# Micromagnetic Simulation of Vortex Development in Magnetic Bi-Material Bow-Tie Structures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

_{L}is defined as parallel to the external magnetic field, while the transverse magnetization component M

_{T}depicts the hysteresis curve perpendicular to it.

## 3. Materials and Methods

_{S,Fe}= 1700 × 10

^{3}A/m (M

_{S,Py}= 796 × 10

^{3}A/m), exchange constant A

_{Fe}= 21 × 10

^{−12}J/m (A

_{Py}= 10.5 × 10

^{−12}J/m), magneto-crystalline anisotropy constant K

_{1,Fe}= 48 × 10

^{3}J/m

^{3}(K

_{1,Py}= 0.5 × 10

^{3}J/m

^{3}). Along the borders, these values were averaged between the values of the single materials.

^{3}for the whole sample. Simulations were typically performed in the range between ± 200 mT in steps of 0.4 mT; if minor loops were found, the field ranges were correspondingly increased to investigate proper full hysteresis loops. All simulations are started from a fully saturated state of the magnetization along the corresponding angle.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Blachowicz, T.; Kosmalska, D.; Döpke, C.; Leiste, H.; Hahn, L.; Ehrmann, A. Varying steps in hysteresis loops of Co square nano-frames. J. Magn. Magn. Mater.
**2019**, 491, 165619. [Google Scholar] [CrossRef] - Ehrmann, A.; Blachowicz, T.; Komraus, S.; Nees, M.-K.; Jakobs, P.-J.; Leiste, H.; Mathes, M.; Schaarschmidt, M. Magnetic properties of square Py nanowires: Irradiation dose and geometry dependence. J. Appl. Phys.
**2015**, 117, 173903. [Google Scholar] [CrossRef] - Pagnanelli, F.; Altimari, P.; Bellagamba, M.; Granata, G.; Moscardini, E.; Schiavi, P.G.; Toro, L. Pulsed electrodeposition of cobalt nanoparticles on copper: Influence of the operating parameters on size distribution and morphology. Electrochim. Acta
**2015**, 155, 228–235. [Google Scholar] [CrossRef] - Romero, M.; Pardo, H.; Faccio, R.; Suescun, L.; Vazquez, S.; Laborda, I.; Fernandez-Werner, L.; Acosta, A.; Castiglioni, J.; Mombru, A.W. A Study on the Polymer Precursor Formation and Microstructure Evolution of Square-Shaped (La
_{0.5}Ba_{0.5})(Mn_{0.5}Fe_{0.5})O_{3}Ceramic Nanoparticles. J. Ceram. Sci. Tech.**2015**, 6, 221–230. [Google Scholar] - Nogués, J.; Sort, J.; Langlais, V.; Skumryev, V.; Surinach, S.; Munoz, J.S.; Baro, M.D. Exchange bias in nanostructures. Phys. Rep.
**2005**, 422, 65. [Google Scholar] [CrossRef] - Mahato, B.K.; Choudhury, S.; Mandal, R.; Barman, S.; Otani, Y.; Barman, A. Tunable configurational anisotropy in collective magnetization dynamics of Ni
_{80}Fe_{20}nanodot arrays with varying dot shapes. J. Appl. Phys.**2015**, 117, 213909. [Google Scholar] [CrossRef] [Green Version] - Thevenard, L.; Zeng, H.T.; Petit, D.; Cowburn, R.P. Macrospin limit and configurational anisotropy in nanoscale permalloy triangles. J. Magn. Magn. Mater.
**2010**, 322, 2152–2156. [Google Scholar] [CrossRef] [Green Version] - Cowburn, R.P.; Adeyeye, A.O.; Welland, M.E. Configurational anisotropy in nanomagnets. Phys. Rev. Lett.
**1998**, 81, 5414. [Google Scholar] [CrossRef] - Wachowiak, A.; Wiebe, J.; Bode, M.; Pietzsch, O.; Morgenstern, M.; Wiesendanger, R. Direct Observation of Internal Spin Structure of Magnetic Vortex Cores. Science
**2002**, 298, 577–580. [Google Scholar] [CrossRef] [PubMed] - Guslienko, K.Y.; Aranda, G.R.; Gonzalez, J.M. Topological gauge field in nanomagnets: Spin-wave excitations over a slowly moving magnetization background. Phys. Rev. B
**2010**, 81, 014414. [Google Scholar] [CrossRef] [Green Version] - Zhang, W.; Haas, S. Phase diagram of magnetization reversal processes in nanorings. Phys. Rev. B
**2010**, 81, 064433. [Google Scholar] [CrossRef] [Green Version] - Zhu, F.Q.; Fan, D.L.; Zhu, X.C.; Zhu, J.G.; Cammarata, R.C.; Chien, C.L. Ultrahigh-Density Arrays of Ferromagnetic Nanorings on Macroscopic Areas. Adv. Mater.
