First-principles study on surface stability and interface magnetic properties of SmFe12

We report the most stable surface of SmFe12 and the interface magnetic properties of SmFe12 with SmCu and bcc Fe as subphases. We find that the (110) surface with the highest exposition of Sm is the most stable surface of SmFe12. Stabilization by the exposition of rare-earth is also confirmed for Nd2Fe14B. Moreover, we also demonstrate that SmCu improves not only magnetic moments and the anisotropy of grain surfaces of SmFe12 but also well suppresses the magnetic interaction between SmFe12 grains.

F erromagnetic rare-earth compounds with the tetragonal ThMn 12 structure have been seen as a promising main phase of permanent magnets becoming a successor to Nd 2 Fe 14 B, due to their magnetic properties comparable with or exceeding those of Nd 2 Fe 14 B. [1][2][3] In particular, among RFe 12 (R: rare-earth element) compounds, the samarium-based compound SmFe 12 with a particularly large anisotropy field 4-6) is a potential candidate for the main phase of permanent magnets. However, the binary SmFe 12 compound has a problem that the bulk phase is thermodynamically unstable. 7,8) Previous experiments reported the stability is ensured by the partial substitution of Fe atoms in SmFe 12 with nonmagnetic third elements such as Ti, V, Mn, Co, and Zn. [8][9][10][11][12][13][14] On the other hand, as effects of microstructures, local magnetic properties including the anisotropy of rare-earth elements in the main phase close to interfaces are greatly affected by the physical properties of subphases, e.g. the crystal structure, the chemical composition, and the magnetism. In particular, those of the interfacial first layer are significantly different from the bulk state. 15,16) Thus, in order to realize the permanent magnets with the magnetic properties superior to the existing ones, it is necessary to understand the local magnetic properties of the interfaces on the atomic scale. Moreover, for the sake of the realization of the high coercivity, the suppression of the magnetic interaction between the main-phase grains by nonmagnetic subphases is needed. 17) The partial substitution of Fe atoms with nonmagnetic elements to stabilize SmFe 12 often causes regrettably the precipitation of ferromagnetic α-Fe. 8,14,[18][19][20] In the case of Sm 2 Fe 17 N 3 , recent investigations have proposed the possibility of avoiding the α-Fe precipitation by introducing Sm-Cu phases, particularly SmCu. 21) These new findings should also suppress the α-Fe precipitation for SmFe 12 , due to the fact that the chemical composition of Sm 2 Fe 17 is close to that of SmFe 12 , and SmCu has been observed in computational study of the Sm-Fe-Cu phase diagram. 22) In this sense, SmCu may be a promising candidate for a subphase of Sm-based permanent magnets with SmFe 12 as the main phase.
As mentioned above, the magnetic properties of the permanent magnets are closely related to the electronic states of interfaces between the main phase and a subphase. In order to reveal the electronic states of interfaces, interfacial atomic configurations are needed. To accomplish this, it is first and foremost required to identify surface structures of main-phase grains, because microstructure interfaces are usually formed by the solidification of a subphase on surfaces of the main phase. However, such surface atomic structures of SmFe 12 as well as of Nd 2 Fe 14 B are yet to be investigated.
In this paper, we first identify the most stable atomic configurations of SmFe 12 surfaces. We find surfaces with the highest exposition of the Sm atoms is found to be most stable. Surfaces of Nd 2 Fe 14 B are also examined, and the same trend is observed. We clarify that this surface stabilization by the exposition of rare-earth atoms comes from weaker chemical bondings of 5d states of rare-earth atoms compared with those of 3d states of Fe atoms, which minimizes the loss of the band energy of d electrons. Second, we investigate the magnetic properties of SmFe 12 (110)/SmCu(100) and SmFe 12 (110)/Fe(001) interfaces. We find that SmCu enhances the magnetic moments of the main-phase Fe atoms located near the interface to approximately 2.49 μ B . The reason for this enhancements is a decrease in the hybridization between 3d states of the Fe atoms compared with that of the SmFe 12 bulk, which is combined with the charge neutrality for 3d states of the Fe atoms. The improvement of the anisotropy of Sm atoms at interfaces are also observed. In addition, we quantitatively evaluated the effective exchange-coupling constant between the main-phase grains. The results show that SmCu well suppresses the magnetic interaction between the main-phase grains compared with ferromagnetic α-Fe.
