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The structural, electronic and magnetic properties of quaternary Heusler alloy TiZrCoIn
Solid State Communications 231-232 (2016) 64–67
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The structural, electronic and magnetic properties of quaternary Heusler alloy TiZrCoIn Peng-Li Yan a, Jian-Min Zhang a,n, Ke-Wei Xu b a b
College of Physics and Information Technology, Shaanxi Normal University, Xian 710119, Shaanxi, PR China College of Physics and Mechanical and Electronic Engineering, Xian University of Arts and Science, Xian 710065, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 October 2015 Received in revised form 30 November 2015 Accepted 8 February 2016 Communicated by: D.D. Sarma Available online 12 February 2016
Employing the first-principles calculations, we have investigated the structural, electronic and magnetic properties of quaternary Heusler alloy TiZrCoIn. The TiZrCoIn alloy with type (I) configuration is predicted to be half-metallic ferromagnet at its equilibrium lattice constant 6.525 Å with an indirect band gap of 0.930 eV in minority spin channel. The total magnetic moment is 2 μB/f.u., following the Slater– Pauling rule μt ¼Zt 18. Moreover, the negative formation energy indicates the thermodynamical stability of this alloy. The band gap of minority spin channel is determined by the bonding (t2g) and antibonding (t1u) states created from the hybridizations of the d states of transition metal atoms Ti, Zr and Co. In addition, the HM, character is kept as hydrostatic strain ranged from 10% to 7.6% and tetragonal strain ranged from 19% to 27%. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Quaternary Heusler alloy C. Half-metallic D. Electronic structures D. Magnetic properties
1. Introduction In spintronic applications, magnetic materials with high spin polarization are crucial for the improvement of the performance of spintronic devices such as the spin filter and the spin valve [1]. Half-metallic (HM) magnets are seen as the most promising candidates of high-spin-polarization materials, because their band structure is metallic in one of the two spin channels and semiconducting or insulating in the other one, which results in complete (100%) spin polarization of electrons at the Fermi level (EF). In 1983, de Groot et al. [2] predicted HM ferromagnetism in NiMnSb. Since then a number of new classes of materials exhibiting half-metallicity have been predicted and some realized experimentally, e.g., metallic oxides such as CrO2 and Fe3 O4 , perovskites and double perovskites such as La1 x Srx MnO3 and Sr2 FeMoO6 , Heusler alloys with Clb and L21 structures such as NiMnSb and Co2 MnSi, zinc-blende compounds such as CrAs and CrSb, sp compounds (without transition metals) with zinc-blende and rocksalt structures such as CaC and SrN, and so on [3–6]. Among these materials, only Heusler-type HM alloys own very high Curie temperature TC [7–10], which yielding them to be one of the most promising candidates for spintronics devices. Therefore, further exploration of HM Heusler alloys is strongly desired. Heusler alloys are categorized as ternary and quaternary alloys. The ternary Heusler alloys have two families, full ternary X2 YZ and n
Corresponding author. Tel.: þ 86 29 81530750. E-mail address: [email protected] (J.-M. Zhang).
http://dx.doi.org/10.1016/j.ssc.2016.02.006 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
half ternary XYZ (X and Y are transition-metal elements, Z is s–p element) [11,12]. The quaternary Heusler alloys have general formula XX0 YZ which is generated by substituting one of the two X atoms in full ternary Heusler alloys X2 YZ for a different transition metal X0 [13,14]. A variety of new researches related to quaternary Heusler alloys show interesting properties such as high spin polarization and spin-gapless semiconductor (SGS) behavior. o€zdo gan et al. studied dozens of quaternary Heusler alloys and found that some of them are HM, and some of them are SGS, and the others are magnetic semiconductors [13]. The Co-based quaternary Heusler alloys CoFeCrZ (Z¼ Al, Si) [14] and CoFeTiZ (Z¼ Si, Ge, Sn) [15] as well as the Zr-based quaternary Heusler alloys ZrCoTiZ (Z ¼Si, Ge, Ga and Al) [16] are all reported to be HM. The equiatomic quaternary Heusler alloy TiZrCoIn can be regarded as the combination of Ti2 CoIn and Zr2 CoIn and has not been investigated either experimentally or theoretically. Thus in present work, we present the results of the structural, electronic and magnetic properties of the novel quaternary Heusler alloy TiZrCoIn by using the first-principles calculations. It is found that TiZrCoIn is HM ferromagnets with a large indirect energy band gap of 0.930 eV. We also discuss the HM stability under hydrostatic strain and tetragonal strain.
