Coercivity limits in nanoscale ferromagnets

Jeotikanta Mohapatra, J. Fischbacher, M. Gusenbauer, M. Y. Xing, J. Elkins, T. Schrefl, and J. Ping Liu

Phys. Rev. B 105, 214431 – Published 24 June 2022

Abstract

It has been a puzzle for a century about how “hard” (coercive) a ferromagnet can be. Seven decades ago, W. Brown gave his famous theorem to correlate coercivity of a ferromagnet to its magnetocrystalline anisotropy field. However, the experimental coercivity values are far below the calculated level given by the theorem, which is called Brown's Coercivity Paradox. The paradox has been considered to be related to the complex microstructures of the magnets in experiments because coercivity is an extrinsic property that is sensitive to any imperfections in the specimens. To date, coercivity cannot be predicted and calculated by quantitative modeling. In this investigation, we carried out a case study on the high magnetic coercivity of Co nanowires exceeding the magnetocrystalline anisotropy field as predicted by Brown's theorem. It is found that the aspect ratio and diameter of the nanocrystals have a strong effect on the coercivity. When the nanocrystals have an increased aspect ratio, the coercivity is significantly higher than the magnetocrystalline anisotropy field of a hcp Co crystal. Micromagnetic simulations give a coercivity aspect-ratio dependence that is well consistent with the experimental results. It is also revealed that a coercivity limit exists based on the geometrical structures of the nanocrystals that govern the demagnetizing process. The quantitative correlation obtained between the structure and coercivity enables material design of advanced permanent magnets in the future.

Received 2 January 2022
Revised 10 March 2022
Accepted 6 June 2022

DOI:https://doi.org/10.1103/PhysRevB.105.214431

Physics Subject Headings (PhySH)

Magnetic anisotropy

Magnetic nanoparticles Nanowires

Magnetization measurements Micromagnetic modeling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jeotikanta Mohapatra ^1,*, J. Fischbacher ^2,3,*, M. Gusenbauer^2,3, M. Y. Xing¹, J. Elkins¹, T. Schrefl ^2,3,†, and J. Ping Liu ^1,‡

¹Department of Physics, University of Texas at Arlington, Arlington, Texas 76019, USA
²Christian Doppler Laboratory for Magnet Design Through Physics Informed Machine Learning, Viktor Kaplan-Straße 2E, 2700 Wiener Neustadt, Austria
³Department for Integrated Sensor Systems, Danube University Krems, Viktor Kaplan-Straße 2E, 2700 Wiener Neustadt, Austria

^*These authors contributed equally to this work.
^†Corresponding author: thomas.schrefl@donau-uni.ac.at
^‡Corresponding author: pliu@uta.edu

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Issue

Vol. 105, Iss. 21 — 1 June 2022

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Images

Figure 1
Aspect ratio dependence coercivity value in Co nanocrystals. (a)–(d) TEM images and (e)–(h) the corresponding hysteresis loops at room temperature of Co nanocrystals with aspect ratios of ∼1, ∼3, ∼5, and ∼10, respectively, where the average diameter of the nanocrystals is about 10 nm. (i) Aspect ratio dependence of coercivity. The full symbols in (i) are the results of a coercivity model, which is based on the micromagnetic simulation of switching over a nonzero energy barrier. The blue dashed line represents the magnetocrystalline anisotropy field of hcp Co.
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Figure 2
Magnetization reversal and coercivity of high-aspect-ratio Co nanocrystals with diameter of 10 nm. (a) The magnetic states along the minimum energy path for Co nanocrystals with $p = 2$ , 4, and 10. (b) Energy along the minimum energy path for Co nanocrystals with aspect ratio $p = 2$ . The external field is applied at 17.6° with respect to the long axis. The field strength is such that the energy barrier is $25 k_{B} T$ . (c) The aspect ratio dependence of the simulated coercivity of Co nanowires with diameter of 10 nm.
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Figure 3
Comparison between the experimental and analytical angular dependence of (a) coercivity and (b) switching field for nanowires with diameter 10 nm and aspect ratio 11. The pink line in (a) and (b) corresponds to the analytical plots calculated using the S-W model, and green lines represent the experimental data obtained from angular-dependent magnetization curves.
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Figure 4
Diameter dependence of magnetic properties in dodecahedral-shaped Co nanocrystals. TEM images, the corresponding histograms of diameter, and the respective room temperature hysteresis loops of Co nanocrystals with average diameter of (a)–(c) 8 nm, (d)–(f) 12 nm, and (g)–(i) 18 nm, respectively.
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Figure 5
Structural characterization of dodecahedral-shaped Co nanocrystals. (a) XRD pattern of the Co nanocrystals with different sizes. (b) High-resolution TEM micrograph of a single Co nanocrystal, and inset the corresponding numerical fast Fourier transform (FFT) pattern.
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Figure 6
Diameter dependence of coercivity in dodecahedral-shaped Co nanocrystals. (a) Comparison of experimental coercivity data obtained from this study and the literature [41, 42, 43, 44, 45, 46] with the simulated coercivity values. The upper dashed red line represents the magnetocrystalline anisotropy field of cobalt materials. (b), (c) The magnetic states show the computed configuration of the magnetization for the saddle point, at which the energy barrier reaches $25 k_{B} T$ .
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