Abstract
Cerium-based intermetallics are currently attracting much interest as a possible alternative to existing high-performance magnets containing scarce heavy rare-earth elements. However, the intrinsic magnetic properties of Ce in these systems are poorly understood due to the difficulty of a quantitative description of the Kondo effect, a many-body phenomenon where conduction electrons screen out the Ce-4f moment. Here, we show that the Ce-4f shell in CeâFe intermetallics is partially Kondo screened. The Kondo scale is dramatically enhanced by nitrogen interstitials suppressing the Ce-4f contribution to the magnetic anisotropy, in striking contrast to the effect of nitrogenation in isostructural intermetallics containing other rare-earth elements. We determine the full temperature dependence of the Ce-4f single-ion anisotropy and show that even unscreened Ce-4f moments contribute little to the room-temperature intrinsic magnetic hardness. Our study thus establishes fundamental constraints on the potential of cerium-based permanent magnet intermetallics.
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Introduction
The last decades have witnessed a rapidly growing demand for energy-efficient technologies, from optimized power generation and conversion devices to new transportation solutions. A key component of motors, generators, sensors, or actuators are high-performance permanent magnets. Identifying and designing improved materials for permanent magnet applications has thus the potential of huge economical and environmental savings1,2. Today, the world market for permanent magnets in the more advanced applications is largely dominated by rare-earth-based materials: the best hard-magnetic materials are based on the Nd-Fe-B â2â14â1â system whose extraordinary hard-magnetic properties are further boosted by partially substituting Nd with heavy rare-earth elements like Dy or Tb. In such rare-earth transition metal intermetallics, the electronic structure of the transition metal sublattice provides a large magnetization and high Curie temperature. The intrinsic magnetic hardness κ is due to the rare-earth local magnetic moment, whichâdue to the large spinâorbit coupling acting on the rare-earth electronic statesâeffectively converts the crystalline anisotropy into single-ion magnetic anisotropy3.
Available resources of heavy rare earths are scarce, thus motivating efforts to replace heavy rare-earth intermetallics with so-called rare-earth balance magnets2 containing more abundant rare-earth elements, in particular Ce1,4,5,6,7. A significant limitation to the magnetic hardness of Ce-transition metal intermetallics is caused by the well-known tendency of Ce-4f electrons to form heavy-electron itinerant bands, as for instance observed in so-called âheavy-fermionâ compounds8. Indeed, in the heavy-fermion regime the Ce local magnetic moment is screened by conduction electrons, which couple to the single Ce-4f electron to form a nonmagnetic singlet state. Such a heavy-fermion behavior has been observed in various Ce intermetallics, for example, in the CeFe2 Laves phase9, as well as in the RCo5 (Râ=ârare earth) and R2Fe14B families of hard-magnetic intermetallics10,11 compromising, especially, the utility of Ce2Fe14B for magnetic applications12.
In order to possess a significant single-ion magnetic anisotropy, an obvious conditio sine qua non is that the 4f electron of Ce needs to be localized and its magnetic moment not significantly Kondo-screened. A reliable quantitative assessment of localized vs. heavy-fermion behavior of the Ce-4f electrons is thus key to a successful theoretical search for prospective hard-magnetic cerium intermetallics. Such an assessment represents a formidable theoretical challenge. Some qualitative insights can be obtained from density functional theory (DFT) calculations treating Ce-4f as itinerant bands13, or estimating the evolution of the Kondo scale14 from the DFT electronic structure. However, the formation of a heavy-fermion state in Ce-based compounds is a genuine non-perturbative many-body phenomenon, the quantitative description of which requires methods beyond DFT or DFTâ+âU techniques15,16. At present, these limitations seriously hamper the possibilities of computational materials design in the field (see e.g., ref. 17).
