Cerium as a possible stabilizer of ThMn$_{12}$-type iron-based compounds: A first-principles study

TL;DR

This study uses first-principles calculations to explore cerium's potential as a stabilizer for ThMn$_{12}$-type iron compounds, highlighting the importance of valency and size in stabilization.

Contribution

It demonstrates that tetravalent cerium can effectively stabilize ThMn$_{12}$-type structures, providing insights into element valency and size effects on stability.

Findings

Tetravalent Ce is a promising stabilizer for ThMn$_{12}$ structure.

Stabilizing effects depend on both valency and atomic size.

Ce's valency influences its stabilizing ability in Fe-based compounds.

Abstract

The structural stability of CeFe$_{12}$ is investigated by using first-principles calculation. The formation energies of CeFe$_{12}$ relative to the Ce$_{2}$Fe$_{17}$ + bcc-Fe phase and to the CeFe$_{2}$ + bcc-Fe phase are calculated with the assumptions of trivalency and tetravalency for Ce. Those values are compared with corresponding results in $R$Fe$_{12}$ for $R=$ Nd, Sm, and Zr. Our results suggest that the tetravalent Ce is a promising stabilizer of the ThMn$_{12}$ structure. We also show that the stabilizing effect of an element depends as much on the valency as on the size of the $R$ element by investigating $R$Fe$_{12}$ where $R$ is assumed to have a hypothetical valency on the basis of first-principles calculation.

Tables

Table 1. Table 1: The calculated lattice constants ( a 𝑎 a and c 𝑐 c ), volume per formula unit ( V 𝑉 V ), and inner coordinates of CeFe 12 are shown for the trivalent and tetravalent Ce. They have the tetragonal ThMn 12 -type structure (space group I ​ 4 / m ​ m ​ m 𝐼 4 𝑚 𝑚 𝑚 I4/mmm , No. 139).

	Ce3+Fe12	Ce4+Fe12
$a$ [Å]	8.579	8.480
$c$ [Å]	4.678	4.673
$V$ [Å3]	172.1	168.0
Ce (2$a$)	(0, 0, 0)	(0, 0, 0)
Fe (8$f$)	(1/4, 1/4, 1/4)	(1/4, 1/4, 1/4)
Fe (8$i$)	(0.360, 0, 0)	(0.360, 0, 0)
Fe (8$j$)	(0.265, 1/2, 0)	(0.270, 1/2, 0)

Table 2. Table 2: The lattice constants ( a 𝑎 a and c 𝑐 c ), volume per formula unit ( V 𝑉 V ), and inner coordinates of Ce 2 Fe 17 are shown for the trivalent and tetravalent Ce. They have the rhombohedral Th 2 Zn 17 -type structure (space group R ​ 3 ¯ ​ m 𝑅 ¯ 3 𝑚 R\bar{3}m , No. 166).

	Ce${}_2{}^{}{}_{}{}^{3+}$Fe17	Ce${}_2{}^{}{}_{}{}^{4+}$Fe17	Exp.Kramp et al. (1996)
$a$ [Å]	8.605	8.459	8.496
$c$ [Å]	12.557	12.513	12.414
$V$ [Å3]	268.4	258.5	258.7
Ce (6$c$)	(0, 0, 0.341)	(0, 0, 0.342)	—
Fe (18$f$)	(0.287, 0, 0)	(0.294, 0, 0)	—
Fe (18$h$)	(0.502, -0.502, 0.158)	(0.500, -0.500, 0.158)	—
Fe (9$d$)	(1/2, 0, 1/2)	(1/2, 0, 1/2)	—
Fe (6$c$)	(0, 0, 0.096)	(0, 0, 0.096)	—

Table 3. Table 3: The lattice constants ( a 𝑎 a ) and volume per formula unit ( V 𝑉 V ) of CeFe 2 are shown for the trivalent and tetravalent Ce. They have the cubic MgCu 2 -type structure (space group F ​ d ​ 3 ¯ ​ m 𝐹 𝑑 ¯ 3 𝑚 Fd\bar{3}m , No. 227).

