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by Rok Dittrich

Thin permalloy disks support a vortex configuration. We study how micromagnetic calculations can be applied to processes that involve a singularity of the magnetization field, namely the Bloch point. The reversal of the core of the vortex under an field applied perpendicularly to the disk plane is investigated. We apply two different procedures to evaluate switching fields and processes: direct micromagnetic time-dependent calculation, and the evaluation of the energy barrier that separates the two orientations of the vortex core in the configuration space, using the nudged elastic band method. Both methods show the occurrence of Bloch points during reversal.

Quasistatic hysteresis
color movie1

Quasistatic calculation of vortex core switching. A field is applied opposite to the vortex core orientation. The vortex core compresses with increasing field. At Hext =540 mT the exchange energy density exceeds the exchange energy density of a BP structure. As a result the vortex core switches by insertion and motion of a BP. The disk diameter was 200 nm, thickness = 50 nm. The magnetization states during vortex core switching are shown as perspective cuts across the thickness through the center of the disk. The reversal of the vortex core is clearly seen in the decrese of the exchange energy. The colour code corresponds to the z-component of the magnetization.

Direct computation of the energy barrier

0mT 400mT
Hext = 0 mT Hext = 400 mT

Minimum energy paths for the reversal of a vortex core in a permalloy disk (50 nm thickness, 200 nm diameter) at different external fields. The field is applied opposite to the initial vortex core orientation (same as above). The displayed plane shows a cut through the disk center. 25 images were used to discretize the path. The number of arrows does not correspond to the number of mesh points.

The animtions show a sequence of 5 magnetizations along the minimum energy path. Note the compressed non-reversed vortex when a field is applied (sequence on the right). In both reversal paths a BP appears on the surface and moves through the vortex core. The energy along minimum paths for different external fields are shown below.

0mT 400mT
Energy along the calculated minimum energy paths for the reversal of the vortex core, for fields applied antiparallel to the core magnetization (induction values are indicated in T). The reference energy is that of the metastable state. The mesh in the central region is 2 nm. The energy barrier decreases with increasing field strength until at a critical field the energy barrier becomes zero. This is the zero temperature switching field strength of the core. Plot of the energy barrier height versus the applied field, for two different core mesh sizes. The barrier decreases under the applied field roughly as a second order polynomial. The extrapolation to zero barrier height from the 4 height field points gives a switching field of 0*Hs = 490 mT (3 nm) and 660 mT (2 nm). The shaded region corresponds to a barrier height below a thermal threshold (25 kBT at room temperature). Mesh refinement increases the calculated barriers and switching fields as seen before. The switching fields obtained at this way agree well with the results from the quasistatic calculation above.

Mesh size dependency


Effect of the mesh refinement on (A) the computed reversal fields by a quasistatic calculation and (B) the energy barrier at zero field. The disk diameter was 200 nm, thickness = 50 nm, permalloy parameters were used.

The calculated switching fields and energy barriers depend on the mesh size in the vortex core region. With decreasing mesh size the exchange energy increases. As a result the obtained energy barriers also increase with decreasing mesh size. The increase of the switching field with decreasing mesh size observed in the quasistatic calculations corresponds to the increase of the energy barrier with decreasing mesh size calculated with the nudged elastic band method. Since the energy barriers increased also the obtained switching field increases with decreasing mesh size. The obtained switching fields are in the order of magnitude of the experimental values if a reasonable mesh size is used. The extrapolation to zero mesh size at room temperature predicts fields that are larger than the experimental values. Thus, the BP insertion is probably assisted by defects in the samples.


[1] Rok Dittrich, "Finite element computation of energy barriers in magnetic systems", Ph.D. thesis, Vienna University of Technology, 2003
[2] A. Thiaville, J.M. Garcia, R. Dittrich, J. Miltat, T. Schrefl "Micromagnetic study of Bloch points nucleated vortex core reversal" Phys. Rev. B. 67 (2003) 094410 Abstract | PDF
[3] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Magnetic Vortex Core Observation in Circular Dots of Permalloy , Science (2000) vol 289

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