- Email: [email protected]
Magnetic properties and atomic ordering of BCC Heusler alloy Fe2MnGa ribbons
Physica B 489 (2016) 51–55
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetic properties and atomic ordering of BCC Heusler alloy Fe2MnGa ribbons Yuepeng Xin, Yuexing Ma, Hongzhi Luo n, Fanbin Meng, Heyan Liu School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2016 Received in revised form 23 February 2016 Accepted 26 February 2016 Available online 27 February 2016
The electronic structure, atomic disorder and magnetic properties of the Heusler alloy Fe2MnGa have been investigated experimentally and theoretically. BCC Fe2MnGa ribbon samples were prepared. Experimentally, a saturation magnetic moment (3.68 μB at 5 K) much larger than the theoretical value (2.04 μB) has been reported. First-principles calculations indicate that the difference is related to the Fe– Mn disorder between A, B sites, as can also be deduced from the XRD pattern. L21 type Fe2MnGa is a ferrimagnet with antiparallel Fe and Mn spin moments. However, when Fe–Mn disorder occurs, part of Mn moments will be parallel to Fe moments, and the Fe moments also clearly increase simultaneously. All this results in a total moment of 3.74 μB, close to the experimental value. & 2016 Elsevier B.V. All rights reserved.
Keywords: Heusler alloys Electronic structure Magnetic properties Anti-site disorder
1. Introduction In recent years, Heusler alloys have attracted considerable attention because of their interesting physical properties like halfmetallicity, ferromagnetic (FM) shape-memory effect, topological insulator and spin gapless semiconductor [1–7]. These properties have potential applications in many fields and are worth further investigation. Generally, Heusler alloys crystallize in a highly-ordered BCC structure and have a stoichiometric composition X2YZ, where X and Y are transition-metal elements and Z is a main-group element. In Heusler alloys, there are the four Wyckoff-positions A (0, 0, 0), B ( 1 , 1 , 1 ), C ( 1 , 1 , 1 ) and D ( 3 , 3 , 3 ), and the transition-metal 4 4 4
2 2 2
4 4 4
elements X, Y enter A, B, C sites while the main group element Z enters the D sites. The site preference of the transition-metal elements can strongly influence the properties of Heusler alloys. Fe2MnGa and alloys with similar compositions are quite interesting because of their diversiform crystal structures, martensitic transformation, giant exchange bias and high spin-polarization ratio of the spin-up and -down conduction electrons at the Fermi level EF. In Fe43Mn28Ga29, close to the stoichiometric 2:1:1 composition, a martensitic transformation from the paramagnetic parent L21 phase to the FM martensite L10 phase has been observed [8]. Zhu et al. have investigated the formation and crystal structure of Fe–Mn–Ga alloys and have reported a single BCC phase for bulk Fe50Mn22.5Ga27.25, together with a field-induced transformation from the paramagnetic parent phase to the FM n
Corresponding author. E-mail address: [email protected] (H. Luo).
http://dx.doi.org/10.1016/j.physb.2016.02.030 0921-4526/& 2016 Elsevier B.V. All rights reserved.
martensite phase [9]. So, Fe–Mn–Ga may be a new family of FM shape-memory alloys. The stoichiometric Fe2MnGa has been reported to crystallize single FCC phase [10,11] or multiphase, consisting of FCC and BCC phases [12]. In FCC Fe2MnGa, an antiferromagnetic to FM transition has been observed, resulting in a giant exchange bias effect [10,13,14]. BCC Fe2MnGa has been predicted to be ferrimagnetic (FIM) with a high spin polarization ratio, which makes it a possible candidate for spintronic applications [15,16]. However, for bulk Fe2MnGa, although firstprinciples calculations suggest that it has a stable L21 structure, a single phase Fe2MnGa with BCC structure has not been reported before the work of Okumura et al. [17]. In 2014, they have fabricated single-phase BCC Fe2MnGa using melt-spinning technique and have found that the crystal structure depends on the quenching conditions. The influence of heat treatment on the crystal structure of Fe2MnGa was investigated in detail. BCC Fe2MnGa ribbon was found to be FM with a Curie temperature TC of 170 K, as determined from the AC susceptibility curve [17]. In Ref. [17], only the paramagnetic magnetization curve of BCC Fe2MnGa at room temperature is presented. Since Fe2MnGa has been predicted to have a high spin polarization ratio and a total moment close to 2 μB [15,16], it is of interest to investigate its low temperature magnetic properties and compare them with theoretical results. Also, knowledge of the magnetic moment at 5 K is important to investigate the atomic ordering in Fe2MnGa ribbons, which is still not quite clear yet. In the present study, we have prepared BCC Fe2MnGa meltspun ribbons. The structural and magnetic properties have been measured and are compared with theoretical results. The influence of atomic disorder on the electronic structure and magnetic properties of the Heusler alloy Fe2MnGa is discussed. The large
52
Y. Xin et al. / Physica B 489 (2016) 51–55
difference between the calculated and the measured saturated magnetic moments is explained by Fe–Mn disorder evolving from the ordered L21 structure.
2. Experimental and computational methods Fe2MnGa ingots were prepared by arc-melting the constituent elements in a high-purity argon atmosphere. The purity of the starting materials was 99.9% or higher. The ingots were melted at least four times, then annealed for homogenization at 1273 K for three days under protection of argon atmosphere. Then, melt-spun ribbons were prepared by a single-wheel technique with the substrate velocity (vs) of 18 m/s, under protection of Ar atmosphere. X-ray powder diffraction (XRD) with Cu-Kα radiation was used to verify the crystal structure and to determine the lattice constants. The magnetization was measured in a SQUID magnetometer. The temperature dependence of AC susceptibility was measured to determine the Curie temperature. The electronic structure and magnetism of Fe2MnGa were calculated by means of the CASTEP code based on the pseudopotential method with a plane-wave basis set [18,19]. The interactions between the atomic core and the valence electrons were described by the ultra-soft pseudopotential [20]. The electronic exchange–correlation energy was treated under the generalized-gradient-approximation (GGA) [21]. To simulate the Fe–Mn or Mn–Ga disorder in Fe2MnGa, we used the supercell method in a 16-atom supercell, which has resulted in the chemical formula Fe8Mn4Ga4. In all cases, a plane-wave basis set cut-off of 500 eV was used. A mesh of 18 18 18 k-points was employed for Brillouin zone integrations during calculations. These parameters ensured good convergences for total energy. The convergence tolerance for the calculations was selected as a difference on total energy within 1 10 6 eV/atom.
