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Electronic structure and magnetism of binary Fe-based half-Heusler alloys Fe2Z (Z=In, Sn, Sb and As)
Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Electronic structure and magnetism of binary Fe-based half-Heusler alloys Fe2Z (Z ¼ In, Sn, Sb and As) Jianqiang Li a, Fanbin Meng a, Guodong Liu a, Xueguang Chen a, Luo Hongzhi a,b,n, Enke Liu b, Guangheng Wu b a b
School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, P R China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P R China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 September 2012 Received in revised form 12 November 2012 Available online 27 November 2012
The electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z ¼ In, Sn, Sb and As) have been studied. It is found that binary Fe2Z has a site preference similar to normal ternary ones. Hybridization between the d states of Fe (A) and Fe (B) leads to the formation of a d–d band gap near EF. This gap is broadened and shifted to low energy as Z varies from In to Sb, which makes Fe2Sb a half-metal. The half-metallicity of Fe2Sb is insensitive to the lattice distortion. The spin polarization ˚ The spin polarization of Fe2As is also high at equilibrium lattice ratio is always 100% from 5.6 A˚ to 6.2 A. constant. With a small expansion of the lattice, it becomes an ideal half-metal. Fe2Z alloys are all ferromagnets with parallel aligned Fe spin moments. & 2012 Elsevier B.V. All rights reserved.
Keywords: Heusler alloy Half-metal Electronic structure
1. Introduction In recent years, half-metallic materials have attracted growing attention for their interesting physical properties and potential applications in spintronic devices [1–3]. One energy band of the half-metal has a semiconductor-like gap at the Fermi level EF, while the other spin band overlaps with EF and shows metallic character. So there is a complete spin polarization of the conduction electrons at the Fermi level position. Half-metals can be used as spin injectors for magnetic random access memories and other spin dependent devices [3,4]. Heusler alloys are a large family of half-metals. For applications in spintronics, they should have high Curie temperatures and be easily prepared as thin films. Heusler alloys can be divided into fullHeusler alloy and half-Heusler alloy. The corresponding chemical formulas for them are X2YZ and XYZ. Here X and Y are transitionmetal elements like Mn, Fe, and Co, and Z is a main-group element like In, Sn and Sb. Half-Heusler alloy XYZ crystallizes in a highlyordered cubic structure, the conventionally stable structure, for which is that the X and Y atoms locate at the A(0,0,0) and the B(1/4,1/4,1/4) Wyckoff positions, respectively, and the Z atom enters the D(3/4,3/4,3/4) position, leaving the C(1/2,1/2,1/2) position unoccupied.
n Corresponding author at: School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, P R China. Tel.: þ86 10 8264 9247; fax: þ 86 10 6256 9068. E-mail addresses: [email protected], [email protected] (L. Hongzhi).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.11.038
In half-Heusler alloys, half-metallic character has been investigated both theoretically and experimentally. The first predicted halfmetal was NiMnSb [1]. Since then, other alloys like PtMnSb, NiCrZ and MnCrSb [5–9] have also been reported. Recently, Fujii et al. reported the half-metallicity in binary alloys Mn2Z (Z¼P, As, Sb and Ge) with half-Heusler structure [10]. Luo et al. also found that Mn2Sn was a half-metallic fully compensated ferrimagnet (HMFCF) [11], in which the Mn moment at A and B sites compensates each other and results in a zero total moment. In Mn2Sn, a transition from half-metal to semi-metal with pressure has been predicted. In ternary half-Heusler alloys, the covalent hybridization between the lower-energy d states of the high-valent transition metal atom X and the higher-energy d states of the low-valent transition metal Y is important for the formation of the halfmetallic gap [12]. Unlike traditional XYZ alloys, the X and Y atoms in binary half-Heusler alloys are the same, so it is interesting to investigate the electronic structure of binary half-Heusler alloys with different compositions. This can help discover new halfmetallic alloys. In this paper, we studied the site preference, electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z¼In, Sn, Sb and As). Some new half-metals have been predicted.
