Thermodynamics and Magnetism of YCo 5 Compound Doped with Fe and Ni: An Ab Initio Study

: YCo 5 permanent magnet exhibits high uniaxial magnetocrystalline anisotropy energy and has a high Curie temperature. These are good properties for a permanent magnet, but YCo 5 has a low energy product, which is notably insu ﬃ cient for a permanent magnet. In order to improve the energy product in YCo 5 , we suggest replacing cobalt with iron, which has a much bigger magnetic moment. With a combination of density-functional-theory calculations and thermodynamic CALculation of PHAse Diagrams (CALPHAD) modeling, we show that a new magnet, YFe 3 (Ni 1-x Co x ) 2 , is thermodynamically stable and exhibits an improved energy product without signiﬁcant detrimental e ﬀ ects on the magnetocrystalline anisotropy energy or the Curie temperature. 2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the SmFe 3 Ni 2 compound) to pure Co (the SmFe 3 Co 2 compound). These calculations show that the pseudo-binary SmFe 3 (Ni 1–x Co x ) 2 alloys could remain stable until almost half of Ni atoms are replaced by Co atoms. heat of formation of YCo 5 equal to ‘ − 12.46 kJ / mole and ‘ − 13.61 kJ / mole at 298 and 1000 K, correspondingly, compared with ‘ − 12.20 ± 0.87 kJ / mole measured between 850 and 1200 K in [85]; (iv) heat of formation of Y 2 Co 17 equal to ‘ − 8.74 kJ / mole and ‘ − 9.47 kJ / mole at 298 and 1000 K, correspondingly, compared with ‘ − 7.6 ± 0.80 kJ / mole measured between 850 and 1200 K in [8].


Introduction
Three principal material parameters define the intrinsic properties of hard magnetic materials: (i) spontaneous (saturation) magnetization, Ms, (ii) Curie temperature, T c , and (iii) magnetocrystalline anisotropy energy (MAE), characterized by its first-order anisotropy coefficient, K 1 [1]. These three parameters all need to be large for an economically effective permanent magnet, i.e., T c~≥ 550 K, M s~≥ 1 MA/m, K 1~≥ 4 MJ/m 3 . By incorporating transition-metal (TM) with rare-earth-metal (RE) atoms in numerous intermetallic compounds [1,2], one produces a material with these fascinating magnetic properties.
In our previous papers [3,4], we suggested a new economically effective permanent magnet, SmFe 3 CoNi, that was developed from the well-known SmCo 5 blueprint. More contemporary neodymium magnets of the Nd-Fe-B type are superior to the SmCo 5 magnet because of their greater maximum energy product emerging from their iron-rich stoichiometry. Our proposed SmFe 3 CoNi magnet, however, eliminated most of the detriment of the SmCo 5 magnet while maintaining its superior high-temperature effectiveness over neodymium magnets.
In regard to the SmCo 5 magnet, the MAE is dominated by the 3D electron contribution from Co, although the localized Sm 4f electrons are believed to influence MAE somewhat [3]. For YCo 5 , on the other hand, magnetic anisotropy is entirely due to the Co 3d electrons because Y has no electrons occupying 4f states [16,18,19,[23][24][25]28]. According to Refs. [16,18,23,25,28,70], the spin-polarized neutron scattering, inelastic spin flip-neuron scattering, nuclear magnetic resonance (NMR), neutron diffraction (ND), and x-ray magnetic circular dichroism (XMCD) measurements of YCo 5 revealed a very large orbital contribution to the total magnetic moment from Co 1 (2c) sites, indicating that the large axial MAE of the YCo 5 magnet arises from the Co 1 (2c) sites. The orbital moment on the Co 2 (3g) sites is smaller than the orbital moment on the Co 1 (2c) sites and contributes to the planar MAE of the YCo 5 magnet [23,25].
Several experimental studies of doping the RECo 5 compound with Fe or Ni [14,31] suggest that iron atoms prefer to occupy 3g sites because they are slightly larger than cobalt atoms and are more easily accommodated in these positions. On the other hand, there are vigorous manifestations that nickel atoms conversely occupy 2c sites. According to ref. [31], the compositional dependency of the lattice constant and the interatomic distances can be satisfactory justified by nickel atoms occupying 2c rather than 3g sites in the RECo 5 compound. Copper atoms choose to occupy 3g sites because the atomic volume of copper is bigger than that of cobalt atoms and even bigger than that of iron atoms [31].
From a price standpoint, it is profitable to replace Co atoms with Fe because of the abundance of Fe in the Earth's crust compared to Co (by a factor of~2000) making a Fe-containing magnet economically more attractive. Furthermore, Fe is a ferromagnetic metal with a very substantial magnetization at room temperature (1.76 MA/m [7]). The YCo 5 magnet doped with iron has been the subject of numerous experimental papers [13,18,20,34,35,41,42,52,72,73,80]. Replacing all cobalt with iron to enhance magnetization in the YFe 5 compound produces a thermodynamically unstable phase, which is not experimentally observed within the equilibrium Y-Fe phase diagram. Contrarily, the Y(Co 1-x Fe x ) 5 compounds (CaCu 5 -type) have been observed, with x ranging from 0.2 to 0.4 [42,52]. Moreover, the Curie temperature of the Y(Co 1-x Fe x ) 5 alloys has been found to increase from~930 K to~1020 K when varying across increasing x from 0.0 to 0.2 [42], contrary to the Y 2 (Co 1-x Fe x ) 17 alloys that display a monotonic reduction in the Curie temperature with rising Fe amount. The orbital moment is also bigger in cobalt compared to iron, and a decrease of the MAE is thus expected for x > 0. The lattice constant and magnetization also enhance with x from 0 to 0.4 [52] in the Y(Co 1-x Fe x ) 5 alloys.
In addition to iron and nickel, copper has also been used to dope the YCo 5 magnet [20,28,31,33,50,52,53]. In contrast to the Y(Co 1-x Fe x ) 5 system, Y(Co 1-x Ni x ) 5 and Y(Co 1-x Cu x ) 5 compounds are stable across the entire composition domain (i.e., x = 0-1). Substituting cobalt atoms with nickel or copper atoms gradually decreases magnetization and magnetic anisotropy. By connecting diffraction techniques and electron microscopy, Colin et al. [66] studied the solubility restriction of Ge in the Y(Co 1-x Ge x ) 5 alloys. A preferential substitution of Ge atoms that were larger than Co atoms on the 3g positions was observed.