3. Results and discussion 3.1. Formation and magnetic properties Fe2MnGa ribbon samples were prepared by melt-spinning method. The XRD pattern of finely powdered Fe2MnGa is shown Fig. 1. In the XRD pattern, the main reflections all belong to the BCC phase except for a small γ-phase reflection at 42.3°. So, the result indicates that single BCC phase Fe2MnGa can be obtained by using the melt-spinning technique [17]. The derived lattice constant for Fe2MnGa is 5.832 Å, which agrees well with the experimental value of 5.830 Å in Ref. [17], but is much larger than the calculated lattice constant of 5.69 Å [15]. In the XRD pattern, two superlattice reflections (111) and (200) are seen, which represent the atomic ordering in Fe2MnGa. It is known that, if we consider the highly-ordered structures only, there are two possible atomic orderings in Heusler alloys: One is the L21 structure (Cu2MnAl-type, space group No. 225), in which the two X atoms occupy A and C positions and the Y, Z atoms enter B and D positions, respectively. The other is the XA structure (Hg2CuTi-type, space group No. 216), in which the two X atoms occupy A and B positions and the Y, Z atoms occupy C and D positions, respectively. This structure is also called “inverse” Heusler alloy [22]. In the literature, Fe2MnGa is predicted to crystallize in the L21 structure [16]. This agrees with the valence-electrons rule in Heusler alloys, which says that “the site preferences of transition metal atoms X and Y are determined by the number of their valence electrons: atoms with more electrons tend to occupy the A and C positions while the atoms with fewer electrons prefer the B position” [23,24]. However, since exceptions have been reported for X2CuAl Heusler alloys [25], it is of interest to check the site preference in Fe2MnGa by experimental and theoretical methods.
Fig. 1. Powder X-ray diffraction pattern of Fe2MnGa. The insets show the (111) and (200) superlattice reflections of the experimental pattern and of simulated patterns of the L21 structure, the XA structure and structures with Fe–Mn disorder or Mn–Ga disorder.
Generally, the highly-ordered Heusler structure is represented by the existence of the superlattice reflections (111) and (200). In the inset of Fig. 1, an enlargement of the experimental patterns between 24° and 32° is presented in which both the (111) and the (200) reflection are clearly seen, indicating an ordered structure of the Fe2MnGa samples. The intensities of lattice reflections are proportional to the square of the structure factor F. The superlattice reflections (111) and (200) are order-dependent, while the principal reflection (220) is order-independent. Therefore, different types of atomic order can be distinguished by comparing the intensity ratios of the different reflections I(111)/I(220) and I(200)/I(220) or I(111)/I (200) [22,26]. In the inset of Fig. 1, the experimental result is compared with simulated patterns for ordered L21 and XA structures and Fe–Mn, Mn–Ga disordered structures. It can be seen that in the pattern of the L21 structure, I(111) is much higher than I(200) but, in the experimental pattern, I(111) and I(200) are similar. So, there may exist a certain degree of antisite disorder based on the L21 structure. Mn (B)–Ga (D) disorder is an usual disorder in Heusler alloys due to which, as shown in Fig. 1, the (111) reflection disappears completely and only the (200) peak remains. This is also quite different from the experimental result. Another usual disorder is between A, B sites. In Fe2MnGa, it means that some Fe (A) and Mn (B) atoms change positions on the L21 lattice. If Mn and Fe change completely, the XA structure will be formed instead of L21. However, with increasing Fe– Mn disorder, I (111) is always visible and higher than I (200) to some extent. This agrees qualitatively with the experimental results. In Mn2NiSb ribbon, huge Ni (A)–Mn (B) disorder introduced by meltspinning has already been reported [27] and similar effects may occur in Fe2MnGa ribbons. However, in view of the possible texture and strain in the melt-spun ribbons, further investigations are necessary to come to proper conclusions. It is known that the magnetic properties of Heusler alloys can differ a lot when magnetic atoms like Fe or Mn occupy different sites. We hope that this may help to determine the atomic ordering in Fe2MnGa. The temperature dependence of the AC susceptibility of a Fe2MnGa ribbon sample measured from 77 K to 380 K is shown in the inset of Fig. 2. A sharp FM-paramagnetic transition is observed at the Curie temperature of about 185 K, which is somewhat higher than the 170 K reported in Ref. [17]. Fig. 2 shows the magnetization of Fe2MnGa ribbon sample measured at 5 K in fields up to 5 T. The saturation magnetization Ms at 5 K is 3.68 μB/f.u. This is a rather large value and about 1.8 times
Y. Xin et al. / Physica B 489 (2016) 51–55
Fig. 2. Magnetization of Fe2MnGa at 5 K in a field up to 5 T. The inset shows the temperature dependence of the AC susceptibility. The Curie temperature TC is indicated by an arrow.
