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Fe-site-resolved anisotropy energies in Nd$_2$Fe$_{14}$B for atomistic spin dynamics

Veronica T. C. Lai, Christopher E. Patrick

Abstract

Nd-Fe-B magnets are the most widely used high performance magnets in the world today, and remain the subject of both experimental and computational research aimed at understanding and optimizing them. Atomistic spin dynamics (ASD) is one technique which has been used in recent years to provide insight into magnetic properties relevant to coercivity, such as domain wall width. Although it is relatively clear how to model magnetocrystalline anisotropy arising from rare-earth atoms in these simulations, the contribution from the transition metal Fe is less obvious, due to the itinerant nature of the magnetism. Here, we examine previous treatments of Fe anisotropy in ASD simulations and identify a discrepancy with previously-published first-principles studies. We derive two models which correct this discrepancy, one based on single-ion theory and the other on anisotropic exchange, and test their performance by comparing to first-principles torque calculations on Y$_2$Fe$_{14}$B. The torque calculations show a contribution which cannot be explained by the single-ion model but arises naturally from (antisymmetric) anisotropic exchange. We propose practical strategies to model Fe anisotropy in future ASD simulations, including a simplified (mean-field) description of anisotropic exchange, which may have applications beyond R$_2$Fe$_{14}$B to the wider class of itinerant magnetic materials.

Fe-site-resolved anisotropy energies in Nd$_2$Fe$_{14}$B for atomistic spin dynamics

Abstract

Nd-Fe-B magnets are the most widely used high performance magnets in the world today, and remain the subject of both experimental and computational research aimed at understanding and optimizing them. Atomistic spin dynamics (ASD) is one technique which has been used in recent years to provide insight into magnetic properties relevant to coercivity, such as domain wall width. Although it is relatively clear how to model magnetocrystalline anisotropy arising from rare-earth atoms in these simulations, the contribution from the transition metal Fe is less obvious, due to the itinerant nature of the magnetism. Here, we examine previous treatments of Fe anisotropy in ASD simulations and identify a discrepancy with previously-published first-principles studies. We derive two models which correct this discrepancy, one based on single-ion theory and the other on anisotropic exchange, and test their performance by comparing to first-principles torque calculations on YFeB. The torque calculations show a contribution which cannot be explained by the single-ion model but arises naturally from (antisymmetric) anisotropic exchange. We propose practical strategies to model Fe anisotropy in future ASD simulations, including a simplified (mean-field) description of anisotropic exchange, which may have applications beyond RFeB to the wider class of itinerant magnetic materials.

Paper Structure

This paper contains 20 sections, 22 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. The standard ASD Hamiltonian
  4. Single-ion Fe anisotropy in previous works
  5. Model 1: Single-ion anisotropy
  6. Model 2: Anisotropic exchange
  7. Torques for a ferromagnetic state
  8. Torque calculations: computational details
  9. Results
  10. The $4e$ sublattice
  11. The $4c$ sublattice
  12. Model 2 for all sublattices
  13. Discussion
  14. Relevance to previous work
  15. Recommendations for future ASD calculations
  16. ...and 5 more sections

Figures (3)

  • Figure 1: DFT-calculated torques (crosses) and Model 1 fits (solid lines) for Fe atoms in the $4e$ sublattice, for calculation sets 1 and 4 (left and right).
  • Figure 2: The equivalent plots to Fig. \ref{['fig.4e']} for Fe atoms in the $4c$ sublattice. The DFT-calculated torques in calculation set 4 (right panel) evidently contain a $\phi$-independent contribution which is not accounted for in Model 1.
  • Figure 3: The comparison of DFT-calculated torques (crosses) and Model 2 fits (solid lines) for Fe atoms in the $16k_2$ sublattice, for all five calculation sets. Set 5 (rightmost plot) shows the 16 atoms split into 8 distinct curves, with each atom degenerate with its inversion partner.