When doping the YCo 5 magnet with copper, the total moment gradually decreases with increasing copper concentration, and finally the moment disappears for the Y(Co 1 Cu 4 ) compound. The calculations performed by Larson and Mazin [61] show that, for the Y(Co 1-x Cu x ) 5 alloys, the total magnetic moment decreases linearly with Cu doping, identical to the results of Ni doping of the Y(Co 1-x Ni x ) 5 alloys [51]. However, in contrast to the Y(Co 1-x Ni x ) 5 compounds, the magnetic moments of the cobalt atoms are reasonably local and barely impacted by Cu doping of the Y(Co 1-x Cu x ) 5 alloys until x~0.5, where the moment promptly falls to a small value before steadily decreasing [61]. The authors [61] concluded that cobalt in the doped YCo 5 compounds can occur either in a high-or a low-spin state, depending on the local surroundings. A similar conclusion (the high spin to the low spin transition) is presented in the paper of Wu et al. [63], where VASP-PAW formalism is applied to study the Y(Co 1-x Ag x ) 5 alloys.
A real-space pseudopotential formalism with SOC incorporated by norm-conserving pseudopotentials has been used to compute MAE of the YCo 5 magnet in different structures [78]. It appears that the authentic CaCu 5 -type structure (Space Group P6/mmm) possesses the largest total magnetic moment m (tot) = 7.13 µ B /f.u. and MAE K 1 = 2.68 MJ/m 3 . The latter is, however, significantly smaller than the experimental value of K 1 = 6.5 MJ/m 3 [6].
Nguyen et al. [79] studied the dependence of MAE of the YCo 5 magnet with intra-atomic Coulomb interaction (the Hubbard U parameter, DFT+U scheme) and strength of SOC within the PAW-VASP method. These calculations showed that, for GGA+U calculations, with 1.4 eV ≤ U ≤ 1.47 eV and J = 0.8 eV, MAE of the YCo 5 compound changes within the experimentally observed range. Similar results are presented in the work of Zhu et al. [70], where LDA+DMFT computational scheme is used to replace the OP correction [37]. The LDA+DMFT calculations suggest that electron correlation effects play an essential role in the formation and the enhancement of orbital moments and magnetic anisotropy in the YCo 5 magnet. ND, XRD measurements, and FPLMTO-DMFT technique have been used [77] to describe an isomorphic (without change of the symmetry) lattice collapse and the electronically topological transition (ETT), which is the first order Lifshitz phase transition in the YCo 5 magnet at pressure about 18 ± 2 GPa first reported in refs. [62,63] based on XRD measurements and FPLO calculations [66]. According to these calculations, the Curie temperature of the YCo 5 magnet, T c ≈ 987 K, is in the range of experimental measurements. These calculations reveal that, at ambient pressure, the orbital moments are 0.22 µ B and 0.18 µ B for 2c and 3g, correspondingly, and at pressure of 7.2 GPa, the orbital moments are 0.18 µ B and 0.15 µ B for 2c and 3g, correspondingly, which is in reasonable agreement with the experimental observation [28] (at ambient pressure) and also in agreement with results of LDA-DMFT calculations [70]. The atom-resolved Co 1 (2c) and Co 2 (3g) orbital moments reach the maximum values for the Coulomb parameter U~2 eV. These results for the orbital moments are consistent with previous XMCD measurements [71], inelastic spin flip-neuron scattering experiments [16], and OP calculations [37,55,56].
The temperature dependence of MAE and the spontaneous magnetization of the YCo 5 magnet has been calculated by utilizing the relativistic KKRASA-CPA-DLM approach [71,[74][75][76]81,82]. The calculated total magnetic moment, m (tot) , is 8.50 µ B /f.u. 8.03 µ B /f.u. at 0 K and around T = 100 K, correspondingly, [71], which is in an fair accord with the experimental data, 8.33 µ B /f.u., at liquid-helium temperature [29], and 7.99 µ B /f.u. at room temperature [28]. Calculated MAE is equal to K 1~4 .63 MJ/m 3 at T ≈ 50 K [71], which is larger than reported in the previous calculations, K 1~2 .83 MJ/m 3 at T = 0 K [58-61] (see Figure 1, [71]), but smaller than the experimental value of K 1~7 .38 MJ/m 3 at T = 4.2 K [24], K 1~6 .40 MJ/m 3 at T = 77 K [15] as well as K 1~6 .03 MJ/m 3 [15], K 1~5 .50 MJ/m 3 [69], and K 1~5 .00 MJ/m 3 [17] at room temperature, T = 293 K. The calculated MAE decreases as temperature rises, for example, K 1~1 .1 MJ/m 3 at 600 K [71]. Calculated Curie temperatures T c~9 65 K [71] and T c~8 85 K 74] compare well with experimental measurements T c~9 20 K [28] and T c~9 87 K [15]. KKRASA-CPA-DLM formalism has been applied to study temperature dependence of magnetization and the Curie temperature of the YCo 5 magnet doped by Ni and Fe [74]. Both experiments and calculations show an increase or a decrease in magnetization with Fe or Ni replacement, correspondingly. The same KKRASA-CPA-DLM formalism has been applied to study the temperature dependence of magnetization and MAE for the Y(Co 1-x-y Fe x Cu y ) 5 magnets [81]. These calculations show that MAE of the YCo 5 magnet could be increased by adding relatively minor amounts of Fe and/or Cu, with the larger MAE field detected for the composition Y(Co 0.82 Fe 0.09 Cu 0.09 ) 5 with the condition of a preferred replacement of Fe and Cu at 3g and 2c, correspondingly. The recent improvement of KKRASA-CPA-DLM methodology allowed Patrick and Staunton [82] to reproduce the experimental MAE of the YCo 5 magnet, K 1~5 .0 MJ/m 3 [17], at room temperature. Recent overviews of the first-principles studies of the doped YCo 5 magnets are presented in refs. [83,84].  Earlier neutron-diffraction studies of the Th(Co1-xFex)5 alloys (also based on the CaCu5-type structure) [14] show that the larger Fe atoms preoccupies the 3g-type sites, whereas the smaller Co atoms choose to occupy the 2c-type sites. This occupational inclination has been affirmed by DFT calculations for YCo5 [67] and SmCo5 [68] compounds. In line with these calculations, the total energy for Fe at the 3g site (E3g) is lower that than for Fe at the 2c site (E2c) by 0.21 eV/f.u. and 0.10 eV/f.u. for YCo5 and SmCo5 magnets, correspondingly. As was mentioned in refs. [19,28], if the YCo5 magnet is doped with Fe and Ni, Fe atoms occupy preferentially 3g sites, while Ni atoms favor 2c sites. Figure 2a shows the heat of formation calculated within the EMTO-CPA technique of the pseudo-binary YFe3(Ni1-xCox)2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the YFe3Ni2 compound) to pure Co (the YFe3Co2 compound). The current calculations show that the YFe3(Ni1-xCox)2 alloys could remain stable until almost all Ni atoms are replaced by Co atoms.
