Abstract
We report on theoretical investigations of intermetallic phases derived from the ThMn_{12}type crystal structure. Our computational highthroughput screening (HTS) approach is extended to an estimation of the anisotropy constant K_{1}, the anisotropy field H_{a} and the energy product (BH)_{max}. The calculation of K_{1} is fast since it is based on the crystal field parameters and avoids expensive totalenergy calculations with many kpoints. Thus the HTS approach allows a very efficient search for hardmagnetic materials for which the magnetization M and the coercive field H_{c} connected to H_{a} represent the key quantities. Besides for NdFe_{12}N which has the highest magnetization we report HTS results for several intermetallic phases based on Cerium which are interesting as alternative hardmagnetic phases because Cerium is a less ressourcecritical element than Neodymium.
Introduction
Due to the resource criticality of rare earth (RE) elements on the world market, especially Nd and Dy, combined with an increasing demand for high performance hard magnets for applications like wind turbines or electric vehicles, new research activities are going on worldwide in the recent years. The goal is to find new hardmagnetic compounds with comparable performance to Nd_{2}Fe_{14}B but with significantly fewer Nd and Dy^{1,2}. The use of more abundant Ce instead of Nd and a significant reduction of the rare earth (RE) content relative to the transition metal (TM) content with respect to the classical Nd_{2}Fe_{14}B would be important steps forward in terms of cost reduction and supply reliabilty.
Recently intermetallic phases with the ThMn_{12} crystal structure have attracted a renewed experimental and theoretical interest^{3,4,5,6,7,8,9} because of the favorable composition ratio RE:TM = 1:12 and the tetragonal crystal structure which is a necessary condition for uniaxial magnetocrystalline anisotropy. A further significant increase of the magnetocrystalline anisotropy energy (MAE) can be achieved by addition of light interstitial elements (IS) like nitrogen leading to a modification of the ThMn_{12} (or 1–12) structure, denoted as ThMn_{12}X (or 1–12X) structure in the following (see Fig. 1)^{6,7}.
The potential of phases based on the ThMn_{12}X structure can be evaluated by a figure of merit, the socalled energy product (BH)_{max} which is given for some important hardmagnetic materials in Table 1. An upper bound is estimated from the magnetization M by which implies the assumption that reasonably about 10% of the polycrystalline microstructure of a bulk magnet consists of nonmagnetic phases^{6}. For the thoroughly studied hardmagnetic materials SmCo_{5} and Nd_{2}Fe_{14}B this estimation is fullfilled quite well. Hence it indicates that NdFe_{12}N and related 1–12 and 1–12X materials are indeed potentially highly interesting.
However, determining the magnetization and hence is not sufficient for the evaluation of the potential of a compound being a good hard magnet since a strong uniaxial magnetic anisotropy field H_{a} is required as well. H_{a} is connected to the anisotropy constant K_{1} and the magnetization M by the formula H_{a} = 2 K_{1}/(μ_{0}M). A fast and approximate determination of K_{1} is implemented in our highthroughput screening (HTS) approach to determine whether a magnetic compound has an uniaxial magnetic anisotropy or not.
For this work 1280 phases derived from the ThMn_{12} structure and the ThMn_{12}X structure are investigated theoretically by a computational HTS based on an approximate but sufficiently fast and accurate density functional theory (DFT) approach^{10}. After comparing the calculated key quantities M, K_{1} and H_{a} for wellknown and important hardmagnetic materials like SmCo_{5}, Sm_{2}Co_{17}, Nd_{2}Fe_{14}B and others with experimental data we turn to the 1–12 and 1–12X structures, respectively decorating the Th site by Ce, Nd or Sm and the Mn sites by Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Al, Si or P. Furthermore, the ThMn_{12}X structure containing light atoms X = B, C or N at the (2b) Wyckoff positions was screened. Besides the discussion of trends numerous RETM intermetallic compounds based on the ThMn_{12} like CeFe_{11}TiX, (X = B, C, N), CeFe_{11}Co_{1}X (X = B, N) CeFe_{8}Co_{4}X, (X = B, C, N) or CeFe_{8}Ni_{4}N with high potential as lowcost hardmagnetic compounds are proposed.