Our first-principles calculations of surfaces and interfaces are based on density function theory using pseudopotentials and pseudo-atomic-orbital basis functions as implemented in the OpenMX code. 23) In all calculations, the Perdew-Burke-Ernzerhof exchange-correlation functional 24) within the generalized-gradient approximation was adopted. As basis sets, s2p2d2 configurations were used for Sm, Nd, Fe, Cu, and B with cutoff radii of 8.0, 8.0, 6.0, 6.0, and 7.0 Bohr, respectively. Semicore orbitals of 3s and 3p in Fe and Cu as well as 5s and 5p in Sm and Nd were treated as valence electrons. Open-core pseudopotentials were used for Sm and Nd atoms, where 4f electrons were treated as spin-polarized core electrons. As for convergence criteria, the maximum force on each atom and the total-energy variation are 10 −4 Hartree/Bohr and 3.1 × 10 −9 Hartree/atom for surface calculations, and 10 −3 Hartree/Bohr and 1.3 × 10 −7 Hartree/atom for interface calculations, respectively. The lattice parameters were set to = = a b 8.54 Å and = c 4.71 Å for SmFe 12 , = = a b 8.79 Å and c = 12.14 Å for Nd 2 Fe 14 B, a = 7.38, = b 4.56 Å, and c = 5.64 Å for FeB type SmCu, and = = = a b c 2.84 Å for bcc Fe obtained by our calculations, respectively. Collinear spin structures neglecting the spin-orbit coupling of the valence electrons are considered with the energy cutoff of 500 Ry. The k-point grids of 8 × 8 × 14 for SmFe 12 and 8 × 8 × 6 for Nd 2 Fe 14 B unit cell were adopted, respectively, where the k-point grids for surfaces and interfaces were scaled according to their lattice parameters. The initial spin configuration has the antiparallel structure between a rare-earth element and Fe.
First, we discuss the energetic stability of rare-earthcompound surfaces. The surfaces are represented as repeated slab models, and separated by 10 Å as the vacuum gap. In the case of SmFe 12 surfaces, the surface-energy density γ is evaluated by  where E slab is the total energy of the surface slab, N i is the number of atoms for the element i within the slab, μ i is the chemical potential of the element i, A is the surface area, and E SmFe 12 is the total energy of the SmFe 12 bulk per formula unit. In the Fe-rich condition, we use m m = Fe bcc Fe meaning that SmFe 12 is equilibrium with bcc Fe. In this case, μ Sm is obtained by Eq. (2). Likewise, the Sm-rich condition is considered by m m = a

Sm
Sm meaning that SmFe 12 is equilibrium with α-Sm.
The surfaces of SmFe 12 with the Miller indices of (110), (100) and (001) are calculated for all possible 14 terminations, and we discuss only the most stable surface for each Miller index. As shown in Fig. 1, it is clear that the (110) surface is most stable. Furthermore, we have identified the trend that the most stable surface termination has the highest exposition of Sm atoms. This stabilization by the exposition of rare-earth atoms is attributed to the fact that the chemical bonding of 5d states of rare-earth atoms is weaker than that of Fe 3d states. The band energy of 5d states of rare-earth atoms which contribute to the chemical bonding is significantly smaller than that of 3d electrons of Fe atoms as depicted in Fig. 1(e) for the SmFe 12 bulk. The Mulliken-population analysis revealed that the number of 5d electrons of the Sm atoms is approximately 1.4 while the number of 3d electrons of the Fe atoms is approximately 6.7 for the SmFe 12 bulk.