2. Calculation method The Vienna ab-initio simulation package (VASP) based on the density functional theory (DFT) [17–20] is employed to perform
P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
65
TiZrCoIn quaternary Heusler alloy with LiMgPbSb-type or Y-type structure as shown in Table 1. Any other alternative site exchange has no resultant change in structure, because of symmetry offered by space group F43m. First, we carry out the structural optimization for the three possible types to get equilibrium lattice structures of TiZrCoIn quaternary Heusler alloy in both nonmagnetic (NM) and ferromagnetic (FM) phases. The calculated total energy as function
electronic structure calculations of TiZrCoIn quaternary Heusler alloy. The electron-ionic core interaction is represented by the projector augmented wave (PAW) potentials [21]. To treat electron exchange and correlation, we chose the Perdew–Burke–Ernzerhof (PBE) [22] formulation of the generalized gradient approximation (GGA), which yields the correct ground-state structure of the alloys. All of the self-consistent loops are iterated until the total energy difference of the systems between the adjacent iterating steps is less than 1 10 4 eV. And the constituent atoms are fully relaxed until the maximum Hellmann–Feynman forces are less than 0.02 eV/Å. The cutoff energy for the plane waves is chosen to be 280 eV. Brillouin zone integration is carried out at 5 5 5 2 2 7 Monkhorst–Pack k-grids. The Ti 3d 4s2 , Zr 4d 5s2 , Co 3d 4s2 and In 2 1 5s 5p electrons are treated as valence electrons.
-94
, -96
,
YI YII
,
YIII
Energy (eV)
-98 3. Results and discussions In general, the quaternary Heusler alloys XX0 YZ have the space group F43m (No. 216) with the four Wyckoff positions: 4a (0, 0, 0), 4b (1/2, 1/2, 1/2), 4c (1/4, 1/4, 1/4), 4d (3/4, 3/4, 3/4) [23,24]. Different atom arrangements result in three possible configurations for
-100
-102
Table 1 The three possible configurations of TiZrCoIn quaternary Heusler alloy with Y-type structure, the corresponding four Wyckoff positions are 4a (0, 0, 0), 4b (1/2, 1/2, 1/ 2), 4c (1/4, 1/4, 1/4) and 4d (3/4, 3/4, 3/4). Types
Ti
Y-type (I) Y-type (II) Y-type (III)
Zr
4a 4a 4a
Co
4c 4b 4c
-104
-106 6.0
In
4b 4c 4d
6.2
6.4
6.6
6.8
7.0
Lattice constant ( Å)
4d 4d 4b
Fig. 1. (Color online) The calculated total energy as a function of lattice constant for three types of atom arrangements in TiZrCoIn quaternary Heusler alloy within NM (red) and FM (black) phases.
Table 2 The total energy Etot (eV), lattice constant a (Å), total magnetic moment μt (μB/f.u.), atomic magnetic moments μatomic (μB/atom), minority-spin band gap Eg (eV) and HM band (eV), as well as the formation energy Ef ormation (eV) for preferred type (I) of TiZrCoIn alloy in FM phase. gap EHM g Compound
TiZrCoIn
Etot (eV)
104.534
μatomic (μB =atom)
μt (μB =f :u:)
a (Å)
6.525
2.000
Ti
Zr
Co
In
1.496
0.512
0.027
0.004
Eg (eV)
(eV) EHM g
Ef ormation (eV)
0.930
0.430
0.576
4 Majority Minority
Minority
Majority
Energy (eV)
2
0
EF
-2
-4
W
L
Γ
Momentum K
X
W K -40 -20
0
20
40 W
DOS (states/eV)
L
Γ
X
W K
Momentum K
Fig. 2. The majority and minority band structures and the corresponding total density of states (DOS) for preferred type (I) of TiZrCoIn alloy in FM phase.
