Ab initio investigation of Fe substitution effect on magnetostructural transition of CoMnGe

In CoMnGe alloy, the structural transformation from a Ni₂In-type hexagonal structure (space group P6₃/mmc, No. 194) to a TiNiSi-type orthorhombic structure (space group Pnma, No. 62) at 375 ∼620 K (T_m) is followed by the magnetic transformation from a paramagnetic state to a ferromagnetic state at 327 ∼355 K (T_C).^1–4 When the crystal structure changes from hexagonal to orthorhombic, there is a large volume change and a large increase in magnetic moment.^1,2,5 If lowering the T_m below the T_C, two, structural and magnetic, transitions can occur simultaneously. For this purpose, there are several experimental studies such as introducing the vacancy,⁶ the fourth elements^4,7–9 in CoMnGe.

Recently, several experimental studies of (Co,Fe)MnGe^3,10–12 and Co(Mn,Fe)Ge^3,13–15 show the Fe substitution produces the decrease of the T_m below the T_C. Thus, these materials have a possibility of a magnetic functional material used as a magnetic-field-driven actuator or a magnetic refrigeration material.^10,16,17 In this paper, we have paid attention to a structural transformation from a hexagonal ferromagnetic structure to a orthorhombic ferromagnetic one.^11,12 We have carried out first-principles calculations for the systems (Co,Fe)MnGe and Co(Mn,Fe)Ge to clarify the effects of Fe substitution for Co or Mn on the hexagonal-structure stabilization from the point of view of an atomic scale or an electronic scale.

To investigate the effects of Fe substitution for Co or Mn on the hexagonal CoMnGe, we calculated an electronic structure of (Co_1−xFe_x)MnGe and Co(Mn_1−xFe_x)Ge (x = 0.25, 0.50), along with the hexagonal CoMnGe, FeMnGe, CoFeGe, and the orthorhombic CoMnGe. For (Co_0.50Fe_0.50)MnGe and Co(Mn_0.50Fe_0.50)Ge, we adopted a cell (space group P1) obtained by the substitution of an Fe atom for one of two Co (or Mn) atoms in the same unit cell as the hexagonal CoMnGe (see Fig. 1). For Co(Mn_0.75Fe_0.25)Ge, we adopted a double-volume cell (space group P1) shown in Fig. 1. For (Co_0.75Fe_0.25)MnGe, we adopted a double-volume cell (space group P$6¯$ m2).

We optimized the lattice constants of the hexagonal and orthorhombic structures by minimizing the total energy. The atomic positions of the hexagonal and orthorhombic structures were optimized by minimizing the forces acting on the atoms. The detailed procedures will be explained elsewhere. The results are listed in Table I. We show atomic positions in supplementary material (Table II).

The first-principles calculations on electronic structure were performed using the full-potential linearized augmented plane wave method.¹⁸ The generalized gradient approximation developed by Perdew et al.¹⁹ was used for the exchange-correlation potential. The plane wave cutoff was RK_max = 10.0, where R is the smallest atomic sphere radius and K_max is the magnitude of the largest K vector. The maximum ℓ value for partial waves used inside atomic spheres was ℓ = 10. For the atomic sphere radius, the following values were used: 2.2 a.u. for the 3d transition metals and 1.9 a.u. for Ge. The used numbers of $k→$ were 14 × 14 × 9 and 9 × 14 × 7 for the hexagonal and orthorhombic structure, respectively.

Our optimized lattice constants of hexagonal and orthorhombic CoMnGe are listed in Table I, along with the calculated magnetic moments for the optimized cell. Our optimized atomic positions of hexagonal (Co,Fe)MnGe and Co(Mn,Fe)Ge are listed in supplementary material (Table II). Concerning lattice constant and magnetic moment, our results are in good agreement with those of other groups. Concerning a volume expansion from ferromagnetic hexagonal to ferromagnetic orthorhombic phases, our value is 5 %, which is close to the experimental values (4.1 %,¹¹ 4.4 %,¹² 5.5 %²²). These values are larger than that of the case from paramagnetic hexagonal to paramagnetic orthorhombic phases (4 %¹). This point will be discussed elsewhere.

The lattice vectors of the hexagonal and orthorhombic structure have relations as $a→o=c→h$⁠, $b→o=a→h$⁠, $c→o=a→h+2b→h$⁠. Thus, the unit-cell volumes of the two phases have the relationship V_o = 2V_h. Here, the subscripts of “h” and “o” mean the quantities corresponding to the hexagonal and orthorhombic structure, respectively. From this fact, we showed the value of V/2 for orthorhombic CoMnGe in Table I.

Fig. 2 shows the composition dependence of lattice constants and volume in hexagonal structures for ferromagnetic (F) and paramagnetic (P) states. It is notable that the dependency for F is quite different from that for P. In the P state, there is also a different tendency between (Co,Fe)MnGe and Co(Mn,Fe)Ge. This could be related to the result that the composition dependence of the transition temperature is different between (Co,Fe)MnGe and Co(Mn,Fe)Ge.³ This point will be discussed elsewhere. Hereafter, we pay attention a ferromagnetic state.

