Calculated Curie temperatures for rare-earth permanent magnets: ab initio inspection on localized magnetic moments in d-electron ferromagnetism

TL;DR

This paper calculates Curie temperatures for rare-earth permanent magnets using ab initio methods and an effective spin model, providing insights into magnetic properties and their dependence on rare-earth elements.

Contribution

It introduces a combined ab initio and spin model approach to predict Curie temperatures in rare-earth magnets, highlighting the trend consistency with experimental data.

Findings

Calculated Curie temperatures align with experimental trends.

The effective spin model's validity is limited by exchange coupling ranges.

Quantitative comparisons reveal the model's accuracy and limitations.

Abstract

We present a data set of calculated Curie temperatures for the main-phase compounds of rare-earth permanent magnets. We employ ab initio electronic structure calculations for the itinerant ferromagnetism and an effective spin model for the finite-temperature magnetism. Curie temperatures are derived on the basis of a classical Heisenberg model mapped via Liechtenstein's formula for atomic-pair-wise exchange couplings. Relative trends with respect to the species of rare-earth elements in calculated Curie temperatures for R2Fe14B are in agreement with experimental trends. Quantitative comparison between calculation and experimental data found in the literature point to an effective range of the exchange couplings imposing a limit on the validity range of the effective spin model.

Full text

I Introduction

Global concern for sustainable energy supply in terms of environmental friendliness, preservation of natural resources puts rare-earth permanent magnets (REPM’s) as one of the most critically important materials for the future of human society. Today’s commercial REPM’s are made of Fe and rare-earth elements like Nd, and issues associated with the usage of REPM at high temperatures have posed a renewed interest in the magnetism of Fe-based materials at finite temperatures. In today’s commercial champion magnet the main-phase compound Nd2Fe14B sagawa_1984 ; croat_1984 ; rmp_1991 comes with a relatively low Curie temperature 585 K, almost only a half of the Curie temperature of elemental Fe at 1043 K. Considering the high-temperature range of practical usage of traction motors of cars up to 450 K, it has been in high demand to find a way to supplement the high-temperature performance of Nd-Fe-B magnets or to design a new champion magnet desirably with higher Curie temperatures with a good structure stability.

The question posed to solid state physics may boil down to finding an optimal material in a space spanned by a) magnetization, b) magnetic anisotropy, and c) Curie temperature, all of which backed-up by d) structure stability JOM_2015 . In the present study we focus on Curie temperature and magnetization which come from the leading-order energy scales in the intrinsic electronic structure of those $4f$-$3d$ intermetallic compounds. The Curie temperature comes from the dominating $3d$-$3d$ direct exchange couplings in the order of 10 meV for each exchange path, which is multiplied by the coordination number in the lattice that is typically from 10 to up to around 20 in the immediate neighborhood of the magnetic atom mm_2016 . Even a tiny amount of coupling in the long-range exchange path could contribute a lot considering the huge coordination number toga_2016 . However, consideration of too long-range exchange path involves a problem in resolving the localized magnetic moment contribution and the obvious itinerant contribution to the intrinsic magnetism. It is an important problem to understand how well we can make ab initio prediction for those important properties a) - d) both for fundamental theory of metallic magnetism and for practical demands in the past and upcoming decades. The leading-order answer to the materials family including the champion magnet compound is presented in Fig. 1.

Clarification of the validity range of a spin model for itinerant magnetism would possibly lead to several possibilities for an optimal compound for permanent magnet utility in the middle of subtle interplay between localized and delocalized nature in the $d$-electron state. Here a possible lesson may be found in the case of an exceptionally high Curie temperature for a Ce-based ferromagnet CeRh3B2 at $120~{}\mbox{K}$ cerh3b2_synthesis ; cerh3b2_Tc that is even higher than that of the Gd-counterpart at $90~{}\mbox{K}$. This problem has been known for almost the same duration of history as for Nd2Fe14B since the early 1980’s and the particular dual nature showing both of localization and delocalization as probed by X-ray absorption spectroscopy imada in $4f$-electron state of Ce has been discussed. Electronic states on the crossover between localized and delocalized electronic states are most fertile both in terms of fundamental solid state physics and practical functionalities and we inspect the problem for $d$-electron ferromagnetism with some additional resolution of the particular exceptional contribution from Ce in Ce2Fe14B.

The present paper is organized as follows. In the next section, our computational methods are described. Main results are shown in Sec. III. Discussions on the difference between Fe-rich limit and Co-rich limit and lattice structure variants are given in Sec. IV. The final section is devoted for conclusions and outlook.