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P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
of lattice constant is displayed in Fig. 1. A clear evidence from Fig. 1 show that the most favorable one is Y-type (I) in FM phase. So we only discuss the TiZrCoIn alloy in Y-type (I) structure for spinpolarized calculations. The optimized results are presented in Table 2, including the total energy Etot (eV), lattice constant a (Å), total magnetic moment per formula unit μt (μB /f :u:), atomic magnetic moments μatomic (μB /atom), minority-spin band gap Eg (eV) (eV), as well as the formation energy and HM band gap EHM g Ef ormation (eV). The total magnetic moment of TiZrCoIn alloy is 2 µB/f.u., following the Slater–Pauling rule μt ¼ Z t 18 because of this case with 20 valance electrons per formula unit (f.u.). This can be also explained by the hybridization of the d orbitals of neighboring atoms, just like the case of Ti2 CoIn and Zr2 CoIn alloys because the quaternary Heusler alloy TiZrCoIn can be regarded as the combination of these two alloys. The nearly zero atom spin magnetic moments are obtained on both Co and In atoms. The magnetic moments of Ti and Zr atoms are paralleled to each other and the former is almost three times of the latter. The integer total EF
60
eg
DOS (states/eV)
40
Total Ti-3d Zr-4d Co-3d In-5p
t1u
t2g
eu
eg
t2g
20
0
-20
eu eg
-40 -4
-3
t2g
-2
-1
eg
t2g
t1u 0
1
2
3
4
Energy (eV) Fig. 3. (Color online) The total density of states (DOS) (black) and partial density of states (DOS) projected onto Ti-3d (red), Zr-4d (magenta), Co-3d (green) and In-5p (blue) for preferred type (I) of TiZrCoIn alloy in FM phase. The positive and negative values represent the majority spin and minority spin channels, respectively. The EF is set at zero energy and indicated by the vertical black dashed line.
magnetic moment and positive atomic spin magnetic moments show that the TiZrCoIn alloy is a typical HM ferromagnet. In order to study the stability of TiZrCoIn alloy, the formation energy Ef ormation is calculated by Ef ormation ¼ Etot ðETi þ EZr þECo þ EIn Þ where Etot is the equilibrium total energy of the unit cell of TiZrCoIn, ETi , EZr , ECo and EIn are the minimized energies per atom of the pure Ti, Zr, Co and In in their individual bulk reference structures respectively. The individual bulk reference structures of Ti, Zr and Co are Hexagonal structures with identical space group P63= MMC (no. 194), while that of In is Body-centered Tetragonal structure with space group I4=MMM (no. 139). As can be seen in Table 2, the Ef ormation of TiZrCoIn is 0.576 eV. In general, a negative value of formation energy indicates that the material is thermodynamically stable and can be synthesized experimentally. The calculated majority and minority band structures and the corresponding total density of states (DOS) for preferred type (I) of TiZrCoIn alloy in FM phase are shown in Fig. 2. It is evident that the minority spin channel opens an indirect band gap (along the Γ X direction) of 0.930 eV while the majority spin channel is metallic. This phenomenon reveals the HM nature of this alloy. The HM gap, which is determined as the minimum between the lowest energy of minority spin conduction band and absolute value of the highest energy of minority spin valance band [16,25–27], is 0.430 eV. In order to discuss the electronic structure in detail, we give in Fig. 3 the calculated total density of states (DOS) (black) and partial density of states (DOS) projected onto Ti-3d (red), Zr-4d (magenta), Co-3d (green) and In-5p (blue) for preferred type (I) of TiZrCoIn alloy in FM phase. The s state of each atom which is located at deep region is not shown in the figure due to the completely filled s orbital with two electrons. Since the In-5p orbitals with only one electron, the total DOS is mainly contributed by the d–d hybridization of transition metals. The main hybridization bonding orbitals eg and t 2g , which contribute the total DOS at the region from 2 to 0.5 eV for both spin channels, mainly occupied by Co-3d electrons. At the EF the situation is markedly different. For majority spin channel, the hybridization orbital t 1u is mainly occupied by Ti-3d electrons together with less Co-3d and Zr-4d electrons. The minority band gap is determined by the lowest unoccupied orbital t 1u and the highest occupied orbital t 2g , which is similar to that of Zr2 CoZ Heusler alloys [28] because the TiZrCoIn is generated by substituting one of the two Zr atoms in Zr2 CoIn by a Ti atom. For unoccupied region, the antibonding hybridization eu (t 1u and eu ) in majority (minority) spin
1.5 VBM CBM Min. band gap
1.0
VBM CBM Min. band gap
Energy (eV)
0.5
0.0
-0.5
-1.0
-1.5 -10
-5
0
5
10
-20
0
20
40
% of strain Fig. 4. Valence band maximum (VBM), conduction band minimum (CBM) and band gap of minority spin channel as a function of (a) hydrostatic strain and (b) tetragonal strain for preferred type (I) of TiZrCoIn alloy in FM phase.