For comparison, we also show ( Figure 2b) the heat of formation calculated within the EMTO-CPA formalism [3,4] of the pseudo-binary SmFe3(Ni1-xCox)2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the SmFe3Ni2 compound) to pure Co (the SmFe3Co2 compound). These calculations show that the pseudo-binary SmFe3(Ni1-xCox)2 alloys could remain stable until almost half of Ni atoms are replaced by Co atoms. In addition to the numerous first-principle studies of the YCo 5 magnet, the phase stability and the thermodynamics of the Y-Co system have been assessed experimentally [85][86][87] and computationally, using the semi-empirical CALPHAD (CALculation of PHAse Diagrams) method [88][89][90]. Experimentally, YCo 5 was reported to melt congruently at T m~1 623 K, possess a small homogeneity range at high temperature, and decompose below T~998 K [86]. In addition, the Y-Co-Ni ternary system evaluation, reported by Gupta [91], revealed a complete solubility from YCo 5 to YNi 5 at 1073 K and 1273 K. This ternary system was assessed using the CALPHAD method [92,93], and the solubility of the Y(Co,Ni) 5 phase has been reproduced [92]. Experimental characterization of the Y-Co-Fe system demonstrated maximum solubility of Fe in Y(Co,Fe) 5 of~20 at.% [94] and 5 at.% [95] at 1323 K and 1073 K, correspondingly. There are no reported CALPHAD assessments (parameters) for Y-Co-Fe and Y-Fe-Ni ternary systems (a maximum solubility of~30 at.% Fe in Y(Fe,Ni) 5 was reported by [95] at 873 K), and no available information exists regarding thermodynamic properties or phase stability of the Y(Co,Fe,Ni) 5 compound.
Therefore, the principal goal of the present research is to examine the stabilizing effects of adding nickel to the Y(Co 1-x Fe x ) 5 alloys. We employ ab initio calculations using two complementary techniques: (i) the fully relativistic exact muffin-tin orbital method (FREMTO) and (ii) the full-potential linear muffin-tin orbital method (FPLMTO). All methods account for all relativistic effects, the so-called spin-orbit coupling (SOC). The implementation of these particular methods ensures that our results are rigorous and independent of technical implementations while employing the most advantageous attributes and durability strengths of each method. Related details of the ab initio [96,97] and the CALPHAD computational methods are outlined in Section 2. Results of the density-functional-theory (DFT) calculations of the ground state properties of the Y(Co-Fe-Ni) 5 alloys connected with CALPHAD calculations are outlined in Section 3. Next, we demonstrate results of the DFT calculations of the magnetic properties of the Y(Co-Fe-Ni) 5 alloys in Section 4. Lastly, discussion and concluding remarks are outlined in Section 5.

Computational Methods
The calculations we denote as EMTO are performed while using the Green's function formalism based on the improved screened Korringa-Kohn-Rostoker technique, where the one-electron potential is defined by the optimized overlapping muffin-tin (OOMT) potential spheres [98,99]. Inside the potential spheres, the potential is spherically symmetric, while it is permanent between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the permanent value in the interstitial region are resolved by minimizing (i) the deviation between the exact and overlapping potentials and (ii) the errors that are caused by the overlap between the spheres. Within the EMTO formalism, the one-electron states are calculated exactly for the OOMT potentials. As a product of the EMTO calculation, one can determine the self-consistent Green's function of the system and the complete, non-spherically symmetric charge density. Lastly, the total energy is calculated while using the full charge-density technique [100]. We consider, as the valence states, 6s, 5p, 5d, and 4f states for Sm and 4s and 3d states for Co and Fe. The corresponding Kohn-Sham orbitals are expanded in terms of spdf exact muffin-tin orbitals, i.e., we select an orbital momentum cutoff l max = 3. The EMTO orbitals, consequently, consist of the spdf partial waves (solutions of the radial Schrödinger equation for the spherical OOMT potential wells) and the spdf screened spherical waves (solutions of the Helmholtz equation for the OOMT muffin-tin zero potential). The completeness of the muffin-tin basis was discussed in detail in ref. [99]. The generalized gradient approximation (GGA-PBE) [101] is selected for the electron exchange and the correlation energy functional. Integration over the Brillouin zone is performed using the special k-point technique [102] with 784 k-points in the irreducible wedge of the zone (IBZ). The moments of the density of states, needed for the kinetic energy and valence charge density, are computed by integrating the Green's function over a complex energy contour with 2.2 Ry diameter while using a Gaussian integration technique with 30 points on a semi-circle enclosing the occupied states. In the case of the implementation of the FR-EMTO formalism, SOC is included through the four-component Dirac equation [103].
In order to treat compositional disorder, the EMTO method is coupled with the coherent potential approximation (CPA) [104,105]. The ground-state properties of the Y(Co 1-x-y Fe x Ni y ) 5 alloys are obtained from EMTO-CPA calculations that include the Coulomb screening potential and energy [106][107][108]. The paramagnetic state of the Y(Co 1-x-y FexNi y ) 5 is modeled within the disordered local moment (DLM) approximation [109,110]. The equilibrium atomic density of the alloy is obtained from a Murnaghan fit to the total energy versus atomic volume curve [111].
The highest level of theory (least approximations) is implemented in a full-potential scheme, i.e., no structural approximations, which includes spin-orbit interaction as explained [112] for early and late lanthanides [96,113]. The full-potential linear muffin-tin orbital (FPLMTO) accomplishes this in ways that are detailed [114]. For Y, we use two energy parameters coupled with each basis function, and these parameters have different values for pseudocore states (4s and 4p) and valence states (5s, 5p, 4d, and 4f ). Test calculations including the 3d states as semi-core states for Y do not significantly change any result. For the TM atom, we use a similar set-up. The spin-orbit interaction and the orbital polarization (OP) operate on the d and f orbital. OP is introduced in FPLMTO in a self-consistent fashion so there are no fixed parameters. Because of the way it is constructed, the orbital polarization often enhances spin-orbit coupling, leading to better orbital moments [114]. Both the generalized gradient and the local-density approximations (GGA-PBE and LDA) [101,115] are applied.
For the computation of the magnetic anisotropy energy, i.e., the difference in total energy between spins pointing in plane (100) or perpendicular to the plane along the z axis (001), one finds small differences. To resolve these small numbers, one has to be very careful calculating the total energy for the two cases as consistent as possible. Here, we decide to remove all crystal symmetry, thus, from a symmetry set-up standpoint, the calculations are identical. This approach is less efficient but in practice more accurate because errors associated with different k points, Brillouin zones, and other numerical approximations are minimized. Brillouin zone integration over 10,000 k points is carried out using Fermi-Dirac broadening corresponding to room temperature.
The CALPHAD method permits the prediction of phase stability and thermodynamics of multi-component systems based on the extrapolation of the phase diagram data and the thermodynamic properties of its constitutive binary and ternary systems [116][117][118]. The CALPHAD formalism consists of modeling Gibbs energy functions for each phase of the system (potentially including all elements) to reproduce the available thermodynamic (e.g., heat of formation) and phase diagram (e.g., melting point) data. Once the parameters of these functions are assessed for the constitutive systems, extrapolation (or prediction) can be made for multi-component systems. Here, we focus on heat of formation (thermodynamic property), phase diagram data (e.g., melting point, solubility limits), and CALPHAD assessment of the YTM 5 compound with TM = Co, Fe, and Ni as a first step toward the establishment of a complete Y-Co-Fe-Ni database. Model parameters optimization and CALPHAD calculations are performed using the Thermo-Calc software package (2019b version) in combination with its PARROT module [119,120].