2. Computational method The electronic structure was calculated by means of the pseudopotential method with a plane-wave basis set based on
J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
density-functional theory [13,14]. The interactions between the atomic core and the valence electrons are described by the ultrasoft pseudopotential [15]. The electronic exchange-correlation energy has been treated under the Generalized Gradient Approximation (GGA) [16]. To ensure good convergence for total energy, a plane-wave basis set cut-off of 500 eV was used and a mesh of 14 14 14 k-points was employed in the irreducible Brillouin zone integration. The calculations were performed based on the theoretical equilibrium lattice parameters. The site preference of 3d elements in Heusler alloys is determined by the number of valence electrons. Atoms with more valence electrons prefer occupying the (A,C) sites while atoms with fewer electrons enter the B site [17]. In Fe2Z, the only 3d atom is Fe, so it is interesting to investigate its site preference. Here we compared the stability of two kinds of crystal models: one is the two Fe atoms entering the A(0,0,0) and the B(1/4,1/4,1/4) sites, which is also the traditional structure of half-Heusler alloys, and the other is the two Fe atoms entering the A(0,0,0) and the C(1/2,1/2,1/2) sites.
3. Results and discussion Structural optimizations on Fe2Z (Z¼ In, Sn, Sb, and As) were performed first to determine the equilibrium lattice constants. In the calculations non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) states were considered. It is found that the structure with Fe entering the A and C sites is about 1 eV higher in energy compared with Fe entering A and B sites, indicating that Fe2Z also prefers crystallizing in the traditional half-Heusler structure like the ternary ones. In Fig. 1, we presented the energy–lattice curves of only the stable structure (with C site unoccupied) for visibility. It is also found that, in Fe2Z, the FM state is lower in energy and more stable: for Fe2In, the AFM state is about 0.3 eV higher compared with the FM one. And for other three Fe2Z alloys, the AFM calculations also converged to the FM ground state finally and had the same total energies compared with the FM calculations.
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The derived equilibrium lattice constants are listed in Table 1. As the Z atom varies from In to As, the lattice constants decrease gradually, which is due to the contraction of the atomic radius of Z. Also, the ˚ which is slightly larger than lattice constant of AFM Fe2In is 5.89 A, that of the FM state. Fig. 2 presents the energy bands of Fe2Z at equilibrium lattice constants. It can be seen that the band structures of these alloys are quite similar around the Fermi level. In the majority spin direction, the energy bands show a metallic character, while in the minority spin, an indirect energy gap near EF is opened. In FM Fe2In, the gap is about 1.0 eV above the Fermi level. In Fe2Sn, the gap is broadened and shifted to þ0.5 eV. This change may have two causes: one is the increasing number of valence electrons, and the other possible cause is the contraction of the cell volume [18,19]. In Fe2Sb, the gap is broadened further and opened around the Fermi level. The L–G gap width is about 0.3 eV. This results in a 100% spin polarization of the conduction electrons and makes Fe2Sb a half-metal. The band structure of Fe2As is also shown in Fig. 2. It can be seen that the energy gap around EF is even broader in Fe2As compared with Fe2Sb. However, the bottom of the minority conduction band overlaps with EF at the X point, so the spin polarization of Fe2As is 96% at equilibrium lattice constant, which is also a high value and meaningful for technical
Table 1 Calculated equilibrium lattice constants a, total and partial magnetic moments M and spin polarization ratio P of Fe2Z (Z ¼In, Sn, Sb, and As). For Fe2In, both the ferromagnetic (FM) and antiferromagnetic (AFM) results are presented. Compounds
˚ a (A)
Mt (mB)
MFe(A) (mB)
MFe(B) (mB)
MZ (mB)
P (%)
Fe2In (AFM) Fe2In (FM) Fe2Sn Fe2Sb Fe2As
5.86 5.89 5.82 5.74 5.44
1.00 4.60 3.95 3.00 2.97
1.52 2.10 1.62 0.54 0.54
2.58 2.54 2.38 2.50 2.42
0.06 0.04 0.06 0.02 0.02
100 – – 100 96
Fig. 1. Calculated total energy for binary half-Heusler alloys Fe2Z (Z¼ In, Sn, Sb, As) as functions of lattice parameters for non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) states.
