Vortex motion in amorphous ferrimagnetic thin film elements

Vortex motion in amorphous ferrimagnetic thin film elements Harald Oezelt, Eugenie Kirk, Phillip Wohlhüter, Elisabeth Müller, Laura Jane Heyderman, Alexander Kovacs, and Thomas Schrefl Center for Integrated Sensor Systems, Danube University Krems, 2700 Wiener Neustadt, Austria Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Laboratory of Biomolecular Research, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland


I. INTRODUCTION
In patterned ferromagnetic thin film elements multidomain states form if the elements are sufficiently large.Most commonly the ground state is a symmetric flux closure pattern.In medium sized square elements the remanent state is a four-domain pattern with straight domain walls and a center vortex. 1 Dietrich and co-workers 2 showed that in permalloy (Ni 81 Fe 19 ) squares a distorted flux closure pattern may arise from the substrate curvature which may lead to curved domain walls.Similarly, the magnetic field created by the tip of a magnetic force microscope can induce twisted flux closure patterns. 3A distortion of the equilibrium domain patterns caused by magnetic fields was also reported by Hertel and co-workers. 4They showed that the demagnetizing field of nano-islands with inclined surface leads to asymmetric magnetic states.
Whereas the above mentioned distortions of the domain patterns are a result of local magnetostatic fields, Heyderman and co-workers 5 reported asymmetric domain configurations in Ni 83 Fe 17 and Co elements when decreasing the thickness to below 17 nm.They attributed the observed patterns to material defects such as edge and surface roughness, but also local magnetocrystalline anisotropy variations, which lead to pinning of domain walls or vortices.For thicker elements they observed a symmetric domain pattern.In this work, we investigate the domain structures in amorphous Fe 64 Gd 36 elements.Due to shape anisotropy the magnetization lies in the plane of the film.Magnetic images are obtained by transmission electron microscopy imaging in the Fresnel mode.In most samples an asymmetric flux closure pattern was observed in zero applied field.Micromagnetic simulations that take into account the random anisotropy distribution 6 of the amorphous film show the same key features of the domain pattern: 1) an off-centered vortex state with 2) curved domain walls at equilibrium, 3) increasing the element thickness leads to a reduced vortex offset, and 4) pinning of the vortex core during application of magnetic fields in the plane of the sample.

II. EXPERIMENTAL METHOD
Amorphous Fe 64 Gd 36 thin film square elements of edge lengths ranging from 3 µm to 10 µm were fabricated by electron beam lithography and subsequent lift-off.The magnetic material with a thickness of 50 nm was deposited by ultra high vacuum (UHV) magnetron sputtering onto silicon nitride membranes and covered by 5 nm Pt to prevent rapid oxidation.For reference measurements permalloy (Py, Ni 81 Fe 19 ) squares of edge length 10 µm and thickness 15 nm were fabricated.
The vortex pinning is imaged with transmission electron microscopy (TEM) in the Fresnel mode.This technique is based on the interaction between an electron beam and the magnetic flux of the sample.Deflection of the electron beam due to the Lorentz force occurs if the magnetization changes in the plane of the sample.In the Fresnel imaging mode, the sample is defocused revealing magnetic contrast.The vortex position is tracked as a function of applied in-plane field, which is created by the remanent field of the TEM objective lens.Tilting the sample inside the objective lens field results in increased in-plane fields.At zero tilt, the in-plane component of the magnetic field is minimal, which is reflected by a center position of the vortex in NiFe.On increasing the tilt, and thus the applied magnetic field, the vortex is displaced from the center.

III. MICROMAGNETIC SIMULATIONS
The micromagnetic treatment of amorphous ferrimagnets such as Fe 64 Gd 36 was pioneered by Mansuripur and co-workers. 7Introducing patches that resemble local disorder of the amorphous film, they were able to explain domain nucleation, domain wall pinning, and coercivity in amorphous rare-earth transition metal films used for magneto-optic recording. 8By means of the patchy structure, they introduced a structural correlation length into their micromagnetic model.Within a patch, which can have arbitrary shape, the direction of the local anisotropy was assumed constant.Fu and coworkers 9 showed that the critical field for the nucleation of reversed domains strongly depends on the patch size, whereas the pinning field of domain walls depends on the patch-to-patch easy axis orientation.They emphasize that the nanoscale patches are magnetic entities and not microstructural features, for example columnar structures or poly-crystalline grains.Mansuripur 10 derived an equation for magnetization dynamics in ferrimagnetic rare earth transition metal alloys.Assuming a strong coupling between the rare earth and transition metal spins, he deduced an equation for the effective magnetization.We adapt this approach for our micromagnetic model as we described in previously published work. 11Patches are created using Voronoi tessellation 12 of the thin film element.The patches are further discretized using a tetrahedral finite element mesh.In order to compute the remanent domain configuration, the sample is initialized with a random magnetization (Fig. 1).Then the system is relaxed by solving the effective Landau-Lifshitz equation with infinite damping. 13Fast convergence of the algorithm is achieved by using the steepest descent method with a modified Barzilai-Borwein step size selection. 14The magneto-static field is computed from a magnetic scalar potential.The parallelepipedic shell transformation 15 is applied to treat the boundary conditions at infinity.

