# Micromagnetic Simulation of Round Ferromagnetic Nanodots with Varying Roughness and Symmetry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{3}which works on GPUs [34].

_{S,Fe}= 1700 × 10

^{3}A/m as magnetization at saturation, A

_{Fe}= 21 × 10

^{−12}J/m as exchange constant and K

_{1,Fe}= 48 × 10

^{3}J/m

^{3}as magneto-crystalline anisotropy constant [35]. The Gilbert damping constant was chosen with a value of α = 0.5 to represent the quasistatic case.

^{3}unless mentioned differently.

_{L}) and transversal magnetization components (M

_{T}), spatially resolved screenshots were taken to investigate the magnetic states and magnetization reversal processes. The simulations were performed at angles θ of the external magnetic field from 0° to 90° at intervals of 15°.

## 3. Results and Discussion

^{3}. This curve again looks quantitatively different from the former curves. Comparing it with both curves simulated with the (92 × 92) mask without edge thinning, it can be stated that no large difference is visible between cell sizes of (5 nm)

^{3}and (2 nm)

^{3}, but reducing the edge roughness quantitatively (not qualitatively) changes the hysteresis loops.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- 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] - 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] - Mejía-López, J.; Altbir, D.; Romero, A.H.; Batlle, X.; Roshchin, I.V.; Li, C.-P.; Ivan, S.K. Vortex state and effect of anisotropy in sub-100-nm magnetic nanodots. J. Appl. Phys.
**2006**, 100, 104319. [Google Scholar] [CrossRef][Green Version] - Noske, M.; Stoll, H.; Fähnle, M.; Gangwar, A.; Woltersdorf, G.; Slavin, A.; Weigand, M.; Dieterle, G.; Förster, J.; Back, C.H.; et al. Three-dimensional character of the magnetization dynamics in magnetic vortex structures: Hybridization of flexure gyromodes with spin waves. Phys. Rev. Lett.
**2016**, 117, 037208. [Google Scholar] [CrossRef] [PubMed][Green Version] - Weigand, M.; van Waeyenberge, B.; Vansteenkiste, A.; Curcic, M.; Sackmann, V.; Stoll, H.; Tyliszczak, T.; Kaznatcheev, K.; Bertwistle, D.; Woltersdorf, G.; et al. Vortex core switching by coherent excitation with single in-plane magnetic field pulses. Phys. Rev. Lett.
**2009**, 102, 077201. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gaididei, Y.; Kravchuk, V.P.; Sheka, D.D.; Mertens, F.G. Multiple vortex-antivortex pair generation in magnetic nanodots. Phys. Rev. B
**2010**, 81, 094431. [Google Scholar] [CrossRef][Green Version] - Otxoa, R.M.; Petit-Watelot, S.; Manfrini, M.; Radu, I.P.; Thean, A.; Kim, J.-V.; Devolder, T. Dynamic influence of vortex-antivortex pairs in magnetic vortex oscillators. J. Magn. Magn. Mater.
**2015**, 394, 292–298. [Google Scholar] [CrossRef][Green Version] - Ehrmann, A.; Blachowicz, T. Influence of tilted fields on magnetization reversal in Fe nanodots. In Proceedings of the IEEE 8th International Conference on Nanomaterials: Applications & Properties, Sumy, Ukraine, 9–11 November 2020. [Google Scholar]
- Li, J.Q.; Wang, Y.; Cao, J.F.; Meng, X.G.; Zhu, F.Y.; Tai, R.Z. Fast control of the polarity of the magnetic vortex for a pair of magnetic nanodots. J. Magn. Magn. Mater.
**2021**, 529, 167841. [Google Scholar] [CrossRef] - Buess, M.; Knowles, T.P.J.; Höllinger, R.; Haug, T.; Krey, U.; Weiss, D.; Pescia, D.; Scheinfein, M.R.; Back, C.H. Excitations with negative dispersion in a spin vortex. Phys. Rev. B
**2005**, 71, 104415. [Google Scholar] [CrossRef][Green Version] - Blachowicz, T.; Ehrmann, A.; Steblinski, P.; Palka, J. Directional-dependent coercivities and magnetization reversal mechanisms in fourfold ferromagnetic systems of varying sizes. J. Appl. Phys.
**2013**, 113, 013901. [Google Scholar] [CrossRef][Green Version] - Janutka, A.; Gawronski, P. Spin-transfer-driven dynamics of magnetic vortices and antivortices in dots with crystalline cubic anisotropy. IEEE Trans. Magn.
**2017**, 53, 4300706. [Google Scholar] [CrossRef] - Pylypovskyi, O.V.; Sheka, D.D.; Kravchuk, V.P.; Gaididei, Y. Effects of surface anisotropy on magnetic vortex core. J. Magn. Magn. Mater.