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J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
Fig. 2. Majority-spin (left column) and minority-spin (right column) band structures for Fe2Z (Z ¼ In, Sn, Sb, and As).
˚ this applications. The lattice constant of Fe2As is only 5.44 A; small lattice constant will help relieve the lattice mismatch when it is grown as thin films on semiconductor substrate. In Fig. 2, we also presented the energy bands of AFM Fe2In for comparison. It is clear that, in the AFM band structure, the minority energy bands move to lower energy compared with the FM one. The minority energy gap is shifted to the Fermi level position and obviously broadened, which makes AFM Fe2In a half-metal. So it can be interesting to investigate the competition of the FM and AFM states in Fe2Z Heusler alloys in further studies. In order to understand the electronic structure of Fe2Z further, the density of states (DOS) is presented in Fig. 3. It can be seen
that the DOS of Fe2Z alloys has a similar structure. The majority DOS is divided into bonding and antibonding parts and is basically below the Fermi level; while in the minority spin, the antibonding peak is shifted high above EF due to the exchange splitting [20]. As the Z atom varies from In to Sb, the minority DOS together with the energy gap moves to low energy. For Fe2Sb, a half-metallic gap is opened around EF. This agrees with preceding discussion on the band structure quite well. The DOS of Fe2Z is mainly determined by the partial DOS (PDOS) of Fe (A) and Fe (B), the d states of Fe (A) and Fe (B); are in the same energy region, indicating that there exists strong hybridization between them. It may also be noticed that the variation of Z atom on the PDOS of Fe (A) and Fe (B) is quite different.
J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
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Fig. 3. Spin-projected total and partial DOS for Fe2Z (Z ¼ In, Sn, Sb, and As).
In the PDOS of Fe2Z, both the majority and minority DOS of Fe (B) move to low energy as Z varies from In to Sb, which keeps the large exchange splitting in the d states of Fe (B). But for Fe (A), the majority DOS moves to high energy and the minority DOS moves to the opposite direction with the increase in number of valence electrons, which will lead to the decrease of Fe (A) moment. Compared with Fe2Sb, the DOS of Fe2As is even lower on energy scale, which is related to the smaller lattice constants of Fe2As. The calculated total and partial spin moments of Fe2Z are presented in Table 1. These alloys are all ferromagnets. The partial moments of Fe (A) and Fe (B) are in parallel alignment. As the Z atom varies from In to Sb, Fe (A) moment decreases obviously
with increasing number of valence electrons, while the change of Fe (B) moment is rather small. This leads to the decrease of the total moment. The calculated total moment of Fe2Sb is 3.00mB, which agrees with the Slater–Pauling curve quite well. In half-metallic Heusler alloys, the Fermi level locates in the minority energy gap, so their magnetic moments are integral values and can be described by the S–P curve of Mt ¼NV 18, where Mt is the total spin magnetic moment per formula unit and NV is the total number of valence electrons [12]. Fe2As also has 21 valence electrons and its spin moment is 2.97mB, close to the ideal value of 3mB. It may also be noticed that the total moment of AFM Fe2In is 1.00mB, which is much smaller than the 4.60mB in FM Fe2In.