Our paper is organized as follows: In section “Computational highthroughput screening” the computational HTS approach is summarized. The following section describes the determination of the anisotropy constant K_{1} which is important for the understanding of the systematic deviations between theoretical and experimental findings. In section “Validation of the HTS for known hardmagnetic compounds” the HTS approach is validated for well known hardmagnetic compounds. Possible fundamental reasons for the deviations between theory and experiment are discussed in section “Discussion of the discrepancy between calculated and experimental results”. In section “Screening results for 1–12 and 1–12X phases” the results of the screening are presented. A concise summary concludes the paper.
Theoretical Approach
Computational highthroughput screening
The applied HTS approach is based the previous work of Drebov et al.^{10}. It allows an automatized generation of new phases by substituting sets of atoms in crystal structures and then determining the formation energy, local magnetic moments and the total magnetization M. The total energies and spin polarized electronic states and densities from which all other quantities are derived are calculated with the tightbinding (TB) linearmuffintinorbital (LMTO) atomicsphere approximation (ASA) method of density functional theory (DFT) using the local spindensity approximation (LSDA)^{11,12,13,14,15}. Compared to other widely employed DFT methods, the TBLMTOASA method has a number of advantages for the search of new hardmagnetic materials: As an allelectron DFT method it is suitable for hardmagnetic intermetallic phases with complex crystal structures composed of transitionmetal and rareearth elements. It can treat the magnetism of localized “opencore” fstates interacting with delocalized valence dstates in a theoretically accurate and consistent manner. The minimal basis of shortranged TBLMTOs and the ASA for crystal potentials and electron densities together make this DFT method computationally fast and efficient at least for metallic phases with topologically closepacked crystal structures. Therefore the TBLMTOASA method is a proper DFT approach for combinatorial highthroughput screening of hardmagnetic materials. More details on this computational approach are given in the following sections.
The determination of the maximum energy product (BH)_{max}, the anisotropy constant K_{1} and anisotropy field H_{a} were added to the HTS analysis. BH_{max} is reasonably estimated by as explained in the introduction.
Further development of work of Fähnle and Hummler^{15} who implemented an evaluation at the crystal field parameters A_{nm} based on TBLMTOASA calculations allows the determination of the anisotropy constant K_{1} and the anisotropy field H_{a}. A short review of the single ion anisotropy approach is given in the following section.
As search criteria for hardmagnetic compounds being promising negative phase formation energies relative to the elemental sources and above 400 kJ/m^{3} are required. 400 kJ/m^{3} is below the already achieved 516 kJ/m^{3} of Nd_{2}Fe_{14}B. However, achieving such high energy products with about 50% less Nd or even without Nd using Ce would be technologically and economically very valuable. As third criterion we require that the theoretically predicted anisotropy field H_{a} should amount to values of at least several Tesla. All our proposed new phases have values above 10 Tesla. However, a quantitatively accurate and reliable prediction is difficult and the reader is referred to section “Discussion of the discrepancy between calculated and experimental results” for the discussion.
Details of the TBLMTOASA calculations
The determination of the basic DFT quantities, total energy and spin dependent electron density, was performed using the framework of the LSDA. The scalarrelativistic approximation of Koelling and Harmon^{29} and the exchangecorrelation functional of von Barth and Hedin^{30} in the parametrization of Moruzzi et al.^{31} were used.
The LSDA is valid only for weakly correlated systems which inhibits a correct treatment of the strongly localized 4f electrons. The problem is overcome according to Brooks et al.^{32} by imposing two constraints for the 4f charge and spin density entering the effective potential in LSDA. The constraint for the charge density fixes the number of electrons in the 4f core to the value of a free RE^{3+} ion. The constraint for the spin density fixes the magnetic spin moment of the 4f core to the value obtained from the standard RusselSaunders coupling scheme for the free ion. In this approach, hybridization of 4f orbitals with other orbitals is forbidden^{33} but the radial distributions of the 4f charge and spin densities are calculated selfconsistently under the above mentioned constraints, is called an “open core states” approach.
