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DFT calculations of magnetocrystalline anisotropy energy with fixed spin moment

Justyn Snarski-Adamski, Joanna Marciniak, Wojciech Marciniak, Justyna Rychły-Gruszecka, Mirosław Werwiński

Abstract

The development of new-generation permanent magnets is based on experimental efforts and innovative theoretical tools for modeling magnetic properties. Magnetocrystalline anisotropy energy (MAE) - one of the main intrinsic properties of permanent magnets - can be calculated using density functional theory (DFT). However, MAEs determined with different exchange-correlation potentials can vary widely. We show how these seemingly contradictory results can be reconciled using the fully relativistic fixed spin moment (FR-FSM) method. This is because the equilibrium pairs [MAE, $m_s$] calculated with different exchange-correlation potentials overlap with the MAE($m_s$) curve determined from the FR-FSM method ($m_s$ denotes the spin magnetic moment). The FR-FSM method also enables the hypothetical maximum MAE value for a given material to be estimated. In the case of magnetic alloys, MAE(FSM) analysis allows the optimal alloying additions to be determined in order to improve the MAE value. Concluding, the framework we describe for MAE versus FSM calculations can be a useful tool in the design of new permanent magnets.

DFT calculations of magnetocrystalline anisotropy energy with fixed spin moment

Abstract

The development of new-generation permanent magnets is based on experimental efforts and innovative theoretical tools for modeling magnetic properties. Magnetocrystalline anisotropy energy (MAE) - one of the main intrinsic properties of permanent magnets - can be calculated using density functional theory (DFT). However, MAEs determined with different exchange-correlation potentials can vary widely. We show how these seemingly contradictory results can be reconciled using the fully relativistic fixed spin moment (FR-FSM) method. This is because the equilibrium pairs [MAE, ] calculated with different exchange-correlation potentials overlap with the MAE() curve determined from the FR-FSM method ( denotes the spin magnetic moment). The FR-FSM method also enables the hypothetical maximum MAE value for a given material to be estimated. In the case of magnetic alloys, MAE(FSM) analysis allows the optimal alloying additions to be determined in order to improve the MAE value. Concluding, the framework we describe for MAE versus FSM calculations can be a useful tool in the design of new permanent magnets.
Paper Structure (4 sections, 1 equation, 3 figures)

This paper contains 4 sections, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Computational details
  3. Results and discussion
  4. Summary and conclusions

Figures (3)

  • Figure 1: Overlap between MAE(FSM) results for various exchange-correlation potentials. Magnetocrystalline anisotropy energies of selected rare-earth-free magnets are denoted by red circles. We used exchange-correlation potentials in the form of: von Barth-Hedin (BH), Perdew-Wang (PW92), Perdew-Burke-Ernzerhof (PBE), and with exchange only. Magenta (LDA) and blue (GGA) plots show the MAE as a function of fixed spin moment. Calculations were made with the FPLO18 code. These data were first published in Refs.: FePt marciniak_dft_2022; CeFe$_{12}$snarski-adamski_effect_2022; FeNi marciniak_magnetic_2024; FeB snarski-adamski_searching_2025; Fe$_2$P casadei_rare-earth-free_2025; MnBi (in preparation).
  • Figure 2: Magnetocrystalline anisotropy energies (MAE) of (Fe$_{1-x}$Co$_{x}$)$_5$SiB$_2$ and (Fe$_{1-x}$Co$_{x}$)$_5$PB$_2$ systems as a function of Co concentration $x$ and total magnetic moment. Calculations were made with the FPLO14 code in the GGA-PBE approach. The Co concentration was modeled using the virtual crystal approximation (VCA). We fixed the magnetic moment using the fixed spin moment (FSM) method. Black circles indicate equilibrium solutions. Data in panel (a) were first published in Ref. werwinski_magnetic_2016.
  • Figure 3: Magnetocrystalline anisotropy energies (MAE) of Fe$_5$SiB$_2$ and (Fe$_{1-x}$Co$_{x}$)$_5$SiB$_2$ as a function of total magnetic moment. Calculations were made with the FPLO14 code in the GGA-PBE approach. The Co concentration was modeled using the virtual crystal approximation (VCA). For the Fe$_5$SiB$_2$ MAE($m$) green plot, we used the fixed spin moment (FSM) method. For MAE as a function of Co concentration ($x$) and volume change ($V/V_0$), magnetic moments were not fixed, but calculated self-consistently and read from the results. MAE versus magnetic moment dependence is preserved whether the FSM or constraint-free self-consistent calculation is used. These data were first published in Ref. werwinski_magnetic_2016.