II Methods

For the champion magnet compound Nd2Fe14B and related ferromagnets, we address the ferromagnetism and the expected Curie temperature on top of it via ab initio electronic structure calculations on the basis of Korringa-Kohn-Rostoker Green’s function method as implemented in “AkaiKKR” AkaiKKR . Our calculations are based on local density approximation (LDA), taking the exchange correlation function following Vosko, Wilk and Nusair vosko_1980 . Presented below are data on the basis of atomic-sphere approximation (ASA). Muffin-tin approximation basically yields the same trends as has been given by ASA for the present purpose to address magnetization and exchange couplings. No further parameters are involved in our first principles calculations: we will see that plain LDA works for $4f$-$3d$ intermetallics which may be compared to the description of the ferromagnetic ground state of bcc-Fe on the basis of LDA.

Electronic state of rare-earth elements are treated within open-core approximation in order to inspect the trends among the exchange couplings between $3d$-electrons that dominate the Curie temperature in the leading order skomski_1998 ; mm_2016 . Trivalent state is assumed for rare-earth elements except for Ce. Ce is set to be in tetravalent state in Ce-Fe/Co intermetallics, due to the exchange splitting of conduction bands spanning the energy width of a few eV’s while the $4f$-electron level in Ce compounds, if localized, sits at a position as shallow as only up to 2 eV below the Fermi level mm_2010 . Thus it is not very likely for the $4f$-electron in Ce to be kept in the localized level when it is put in the exchange-split conduction band made of $3d$-electrons coming from Fe or Co.

Recently, construction of spin models for finite-temperature magnetism of Nd2Fe14B has been attempted by several groups toga_2016 ; nishino_2017 ; toga_2018 ; westmoreland_2018 ; tsuchiura_2018 ; gong_prb_2019 ; gong_prmater_2019 ; gong_2020 . We employ most simplified form of them to give an overview of the trend in the calculated Curie temperatures. Non-relativistic calculations are employed which should be sufficiently precise to extract the trends in magnetization and exchange couplings. For a converged ferromagnetic state of each target material, inter-atomic exchange couplings are determined following Liechtenstein et al. liechtenstein_1987 . An effective spin Hamiltonian which goes like the following ${\cal H}=-\sum_{\left<i,j\right>}2J_{i,j}S_{i}S_{j}\mathbf{e}_{i}\cdot\mathbf{e}_{j}$ is used to survey the trends in the Curie temperature on the basis of mean-field approximation (MFA). Here the site index $i$ specifies an atom on which localized magnetic moment of the magnitude $S_{i}$ and the direction as pointed by the unit vector $\mathbf{e}_{i}$ is assumed. Summation over the exchange path between sites $i$ and $j$ runs only once on each path $\left<i,j\right>$. The range of the exchange couplings is incorporated without any cutoff and all of the exchange couplings in the metallic ferromagnetism are directly plugged into the mean-field approximation on the basis of KKR Green’s function method. Only $3d$-electrons from Fe or Co and $5d$-electrons from rare earth elements are considered in the spin Hamiltonian, on the basis of the open-core approximation for rare-earth elements. Systematic error originating in MFA can be eliminated by a numerically exact solution of the spin Hamiltonian via Monte Carlo methods jaswal ; mm_2016 ; toga_2016 while more fundamental error between theory and experiment can come in from the discrepancy between itinerant ferromagnetism and localized spin model. This is what we wish to address through the inspection of the trends between the MFA data on the basis of ab initio electronic structure and the experimental data.

III Results

Calculated magnetization for R2Fe14B is shown in Fig. 2. Taken together with the results shown in Fig. 1, experimental trends of the intrinsic ferromagnetism in the 2:14:1 material family with respect to the species of rare-earth (RE) elements are reasonably well reproduced on the basis of our electronic structure calculations involving only $d$-electrons. Intrinsic ferromagnetism concerning the exchange couplings has been well described up to the leading-order, on the basis of the precision for the total magnetization where the $4f$-electron contribution can be restored on top of the results from open-core calculations. Results for Co-analogues are shown in Fig. 3. Overall parallel trends between theory and experiments again hold but with even a better quantitative precision.

The decreasing trend in the Curie temperature of R2Fe14B with R being the heavy rare-earth elements can be attributed to lanthanide contraction which seems to trigger the lattice contraction as we overlap another calculated Curie temperatures with the lattice fixed to be that of Nd2Fe14B and dependence on the chemical composition $R$ is monitored. The results for this line of reasoning is presented in Fig. 4. The particular drop of the Curie temperature for Ce2Fe14B in the overall trend in R2Fe14B can be attributed to the exceptional tetra-valent state and the corresponding lattice shrinkage. As seen in Fig. 4, fictitiously restoring trivalent state for Ce brings up the Curie temperature and putting Ce${}^{3+}_{2}$Fe14B on the same fixed lattice of Nd2Fe14B brings about the smooth trend in calculated Curie temperatures, in strong contrast to the parallel trends of calculated Curie temperatures and experimental ones in Fig. 1. Inspecting the difference in the density of states (DOS) for R2Fe14B on the fixed lattice and that on the experimental lattice with the lanthanide contraction, say for R=Yb3+ as shown in Fig. 5, we see that the lanthanide contraction has made the $d$-electron magnetism to weak ferromagnetism in contrast to the strong ferromagnetism found for Nd2Fe14B kitagawa_and_asari_2010 .