In the ASA the crystal volume is subdivided into atomic spheres such that the sum of volumes of the partially overlapping spheres is equal to the total crystal volume. For the ratio of the atomic sphere radii r we rely on the well tested choice of r(RE)/r(TM)/r(IS) = 1.35/1/0.7 of previous work^{34}.
Our TBLMTOASA calculations included s, p and d orbitals (in addition to the 4f “open core states” as wavefunctions) and the combinedcorrection (cc) term^{35}. For the kpoint sampling of the Brillouinzone integrals the linear tetrahedron method with MonkhorstPack meshes of 6 × 6 × 10 kpoints was used.
Crystal structure models
The ThMn_{12} structure has the space group #139 (I4/mmm). For the internal structural parameters the values determined experimentally by Isnard et al. (see Table 4) were taken^{36}. Additional interstitial atoms like B, C and N were inserted at the (2b) Wyckoff positions leading to the 1–12X structure. The tetragonal crystal unit cells of the 1–12 and 1–12X structure contain 26 atoms and 28 atoms, respectively. They are displayed in Fig. 1.
Tetragonal lattice constants a = 8.566 Å and c = 4.802 Å of SmFe_{11}Ti^{36} were taken for our screening of all the 1–12 and 1–12X phases.
For CeFe_{11}Ti Isnard et al. have found that Ti occupies about one quarter of the (8i) Wyckoff positions. We assume that the Ti atoms exclusively sit on these positions for all our screening of RETM_{11}Ti and related compounds.
Of course, there are variations of the lattice constants and internal parameters when different RE, TM and IS atoms are considered. However, the experimental results are somehow contradictive. On the one hand Yang et al.^{37,38} report a shortening of the lattice parameter c upon nitrogenation of REFe_{11}TiN_{x} with RE = Nd or Sm. On the other hand Liao et al.^{39} report a slight increase of about 1% and Akayama et al. have found that the c value increases even more^{21}.
Our test calculations showed that changes of the order of 1% in structural parameters had little influence on the formation energies and magnetic moments. Beuerle et al.^{40} showed that taking the experimental latticeconstant data generally leads to better results for the magnetic moments than taking optimized LDA values. The influence of variations in the structural parameters on the anisotropy constant K_{1} is much bigger than few percent as can be seen by comparing work of Miyake et al.^{7} and Harashima et al.^{8}. It is not clear whether experimental, LDA or GGArelaxed structure parameters lead to better results. We suppose that only strong changes in the lattice constants a and c due to interstitial atoms require a new relaxation of the internal structural parameters. Since for 1–12 and 1–12X no such strong changes occur keeping all structural parameters constant only leads to small quantitative errors.
Calculation of the magnetocrystalline anisotropy energy
For ferromagnetic crystals with uniaxial symmetry the magnetrocrystalline anisotropy energy is given in lowest order as:
K_{1} is the firstorder magnetrocrystalline anisotropy constant and has the energydensity unit J/m^{3}. θ is the angle of the magnetization vector relative to the crystal axis of highest symmetry. Negative K_{1} values imply an easy plane whereas positive K_{1} values indicate an easy axis. For good hardmagnetic materials large positive values of several MJ/m^{3} for K_{1} are required.
For RETM compounds the MAE is dominated by the the RE contribution^{15}. The crystal field formed by the surrounding atoms breaks the spherical symmetry for the RE atoms which implies a preferred arrangement for the RE charge. Since the spinorbit coupling scales with the 4^{th} power of the nuclear charge, Z^{4}, for heavy atoms like the RE element the coupling of charge and spin degrees of freedom is important. Thus the preferred arrangement of the charge is transferred to a preferred direction of the spin leading to easyaxis or easy plane aligment of the magnetization. Clear and concise reviews of this model are given in refs 15 and 16 and in more detail in ref. 17 which show how the interaction energy of the 4f electrons with the surrounding electron distribution can be expanded in spherical harmonics and that K_{1} is linear in lowest order approximation to the crystal field parameter A_{20}:
J is the total (orbital + spin) angular momentum and well defined for the 4f electrons. For all RE elements an ionic 3^{+} valence configuration for the 4f states is assumed. This is well fullfilled for most RE elements but leads to problems especially for the mixedvalence element Cerium as we will show when we discuss the numerical results. α_{J} is the so called Stevens factor^{18} which accounts for the shape of the charge distribution. The shapes of the 4f charge clouds are depicted for example in the book of Coey (chapter 4.4 singleion anisotropy)^{19}. is the expectation value of the squared radius at the RE site.