As x increases, the volume decreases when either Co or Mn was substituted with Fe. This tendency is consistent with the experimental results.^10,13,15 As x increases, the lattice constant a increases for (Co_1−xFe_x)MnGe and decreases for Co(Mn_1−xFe_x)Ge, but the lattice constant c decreases for both. The composition dependence of a and c for (Co_1−xFe_x)MnGe is consistent with the observation results for (Ni_1−xFe_x)MnGe.²³ The composition dependence of a and c for Co(Mn_1−xFe_x)Ge is consistent with the observation results.^13,15

We showed that the volume of hexagonal (Co,Fe)MnGe and Co(Mn,Fe)Ge decreases as an amount of Fe increases. These facts indicate that the hexagonal structure is more stabilized by Fe substitution (in other words, the orthorhombic structure is less stabilized) because the volume reduction by Fe substitution is an obstacle to a volume expansion. Furthermore, the fact that the volume reduction of Co(Mn_1−xFe_x)Ge is larger than that of (Co_1−xFe_x)MnGe indicates that the stabilization of the hexagonal structure is more effective for Co(Mn_1−xFe_x)Ge than for (Co_1−xFe_x)MnGe, that is, a decrease in T_m is larger for Co(Mn_1−xFe_x)Ge than for (Co_1−xFe_x)MnGe. This is consistent with the previous research.^3,24

In this section, we investigated the volume reduction by Fe substitution from the point of view of an atomic scale.

Fig. 3(a) shows the change of an interatomic distance (CID) between atoms on the (110) plane of (Co_0.75Fe_0.25)MnGe in comparison with CoMnGe. Remarkable changes are recognized in d(Mn-Mn), d(Fe-Ge), where d(X-Y) means the distance between X and Y atoms. The rates of change are -3.1% for d(Mn-Mn), and -1.1% for d(Fe-Ge). Fig. 3(b) shows the CID between atoms on the (110) plane of (Co_0.50Fe_0.50)MnGe in comparison with CoMnGe. Remarkable changes are recognized in d(Mn-Mn), d(Co-Ge), and d(Fe-Ge), where d(X-Y) means the distance between X and Y atoms. The rates of change are -3.4% for d(Mn-Mn), -1.3% for d(Co-Ge), and -1.3% for d(Fe-Ge).

Fig. 3(c) shows the CID between atoms on the (110) plane of Co(Mn_0.75Fe_0.25)Ge in comparison with CoMnGe. Remarkable changes are recognized in d(Mn-Fe), d(Co-Ge), and the rates of change are -1.6% for d(Mn-Fe), -1.5% for d(Co-Ge). Fig. 3(d) shows the CID between atoms on the (110) plane of Co(Mn_0.50Fe_0.50)Ge in comparison with CoMnGe. Remarkable changes are recognized in d(Mn-Fe), d(Co-Ge), and the rates of change are -1.0% for d(Mn-Fe), -1.7% for d(Co-Ge).

These results show that the substitution of Co or Mn with Fe produces a reduction in the distance between adjacent atoms along the direction of the c axis.

In this section, we investigated the effect of substitution of Co or Mn with Fe from the point of view of an electronic scale.

Figs. 4(a), 4(b), and 4(c) show the charge density distribution (CDD) around the atoms on the c axis. For the case of (b) and(c), the charge between Mn and Mn (referred to as ρ(Mn-Mn)) increases in comparison with CoMnGe. This is because of the reduction of d(Mn-Mn). Hereafter, we use the symbol ρ(X-Y) that means the charge between X and Y atoms.

Figs. 4(d), 4(e), and 4(f) show the CDD around the atoms on the (110) plane. In comparison with CoMnGe, only ρ(Fe-Ge) increases in (e). In (f), both of ρ(Co-Ge) and ρ(Fe-Ge) increase.

The results of the increase of ρ(Mn-Mn) and ρ(Fe-Ge) in (f) are consistent with those in (Ni_0.50Fe_0.50)MnGe.²³ 

We have carried out first-principles calculations for the ferromagnetic systems (Co_1−xFe_x)MnGe and Co(Mn_1−xFe_x)Ge to clarify the effects of Fe substitution for Co or Mn on the hexagonal-structure stabilization from the point of view of an atomic scale or an electronic scale. Our results show the cell volume decreases as an amount of Fe increases mainly because of the reduction of the lattice constant c. The volume decreasing is larger in Co(Mn_1−xFe_x)Ge than in (Co_1−xFe_x)MnGe. The substitution of Co or Mn with Fe produces a reduction in the distance between adjacent atoms along the direction of the c axis. This reduction produces an increase in charge density between those atoms. That fact indicates that the hexagonal structure is more stabilized (or a decrease of T_m) by this enhancement of the covalent bond.

See supplementary material for our optimized atomic positions of hexagonal (Co,Fe)MnGe and Co(Mn,Fe)Ge.

This work was partly supported by JSPS KAKENHI grant number JP17K06840.