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Electronic structure and magnetic property of MnSn: Prediction of half-metallic ferromagnetism in zinc-blende structure
Solid State Communications 144 (2007) 18–22 www.elsevier.com/locate/ssc
Electronic structure and magnetic property of MnSn: Prediction of half-metallic ferromagnetism in zinc-blende structure L.H. Yu a,∗ , K.L. Yao a,b , Z.L. Liu a , Y.S. Zhang a a Department of Physics, Huazhong University of Science and Technology, WuHan 430074, China b International Center of Material Physics, Chinese Academy of Science, ShengYang 110015, China
Received 14 January 2007; received in revised form 19 April 2007; accepted 26 July 2007 by S. Scandolo Available online 8 August 2007
Abstract The electronic and magnetic properties of the transition-metal compound MnSn were studied, using the full potential augmented plane wave plus local orbitals method within density functional theory. The total energy calculations show, amongst four investigated structures: zinc-blende, NaCl, CsCl, and NiAs, MnSn prefers the antiferromagnetic NiAs structure at the ground state. It was found that only the zinc-blende metastable phase is a half-metallic ferromagnet with a magnetic moment of 3.00µB per formula unit, the ferromagnetic configuration is more stable than the antiferromagnetic one energetically. Its half-metallic gap reaches 0.24 eV at the equilibrium volume, and keep non-zero until compressed by 16% in the relative volume. The details of the electronic structure of zinc-blende MnSn were examined, the half-metallicity is attributed to the large exchange splitting and bonding–antibonding splitting. The volume compression results in the upward movement of the Fermi level, due to the increase in bandwidths, under the volume expansion an opposite behavior was found. c 2007 Elsevier Ltd. All rights reserved.
PACS: 71.20.Be; 75.90+w; 75.50.Pp; 71.70.Gm Keywords: A. Magnetically ordered materials; D. Electronic band structure
1. Introduction In the half-metallic ferromagnets, one spin band is semiconducting with a gap around the Fermi level, but another spin band is metallic, leading to 100% spin polarization at the Fermi level. Such a behavior is believed to be promising in the spin-based electronics (also known as spintronics) applications [1]. Based on band-structure calculations, de Groot et al. [2] first predicted the half-metallic behavior of C1b type half-Heusler alloy NiMnSb. Since then, such a property has been theoretically predicted in the transition-metal oxide CrO2 [3], magnetite (Fe3 O4 ) [4], some perovskite structures [5], several rare-earth nitrides [6], some Heusler alloys [7–12], and the zinc-blende transition-metal pnictides [13–22], chalcogenides [20,23,24], and the transitionmetal IV-group compounds (TmC [19,24,25], TmSi [24], and
∗ Corresponding author.
E-mail address: [email protected] (L.H. Yu). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.037
TmGe [24], where Tm is the transition-metal element). The calculations of the total energy and structural optimization show that these transition-metal IV-, V-, VI-group compounds do not crystallize in the zinc-blende structure at the ground state. In most cases, the NiAs structure of these compounds has the lowest energy [13–15,17–19,22,23], but this structure has not the half-metallicity. However, it is possible that these compounds crystallize in the zinc-blende metastable phase as thin film on the common zinc-blende semiconductor substrates. On the experimental side, the half-metallic ferromagnetism has been found in the transition-metal oxide CrO2 [26], Fe3 O4 [27], double-perovskite Sr2 CrReO6 [28], perovskite structure La0.7 Sr0.3 MnO3 [29]. In 2000, Akinaga et al. [30] first grew a thin film of the zinc-blende CrAs on top of GaAs, it was shown that the total spin magnetic moment is 3µB (Bohr magneton), which is consistent with the half-metallic property predicted by the electronic structure calculations published in the same paper. The sample also shows the desirable high Curie temperature TC , exceeding 400 K. This discovery intrigued the interest from both experimental and theoretical sides.