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This is mainly related to the antiparallel aligned Fe spin moments. Though the AFM Fe2In is higher on energy scale compared with the FM one, we may still expect that by doping and alloying in Fe2In, the total energy of AFM state can be lowered and then magnetic transition between FM and AFM states may occur. This can be important in searching for materials with high magnetocaloric effect. As has been discussed above, Fe2As is not an ideal half-metal as the bottom of its conduction band overlaps with EF. It is also known that a slight change of the lattice can influence the electronic properties around the Fermi level obviously [21,22]. In experimental preparations, the strain in the sample may be large and make the lattice constant to deviate from the equilibrium lattice constant. Meanwhile, for application in spintronics, half-metals are usually prepared as thin films. The lattice constant of the thin films is strongly influenced by the lattice of the substrate. So it is meaningful
to study the variation of half-metallicity of Fe2As and Fe2Sb with different lattice constants. Fe2As becomes a half-metal as the lattice constant is larger ˚ As an example, we present the energy band of Fe2As at than 5.6 A. 5.6 A˚ in Fig. 4. It is clear that with increasing lattice constant, the bottom of the minority conduction band moves to high energy and a real gap is opened around EF. In Fig. 5, variations of total spin moment and spin polarization ratio of Fe2As and Fe2Sb as functions of lattice constants are presented from 5.4 A˚ to 6.2 A˚ ˚ For Fe2Sb, the spin polarization ratio remains with a step of 0.1 A. ˚ 100% and the total moment remains 3mB between 5.6 A˚ and 6.2 A, which is a rather wide range. So we can say that the halfmetallicity of Fe2Sb is stable under uniform lattice distortion. This is related to the wide minority energy gap. At equilibrium lattice constant, the Fermi level is located in the middle of the gap, and moves within the gap during a moderate lattice distortion, which helps to keep the 100% spin polarization ratio. For ˚ the spin Fe2As, when the lattice constant increases to 5.6 A, ˚ and the total spin moment polarization becomes 100% till 6.05 A, is always 3.00mB within this range.
4. Conclusion
˚ Fig. 4. Band structure of Fe2As calculated at 5.6 A.
The site preference, electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z¼ In, Sn, Sb and As) have been studied. Just like ternary Heusler alloys, in Fe2Z, the two Fe atoms prefer entering the A and B sites and leaving the C sites unoccupied. The hybridization between the d states of Fe (A) and Fe (B) formats a d–d energy gap near EF. As Z varies from In to Sb, this gap is broadened and shifted to lower energy. In Fe2Sb, EF is located just in the middle of the gap, which makes it a half-metal. The half-metallicity of Fe2Sb is insensitive to the lattice constant and the spin polarization ratio is always 100% in ˚ With a small expansion of the the range from 5.6 A˚ to 6.2 A. lattice, Fe2As can also be half-metallic. Fe2Z alloys are all ferromagnets with parallel aligned Fe spin moments.
Fig. 5. Total and partial magnetic moments and spin polarization ratio P as functions of the lattice constant for Heusler alloys Fe2As and Fe2Sb. The calculation ˚ was performed within the range of 5.4–6.2 A.
J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 50901028 and 51171056, and the Natural Science Foundation of Hebei under Grant no. E2012202009. References [1] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Physical Review Letters 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. [3] M. Julliere, Physics Letters 54A (1975) 225. [4] I. Zutic, J. Fabian, S. Das Sarma, Reviews of Modern Physics 76 (2004) 323. [5] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Bushow, Journal of Applied Physics 55 (1990) 2151. [6] E. S- as-ıo˘glu, L.M. Sandratskii, P. Bruno, Journal of Applied Physics 98 (2005) 063523. [7] B.R.K. Nanda, I. Dasgupta, Computational Materials Science 36 (2006) 96. [8] R.A. de Groot, Physica B 172 (1991) 45. [9] F. Casper, T. Graf, S. Chadov, B. Balke, C. Felser, Semiconductor Science and Technology 27 (2012) 063001.
