Structures of Sm–Cu intermetallics with Fe as subphase candidates in SmFe12-based permanent magnets studied by first-principles thermodynamics

Since crystalline grain-boundary subphases are expected in SmFe12-based permanent magnets, stable crystal structures dependent on temperatures and compositions are examined by first-principles thermodynamics. The free-energy landscape of ternary Sm–Cu–Fe is constructed with self-consistent phonon theory to overcome the dynamical instability problem, where a barrier between binary B2 Sm–Cu and bcc Fe is found. Cu-rich B2 Sm–Cu is substantially stabilized by phonons as well as by configurational entropy. Furthermore, Fe atoms in the Cu sublattice of B27 SmCu contribute to the relative phase stabilization. These results indicate the existence of B2–B27 two-phase equilibria expanding the stable composition region of nonmagnetic SmCu-based intermetallics.

S mFe 12 with the tetragonal ThMn 12 structure is expected to be a candidate for the main phase of strong rare-earth permanent magnets due to favorable magnetic properties of large magnetization and a large anisotropy field that are superior to those of Nd 2 Fe 14 B at high temperatures. [1][2][3][4][5] The bulk phase of SmFe 12 is attempted to be stabilized by partial substitutions of nonmagnetic elements for Fe. [6][7][8][9][10][11][12][13] On the other hand, to improve the performance of permanent magnets, it is necessary to design microstructures with nonmagnetic grain-boundary subphases. 5,[13][14][15][16][17][18] In fact, however, ferromagnetic bcc Fe often precipitates by substituting nonmagnetic elements for some of Fe atoms in SmFe 12 . 1,5,[13][14][15][16] Recent studies have shown that the precipitation of bcc Fe can be suppressed by introducing an Sm-Cu subphase in Sm 2 Fe 17 N 3 -based magnets. 19) Since the chemical composition of Sm 2 Fe 17 is close to that of SmFe 12 , the bcc-Fe precipitation may also be suppressed in the development of SmFe 12 -based magnets. Recently, it has been reported experimentally that a Cu-41 at%Sm alloy equilibrates with SmFe 11 Ti. 20) Stoichiometric SmCu has been examined as a prototype subphase for SmFe 12 -based magnets by first-principles calculations, where the most stable B27 structure was used for SmCu intermetallics. 21) Binary SmCu intermetallics have two possible crystal structures: the B2 structure (CsCl type,Pm m 3 ) and the B27 structure (FeB-b type, Pnma). [21][22][23][24][25][26] Experimentally, the B2 structure can be obtained by rapid quenching of the Sm-Cu liquid phase, 22) while the B27 structure is available by another heat treatment. 23) However, the dependences of the temperature and the composition including impurities on the phase stability are yet to be clarified. In particular, such information on the SmCu phase stability in off-stoichiometric composition regions should help avoid multiphase equilibria including SmFe 2 , SmFe 3 , and bcc Fe in order to make grainboundary phases nonmagnetic.
The phase stability at finite temperatures is governed by the Gibbs free energy that contains effects of phonons and the configurational entropy as main components. 27) First-principles evaluations of phonon free energies have large computational costs for systems with disorders such as random alloys and paramagnetic configurations. 28,29) Furthermore, computations of free energies for a wide composition range have another potential difficulty, i.e. the lattice instability.
The dynamical instability at zero temperature is cured by taking anharmonic lattice vibrations, in particular, the fourthorder anharmonicity into account. Self-consistent phonon (SCPh) theory [30][31][32] is a practical scheme to incorporate the fourth-order anharmonicity into temperature-dependent harmonic phonons effectively. Thus, the SCPh approach is expected to be suitable for evaluating the free-energy landscape.
In this study, we report first-principles phonon calculations for several compositions of binary Sm-Cu and ternary Sm-Cu-Fe with the B2 and B27 structures. We considered the fourth-order anharmonicity through the SCPh method for systems with dynamical instability to eliminate imaginary phonons due to the harmonic approximation. We show the existence of a free-energy barrier between binary B2 Sm-Cu and bcc Fe. In the vicinity of compositions close to stoichiometric SmCu, it is found that Cu-rich B2 Sm-Cu is significantly stabilized by phonon effects in addition to the configurational entropy. Furthermore, B27 Sm(Cu, Fe) is identified as more stable than the B2 one. These results indicate composition regions of two-phase equilibria between the B2 and B27 phases, which suggest more degrees of freedom in the permanent-magnet design containing Fe-added Sm-Cu intermetallics as a nonmagnetic grainboundary subphase excluding other ferromagnetic subphases.
