# Crystal Electric Fields in Rare-Earth-Iron-Boron Permanent Magnets

IEEE International Magnetics Conference

Takuya Yoshioka、Hiroki Tsuchiura、Naoki Miura ( Tohoku University )

Pavel Novak ( ASCR )

### 概要/Abstract

Because of the recent developments in renewable and sustainable technology, there has been emerging interest inimproving the coercivity of Nd-Fe-B permanent magnets, and in revealing its mechanism. In practice, the coercivitymechanisms in the permanent magnets are very much complicated and affected by many factors. It has been widelyaccepted, however, that the large magnetic anisotropic energy due to the Nd ions is the dominant factor of the coercivityof Nd-Fe-B magnets. For deeper understanding of the coercivity mechanism, it is necessary to study the crystallineelectric field around the Nd ions which is the origin of the magnetic anisotropy of Nd. The experimental determination ofcrystal field parameters (CFPs) requires a multi-parameter fit to experimental data, which involves inevitable ambiguity.Although first-principles calculations can be applied to estimate CFPs without such ambiguity, the validity of the results isnot evident. Thus in this study, we thoroughly investigate the CFPs in R2Fe14B systems by using first-principlescalculations, and construct crystal field Hamiltonians for the rare-earth ions using the obtained CFPs. We can determinethe most stable direction of the magnetization by using the crystal field Hamiltonians, and can see whether they exhibit theso-called spin-reorientation phenomena or not. It has been known experimentally that R2Fe14B systems undergo the spin-reorientation transition when R=Nd, Ho, Er, Tm Yb. Therefore, to check the R-dependence of the spin-reorientation can bea reasonable criterion for the validity of the first-principles CFP estimation. analyze the higher order CEF parameters for light (R=Pr and Nd) and heavy (R=Tb, Dy, and Ho) R ions in R2Fe14B crystalsand discuss the spin reorientation phenomena investigated in the Nd2Fe14B3 and Ho2Fe14B4, where the angle of themagnetization Δθ from the [001] to the [110] direction tilts at about 32° and 22°, respectively. The magnetic anisotropy energy of R ions is described by the Hamiltonian consisting of spin-orbit interaction, magneticexchange, and CEF interactions. The CEF interaction energy depends on the orientation of 4f magnetic moment, because arotation of magnetic moment leads to a rotation of the 4f aspherical charge cloud due to strong spin-orbit interaction. Thusthe 4f anisotropic magnetic energy depends on the CEF energy described by the standard model Hamiltonian, HCEF =ΣlmΘlmAlm⟨rl⟩Ôlm, where Ôlm are the Stevens operator equivalents and Θlm are the reduced matrix elements5,6. The CEFparameters Alm⟨rl⟩ originating from the aspherical part of the total single particle discrete Fourier transform potential in thecrystal can be obtained from the radial part of the 4f-orbital wave function R4f(r) and the components of the totalpotential. To obtain the total potential, we use the full-potential linearized augmented plane wave plus local orbitals(APW+lo) method implemented in the WIEN2k code.7 The Kohn-Sham equations are solved within the generalized-gradientapproximation (GGA). To simulate localized 4f states in the R2 Fe14B systems, we switch off the hybridization between 4fand valence states, and treat the 4f states in the spherical part of the potential as atomic-like core states (so-called opencore treatment). Here, R4f(r) is obtained by performing separate atomic calculations of the electronic structure of anisolated R atom. In these calculations the correction for the self-interaction (SIC) was included, which leads to betterapproximation for the single electron densities. This approach8 was found to provide a 4f charge density being very close tothat obtained from more rigorous SIC-DFT band calculations.9 We study the magnetic structures of R2Fe14B (R=Pr, Nd, Tb, Dy, and Ho) using APW+lo method with open core treatment.As the first step, we show the calculated local spin magnetic moments for the non-equivalent Fe sites and their meanvalues in TABLE I, in comparison with previous studies for R=Nd using the full-potential linear-muffin-tin-orbital (FL-LMTO) method,10 and for R=Tb, Dy, and Ho using the LMTO theory in atomic-sphere approximation (LMTO-ASA).11 Wecan confirm that our series of local magnetic moments of Fe ions for these crystals denote same tendency of previoustheoretical calculations, however, the mean values are consistently larger than the previous ones and close in value toexperiment in case of R=Nd.12 Next, we show the CEF parameters Alm⟨rl⟩ in TABLE II for 2f1 and 2g1 sites of R ions in thecrystalline R2Fe14B (see FIG. 1). Here, we calculate the CEF parameters from Coulomb potential inside the atomic sphereof the R ions with the muffin-tin sphere radius, RMT=3.2. To check the replicability of experiment, we calculate theanisotropic magnetic energy of R ions and the angle of magnetization Δθ from the [001] to the [110] direction at absolutezero. In case of heavy R=Tb, Dy, and Ho, we have obtained the results Δθ=0, 0, and 15°, respectively, which arecomparable with experiment.7 On the other hand, in both heavy R=Pr and Nd cases, Δθ=0. The difference of accuracybetween light and heavy R seems to be originating from the radial density of 4f electron decreases fast in heavy R ascompared with light R. So we have to some sort of improvement for light R in R2Fe14B. In this talk, we will discuss thetemperature dependence of the tilting angles in R2Fe14B (R=Nd and Ho) crystals.

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