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Magnetic properties and origin of the half-metallicity of Ti2MnZ (Z=Al, Ga, In, Si, Ge, Sn) Heusler alloys with the Hg2CuTi-type structure
Journal of Magnetism and Magnetic Materials 349 (2014) 104–108
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Magnetic properties and origin of the half-metallicity of Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) Heusler alloys with the Hg2CuTi-type structure Qing-Long Fang a, Jian-Min Zhang a,n, Ke-Wei Xu b a b
College of Physics and Information Technology, Shaanxi Normal University, Xian 710062, Shaanxi, PR China State Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University, Xian 710049, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 May 2013 Received in revised form 1 August 2013 Available online 31 August 2013
Using the first-principles calculations within density functional theory, we investigate the magnetic properties and electronic structures of the Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure. The Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn) are found to be half-metallic ferrimagnets. The total magnetic moments (mt) of the Ti2MnZ alloys are calculated to be 0 for Z¼ Al, Ga, In and 1 for Z¼ Si, Ge, Sn, linearly scaled with the total number of valence electrons (Zt) by mt ¼Zt 18. The origin of the band gap for these half-metallic alloys is well understood. We expect our results to trigger further experimental interest in these alloys. & 2013 Elsevier B.V. All rights reserved.
Keywords: Heusler alloy Half-metallic Magnetic properties First-principles
1. Introduction Increased interest in the field of magnetoelectronics or spin electronics during the last decade has intensified research on the so-called half-metallic materials (HMM) which are metallic for one spin direction while at the same time semiconducting for the other spin direction and thus exhibit a complete spin polarization at the Fermi level and the realistic applications for spintronic devices [1–3]. This offers opportunities for a new generation of devices combining standard microelectronic with spin-dependent effects such as nonvolatile magnetic random access memories and magnetic sensors [4]. Half-metallicity was first predicted by de Groot and collaborators in a half-Heusler alloy [5]. Since then, more and more Heusler alloys have been initially predicted with half-metallicity by ab initio calculations and later experimental verification [6–13]. Heusler alloys can be classified into two main groups, i.e., halfHeusler XYZ alloys and full-Heusler X2YZ alloys. Here, X and Y denote transition metal elements, and Z is an s–p element [14,15]. Full-Heusler X2YZ alloys generally have two types of structures, Cu2MnAl and Hg2CuTi. Usually, the Heusler structure can be looked as four interpenetrating face-centered-cubic (FCC) lattices and has four unique crystal sites namely A(0,0,0), B(1/4,1/4,1/4), C(1/2,1/2,1/2), D(3/4,3/4,3/4) in Wyckoff coordinates. It is found that the site preference of the X and Y atoms is strongly influenced
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by the number of their valence electrons [16]. The Cu2MnAl-type structure, where two X atoms are located at A and C sites, whereas the Y and Z atoms are placed at B and D sites, respectively. On the other hand, for the Hg2CuTi-type structure, two X atoms are located at the A and B sites, whereas the Y and Z atoms are located at the C and D sites, respectively. Ti-based Heusler alloy Ti2YZ with the Hg2CuTi-type structure, such as Ti2CoAl [17], Ti2NiAl [18] and Ti2CoGa [19], the neighbor Ti atoms occupy A (0,0,0) and B(1/4,1/4,1/4) sites, and the remaining Y enters C(1/2,1/ 2,1/2) and Z at D(3/4,3/4,3/4) [16]. Due to different surroundings of the neighbor Ti atoms, in Ti2YZ alloy the d states of the neighbor Ti atoms should be hybridized first, then the hybridized Ti–Ti orbitals continue to couple with d states of Y atom [20], finally resulting in a similar band structure and magnetic behavior to the half-Heusler alloy. In this work, we present a systemic study the structure, electronic and magnetic properties of a series Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure by using the first-principles projector augmented wave (PAW) potential within the generalized gradient approximation (GGA). Up to now, no reports have been found on either theory or experiment investigations of the structure, electronic and magnetic properties of these alloys except Ti2MnAl [20]. The paper is organized as follows. In Section 2, the computational method is described detailly. In Section 3.1, the optimized lattice constants and magnetic properties are discussed. In Section 3.2, we discuss the origin of the band gap in half-metallic alloys. Section 3.3 presents the effect of lattice constant on the gap width. Finally in Section 4, we summarize our results and conclusions.
Q.-L. Fang et al. / Journal of Magnetism and Magnetic Materials 349 (2014) 104–108
2. Computational method The calculations are performed using the Vienna ab initio simulation package (VASP) based on the density function theory (DFT) [21–24]. The electron-ionic core interaction is represented by the projector augmented wave (PAW) potentials [25] which are more accurate than the ultra-soft pseudopotentials. To treat electron exchange and correlation, we chose the Perdew–Burke– Ernzerhof (PBE) [26] formulation of the generalized gradient approximation (GGA). A conjugate-gradient algorithm is used to relax the ions into their ground states, and the energies and the forces on each ion are converged within 1.0 10 5 eV/atom and 0.01 eV/Å, respectively. The cutoff energy for the plane-waves is chosen to be 400 eV. A 13 13 13 Monkhorst–Pack grid for k-point sampling is adopted for Brillouin zone integration, together with a Gaussian smearing broadening of 0.2 eV.