**2014**, 361, 201–205. [Google Scholar] [CrossRef][Green Version] - Sudsom, D.; Ehrmann, A. Micromagnetic simulations of Fe and Ni nanodot arrays surrounded by magnetic or non-magnetic matrices. Nanomater.
**2021**, 11, 349. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef][Green Version] - Sellarajan, B.; Saravanan, P.; Ghosh, S.K.; Nagaraja, H.S.; Barshilia, H.C.; Chowdhury, P. Shape induced magnetic vortex state in hexagonal ordered CoFe nanodot arrays using ultrathin alumina shadow mask. J. Magn. Magn. Mater.
**2018**, 451, 51–56. [Google Scholar] [CrossRef] - Prejbeanu, I.L.; Natali, M.; Buda, L.D.; Ebels, U.; Lebib, A.; Chen, Y.; Ounadjela, K. In-plane reversal mechanism in circular Co dots. J. Appl. Phys.
**2002**, 91, 7343–7345. [Google Scholar] [CrossRef] - Vavassori, P.; Zaluzec, N.; Metlushko, V.; Novosad, V.; Ilic, B.; Grimsditch, M. Magnetization reversal via single and bouble vortex states in submicron Permalloy ellipses. Phys. Rev. B
**2004**, 69, 214404. [Google Scholar] [CrossRef][Green Version] - Prosandeev, S.; Bellaiche, L. Controlling double vortex states in low-dimensional dipolar systems. Phys. Rev. Lett.
**2008**, 101, 097203. [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] - Zhang, W.; Haas, S. Phase Diagram of Magnetization Reversal Processes in Nanorings. Phys. Rev. B
**2010**, 81, 064433. [Google Scholar] [CrossRef][Green Version] - Yoo, Y.G.; Kläui, M.; Vaz, C.A.F.; Heyderman, L.J.; Bland, J.A.C. Switching field phase diagram of Co nanoring magnets. Appl. Phys. Lett.
**2003**, 82, 2470–2472. [Google Scholar] [CrossRef] - Vaz, C.A.F.; Hayward, T.J.; Llandro, J.; Schackert, F.; Morecroft, D.; Bland, J.A.C.; Kläui, M.; Laufenberg, M.; Backes, D.; Rüdiger, U.; et al. Ferromagnetic nanorings. J. Phys. Condens. Matter
**2007**, 19, 255207. [Google Scholar] [CrossRef][Green Version] - Park, M.H.; Hong, Y.K.; Choi, B.C.; Donahue, M.J.; Han, H.; Gee, S.H. Vortex head-to-head domain walls and their formation in onion-state ring elements. Phys. Rev. B
**2006**, 73, 094424. [Google Scholar] [CrossRef][Green Version] - Muscas, G.; Menniti, M.; Brucas, R.; Jönsson, P.E. Mesoscale Magnetic Rings: Complex magnetization reversal uncovered by FORC. J. Magn. Magn. Mater.
**2020**, 502, 166559. [Google Scholar] [CrossRef] - Fernandez, E.; Tu, K.-H.; Ho, P.; Ross, C.A. Thermal stability of L1
_{0}-FePt nanodots patterned by self-assembled block copolymer lithography. Nanotechnology**2018**, 29, 465301. [Google Scholar] [CrossRef][Green Version] - Bryan, M.T.; Atkinson, D.; Cowburn, R.P. Experimental study of the influence of edge roughness on magnetization switching in Permalloy nanostructures. Appl. Phys. Lett.
**2004**, 85, 3510. [Google Scholar] [CrossRef] - Zhu, F.Q.; Shang, Z.; Monet, D.; Chien, C.L. Large enhancement of coercivity of magnetic Co/Pt nanodots with perpendicular anisotropy. J. Appl. Phys.
**2007**, 101, 09J101. [Google Scholar] [CrossRef] - Li, X.; Leung, C.W.; Chiu, C.-C.; Lin, K.-W.; Chan, M.S.; Zhou, Y.; Pong, P.W.T. Reduced magnetic coercivity and switching field in NiFeCuMo/Ru/NiFeCuMo synthetic-ferrimagnetic nanodots. Appl. Surf. Sci.
**2017**, 410, 479–484. [Google Scholar] [CrossRef] - Madami, M.; Gubbiotti, G.; Tacchi, S.; Carlotti, G. Magnetization dynamics of single-domain nanodots and minimum energy dissipation during either irreversible or reversible switching. J. Phys. D. Appl. Phys.
**2017**, 50, 453002. [Google Scholar] [CrossRef] - Tu, K.-H.; Bai, W.B.; Liontos, G.; Ntetsikas, K.; Avgeropoulos, A.; Ross, C.A. Universal pattern transfer methods for metal nanostructures by block copolymer lithography. Nanotechnology
**2015**, 26, 375301. [Google Scholar] [CrossRef][Green Version] - 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. [Google Scholar]
- Gilbert, T.L. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Magn.
**2004**, 40, 3443–3449. [Google Scholar] [CrossRef] - Leliaert, 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] - Kneller, E.F.; Hawig, R. The exchange-spring magnet: A new material principle for permanent magnets. IEEE Trans. Magn.
**1991**, 27, 3588–3560. [Google Scholar] [CrossRef] - Ehrmann, A.; Blachowicz, T. Asymmetric hysteresis loops in Co thin films. Condens. Matter.
**2020**, 5, 71. [Google Scholar] [CrossRef]