Thermodynamics Properties of the Y(Co-Fe-Ni) 5 Alloys
The YCo 5 compound crystallizes in the hexagonal CaCu 5 -type structure with three non-equivalent atomic sites: Y1-(1a), Co1-(2c), and Co2-(3g) (see Figure 1) with six atoms per formula unit and per computational cell. Table 1 shows the equilibrium formula unit (f.u.) volume, bulk modulus, and the pressure derivative of the bulk modulus for the YCo 5 compound calculated using each of the ab initio techniques. The results of the EMTO calculations are presented both without ("SREMTO", "SR" means scalar-relativistic) and with ("FREMTO") inclusion of SOC. The experimental value of the equilibrium f.u. volume is taken from ref. [29,55]-see further discussion below. Notice that the equilibrium volume, calculated by FPLMTO method, agrees very well with the experimental data and that incorporating the relativistic SOC in the calculation has only a minor effect. Earlier neutron-diffraction studies of the Th(Co 1-x Fe x ) 5 alloys (also based on the CaCu 5 -type structure) [14] show that the larger Fe atoms preoccupies the 3g-type sites, whereas the smaller Co atoms choose to occupy the 2c-type sites. This occupational inclination has been affirmed by DFT calculations for YCo 5 [67] and SmCo 5 [68] compounds. In line with these calculations, the total energy for Fe at the 3g site (E 3g ) is lower that than for Fe at the 2c site (E 2c ) by 0.21 eV/f.u. and 0.10 eV/f.u. for YCo 5 and SmCo 5 magnets, correspondingly. As was mentioned in refs. [19,28], if the YCo 5 magnet is doped with Fe and Ni, Fe atoms occupy preferentially 3g sites, while Ni atoms favor 2c sites. Figure 2a shows the heat of formation calculated within the EMTO-CPA technique of the pseudo-binary YFe 3 (Ni 1-x Co x ) 2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the YFe 3 Ni 2 compound) to pure Co (the YFe 3 Co 2 compound). The current calculations show that the YFe 3 (Ni 1-x Co x ) 2 alloys could remain stable until almost all Ni atoms are replaced by Co atoms.
For comparison, we also show ( Figure 2b) the heat of formation calculated within the EMTO-CPA formalism [3,4] of the pseudo-binary SmFe 3 (Ni 1-x Co x ) 2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the SmFe 3 Ni 2 compound) to pure Co (the SmFe 3 Co 2 compound). These calculations show that the pseudo-binary SmFe 3 (Ni 1-x Co x ) 2 alloys could remain stable until almost half of Ni atoms are replaced by Co atoms. In order to understand the difference between Figure 2a,b, one needs to take into consideration several of the following arguments. Nickel metal forms the stable CaCu5-type compounds with both yttrium and samarium metals. Calculated within EMTO formalism, the heat of formation of SmNi5 and YNi5 compounds (in the CaCu5-type structure) is −18.95 mRy/atom and −22.91 mRy/atom, correspondingly, which is in accord with the experimental measurements of −23.08 mRy/atom (−30.3 kJ/mole, SmNi5, [32]) and −25.98 mRy/ atom (−34.1 kJ/mole, YNi5, [32]). As we mentioned in the Introduction, the YFe5 compound as well as the SmFe5 compound (see refs. [3,4]) do not exist in the equilibrium Y-Fe and Sm-Fe phase diagrams, correspondingly, thus no experimental information about the heat of formation of these hypothetical compounds is available. However, the EMTO calculations show that that the heat of formation of the YFe5 compound is positive, +6.46 mRy/atom, and is half of the calculated heat of formation of the SmFe5 compound, + 12.68 mRy/atom. As a result, the calculated heat of formation of the YFe3Co2 compound, +0.09 mRy/atom, appears to be smaller than the calculated heat of formation of the SmFe3Co2 compound, +2.15 mRy/atom [4]. The computed heats of formation of the YFe3Ni2 and the SmFe3Ni2 compounds are both negative (stable compounds), however, the absolute value of the calculated heat of formation of the YFe3Ni2 compound,│−6.20│mRy/atom, is more than twice as high as the absolute value of the heat of formation of the SmFe3Ni2 compound,│−2.72│mRy/atom [4]. As a result, the region of stability of the pseudo-binary YFe3(Ni1-xCox)2 alloys appears to be almost twice as wide as the region of stability of the pseudo-binary SmFe3(Ni1-xCox)2 alloys.
The promising phase stability of the YFe3CoNi compound highlighted by the EMTO-predicted heat of formation of the pseudo-binary YFe3(Ni1-xCox)2 alloys in Figure 2a sparks interest in studying the thermodynamics and the stability of the Y(Co,Fe,Ni)5 compounds versus temperature in the broader Y-Co-Fe-Ni composition space. As mentioned in Section 2, a full CALPHAD assessment of the Co-Fe-Ni-Y database requires evaluation of the constitutive binary (i.e., Co-Fe, Co-Ni, Co-Y, Fe-Ni, Fe-Y, Ni-Y) and ternary (i.e., Co-Fe-Ni, Y-Co-Fe, Y-Co-Ni, Y-Fe-Ni) systems. This study is focused on the Y-containing binary systems to leverage and confirm the ab initio findings and serve as foundation for an upcoming CALPHAD-centric study.
The first Y-Co CALPHAD assessment, in which all intermetallic compounds were evaluated as stoichiometric compounds, was reported by Du et al. [88]. This assessment was revisited in [89] to consider solubility range in YCo5 and Y2Co17. As both structures of Y2Co17 (Th2Ni17-type structure at high temperature and Th2Zn17-type structure at low temperature) are similar to the hexagonal CaCu5type structure (YCo5), Du and Lü [89] treated both Y2Co17 and YCo5 compounds as one phase with the formula (Y,Co2)1(Y,Co2)2Co15. The reasoning behind the use of a one-phase three-sublattice model is that homogeneity regions of Y2Co17 and YCo5 could be described by a CaCu5-type lattice in which In order to understand the difference between Figure 2a,b, one needs to take into consideration several of the following arguments. Nickel metal forms the stable CaCu 5 -type compounds with both yttrium and samarium metals. Calculated within EMTO formalism, the heat of formation of SmNi 5 and YNi 5 compounds (in the CaCu 5 -type structure) is −18.95 mRy/atom and −22.91 mRy/atom, correspondingly, which is in accord with the experimental measurements of −23.08 mRy/atom (−30.3 kJ/mole, SmNi 5 , [32]) and −25.98 mRy/atom (−34.1 kJ/mole, YNi 5 , [32]). As we mentioned in the Introduction, the YFe 5 compound as well as the SmFe 5 compound (see refs. [3,4]) do not exist in the equilibrium Y-Fe and Sm-Fe phase diagrams, correspondingly, thus no experimental information about the heat of formation of these hypothetical compounds is available. However, the EMTO calculations show that that the heat of formation of the YFe 5 compound is positive, +6.46 mRy/atom, and is half of the calculated heat of formation of the SmFe 5 compound, +12.68 mRy/atom. As a result, the calculated heat of formation of the YFe 3 Co 2 compound, +0.09 mRy/atom, appears to be smaller than the calculated heat of formation of the SmFe 3 Co 2 compound, +2.15 mRy/atom [4]. The computed heats of formation of the YFe 3 Ni 2 and the SmFe 3 Ni 2 compounds are both negative (stable compounds), however, the absolute value of the calculated heat of formation of the YFe 3 Ni 2 compound, |−6.20| mRy/atom, is more than twice as high as the absolute value of the heat of formation of the SmFe 3 Ni 2 compound, |−2.72| mRy/atom [4]. As a result, the region of stability of the pseudo-binary YFe 3 (Ni 1-x Co x ) 2 alloys appears to be almost twice as wide as the region of stability of the pseudo-binary SmFe 3 (Ni 1-x Co x ) 2 alloys.