Our first-principles calculations were based on density functional theory within the generalized gradient approximation by the Perdew-Burke-Ernzerhof exchange-correlation energy functional. 33,34) The Kohn-Sham equations were selfconsistently solved by the total-energy minimization using the frozen-core all-electron projector-augmented wave (PAW) method as implemented in the VASP code. [35][36][37] An open-core PAW potential was used for Sm atoms, where 4f electrons were put in the core. As for convergence criteria, the maximum force on each atom and the unit cell were 10 −3 eV Å −1 , while the total-energy variation was set as within 10 −6 eV. The cutoff energy was 296 eV except for 385 eV in the structural optimization.
Crystal structures of B2 and B27 SmCu are shown in Figs. 1(a) and 1(b). The supercell approach was employed in phonon calculations as well as total-energy calculations of off-stoichiometric intermetallics with a randomly configured sublattice. In this approach, the B2 structure was calculated using the 3 × 3 × 3 supercell with 54 atoms, while the 2 × 2 × 2 supercell (64 atoms) was used for the B27 structure unless otherwise stated. The Γ-centered k-point grid for these supercells was set as 4 × 4 × 4 for the B2 structure and 5 × 8 × 6 for the B27 structure in total-energy calculations. To obtain force-displacement data sets, we displaced atoms from their equilibrium positions by 0.01 Å, where forces acting on atoms were calculated for each displaced configuration. Phonon free energies were evaluated using Γcentered q-point grids of at least 18 × 18 × 18 by the ALAMODE code. 38) We first calculated phonons with the harmonic approximation. For the cases of the presence of imaginary phonons, SCPh calculations were performed additionally by solving the SCPh equation, where reduced q samplings were made combined with interpolations for terms related to the anharmonicity. 31) Harmonic force constants were calculated for all possible points and pairs, while anharmonic ones were evaluated within cutoff radii of 3 to 7 Å depending on systems. Off-stoichiometric compounds with compositions of (Sm 1−y Cu y )Cu, Sm(Cu 1−y Sm y ), Sm(Cu 1−y Fe y ), and (Sm 1−y Fe y )Cu were treated by one-atom substitution from stoichiometric SmCu in the supercells, where y is the sublattice concentration of substitutional defects. Hence, we used y = 3.70 at% for the B2 structure and y = 3.13 at% for the B27 structure. As an exception, B2 Sm 1−y CuFe y has y = 1.56 at% with the 4 × 4 × 4 supercell to retain the dynamical stability. We considered the configurational entropy per atom within the Bragg-Williams approximation, where k B is the Boltzmann constant. First, we discuss the stable structures of stoichiometric SmCu intermetallics. Both the B2 and B27 phases were found to be dynamically stable. Since off-stoichiometric binary SmCu has not been reported experimentally, [22][23][24][25][26] we calculated first the phase-transition temperature by comparing free energies of the stoichiometric B2 and B27 phases. The Gibbs phase rule will be taken into account in the later discussion. In the present study, we quantified the free energy of formation that is the summation of the formation energy E form relative to the stable pure substances, viz., α Sm, fcc Cu, and bcc Fe, the phonon free energy F ph , and the term coming from the configurational entropy −TS conf . Considering S conf = 0 for the stoichiometric intermetallics, we show E form + F ph in Fig. 1(c). It is clearly seen that the phase transition between the high-temperature B2 phase and the low-temperature B27 phase occurs at 749 K. This value is sufficiently lower than reported melting temperatures of SmCu, approximately 1000 to 1060 K. [22][23][24][25][26] Even though some of the metals have a tendency that a more close-packed phase becomes the low-temperature phase, it is not the case for B2 and B27. The optimized lattice constants are a = 3.532 Å for B2 SmCu and a = 7.239 Å, b = 4.502 Å, c = 5.536 Å for B27 SmCu. The volume of the B2 phase, 44.1 Å 3 , is slightly smaller than that of the B27 phase, 45.1 Å 3 , per formula unit. Figure 1(d) shows the phonon density of states (DOS) of B2 and B27 SmCu. Higher symmetry of the B2 phase seems to contribute to isotropic vibrations resulting in the stabilization due to higher phonon DOS at lower frequencies compared with B27 SmCu.
In contrast to SmCu, we found that B2 FeCu, B2 SmFe, and bcc Sm are dynamically unstable. As an example, we show effective phonon dispersions of B2 SmFe at 300 K obtained by SCPh calculations in Fig. 2, where imaginary phonons in the harmonic approximation are compared. With the elimination of imaginary phonons, we evaluated the phonon free energy. Then, we calculated the free energy of formation for the ordered B2 phase of SmCu, SmFe, and FeCu together with Sm, Cu, and Fe in the bcc structure. Figure 3(a) shows a very rough estimate of free-energy landscape of B2 Sm-Cu-Fe constructed from these six compounds or pure substances as a ternary contour plot. 39) The stability of B2 SmCu is clearly seen. In addition, a freeenergy barrier lies between B2 Sm-Cu and bcc Fe.