Due to the localized nature of the 4f electrons the quantities J, α_{J} and also are interatomic properties quasi independent of the crystal structure and close to the values for isolated ions. Connected to the crystal structure are n_{RE} which is the number of RE atoms per volume and A_{20} which contains the interaction of the RE charge density ρ_{4f}(r) with the remaining crystal charge density ρ(r):
where r_{<}(r ^{>}) denotes the smaller (larger) of the variables r and R. These integrals have to be cut off for numerical evaluation at some radius r_{c}. They were evaluated differently by others groups including point charge models for the large R contributions^{20}. In our work we follow the approach of Fähnle et al.^{15} and take the sphere radius of the ASA r_{ASA} as cutoff radius. For RE atoms r_{ASA} is typically about 1.8 Å. Harashima et al.^{8} have taken a smaller cutoff radius r_{c} = 1.005 Å. In order to evaluate which cutoff works better the theoretical results are compared below to the experimentally obtained values of K_{1} and H_{a} of some long known and well studied hardmagnetic materials (see Table 2).
Finally, since it is relevant for the interpretation of our results, we want to note that the REion anisotropy approach is only a firstorder perturbation treatment. When turning the charge cloud of the 4f electrons away from the easy axis the reaction on the surrounding charge density in the crystal is not taken into account. A change of the 4f charge cloud of course modifies as well the wavefunctions (and densities) of all other electrons. This is already of second order in perturbation theory and neglected in the presented approach. A thorough critical discussion of limitations of the REion anisotropy model has been given by Fähnle et al.^{15}.
Results and Discussion
Validation of the HTS for known hardmagnetic compounds
In Table 2 a summary of several well studied hardmagnetic materials is given by listing data for the magnetization M, the anisotropy constant K_{1} and the anisotropy field H_{a}.
Typically, the magnetization is well reproduced by the TBLMTOASA method. The experimental and theoretical values deviate only by few percent. Only for NdFe_{12}N the theoretical value clearly deviates from the experimental one. A possible reason may be that until now the NdFe_{12}N phase could only be stabilized experimentally in thin films up to 360 nm thickness and not yet as a bulk material^{6}.
The anisotropy represented by K_{1} or H_{a} is more difficult to calculate. Only numbers without digits after the decimal point are given since the approach cannot deliver data of higher significance. In Table 2 we list experimental values for K_{1} and H_{a} at room temperature. A comparison of the calculated values (T = 0 K) with experimental low temperature values (T ≈ 4 K) would of course be more straightforward. However, we have found less experimental data for low temperatures or extrapolated values from T = 70 K and above with an inherent uncertainty of the applied extrapolation schemes. Our intention is to identify robust general ratios between the experimental and theoretical results which allow an assessment of the screening results. Ratios of theoretical zerotemperature values to experimental roomtemperature values are even more desirable with respect to permanent magnets used in practical applications.
The model calculations predict correctly the signs of K_{1} and H_{a} for all considered cases. Sm_{2}Fe_{14}B has a strongly negative K_{1} and thus no easyaxis. Y_{2}Co_{17} has a vanishing K_{1} hence it has no electrons in partially filled shells. All other compounds have large positive K_{1} indicating their hardmagnetic behavior. The nonzero experimental value for Y_{2}Co_{17} is due to the TM contribution to the magnetocrystalline anisotropy which is not included in the model used. The major part of the anisotropy originates from the RE atoms as explained in preceding section. The example Y_{2}Co_{17} confirms that the TM contribution is rather small in RETM compounds^{15}.