Here, we address this challenge by developing an ab initio approach to the magnetocrystalline anisotropy of Ce-based intermetallics. We describe the electronic structure of magnetic Ce-transition metal intermetallics, using advanced many-body electronic structure methods that take into account the effect of the Kondo screening. Furthermore, we supplement this first-principle framework with a variational approach to evaluate the magnitude of the Ce single-ion magnetic anisotropy explicitly taking into account the impact of the Kondo effect. In contrast to previous zero-temperature DFT calculations17,18,19, the present study investigates the full temperature dependence of the Ce magnetic anisotropy. Our theoretical predictions are compared to the experimental magnetic anisotropy that we measure in the same wide temperature range.
We focus on the CeâFe â1â12â material system4,7,17,20,21, which has recently been under scrutiny as a potential hard magnet16,22,23. Our calculations show a significant reduction of the Ce magnetic moment due to the Kondo effect. This Kondo screening is found to be very sensitive to substitutions and small-atom insertions. In particular, interstitial nitrogen is predicted to suppress the Ce contribution to the magnetization and magnetic anisotropy. This behavior is in drastic contrast to the one observed in â1â12â intermetallics with localized rare earths, like Nd, where nitrogenation has been found to significantly enhance the uniaxial magnetic anisotropy24. Indeed, the unexpected suppression of the Ce magnetization by N interstitials is absent if the Ce moment is treated as purely local and stems from a large enhancement of the Kondo scale by nitrogenation, as schematically shown in Fig. 1bâd. It is confirmed by our measurements of the anisotropy in polycrystalline samples with and without nitrogenation. Such interstitial atoms effectively controlling the state of the Ce-4f shell could be used for fine-tuning and probing the complex many-electron physics of Ce. Most importantly, our calculations establish the salient fact that even in the absence of Kondo screening, the small size of the Ce-4f spin moment necessarily leads to rather small values of the magnetic anisotropy at elevated temperatures. Our study, therefore, reveals significant limitations for the Ce contribution to the magnetic hardness at temperatures relevant for applications.
Results
Kondo screening of the Ce magnetic moment in â1â12â systems
The â1â12â CeâFe intermetallics crystallize in the tetragonal lattice structure shown in Fig. 1a. A partial substitution of Fe with another transition metal (usually Ti) is necessary to stabilize this â1â12â phase in the bulk. Hence, we consider the realistic CeFe11Ti composition and, for the sake of comparison, also parent thermodynamically unstable CeFe12. For the latter, we assume the same crystal structure. The magnetic properties of â1â12â rare-earth transition metal intermetallics can be modified by small-atom interstitials, in particular nitrogen. Nitrogenation has been found to significantly enhance the uniaxial magnetic anisotropy in NdFe12N and NdFe11TiN (ref. 24). Nitrogenated CeâFe â1â12â compounds have been patented as prospective high-performance magnets5, with N interstitials found to significantly increase their Curie temperature and magnetization5,20,21. Therefore, we also consider nitrogenated CeFe11Ti with N interstitials in the experimentally determined Wyckoff 2b position as shown in Fig. 1a.
The ordered Ce-4f magnetic moment Mz in CeFe12, CeFe11Ti, and CeFe11TiN as a function of temperature T is displayed in Fig. 2a as symbols. This Ce moment is induced by the exchange field Bex generated by the ferromagnetically ordered iron sublattice, as schematically shown in Fig. 1bâd. In our ab initio approach, abbreviated below as DMFT(QMC), we take into account the magnetization of Fe within the local spin-density approximation (LSDA), while many-electron effects on the Ce-4f shell are treated within dynamical mean-field theory (DMFT) in conjunction with a numerically exact quantum Monte-Carlo (QMC) approach25. With the Fe ferromagnetism included within LSDA, the exchange field Bex acting on the Ce-4f shell is virtually temperature independent (this approximation is justified since we focus on the temperature range up to 300âK, which is well below the Curie temperatures of 485 and 710âK for CeFe11Ti and CeFe11TiN, respectively). Hence, the Mz vs. T curve reflects the temperature dependence of the Ce-4f magnetic susceptibility. One observes a weak T dependence for all three compounds; it is very different from a Curie-like behavior expected for Ce-4f local moments subjected to an exchange field. The obtained values are well below the saturated Russel-Saunders value Mzâ=âgJâ Jâ=â2.14μB for the Ce f5/2 manifold. In particular, in CeFe11TiN the 4f moment is tiny (0.04âμB). A small increase of the Ce-4f magnetic moment with respect to pure CeFe12 is induced by Ti substitution. Overall, the magnitude and temperature dependence of the Ce-4f magnetic moment in the investigated â1â12â intermetallics correspond to a Kondo-screened Fermi-liquid Ce-4f state.