L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
Afterwards, Zhao et al. [31] and Deng et al. [32] have grown zinc-blende CrSb films by molecular beam epitaxy, and found the ferromagnetic transition temperature (beyond 400 K) to be similar to zinc-blende CrAs. However, it is difficult to grow the zinc-blende half-metallic films to be thick enough for the device fabrication [23,32]. Although many half-metallic ferromagnets have been theoretically predicted and some of them have been confirmed in experiments, it is important to explore new half-metallic ferromagnetic materials, which are compatible with the common semiconductors, and at the same time the halfmetallic gap is large enough. An important aspect on the applications of the half-metallic ferromagnets is the stability of the ferromagnetism and the half-metallic gap against the variation of the lattice parameter. In this work, we have performed the full potential first-principle calculations based on density functional theory for the manganese IV-group compound (MnSn). Based on band structure calculations, the half-metallic ferromagnetism of zinc-blende MnSn is predicted, the influences of lattice parameter on the electronic and magnetic properties are discussed. 2. Computational details We have calculated the structural, electronic, and magnetic properties of MnSn in several structural forms such as NiAs, zinc-blende, NaCl, and CsCl structures, using the full potential augmented plane wave plus local orbitals method (APW+lo) within density functional theory, as implemented in the Wien2k code [33]. In the APW+lo method, the Kohn–Sham orbitals are expanded into atom-like orbitals inside the non-overlapping muffin-tin (MT) spheres and into plane waves in the interstitial region [34–37]. For all four structures, we chose the muffintin sphere radii RMT = 2.1 bohr for Mn atom, and RMT = 2.3 bohr for Sn atom, respectively. The cutoff parameter RMT ∗ K MAX is equal to 7.5, where RMT is the smallest atomic sphere radius (i.e. Mn sphere radius) and K MAX (3.57/bohr) is the largest reciprocal lattice vector used in the plane wave expansion. The basis sets inside each MT sphere are split into core and valence states. For Mn atom 3 p, 3d, 4s states, and for Sn atom 4d, 5s, 5 p states are considered as valence states, respectively, others as core states. The core states are treated in a fully relativistic manner, for the valence states the relativistic effects are included at the scalar-relativistic level. For low-lying Mn 3 p and Sn 4d valence states (usually called semi-core states), additional local orbitals (LOs) [38, 39] are added to the basis sets to improve the variational flexibility. The exchange–correlation effects are treated within the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof [40]. The calculations of Continenza et al. [14] for Mn pnictides (MnSb, MnAs, and MnP) show that the local spin density approximation (LSDA) severely underestimates the equilibrium volume, but GGA obtains accurate equilibrium volume and magnetic moments. For the k-point sampling, we used a 12 × 12 × 12 k-points mesh in the first Brillouin zone for zinc-blende, NaCl, and CsCl structures,
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a 14×14×9 k-points mesh for NiAs structure, respectively. The self-consistency was achieved by demanding a convergence of and the total energy to be smaller than 5 × 10−5 Ry/MnSn, R at the same time the integrated charge difference ( |ρn (r ) − ρn−1 (r )|dr ) between last 2 iterations to be smaller than 1×10−4 electron. Since our ferromagnetic (FM) calculations revealed only the zinc-blende structure has the half-metallic property amongst four investigated structures, and the NiAs structure has the lowest energy, we also carried out the calculations of antiferromagnetic (AFM) ordering in the zinc-blende and NiAs structures. The zinc-blende structure belongs to the cubic facecentered structure with the space group F43m (No. 216), and has Mn atom at 4a site (0, 0, 0), Sn atom at 4c site (1/4, 1/4, 1/4) in the Wyckoff’s notation, respectively. The AFM configuration chosen by us has the + - + - sequence of ferromagnetically coupled Mn atom layers along z axis, the AFM interaction occurs between the Mn atom layers at z = 0 and z = 1/2. A tetragonal structure was chosen for the zincblende AFM calculation, this structure has the space group P4m2 (No. 115), with two Mn atom sites at 1a (0, 0, 0) and 1c (1/2, 1/2, 1/2), and one Sn atom site at 2g (0, 1/2, 