The promising phase stability of the YFe 3 CoNi compound highlighted by the EMTO-predicted heat of formation of the pseudo-binary YFe 3 (Ni 1-x Co x ) 2 alloys in Figure 2a sparks interest in studying the thermodynamics and the stability of the Y(Co,Fe,Ni) 5 compounds versus temperature in the broader Y-Co-Fe-Ni composition space. As mentioned in Section 2, a full CALPHAD assessment of the Co-Fe-Ni-Y database requires evaluation of the constitutive binary (i.e., Co-Fe, Co-Ni, Co-Y, Fe-Ni, Fe-Y, Ni-Y) and ternary (i.e., Co-Fe-Ni, Y-Co-Fe, Y-Co-Ni, Y-Fe-Ni) systems. This study is focused on the Y-containing binary systems to leverage and confirm the ab initio findings and serve as foundation for an upcoming CALPHAD-centric study.
The first Y-Co CALPHAD assessment, in which all intermetallic compounds were evaluated as stoichiometric compounds, was reported by Du et al. [88]. This assessment was revisited in [89] to consider solubility range in YCo 5 and Y 2 Co 17 . As both structures of Y 2 Co 17 (Th 2 Ni 17 -type structure at high temperature and Th 2 Zn 17 -type structure at low temperature) are similar to the hexagonal CaCu 5 -type structure (YCo 5 ), Du and Lü [89] treated both Y 2 Co 17 and YCo 5 compounds as one phase with the formula (Y,Co 2 ) 1 (Y,Co 2 ) 2 Co 15 . The reasoning behind the use of a one-phase three-sublattice model is that homogeneity regions of Y 2 Co 17 and YCo 5 could be described by a CaCu 5 -type lattice in which part of the Ca sites is occupied by pairs of Co atoms [89]. However, as mentioned by Golumbfskie and Liu [90], modeling the YCo 5 and the Y 2 Co 17 compounds separately allows for the extension of this binary system to be more readily implemented into a multicomponent databases. Thus, YCo 5 and Y 2 Co 17 were modeled in [90] using (Y,Co 2 ) 1 (Co) 4 (Co,Va) and (Y,Co 2 ) 1 (Y,Co 2 ) 2 (Co) 15 formulas, correspondingly. Note that vacancies (Va) were introduced, and first-principles calculations for the end members of the YCo 5 compound were used to parameterize the model. Finally, Golumbfskie and Liu [90] underlined that the Y 2 Co 17 phase should be produced via a congruent melting reaction (as experimentally accepted) rather than the result of the peritectic reaction (as produced by the assessment of [89]). The most recent assessment [90] was used as a starting point to re-assess the Y-Co phase diagram using simpler (Y) 2 (Co) 17 and (Y) 1 (Co) 5 two-sublattice models for Y 2 Co 17 and YCo 5 compounds, correspondingly, to ease the extension of the database to multicomponent systems while conserving the main features of the system.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 21 part of the Ca sites is occupied by pairs of Co atoms [89]. However, as mentioned by Golumbfskie and Liu [90], modeling the YCo5 and the Y2Co17 compounds separately allows for the extension of this binary system to be more readily implemented into a multicomponent databases. Thus, YCo5 and Y2Co17 were modeled in [90] using (Y,Co2)1(Co)4(Co,Va) and (Y,Co2)1(Y,Co2)2(Co)15 formulas, correspondingly. Note that vacancies (Va) were introduced, and first-principles calculations for the end members of the YCo5 compound were used to parameterize the model. Finally, Golumbfskie and Liu [90] underlined that the Y2Co17 phase should be produced via a congruent melting reaction (as experimentally accepted) rather than the result of the peritectic reaction (as produced by the assessment of [89]). The most recent assessment [90] was used as a starting point to re-assess the Y-Co phase diagram using simpler (Y)2(Co)17 and (Y)1(Co)5 two-sublattice models for Y2Co17 and YCo5 compounds, correspondingly, to ease the extension of the database to multicomponent systems while conserving the main features of the system. The re-assessed Y-Co phase diagram is presented in Figure 3 (with model parameters reported in Table 2) and exhibits good agreement with experiments: (i) congruent melting of Y2Co17 and YCo5 at 1630 K (exp.: 1630 K [86]) and 1615 K (exp.: 1623 K [86]), correspondingly; (ii) decomposition temperature of YCo5 at 914 K (exp.: ~998 K [86]); (iii) heat of formation of YCo5 equal to '−12.46 kJ/mole and '−13.61 kJ/mole at 298 and 1000 K, correspondingly, compared with '−12.20 ± 0.87 kJ/mole measured between 850 and 1200 K in [85]; (iv) heat of formation of Y2Co17 equal to '−8.74 kJ/mole and '−9.47 kJ/mole at 298 and 1000 K, correspondingly, compared with '−7.6 ± 0.80 kJ/mole measured between 850 and 1200 K in [8]. The first CALPHAD assessment of the Y-Fe system was reported by Du et al. [121] and later refined by Lü et al. [122] to better replicate the ternary phase equilibria in the Y-Fe-Ni system. However, as mentioned by Konar et al. [123], the Gibbs energy of compounds reported in [121] was distinct from the experimental data, and neither the thermodynamic parameters for liquid or solid phases nor the invariant reactions were given in the updated assessment [122], making it impossible to evaluate. The Y-Fe system was re-assessed by Kardellass et al. [124,125] considering homogeneity range in the Y6Fe23 and the YFe2 phases compared to the stoichiometric compounds previously considered. Then, Konar et al. [123] published a new thermodynamic assessment of the Y-Fe system. However, the use of the modified quasichemical model for the liquid phase renders this work incompatible with the substitutional model used in other assessments. Thus, the latest assessment by Saenko et al. [126], which is the most comprehensive and compatible one, is used in the present study. In addition to the four reported compounds (i.e., Y2Fe17, Y6Fe23, YFe3, YFe2), we introduce the metastable YFe5 compound based on the ab initio heat of formation (+6.46 mRy/atom = +8.48 kJ/mole). The first CALPHAD assessment of the Y-Fe system was reported by Du et al. [121] and later refined by Lü et al. [122] to better replicate the ternary phase equilibria in the Y-Fe-Ni system. However, as mentioned by Konar et al. [123], the Gibbs energy of compounds reported in [121] was distinct from the experimental data, and neither the thermodynamic parameters for liquid or solid phases nor the invariant reactions were given in the updated assessment [122], making it impossible to evaluate. The Y-Fe system was re-assessed by Kardellass et al. [124,125] considering homogeneity range in the Y 6 Fe 23 and the YFe 2 phases compared to the stoichiometric compounds previously considered. Then, Konar et al. [123] published a new thermodynamic assessment of the Y-Fe system. However, the use of the modified quasichemical model for the liquid phase renders this work incompatible with the substitutional model used in other assessments. Thus, the latest assessment by Saenko et al. [126], which is the most comprehensive and compatible one, is used in the present study. In addition to the four reported compounds (i.e., Y 2 Fe 17 , Y 6 Fe 23 , YFe 3 , YFe 2 ), we introduce the metastable YFe 5 compound based on the ab initio heat of formation (+6.46 mRy/atom = +8.48 kJ/mole). The introduction of this end-member in the (Y) 1 (Co,Fe,Ni) 5 model (Table 2) does not change the Y-Fe phase diagram but permits us to calculate the heat of formation of YFe 3 (Ni 1-x Co x ) 2 alloys when combined with the Y-Ni system. Table 2.