3. Results and discussions 3.1. The optimized lattice constants and magnetic properties There is no available experimental lattice constant for Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn) alloys. Consequently, the cubic lattice parameters for the alloys with the Hg2CuTi-type structure are optimized by minimizing the total energy with respect to the lattice parameter variation. For all studied alloys, the ferromagnetic state was found to be more stable than the paramagnetic state. As listed in Table 1, the optimized lattice constants of the Ti2MnZ alloys are 6.138, 6.199, 6.248, 5.997, 6.076, 6.317 Å for Z¼Al, Ga, In, Si, Ge, Sn, respectively. The optimized lattice constants for Ti substituting Mn at C(1/2,1/2,1/2) site, i.e., Ti2TiZ (Z ¼Al, Ga, In, Si, Ge, Sn), and Mn vacancy i.e., Ti2Z (Z¼Al, Ga, In, Si, Ge, Sn) are also listed in Table 1 for comparison. The optimized lattice constants of 6.138 and 6.281 Å for Ti2MnAl and Ti2TiSi alloys agree well with previous works of 6.24 and 6.29 Å, respectively [27]. We can see that with the same Z (Z¼ Al, Ga, In, Si, Ge, Sn) atom, the optimized lattice constant increases according to the order of Ti2MnZ, Ti2Z and Ti2TiZ. Combined with the values and orientations of the atomic magnetic moments listed in Table 1as well, the lattice constant of the Ti2MnZ is smaller than that of the Ti2TiZ may be resulted from not only the atomic radius of 1.79 Å for Mn atom is smaller than that of 2.00 Å for Ti atom but also the Table 1 Optimized lattice constant a (Å), partial magnetic moments m (mB) of the atoms at different sites, total magnetic moment mt (mB) and gap width Eg (eV) for Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys together with the results of Ti2TiZ (Z¼ Al, Ga, In, Si, Ge, Sn) and Ti2Z (Z¼ Al, Ga, In, Si, Ge, Sn) for comparison.
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additional attraction from the antiferromagnetic coupling between Mn and Ti atoms. The lattice constant of the Ti2MnZ is even smaller than that of the Ti2Z with Mn vacancy can be attributed to the additional repulsive force from the larger ferromagnetic coupling between two neighbor Ti atoms of the Ti2Z. It is also noted that from Table 1, For Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys, the magnetic moments of Mn atom are antiparallel to those of two neighbor Ti atoms. For Ti2TiZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, the magnetic moment of the Ti atom at C site is identical to that of the Ti atom at A site due to the same atomic circumstance. Two relative larger positive magnetic moments are observed for two neighbor Ti atoms in Ti2Z (Z ¼Al, Ga, In, Si, Ge, Sn). For all alloys studied here, the magnetic moments of the Ti1 and Ti2 atoms at A and B sites respectively are not equal due to the different atomic surrounding and the magnetic moments of the s–p elements are neglectable small. Previous first-principles studies predicted that many fullHeusler alloys with L21 structure were half-metallic ferromagnets. The total magnetic moments (mt) of the full-Heusler alloys follow the Slater-Pauling rule mt ¼Zt 24, where Zt is the total number of valence electrons in the unit cell. A similar behavior can also be found in half-Heusler alloys, but for these alloys the total magnetic moments follow the relation mt ¼ Zt 18 [7,28,29]. Recent research reveals that many Ti2-based full-Heusler alloys with the Hg2CuTitype structure can also belong to the family of half-metallic material, and the total magnetic moments of these alloys follow the mt ¼Zt 18 rule instead of the mt ¼Zt 24 rule [17,27,30]. Our integrated total moment mt is integer 0 with respect to total valence electrons 18 of the half-metallic Ti2MnZ (Z¼ Al, Ga, In) alloys, as well as integer 1 with respect to total valence electrons 19 of the half-metallic Ti2MnZ (Z ¼Si, Ge, Sn) alloys. All of these alloys are found to be the half-metallic ferrimagnets, their gap widths in spin-down channel (see next section) are listed in the last column of Table 1. In detail, the Ti2MnZ (Z¼ Al, Ga, In) alloys are half-metallic ferrimagnets with zero total magnetic moments, and are thus known as the half-metallic antiferromagnets or fully compensated ferrimagnets [31,32]. Half-metallic antiferromagnet differs from the conventional antiferromagnet in nature. In conventional antiferromagnet, the electronic structures are identical for both spin channels, as the result, the spin-polarization is zero, whereas in the half-metallic antiferromagnet the electronic structures are completely asymmetric, resulting in 100% spin polarization. Half-metallic antiferromagnet has an advantage in spintronics since the materials do not give rise to stray flux. We can see that, after a Ti substituting Mn atom or a Mn vacancy, the total magnetic moment mt is increased especially for a Mn vacancy case.