1/4), the magnetic moments of two Mn atoms are antiparallel. The lattice of this tetragonal structure are a 0 = b0 = √ constants 0 a/ 2, c = a, where a is the lattice constant of zinc-blende. The hexagonal NiAs structure has the space group P63 /mmc (No. 194), with Mn atom at 2a site (0, 0, 0), Sn atom at 2c site (1/3, 2/3, 1/4). In the AFM-NiAs configure chosen by us, the AFM interaction occurs between the Mn atom layers at z = 0 and z = 1/2. We used a hexagonal structure for AFM-NiAs calculation, which has the space group P3m1 (No. 164), with two Mn atom sites at 1a (0, 0, 0) and 1b (0, 0, 1/2), and one Sn atom site at 2d (1/3, 2/3, 1/4), the magnetic moments of two Mn atoms are antiparallel. For the AFM zinc-blende (P4m2) and NiAs (P3m1) calculations, 14 × 14 × 10 and 14 × 14 × 9 kpoints meshes were used, respectively, but other calculation parameters are the same as the FM calculations. In order to check the numerical precision, we also calculated the FM state of the tetragonal structure (P4m2, two Mn atoms have parallel moments). By comparison of the total energies between AFM and FM, we try to find the magnetic ordering of ground state. 3. Results and discussions The volumes of the zinc-blende, NaCl, CsCl, and NiAs structures and the c/a ratio of the hexagonal NiAs structure were optimized by the total energy minimization. Fig. 1 depicts the total energy–volume and magnetic moment–volume curves of MnSn in the different structures. It is clear that the ground state phase of MnSn is in the AFM-NiAs structure, the FM-NiAs structure has a little (0.04 eV/MnSn) higher energy than the AFM-NiAs structure, this property also was found in the similar compound CrSb (FM state has 0.066 eV higher energy [17]). The zinc-blende structure has the highest energy amongst four structures, so the zinc-blende MnSn should not exist as bulk crystal. The zinc-blende MnSn can be achieved only when the material is grown on top of a
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L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
Fig. 2. The spin-dependent band structures of zinc-blende MnSn at the predicted equilibrium lattice constant. The deeper Sn 5s valence bands are not shown in the figure. The Fermi level is set to 0 eV.
Fig. 1. The total energy and magnetic moment per formula unit, as a function of volume. E tot is the total energy for different structures, E eq (AFM-NiAs) is the equilibrium total energy of AFM-NiAs structure. The perpendicular dashed lines indicate the equilibrium volumes for corresponding structures. Table 1 The calculated equilibrium lattice constants, magnetic moments M, and the relative energies Er , per formula unit
Zinc-blende NaCl CsCl NiAs
˚ Lattice constant (A)
M (µB )
Er (eV)
FM: a = 6.145 FM: a = 5.499 FM: a = 3.333 FM: a = 4.081, c/a = 1.36 AFM: a = 4.178, c/a = 1.27
3.00 3.16 3.08 2.79 0
1.41 0.43 0.37 0.04 0
compatible semiconductor substrate. Table 1 lists the calculated equilibrium lattice constants, magnetic moments, and the total energies relative to the AFM-NiAs structure for four investigated structures. It can be seen from Fig. 1 that when increasing the volume from the equilibrium volume of NiAs to that of zinc-blende, the difference in total energies between NiAs and zinc-blende decreases. The calculated equilibrium lattice constant of FM zinc-blende MnSn is quite close to that of the zinc-blende ˚ [19], 6.14 A ˚ [17]) half-metallic ferromagnets CrSb (6.15 A ˚ and MnSb (6.19 A [19]). The total energy–volume curves (the lower panel of Fig. 1) reveal that for the zinc-blende structure the FM state is energetically preferable compared to the AFM one in a large range of volume, the AFM ordering makes its equilibrium total energy per formula unit increase by 0.27 eV. The discrepancy of equilibrium total energy between FM zinc-blende and FM tetragonal structures is about 2 × 10−4 Ry/MnSn, which demonstrates the validity of our self-consistent calculations. The magnetic moment–volume relations (the upper panel of Fig. 1) show the moments in all four structures increase with volume. However, the moments of zinc-blende structure remain 3.00µB between −16% and at least 29% in the relative volume, which implies that the
Fig. 3. The partial density of states (DOS) of zinc-blende MnSn at the predicted equilibrium lattice constant. The positive and negative values are corresponding to the spin-up and spin-down states, respectively. The inset shows the hybridization between Sn p and Mn d-t2g above Fermi level.
