transfer times. For our range of K$_σ$ values we get a drop in the values of $τ$ of up to about 70$\%$. From the lagrangian of the system we obtained th…
Electric control of magnetic vortex dynamics in a reproducible way and on an ultrafast time scale is key element in the quest for efficient spintronic devices with low-energy consumption. To be useful the control scheme should ideally be swift, scalable, non-invasive, and resulting in reliable magnetic switching. Such requirements and in particular the reproducibility of altering the vortex chirality and/or polarity are not yet met by magnetic vortex switching via external magnetic fields, spin-polarized currents, spin waves, or laser pulses. Here we demonstrate a novel packaged-skyrmion media…
Magnetic vortex, as one of the most interesting magnetic solitons, has attracted great interests in the past two decades. A fast and reliable method to switch vortex polarity and chirality is one of the key issues for various applications. Based on micromagnetic simulation, here we report a fast, low energy cost and deterministic switching of vortex core, through the designing of a notch structure in disks and the use of out-of-plane current geometry. We demonstrate that with such design, the multiple switching problems found in notch disk system can be avoided. Furthermore, the switching time…
We present a theoretical discovery of an unconventional mechanism of inverse Faraday effect (IFE) which acts selectively on topological magnetic structures. The effect, topological inverse Faraday effect (TIFE), is induced by spin Berry's phase of the magnetic structure when a circularly polarized light is applied. Thus a spin-orbit interaction is not necessary unlike in the conventional IFE. We demonstrate by numerical simulation that TIFE realizes ultrafast switching of a magnetic vortex within a switching time of 150 ps without magnetic field.
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformationof pillow- or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the sw…
We have investigated the motion of vortex domain walls passing across non symmetric triangular notches in single Permalloy nanowires. We have measured hysteresis cycles using the focused magneto-optical Kerr effect before and beyond the notch, which allowed to probe beyond the notch the occurrence probability of clockwise (CW) and counter-clockwise (CCW) walls in tail-to-tail (TT) and head-to-head (HH) configurations. We present experimental evidence of chirality flipping provided by the vortex -- notch interaction. With a low exit angle the probability of chirality flipping increases and here…
Symmetry classification of the magnetic vortices and skyrmions has been suggested. Relation between symmetry based predictions and direct calculation has been shown. It was shown, that electric dipole moment of the vortex is located inside the small vortex core. The antivortices and antiskyrmions do not carry the total core electric dipole induced by the flexomagnetoelectric interaction in the hexoctahedral cubic crystal. The volumetric bound electric charge is distributed around the core. Switching of the core electric dipole direction produces the switching of the core magnetization or vorte…
Nanodots with magnetic vortices have many potential applications, such as magnetic memories (VRAMs) and spin transfer nano-oscillators (STNOs). Adding a perpendicular anisotropy term to the magnetic energy of the nanodot it becomes possible to tune the vortex core properties. This can be obtained, e.g., in Co nanodots by varying the thickness of the Co layer in a Co/Pt stack. Here we discuss the spin configuration of circular and elliptical nanodots for different perpendicular anisotropies; we show for nanodisks that micromagnetic simulations and analytical results agree. Increasing the perpen…
Scanning transmission x-ray microscopy is employed to investigate experimentally the reversal of the magnetic vortex core polarity in cylindrical Ni81Fe19 nanodisks triggered by two orthogonal monopolar magnetic field pulses with peak amplitude $B_0$, pulse length $τ$=60 ps and delay time $Δ$t in the range from -400 ps to +400 ps between the two pulses. The two pulses are oriented in-plane in the x- and y-direction. We have experimentally studied vortex core reversal as function of $B_0$ and $Δ$t. The resulting phase diagram shows large regions of unidirectional vortex core switching where the…
In a nanopillar with dipolarly coupled vortices, we present an experimental and simulation study to understand how the interplay between the bias field and spin transfer torque impacts reversal of the vortex cores. We find that, depending on the current values, vortex cores might experience different physical mechanisms for their reversal, namely a static or a dynamic switching. We believe that our results might be useful in the context of vortex based non volatile memories, as a current controlled selective core switching is proposed.
We study the influence of the spin torque, which depends on the space and time derivative of magnetization, on magnetization reversal time in a ferromagnetic fine particle. The spin torque operates to dissipate the angular momentum of the magnetization precession, and the torque increases in a spin vortex structure. We calculate the magnetization reversal time under a DC magnetic field using the Landau-Lifshitz-Gilbert equation containing a spin torque term. We found that the spin torque changes the magnetization switching speed significantly during the reversal process by maintaining a spin v…
Switching magnetic texture via in-plane magnetic field in noncentrosymmetric dipolar magnets: From skyrmions to antiskyrmions and nontopological magnetic bubbles
Micromagnetic sensors play a major role towards the miniaturization in the industrial society. The adoption of new and emerging sensor technologies like anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) sensors are mainly driven by their integrability and enhanced sensitivity. At the core of such sensors, a microstructured ferromagnetic thin film element transduces the magnetic signal. Such elements usually switch via multi-domain, C- or S-shaped magnetization states and, therefore, often exhibit an open non-linear hysteresis curve. Linearity…