P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
channel is located at the region from 0.5 to 1.5 eV (0.5–2.3 eV). And the unoccupied sates eg and t 2g are located at the region from 2.3 to 3.0 eV (2.3–3.5 eV) in majority (minority) spin channel. Finally, we study the effect of hydrostatic strain and tetragonal strain (keeping the unit-cell volume the same as the equilibrium bulk volume) on the half-metallicity. The variation of valence band maximum (VBM), conduction band minimum (CBM) and band gap of minority spin channel with respect to the hydrostatic strain and tetragonal strain are plotted in Fig. 4(a) and (b), respectively. In Fig. 4(a), we find that the HM character is kept as hydrostatic strain ranged from 10% to 7.6%. As the hydrostatic strain increasing, the band gap of minority spin channel decreases. And the TiZrCoIn lost its HM nature when the VBM and CBM overlap. But for the influence of tetragonal strain shown in Fig. 4(b), the results are obviously different. The HM character sustains for relatively larger values of tetragonal strain from 19% to 27%. The VBM, CBM, and band gap of minority spin channel are maximum at the equilibrium lattice constant and the absolute value of them decreases monotonically with both positive and negative tetragonal strain.
4. Conclusions For the equiatomic quaternary Heusler alloy TiZrCoIn, the structural, electronic and magnetic properties have been studied by using the first-principles calculations. It is shown that at the equilibrium lattice constant the TiZrCoIn alloy with type (I) configuration is HM ferromagnet with an indirect band gap and a HM gap of 0.930 eV and 0.430 eV, respectively. And the total magnetic moment in unit cell is an integer of 2 μB , which is following the Slater–Pauling rule μt ¼ Z t 18. The negative formation energy indicates the thermodynamical stability of this alloy. The band gap of minority spin channel is determined by the bonding (t 2g ) and antibonding (t 1u ) states created from the hybridizations of the d states of transition metal atoms Ti, Zr and Co. Moreover, the HM character is kept as hydrostatic strain ranged from 10% to 7.6% and tetragonal strain ranged from 19% to 27%.
67
Acknowledgments The authors would like to acknowledge the National Natural Science Foundation of China (Grant no. 51071098) and the State Key Development for Basic Research of China (Grant no. 2010CB631002) for providing financial support for this research.
References [1] K. Inomata, N. Ikeda, N. Tezuka, R. Goto, S. Sugimoto, M. Wojcik, E. Jedryka, J. Phys. D 9 (2008) 014101. [2] R.A. de Groot, F.M. Mueller, P.G. Van Engen, K.J.H. Buschow, Phys. Rev. Lett. 50 (1983) 2024. [3] M.I. Katsnelson, V.Yu Irkhin, L. Chioncel, A.I. Lichtenstein, R.A. de Groot, Rev. Mod. Phys. 80 (2008) 315. [4] C. Felser, G.H. Fecher, B. Balke, Angew. Chem. Int. Ed. 46 (2007) 668. [5] T. Graf, C. Felsler, S.P.P. Parkin, Prog. Solid State Chem. 39 (2011) 1. [6] O. Volnianska, P. Boguslawski, J. Phys. Condens. Matter 22 (2010) 073202. [7] I. Galanakis, E. Şaşıoğlu, Appl. Phys. Lett. 99 (2011) 052509. [8] I. Galanakis, K. Özdoğan, E. Şaşıoğlu, Appl. Phys. Lett. 103 (2013) 142404. [9] G.H. Fecher, H.C. Kandpal, S. Wurmehl, C. Felser, G. Schonhense, J. Appl. Phys. 99 (2006) 08J106. [10] I. Galanakis, K. Özdoğan, E. Şaşıoğlu, J. Phys.: Condens. Matter 26 (2014) 086003. [11] M. Singh, H.S. Saini, J. Thakur, A.H. Reshak, M.K. Kashyap, J. Alloy. Compd. 580 (2013) 201. [12] L. Bainsla, K.G. Suresh, A.K. Nigam, M. Manivel Raja, B.S.D. Ch., S. Varaprasad, Y.K. Takahashi, K. Hono, J. Appl. Phys. 116 (2014) 203902. [13] K. Özdoğan, E. Şaşıoğlu, I. Galanakis, J. Appl. Phys. 113 (2013) 193903. [14] G.Y. Gao, L. Hu, K.L. Yao, B. Luo, N. Liu, J. Alloy. Compd. 551 (2013) 539. [15] Y.J. Zhang, Z.H. Liu, G.T. Li, X.Q. Ma, G.D. Liu, J. Alloy. Compd. 