	Ce3+Fe2	Ce4+Fe2	Exp.Duc and Hien (1995)
$a$ [Å]	7.501	7.361	7.302
$V$ [Å3]	105.5	99.7	97.3
Ce (8$a$)	(3/8, 3/8, 3/8)	(3/8, 3/8, 3/8)	—
Fe (16$d$)	(0, 0, 0)	(0, 0, 0)	—

Equations

$$\Delta E\equiv E[{\text{CeFe}}_{12}]-\left(\frac{1}{2}E[{\text{Ce}}_2{\text{Fe}}_{17}]+\frac{7}{2}E[\text{Fe}]\right),$$

$$\Delta E^{{\text{CeFe}}_{12}}\equiv E[{\text{CeFe}}_{12}]-\left(E[\text{Ce}]+12E[\text{Fe}]\right),$$

$$\Delta E^{{\text{Ce}}_2{\text{Fe}}_{17}}\equiv \frac{1}{2}E[{\text{Ce}}_2{\text{Fe}}_{17}]-\left(E[\text{Ce}]+\frac{17}{2}E[\text{Fe}]\right).$$

$$E[{\text{CeFe}}_{12}]-\left(E[\text{Ce}{\text{Fe}}_2]+10E[\text{Fe}]\right).$$

Full text

The permanent magnet is one of the most important materials in industries. The range of its application is widespread: the motors in electric vehicles, wind turbine generators, hard disks, and so on. Currently, the most widely used high-performance permanent magnet is NdFeB-based magnets, the main phase of which is Nd2Fe14B. Performance of a permanent magnet is determined by the intrinsic properties of the main phase, and by the microstructure formed by the subphases. Intensive research over three decades has improved the performance of the NdFeB-based magnet by controlling its microstructure. However, the performance seems to reach near to its theoretical limit today, hence a new magnet compound for the main phase is required.

Rare-earth compounds with the ThMn12 structure have been regarded as potential strong magnet compounds since iron-rich phases, such as SmFe11Ti, were synthesized in the late 1980’s. Ohashi et al. (1988); Buschow (1988); Yang et al. (1988) Compounds with high iron content, which leads to large magnetization, has been investigated intensely these days. Hirayama et al. have succeeded in synthesizing NdFe12N Hirayama et al. (2015), stimulated by a theoretical work on NdFe12 and NdFe12N Miyake et al. (2014). They measured intrinsic magnetic properties, and found that its saturation magnetization, anisotropy field and Curie temperature are higher than those of Nd2Fe14B.

The thermodynamic stability has also been one of the central issues in exploration of high-performance ThMn12 compounds. Unfortunately, NdFe12N (and its mother compound, NdFe12) exists only as a film due to its instability. Introducing another element can stabilize the ThMn12 structure, and it affects the magnetic properties significantly. For example, it has already been known in the 1980’s that substitution of Ti for one of 12 Fe sites improves the stability, but it significantly reduces the saturation magnetization. Friedel (1958); Verhoef et al. (1988); Miyake et al. (2014); Harashima et al. (2016) Partial substitution of Zr for the rare-earth element has eagerly investigated recently with the expectation of improvement in stability and of small harm in its magnetism. Suzuki et al. (2014, 2016); Kuno et al. (2016); Sakuma et al. (2016); Gjoka et al. (2016); Gabay and Hadjipanayis (2016, 2017); Kobayashi et al. (2017) In a previous work, we reported first-principle calculation of $R$Fe12 for 14 $R$ elements: 12 rare-earth elements (La, Pr, Nd, Sm, Gd, Dy, Ho, Er, Tm, Lu, Y, Sc) and 2 group IV elements (Zr, Hf). Harashima et al. (2018) On the basis of the calculated formation energy relative to the $R_{2}$Fe17 + bcc-Fe phase, we showed that some elements has potential in stabilizing the ThMn12 structure.

In the present work, we investigate potential of Ce as a stabilizer of the ThMn12 structure. Cerium is an abundant element, and often called as a “free” rare-earth because it is much more inexpensive than the other rare-earths. It has been attempted to replace other rare-earth elements in the ThMn12-type structure with Ce. Gabay et al. (2016); Gabay and Hadjipanayis (2016); Martin-Cid et al. (2018) Cerium is usually tetravalent Buschow and van Wieringen (1970) or in a mixed valence state Coey et al. (1993) in rare-earth–iron compounds. This is quite distinct from other rare-earth elements: Most of rare-earth elements are trivalent, and some of them, e.g. Eu and Yb, are divalent, but none of them are tetravalent in $R$–Fe compounds. Although the magnetism of Ce-doped ThMn12-type compounds has been studied well especially in experiments, there is no study on the potential of Ce as a stabilizer of the ThMn12 structure to the best of our knowledge.