To examine the existence of the barrier more precisely, we evaluate E form − TS conf of (Sm y Cu 1−y )Fe for random configurations of the Sm-Cu sublattice using the clusterexpansion method. As shown in Fig. 3(b), the existence of the free-energy barrier for E form − TS conf is evident. Even though we expect that phonon free energies of (Sm y Cu 1−y )Fe will lower the barrier at most 0.1 eV at 800 K judging from phonon free energies of B2 SmCu, B2 SmFe, B2 FeCu, and bcc Fe, this effect should not cause the disappearance of the barrier.
Next, alloy compositions in the vicinity of stoichiometric SmCu are focused on due to practical relevance to the permanent-magnet material design. It is one of particular interest if SmCu can be made Cu-rich in the solid state, to design alloys with nonmagnetic grain-boundary phases. Figure 4 shows the dependences of energies and free energies on the composition of Sm 1−y Cu 1+y , Sm 1+y Cu 1−y , SmCu 1−y Fe y , and Sm 1−y Fe y Cu in the B2 phase, where y is either 3.7 or 1.6 at%. For all cases, E form is higher than that of stoichiometric B2 SmCu. This situation remains unchanged for the case, where F ph and S conf are taken into account at 800 K, except for Sm 1−y Cu 1+y . In Sm 1−y Cu 1+y , the free energy becomes lower by both F ph and −TS conf . These two contributions are quantitatively comparable. As a result, The Japan Society of Applied Physics by IOP Publishing Ltd E form + F ph − TS conf of B2 Sm 1−y Cu 1+y is substantially lower than that of B2 stoichiometric SmCu. Thus, this stabilization implies the stable existence of Cu-rich B2 Sm-Cu alloys in equilibrium.
We examine phase equilibria of Sm-Cu between the B2 and B27 phases considering off-stoichiometric effects. Figure 5(a) shows differences in the free energy E form + F ph − TS conf with respect to stoichiometric B27   SmCu as functions of the temperature. For temperatures higher than room temperature, Cu-rich B2 Sm-Cu is more stable than the stoichiometric one. In contrast, E form of binary off-stoichiometric B27 Sm-Cu is significantly higher than that of the stoichiometric one as shown in Fig. 5(b). Thus, Cu-rich Sm-Cu is more stable for the B2 phase, while stoichiometric SmCu is the most stable for the B27 phase within the temperature range of ∼300 to ∼1000 K, i.e. approximately the melting temperature of SmCu. This composition difference between the B2 and B27 phases results in two-phase equilibria governed by the so-called common-tangent rule for temperatures between, very roughly speaking, ∼640 to 749 K. Also for ternary Sm-Cu-Fe, we examine two-phase equilibria focusing on Sm(Cu 1−y Fe y ) having relatively low E form for the B27 phase as in Fig. 5(b). Figure 5(c) shows free energies of B2 and B27  Sm(Cu 1−y Fe y ) together with their common tangent. Depending on the Fe concentration, the Sm(Cu 1−y Fe y ) phase diagram has composition regions of the B2 single phase, the B2-B27 two-phase coexistence, and the B27 single phase. These two-phase equilibria between the B2 and B27 phases for Sm 1−y Cu 1+y and Sm(Cu 1−y Fe y ) provide more degrees of freedom in phase equilibria with other phases compared with the case of the SmCu stoichiometric stability only. This is of significant merit, because it opens more possibilities of the microstructure design of SmFe 12 -based permanent magnets with nonmagnetic grain-boundary subphases avoiding ferromagnetic SmFe 2 , SmFe 3 , and Fe.
On the basis of first-principles thermodynamics with the help of the SCPh method, we examined the stability of Fedoped Sm-Cu in the B2 and B27 phases. Due to the effects of both atomic vibrations and the configurational entropy, Cu-rich B2 Sm-Cu becomes more stable than stoichiometric B2 SmCu, while it is not the case for the B27 phase. In addition, free energies of B2 and B27 Sm(Cu, Fe) have a common tangent. These B2-B27 two-phase equilibria suggest possibilities of the alloy design utilizing composition ranges wider than the case considering only the B2 phase in order to avoid ferromagnetic grain-boundary subphases. Such more degrees of freedom should enable the composition of the grain-boundary phase similar to those of the liquid phase without significant diffusion upon cooling.