For comparison, the ordered Ce-4f magnetic moment \({M}_{z}^{{\rm{at}}}\) has also been calculated within a quasi-atomic (Hubbard-I) approximation26,27 to the Ce-4f quantum impurity problem (solid lines in Fig. 2a). In this approach, abbreviated below as DFTâ+âHubI, the Kondo effect is neglected (as schematically shown in Fig. 1b). The Ce magnetic moment evaluated within DFTâ+âHubI exhibits the expected Curie behavior and is significantly larger than the Kondo-screened one at low temperatures. Above Tâ~â400âK, the magnitude of the Ce magnetic moments in CeFe12 and CeFe11Ti agree reasonably well between DFTâ+âHubI and DMFT(QMC), indicating this temperature to be the Kondo scale of these compounds. Within the quasi-atomic picture one would expect the largest \({M}_{z}^{{\rm{at}}}\) for CeFe11TiN (blue solid line in Fig. 2a), for which the DMFT(QMC) moment remains, however, heavily screened in the whole temperature range up to 600âK.
Magnetic anisotropy of Ce in the quasi-atomic picture
In the quasi-atomic approximation (schematically shown in Fig. 1b), the single-ion anisotropy due to the Ce-4f magnetic moments is determined by Bex stemming from the Fe-sublattice and the crystal field (CF) splitting on the Ce-4f shell28. While the magnitude of Bex is similar in all three investigated compounds, the CF splitting turns out to be very different (see the crystal field parameters \({A}_{2}^{0}\langle {r}^{2}\rangle\) in Table 1). Of particular interest for the Ce-4f magnetic moment at low temperature is the composition of the ground state levels. The CF ground state in CeFe11TiN is well separated from excited states and consists almost exclusively of the \(\left|\!\pm \!5/2\right\rangle\) eigenstates, which have the largest \(| {M}_{z}^{{\rm{at}}}|\) among the jâ=â5/2 manifold. CeFe11TiN is thus expected to exhibit a strong easy-axis magnetic anisotropy in the quasi-atomic picture. In CeFe11Ti, the contribution of the \(\left|\!\pm \!5/2\right\rangle\) eigenstates to the CF ground state is reduced leading to a smaller easy-axis anisotropy. In CeFe12, the CF ground state is \(\left|\!\pm \!1/2\right\rangle\) corresponding to an easy-plane anisotropy (see Supplementary Fig. 1 for the full CF levels scheme). The peak of the CeFe12 magnetic moment at ~40âK (Fig. 2a) can be explained by the fact that the applied exchange field oriented along the hard axis of CeFe12 is not sufficient to saturate the ground state moment. Thus, with increasing temperature excited states with a larger moment along the hard axis become thermally populated before the usual Curie decay of the magnetization sets in.