3.2. Origin of the band gap Alloys
a (Å)
mX1 (mB)
mX2 (mB)
mY (mB)
mZ (mB)
mt (mB)
Eg (eV)
Ti2MnAl Ti2TiAl Ti2Al Ti2MnGa Ti2TiGa Ti2Ga Ti2MnIn Ti2TiIn Ti2In Ti2MnSi Ti2TiSi Ti2Si Ti2MnGe Ti2TiGe Ti2Ge Ti2MnSn Ti2TiSn Ti2Sn
6.138 6.458 6.310 6.199 6.401 6.231 6.248 6.637 6.505 5.997 6.281 6.135 6.076 6.350 6.199 6.317 6.597 6.501
1.245 0.361 1.443 1.345 0.266 1.420 1.473 0.322 1.374 1.126 0.260 1.694 1.272 0.256 1.737 1.498 0.247 1.681
1.081 1.265 1.588 1.232 1.180 1.670 1.372 1.282 1.626 0.679 1.024 1.952 0.856 1.019 1.994 1.119 1.084 1.917
2.373 0.361
0.051 0.036 0.077 0.024 0.036 0.107 0.046 0.043 0.082 0.052 0.033 0.041 0.032 0.034 0.074 0.056 0.041 0.047
0.005 1.951 2.954 0.020 1.676 2.983 0.095 1.883 2.918 1.050 1.511 3.605 1.030 1.497 3.657 0.955 1.537 3.551
0.538
2.620 0.266 2.987 0.322 0.810 0.260 1.129 0.256 1.719 0.247
0.592
0.526
0.455
0.504
0.538
Since the magnetisms of the Heusler alloys are mainly determined by the not completely filled d orbitals of the 3d transition metals, the partial density of states (PDOS) projected onto d states of transition metal atoms Ti1, Ti2 and Mn at A, B and C sites, respectively, in Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys are shown in Fig. 1 together with those projected onto not completely filled p orbitals of Z atom at D site. The PDOS projected onto the completely filled s orbitals of each atom with two electrons are not plotted here because they are located in deep energy region around 6 eV. Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green lines. The origin of the band gap is usually distinguished into three categories: (1) covalent band gap, (2) d–d band gap, and (3) charge transfer band gap [33,34]. The covalent band gap has been found to exist in half-Heusler alloys with C1b structure. The d–d band gap is responsible for the half-metallicity of the full-Heusler alloys with
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4 2
PDOS (states/eV)
0 -2 -4
Ti2MnAl
Ti2MnGa
Ti2MnIn
Ti2MnSi
Ti2MnGe
Ti2MnSn
4 2 0 -2 -4
-6
-4
-2
0
2
4 -6
-4
-2 0 Energy (eV)
2
4 -6
-4
-2
0
2
4
Fig. 1. Partial density of states (PDOS) projected onto Ti1-d (black line), Ti2-d (red line), Mn-d (blue line) and Z–p (cyan line) for Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) Heusler alloys. Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3xtu 2xeu
3xtu 3xt2g
2xeu
3xt2g
2xeg
3xt2g
2xeg
3xtu 2xeu 2xe* 3xt* 3xt
3xt2g
3xt2g
EF 2xeg
2xeg
2xeg 2xe
Ti1
Ti2
Ti1-Ti2
Mn
Fig. 2. Possible hybridizations between d states of transition metal atoms in the Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys. The interactions between the two inequivalent Ti atoms are first taken into account (a), and subsequently consider the interaction between the Ti1–Ti2 coupling and Mn atom (b).
L21 structure. The charge transfer band gap is usually existed in CrO2 and double perovskites [34–36]. In our Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys, the situation is more complex due to the special crystallized structure. The Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys have a space group similar to the half-Heusler but the empty site was alternated by the additional Ti atom. Different from the full-Heusler alloy with L21 structure where two X atoms are located at A(0,0,0) and C(1/2,1/2,1/2) sites with identical surrounding, in Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, two Ti atoms occupy two neighbor sublattices A(0,0,0) and B(1/4,1/4,1/4) with different surrounding. Therefore, we must pay more attention to the effects of these two Ti atoms. To discuss the possible hybridization between the d states of the transition metal atoms, let us observe PDOS shown in Fig. 1. It can be seen that the d states of transition metal atoms have the similar feature of PDOS for Ti2MnAl, Ti2MnGa and Ti2MnIn alloys, while the other similar feature of PDOS in Ti2MnSi, Ti2MnGe and Ti2MnSn alloys. But for all of these alloys, the calculated results show a band gap in spin-down channel but a metal character in spin-up channel. The total density of states (TDOS) shown in Fig. 3 with black lines can be characterized by the large gap at the Fermi level and the occupied bonding states mainly present Mn atom characters below the Fermi level. It is evident that the d–d orbitals hybridization between transition metals is rather intensive. The Ti1 and Ti2 atoms are nearest-neighbors to each other. Thus, the interactions between the two inequivalent Ti atoms are
first taken into account. If the Mn and Z atoms are neglected, the two Ti sites form a diamond structure belonging to the octahedral symmetry (Oh). Since the current studied Heusler alloys have a tetrahedral Td symmetry which is a subgroup of Oh. Therefore, there could be states obeying the Oh being exclusively localized at the Ti atom. The possible hybridizations between the transition metal atoms are schematically shown in Fig. 2. The d orbitals of each Ti atom split into the double-degenerated 2xeg (dx2 y2 and dz2 ) states and triple-degenerated 3xt2g (dxy , dyz and dzx ) states. The names of the orbitals and subscript follow the nomenclature used in the documents and the coefficient represents the degeneracy of each orbital. There are strong hybridizations of the 3xt2g (2xeg) states between two neighbor Ti atoms, resulting in bonding 3xt2g (2xeg) and antibonding 3xtu (2xeu) states. Subsequently, the interactions between the Ti1–Ti2 coupling and Mn atom are considered. The Mn atom has a Td symmetry, and its d orbital is also split into the double-degenerated 2xeg (dx2 y2 and dz2 ) states and triple-degenerated 3xt2g (dxy , dyz and dzx ) states. The antibonding 3xtu and 2xeu states of Ti1–Ti2 coupling cannot hybridize to the 3xt2g and 2xeg states of the Mn atom, respectively, because of the lack of symmetry. While the bonding 2xeg and 3xt2g states of Ti1–Ti2 coupling also transform with the 2xeg and 3xt2g representations in Td group, respectively. So the 2xeg and 3xt2g states of the Ti1–Ti2 coupling are further hybridized with the 2xeg and 3xt2g states of Mn atom, giving rise to the bonding 2xe and 3xt as well as antibonding 2xen and 3xtn states.