IV. RESULTS AND DISCUSSION
The remanent Fresnel images of the specimens show the well-known four-domain flux closure pattern.However, in thin ferrimagnetic specimens, for example the 20 nm Fe 64 Gd 36 square in Fig. 2c it can be seen that the vortex is displaced from the center of the square element and the domain walls are curved.In the reference measurement on 15 nm thick Ni 81 Fe 19 , the vortex core is centered as expected and the domain walls are almost straight (Fig. 2a).Further measurements revealed that with increased thickness of the FeGd squares the vortex core moves towards the center of the square and therefore closer to the ideal symmetric pattern.
In the micromagnetic simulations we compare a continuous element to a patchy element.The continuous element has no structural features.For both elements the saturation polarization was set to J s = 1 T and the exchange stiffness constant to A x = 10 pJ/m.The patchy element was given a mean magnetocrystalline anisotropy of Ku = 0.1 MJ/m 3 with standard deviation σ K = 20% across the patches, but with random anisotropic easy axis for each patch.The patchy structure was deeply investigated by Mansuripur for amorphous films in magneto optical recording. 10Nucleation fields and domain wall pinning fields computed with the assumption of random anisotropy fluctuation compare well with experimental data.The mean anisotropy field assigned to the patches of our micromagnetic model corresponds to the experimentally measured one in Fe 64 Gd 36 . 16For comparison we modelled a continuous film with zero magneto-crystalline anisotropy.The exchange constant and the magnetization of this film match the values for NiFe.The uniaxial anisotropy of NiFe is essentially zero unless the film shows a stress induced magneto-elastic anisotropy.By this comparison we want to show that the presence of randomness in the magneto-crystalline anisotropy creates pinning sites for domain walls and vortex cores.In GdFe structural randomness is more important than in permalloy because the magneto-crystalline anisotropy in GdFe is several orders of magnitude higher than in NiFe.For the ease of computation the lateral extension of the square elements was reduced to 300 nm × 300 nm.In Fig. 2 the simulated remanent states are compared to the measurement.
In the continuous case the domain pattern is perfectly symmetric, thus in Fig. 2b the gray ribbon representing the domain wall is perpendicular and can not be distinguished from the dotted white line.In the patchy sample the random anisotropy introduced through the patches leads to an asymmetric closure domain pattern.The vortex is shifted and the domain walls are curved.This is in agreement with the experimental results.When we double the film thickness from 20 nm to 40 nm of the patchy element the vortex moves towards the center.This is shown in Fig. 2e where the vortex is located close to the center.However, the randomness still can be observed by the twist in the domain wall.The now symmetric pattern may be attributed either to the more dominant magnetostatic energy in the thicker film or the averaging of the anisotropy fluctuations throughout the thickness of the film.
Starting from the remanent state, we now apply an increasing in-plane field.In TEM this is done by tilting the sample inside the objective lens field.This allows us to observe how the vortex core gets pushed away from its original position due to the growth of domains parallel to the applied field.In a material with little pinning such as NiFe, the vortex is displaced reversibly with the applied field, which can be also observed in the simulated continuous model.In case of ferrimagnetic Fe 64 Gd 36 , the behavior is more complex: at low fields, the vortex displacement is exponential; at intermediate fields, the displacement is hysteretic; and at high fields, the displacement is linear (Fig. 3a).In the simulations with the patchy model we can reproduce the hysteresis in the magnetization curve (Fig. 3b).

V. CONCLUSION
Asymmetric four-domain patterns in remanent state were observed in thin ferrimagnetic FeGd square elements by Fresnel imaging in a transmission electron microscope.When an increasing inplane field is applied, nonlinear vortex movements are detected.These findings were investigated by micromagnetic simulations.By dividing the amorphous ferrimagnet into patches with varying anisotropic properties structural inhomogeneities are introduced.While possible sources such as magnetostatic fields can not be ruled out, the introduced randomness is sufficient to explain several features observed in the measurements: 1) off-center vortex core and 2) curved domain walls in remanent state, 3) increasing thickness of the square decreases the vortex offset, and 4) pinning and depinning of the vortex core during magnetization by an external in-plane field.All these features could not be seen in the reference measurements and simulations of ferromagnetic NiFe squares.

FIG. 1 .
FIG. 1. Schematic of the micromagnetic model used to simulate the FeGd element: It is divided into patches which in turn are subdivided into tetrahedral finite elements.Random magnetization across the nodes of the mesh are used as the initial state.

FIG. 2 .
FIG.2.The remanent domain pattern of a permalloy square with edge length 10 µm shows an almost ideal flux-closure pattern and straight domain walls in the TEM image a) and also in the simulation b).The ferrimagnetic FeGd squares with edge lengths 10 µm clearly show a displaced vortex core and curved domain walls in the measurement c) as well as in the simulation with a patchy microstructure d).When the thickness of the simulated element is doubled e) the vortex core is centered again, but the randomness can still be recognized by the twist in the domain walls.The white dotted lines are a guide to the eye to mark the ideal four-domain flux closure pattern.The domain walls in the simulations are shown as gray ribbons and the magnetic moments as gray arrows.The in-plane angle of the magnetic moments is represented according to the color-map.