We extract the values of the CF parameters, as well as Bex from our ab initio DFTâ+âHubI calculations and then evaluate the Ce single-ion anisotropy constants. The resulting constants \({K}_{1}^{{\rm{at}}}\) (in K per f.u.) are listed in Table 1. At liquid helium temperature, \({K}_{1}^{{\rm{at}}}\) is easy-plane in CeFe12, and easy-axis in CeFe11Ti and CeFe11TiN. For Tâ=â300âK, CeFe11TiN exhibits a substantially stronger easy-axis anisotropy as compared to CeFe11Ti due to a larger value of its low-rank â20â crystal field parameter (see Supplementary Table I for values). Both compounds exhibit a strong reduction of the anisotropy with increasing temperature. The calculated room-temperature anisotropy constants \({K}_{1}^{{\rm{at}}}\) are in very good agreement with the high-T limit analytical expression from ref. 28, which reads
with \(x=\frac{2J({g}_{J}-1){B}_{{\rm{ex}}}}{{k}_{\mathrm{B}}T}\), where gJâ=â6/7 is the gyromagnetic ratio for the Ce-4f1 Jâ=â5/2 shell. Using this analytical expression, one may understand the very rapid suppression of the Ce single-ion anisotropy apparent in Table 1. The Zeeman energy 2Jâ£gJâââ1â£Bex associated to the exchange field Bexâ~â300âT gives a characteristic temperature of 2Jâ£gJâââ1â£Bex/kB ~ 140âK, explaining the rapid suppression of the \({K}_{1}^{{\rm{at}}}\) values with temperature and the small value at room temperature. For comparison, we note that the same simple estimate for Sm3+, with gJâ=â2/7 gives a five times higher characteristic temperature (700âK), resulting in a 25 times slower reduction of \({K}_{1}^{{\rm{at}}}\) vs. T at high temperatures. Hence, as the small magnitude of the spin represents an intrinsic feature of the Ce3+ ion, i.e., not sensitive to the crystalline environment, the Ce-4f contribution to the magnetic hardness at room temperature and above is also expected to be intrinsically weak.
Figure 2b shows our measured anisotropy fields obtained from hard-axis measurements of the textured samples. Our results are consistent with the measurements at room temperature previously reported in refs. 4,20,21. The anisotropy field is already relatively small at low temperature, and then exhibits a rather slow decay with temperature.
In order to compare theory and experiment, one needs to add the contribution from the Fe-sublattice to the calculated single-ion \({K}_{1}^{{\rm{at}}}\) of Ce. As in this work, we do not aim at calculating the Fe-sublattice magnetic anisotropy, we extract its value and temperature dependence from measurements performed on YFe11Ti (ref. 29), using the experimental dependence \({K}_{1}^{{\rm{Fe}}}(T)=23.3-28T/{T}_{{\rm{c}}}\) (K/f.u.) measured in ref. 29 together with our measured Curie temperatures. At low T, the iron contribution is much smaller than the Ce one and does not affect the picture qualitatively.
Using the relation \({H}_{{\rm{A}}}\approx \frac{2{K}_{1}^{{\rm{tot}}}}{{M}_{{\rm{S}}}}\), where HA is the anisotropy field, MS the total magnetic moment of the compound saturated in the direction of HA, and \({K}_{1}^{{\rm{tot}}}={K}_{1}^{{\rm{Fe}}}+{K}_{1}^{{\rm{at}}}\), we calculate the corresponding values of HA in the quasi-atomic approximation listed in Table 2 together with the experimental values. The quasi-atomic approximation predicts a strong anisotropy that decreases quickly with temperature, while experimentally one finds only a moderate decrease with T, in particular in CeFe11TiN, which bears little resemblance to the predictions of DFTâ+âHubI. Notice that the DFT calculations of ref. 18 treating Ce-4f as delocalized band-like states predict the zero-temperature HA of CeFe11Ti to be reduced by nitrogenation by ~0.9âT, in disagreement with our measurements and previous experiments20,21 finding the difference in HA between CeFe11Ti and CeFe11TiNx to be in the range of 2.5â3âT (see Table 2). The low-T anisotropy of CeFe11Ti seems to be significantly underestimated by DFT18,19. Conversely, by treating Ce-4f as an open core Körner et al.17 obtained HAâ=â18 T for CeFe11Ti, which is, similarly to our DFTâ+âHubI result, strongly overestimated compared to experimental values. Hence, neither DFT nor the fully localized treatment of 4f is able to capture the impact of interstitials on the Ce magnetic anisotropy. As we show below, the theoretical picture is drastically modified when the impact of Kondo screening on the Ce-4f magnetic anisotropy is taken into account within a full many-body framework.