Q.-L. Fang et al. / Journal of Magnetism and Magnetic Materials 349 (2014) 104–108
107
20 10
TDOS (states/eV)
0 -10 -20 20
Z=Al
Z=Ga
Z=In
Z=Ge
Z=Sn
10 0 -10 Z=Si -20 -10 -8 -6 -4 -2
0
2
4
6 -10 -8 -6 -4 -2 0 2 Energy (eV)
4
6 -10 -8 -6 -4 -2
0
2
4
6
Fig. 3. Total density of states (TDOS) of the Ti2MnZ (black line), Ti2TiZ (red line) and Ti2Z (blue line) alloys (Z ¼Al, Ga, In, Si, Ge, Sn). Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.3. Variation of the gap with lattice constant The half-metallic character is closely related to the lattice constant of the alloys since the lattice constant significantly influences the width of the band gap at the Fermi level. In Fig. 4, we present the dependence of the gap width on the lattice constant for Ti2MnZ (Z¼ Al, Ga, In) alloys with gap in spin-up (spin-down) as examples. It can be seen that the gap width increases slightly with increasing the lattice constant for Ti2MnAl and Ti2MnGa alloys, while the gap width of Ti2MnIn alloy decreases with increasing the lattice constant. The change of gap width can be traced back to two combined effects: (1) p–d orbital
0.6 0.5 Ti2MnAl
0.4
Gap width (eV)
As elucidated above, the band gap of Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure are suggested to be determined mainly by the bonding 3xt and antibonding 3xt* states created from the hybridizations of the d states between the Ti1–Ti2 coupling and Mn atom, as well as the Fermi level just located in the band gap. The states near the Fermi level can be well ascribed to covalent hybridization and d–d orbitals hybridization between transition metal atoms. In addition, as can be seen from the PDOS patterns, the p states of the Z atoms play an important role in the Heusler alloys, although they do not directly form the band gap due to their zero value around the Fermi level, the hybridization between the p states of the Z atoms and d states of the transition metal atoms determines the degree of occupation of the p–d orbitals. Thus, the hybridization between p–d electrons affects the formation and width of the energy gap. In conclusion, it should be emphasized that there exist two mechanisms in the formation of the band gap, that is covalent band gap and d–d band gap, but it is mainly the d–d band gap that characterizes the half-metallicity in Ti2-based alloys. The calculated total densities of states (TDOS) are compared in Fig. 3 for Ti2MnZ (black lines), Ti2TiZ (red lines) and Ti2Z (blue lines) alloys (Z¼Al, Ga, In, Si, Ge, Sn). Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green lines. We can see that, after a Ti or vacancy substituting Mn atom at C site, the spin-down band gaps and thus the half-metallic character disappear in Ti2TiZ and Ti2Z (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, and below the Fermi level the occupied DOS in both channels shifts to the high energy region compared with Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn).
0.6 0.5 Ti2MnGa
0.4 0.6 0.5
Ti2MnIn
0.4 5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
Lattice constant (Å) Fig. 4. The gap width changes with the lattice constant in the range of 5.9–6.6 Å.
hybridization, which obviously depends on the p states of s–p element. (2) their different lattice constants. In Fig. 4, we can also see clearly the changes of width of the gap with different lattice constants. In fact, the contraction or expansion of the lattice constant has an influence on the delocalized p electrons of the s–p elements but not well localized d electrons of the transition metal atoms.
4. Conclusions The magnetic properties, electronic structures, and halfmetallicity of full-Heusler Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure have been studied by using the first-principles projector augmented wave (PAW) potential within the generalized gradient approximation (GGA). The following conclusions are obtained:
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(1) The Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys are found to be half-metallic ferrimagnets and potential applications in spintronic devices. (2) The total magnetic moments of the Heusler alloys Ti2MnZ follow the mt ¼Zt 18 rule and agree with the Slater–Pauling curve quite well. For Ti2MnZ (Z ¼Al, Ga, In) alloys the total magnetic moment per unit cell is 0, together with the spindown band gaps of 0.538, 0.592, 0.526 eV for Z¼ Al, Ga, In, respectively. While for Ti2MnZ (Z¼Si, Ge, Sn) alloys, the total magnetic moment per unit cell is 1mB and their spin-down band gaps are 0.455, 0.504, 0.538 eV, respectively. (3) The band gaps are mainly determined by the bonding and antibonding states created from the hybridizations of the d states between the Ti1–Ti2 coupling and Mn atom. (4) The s–p elements play an important role in the half-metallicity of these Heusler alloys. In addition, for all Ti2-based Heusler alloys, the localized magnetic moment carried by the Ti and Mn atoms is restricted by the s–p element even though they have a small contribution to the total spin magnetic moment.