616 (2014) 449. [16] S. Berri, M. Ibrir, D. Maouche, M. Attallah, Comput. Condens. Matter 1 (2014) 26. [17] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [18] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [19] G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15. [20] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [21] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [22] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [23] J. Drews, U. Eberz, H. Schuster, J. Less Common Met. 116 (1986) 271. [24] X. Dai, G. Liu, G.H. Fecher, C. Felser, Y. Li, H. Liu, J. Appl. Phys. 105 (2009) 07E901. [25] S. Berri, M. Ibrir, D. Maouche, M. Attallah, J. Magn. Magn. Mater. 371 (2014) 106. [26] Y.Q. Xu, B.-G. Liu, D.G. Pettifor, Phys. Rev. B 66 (2002) 184435. [27] N.S. Xing, H. Li, J.M. Dong, R. Long, C.W. Zhang, Comput. Mater. Sci. 42 (2008) 600. [28] P.L. Yan, J.M. Zhang, K.W. Xu, J. Magn. Magn. Mater. 391 (2015) 43.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The structural, electronic and magnetic properties of quaternary Heusler alloy TiZrCoIn Peng-Li Yan a, Jian-Min Zhang a,n, Ke-Wei Xu b a b
College of Physics and Information Technology, Shaanxi Normal University, Xian 710119, Shaanxi, PR China College of Physics and Mechanical and Electronic Engineering, Xian University of Arts and Science, Xian 710065, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 October 2015 Received in revised form 30 November 2015 Accepted 8 February 2016 Communicated by: D.D. Sarma Available online 12 February 2016
Employing the first-principles calculations, we have investigated the structural, electronic and magnetic properties of quaternary Heusler alloy TiZrCoIn. The TiZrCoIn alloy with type (I) configuration is predicted to be half-metallic ferromagnet at its equilibrium lattice constant 6.525 Å with an indirect band gap of 0.930 eV in minority spin channel. The total magnetic moment is 2 μB/f.u., following the Slater– Pauling rule μt ¼Zt 18. Moreover, the negative formation energy indicates the thermodynamical stability of this alloy. The band gap of minority spin channel is determined by the bonding (t2g) and antibonding (t1u) states created from the hybridizations of the d states of transition metal atoms Ti, Zr and Co. In addition, the HM, character is kept as hydrostatic strain ranged from 10% to 7.6% and tetragonal strain ranged from 19% to 27%. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Quaternary Heusler alloy C. Half-metallic D. Electronic structures D. Magnetic properties
1. Introduction In spintronic applications, magnetic materials with high spin polarization are crucial for the improvement of the performance of spintronic devices such as the spin filter and the spin valve [1]. Half-metallic (HM) magnets are seen as the most promising candidates of high-spin-polarization materials, because their band structure is metallic in one of the two spin channels and semiconducting or insulating in the other one, which results in complete (100%) spin polarization of electrons at the Fermi level (EF). In 1983, de Groot et al. [2] predicted HM ferromagnetism in NiMnSb. Since then a number of new classes of materials exhibiting half-metallicity have been predicted and some realized experimentally, e.g., metallic oxides such as CrO2 and Fe3 O4 , perovskites and double perovskites such as La1 x Srx MnO3 and Sr2 FeMoO6 , Heusler alloys with Clb and L21 structures such as NiMnSb and Co2 MnSi, zinc-blende compounds such as CrAs and CrSb, sp compounds (without transition metals) with zinc-blende and rocksalt structures such as CaC and SrN, and so on [3–6]. Among these materials, only Heusler-type HM alloys own very high Curie temperature TC [7–10], which yielding them to be one of the most promising candidates for spintronics devices. Therefore, further exploration of HM Heusler alloys is strongly desired. Heusler alloys are categorized as ternary and quaternary alloys. The ternary Heusler alloys have two families, full ternary X2 YZ and n
Corresponding author. Tel.: þ 86 29 81530750. E-mail address: [email protected] (J.-M. Zhang).