larger than the theoretical value of about 2 μB reported in literature [15,16]. It is known that, in Fe2Mn-based Heusler alloys, Mn enters the B site and its spin moment is antiparallel to the Fe moments at A, C sites which leads to a relatively small total moment. For example, Ms of Fe2MnAl at 5 K is 1.58 μB [28]. So it is quite interesting to investigate the origin of the large Ms in Fe2MnGa. 3.2. Electronic structure As has been discussed above, there may exist a certain degree of Fe (A)–Mn (B) disorder in Fe2MnGa ribbons. When Fe enters the B site, it is located in a BCC sublattice and its densities of states (DOS) will show large large exchange splitting due to the crystal field effect. Thus its spin moment will become large and be parallel to the Fe moments at A, C sites. So this disorder can enhance the total moment of Fe2MnGa compared with the L21 type Fe2MnGa. In order to investigate the influence of atomic disorder on the magnetic properties of Fe2MnGa further, we have calculated the electronic structure and magnetic properties of L21- and XA-type ordered Fe2MnGa and of alloys with Fe (A)–Mn (B) or Mn (B)–Ga (D) disorder. In the calculations, we have considered FM and FIM states for the two alloys. In the FIM state, antiparallel coupled moments between Fe (B)–Mn (A) or Fe (C)–Mn (A) in the XA structure and between Fe (A, C)–Mn (B) in the L21 structure have been considered. For the Fe (A)–Mn (B) disordered structure, we have also considered antiparallel coupled Mn spin moments at A and B sites. The structure with a certain degree of Fe (A)–Mn (B) disorder can be looked at as an intermediate state between the XA and the L21 structure. We will first discuss the properties of the ordered XA and L21 structures. The total energies of Fe2MnGa with the XA and the L21 structure have been calculated as a function of the lattice constant and are shown in Fig. 3. It is clear that L21 type Fe2MnGa in the FIM state with antiparallel coupling between Fe and Mn spin moments has the lowest total energy and, therefore, is the most stable state. The energy difference between the FIM state and the FM state is 0.29 eV. This is a relatively large value and ensures the high stability of the FIM state in the L21 type Fe2MnGa. Unlike in the L21 structure, the FM state is more stable in XA type Fe2MnGa, which is located 0.1 eV above the most stable L21 curve. In Fig. 3, we can see that the XA structure tends to have a larger lattice constant than the L21 structure and also that the FM state has a larger lattice constant in both the XA and the L21 structure. The derived equilibrium lattice constants for L21 and XA type Fe2MnGa are listed in Table 1.
53
Fig. 3. The calculated total energies as a function of the lattice constant for L21 and XA type Fe2MnGa in the FM and the FIM state. Here, the total energy of the FIM L21 phase was set as zero for comparison.
The calculated total and partial spin moments for Fe2MnGa with the L21 and the XA structure are also presented in Table 1. In L21 type Fe2MnGa, the Fe atoms at A, C sites have a small and negative spin moment of 0.52 μB. This leads to a relatively small total moment of 2.04 μB, although the Mn moment is as high as 3.08 μB. Similar results have been reported in Refs. [15] and [16]. However, in XA type Fe2MnGa, the Fe (C), Fe (B) and Mn (A) spin moments are parallel to each other and quite large, which leads to a large total moment. Then, if a certain degree of Fe (A)–Mn (B) disorder is introduced in the ordered L21 structure by melt spinning, the antiparallel coupling between Fe and Mn spin moments can be relieved and the total moment can increase substantially. We will discuss this in detail in the next section. In Fig. 4, the total and partial DOS of L21 and XA type Fe2MnGa are presented. The DOS of L21 type Fe2MnGa looks like a spin gapless semiconductor (SGS) and, for both spin directions, the Fermi level EF is located in an energy gap or valley. This results in rather low density of states N(EF) at EF and may help to increase the stability of the L21 structure [29]. The origin of the gap can be understood as follows: Fe2MnGa has 26 valence electrons, 14 of them tend to enter the majority spin band and the other 12 the minority spin band. This will introduce energy gaps or valleys at EF for both spin directions. Similar results have also been observed in Mn2CoAl, which also contains 26 valence electrons and is the first-reported SGS Heusler alloy [7]. The detail mechanism can be found in Refs. [25] and [30]. The DOS structure of XA type Fe2MnGa is quite different from the L21 DOS. There is a sharp DOS peak at EF in the minority spin. This clearly increases N(EF) which is not favorable for the stability of the XA structure. There are also clear differences between the PDOS of Fe in the L21 and the XA structure. In the XA DOS, the majority Fe states are basically below 0.6 eV while a part of the minority Fe states is located far above EF. This large exchange splitting can give rise to large partial moments on the Fe (A) and Fe (B) sites in the XA structure. However, in the L21 DOS, the majority Fe states are shifted to higher energy and there is a pronounced majority DOS peak at 0.45 eV. This weakens the exchange splitting and lowers the Fe moments in the L21 structure. 3.3. Influence of atomic disorder In this section, we will discuss the influence of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa. In Fig. 5, the structural optimization results for the disordered system are presented. It can be seen that, compared
54
Y. Xin et al. / Physica B 489 (2016) 51–55
Table 1 Total-energy difference ΔE between the XA and L21 structures, lattice constants a and magnetic moments for L21 and XA type Fe2MnGa. Here aexp and Ms are the experimental lattice constant at room temperature and the saturated magnetic moment at 5 K, respectively. Structure
ΔE (eV)
a (Å)
aexp (Å)
Mt (μB)
L21 XA
0.1
5.69 5.83
5.832
2.04 7.48
MFe(A)/(A,C) (μB) 0.52 2.22
MFe(B) (μB)
MMn(B) (μB)
MMn(C) (μB)
Ms (μB)
TC (K)
– 2.24
3.08 –
– 3.22
3.68
185
Fig. 4. Spin-projected total and partial DOS for L21 and XA type Fe2MnGa.
with ordered L21 Fe2MnGa, both Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder can lead to an expansion of the lattice: the equilibrium lattice constant is 5.76 Å for the former and 5.71 Å for the latter, both of them being larger than the 5.69 Å for the L21 structure. At the same time, the 5.76 Å for Fe (A)–Mn (B) disorder is more consistent with the experimental lattice constant of 5.832 Å. It can also be seen that the Fe (A)–Mn (B) disorder has a lower total energy compared with Ga (D)–Mn (B) disorder, so that the former is more likely to occur in Fe2MnGa ribbons. The influences of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa are summarized in Table 2. The total moment of Fe2MnGa with Ga (D)–Mn (B) disorder is 2.14 μB and quite close to the 2.04 μB for the L21 structure. Therefore, it can not cause the large saturation magnetization observed at 5 K. Fe (A)–Mn (B) disorder has a quite different effect, the calculated total moment being 3.74 μB, which is close to the experimental value of 3.68 μB. This further supports the assumption that there exists huge Fe–Mn antisite disorder in Fe2MnGa ribbons. It is also known that the disorder between A, B sites usually can lower the spin polarization ratio and destroy the half-metallicity. Then, the application of BCC Fe2MnGa in spintronics may be limited. In Table 2, we can see that, when Fe–Mn disorder occurs, the Mn (A) and Mn (B) partial moments are antiparallel to each other, and one of them has FM coupling with the Fe spin moments while, in the ordered L21 structure, the Fe and Mn moments are strictly
Fig. 5. Structural optimization results for Fe2MnGa with Fe–Mn and Mn–Ga disorder.
antiparallel aligned. At the same time, the spin moments of Fe also increase clearly with Fe–Mn disorder, especially the antisite Fe entering B site, due to the different chemical surroundings of Fe compared with that in the ordered L21 structure. All this leads to the large saturation magnetic moment of Fe2MnGa ribbons.