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[10] S. Fujii, S. Ishida, S. Asano, Journal of the Physical Society of Japan 79 (2010) 124702. [11] H.Z. Luo, G.D. Liu, F.B. Meng, W.H. Wang, G.H. Wu, X.X. Zhu, C.B. Jiang, Physica B 406 (2011) 4245. [12] I. Galanakis, Ph. Mavropoulos, P.H. Dederichs, Journal of Physics D: Applied Physics 39 (2006) 765. [13] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoolous, Reviews of Modern Physics 64 (1992) 1065. [14] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M.C. Payne, Journal of Physics: Condensed Matter 14 (2002) 2717. [15] D. Vanderbilt, Physical Review B 41 (1990) 7892. [16] J.P. Perdew, K. Burke, M. Ernzerhof, Physical Review Letters 77 (1996) 3865. [17] H.C. Kandpal, G.H. Fecher, C. Felser, Journal of Physics D: Applied Physics 40 (2007) 1507. [18] H.Z. Luo, H.W. Zhang, Z.Y. Zhu, L. Ma, S.F. Xu, G.H. Wu, X.X. Zhu, C.B. Jiang, H.B. Xu, Journal of Applied Physics 103 (2008) 083908. [19] T. Block, M.J. Carey, B.A. Gurney, O. Jepsen, Physical Review B 70 (2004) 205114. [20] B.R.K. Nanda, I. Dasgupta, Journal of Physics: Condensed Matter 15 (2003) 7307. [21] T. Block, M.J. Carey, B.A. Gurney, O. Jepsen, Physical Review B 70 (2004) 205114. ¨ zdo˘gan, E. S- as- ıo˘glu, B. Aktas-, Physical Review B 75 (2007) [22] I. Galanakis, K. O 172405.
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Electronic structure and magnetism of binary Fe-based half-Heusler alloys Fe2Z (Z ¼ In, Sn, Sb and As) Jianqiang Li a, Fanbin Meng a, Guodong Liu a, Xueguang Chen a, Luo Hongzhi a,b,n, Enke Liu b, Guangheng Wu b a b
School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, P R China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P R China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 September 2012 Received in revised form 12 November 2012 Available online 27 November 2012
The electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z ¼ In, Sn, Sb and As) have been studied. It is found that binary Fe2Z has a site preference similar to normal ternary ones. Hybridization between the d states of Fe (A) and Fe (B) leads to the formation of a d–d band gap near EF. This gap is broadened and shifted to low energy as Z varies from In to Sb, which makes Fe2Sb a half-metal. The half-metallicity of Fe2Sb is insensitive to the lattice distortion. The spin polarization ˚ The spin polarization of Fe2As is also high at equilibrium lattice ratio is always 100% from 5.6 A˚ to 6.2 A. constant. With a small expansion of the lattice, it becomes an ideal half-metal. Fe2Z alloys are all ferromagnets with parallel aligned Fe spin moments. & 2012 Elsevier B.V. All rights reserved.
Keywords: Heusler alloy Half-metal Electronic structure
1. Introduction In recent years, half-metallic materials have attracted growing attention for their interesting physical properties and potential applications in spintronic devices [1–3]. One energy band of the half-metal has a semiconductor-like gap at the Fermi level EF, while the other spin band overlaps with EF and shows metallic character. So there is a complete spin polarization of the conduction electrons at the Fermi level position. Half-metals can be used as spin injectors for magnetic random access memories and other spin dependent devices [3,4]. Heusler alloys are a large family of half-metals. For applications in spintronics, they should have high Curie temperatures and be easily prepared as thin films. Heusler alloys can be divided into fullHeusler alloy and half-Heusler alloy. The corresponding chemical formulas for them are X2YZ and XYZ. Here X and Y are transitionmetal elements like Mn, Fe, and Co, and Z is a main-group element like In, Sn and Sb. Half-Heusler alloy XYZ crystallizes in a highlyordered cubic structure, the conventionally stable structure, for which is that the X and Y atoms locate at the A(0,0,0) and the B(1/4,1/4,1/4) Wyckoff positions, respectively, and the Z atom enters the D(3/4,3/4,3/4) position, leaving the C(1/2,1/2,1/2) position unoccupied.