http://dx.doi.org/10.1016/j.ssc.2016.02.006 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
half ternary XYZ (X and Y are transition-metal elements, Z is s–p element) [11,12]. The quaternary Heusler alloys have general formula XX0 YZ which is generated by substituting one of the two X atoms in full ternary Heusler alloys X2 YZ for a different transition metal X0 [13,14]. A variety of new researches related to quaternary Heusler alloys show interesting properties such as high spin polarization and spin-gapless semiconductor (SGS) behavior. o€zdo gan et al. studied dozens of quaternary Heusler alloys and found that some of them are HM, and some of them are SGS, and the others are magnetic semiconductors [13]. The Co-based quaternary Heusler alloys CoFeCrZ (Z¼ Al, Si) [14] and CoFeTiZ (Z¼ Si, Ge, Sn) [15] as well as the Zr-based quaternary Heusler alloys ZrCoTiZ (Z ¼Si, Ge, Ga and Al) [16] are all reported to be HM. The equiatomic quaternary Heusler alloy TiZrCoIn can be regarded as the combination of Ti2 CoIn and Zr2 CoIn and has not been investigated either experimentally or theoretically. Thus in present work, we present the results of the structural, electronic and magnetic properties of the novel quaternary Heusler alloy TiZrCoIn by using the first-principles calculations. It is found that TiZrCoIn is HM ferromagnets with a large indirect energy band gap of 0.930 eV. We also discuss the HM stability under hydrostatic strain and tetragonal strain.
2. Calculation method The Vienna ab-initio simulation package (VASP) based on the density functional theory (DFT) [17–20] is employed to perform
P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
65
TiZrCoIn quaternary Heusler alloy with LiMgPbSb-type or Y-type structure as shown in Table 1. Any other alternative site exchange has no resultant change in structure, because of symmetry offered by space group F43m. First, we carry out the structural optimization for the three possible types to get equilibrium lattice structures of TiZrCoIn quaternary Heusler alloy in both nonmagnetic (NM) and ferromagnetic (FM) phases. The calculated total energy as function
electronic structure calculations of TiZrCoIn quaternary Heusler alloy. The electron-ionic core interaction is represented by the projector augmented wave (PAW) potentials [21]. To treat electron exchange and correlation, we chose the Perdew–Burke–Ernzerhof (PBE) [22] formulation of the generalized gradient approximation (GGA), which yields the correct ground-state structure of the alloys. All of the self-consistent loops are iterated until the total energy difference of the systems between the adjacent iterating steps is less than 1 10 4 eV. And the constituent atoms are fully relaxed until the maximum Hellmann–Feynman forces are less than 0.02 eV/Å. The cutoff energy for the plane waves is chosen to be 280 eV. Brillouin zone integration is carried out at 5 5 5 2 2 7 Monkhorst–Pack k-grids. The Ti 3d 4s2 , Zr 4d 5s2 , Co 3d 4s2 and In 2 1 5s 5p electrons are treated as valence electrons.
-94
, -96
,
YI YII
,
YIII
Energy (eV)
-98 3. Results and discussions In general, the quaternary Heusler alloys XX0 YZ have the space group F43m (No. 216) with the four Wyckoff positions: 4a (0, 0, 0), 4b (1/2, 1/2, 1/2), 4c (1/4, 1/4, 1/4), 4d (3/4, 3/4, 3/4) [23,24]. Different atom arrangements result in three possible configurations for
-100
-102
Table 1 The three possible configurations of TiZrCoIn quaternary Heusler alloy with Y-type structure, the corresponding four Wyckoff positions are 4a (0, 0, 0), 4b (1/2, 1/2, 1/ 2), 4c (1/4, 1/4, 1/4) and 4d (3/4, 3/4, 3/4). Types
Ti
Y-type (I) Y-type (II) Y-type (III)
Zr
4a 4a 4a
Co
4c 4b 4c
-104
-106 6.0
In
4b 4c 4d
6.2
6.4
6.6
6.8
7.0
Lattice constant ( Å)
4d 4d 4b
Fig. 1. (Color online) The calculated total energy as a function of lattice constant for three types of atom arrangements in TiZrCoIn quaternary Heusler alloy within NM (red) and FM (black) phases.