Y. Xin et al. / Physica B 489 (2016) 51–55
Table 2 Influence of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa. Disorder
Mt (μB)
MFe(A) (μB)
Fe (A)–Mn 3.74 (B) Ga (D)–Mn 2.14 (B)
1.74
2.02
2.54
0.60
0.60
–
MFe(C) (μB)
MFe(B) (μB)
MMn(A) (μB) 2.94 –
MMn(B) (μB)
MMn(D) (μB)
2.52
–
3.28
3.28
4. Conclusions Single-phase Fe2MnGa Heusler alloy can be prepared by the meltspinning technique. The Curie temperature of BCC Fe2MnGa ribbon is about 185 K and the saturation moment at 5 K is 3.68 μB/f.u., which is much larger than the total moment of about 2 μB calculated in literature as well as in the present work for Fe2MnGa with the ordered L21 structure. This difference can be explained by the Fe–Mn disorder on A, B sites introduced by melt spinning. The XRD pattern also reveals the possible existence of this kind of disorder. L21 type Fe2MnGa is a ferrimagnet with strictly antiparallel Fe and Mn spin moments. However, when Fe–Mn disorder occurs, part of Mn moments will be parallel to Fe moments and the spin moments of Fe also increase clearly at the same time. All this results in a total moment of 3.74 μB, which agrees quite well with the experimental value.
Acknowledgments This work is supported by the National Natural Science Foundation of China in Grant no. 11474343 and 51371075, by the Foundation of Hebei Provincial Education Department in Grant no. BJ2014012 and also by Program for Changjiang Scholars and Innovative Research Team in University in Grant no. IRT13060.
References [1] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Phys. Rev. Lett. 50 (1983) 2024. [2] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar, M. L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488.
55
[3] Y. Sakuraba, K. Izumi, T. Iwase, S. Bosu, K. Saito, K. Takanashi, Y. Miura, K. Futatsukawa, K. Abe, M. Shirai, Phys. Rev. B 82 (2010) 094444. [4] K. Ullakko, J.K. Huang, C. Kanter, V.V. Kokorin, R.C. O’Handley, Appl. Phys. Lett. 69 (1996) 1966. [5] R. Kainuma, Y. Imano, W. Ito, Y. Sutou, H. Morito, S. Okamoto, O. Kitakami, K. Oikawa, A. Fujita, T. Kanomata, K. Ishida, Nature 439 (2006) 957. [6] S. Chadov, X.L. Qi, J. Kübler, G.H. Fecher, C. Felser, S.C. Zhang, Nat. Mater. 9 (2010) 541. [7] S. Ouardi, G.H. Fecher, C. Felser, Phys. Rev. Lett. 110 (2013) 100401. [8] T. Omori, K. Watanabe, R. Umetsu, R. Kainuma, K. Ishida, Appl. Phys. Lett. 95 (2009) 082508. [9] W. Zhu, E.K. Liu, L. Feng, X.D. Tang, J.L. Chen, G.H. Wu, H.Y. Liu, F.B. Meng, H. Z. Luo, Appl. Phys. Lett. 95 (2009) 222512. [10] X.D. Tang, W.H. Wang, W. Zhu, E.K. Liu, G.H. Wu, F.B. Meng, H.Y. Liu, H.Z. Luo, Appl. Phys. Lett. 97 (2010) 242513. [11] M. Yin, P. Nash, Intermetallics 57 (2015) 34. [12] Y.V. Kudryavtsev, N.V. Uvarov, V.N. Iermolenko, I.N. Glavatskyy, J. Dubowik, Acta Mater. 60 (2012) 4780. [13] T. Gasi, A.K. Nayk, M. Nicklas, C. Felser, J. Appl. Phys. 113 (2013) 17E301. [14] E.C. Passamani, C. Larica, G. Viali, J.R. Andrez, A.Y. Takeuchi, V.P. Nascimento, V. A.P. Rodriguez, C. Rojas-Ayala, E. Baggio-Saitovitch, J. Alloy. Compd. 628 (2015) 164. [15] A.T. Zayak, P. Entel, K.M. Rabe, W.A. Adeagbo, M. Acet, Phys. Rev. B 72 (2005) 054113. [16] X.H. Liu, J.B. Lin, Y.H. Liu, Y.J. Jin, Acta Phys. Sin. 60 (2011) 107104. [17] H. Okumura, E. Hashiba, K. Nagata, Intermetallics 49 (2014) 65. [18] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1065. [19] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M. C. Payne, J. Phys. : Condens. Matter 14 (2002) 2717. [20] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [21] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [22] T. Graf, C. Felser, S.S. Parkin, Prog. Solid State Chem. 39 (2011) 1. [23] T.J. Burch, T. Litrenka, Phys. Rev. Lett. 33 (1974) 421. [24] H.C. Kandpal, G.H. Fecher, C. Felser, J. Phys. D: Appl. Phys. 40 (2007) 1507. [25] H.Z. Luo, Y.P. Xin, B.H. Liu, F.B. Meng, H.Y. Liu, E.K. Liu, G.H. Wu, J. Alloy. Compd. 665 (2016) 180. [26] K. Endo, T. Kanomata, H. Nishihara, K.R.A. Ziebeck, J. Alloy. Compds 510 (2012) 1. [27] H.Z. Luo, W. Zhu, L. Ma, G.D. Liu, Y.X. Li, X.X. Zhu, C.B. Jiang, H.B. Xu, G.H. Wu, J. Phys. D: Appl. Phys. 42 (2009) 095001. [28] Z.H. Liu, X.Q. Ma, F.B. Meng, G.H. Wu, J. Alloy. Compd. 509 (2011) 3219. [29] T. Hong, T.J. Watson-Yang, A.J. Freeman, T. Oguchi, J.H. Xu, Phys. Rev. B 41 (1990) 12462. [30] G.Z. Xu, E.K. Liu, Y. Du, G.J. Li, G.D. Liu, W.H. Wang, G.H. Wu, EPL 102 (2013) 17007.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetic properties and atomic ordering of BCC Heusler alloy Fe2MnGa ribbons Yuepeng Xin, Yuexing Ma, Hongzhi Luo n, Fanbin Meng, Heyan Liu School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2016 Received in revised form 23 February 2016 Accepted 26 February 2016 Available online 27 February 2016
The electronic structure, atomic disorder and magnetic properties of the Heusler alloy Fe2MnGa have been investigated experimentally and theoretically. BCC Fe2MnGa ribbon samples were prepared. Experimentally, a saturation magnetic moment (3.68 μB at 5 K) much larger than the theoretical value (2.04 μB) has been reported. First-principles calculations indicate that the difference is related to the Fe– Mn disorder between A, B sites, as can also be deduced from the XRD pattern. L21 type Fe2MnGa is a ferrimagnet with antiparallel Fe and Mn spin moments. However, when Fe–Mn disorder occurs, part of Mn moments will be parallel to Fe moments, and the Fe moments also clearly increase simultaneously. All this results in a total moment of 3.74 μB, close to the experimental value. & 2016 Elsevier B.V. All rights reserved.