n Corresponding author at: School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, P R China. Tel.: þ86 10 8264 9247; fax: þ 86 10 6256 9068. E-mail addresses: [email protected], [email protected] (L. Hongzhi).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.11.038
In half-Heusler alloys, half-metallic character has been investigated both theoretically and experimentally. The first predicted halfmetal was NiMnSb [1]. Since then, other alloys like PtMnSb, NiCrZ and MnCrSb [5–9] have also been reported. Recently, Fujii et al. reported the half-metallicity in binary alloys Mn2Z (Z¼P, As, Sb and Ge) with half-Heusler structure [10]. Luo et al. also found that Mn2Sn was a half-metallic fully compensated ferrimagnet (HMFCF) [11], in which the Mn moment at A and B sites compensates each other and results in a zero total moment. In Mn2Sn, a transition from half-metal to semi-metal with pressure has been predicted. In ternary half-Heusler alloys, the covalent hybridization between the lower-energy d states of the high-valent transition metal atom X and the higher-energy d states of the low-valent transition metal Y is important for the formation of the halfmetallic gap [12]. Unlike traditional XYZ alloys, the X and Y atoms in binary half-Heusler alloys are the same, so it is interesting to investigate the electronic structure of binary half-Heusler alloys with different compositions. This can help discover new halfmetallic alloys. In this paper, we studied the site preference, electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z¼In, Sn, Sb and As). Some new half-metals have been predicted.
2. Computational method The electronic structure was calculated by means of the pseudopotential method with a plane-wave basis set based on
J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
density-functional theory [13,14]. The interactions between the atomic core and the valence electrons are described by the ultrasoft pseudopotential [15]. The electronic exchange-correlation energy has been treated under the Generalized Gradient Approximation (GGA) [16]. To ensure good convergence for total energy, a plane-wave basis set cut-off of 500 eV was used and a mesh of 14 14 14 k-points was employed in the irreducible Brillouin zone integration. The calculations were performed based on the theoretical equilibrium lattice parameters. The site preference of 3d elements in Heusler alloys is determined by the number of valence electrons. Atoms with more valence electrons prefer occupying the (A,C) sites while atoms with fewer electrons enter the B site [17]. In Fe2Z, the only 3d atom is Fe, so it is interesting to investigate its site preference. Here we compared the stability of two kinds of crystal models: one is the two Fe atoms entering the A(0,0,0) and the B(1/4,1/4,1/4) sites, which is also the traditional structure of half-Heusler alloys, and the other is the two Fe atoms entering the A(0,0,0) and the C(1/2,1/2,1/2) sites.
3. Results and discussion Structural optimizations on Fe2Z (Z¼ In, Sn, Sb, and As) were performed first to determine the equilibrium lattice constants. In the calculations non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) states were considered. It is found that the structure with Fe entering the A and C sites is about 1 eV higher in energy compared with Fe entering A and B sites, indicating that Fe2Z also prefers crystallizing in the traditional half-Heusler structure like the ternary ones. In Fig. 1, we presented the energy–lattice curves of only the stable structure (with C site unoccupied) for visibility. It is also found that, in Fe2Z, the FM state is lower in energy and more stable: for Fe2In, the AFM state is about 0.3 eV higher compared with the FM one. And for other three Fe2Z alloys, the AFM calculations also converged to the FM ground state finally and had the same total energies compared with the FM calculations.