**2004**, 16, 2155. [Google Scholar] [CrossRef] - Mejia-López, J.; Altbir, D.; Romero, A.H.; Batlle, X.; Roshchin, I.V.; Li, C.-P.; Schuller, I.K. Vortex state and effect of anisotropy in sub-100-nm magnetic nanodots. J. Appl. Phys.
**2006**, 100, 104319. [Google Scholar] [CrossRef] [Green Version] - Ehrmann, A.; Blachowicz, T. Influence of shape and dimension on magnetic anisotropies and magnetization reversal of Py, Fe, and Co nano-objects with four-fold symmetry. AIP Adv.
**2015**, 5, 097109. [Google Scholar] [CrossRef] [Green Version] - Subramani, A.; Geerpuram, D.; Domanowski, A.; Baskaran, V.; Metlushko, V. Vortex state in magnetic rings. Phys. C
**2004**, 404, 241. [Google Scholar] [CrossRef] - Wang, J.; Adeyeye, A.O.; Singh, N. Magnetostatic interactions in mesoscopic Ni
_{80}Fe_{20}ring arrays. Appl. Phys. Lett.**2005**, 87, 262508. [Google Scholar] [CrossRef] - Gao, X.S.; Adeyeye, A.O.; Goolaup, S.; Singh, N.; Jung, W.; Castano, F.J.; Ross, C.A. Inhomogeneities in spin states and magnetization reversal of geometrically identical elongated Co rings. J. Appl. Phys.
**2007**, 101, 09F505. [Google Scholar] [CrossRef] - Vavassori, P.; Grimsditch, M.; Novosad, V.; Metlushko, V.; Ilic, B. Metastable states during magnetization reversal in square permalloy rings. Phys. Rev. B
**2003**, 67, 134429. [Google Scholar] [CrossRef] - Vavassori, P.; Bovolenta, R.; Metlushko, V.; Ilic, B. Vortex rotation control in Permalloy disks with small circular voids. J. Appl. Phys.
**2006**, 99, 053902. [Google Scholar] [CrossRef] [Green Version] - Leong, T.G.; Zarafshar, A.M.; Gracias, D.H. Three-dimensional fabrication at small size scales. Small
**2010**, 6, 792. [Google Scholar] [CrossRef] - Amaladass, E.; Ludescher, B.; Schütz, G.; Tyliszczak, T.; Lee, M.-S.; Eimüller, T. Nanospheres generate out-of-plane magnetization. J. Appl. Phys.
**2010**, 107, 053911. [Google Scholar] [CrossRef] - Blachowicz, T.; Ehrmann, A.; Steblinski, P.; Pawela, L. Magnetization reversal in magnetic half-balls influenced by shape perturbations. J. Appl. Phys.
**2010**, 108, 123906. [Google Scholar] [CrossRef] [Green Version] - Ehrmann, A.; Blachowicz, T. Vortex and double-vortex nucleation during magnetization reversal in Fe nanodots of different dimensions. J. Magn. Magn. Mater.
**2019**, 475, 727–733. [Google Scholar] [CrossRef] - Lyberatos, A.; Komineas, S.; Papanicolaou, N. Precessing vortices and antivortices in ferromagnetic elements. J. Appl. Phys.
**2011**, 109, 023911. [Google Scholar] [CrossRef] [Green Version] - Ehrmann, A.; Blachowicz, T. Systematic study of magnetization reversal in square Fe nanodots of varying dimensions in different orientations. Hyperfine Interact.
**2018**, 239, 48. [Google Scholar] [CrossRef] - Mohler, G.; Harter, A.W. Micromagnetic investigation of resonance frequencies in ferromagnetic particles. J. Appl. Phys.
**2005**, 97, 10E313. [Google Scholar] [CrossRef] - Kanso, H.; Pate, R.; Baltz, V.; Ledue, D. Influence of finite-size and edge effects on the exchange-bias properties of ferromagnetic/antiferromagnetic nanodots: Granular Monte Carlo investigation. Phys. Rev. B
**2019**, 99, 054410. [Google Scholar] [CrossRef] [Green Version] - Depondt, P.; Levy, J.