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Magnetic properties and origin of the half-metallicity of Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) Heusler alloys with the Hg2CuTi-type structure Qing-Long Fang a, Jian-Min Zhang a,n, Ke-Wei Xu b a b
College of Physics and Information Technology, Shaanxi Normal University, Xian 710062, Shaanxi, PR China State Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University, Xian 710049, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 May 2013 Received in revised form 1 August 2013 Available online 31 August 2013
Using the first-principles calculations within density functional theory, we investigate the magnetic properties and electronic structures of the Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure. The Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn) are found to be half-metallic ferrimagnets. The total magnetic moments (mt) of the Ti2MnZ alloys are calculated to be 0 for Z¼ Al, Ga, In and 1 for Z¼ Si, Ge, Sn, linearly scaled with the total number of valence electrons (Zt) by mt ¼Zt 18. The origin of the band gap for these half-metallic alloys is well understood. We expect our results to trigger further experimental interest in these alloys. & 2013 Elsevier B.V. All rights reserved.
Keywords: Heusler alloy Half-metallic Magnetic properties First-principles
1. Introduction Increased interest in the field of magnetoelectronics or spin electronics during the last decade has intensified research on the so-called half-metallic materials (HMM) which are metallic for one spin direction while at the same time semiconducting for the other spin direction and thus exhibit a complete spin polarization at the Fermi level and the realistic applications for spintronic devices [1–3]. This offers opportunities for a new generation of devices combining standard microelectronic with spin-dependent effects such as nonvolatile magnetic random access memories and magnetic sensors [4]. Half-metallicity was first predicted by de Groot and collaborators in a half-Heusler alloy [5]. Since then, more and more Heusler alloys have been initially predicted with half-metallicity by ab initio calculations and later experimental verification [6–13]. Heusler alloys can be classified into two main groups, i.e., halfHeusler XYZ alloys and full-Heusler X2YZ alloys. Here, X and Y denote transition metal elements, and Z is an s–p element [14,15]. Full-Heusler X2YZ alloys generally have two types of structures, Cu2MnAl and Hg2CuTi. Usually, the Heusler structure can be looked as four interpenetrating face-centered-cubic (FCC) lattices and has four unique crystal sites namely A(0,0,0), B(1/4,1/4,1/4), C(1/2,1/2,1/2), D(3/4,3/4,3/4) in Wyckoff coordinates. It is found that the site preference of the X and Y atoms is strongly influenced
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by the number of their valence electrons [16]. The Cu2MnAl-type structure, where two X atoms are located at A and C sites, whereas the Y and Z atoms are placed at B and D sites, respectively. On the other hand, for the Hg2CuTi-type structure, two X atoms are located at the A and B sites, whereas the Y and Z atoms are located at the C and D sites, respectively. Ti-based Heusler alloy Ti2YZ with the Hg2CuTi-type structure, such as Ti2CoAl [17], Ti2NiAl [18] and Ti2CoGa [19], the neighbor Ti atoms occupy A (0,0,0) and B(1/4,1/4,1/4) sites, and the remaining Y enters C(1/2,1/ 2,1/2) and Z at D(3/4,3/4,3/4) [16]. Due to different surroundings of the neighbor Ti atoms, in Ti2YZ alloy the d states of the neighbor Ti atoms should be hybridized first, then the hybridized Ti–Ti orbitals continue to couple with d states of Y atom [20], finally resulting in a similar band structure and magnetic behavior to the half-Heusler alloy. In this work, we present a systemic study the structure, electronic and magnetic properties of a series Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure by using the first-principles projector augmented wave (PAW) potential within the generalized gradient approximation (GGA). Up to now, no reports have been found on either theory or experiment investigations of the structure, electronic and magnetic properties of these alloys except Ti2MnAl [20]. The paper is organized as follows. In Section 2, the computational method is described detailly. In Section 3.1, the optimized lattice constants and magnetic properties are discussed. In Section 3.2, we discuss the origin of the band gap in half-metallic alloys. Section 3.3 presents the effect of lattice constant on the gap width. Finally in Section 4, we summarize our results and conclusions.
Q.-L. Fang et al. / Journal of Magnetism and Magnetic Materials 349 (2014) 104–108
2. Computational method The calculations are performed using the Vienna ab initio simulation package (VASP) based on the density function theory (DFT) [21–24]. The electron-ionic core interaction is represented by the projector augmented wave (PAW) potentials [25] which are more accurate than the ultra-soft pseudopotentials. To treat electron exchange and correlation, we chose the Perdew–Burke– Ernzerhof (PBE) [26] formulation of the generalized gradient approximation (GGA). A conjugate-gradient algorithm is used to relax the ions into their ground states, and the energies and the forces on each ion are converged within 1.0 10 5 eV/atom and 0.01 eV/Å, respectively. The cutoff energy for the plane-waves is chosen to be 400 eV. A 13 13 13 Monkhorst–Pack grid for k-point sampling is adopted for Brillouin zone integration, together with a Gaussian smearing broadening of 0.2 eV.