Comparing the theoretical values of K_{1} and H_{a} obtained with a cutoff radius r_{c} = r_{ASA} reveals an overestimation by a factor 3 to 10 of the experimental values (except for case of Ce_{2}Fe_{14}B). One reason for the overestimation is that at room temperature experimental K_{1} and H_{a} values are in general lower by a factor 2 to 3 than at zero temperature. H_{a} decreases from 7.0 Tesla at 4.2 K to 2.3 Tesla at 300 K for CeFe_{11}Ti, e.g.^{21}. For Nd_{2}Fe_{14}B the anisotropy field H_{a} decreases from 17.0 Tesla at 4 K to 7.3 Tesla at 295 K^{22}. Calculating the ratio of the experimental low temperature values and the TBLMTOASA values then results in percentages of 60% to 70%. In the following section we will give arguments why such an overestimation has to occur and is not a failure of the theoretical approach.
The exceptional case of Ce_{2}Fe_{14}B for which the experimentally found K_{1} value is only 2.8% of the calculated value is a complicated case because the valence state of Cerium in RETM phases is not strictly 3^{+} as assumed in our model. Capehart et al.^{23} found a mixed valence state of 3.44^{+} for Ce_{2}Fe_{14}B which implies that the theoretical model used overestimates K_{1} for Cerium containing compounds even more. In the REion anisotropy model Cerium has an assumed ionic valence state of 3^{+}. A valence of 4^{+} for Cerium would lead to a vanishing anisotropy. Hence the valence state depends on the RETM phase considered. For CeFe_{11}Ti the / ratio is 13% and thus in the range seen also for other Nd and Smcontaining examples. We have not found any experimental or theoretical information on the valence state of Ce in 1–12 and 1–12X structures, respectively. Therefore the predicted K_{1} values based on the assumed valence are an upper bound for K_{1}.
The values obtained with r_{c} = 1.005 Å used in ref. 8 appear to be much better at a first glance since they are in general smaller and thus closer to the experimentally determined values. However, the relative hierarchy of the theoretical anisotropy values of several examples is not as well reproduced. SmCo_{5} has about the same K_{1} value as Sm_{2}Fe_{17}N_{3} and Nd_{2}Fe_{12}N although K_{1} of SmCo_{5} should be much higher.
Our working hypothesis is that a massive overestimation of the anisotropy does not hinder a screening approach as long as the relative hierarchy is conserved and a conversion factor for the prediction of experimentally achievable values can be estimated like in the case of (BH)_{max} (Table 1). For the very well investigated and optimized phases SmCo_{5}, Sm_{2}Fe_{17}N_{3} and Nd_{2}Fe_{14}B for example a factor of about 1/4 would bring close the experimental K_{1} since the ratio / is around 25%.
For Cerium the calculated K_{1} anisotropy coefficient for the 1–12 phase CeFe_{11}Ti overestimates the experimental one by about a factor of 10. For Ce_{2}Fe_{14}B the room temperature K_{1} is overestimated 35 times. The division of by 35 can be seen as a conservative estimate when estimating experimentally achievable values. The division of by 10 may be an optimistic estimate.
Discussion of the discrepancy between calculated and experimental results
There is a large discrepancy between the experimental coercive field H_{c} of real bulk magnetic materials and the theoretical anisotropy field determined by H_{a} = 2K_{1}/(μ_{0}M). This is known as Brown’s paradox^{24}. Its origin is the idealization of theory. Real materials are structurally, chemically and magnetically never perfectly homogeneous due to point defects and extended defects in the crystal structure and at least equally important, surface asperities which cause strong demagnetization fields^{19,25}. The reversal of magnetization is strongly connected to small nucleation or pinning centers in the material. Our anisotropy model described above is not containing any inhomogenities or surface effects. Furthermore, in theory it is assumed implicitly that magnetization is rotated in a coherent mode which means that magnetization remains uniform everywhere and rotates unison.
Real magnetic single crystals are never free of point defects, have surfaces or interfaces and are not in perfect singledomain configurations like assumed in the model calculations. This implies that experimentally determined K_{1} values for single crystals are very likely lower than the theoretically determined values. In experiments extremely high fields would be necessary to obtain completely saturated magnetic singledomain single crystals. For example in Fig. 1 of ref. 26 one can see that K_{1} differs with the applied external field. But even in an extremely high external field a magnet of one pure phase contains point defects, dislocations and little surface asperities which can counteract to the single domain state. Thus the critical threshold for rotating the magnetization is lower than for an ideal magnetic crystal.