Ce magnetic anisotropy in the presence of Kondo screening
A quantitative evaluation of the magnetic anisotropy of a partially Kondo-screened magnetic moment is a formidable problem whichâto the best of our knowledgeâhas not been addressed even for simple model systems so far. In the context of the present work on complex realistic systems, we have developed a variational approach to meet this challenge. In order to capture the interplay of the Kondo interaction (HK) with the exchange (HB) and crystal (HCF) fields in the total impurity Hamiltonian Hâ=âHcâ+âHKâ+âHBâ+âHCF for the Ce-4f shell, we assume the ground state of the Ce-4f shell to be given by the following variational many-electron wave function:
where ΨS is the singlet formed by the Ce-4f total moment coupled to the conduction-electron sea defined by Hc; the second term ΨJ in Eq. (2) represents the 4f ground state in the absence of Kondo screening, i.e., determined by HBâ+âHCF; α is a variational coefficient. Only the second term, due to its nonzero angular momentum, couples to the exchange field
As shown in the Supplementary Note 3, the lowest-order anisotropy constant obtained with the variational ansatz Eq. (2) then reads
where \({K}_{1}^{{\rm{at}}}\) is the anisotropy constant in the quasi-atomic approximation (DFTâ+âHubI), as detailed above. The value of α2 is given in accordance with Eq. (3) by \({M}_{z}(T=0)/{M}_{z}^{{\rm{at}}}(T=0)\) and can be extracted by extrapolating the curves in Fig. 2a to zero temperature. The denominator 1â+âδ in Eq. (4), where \(\delta ={K}_{1}^{{\rm{at}}}/A\), is due to the change of α upon the rotation of Bex from the uniaxial \({\overrightarrow{B}}_{{\rm{ex}}}| | c\) to in-plane direction and can be evaluated from the dependence of α2 vs. Bexâ£â£z, calculated by DMFT(QMC) (see Supplementary Note 3 for details).
By multiplying the quasi-atomic \({K}_{1}^{{\rm{at}}}\) for Ce with the corresponding reduction factors due to the Kondo screening (Eq. (4)) and adding the contribution of the Fe-sublattice \({K}_{1}^{{\rm{Fe}}}(T)\), we obtain the total DMFT(QMC) anisotropy field listed in Table 2. Strictly speaking, the above variational treatment is applicable only at zero temperature. However, close to room temperature, the cerium contribution is overruled by the iron one, and the final value becomes relatively insensitive to the precise estimate of the Kondo reduction factor. At low temperature, instead, the Kondo reduction is very significant and the Ce contribution to HA is completely suppressed in the nitrogenated compound, but still relatively important in CeFe11Ti. The values of the anisotropy field estimated within DMFT(QMC) are in good agreement with our experimental results and previous values reported in refs. 4,20,21, see Fig. 2b. In particular, the dramatic reduction of the low-T anisotropy field upon nitrogenation predicted by our calculations (by 45%, i.e., 2.6âT) is in quantitative agreement with the reduction observed in experiment. Our prediction for the low-T HA of CeFe11Ti (5.8âT) is in the middle of the measured range (5.1â7.4âT), in contrast to the itinerant 4f-in-valence18,19 and localized 4f-in-core17 DFT limits, which are not able to capture the intermediate partially Kondo-screened state in this system. The itinerant picture with 4f-in-valence18 performs, however, reasonably well for the strongly screened Ce-4f states in CeFe11TiN.
Electronic structure and Ce-4f hybridization
As demonstrated above, the magnetic moment and anisotropy of Ce-4f in the CeFe11MX intermetallics is heavily influenced by the Kondo effect. Here, we analyze its imprint on the electronic structure of CeFe11Ti(N) and identify the origin of the enhanced Kondo screening in the nitrogenated system.