3. Results and discussions 3.1. The optimized lattice constants and magnetic properties There is no available experimental lattice constant for Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn) alloys. Consequently, the cubic lattice parameters for the alloys with the Hg2CuTi-type structure are optimized by minimizing the total energy with respect to the lattice parameter variation. For all studied alloys, the ferromagnetic state was found to be more stable than the paramagnetic state. As listed in Table 1, the optimized lattice constants of the Ti2MnZ alloys are 6.138, 6.199, 6.248, 5.997, 6.076, 6.317 Å for Z¼Al, Ga, In, Si, Ge, Sn, respectively. The optimized lattice constants for Ti substituting Mn at C(1/2,1/2,1/2) site, i.e., Ti2TiZ (Z ¼Al, Ga, In, Si, Ge, Sn), and Mn vacancy i.e., Ti2Z (Z¼Al, Ga, In, Si, Ge, Sn) are also listed in Table 1 for comparison. The optimized lattice constants of 6.138 and 6.281 Å for Ti2MnAl and Ti2TiSi alloys agree well with previous works of 6.24 and 6.29 Å, respectively [27]. We can see that with the same Z (Z¼ Al, Ga, In, Si, Ge, Sn) atom, the optimized lattice constant increases according to the order of Ti2MnZ, Ti2Z and Ti2TiZ. Combined with the values and orientations of the atomic magnetic moments listed in Table 1as well, the lattice constant of the Ti2MnZ is smaller than that of the Ti2TiZ may be resulted from not only the atomic radius of 1.79 Å for Mn atom is smaller than that of 2.00 Å for Ti atom but also the Table 1 Optimized lattice constant a (Å), partial magnetic moments m (mB) of the atoms at different sites, total magnetic moment mt (mB) and gap width Eg (eV) for Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys together with the results of Ti2TiZ (Z¼ Al, Ga, In, Si, Ge, Sn) and Ti2Z (Z¼ Al, Ga, In, Si, Ge, Sn) for comparison.
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additional attraction from the antiferromagnetic coupling between Mn and Ti atoms. The lattice constant of the Ti2MnZ is even smaller than that of the Ti2Z with Mn vacancy can be attributed to the additional repulsive force from the larger ferromagnetic coupling between two neighbor Ti atoms of the Ti2Z. It is also noted that from Table 1, For Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys, the magnetic moments of Mn atom are antiparallel to those of two neighbor Ti atoms. For Ti2TiZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, the magnetic moment of the Ti atom at C site is identical to that of the Ti atom at A site due to the same atomic circumstance. Two relative larger positive magnetic moments are observed for two neighbor Ti atoms in Ti2Z (Z ¼Al, Ga, In, Si, Ge, Sn). For all alloys studied here, the magnetic moments of the Ti1 and Ti2 atoms at A and B sites respectively are not equal due to the different atomic surrounding and the magnetic moments of the s–p elements are neglectable small. Previous first-principles studies predicted that many fullHeusler alloys with L21 structure were half-metallic ferromagnets. The total magnetic moments (mt) of the full-Heusler alloys follow the Slater-Pauling rule mt ¼Zt 24, where Zt is the total number of valence electrons in the unit cell. A similar behavior can also be found in half-Heusler alloys, but for these alloys the total magnetic moments follow the relation mt ¼ Zt 18 [7,28,29]. Recent research reveals that many Ti2-based full-Heusler alloys with the Hg2CuTitype structure can also belong to the family of half-metallic material, and the total magnetic moments of these alloys follow the mt ¼Zt 18 rule instead of the mt ¼Zt 24 rule [17,27,30]. Our integrated total moment mt is integer 0 with respect to total valence electrons 18 of the half-metallic Ti2MnZ (Z¼ Al, Ga, In) alloys, as well as integer 1 with respect to total valence electrons 19 of the half-metallic Ti2MnZ (Z ¼Si, Ge, Sn) alloys. All of these alloys are found to be the half-metallic ferrimagnets, their gap widths in spin-down channel (see next section) are listed in the last column of Table 1. In detail, the Ti2MnZ (Z¼ Al, Ga, In) alloys are half-metallic ferrimagnets with zero total magnetic moments, and are thus known as the half-metallic antiferromagnets or fully compensated ferrimagnets [31,32]. Half-metallic antiferromagnet differs from the conventional antiferromagnet in nature. In conventional antiferromagnet, the electronic structures are identical for both spin channels, as the result, the spin-polarization is zero, whereas in the half-metallic antiferromagnet the electronic structures are completely asymmetric, resulting in 100% spin polarization. Half-metallic antiferromagnet has an advantage in spintronics since the materials do not give rise to stray flux. We can see that, after a Ti substituting Mn atom or a Mn vacancy, the total magnetic moment mt is increased especially for a Mn vacancy case.