zinc-blende MnSn has the half-metallic ferromagnetism. The band structure calculations show the NiAs (both AFM and FM states), NaCl, and CsCl structures of MnSn are metallic. The band structures of zinc-blende phase are shown in Fig. 2. It is clear that the spin-up bands (majority spin electrons) are metallic, but there is a gap of 0.74 eV around the Fermi level for spin-down bands (minority spin electrons), where the bottom of conduction bands is at the X -point (0.50 eV) of the Brillouin zone, the top of valence bands is at the X - or K -point (−0.24 eV, the energy difference in these two points is smaller than 0.001 eV). In the spin-down bands, the minimal gaps for creating hole and electron are 0.24 and 0.50 eV, respectively, so the minimal energy gap for a spin excitation (half-metallic gap) is 0.24 eV, the smaller one of these two gaps. From the band structures (Fig. 2) and partial density of states (as shown in Fig. 3) it can be seen that the two quite narrow bands are mainly formed by the Mn d-eg orbitals, around −2 eV for spin-up bands and +1.3 eV for spin-down bands, reflecting the large exchange splitting. Their hybridization with the states of Sn neighbors is weak, especially, at Γ -point (doubly degenerate Γ12 ) even zero, due to the tetrahedral coordination environment. If the exchange splitting is expressed as 1E x ≈ IM [41], we obtain I ≈ 1.1 eV for the exchange integral, where M = 3 for the spin moment. These flat bands can accommodate two electrons per spin. On the other hand, the
L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
tetrahedral environment allows the Mn d-t2g states to hybridize with the p states of the four Sn neighbors. Below the eg bands, there are three symmetry-induced p–d hybridization bonding-bands (at k = 0, triply degenerate Γ15 ), with a large dispersion due to the p–d interaction. These full-filled bands do not contribute to the total moment. Above the eg bands, the substantially wider antibonding p–d hybridization bands appear, crossing the Fermi level for spin-up bands and above the Fermi level for spin-down bands. Their large bandwidths also can be attributed to the strong p–d hybridization. An evident bonding–antibonding splitting can be seen from Fig. 2. The bonding–antibonding splitting and large exchange splitting together drive the half-metallic ferromagnetism to appear, this physical mechanism involved is similar to the V-, Cr-, Mnpnictides and chalcogenides [20,19,15]. The Mn and Sn atoms have seven and four valence electrons, respectively. In Fig. 2 five spin-up bands and three spin-down bands are fully filled. The s bands of Sn are very low in energy and omitted from the Fig. 2, where two Sn 5s electrons are accommodated (one per spin). The hybridization bondingbands (below the d-eg bands) can accommodate six electrons (three per spin). After eight of the valence electrons fully fill these four bands, the remaining electrons will fully fill the spinup d-eg bands (creating the magnetic moment of 2µB ), two partly filled spin-up bands together contribute 1µB to the total moment. Consequently, an integer total moment will be M = (Z tot − 8)µB = 3.0µ B /MnSn, where Z tot is the total number of valence electrons in the primitive cell (per formula unit). This similar Slater–Pauling behavior was previously found in the half-metallic full-Heusler [8] and half-Heusler [7] alloys (as the rule: M = (Z tot − 24)µB or M = (Z tot − 18)µB , respectively). The Sn atom has an induced local magnetic moment, which is antiparallel with the moment of the Mn atom, the magnetic moments in the Mn, Sn MT spheres used by us, and the interstitial regions are 3.35, −0.24, and −0.11µB , respectively. This property was also found in the other zinc-blende halfmetallic compounds [14,20]. We plot the total density of states at the calculated equilibrium volume and the compressed, expanded volumes in the Fig. 4. The volume compression causes a stronger hybridization, and increases the bandwidths, so some states of the spin-up bands near the Fermi level will be shifted to higher energies, the Fermi level must move upwards to conserve the total number of electrons (accommodate the integer electrons). When the volume is expanded, an opposite movement of the Fermi level occurs. Under the large volume compression or expansion, the Fermi level can move into the conduction or valence bands of spin-down bands, the half-metallicity will be lost. In summary, we have applied the full potential APW+lo method to investigate the electronic and magnetic properties of the transition-metal compound MnSn, and found that the zincblende phase is a half-metallic ferromagnet with a magnetic moment of 3.00µB per formula unit. Its half-metallic gap reaches 0.24 eV, and keep non-zero over a large volume range. Amongst four investigated structure: zinc-blende, NaCl, CsCl, and NiAs, only the zinc-blende phase exhibits the half-
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Fig. 4. The total spin-dependent DOS for the zinc-blende MnSn at the compressed (a), expanded (b), and the equilibrium (c) volumes.