To investigate the possibility, we perform first-principles calculation of the formation energy of CeFe12, and compare the result for other $R$Fe12 compounds. We analyze the result in terms of the atomic radius and valency of the $R$ element, referring to the results of our previous study. We also provide new calculational results in which $R$ is assumed to be tetravalent for the analysis.

We use QMAS, a first-principles calculation code, which is constructed on the basis of density functional theory Hohenberg and Kohn (1964); Kohn and Sham (1965) and the projector augmented-wave method Blöchl (1994); Kresse and Joubert (1999). The exchange-correlation functional is approximated by the generalized gradient approximation of the Perdew–Burke–Ernzerhof (PBE) formula Perdew et al. (1996). 8 $\times$ 8 $\times$ 8 $k$-points are sampled, and the cutoff energy for the plane wave basis is set to 40.0 Ry. The 4$f$ electrons of rare-earth elements are treated as open-core states. Occupation of those open-core states satisfies Hund’s first rule: The minority spin channel is first occupied, then, if electrons remain, the majority spin channel is occupied. For detailed discussion of the reliability of the open-core treatment in calculation of the formation energy, we refer readers to Ref. Harashima et al., 2018.

First, we show the lattice parameters for CeFe12 obtained by the numerical optimization, and also those for Ce2Fe17 and CeFe2 for comparison. Table 1 shows the optimized structures of CeFe12 calculated with the assumption of the Ce3+ and Ce4+ valency, where the number of occupied 4$f$ electrons are assumed to be one and zero, respectively; Table 2 shows the result for the rhombohedral Ce2Fe17; Table 3 shows the result for the cubic CeFe2.

In all phases, the volume is significantly smaller with Ce4+ than with Ce3+. It can be attributed to the additional delocalized electron in the Ce4+ systems that is fixed to an $f$-state in the Ce3+ systems. The extended electron contributes to cohesion, and makes the difference in volume. The experimental lattice parameters of $a=$ 8.496 Å  and $c=$ 12.414 Å  have been reported for Ce2Fe17 Kramp et al. (1996) (although the structure is annotated to be ‘hexagonal’ in the literature, we take its Bravais lattice as rhombohedral, which has a hexagonal conventional cell, because the given parameters are unreasonably different from other compounds having a hexagonal Bravais lattice). The experimental value of $a=$ 7.302 Å  for CeFe2 have also been reported. Duc and Hien (1995) Our calculated lattice constant with Ce4+ agrees well with the experimental values while those with Ce3+ deviate significantly from the experimental values. Gabay et al. have reported that the volume of CeFe9Co2Ti having the ThMn12 structure is slightly smaller than that of SmFe9Co2Ti. Gabay et al. (2016) This tendency agrees with our calculation of CeFe12 and SmFe12 ($a=$ 8.497 Å  and $c=$ 4.687 ÅHarashima et al. (2018)) when the tetravalency of Ce is assumed.

Next, we calculate the formation energy of CeFe12. We take Ce2Fe17 + bcc-Fe as a reference system because the closest compound to CeFe12 is Ce2Fe17 in the phase diagram. The definition of the formation energy is

[TABLE]

where $E[\cdot]$ denotes the total energy per chemical formula of the system in the bracket. This can be also expressed in terms of the formation energy of CeFe12 and Ce2Fe17 from simple substances by $\Delta E=\Delta E^{\text{CeFe}_{12}}-\Delta E^{\text{Ce}_{2}\text{Fe}_{17}}$ where they are defined as

[TABLE]

We consider both the Ce3+ and Ce4+ valency in the following, but the valency of Ce in CeFe12 and Ce2Fe17 are assumed to be the same to each other in Eq. (1). We also take account of the CeFe2 phase, another competing phase which appears in the phase diagram as the next nearest phase of CeFe12, with calculation of the formation energy of CeFe12 from CeFe2:

[TABLE]

In Fig. 1, the formation energies for CeFe12 are plotted and the corresponding results for $R$Fe12 ($R=$ Zr, Nd, and Sm) are shown for comparison. Nd and Sm are typical rare-earths used in permanent magnet compounds, and Zr is recently suggested as a stabilizer of the ThMn12-type structure. The values for Ce4+Fe12 are significantly less than those for NdFe12, SmFe12 and ZrFe12. Although the formation energy against Ce4+Fe2 is not negative, it is also much less than the values of the other $R$Fe12 systems against $R$Fe2. Providing that the CeFe2 phase can be suppressed in a process, which is achieved in synthesis of Ce2Fe17 or Sm2Fe17 in practice, our result suggest the potential of Ce4+ in stabilizing the CeFe12 phase.