3.2. Origin of the band gap Alloys
a (Å)
mX1 (mB)
mX2 (mB)
mY (mB)
mZ (mB)
mt (mB)
Eg (eV)
Ti2MnAl Ti2TiAl Ti2Al Ti2MnGa Ti2TiGa Ti2Ga Ti2MnIn Ti2TiIn Ti2In Ti2MnSi Ti2TiSi Ti2Si Ti2MnGe Ti2TiGe Ti2Ge Ti2MnSn Ti2TiSn Ti2Sn
6.138 6.458 6.310 6.199 6.401 6.231 6.248 6.637 6.505 5.997 6.281 6.135 6.076 6.350 6.199 6.317 6.597 6.501
1.245 0.361 1.443 1.345 0.266 1.420 1.473 0.322 1.374 1.126 0.260 1.694 1.272 0.256 1.737 1.498 0.247 1.681
1.081 1.265 1.588 1.232 1.180 1.670 1.372 1.282 1.626 0.679 1.024 1.952 0.856 1.019 1.994 1.119 1.084 1.917
2.373 0.361
0.051 0.036 0.077 0.024 0.036 0.107 0.046 0.043 0.082 0.052 0.033 0.041 0.032 0.034 0.074 0.056 0.041 0.047
0.005 1.951 2.954 0.020 1.676 2.983 0.095 1.883 2.918 1.050 1.511 3.605 1.030 1.497 3.657 0.955 1.537 3.551
0.538
2.620 0.266 2.987 0.322 0.810 0.260 1.129 0.256 1.719 0.247
0.592
0.526
0.455
0.504
0.538
Since the magnetisms of the Heusler alloys are mainly determined by the not completely filled d orbitals of the 3d transition metals, the partial density of states (PDOS) projected onto d states of transition metal atoms Ti1, Ti2 and Mn at A, B and C sites, respectively, in Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys are shown in Fig. 1 together with those projected onto not completely filled p orbitals of Z atom at D site. The PDOS projected onto the completely filled s orbitals of each atom with two electrons are not plotted here because they are located in deep energy region around 6 eV. Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green lines. The origin of the band gap is usually distinguished into three categories: (1) covalent band gap, (2) d–d band gap, and (3) charge transfer band gap [33,34]. The covalent band gap has been found to exist in half-Heusler alloys with C1b structure. The d–d band gap is responsible for the half-metallicity of the full-Heusler alloys with
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4 2
PDOS (states/eV)
0 -2 -4
Ti2MnAl
Ti2MnGa
Ti2MnIn
Ti2MnSi
Ti2MnGe
Ti2MnSn
4 2 0 -2 -4
-6
-4
-2
0
2
4 -6
-4
-2 0 Energy (eV)
2
4 -6
-4
-2
0
2
4
Fig. 1. Partial density of states (PDOS) projected onto Ti1-d (black line), Ti2-d (red line), Mn-d (blue line) and Z–p (cyan line) for Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) Heusler alloys. Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3xtu 2xeu
3xtu 3xt2g
2xeu
3xt2g
2xeg
3xt2g
2xeg
3xtu 2xeu 2xe* 3xt* 3xt
3xt2g
3xt2g
EF 2xeg
2xeg
2xeg 2xe
Ti1
Ti2
Ti1-Ti2
Mn
Fig. 2. Possible hybridizations between d states of transition metal atoms in the Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys. The interactions between the two inequivalent Ti atoms are first taken into account (a), and subsequently consider the interaction between the Ti1–Ti2 coupling and Mn atom (b).
L21 structure. The charge transfer band gap is usually existed in CrO2 and double perovskites [34–36]. In our Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys, the situation is more complex due to the special crystallized structure. The Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys have a space group similar to the half-Heusler but the empty site was alternated by the additional Ti atom. Different from the full-Heusler alloy with L21 structure where two X atoms are located at A(0,0,0) and C(1/2,1/2,1/2) sites with identical surrounding, in Ti2MnZ (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, two Ti atoms occupy two neighbor sublattices A(0,0,0) and B(1/4,1/4,1/4) with different surrounding. Therefore, we must pay more attention to the effects of these two Ti atoms. To discuss the possible hybridization between the d states of the transition metal atoms, let us observe PDOS shown in Fig. 1. It can be seen that the d states of transition metal atoms have the similar feature of PDOS for Ti2MnAl, Ti2MnGa and Ti2MnIn alloys, while the other similar feature of PDOS in Ti2MnSi, Ti2MnGe and Ti2MnSn alloys. But for all of these alloys, the calculated results show a band gap in spin-down channel but a metal character in spin-up channel. The total density of states (TDOS) shown in Fig. 3 with black lines can be characterized by the large gap at the Fermi level and the occupied bonding states mainly present Mn atom characters below the Fermi level. It is evident that the d–d orbitals hybridization between transition metals is rather intensive. The Ti1 and Ti2 atoms are nearest-neighbors to each other. Thus, the interactions between the two inequivalent Ti atoms are
first taken into account. If the Mn and Z atoms are neglected, the two Ti sites form a diamond structure belonging to the octahedral symmetry (Oh). Since the current studied Heusler alloys have a tetrahedral Td symmetry which is a subgroup of Oh. Therefore, there could be states obeying the Oh being exclusively localized at the Ti atom. The possible hybridizations between the transition metal atoms are schematically shown in Fig. 2. The d orbitals of each Ti atom split into the double-degenerated 2xeg (dx2 y2 and dz2 ) states and triple-degenerated 3xt2g (dxy , dyz and dzx ) states. The names of the orbitals and subscript follow the nomenclature used in the documents and the coefficient represents the degeneracy of each orbital. There are strong hybridizations of the 3xt2g (2xeg) states between two neighbor Ti atoms, resulting in bonding 3xt2g (2xeg) and antibonding 3xtu (2xeu) states. Subsequently, the interactions between the Ti1–Ti2 coupling and Mn atom are considered. The Mn atom has a Td symmetry, and its d orbital is also split into the double-degenerated 2xeg (dx2 y2 and dz2 ) states and triple-degenerated 3xt2g (dxy , dyz and dzx ) states. The antibonding 3xtu and 2xeu states of Ti1–Ti2 coupling cannot hybridize to the 3xt2g and 2xeg states of the Mn atom, respectively, because of the lack of symmetry. While the bonding 2xeg and 3xt2g states of Ti1–Ti2 coupling also transform with the 2xeg and 3xt2g representations in Td group, respectively. So the 2xeg and 3xt2g states of the Ti1–Ti2 coupling are further hybridized with the 2xeg and 3xt2g states of Mn atom, giving rise to the bonding 2xe and 3xt as well as antibonding 2xen and 3xtn states.