Keywords: Heusler alloys Electronic structure Magnetic properties Anti-site disorder
1. Introduction In recent years, Heusler alloys have attracted considerable attention because of their interesting physical properties like halfmetallicity, ferromagnetic (FM) shape-memory effect, topological insulator and spin gapless semiconductor [1–7]. These properties have potential applications in many fields and are worth further investigation. Generally, Heusler alloys crystallize in a highly-ordered BCC structure and have a stoichiometric composition X2YZ, where X and Y are transition-metal elements and Z is a main-group element. In Heusler alloys, there are the four Wyckoff-positions A (0, 0, 0), B ( 1 , 1 , 1 ), C ( 1 , 1 , 1 ) and D ( 3 , 3 , 3 ), and the transition-metal 4 4 4
2 2 2
4 4 4
elements X, Y enter A, B, C sites while the main group element Z enters the D sites. The site preference of the transition-metal elements can strongly influence the properties of Heusler alloys. Fe2MnGa and alloys with similar compositions are quite interesting because of their diversiform crystal structures, martensitic transformation, giant exchange bias and high spin-polarization ratio of the spin-up and -down conduction electrons at the Fermi level EF. In Fe43Mn28Ga29, close to the stoichiometric 2:1:1 composition, a martensitic transformation from the paramagnetic parent L21 phase to the FM martensite L10 phase has been observed [8]. Zhu et al. have investigated the formation and crystal structure of Fe–Mn–Ga alloys and have reported a single BCC phase for bulk Fe50Mn22.5Ga27.25, together with a field-induced transformation from the paramagnetic parent phase to the FM n
Corresponding author. E-mail address: [email protected] (H. Luo).
http://dx.doi.org/10.1016/j.physb.2016.02.030 0921-4526/& 2016 Elsevier B.V. All rights reserved.
martensite phase [9]. So, Fe–Mn–Ga may be a new family of FM shape-memory alloys. The stoichiometric Fe2MnGa has been reported to crystallize single FCC phase [10,11] or multiphase, consisting of FCC and BCC phases [12]. In FCC Fe2MnGa, an antiferromagnetic to FM transition has been observed, resulting in a giant exchange bias effect [10,13,14]. BCC Fe2MnGa has been predicted to be ferrimagnetic (FIM) with a high spin polarization ratio, which makes it a possible candidate for spintronic applications [15,16]. However, for bulk Fe2MnGa, although firstprinciples calculations suggest that it has a stable L21 structure, a single phase Fe2MnGa with BCC structure has not been reported before the work of Okumura et al. [17]. In 2014, they have fabricated single-phase BCC Fe2MnGa using melt-spinning technique and have found that the crystal structure depends on the quenching conditions. The influence of heat treatment on the crystal structure of Fe2MnGa was investigated in detail. BCC Fe2MnGa ribbon was found to be FM with a Curie temperature TC of 170 K, as determined from the AC susceptibility curve [17]. In Ref. [17], only the paramagnetic magnetization curve of BCC Fe2MnGa at room temperature is presented. Since Fe2MnGa has been predicted to have a high spin polarization ratio and a total moment close to 2 μB [15,16], it is of interest to investigate its low temperature magnetic properties and compare them with theoretical results. Also, knowledge of the magnetic moment at 5 K is important to investigate the atomic ordering in Fe2MnGa ribbons, which is still not quite clear yet. In the present study, we have prepared BCC Fe2MnGa meltspun ribbons. The structural and magnetic properties have been measured and are compared with theoretical results. The influence of atomic disorder on the electronic structure and magnetic properties of the Heusler alloy Fe2MnGa is discussed. The large
52
Y. Xin et al. / Physica B 489 (2016) 51–55
difference between the calculated and the measured saturated magnetic moments is explained by Fe–Mn disorder evolving from the ordered L21 structure.
2. Experimental and computational methods Fe2MnGa ingots were prepared by arc-melting the constituent elements in a high-purity argon atmosphere. The purity of the starting materials was 99.9% or higher. The ingots were melted at least four times, then annealed for homogenization at 1273 K for three days under protection of argon atmosphere. Then, melt-spun ribbons were prepared by a single-wheel technique with the substrate velocity (vs) of 18 m/s, under protection of Ar atmosphere. X-ray powder diffraction (XRD) with Cu-Kα radiation was used to verify the crystal structure and to determine the lattice constants. The magnetization was measured in a SQUID magnetometer. The temperature dependence of AC susceptibility was measured to determine the Curie temperature. The electronic structure and magnetism of Fe2MnGa were calculated by means of the CASTEP code based on the pseudopotential method with a plane-wave basis set [18,19]. The interactions between the atomic core and the valence electrons were described by the ultra-soft pseudopotential [20]. The electronic exchange–correlation energy was treated under the generalized-gradient-approximation (GGA) [21]. To simulate the Fe–Mn or Mn–Ga disorder in Fe2MnGa, we used the supercell method in a 16-atom supercell, which has resulted in the chemical formula Fe8Mn4Ga4. In all cases, a plane-wave basis set cut-off of 500 eV was used. A mesh of 18 18 18 k-points was employed for Brillouin zone integrations during calculations. These parameters ensured good convergences for total energy. The convergence tolerance for the calculations was selected as a difference on total energy within 1 10 6 eV/atom.