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The derived equilibrium lattice constants are listed in Table 1. As the Z atom varies from In to As, the lattice constants decrease gradually, which is due to the contraction of the atomic radius of Z. Also, the ˚ which is slightly larger than lattice constant of AFM Fe2In is 5.89 A, that of the FM state. Fig. 2 presents the energy bands of Fe2Z at equilibrium lattice constants. It can be seen that the band structures of these alloys are quite similar around the Fermi level. In the majority spin direction, the energy bands show a metallic character, while in the minority spin, an indirect energy gap near EF is opened. In FM Fe2In, the gap is about 1.0 eV above the Fermi level. In Fe2Sn, the gap is broadened and shifted to þ0.5 eV. This change may have two causes: one is the increasing number of valence electrons, and the other possible cause is the contraction of the cell volume [18,19]. In Fe2Sb, the gap is broadened further and opened around the Fermi level. The L–G gap width is about 0.3 eV. This results in a 100% spin polarization of the conduction electrons and makes Fe2Sb a half-metal. The band structure of Fe2As is also shown in Fig. 2. It can be seen that the energy gap around EF is even broader in Fe2As compared with Fe2Sb. However, the bottom of the minority conduction band overlaps with EF at the X point, so the spin polarization of Fe2As is 96% at equilibrium lattice constant, which is also a high value and meaningful for technical
Table 1 Calculated equilibrium lattice constants a, total and partial magnetic moments M and spin polarization ratio P of Fe2Z (Z ¼In, Sn, Sb, and As). For Fe2In, both the ferromagnetic (FM) and antiferromagnetic (AFM) results are presented. Compounds
˚ a (A)
Mt (mB)
MFe(A) (mB)
MFe(B) (mB)
MZ (mB)
P (%)
Fe2In (AFM) Fe2In (FM) Fe2Sn Fe2Sb Fe2As
5.86 5.89 5.82 5.74 5.44
1.00 4.60 3.95 3.00 2.97
1.52 2.10 1.62 0.54 0.54
2.58 2.54 2.38 2.50 2.42
0.06 0.04 0.06 0.02 0.02
100 – – 100 96
Fig. 1. Calculated total energy for binary half-Heusler alloys Fe2Z (Z¼ In, Sn, Sb, As) as functions of lattice parameters for non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) states.
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J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
Fig. 2. Majority-spin (left column) and minority-spin (right column) band structures for Fe2Z (Z ¼ In, Sn, Sb, and As).
˚ this applications. The lattice constant of Fe2As is only 5.44 A; small lattice constant will help relieve the lattice mismatch when it is grown as thin films on semiconductor substrate. In Fig. 2, we also presented the energy bands of AFM Fe2In for comparison. It is clear that, in the AFM band structure, the minority energy bands move to lower energy compared with the FM one. The minority energy gap is shifted to the Fermi level position and obviously broadened, which makes AFM Fe2In a half-metal. So it can be interesting to investigate the competition of the FM and AFM states in Fe2Z Heusler alloys in further studies. In order to understand the electronic structure of Fe2Z further, the density of states (DOS) is presented in Fig. 3. It can be seen
that the DOS of Fe2Z alloys has a similar structure. The majority DOS is divided into bonding and antibonding parts and is basically below the Fermi level; while in the minority spin, the antibonding peak is shifted high above EF due to the exchange splitting [20]. As the Z atom varies from In to Sb, the minority DOS together with the energy gap moves to low energy. For Fe2Sb, a half-metallic gap is opened around EF. This agrees with preceding discussion on the band structure quite well. The DOS of Fe2Z is mainly determined by the partial DOS (PDOS) of Fe (A) and Fe (B), the d states of Fe (A) and Fe (B); are in the same energy region, indicating that there exists strong hybridization between them. It may also be noticed that the variation of Z atom on the PDOS of Fe (A) and Fe (B) is quite different.
J. Li et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 82–87
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Fig. 3. Spin-projected total and partial DOS for Fe2Z (Z ¼ In, Sn, Sb, and As).