Table 2 The total energy Etot (eV), lattice constant a (Å), total magnetic moment μt (μB/f.u.), atomic magnetic moments μatomic (μB/atom), minority-spin band gap Eg (eV) and HM band (eV), as well as the formation energy Ef ormation (eV) for preferred type (I) of TiZrCoIn alloy in FM phase. gap EHM g Compound
TiZrCoIn
Etot (eV)
104.534
μatomic (μB =atom)
μt (μB =f :u:)
a (Å)
6.525
2.000
Ti
Zr
Co
In
1.496
0.512
0.027
0.004
Eg (eV)
(eV) EHM g
Ef ormation (eV)
0.930
0.430
0.576
4 Majority Minority
Minority
Majority
Energy (eV)
2
0
EF
-2
-4
W
L
Γ
Momentum K
X
W K -40 -20
0
20
40 W
DOS (states/eV)
L
Γ
X
W K
Momentum K
Fig. 2. The majority and minority band structures and the corresponding total density of states (DOS) for preferred type (I) of TiZrCoIn alloy in FM phase.
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P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
of lattice constant is displayed in Fig. 1. A clear evidence from Fig. 1 show that the most favorable one is Y-type (I) in FM phase. So we only discuss the TiZrCoIn alloy in Y-type (I) structure for spinpolarized calculations. The optimized results are presented in Table 2, including the total energy Etot (eV), lattice constant a (Å), total magnetic moment per formula unit μt (μB /f :u:), atomic magnetic moments μatomic (μB /atom), minority-spin band gap Eg (eV) (eV), as well as the formation energy and HM band gap EHM g Ef ormation (eV). The total magnetic moment of TiZrCoIn alloy is 2 µB/f.u., following the Slater–Pauling rule μt ¼ Z t 18 because of this case with 20 valance electrons per formula unit (f.u.). This can be also explained by the hybridization of the d orbitals of neighboring atoms, just like the case of Ti2 CoIn and Zr2 CoIn alloys because the quaternary Heusler alloy TiZrCoIn can be regarded as the combination of these two alloys. The nearly zero atom spin magnetic moments are obtained on both Co and In atoms. The magnetic moments of Ti and Zr atoms are paralleled to each other and the former is almost three times of the latter. The integer total EF
60
eg
DOS (states/eV)
40
Total Ti-3d Zr-4d Co-3d In-5p
t1u
t2g
eu
eg
t2g
20
0
-20
eu eg
-40 -4
-3
t2g
-2
-1
eg
t2g
t1u 0
1
2
3
4
Energy (eV) Fig. 3. (Color online) The total density of states (DOS) (black) and partial density of states (DOS) projected onto Ti-3d (red), Zr-4d (magenta), Co-3d (green) and In-5p (blue) for preferred type (I) of TiZrCoIn alloy in FM phase. The positive and negative values represent the majority spin and minority spin channels, respectively. The EF is set at zero energy and indicated by the vertical black dashed line.