-C.S.; Mamica, S. Vortex polarization dynamics in a square magnetic nanodot. J. Phys. Cond. Matter
**2013**, 25, 466001. [Google Scholar] [CrossRef] - Ivanov, Y.P.; Nefedev, K.V.; Iljin, A.I.; Pustovalov, E.V.; Chebotkevich, L.A. Magnetization reversal of nanodots with different magnetic anisotropy and magnetostatic energy. J. Phys. Conf. Ser.
**2011**, 266, 012117. [Google Scholar] [CrossRef] - Song, S.Y.; Zhang, G.F.; Song, W.B.; Guo, G.H. Dynamical reversal of rectangular nanodot studied by micromagnetics. Act. Phys. Sin.
**2009**, 58, 5757–5762. [Google Scholar] - Tillmanns, A.; Blachowicz, T.; Fraune, M.; Güntherodt, G.; Schuller, I.K. Anomalous magnetization reveral mechanism in unbiased Fe/FeF
_{2}investigated by means of the magneto-optic Kerr effect. J. Magn. Magn. Mater.**2009**, 321, 2932–2935. [Google Scholar] [CrossRef] - Nasirpouri, F.; Engbarth, M.A.; Bending, S.J.; Peter, L.M.; Knittel, A.; Fangohr, H.; Milosevic, M.V. Three-dimensional ferromagnetic architectures with multiple metastable states. Appl. Phys. Lett.
**2011**, 98, 222506. [Google Scholar] [CrossRef] [Green Version] - Mulkers, J.; Milosevic, M.V.; van Waeyenberge, B. Cycloidal versus skyrmionic states in mesoscopic chiral magnets. Phys. Rev. B
**2016**, 93, 214405. [Google Scholar] [CrossRef] [Green Version] - Mulkers, J.; van Waeyenberge, B.; Milosevic, M.V. Effects of spatially engineered Dzyaloshinskii-Moriya interaction in ferromagnetic films. Phys. Rev. B
**2017**, 95, 144401. [Google Scholar] [CrossRef] [Green Version] - Mulkers, J.; van Waeyenberge, B.; Milosevic, M.V. Tunable Snell’s law for spin waves in heterochiral magnetic films. Phys. Rev. B
**2018**, 97, 104422. [Google Scholar] [CrossRef] [Green Version] - Menezes, R.M.; Mulkers, J.; de Souza Silva, C.C.; Milosevic, M.V. Deflection of ferromagnetic and antiferromagnetic skymions at heterochiral interfaces. Phys. Rev. B
**2019**, 99, 104409. [Google Scholar] [CrossRef] [Green Version] - Wang, S.Y.; Zhan, Q.W. Modified bow-tie antenna with strong broadband field enhancement for RF photonic applications. Proc. SPIE
**2013**, 8806, 88061V. [Google Scholar] - Ehrmann, A.; Blachowicz, T. Influence of the distance between nanoparticles in clusters on the magnetization reversal process. J. Nanomater.
**2017**, 2017, 5046076. [Google Scholar] [CrossRef] [Green Version] - Ehrmann, A.; Blachowicz, T. Interaction between magnetic nanoparticles in clusters. AIMS Mater. Sci.
**2017**, 4, 383–390. [Google Scholar] [CrossRef] - Donahue, M.J.; Porter, D.G. OOMMF User’s Guide, Version 1.0; Interagency Report NISTIR 6376; National Institute of Standards and Technology: Gaithersburg, MD, USA, 1999.
- Gilbert, T.L. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Magn.
**2004**, 40, 3443. [Google Scholar] [CrossRef] - Leliart, J.; Dvornik, M.; Mulkers, J.; de Clercq, J.; Milosevic, M.V.; van Waeyenberge, B. Fast micromagnetic simulations on GPU—Recent advances made with mumax
^{3}. J. Phys. D Appl. Phys.**2018**, 51, 123002. [Google Scholar] [CrossRef] - Oezelt, H.; Kirk, E.; Wohlhüter, P.; Müller, E.; Heyerman, L.J.; Kowacs, A.; Schrefl, T. Vortex motion in amorphous ferrimagnetic thin film elements. AIP Adv.
**2017**, 7, 056001. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Sketch of the bow-tie structure under examination consisting of iron (