3. Results and discussion 3.1. Formation and magnetic properties Fe2MnGa ribbon samples were prepared by melt-spinning method. The XRD pattern of finely powdered Fe2MnGa is shown Fig. 1. In the XRD pattern, the main reflections all belong to the BCC phase except for a small γ-phase reflection at 42.3°. So, the result indicates that single BCC phase Fe2MnGa can be obtained by using the melt-spinning technique [17]. The derived lattice constant for Fe2MnGa is 5.832 Å, which agrees well with the experimental value of 5.830 Å in Ref. [17], but is much larger than the calculated lattice constant of 5.69 Å [15]. In the XRD pattern, two superlattice reflections (111) and (200) are seen, which represent the atomic ordering in Fe2MnGa. It is known that, if we consider the highly-ordered structures only, there are two possible atomic orderings in Heusler alloys: One is the L21 structure (Cu2MnAl-type, space group No. 225), in which the two X atoms occupy A and C positions and the Y, Z atoms enter B and D positions, respectively. The other is the XA structure (Hg2CuTi-type, space group No. 216), in which the two X atoms occupy A and B positions and the Y, Z atoms occupy C and D positions, respectively. This structure is also called “inverse” Heusler alloy [22]. In the literature, Fe2MnGa is predicted to crystallize in the L21 structure [16]. This agrees with the valence-electrons rule in Heusler alloys, which says that “the site preferences of transition metal atoms X and Y are determined by the number of their valence electrons: atoms with more electrons tend to occupy the A and C positions while the atoms with fewer electrons prefer the B position” [23,24]. However, since exceptions have been reported for X2CuAl Heusler alloys [25], it is of interest to check the site preference in Fe2MnGa by experimental and theoretical methods.
Fig. 1. Powder X-ray diffraction pattern of Fe2MnGa. The insets show the (111) and (200) superlattice reflections of the experimental pattern and of simulated patterns of the L21 structure, the XA structure and structures with Fe–Mn disorder or Mn–Ga disorder.
Generally, the highly-ordered Heusler structure is represented by the existence of the superlattice reflections (111) and (200). In the inset of Fig. 1, an enlargement of the experimental patterns between 24° and 32° is presented in which both the (111) and the (200) reflection are clearly seen, indicating an ordered structure of the Fe2MnGa samples. The intensities of lattice reflections are proportional to the square of the structure factor F. The superlattice reflections (111) and (200) are order-dependent, while the principal reflection (220) is order-independent. Therefore, different types of atomic order can be distinguished by comparing the intensity ratios of the different reflections I(111)/I(220) and I(200)/I(220) or I(111)/I (200) [22,26]. In the inset of Fig. 1, the experimental result is compared with simulated patterns for ordered L21 and XA structures and Fe–Mn, Mn–Ga disordered structures. It can be seen that in the pattern of the L21 structure, I(111) is much higher than I(200) but, in the experimental pattern, I(111) and I(200) are similar. So, there may exist a certain degree of antisite disorder based on the L21 structure. Mn (B)–Ga (D) disorder is an usual disorder in Heusler alloys due to which, as shown in Fig. 1, the (111) reflection disappears completely and only the (200) peak remains. This is also quite different from the experimental result. Another usual disorder is between A, B sites. In Fe2MnGa, it means that some Fe (A) and Mn (B) atoms change positions on the L21 lattice. If Mn and Fe change completely, the XA structure will be formed instead of L21. However, with increasing Fe– Mn disorder, I (111) is always visible and higher than I (200) to some extent. This agrees qualitatively with the experimental results. In Mn2NiSb ribbon, huge Ni (A)–Mn (B) disorder introduced by meltspinning has already been reported [27] and similar effects may occur in Fe2MnGa ribbons. However, in view of the possible texture and strain in the melt-spun ribbons, further investigations are necessary to come to proper conclusions. It is known that the magnetic properties of Heusler alloys can differ a lot when magnetic atoms like Fe or Mn occupy different sites. We hope that this may help to determine the atomic ordering in Fe2MnGa. The temperature dependence of the AC susceptibility of a Fe2MnGa ribbon sample measured from 77 K to 380 K is shown in the inset of Fig. 2. A sharp FM-paramagnetic transition is observed at the Curie temperature of about 185 K, which is somewhat higher than the 170 K reported in Ref. [17]. Fig. 2 shows the magnetization of Fe2MnGa ribbon sample measured at 5 K in fields up to 5 T. The saturation magnetization Ms at 5 K is 3.68 μB/f.u. This is a rather large value and about 1.8 times
Y. Xin et al. / Physica B 489 (2016) 51–55
Fig. 2. Magnetization of Fe2MnGa at 5 K in a field up to 5 T. The inset shows the temperature dependence of the AC susceptibility. The Curie temperature TC is indicated by an arrow.
larger than the theoretical value of about 2 μB reported in literature [15,16]. It is known that, in Fe2Mn-based Heusler alloys, Mn enters the B site and its spin moment is antiparallel to the Fe moments at A, C sites which leads to a relatively small total moment. For example, Ms of Fe2MnAl at 5 K is 1.58 μB [28]. So it is quite interesting to investigate the origin of the large Ms in Fe2MnGa. 3.2. Electronic structure As has been discussed above, there may exist a certain degree of Fe (A)–Mn (B) disorder in Fe2MnGa ribbons. When Fe enters the B site, it is located in a BCC sublattice and its densities of states (DOS) will show large large exchange splitting due to the crystal field effect. Thus its spin moment will become large and be parallel to the Fe moments at A, C sites. So this disorder can enhance the total moment of Fe2MnGa compared with the L21 type Fe2MnGa. In order to investigate the influence of atomic disorder on the magnetic properties of Fe2MnGa further, we have calculated the electronic structure and magnetic properties of L21- and XA-type ordered Fe2MnGa and of alloys with Fe (A)–Mn (B) or Mn (B)–Ga (D) disorder. In the calculations, we have considered FM and FIM states for the two alloys. In the FIM state, antiparallel coupled moments between Fe (B)–Mn (A) or Fe (C)–Mn (A) in the XA structure and between Fe (A, C)–Mn (B) in the L21 structure have been considered. For the Fe (A)–Mn (B) disordered structure, we have also considered antiparallel coupled Mn spin moments at A and B sites. The structure with a certain degree of Fe (A)–Mn (B) disorder can be looked at as an intermediate state between the XA and the L21 structure. We will first discuss the properties of the ordered XA and L21 structures. The total energies of Fe2MnGa with the XA and the L21 structure have been calculated as a function of the lattice constant and are shown in Fig. 3. It is clear that L21 type Fe2MnGa in the FIM state with antiparallel coupling between Fe and Mn spin moments has the lowest total energy and, therefore, is the most stable state. The energy difference between the FIM state and the FM state is 0.29 eV. This is a relatively large value and ensures the high stability of the FIM state in the L21 type Fe2MnGa. Unlike in the L21 structure, the FM state is more stable in XA type Fe2MnGa, which is located 0.1 eV above the most stable L21 curve. In Fig. 3, we can see that the XA structure tends to have a larger lattice constant than the L21 structure and also that the FM state has a larger lattice constant in both the XA and the L21 structure. The derived equilibrium lattice constants for L21 and XA type Fe2MnGa are listed in Table 1.