In Fig. 3a, we show the real part of the DMFT self-energy ReâΣ(Ï) in CeFe11Ti and CeFe11TiN, respectively, averaged over the six (degeneracy of the Jâ=â5/2 multiplet) Ce-4f orbital contributions with the largest occupation number. The slope of ReΣ(Ï) at Ïâ=â0 is a direct proxy for the strength of electronic correlations and determines the quasiparticle weight: \(Z={\left[1-\frac{\mathrm{d}\left[{\mathrm{Re}} \Sigma (\omega )\right]}{\mathrm{d}\omega }{| }_{\omega = 0}\right]}^{-1}\). The obtained quasiparticle weight is higher in CeFe11TiN, as compared to CeFe11Ti, indicating a more delocalized character of the Ce-4f states in the nitrogenated compound. This can also be seen from the low occupation of nâ=â0.86 electrons on the Ce-4f shell in CeFe11TiN, deviating stronger from the localized limit with nâ=â1 electron as compared to nâ=â0.94 in CeFe11Ti. For a further comparison, in CeFe12 the Ce-4f occupancy is nâ=â0.93, a bit lower than in CeFe11Ti and thus indicating that the necessary Ti substitution slightly favors the localization of the Ce-4f states (for the spectral function of CeFe12, we refer the reader to the Supplementary Fig. 4). The DMFT hybridization function shown in Fig. 3b quantifies the hybridization of the Ce-4f shell with other states of the system. Especially at low energies in the vicinity of the Fermi level, the hybridization function is much larger in CeFe11TiN, indicating an increased mixing of the Ce-4f states with neighboring nitrogen states, e.g., N-2p, which is at the origin of the more delocalized behavior of the Ce-4f electrons.
The corresponding DMFT(QMC) spectral functions are displayed in Fig. 3c, d. In the characteristic three-peak structure of the Ce-4f spectral function the Kondo peak at the Fermi level is further split into two sub-peaks due to the spinâorbit splitting. The position of the localized features, i.e., the lower and upper Hubbard bands centered at â2 and 5âeV, respectively, is in good agreement with the photoemission (PES) spectra of CeFe2 (ref. 9; we are not aware of any experimental PES for the CeFe12 materials family). The more delocalized character of the Ce-4f states in CeFe11TiN is also reflected in the spectral function, Fig. 3d: a more prominent Kondo peak and less pronounced Hubbard bands signal a more delocalized behavior.
Discussion
This work introduces a DMFT-based approach to the Ce magnetization and magnetic anisotropy in ferromagnetic intermetallics. The intermediate valence behavior of Ce in the investigated intermetallics can be captured neither by pure DFT nor by approaches treating the Ce-4f states as purely localized. Indeed, in our work, we show that a fully localized treatment of the Ce-4f states within the quasi-atomic approximation strongly overestimates the Ce contribution to the anisotropy at low temperatures. The problem at hand thus calls for sophisticated many-body approaches, such as DMFT.
We derive a simple formula (Eq. (4)) relating the reduction of the magnetic anisotropy due to the Kondo screening to that of the ordered Ce moment; the latter being more easily accessible within DMFT. Moreover, this formula is of general applicability and can be used, for example, to extract the reduction of the anisotropy from the measured Ce magnetization. Our approach quantitatively captures the impact of Kondo screening on the intrinsic magnetic hardness κ of realistic Ce systems, thus opening the possibility for a theoretical search of Ce-based intermetallics with enhanced κ.
Our calculations reveal a partially Kondo-screened state of the Ce-4f magnetic moment in CeâFe intermetallics of the â1â12â family. Nitrogen interstitials strongly enhance the screening of the 4f moment, thus completely suppressing the Ce single-ion contribution to the anisotropy both at liquid helium and room temperature. These theoretical predictions are validated by our experimental measurements finding the anisotropy field in CeFe11Ti to be strongly suppressed by nitrogenation, in particular at low temperatures where the Ce contribution is the most significant. The effect of the dramatic Kondo-scale enhancement by nitrogen in the CeâFe â1â12â system indicates an unexpected sensitivity of this scale to interstitials in hard-magnetic intermetallics. The Kondo effect at low temperatures might thus be effectively controlled by such interstitials. Therefore, one may systematically search for interstitial elements that, in contrast to nitrogen in CeâFe â1â12â, suppress the Kondo screening of the 4f moment. The present theoretical framework can be employed in this search.