To analyze the stabilization by a tetravalent element systematically, we consider hypothetical $R$Fe12 systems for $R=$ Pr, Nd, Sm, Gd, Dy, Ho, Er, Tm and Lu, with the valency of $R$ assumed to be tetravalent. We show the formation energy of $R$Fe12 from $R_{2}$Fe17 + bcc-Fe as a function of $r_{R}^{\text{voronoi}}$ in Fig. 2. The results we have reported in Ref. Harashima et al., 2018 are also plotted in the figure for comparison. $r_{R}^{\text{voronoi}}$ is defined as the radius of a sphere whose volume is equal to the Voronoi cell volume of the $R$ site in $R$Fe12, which can be regarded as an estimation of the atomic radius of $R$. The data points form two distinct curves, which is differentiated by the valency of the $R$ element. Interestingly, Hf and Zr, which are tetravalent and non-rare-earth elements, also follow this trend. The formation energies become minima around $r_{R}^{\text{voronoi}}\approx$ 1.62 Å , and the tetravalent elements tend to yield lower formation energy than the trivalent elements. The valency of the rare-earth elements significantly affects the stability, and is important as well as the atomic size.

We also investigate stability of the tetravalent phase over the trivalent phase. Figure 3 shows the total energy difference per $R$ atom between $R^{3+}$Fe12 and $R^{4+}$Fe12 (red) and that between $R^{3+}_{2}$Fe17 and $R^{4+}_{2}$Fe17 (blue). The values at $R=$ Ce are admittedly positive, which means that the tetravalent state is less stable than the trivalent state. This contradicts the experimental observation of Ce2Fe17 and our expectation for CeFe12. However, the energy difference systematically decreases as a function of the atomic number of $R$ decreases. This means that there is a trend toward a tetravalent state when $R$ approaches Ce, which is in agreement with the expectation and the experiment. The deviation may be ascribed to theoretical errors coming from our calculational framework using GGA and the open-core treatment, solution of which is an open question at the present stage.

Finally, we investigate the magnetization of CeFe12. Figure 4 shows the magnetizations of CeFe12 and other $R$Fe12 compounds. All systems has a similar contribution to the magnetization from the valence electrons, and the difference comes mainly from the magnetization of the $f$-electrons. Ce4+Fe12 has a lower magnetization than Ce3+Fe12, large part of which is attributed to the local magnetic moment of Ce-4$f$ electrons. Difference in hybridization between Ce and Fe also has a minor but discernible effect on the magnetization.

In conclusion, we performed first-principles calculation for CeFe12 with the trivalent and tetravalent Ce. The formation energies of CeFe12 from Ce2Fe17 + bcc-Fe and CeFe2 + bcc-Fe are calculated. By comparison with the formation energies of $R$Fe12 for $R=$ Nd, Sm, and Zr, it is found that the value for Ce4+Fe12 is lower than the other $R$Fe12 systems. This result indicates that the tetravalent Ce is a promising stabilizer for the ThMn12-type compounds. Our analysis revealed that the tetravalency of Ce is important in its functionality as a stabilizer. We also calculated the magnetization of CeFe12. Ce4+Fe12 has a smaller magnetization than NdFe12 and SmFe12, but its potential in stabilizing the ThMn12 phase is still appealing. We expect fractional doping of Ce can improve the stability of the $R$Fe12 phase with a small loss of the magnetization. However, the amount of Ce is necessary to be tuned to balance the magnetization and stability, which cannot be predicted by a simple interpolation in general even when the concentration of the dopant is small,Fukazawa et al. (2018) in practical use. Finding the optimum concentration of Ce is an important open problem, in which we should take account of other stabilizers. We refer those who are interested in such a composition-optimization problem to Ref. Fukazawa et al., 2019, one of our previous study with a machine learning technique.

We thank Kiyoyuki Terakura, Hisazumi Akai and Satoshi Hirosawa for fruitful discussion. This work was supported by the Elements Strategy Initiative Project under the auspices of MEXT, and by MEXT as a social and scientific priority issue (Creation of new functional Devices and high-performance Materials to Support next-generation Industries; CDMSI) to be tackled by using the post-K computer. The computation was partly conducted using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, and the supercomputer of ACCMS, Kyoto University. This research also used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID:hp180206).