53
Fig. 3. The calculated total energies as a function of the lattice constant for L21 and XA type Fe2MnGa in the FM and the FIM state. Here, the total energy of the FIM L21 phase was set as zero for comparison.
The calculated total and partial spin moments for Fe2MnGa with the L21 and the XA structure are also presented in Table 1. In L21 type Fe2MnGa, the Fe atoms at A, C sites have a small and negative spin moment of 0.52 μB. This leads to a relatively small total moment of 2.04 μB, although the Mn moment is as high as 3.08 μB. Similar results have been reported in Refs. [15] and [16]. However, in XA type Fe2MnGa, the Fe (C), Fe (B) and Mn (A) spin moments are parallel to each other and quite large, which leads to a large total moment. Then, if a certain degree of Fe (A)–Mn (B) disorder is introduced in the ordered L21 structure by melt spinning, the antiparallel coupling between Fe and Mn spin moments can be relieved and the total moment can increase substantially. We will discuss this in detail in the next section. In Fig. 4, the total and partial DOS of L21 and XA type Fe2MnGa are presented. The DOS of L21 type Fe2MnGa looks like a spin gapless semiconductor (SGS) and, for both spin directions, the Fermi level EF is located in an energy gap or valley. This results in rather low density of states N(EF) at EF and may help to increase the stability of the L21 structure [29]. The origin of the gap can be understood as follows: Fe2MnGa has 26 valence electrons, 14 of them tend to enter the majority spin band and the other 12 the minority spin band. This will introduce energy gaps or valleys at EF for both spin directions. Similar results have also been observed in Mn2CoAl, which also contains 26 valence electrons and is the first-reported SGS Heusler alloy [7]. The detail mechanism can be found in Refs. [25] and [30]. The DOS structure of XA type Fe2MnGa is quite different from the L21 DOS. There is a sharp DOS peak at EF in the minority spin. This clearly increases N(EF) which is not favorable for the stability of the XA structure. There are also clear differences between the PDOS of Fe in the L21 and the XA structure. In the XA DOS, the majority Fe states are basically below 0.6 eV while a part of the minority Fe states is located far above EF. This large exchange splitting can give rise to large partial moments on the Fe (A) and Fe (B) sites in the XA structure. However, in the L21 DOS, the majority Fe states are shifted to higher energy and there is a pronounced majority DOS peak at 0.45 eV. This weakens the exchange splitting and lowers the Fe moments in the L21 structure. 3.3. Influence of atomic disorder In this section, we will discuss the influence of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa. In Fig. 5, the structural optimization results for the disordered system are presented. It can be seen that, compared
54
Y. Xin et al. / Physica B 489 (2016) 51–55
Table 1 Total-energy difference ΔE between the XA and L21 structures, lattice constants a and magnetic moments for L21 and XA type Fe2MnGa. Here aexp and Ms are the experimental lattice constant at room temperature and the saturated magnetic moment at 5 K, respectively. Structure
ΔE (eV)
a (Å)
aexp (Å)
Mt (μB)
L21 XA
0.1
5.69 5.83
5.832
2.04 7.48
MFe(A)/(A,C) (μB) 0.52 2.22
MFe(B) (μB)
MMn(B) (μB)
MMn(C) (μB)
Ms (μB)
TC (K)
– 2.24
3.08 –
– 3.22
3.68
185
Fig. 4. Spin-projected total and partial DOS for L21 and XA type Fe2MnGa.
with ordered L21 Fe2MnGa, both Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder can lead to an expansion of the lattice: the equilibrium lattice constant is 5.76 Å for the former and 5.71 Å for the latter, both of them being larger than the 5.69 Å for the L21 structure. At the same time, the 5.76 Å for Fe (A)–Mn (B) disorder is more consistent with the experimental lattice constant of 5.832 Å. It can also be seen that the Fe (A)–Mn (B) disorder has a lower total energy compared with Ga (D)–Mn (B) disorder, so that the former is more likely to occur in Fe2MnGa ribbons. The influences of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa are summarized in Table 2. The total moment of Fe2MnGa with Ga (D)–Mn (B) disorder is 2.14 μB and quite close to the 2.04 μB for the L21 structure. Therefore, it can not cause the large saturation magnetization observed at 5 K. Fe (A)–Mn (B) disorder has a quite different effect, the calculated total moment being 3.74 μB, which is close to the experimental value of 3.68 μB. This further supports the assumption that there exists huge Fe–Mn antisite disorder in Fe2MnGa ribbons. It is also known that the disorder between A, B sites usually can lower the spin polarization ratio and destroy the half-metallicity. Then, the application of BCC Fe2MnGa in spintronics may be limited. In Table 2, we can see that, when Fe–Mn disorder occurs, the Mn (A) and Mn (B) partial moments are antiparallel to each other, and one of them has FM coupling with the Fe spin moments while, in the ordered L21 structure, the Fe and Mn moments are strictly
Fig. 5. Structural optimization results for Fe2MnGa with Fe–Mn and Mn–Ga disorder.
antiparallel aligned. At the same time, the spin moments of Fe also increase clearly with Fe–Mn disorder, especially the antisite Fe entering B site, due to the different chemical surroundings of Fe compared with that in the ordered L21 structure. All this leads to the large saturation magnetic moment of Fe2MnGa ribbons.