Phase
Model Reference The Y-Ni system was assessed by Du and Zhang [127] considering all intermetallics as line compounds. This work was slightly modified by Mattern et al. [128] to reproduce Y-Nb-Ni data. Later on, Du and Lü [94] re-assessed the Y-Ni system to make it compatible with the three sublattice model used for the Y-Co assessment. This last work was also modified by Huang et al. [129] to fit the Y-Al-Ni system. Finally, Mezbahul-Islam and Medraj [130] assessed the Y-Ni system using the modified quasichemical model to consider the prevalence of short-range ordering in the liquid phase. In the present study, the initial assessment of Du and Zhang [127] is considered as it agrees with most experimental data [32,131,132]. In particular, the heat of formation of YNi 5 (−29.58 kJ/mole and −31.05 kJ/mole at 298 and 1000 K, correspondingly) [127] is in accord with the experimental data of −34.1 kJ/mole (298 K) [32] and −21.28 ± 0.42 kJ/mole (average between 887 and 1224 K) [131] and the calculated value of −30.09 kJ/mole (= −22.91 mRy/atom, present study).
The resulting CALPHAD model presented in Table 2 for the YTM 5 phase can now be used to calculate the heat of formation of the pseudo-binary YFe 3 (Ni 1-x Co x ) 2 alloys with reference to hcp-Y, hcp-Co, bcc-Fe, and fcc-Ni at 298 K. As observed in Figure 2a, the CALPHAD results at 298 K are in excellent agreement with the ab initio predicted values at 0 K. Considering that the CALPHAD database is a combination from various sources (i.e., Y-Co re-assessed from [92], Y-Fe from [126] with new ab inito input, and Y-Ni from [127]), this agreement highlights both the predictive capability of ab initio methods and the flexibility and reliability of the CALPHAD method. The CALPHAD work performed in the present study on the Y-containing binary systems reveals a negative heat of formation across almost the entire YFe 3 (Ni 1-x Co x ) 2 pseudo-binary, confirming the great potential of stabilizing a YFe 3 (Ni 1-x Co x ) 2 magnet. However, additional assessments are required to include the experimental data gathered on the ternary Y-Co-Ni [91,95], Y-Fe-Ni [95], and Y-Co-Fe [94,95] systems to consistently represent the Gibbs energy landscape of the entire Y-Co-Fe-Ni composition space and propose alloy compositions and thermal treatments to stabilize the Fe-rich Y(Co,Ni,Fe) 5 magnet. The goal of the current paper is to examine the stabilizing outcomes (enthalpy) of adding nickel to the Y(Co 1-x Fe x ) 5 alloys and its impact on magnetic properties (Section 4); this additional CALPHAD work will be presented in a follow up paper. Table 3 presents the results of the FREMTO calculations of the site-projected spin, m (s) , and orbital, m (o) , moments of the YCo 5 compound. According to [37], Y and Co spins should align in an antiparallel fashion (AF) that is predicted in the present self-consistent calculations. The calculated total moment, m (tot) = 7.82 µ B /f.u., is slightly smaller than the experimentally reported value of 8.30 µ B /f.u. [29]. The calculated spin moments are 1.55 µ B and 1.47 µ B for 2c and 3g sites, correspondingly, which are larger than the recorded experimental values of 1.44 µ B and 1.31 µ B [16,28]. The calculated orbital moments are 0.14 µ B and 0.11 µ B for 2c and 3g sites, correspondingly, which are smaller than recorded experimental data of 0.26 µ B and 0.24 µ B [16]. It is important to mention that the present FREMTO calculations reflect the experimental (spin flip-neuron scattering) observation; for the YCo 5 compound, the orbital moment of Co1(2c) atoms is bigger than the orbital moment of Co2(3g) atoms [16,19,23,25], which was confirmed by the previous calculations [37,56,71]. The large MAE of the YCo 5 compound comes from a big orbital allowance from Co 1 (2c) sites, which are located in the same plane as Y 1 (1a) sites [19,23,25]. Appropriately, Co 1 (2c) atoms have a big positive MAE allowance, while Co 2 (3g) atoms have a small negative MAE allowance. In line with ref. [71], the axial (positive) MAE of the YCo 5 magnet can be achieved only if orbital moments on the Co 1 (2c) atoms are bigger than the orbital moments of the Co 2 (3g) atoms. The same outcome was reached in a past study [3,4]; in the SmCo 5 magnet, the orbital moment on Co 1 (2c) atom is bigger than on Co 2 (3g) atom. Calculations [3] anticipated MAE of the SmCo 5 magnet would be K 1 = 19.63 MJ/m 3 , which is actually close to the experimental data K 1 = 17.2 MJ/m 3 [3]. A mean-field treatment for the Curie temperature, T c , can be formulated as [133,134]:

Magnetic Properties of the Y(Co-Fe-Ni) 5 Alloys
where E DLM tot − E AF tot is the difference among the ground state total energies of the DLM and the AF state, and k B is the Boltzmann constant. Principally, an assessment of the Curie temperature can be achieved from the total energy difference between the ferromagnetic (or antiferromagnetic) and the paramagnetic states. However, in line with {133], the difference between the total energies can be substituted by the difference between the effective single-particle (one atomic specie) energies, which are directly associated with AF and DLM states (the so-called mean-field treatment). In the present work, E DLM tot and E AF tot are calculated at the equilibrium volumes for DLM and AF states, correspondingly. According to the present EMTO-DLM and EMTO-AF calculations, T c = 891.8 K for the YCo 5 magnet, which is in good accord with the experimental data T c = 920 K [28], which is essentially bigger than that of the commonly used Nd 2 Fe 14 B magnet (T c = 588 K [7]). The similar EMTO calculations reveal T c = 1149.3 K for the YFe 5 compound, although, as was mentioned in the Introduction, this compound does not exist in the Y-Fe phase diagram. However, as was also mentioned in the Introduction, there is an experimentally observed [42] tendency of the Curie temperature to increase with Fe doping of the YCo 5 magnet, i.e., from T c = 930 K (the YCo 5 compound) to T c = 1020 K (the Y(Co 0.8 Fe 0.2 ) 5 compound). Our calculations qualitatively reflect this tendency. Figure 4 shows the Curie temperature calculated within the EMTO-CPA technique of the pseudo-binary YFe 3 (Ni 1-x Co x ) 2 alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the YFe 3 Ni 2 compound) to pure Co (the YFe 3 Co 2 compound). The dotted line corresponds to the calculated Curie temperature of the YCo 5 magnet, T c = 891.8 K. The calculated Curie temperature is equal to 572.6 K and 1097.7 K for the YFe 3 Ni 2 and YFe 3 Co 2 magnets, correspondingly. Although the calculated Curie temperature of the YFe 3 Ni 2 compound lies about 320 K below of the Curie temperature of the YCo 5 magnet, this deficiency can be removed by substituting 70 at.