At the same time, the small magnetic anisotropy of a purely local Ce-4f shell at elevated temperatures found in the present work limits the direct use of cerium-based â1â12â systems as hard-magnetic materials. Cerium can still be successfully used as a partial substitution element for the critical heavy rare-earth elements. In this case, the essential magnetic hardness is provided by heavy rare earths, but their concentration may in some cases be reduced without compromising the magnetic properties as recently shown in ref. 6 on the example of a mixture of La and Ce reducing the material cost of Nd2Fe14B-type hard-magnetic materials. However, the present study shows that cerium itself should not be expected to provide an essential contribution to the magnetic anisotropy at room temperature and above. Though in the present work we considered only the â1â12â family of CeâFe intermetalics, this conclusion based on the intrinsic properties of the Ce-4f1 shell should be rather independent of the crystalline environment. It is expected thus to be of general validity for magnetic Ce intermetallics, unless one finds aâpossibly exoticâmechanism allowing the hybridization to enhance the Ce-4f anisotropy in the high-temperature local moment regime (i.e., above the Kondo temperature). Such an enhancement would manifest itself in a larger value of the Ce easy-axis magnetic moment obtained within a full DMFT treatment, as compared to the value obtained in the quasi-atomic approximation. In the current â1â12â systems, such an effect is not observed. Given the environmental and economical impact that is at stake if improved permanent magnet materials are found, it might however be worthwhile to systematically search for it.
Methods
Theory
Our ab initio calculationsâcombined DFT and DMFT27,30,31,32âdescribe many-body effects in the Ce-4f shell taking into account on equal footing the key competing energy scales of the problem: the CF and exchange field induced by the spin-polarized electronic structure, as well as the spinâorbit coupling. We start from a charge-self-consistent spin-polarized DFTâ+âDMFT calculation33,34,35,36,37 of our target compounds, where we take into account the magnetization of Fe within the LSDA, with the Fe moment aligned along the câ£â£z axis, while we treat the Ce-4f magnetic moment within the quasi-atomic Hubbard-I approximation26,27. In our self-consistent DFTâ+âDMFT calculations within the Hubbard-I approximation, abbreviated by DFTâ+âHubI, we suppress the self-interaction contribution to the CF splitting, as well as the 4f contribution to the exchange field Bex on the Ce site. In result, the DFT exchange-correlation potential is calculated using a non-spin-polarized and spherical Ce-4f electronic density, while the nonspherical contributions and magnetization density of other states (including non-4f Ce orbitals like Ce-5d) are fully taken into account. This approach provides reliable CFs and Bex in hard-magnetic intermetallics, as previously demonstrated in ref. 16. We employ the fully localized limit double-counting correction, in conjunction with the nominal nâ=â1 occupancy for the Ce-4f shell38. Since we obtain the Fe exchange field Bex within the DFTâ+âHubI calculations, it is temperature independent. This is a reasonable approximation for the temperature range <300âK that we focus on. The experimental Curie temperatures Tc for CeFe11Ti and CeFe11TiN of 485 and 710âK, respectively, are significantly >300âK. Our tests within the quasi-atomic treatment of the Ce-4f shell show that including the finite-temperature decay Bex leads to a rather small (<10% at 300âK) additional reduction of the Ce moment, as compared to the dependence (Fig. 2a) obtained with the fixed zero-T value of Bex.