IV Discussions

IV.1 Inspection on the range of exchange interaction

We have seen that the effective Heisenberg model can describe the trends of magnetization and Curie temperature for R2T14B with respect to R while the quantitative levels achieved for T=Fe and T=Co look quite different. To see how a spin model can be acceptable for the itinerant ferromagnetism, we come back to the reference elemental case studies with bcc-Fe ($a=5.417$ a.u.) and hcp-Co ($a=4.7377$ a.u. and $c/a=1.623$). Calculated Curie temperature is 1200 K for bcc-Fe and 1411 K for hcp-Co, in a reasonable agreement with experimental data 1043 K for bcc-Fe and 1394 K for hcp-Co on the basis of MFA. The exchange couplings in the spin Hamiltonian as calculated following Liechtenstein et al’s approach liechtenstein_1987 is shown in Fig. 6 together with the corresponding data for Nd2Fe14B and Nd2Co14B. The range of Fe-Fe exchange couplings is longer than Co-Co exchange couplings. Long-range couplings reflect the inherently itinerant electronic states PRL_2016 . The trend from weak ferromagnetism to strong ferromagnetism goes like bcc-Fe, Nd2Fe14B, Nd2Co14B and hcp-Co as can be seen in the calculated DOS shown in Fig. 7. Long-range exchange couplings comes from the metallic nature in the electronic state. The overall trend is that the less metallic the better description has been achieved on the basis of the spin model. The description of bcc-Fe could have been worse if the minority spin band had more DOS around the Fermi level. An apparent reasonable description of bcc-Fe on the basis of the simplified spin model might depend on the particular density of states where the diminishing DOS in the minority-spin band renders the system less metallic. This may compensate for the weak ferromagnetic nature with significant DOS in the majority-spin band, out of which yielded ferromagnetism may not have been very suited for the description based on localized spins.

Strictly speaking, a spin-only model for itinerant magnetism is invalid by construction for quantitative studies while discussions focusing on relative trends in a restricted range of the chemical composition and the lattice structure may well be justified mm_2016 ; toga_2016 up to the leading order. Besides, anomalous temperature dependence of the lattice constants andreev_1995 observed in R2Fe14B has been completely dropped from our present description of Curie temperature. Proper incorporation of finite-temperature lattice dynamics should improve the calculated Curie temperature that is dominated by itinerant $d$-electron magnetism. Incorporation of the lattice effect in the description of finite-temperature magnetism on elemental Fe and Co has been done recently and relevance of lattice effects especially for bcc-Fe has been shown ruban_2018 . Common physics may well be at work in Nd2Fe14B.

IV.2 Crystal structure trends

For Nd2Fe17 our calculated Curie temperature is $T_{\rm Curie}=670$ K. Compared with the experimental data 326 K isnard_et_al_1992 , the systematic overestimate is coming by a factor of 2.1 which is on a par with what we got for Nd2Fe14B where $T^{\rm calc.}_{\rm Curie}/T^{\rm expt.}_{\rm Curie}\simeq 1.7$. For Sm2Co17, our calculated Curie temperature is $T_{\rm Curie}=1173$ K which compares well with the experimental number for 1200K. Calculated Curie temperature is strongly influenced by the chemical composition: in the overall virtual parameter space spanned by the chemical composition and crystal structure, the composition of $3d$-metals plays a dominant role. In a “cross section” specified by a fixed chemical composition the relative trends with respect to RE elements can be satisfactorily addressed.

V Conclusions and outlook

We have shown that the leading-order trends of the Curie temperature in the champion magnet compound family with respect to the species of rare-earth elements can be described from first principles focusing on the dominating $d$-electrons. Care must be taken in addressing the trend of calculated Curie temperatures in Fe-Co alloy on the basis of the effective spin model. The validity range of the spin-only description quantitatively differs between the all-Fe limit and all-Co limit. The strong ferromagnetism region in the Co-rich side characterized by the robust localized magnetic moment seems to be better suited for the spin model for a quantitative description. Within a subspace of fixed chemical composition, the relative trends in the intrinsic magnetism can be satisfactorily addressed on the basis of the effective spin model.

Having clarified the validity range of the spin model written in terms of localized degrees of freedom for the intrinsically delocalized ferromagnetism of $4f$-$3d$ intermetallics, it is hoped that several possibilities for an optimal ferromagnetism for permanent magnet utility can be identified in the middle of subtle interplay between localized and delocalized nature in the $d$-electron state.

Acknowledgements.

This work was supported by Toyota Motor Corporation and the Elements Strategy Initiative Center for Magnetic Materials (ESICMM) under the outsourcing project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Helpful discussions with collaborators in related projects are gratefully acknowledged.