We think that the main reason for the discrepancy is that the charge distribution of all electrons needs to be determined selfconsistently for the parallel and perpendicular arrangment. As mentioned before in the REion anisotropy approach only the firstorder perturbation is taken into account which is the change of the energy when rotating the charge cloud of the 4f electrons with respect to the surrounding charge cloud of all other electrons. The reaction on the charge density, i.e. a modification of the wavefunctions (second order perturbation) is neglected. Selfconsistent MAE calculations based on differences of total energies (for the magnetization parallel and perpendicular, respectively, to the easy axis of the crystal) take this reaction into account. This and the inclusion of the TM contribution to the MAE are clearly advantages of the totalenergy calculations. However, in order to obtain quantitatively accurate values for K_{1} and H_{a} dense kmeshes of several thousands of kpoints are needed for the Brillouinzone integrals^{27,28}. This make totalenergy calculations for HTS rather disadvantageous.
Screening results for 1–12 and 1–12X phases
In Table 3 a selection of the 1280 investigated phases are listed. The upper part contains a systematic collection of phases which allows to see trends when changing from 1–12 to 1–12X, from REFe_{12} to REFe_{11}Ti, or exchanging the interstitials B, C and N. The lower part of the table proposes some further promising compounds.
Trends
Since the TBLMTOASA approach is able to calculate the magnetization M accurately and reliably (see Table 2), we interpret discrepancies between theory and experiment as a hint to further optimization potential. NdFe_{12}N was successfully synthesized in thin films with a reported magnetization of 1.66 T^{6}. Our value of 2.06 T indicates an upper bound for a bulk material. From the magnetization M the energy products in the first column of Table 3 are derived according to .
Concerning the anisotropy we find that Ce and Nd, both having oblate 4f charge clouds, lead to easyaxis magnets for all named 1–12 and 1–12X phases whereas Sm with its prolate 4f charge cloud always leads to easyplane magnets. These results are partially at variance to the work of Harashima et al. that report a positive K_{1} for SmFe_{12}. Also they predict a negative K_{1} for NdFe_{12} which contradicts our prediction of an easy axis. Taking exactly their structural parameters and redoing the K_{1} calculation did not change the sign for our results but has led to values closer to zero. Probably one can say that these K_{1} values are very small anyway and should be considered to be approximately zero and differences in sign are thus uncritical.
A clear trend which was already found and discussed by Miyake et al.^{7} is the strong increase in K_{1} when going from 1–12 to 1–12X phases. At least for the doping with nitrogen this was already observed in experiments^{6,21}. According to our model calculations the increase can be about one order of magnitude irrespective of the interstitial element used. Doping with B always leads to smaller values than doping with C or N. As a general argument we assume that B changes least the charge distribution around the RE atoms and thus has the lowest impact on the RE anisotropy.
Going from REFe_{12}X phases to REFe_{11}TiX phases clearly reduces the magnetization and increases the anisotropy. The above mentioned results for NdFe_{12}N and NdFe_{11}TiN are apparently in good agreement with experimental findings^{6}. Already going from REFe_{12} to REFe_{11}Ti reduces the magnetization and increases the anisotropy. But REFe_{12} compounds are experimentally unstable^{6}, as reflected in high formation energies in the calculations.
For NdFe_{11}TiX (X = B, C, N) our results deviate from the theoretical findings of Harashima et al.^{9}. Besides the chemical change they take the structural change by individual relaxations of the respective structures into account. Experiments will show whether such big differences in the anisotropy between the NdFe_{11}TiX phases (X = B, C, N) exist or whether the change is rather comparable, similar as the change seen when going from Nd_{2}Fe_{14}B to Nd_{2}Fe_{14}C (see Table 2).
Substituting Ti on the 8i positions by V, Cr, Cu, Zn, Al, Si, or P has not lead to promising compounds. In general this formation energies are increased making their existence less likely and the magnetizations and anisotropies are lower than for Ti containing compounds. Therefore none of these screened compounds entered Table 3.