Having converged the self-consistent DFTâ+âHubI calculations, we use the resulting charge density to produce the KohnâSham one-electron part of the Hamiltonian that includes the correct CF and Bex acting on the Ce-4f shell. We then perform an additional DMFT run using a hybridization-expansion continuous-time QMC solver in the segment picture25,39. We abbreviate this approach as DMFT(QMC). By using in DMFT(QMC) the charge and magnetization density obtained in DFTâ+âHubI, we neglect the feedback of possible Ce-4f delocalization to the Fe magnetization and the exchange field Bex. This treatment is in-line with the usual assumption that the 4f magnetism in the TM-RE hard magnets has a negligible influence on the TM-sublattice magnetism.
In the Ce-4f quantum impurity problem, we employ the basis of f1 eigenstates of the atomic Hamiltonian, which represents two manifolds, jâ=â5/2 and jâ=â7/2, split by the spinâorbit coupling, with additional smaller splittings within the manifolds due to the CF and EF. We treat all 14 Ce-4f orbitals as correlated and employ the densityâdensity approximation for the corresponding Coulomb vertex. We calculate the ordered Ce-4f magnetic moment \({M}_{z}={\rm{Tr}}\left[({\hat{L}}_{z}+2{\hat{S}}_{z})\rho \right]\) from the converged DMFT density matrix Ï.
The Ce single-ion anisotropy constant within the atomic approximation, \({K}_{1}^{{\rm{at}}}\), is evaluated from the dependence of the anisotropy energy \({E}_{a}^{{\rm{at}}}\) on the azimutal angle θ of the exchange field Bex by fitting it to the form \({E}_{a}^{{\rm{at}}}={K}_{1}^{{\rm{at}}}{\sin }^{2}\theta\). The dependence \({E}_{a}^{{\rm{at}}}\) vs. θ is calculated by a numerical diagonalization of the Ce-4f CF Hamiltonian under Bex applied in the corresponding direction. Please refer to the Supplementary Note 2 for further details regarding our ab initio calculations and the calculation of the magnetic anisotropy.
Experiment
Polycrystalline CeFe11Ti samples are produced by the suction casting method using commercial purity elements (>99.99%). The as-cast samples are wrapped in Mo foil, sealed in a quartz ampule under argon atmosphere and annealed at 850â°C for 12âh followed by quenching in water. Part of the suction-cast plates are ground into powder <20âμm to carry out the nitrogenation. The determination of the crystal structures is done using a room-temperature x-ray powder diffractometer (XRD) and the XRD patterns are fitted with a ThMn12-type (I4/mmm, space group no. 139) structure. Thermomagnetic measurements are carried out for the determination of the Curie temperatures. The anisotropy fields are measured on the polycrystalline CeFe11Ti and CeFe11TiN samples by using the method suggested by Durst and Kronmüller40. Applications of this model to ThMn12 systems have been reported recently for similar material systems41. Further details of the sample production and characterization can be found in the Supplementary Note 1.
Data availability
The data generated and analyzed during this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Tilmann Hickel, Takashi Miyake, and Halil Sözen for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft and French Agence Nationale de la Recherche in the framework of the international collaborative DFG-ANR project RE-MAP (Project No. 316912154), the European Research Council under its Consolidator Grant scheme (Project No. 617196), and IDRIS/GENCI Orsay under Project No. t2020091393. A.G. acknowledges support through Schrödinger fellowship J-4267 of the Austrian Science Fund (FWF). L.V.P. acknowledges the support by the future pioneering program MagHEM, grant number JPNP14015, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). We thank the computer team at CPHT for support.
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A.G. performed the electronic structure calculations, L.V.P. did the analytical derivations; S.E., F.M., I.D., and K.P.S. performed the experiments. L.V.P., S.B., and O.G. led the project. A.G., L.V.P., and S.E. drafted the manuscript. All authors discussed the results and contributed to the final version of the manuscript.
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Galler, A., Ener, S., Maccari, F. et al. Intrinsically weak magnetic anisotropy of cerium in potential hard-magnetic intermetallics. npj Quantum Mater. 6, 2 (2021). https://doi.org/10.1038/s41535-020-00301-6
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DOI: https://doi.org/10.1038/s41535-020-00301-6