Since we expect this trend is universal, we also examine another compound with a rare-earth element and a transition metal, Nd 2 Fe 14 B. In the case of Nd 2 Fe 14 B surfaces, the surface-energy density γ is evaluated by is the total energy of the Nd 2 Fe 14 B bulk per formula unit. By considering all possible 19 terminations, we also identified the most stable termination for the Nd 2 Fe 14 B(100) surface. Also for this surface, the highest exposition of the rare-earth element, Nd, is seen as depicted in Fig. 2(a). Thus, the (001) and (110) surfaces of Nd 2 Fe 14 B were examined for only the termination with the highest exposition of Nd atoms. As is clear from Fig. 2(b), the (001) surface is most stable for Nd 2 Fe 14 B, whereas the next stable surface index is (100).
Next, we discuss local magnetic properties of SmFe 12 (110)/SmCu(100) and SmFe 12 (110)/Fe(001) interfaces. In the case of the interface calculations, we relaxed the lattice parameters for the interface-perpendicular direction in addition to atomic coordinates. For the SmFe 12 (110)/SmCu(100) interface,´2 2 1 SmFe 12 and 1 × 1 × 2 SmCu were adopted, where the lattice mismatch is approximately 6.4% for the x-axis and 3.3% for the y-axis, respectively. For the SmFe 12 (110)/Fe(100) interface,2 2 3 SmFe 12 and 4 × 5 × 2 Fe were adopted, where the lattice mismatch is approximately 5.4% for the xaxis and 0.7% for the y-axis, respectively. Our interface calculations contain up to approximately 550 atoms in the supercell. We evaluate the effective exchange-coupling constant J ex by where and E tot APMA and E tot PMA are the total energy of the optimized interface structures with the antiparallel magnetization alignment (APMA) and the parallel magnetization alignment (PMA) between SmFe 12 grains, respectively. The total energy E tot APMA can be calculated by a supercell doubled in the interface-perpendicular direction as depicted in Fig. 3.
Magnetic moments of the main-phase Fe atoms of each interface and the bulk are shown in Fig. 4(a). In the case of SmFe 12 (110)/SmCu(110) interface, the magnetic moments of   the Fe atoms at the first interface layer (z = 0 and z = 12.1 Å) greatly increase compared with that of the SmFe 12 bulk. This enhancement comes from a decrease in the hybridization between 3d minority-spin states of the main-phase Fe atoms, which is seen as weakened splitting between bonding and antibonding states in Fig. 4(b). To keep the charge neutrality, the majority spin increases by the amount of the decrease in the minority spin. In addition, the effective exchangecoupling constant J ex is listed in Table I. It is clear that SmCu well suppresses the magnetic interaction between the main-phase grains compared with bcc Fe.
The magnetocrystalline anisotropy constant K 1 of the main-phase Sm atoms is evaluated up to the first order by where J, α J , á ñ r 2 , and A 2 0 are the total angular momentum quantum number, the first Stevens factor, the spatial extent of the 4f orbitals, and the second-order crystal field coefficient, 25,26) respectively. For Sm 3+ , J = 5/2 (L = 5, S = 5/2) and α J = 13/315 were adopted. Electronic states of a specific crystal structure determine á ñ A r 2 0 2 through the expanded radial component of the single-particle effective potential for the spherical harmonics Y 2 0 . In evaluating K 1 , the quantization axis is set as the [001] direction of SmFe 12 , not the direction perpendicular to the interface denoted as z. As shown in Fig. 5, the stable magnetic direction is always [001] of the main phase (K 1 > 0) that is parallel to the (110) interface. The uniaxial anisotropy enhances at the interfaces compared with that of the bulk.
In summary, the most stable surface of SmFe 12 was determined to be the (110) surface with the highest exposition of Sm atoms from first principles. This trend of the highest exposition of rare-earth atoms, which comes from weaker chemical bonding of rare-earth 5d states compared with that of Fe 3d states, was universally observed for Nd 2 Fe 14 B surfaces as well. Moreover, analyzing the magnetic properties of SmFe 12 -based interfaces, we conclude that SmCu is preferred as a subphase of the SmFe 12 -based permanent magnets. Our new findings in this paper will hopefully promote further development of new permanent magnets.