**Figure 1.**Sketches of the five cases under examination: (

**a**) case 1; (

**b**) case 2; (

**c**) case 3; (

**d**) case 4; and (

**e**) case 5; (

**f**) definition of angles in this study. The colour of the snapshots depends on the orientation of the magnetization: red = magnetization pointing to the right, blue = magnetization pointing to the left, white = magnetization pointing from top to bottom or vice versa.

**Figure 2.**(

**a**,

**c**,

**e**) Hysteresis loops and (

**b**,

**d**,

**f**) snapshots of the magnetization reversal from positive to negative saturation and back for case 1, simulated for the angles θ depicted in the graphs.

**Figure 3.**(

**a**) Hysteresis loops, simulated for case 1 at θ = 0° with different random orientations of anisotropy axes and different edge definitions; (

**b**) comparison between different cells sizes and edge constructions; (

**c**) comparison between different simulation ΔH; (

**d**) snapshots for the magnetization reversal with “smoothed” edges.

**Figure 4.**(

**a**,

**c**) Hysteresis loops and (

**b**,

**d**) snapshots of the magnetization reversal from positive to negative saturation and back for case 2, simulated for the angles depicted in the graphs.

**Figure 5.**(

**a**,

**c**,

**e**) Hysteresis loops and (

**b**,

**d**,

**f**) snapshots of the magnetization reversal from positive to negative saturation and back for case 3, simulated for the angles depicted in the graphs.

**Figure 6.**(

**a**) Hysteresis loop and (

**b**) snapshots of the magnetization reversal from positive to negative saturation and back for case 4, simulated for θ = 60°.

**Figure 7.**(

**a**,

**c**,

**e**) Hysteresis loops and (

**b**,

**d**,

**f**) snapshots of the magnetization reversal from positive to negative saturation and back for case 5, simulated for the angles depicted in the graphs.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Steinmetz, P.; Ehrmann, A. Micromagnetic Simulation of Round Ferromagnetic Nanodots with Varying Roughness and Symmetry. *Condens. Matter* **2021**, *6*, 19.
https://doi.org/10.3390/condmat6020019

**AMA Style**

Steinmetz P, Ehrmann A. Micromagnetic Simulation of Round Ferromagnetic Nanodots with Varying Roughness and Symmetry. *Condensed Matter*. 2021; 6(2):19.
https://doi.org/10.3390/condmat6020019

**Chicago/Turabian Style**

Steinmetz, Pia, and Andrea Ehrmann. 2021. "Micromagnetic Simulation of Round Ferromagnetic Nanodots with Varying Roughness and Symmetry" *Condensed Matter* 6, no. 2: 19.
https://doi.org/10.3390/condmat6020019