magnetic moment and positive atomic spin magnetic moments show that the TiZrCoIn alloy is a typical HM ferromagnet. In order to study the stability of TiZrCoIn alloy, the formation energy Ef ormation is calculated by Ef ormation ¼ Etot ðETi þ EZr þECo þ EIn Þ where Etot is the equilibrium total energy of the unit cell of TiZrCoIn, ETi , EZr , ECo and EIn are the minimized energies per atom of the pure Ti, Zr, Co and In in their individual bulk reference structures respectively. The individual bulk reference structures of Ti, Zr and Co are Hexagonal structures with identical space group P63= MMC (no. 194), while that of In is Body-centered Tetragonal structure with space group I4=MMM (no. 139). As can be seen in Table 2, the Ef ormation of TiZrCoIn is 0.576 eV. In general, a negative value of formation energy indicates that the material is thermodynamically stable and can be synthesized experimentally. The calculated majority and minority band structures and the corresponding total density of states (DOS) for preferred type (I) of TiZrCoIn alloy in FM phase are shown in Fig. 2. It is evident that the minority spin channel opens an indirect band gap (along the Γ X direction) of 0.930 eV while the majority spin channel is metallic. This phenomenon reveals the HM nature of this alloy. The HM gap, which is determined as the minimum between the lowest energy of minority spin conduction band and absolute value of the highest energy of minority spin valance band [16,25–27], is 0.430 eV. In order to discuss the electronic structure in detail, we give in Fig. 3 the calculated total density of states (DOS) (black) and partial density of states (DOS) projected onto Ti-3d (red), Zr-4d (magenta), Co-3d (green) and In-5p (blue) for preferred type (I) of TiZrCoIn alloy in FM phase. The s state of each atom which is located at deep region is not shown in the figure due to the completely filled s orbital with two electrons. Since the In-5p orbitals with only one electron, the total DOS is mainly contributed by the d–d hybridization of transition metals. The main hybridization bonding orbitals eg and t 2g , which contribute the total DOS at the region from 2 to 0.5 eV for both spin channels, mainly occupied by Co-3d electrons. At the EF the situation is markedly different. For majority spin channel, the hybridization orbital t 1u is mainly occupied by Ti-3d electrons together with less Co-3d and Zr-4d electrons. The minority band gap is determined by the lowest unoccupied orbital t 1u and the highest occupied orbital t 2g , which is similar to that of Zr2 CoZ Heusler alloys [28] because the TiZrCoIn is generated by substituting one of the two Zr atoms in Zr2 CoIn by a Ti atom. For unoccupied region, the antibonding hybridization eu (t 1u and eu ) in majority (minority) spin
1.5 VBM CBM Min. band gap
1.0
VBM CBM Min. band gap
Energy (eV)
0.5
0.0
-0.5
-1.0
-1.5 -10
-5
0
5
10
-20
0
20
40
% of strain Fig. 4. Valence band maximum (VBM), conduction band minimum (CBM) and band gap of minority spin channel as a function of (a) hydrostatic strain and (b) tetragonal strain for preferred type (I) of TiZrCoIn alloy in FM phase.
P.-L. Yan et al. / Solid State Communications 231-232 (2016) 64–67
channel is located at the region from 0.5 to 1.5 eV (0.5–2.3 eV). And the unoccupied sates eg and t 2g are located at the region from 2.3 to 3.0 eV (2.3–3.5 eV) in majority (minority) spin channel. Finally, we study the effect of hydrostatic strain and tetragonal strain (keeping the unit-cell volume the same as the equilibrium bulk volume) on the half-metallicity. The variation of valence band maximum (VBM), conduction band minimum (CBM) and band gap of minority spin channel with respect to the hydrostatic strain and tetragonal strain are plotted in Fig. 4(a) and (b), respectively. In Fig. 4(a), we find that the HM character is kept as hydrostatic strain ranged from 10% to 7.6%. As the hydrostatic strain increasing, the band gap of minority spin channel decreases. And the TiZrCoIn lost its HM nature when the VBM and CBM overlap. But for the influence of tetragonal strain shown in Fig. 4(b), the results are obviously different. The HM character sustains for relatively larger values of tetragonal strain from 19% to 27%. The VBM, CBM, and band gap of minority spin channel are maximum at the equilibrium lattice constant and the absolute value of them decreases monotonically with both positive and negative tetragonal strain.
4. Conclusions For the equiatomic quaternary Heusler alloy TiZrCoIn, the structural, electronic and magnetic properties have been studied by using the first-principles calculations. It is shown that at the equilibrium lattice constant the TiZrCoIn alloy with type (I) configuration is HM ferromagnet with an indirect band gap and a HM gap of 0.930 eV and 0.430 eV, respectively. And the total magnetic moment in unit cell is an integer of 2 μB , which is following the Slater–Pauling rule μt ¼ Z t 18. The negative formation energy indicates the thermodynamical stability of this alloy. The band gap of minority spin channel is determined by the bonding (t 2g ) and antibonding (t 1u ) states created from the hybridizations of the d states of transition metal atoms Ti, Zr and Co. Moreover, the HM character is kept as hydrostatic strain ranged from 10% to 7.6% and tetragonal strain ranged from 19% to 27%.
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Acknowledgments The authors would like to acknowledge the National Natural Science Foundation of China (Grant no. 51071098) and the State Key Development for Basic Research of China (Grant no. 2010CB631002) for providing financial support for this research.
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