**black**) and permalloy (

**green**) as well as orientation of the external magnetic field angles.

**Figure 2.**Snapshots of the magnetization reversal process and hysteresis loops of pure iron nanoparticles, simulated for an external field orientation of 45°, for a thickness of (

**a,b**) 15 nm or (

**c,d**) 30 nm, respectively. In all graphs, the longitudinal magnetization M

_{L}is detected parallel to the magnetic field, while the transverse magnetization M

_{T}is perpendicular to it.

**Figure 3.**Snapshots of the magnetization reversal process and hysteresis loops of pure permalloy nanoparticles, simulated for an external field orientation of 30°, for a thickness of (

**a,b**) 15 nm or (

**c,d**) 30 nm, respectively.

**Figure 4.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 30° and a thickness of 15 nm.

**Figure 5.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 30° and a thickness of 30 nm.

**Figure 6.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 45° and a thickness of 15 nm.

**Figure 7.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 45° and a thickness of 30 nm.

**Figure 8.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 60° and a thickness of 15 nm.

**Figure 9.**(

**a**) Snapshots of the magnetization reversal process and (

**b**) hysteresis loops of bow-tie nanoparticles, simulated for an external field orientation of 60° and a thickness of 30 nm.

**Figure 10.**Single-material bow tie structures (

**a**) as layout and (

**b**) produced with an edge length of 200 nm with negative resist AR-N.

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**MDPI and ACS Style**

Sudsom, D.; Juhász Junger, I.; Döpke, C.; Blachowicz, T.; Hahn, L.; Ehrmann, A.
Micromagnetic Simulation of Vortex Development in Magnetic Bi-Material Bow-Tie Structures. *Condens. Matter* **2020**, *5*, 5.
https://doi.org/10.3390/condmat5010005

**AMA Style**

Sudsom D, Juhász Junger I, Döpke C, Blachowicz T, Hahn L, Ehrmann A.
Micromagnetic Simulation of Vortex Development in Magnetic Bi-Material Bow-Tie Structures. *Condensed Matter*. 2020; 5(1):5.
https://doi.org/10.3390/condmat5010005

**Chicago/Turabian Style**

Sudsom, Devika, Irén Juhász Junger, Christoph Döpke, Tomasz Blachowicz, Lothar Hahn, and Andrea Ehrmann.
2020. "Micromagnetic Simulation of Vortex Development in Magnetic Bi-Material Bow-Tie Structures" *Condensed Matter* 5, no. 1: 5.
https://doi.org/10.3390/condmat5010005