% of Ni by Co. The Curie temperature of the YFe 3 (Ni 0 . 3 Co 0 . 7 ) 2 magnet is equal to 899.9 K.  According to the present calculations, the YFe3(Ni0.3Co0.7)2 magnet shows an enormous total moment of m (tot) ~9.79 B, essentially due to the iron atoms that each contribute with 2.45B. The total moment of the YFe3(Ni0.3Co0.7)2 magnet is thus essentially bigger than that of the traditional YCo5 magnet that has a calculated total moment of m (tot) ~7.82 B. The experimental values of saturation magnetization (Ms) and the maximum energy product ((BH)max) for the YCo5 magnet are 0.85 MA/m and 224 kJ/m 3 , correspondingly [6,10,11]. Because saturation magnetization and magnetic moment are approximately proportional, Ms ~m (tot) , and the maximum energy product is approximately proportional to the square of the saturation magnetization, (BH)max ~(Ms) 2 [135], one can evaluate that saturation magnetization for the YFe3(Ni0.3Co0.7)2 magnet is proportional to: which is ~69% of the record maximum energy product of the Nd2Fe14B magnet, (BH)max = 512 kJ/m 3 [7]. Particularly, the YFe3(Ni0.3Co0.7)2 magnet, which has a Curie temperature similar to the YCo5 magnet, is a substantially steadier magnet than the YCo5 magnet (its maximum energy According to the present calculations, the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet shows an enormous total moment of m (tot)~9 .79 µ B , essentially due to the iron atoms that each contribute with 2.45 µ B . The total moment of the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet is thus essentially bigger than that of the traditional YCo 5 magnet that has a calculated total moment of m (tot)~7 .82 µ B . The experimental values of saturation magnetization (M s ) and the maximum energy product ((BH) max ) for the YCo 5 magnet are 0.85 MA/m and 224 kJ/m 3 , correspondingly [6,10,11]. Because saturation magnetization and magnetic moment are approximately proportional, M s~m (tot) , and the maximum energy product is approximately proportional to the square of the saturation magnetization, (BH) max~( M s ) 2 [135], one can evaluate that saturation magnetization for the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet is proportional to: which is~69% of the record maximum energy product of the Nd 2 Fe 14 B magnet, (BH) max = 512 kJ/m 3 [7]. Particularly, the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet, which has a Curie temperature similar to the YCo 5 magnet, is a substantially steadier magnet than the YCo 5 magnet (its maximum energy product should bẽ 57% larger). As has been mentioned in the Introduction, the magnetic anisotropy energy (MAE) is one of the more important properties of an efficient magnet. In the quest to increase the saturation magnetic moment or energy product, by substituting cobalt for iron, one has to review the impact of the doping on the MAE as well.
In Table 4 we collect calculated MAEs for the YCo 5 -type magnets alloyed with iron, nickel, or both. Here, for the YFe 3 Co 2 and the YFe 3 CoNi magnets, we keep the iron atoms on the energetically favorably 3g sites as discussed above. In the case of YFe 3 CoNi, we model Co and Ni on the 2c sites as two average atoms consistent with our modeling of SmFe 3 CoNi [3]. In all calculations, we relax the atomic volume and the c/a axial ratio of the hexagonal phase. We find some sensitivity (not shown) of the MAE to the axial ratios, suggesting that the structural relaxation is important. According to the values calculated in Table 4, we realize that both YFe 5 and YNi 5 have relatively small magnetic anisotropy and, for that reason alone, they are not particularly good magnets. YFe 5 is really only included in the table to provide context to the other magnets, as it does not exist in the hexagonal phase. YCo 5 contrarily exists and is predicted to have significant magnetic anisotropy. We primarily rely on DFT-GGA calculations for these magnetic compounds because GGA performs better for the magnetic 3d transition metals relative to the LDA or even more modern approximations [136]. It is also a well-known fact that GGA reproduces the proper magnetic ground state of iron, as opposed to the LDA. Our GGA calculations reproduce the experimental atomic volume very well but overestimate the MAE for YCo 5 relative to experimental data. DFT-GGA (T = 0 K) gives the unit cell volume V cell = 82.65 Å 3 and anisotropy K 1 = 9.89 meV/cell (19.2 MJ/m 3 ). These numbers shall be compared to experimental data at Т = 4.1 K, V cell = 82.50 Å 3 [29,55], K 1 = 3.80 meV/cell (7.38 MJ/m 3 ) [29,56] and at T = 293 K, V cell = 83.99 Å 3 [33], K 1 = 3.04 meV/cell (5.80 MJ/m 3 ) [29,56]. Here, the unit cell volume at T = 4.1 K, V cell = 82.50 Å 3 is identified using the experimental value of the MAE coefficient, K 1 , presented in the units of (MJ/m 3 ), [29,56], and (meV/cell) [55]. Steinbeck et al. [56], on the other hand, calculated K 1 = 4.4 meV/cell using LDA and orbital polarization in better agreement with experimental data. However, their calculation was performed at the experimental cell volume and not the LDA volume, which is drastically smaller. Their agreement with experiment is thus somewhat coincidental. We believe the difference in the calculated K 1 numbers is mostly due to electron exchange and correlation approximation. In Table 4, we also show our LDA results, and for YCo 5 (6.44 meV/cell), it is closer to experimental data (3.8 meV/cell) [29,55] than our GGA result.