Y. Xin et al. / Physica B 489 (2016) 51–55
Table 2 Influence of Fe (A)–Mn (B) and Ga (D)–Mn (B) disorder on the magnetic properties of Fe2MnGa. Disorder
Mt (μB)
MFe(A) (μB)
Fe (A)–Mn 3.74 (B) Ga (D)–Mn 2.14 (B)
1.74
2.02
2.54
0.60
0.60
–
MFe(C) (μB)
MFe(B) (μB)
MMn(A) (μB) 2.94 –
MMn(B) (μB)
MMn(D) (μB)
2.52
–
3.28
3.28
4. Conclusions Single-phase Fe2MnGa Heusler alloy can be prepared by the meltspinning technique. The Curie temperature of BCC Fe2MnGa ribbon is about 185 K and the saturation moment at 5 K is 3.68 μB/f.u., which is much larger than the total moment of about 2 μB calculated in literature as well as in the present work for Fe2MnGa with the ordered L21 structure. This difference can be explained by the Fe–Mn disorder on A, B sites introduced by melt spinning. The XRD pattern also reveals the possible existence of this kind of disorder. L21 type Fe2MnGa is a ferrimagnet with strictly antiparallel Fe and Mn spin moments. However, when Fe–Mn disorder occurs, part of Mn moments will be parallel to Fe moments and the spin moments of Fe also increase clearly at the same time. All this results in a total moment of 3.74 μB, which agrees quite well with the experimental value.
Acknowledgments This work is supported by the National Natural Science Foundation of China in Grant no. 11474343 and 51371075, by the Foundation of Hebei Provincial Education Department in Grant no. BJ2014012 and also by Program for Changjiang Scholars and Innovative Research Team in University in Grant no. IRT13060.
References [1] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Phys. Rev. Lett. 50 (1983) 2024. [2] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar, M. L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488.
55
[3] Y. Sakuraba, K. Izumi, T. Iwase, S. Bosu, K. Saito, K. Takanashi, Y. Miura, K. Futatsukawa, K. Abe, M. Shirai, Phys. Rev. B 82 (2010) 094444. [4] K. Ullakko, J.K. Huang, C. Kanter, V.V. Kokorin, R.C. O’Handley, Appl. Phys. Lett. 69 (1996) 1966. [5] R. Kainuma, Y. Imano, W. Ito, Y. Sutou, H. Morito, S. Okamoto, O. Kitakami, K. Oikawa, A. Fujita, T. Kanomata, K. Ishida, Nature 439 (2006) 957. [6] S. Chadov, X.L. Qi, J. Kübler, G.H. Fecher, C. Felser, S.C. Zhang, Nat. Mater. 9 (2010) 541. [7] S. Ouardi, G.H. Fecher, C. Felser, Phys. Rev. Lett. 110 (2013) 100401. [8] T. Omori, K. Watanabe, R. Umetsu, R. Kainuma, K. Ishida, Appl. Phys. Lett. 95 (2009) 082508. [9] W. Zhu, E.K. Liu, L. Feng, X.D. Tang, J.L. Chen, G.H. Wu, H.Y. Liu, F.B. Meng, H. Z. Luo, Appl. Phys. Lett. 95 (2009) 222512. [10] X.D. Tang, W.H. Wang, W. Zhu, E.K. Liu, G.H. Wu, F.B. Meng, H.Y. Liu, H.Z. Luo, Appl. Phys. Lett. 97 (2010) 242513. [11] M. Yin, P. Nash, Intermetallics 57 (2015) 34. [12] Y.V. Kudryavtsev, N.V. Uvarov, V.N. Iermolenko, I.N. Glavatskyy, J. Dubowik, Acta Mater. 60 (2012) 4780. [13] T. Gasi, A.K. Nayk, M. Nicklas, C. Felser, J. Appl. Phys. 113 (2013) 17E301. [14] E.C. Passamani, C. Larica, G. Viali, J.R. Andrez, A.Y. Takeuchi, V.P. Nascimento, V. A.P. Rodriguez, C. Rojas-Ayala, E. Baggio-Saitovitch, J. Alloy. Compd. 628 (2015) 164. [15] A.T. Zayak, P. Entel, K.M. Rabe, W.A. Adeagbo, M. Acet, Phys. Rev. B 72 (2005) 054113. [16] X.H. Liu, J.B. Lin, Y.H. Liu, Y.J. Jin, Acta Phys. Sin. 60 (2011) 107104. [17] H. Okumura, E. Hashiba, K. Nagata, Intermetallics 49 (2014) 65. [18] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1065. [19] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M. C. Payne, J. Phys. : Condens. Matter 14 (2002) 2717. [20] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [21] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [22] T. Graf, C. Felser, S.S. Parkin, Prog. Solid State Chem. 39 (2011) 1. [23] T.J. Burch, T. Litrenka, Phys. Rev. Lett. 33 (1974) 421. [24] H.C. Kandpal, G.H. Fecher, C. Felser, J. Phys. D: Appl. Phys. 40 (2007) 1507. [25] H.Z. Luo, Y.P. Xin, B.H. Liu, F.B. Meng, H.Y. Liu, E.K. Liu, G.H. Wu, J. Alloy. Compd. 665 (2016) 180. [26] K. Endo, T. Kanomata, H. Nishihara, K.R.A. Ziebeck, J. Alloy. Compds 510 (2012) 1. [27] H.Z. Luo, W. Zhu, L. Ma, G.D. Liu, Y.X. Li, X.X. Zhu, C.B. Jiang, H.B. Xu, G.H. Wu, J. Phys. D: Appl. Phys. 42 (2009) 095001. [28] Z.H. Liu, X.Q. Ma, F.B. Meng, G.H. Wu, J. Alloy. Compd. 509 (2011) 3219. [29] T. Hong, T.J. Watson-Yang, A.J. Freeman, T. Oguchi, J.H. Xu, Phys. Rev. B 41 (1990) 12462. [30] G.Z. Xu, E.K. Liu, Y. Du, G.J. Li, G.D. Liu, W.H. Wang, G.H. Wu, EPL 102 (2013) 17007.