Promising Compounds
From the upper part of Table 3 NdFe_{12}X, NdFe_{11}TiX, CeFe_{12}X and CeFe_{11}TiX (X = B, C, N) provide good key data. Nd containing compounds achieve higher magnetizations relative to Ce containing compounds due to the local magnetic moments of Nd atoms. Also we expect the anisotropy for Nd to be higher than for Ce. A division of the theoretical and values of Nd compounds by about 4 should indicate an upper bound for the experimentally achievable values (see discussion of Table 2). For Ce_{2}Fe_{14}B only about 3% of the values determined by TBLMTOASA is achieved, so a division by 35 of the and values Cerium compounds is a conservative estimate for the potential (cf. discussion in Section 2). However, CeFe_{11}Ti achieved more than 10% of the idealized and values. Thus in case that the mixed valence of Cerium in 1–12 and 1–12X phases is less dominant than in 2141 phases the model assumptions work better and the overestimation by theory is lower.
The substitution of Mn on Fe positions in the 1–12 and 1–12X structures also did not give any promising compounds since the magnetization decreased in most cases significantly (cf. ref. 10 for 1142 phases). However, the partial substitution of Co or Ni for Fe leads to quaternary phases like CeFe_{11}Co_{1}X (X = B, N), CeFe_{8}Co_{4}X (X = B, C, N) or CeFe_{8}Ni_{4}N with rather high hardmagnetic potentials. They are expected to have anisotropy fields of about 20 Tesla and higher. Although, this is only half of the anisotropy field of SmCo_{5}, but in combination with much higher magnetizations much higher energy products should be achievable. Furthermore, the expensive Sm would be replaced by the uncritical Ce and the Co may lead to rather high Curie temperatures.
Summary
We have studied the magnetic properties of 1–12 and 1–12X phase on the search for promising hardmagnetic phases by HTS calculations based on a fast TBLMTOASA approach which predicts the magnetization in good accuracy. This allows an estimation of the maximum energy product (BH)_{max} which is an important figure of merit for permanent magnets. We determined the anisotropy for the single domain state of the perfect single crystal. Although our values overestimate the experimentally obtained values of wellknown magnets it was shown that at least a trend of the anisotropy constant K_{1} and the anisotropy field H_{a} can be reproduced. Thus by scaling down the theoretical result by a factor 4 for Nd and Sm and a factor 10 to 35 for Cerium leads to estimates for upper bounds of K_{1} and H_{a} that are achievable in experiments. Our screening of 1280 phases has lead to several promising phases like NdFe_{12}X or NdFe_{11}TiX (X = B, C, N) with energy products (BH)_{max} up to 600 kJ/m^{3} and anisotropy fields up to 10 Tesla. Ce containing compounds like CeFe_{11}TiX, (X = B, C, N), CeFe_{11}Co_{1}X (X = B, N) CeFe_{8}Co_{4}X (X = B, C, N), or CeFe_{8}Ni_{4}N have lower energy products but have the advantage of being less resourcecritical due to the avoidance of Nd.
We recommend the above proposed RETMIS compounds as worth and promising to be investigated further experimentally.
Additional Information
How to cite this article: Körner, W. et al. Theoretical screening of intermetallic ThMn_{12}type phases for new hardmagnetic compounds with low rare earth content. Sci. Rep. 6, 24686; doi: 10.1038/srep24686 (2016).
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Acknowledgements
Financial support for this work was provided by the Fraunhofer lighthouse project Criticality of Rare Earths and by the cooperative project Neue Supermagnete  Exzellent in Leistung und REEffizienz (NEXT) funded by the Ministry of Finance and Economics (MFW) of the German Federal State of BadenWürttemberg. We thank Professor Gutfleisch and his coworkers (TU Darmstadt and Fraunhofer IWKS) and Professor Goll (HS Aalen)I for valuable discussions.
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This work was planned, the results were discussed and this report was completed by all three authors together. The calculations and analysis was mainly done W.K. and G.K. The main text was formulated by W.K.
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Körner, W., Krugel, G. & Elsässer, C. Theoretical screening of intermetallic ThMn_{12}type phases for new hardmagnetic compounds with low rare earth content. Sci Rep 6, 24686 (2016). https://doi.org/10.1038/srep24686
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DOI: https://doi.org/10.1038/srep24686
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