Unfortunately, we realize that the MAE for these magnetic systems can be very sensitive to the particular DFT approximation (GGA or LDA) that is applied. Furthermore, there does not appear to be a simple systematic explanation for the difference in the MAE when comparing GGA and LDA methods. The results are very consistent for YFe 5 , YNi 5 (small MAE), and YFe 3 CoNi (intermediate MAE~4.5-5 meV/cell), while for YCo 5 , the GGA produces a value significantly greater than the LDA (9.89 and 6.44 meV/cell, correspondingly). For YFe 3 Co 2 and YFe 3 Ni 2 , the LDA values are about three and two times larger than the GGA number, correspondingly. Apparently, the particular electron exchange and the correlation approximation are important for the anisotropy, and to explore this further, we applied several other popular formulations for YFe 3 Co 2 ; the resulting MAE varied very sensitively. This remarkable sensitivity has not been recognized before for these types of magnets and certainly warrants further studies.

Discussion and Conclusions
According to our past calculations [3,4], replacing most of Co with Fe in the SmCo 5 magnet and using Ni as a thermodynamic mediator results in an SmFe 3 CoNi magnet that has extraordinary magnetic properties, such as a very high Curie temperature, robust magnetic anisotropy about twice that of the Nd 2 Fe 14 B magnet, and a big maximum energy product (an estimated 70.5% of the record maximum energy product of the Nd 2 Fe 14 B magnet), which should be about 56% larger than that of the SmCo 5 precursor. However, even if we substitute a relatively expensive metal Co (283 RMB/kg) by cheaper Ni (99 RMB/kg) and Fe (3 RMB/kg), the cost of the suggested SmFe 3 CoNi magnet is still mostly defined by the relatively expensive RE samarium metal (450 RMB/kg). Thus, in this paper, we expand on the fundamental idea of refs. [3,4] by using nickel metal as the stabilizing material in the YCo 5 magnet (Y is almost half the price of Sm [5]) in order to accommodate the maximum amount of iron metal to favor a very high magnetization.
We find a significant difference between the regions of stability of the YFe 3 (Ni 1-x Co x ) 2 and SmFe 3 (Ni 1-x Co x ) 2 alloys. In order to stabilize the SmFe 3 (Ni 1-x Co x ) 2 alloys, one should have at least half of the 2c-type sites to be occupied by Ni atoms. The extreme case is the SmFe 3 CoNi magnet, where Fe atoms occupy all 3g-type sites, and Ni and Co atoms occupy the 2c-type sites with equal probability, which has been suggested in refs [3,4]. For the YFe 3 (Ni 1-x Co x ) 2 alloys, it is possible to have the stable solutions until approximately all Ni atoms are substituted by Co atoms. Our ab initio heat of formation predictions are confirmed by CALPHAD modeling at 298 K (Figure 2a). The combination of negative heat of formation and extended solubility limits experimentally observed in the YTM 5 (TM = Co, Fe, Ni) magnets (i.e., complete solubility from YCo 5 to YNi 5 at 1073 K and 1273 K; solubility of~20 at.% Fe in Y(Co,Fe) 5 at 1323 K [94]; solubility of~30 at.% Fe in Y(Fe,Ni )5 at 873 K [95]) is promising for experimentalists who would like to synthesize these new magnets. Based on a thorough literature review, theoretical calculations, and preliminary CALPHAD works presented in this study, an in-depth CALPHAD assessment of the Y-Co-Fe-Ni system has been initiated and will provide guidance on how to make these new magnets. Meanwhile, a specific example from this work is the YFe 3 Co 2 magnet for which we calculated the Curie temperature, Tc, which is equal to 1097.7 K and the maximum energy product, (BH) max (YFe 3 Co 2 ), as~365 kJ/m 3 , which is~71% of the record maximum energy product of the Nd 2 Fe 14 B magnet, (BH) max = 512 kJ/m 3 . Here, the maximum energy products of YFe 3 Co 2 and YFe 3 CoNi magnets are estimated using the calculated total magnetic moments of YFe 3 Co 2 , YFe 3 CoNi, and YCo 5 magnets as well as the experimental values of the saturated magnetization and the maximum energy product of the YCo 5 magnet. Calculations are performed in the same fashion as for the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet (see Section 4).
According to our calculations, the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet has a Curie temperature T c~9 00 K that is relatively close to the calculated Curie temperature of the YCo 5 magnet, T c ≈ 892 K. In addition, the maximum energy product of the YFe 3 (Ni 0.3 Co 0.7 ) 2 magnet is significantly improved compared to the YCo 5 magnet (~57% larger).
The calculated intrinsic properties of the magnets suggested in ref. [3,4] and in this paper are reported in Table 5 in conjunction with the experimental data of Nd 2 Fe 14 B, SmCo 5 , and YCo 5 magnets for comparison. All four suggested permanent magnets have a Curie temperature significantly higher than that of the Neomax (Nd 2 Fe 14 B), T c ≈ 588 K, spanning from 785 K to 1103 K. In addition, their maximum energy products are significantly higher than that of the commercially used SmCo 5 and YCo 5 magnets (231 kJ/m 3 and 224 kJ/m 3 , correspondingly), reaching a maximum value of 365 kJ/m 3 for YFe 3 Co 2 . Our calculated (Table 4, LDA) MAEs for YFe 3 CoNi magnet is not much smaller than that of YCo 5 magnet (10.6 MJ/m 3 and 13.5 MJ/m 3 , correspondingly). Although the cobalt price plunged from more than 630 RMB/kg (May 2018) to 282 RMB/kg (June 2020) during the last two years, it is unlikely it will reach its record low value of approximately 157 RMB/kg (June 2016) as the global electric car market (production) continues to grow at an accelerated pace. Indeed, cobalt remains a necessary component in electric vehicle lithium-ion batteries, e.g., LiCoO 2 , and cobalt costs will probably rise as a result of anticipated demands for lithium-ion batteries, which consume more than 70% of total cobalt mining. Considering SmFe 3 CoNi and YFe 3 CoNi magnets comprise 80% less Co than their SmCo 5 and YCo 5 precursors, maturing of these magnets becomes even more captivating from the current economic viewpoint.
In conclusion, we showed that replacing part of cobalt with iron in SmCo 5 and YCo 5 magnets stabilized with a small portion of nickel results in novel permanent magnets that we consider to be conceivably synthesized. They are anticipated to have outstanding magnetic properties, a big maximum energy product, a strong magnetic anisotropy, and an exceptionally high Curie temperature. However, it is very important to emphasize that, although these three intrinsic parameters are necessary to be large for an efficient permanent magnet, this is not a sufficient condition. It is also necessary to optimize the microstructure of the alloys in order to optimize the extrinsic magnetic properties.