In the PDOS of Fe2Z, both the majority and minority DOS of Fe (B) move to low energy as Z varies from In to Sb, which keeps the large exchange splitting in the d states of Fe (B). But for Fe (A), the majority DOS moves to high energy and the minority DOS moves to the opposite direction with the increase in number of valence electrons, which will lead to the decrease of Fe (A) moment. Compared with Fe2Sb, the DOS of Fe2As is even lower on energy scale, which is related to the smaller lattice constants of Fe2As. The calculated total and partial spin moments of Fe2Z are presented in Table 1. These alloys are all ferromagnets. The partial moments of Fe (A) and Fe (B) are in parallel alignment. As the Z atom varies from In to Sb, Fe (A) moment decreases obviously
with increasing number of valence electrons, while the change of Fe (B) moment is rather small. This leads to the decrease of the total moment. The calculated total moment of Fe2Sb is 3.00mB, which agrees with the Slater–Pauling curve quite well. In half-metallic Heusler alloys, the Fermi level locates in the minority energy gap, so their magnetic moments are integral values and can be described by the S–P curve of Mt ¼NV 18, where Mt is the total spin magnetic moment per formula unit and NV is the total number of valence electrons [12]. Fe2As also has 21 valence electrons and its spin moment is 2.97mB, close to the ideal value of 3mB. It may also be noticed that the total moment of AFM Fe2In is 1.00mB, which is much smaller than the 4.60mB in FM Fe2In.
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This is mainly related to the antiparallel aligned Fe spin moments. Though the AFM Fe2In is higher on energy scale compared with the FM one, we may still expect that by doping and alloying in Fe2In, the total energy of AFM state can be lowered and then magnetic transition between FM and AFM states may occur. This can be important in searching for materials with high magnetocaloric effect. As has been discussed above, Fe2As is not an ideal half-metal as the bottom of its conduction band overlaps with EF. It is also known that a slight change of the lattice can influence the electronic properties around the Fermi level obviously [21,22]. In experimental preparations, the strain in the sample may be large and make the lattice constant to deviate from the equilibrium lattice constant. Meanwhile, for application in spintronics, half-metals are usually prepared as thin films. The lattice constant of the thin films is strongly influenced by the lattice of the substrate. So it is meaningful
to study the variation of half-metallicity of Fe2As and Fe2Sb with different lattice constants. Fe2As becomes a half-metal as the lattice constant is larger ˚ As an example, we present the energy band of Fe2As at than 5.6 A. 5.6 A˚ in Fig. 4. It is clear that with increasing lattice constant, the bottom of the minority conduction band moves to high energy and a real gap is opened around EF. In Fig. 5, variations of total spin moment and spin polarization ratio of Fe2As and Fe2Sb as functions of lattice constants are presented from 5.4 A˚ to 6.2 A˚ ˚ For Fe2Sb, the spin polarization ratio remains with a step of 0.1 A. ˚ 100% and the total moment remains 3mB between 5.6 A˚ and 6.2 A, which is a rather wide range. So we can say that the halfmetallicity of Fe2Sb is stable under uniform lattice distortion. This is related to the wide minority energy gap. At equilibrium lattice constant, the Fermi level is located in the middle of the gap, and moves within the gap during a moderate lattice distortion, which helps to keep the 100% spin polarization ratio. For ˚ the spin Fe2As, when the lattice constant increases to 5.6 A, ˚ and the total spin moment polarization becomes 100% till 6.05 A, is always 3.00mB within this range.
4. Conclusion
˚ Fig. 4. Band structure of Fe2As calculated at 5.6 A.
The site preference, electronic structure and magnetic properties of Fe-based binary half-Heusler alloys Fe2Z (Z¼ In, Sn, Sb and As) have been studied. Just like ternary Heusler alloys, in Fe2Z, the two Fe atoms prefer entering the A and B sites and leaving the C sites unoccupied. The hybridization between the d states of Fe (A) and Fe (B) formats a d–d energy gap near EF. As Z varies from In to Sb, this gap is broadened and shifted to lower energy. In Fe2Sb, EF is located just in the middle of the gap, which makes it a half-metal. The half-metallicity of Fe2Sb is insensitive to the lattice constant and the spin polarization ratio is always 100% in ˚ With a small expansion of the the range from 5.6 A˚ to 6.2 A. lattice, Fe2As can also be half-metallic. Fe2Z alloys are all ferromagnets with parallel aligned Fe spin moments.
Fig. 5. Total and partial magnetic moments and spin polarization ratio P as functions of the lattice constant for Heusler alloys Fe2As and Fe2Sb. The calculation ˚ was performed within the range of 5.4–6.2 A.
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