Q.-L. Fang et al. / Journal of Magnetism and Magnetic Materials 349 (2014) 104–108
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20 10
TDOS (states/eV)
0 -10 -20 20
Z=Al
Z=Ga
Z=In
Z=Ge
Z=Sn
10 0 -10 Z=Si -20 -10 -8 -6 -4 -2
0
2
4
6 -10 -8 -6 -4 -2 0 2 Energy (eV)
4
6 -10 -8 -6 -4 -2
0
2
4
6
Fig. 3. Total density of states (TDOS) of the Ti2MnZ (black line), Ti2TiZ (red line) and Ti2Z (blue line) alloys (Z ¼Al, Ga, In, Si, Ge, Sn). Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.3. Variation of the gap with lattice constant The half-metallic character is closely related to the lattice constant of the alloys since the lattice constant significantly influences the width of the band gap at the Fermi level. In Fig. 4, we present the dependence of the gap width on the lattice constant for Ti2MnZ (Z¼ Al, Ga, In) alloys with gap in spin-up (spin-down) as examples. It can be seen that the gap width increases slightly with increasing the lattice constant for Ti2MnAl and Ti2MnGa alloys, while the gap width of Ti2MnIn alloy decreases with increasing the lattice constant. The change of gap width can be traced back to two combined effects: (1) p–d orbital
0.6 0.5 Ti2MnAl
0.4
Gap width (eV)
As elucidated above, the band gap of Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure are suggested to be determined mainly by the bonding 3xt and antibonding 3xt* states created from the hybridizations of the d states between the Ti1–Ti2 coupling and Mn atom, as well as the Fermi level just located in the band gap. The states near the Fermi level can be well ascribed to covalent hybridization and d–d orbitals hybridization between transition metal atoms. In addition, as can be seen from the PDOS patterns, the p states of the Z atoms play an important role in the Heusler alloys, although they do not directly form the band gap due to their zero value around the Fermi level, the hybridization between the p states of the Z atoms and d states of the transition metal atoms determines the degree of occupation of the p–d orbitals. Thus, the hybridization between p–d electrons affects the formation and width of the energy gap. In conclusion, it should be emphasized that there exist two mechanisms in the formation of the band gap, that is covalent band gap and d–d band gap, but it is mainly the d–d band gap that characterizes the half-metallicity in Ti2-based alloys. The calculated total densities of states (TDOS) are compared in Fig. 3 for Ti2MnZ (black lines), Ti2TiZ (red lines) and Ti2Z (blue lines) alloys (Z¼Al, Ga, In, Si, Ge, Sn). Positive (negative) value denotes spin-up (spin-down) channel. The Fermi level is set at zero energy and indicated by vertical green lines. We can see that, after a Ti or vacancy substituting Mn atom at C site, the spin-down band gaps and thus the half-metallic character disappear in Ti2TiZ and Ti2Z (Z¼ Al, Ga, In, Si, Ge, Sn) alloys, and below the Fermi level the occupied DOS in both channels shifts to the high energy region compared with Ti2MnZ (Z ¼Al, Ga, In, Si, Ge, Sn).
0.6 0.5 Ti2MnGa
0.4 0.6 0.5
Ti2MnIn
0.4 5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
Lattice constant (Å) Fig. 4. The gap width changes with the lattice constant in the range of 5.9–6.6 Å.
hybridization, which obviously depends on the p states of s–p element. (2) their different lattice constants. In Fig. 4, we can also see clearly the changes of width of the gap with different lattice constants. In fact, the contraction or expansion of the lattice constant has an influence on the delocalized p electrons of the s–p elements but not well localized d electrons of the transition metal atoms.
4. Conclusions The magnetic properties, electronic structures, and halfmetallicity of full-Heusler Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys with the Hg2CuTi-type structure have been studied by using the first-principles projector augmented wave (PAW) potential within the generalized gradient approximation (GGA). The following conclusions are obtained:
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(1) The Ti2MnZ (Z¼Al, Ga, In, Si, Ge, Sn) alloys are found to be half-metallic ferrimagnets and potential applications in spintronic devices. (2) The total magnetic moments of the Heusler alloys Ti2MnZ follow the mt ¼Zt 18 rule and agree with the Slater–Pauling curve quite well. For Ti2MnZ (Z ¼Al, Ga, In) alloys the total magnetic moment per unit cell is 0, together with the spindown band gaps of 0.538, 0.592, 0.526 eV for Z¼ Al, Ga, In, respectively. While for Ti2MnZ (Z¼Si, Ge, Sn) alloys, the total magnetic moment per unit cell is 1mB and their spin-down band gaps are 0.455, 0.504, 0.538 eV, respectively. (3) The band gaps are mainly determined by the bonding and antibonding states created from the hybridizations of the d states between the Ti1–Ti2 coupling and Mn atom. (4) The s–p elements play an important role in the half-metallicity of these Heusler alloys. In addition, for all Ti2-based Heusler alloys, the localized magnetic moment carried by the Ti and Mn atoms is restricted by the s–p element even though they have a small contribution to the total spin magnetic moment.
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