metallicity. Although the calculated total energies show that the zinc-blende phase is unstable relative to its ground state AFM-NiAs phase, in view of recent experimental success to grow analogous such compounds on semiconductors, we hope it may be possible to grow zinc-blende MnSn films epitaxially on appropriate semiconductor substrates for the spintronics applications. Acknowledgment We acknowledge the support from the National Natural Science Foundation of China under the Grant No. 10574047, 10574048 and 20490210. References [1] 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. [2] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Phys. Rev. Lett. 50 (1983) 2024. [3] K. Schwarz, J. Phys. F: Met. Phys. 16 (1986) L211; M.A. Korotin, V.I. Anisimov, D.I. Khomskii, G.A. Sawatzky, Phys. Rev. Lett. 80 (1998) 4305. [4] A. Yanase, K. Siratori, J. Phys. Soc. Japan 53 (1984) 312. [5] K.I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, Y. Tokura, Nature 395 (1998) 677; Z. Szotek, W.M. Temmerman, A. Svane, L. Petit, H. Winter, Phys. Rev. B 68 (2003) 104411. [6] C.M. Aerts, P. Strange, M. Horne, W.M. Temmerman, Z. Szotek, A. Svane, Phys. Rev. B 69 (2004) 045115; C.G. Duan, R.F. Sabiryanov, J.J. Liu, W.N. Mei, P.A. Dowben, J.R. Hardy, Phys. Rev. Lett. 94 (2005) 237201. [7] I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev. B 66 (2002) 134428. [8] I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev. B 66 (2002) 174429. [9] I. Galanakis, J. Phys.: Condens. Matter. 16 (2004) 3089.
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Electronic structure and magnetic property of MnSn: Prediction of half-metallic ferromagnetism in zinc-blende structure L.H. Yu a,∗ , K.L. Yao a,b , Z.L. Liu a , Y.S. Zhang a a Department of Physics, Huazhong University of Science and Technology, WuHan 430074, China b International Center of Material Physics, Chinese Academy of Science, ShengYang 110015, China
Received 14 January 2007; received in revised form 19 April 2007; accepted 26 July 2007 by S. Scandolo Available online 8 August 2007
Abstract The electronic and magnetic properties of the transition-metal compound MnSn were studied, using the full potential augmented plane wave plus local orbitals method within density functional theory. The total energy calculations show, amongst four investigated structures: zinc-blende, NaCl, CsCl, and NiAs, MnSn prefers the antiferromagnetic NiAs structure at the ground state. It was found that only the zinc-blende metastable phase is a half-metallic ferromagnet with a magnetic moment of 3.00µB per formula unit, the ferromagnetic configuration is more stable than the antiferromagnetic one energetically. Its half-metallic gap reaches 0.24 eV at the equilibrium volume, and keep non-zero until compressed by 16% in the relative volume. The details of the electronic structure of zinc-blende MnSn were examined, the half-metallicity is attributed to the large exchange splitting and bonding–antibonding splitting. The volume compression results in the upward movement of the Fermi level, due to the increase in bandwidths, under the volume expansion an opposite behavior was found. c 2007 Elsevier Ltd. All rights reserved.
PACS: 71.20.Be; 75.90+w; 75.50.Pp; 71.70.Gm Keywords: A. Magnetically ordered materials; D. Electronic band structure
1. Introduction In the half-metallic ferromagnets, one spin band is semiconducting with a gap around the Fermi level, but another spin band is metallic, leading to 100% spin polarization at the Fermi level. Such a behavior is believed to be promising in the spin-based electronics (also known as spintronics) applications [1]. Based on band-structure calculations, de Groot et al. [2] first predicted the half-metallic behavior of C1b type half-Heusler alloy NiMnSb. Since then, such a property has been theoretically predicted in the transition-metal oxide CrO2 [3], magnetite (Fe3 O4 ) [4], some perovskite structures [5], several rare-earth nitrides [6], some Heusler alloys [7–12], and the zinc-blende transition-metal pnictides [13–22], chalcogenides [20,23,24], and the transitionmetal IV-group compounds (TmC [19,24,25], TmSi [24], and
∗ Corresponding author.
E-mail address: [email protected] (L.H. Yu). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.07.037
TmGe [24], where Tm is the transition-metal element). The calculations of the total energy and structural optimization show that these transition-metal IV-, V-, VI-group compounds do not crystallize in the zinc-blende structure at the ground state. In most cases, the NiAs structure of these compounds has the lowest energy [13–15,17–19,22,23], but this structure has not the half-metallicity. However, it is possible that these compounds crystallize in the zinc-blende metastable phase as thin film on the common zinc-blende semiconductor substrates. On the experimental side, the half-metallic ferromagnetism has been found in the transition-metal oxide CrO2 [26], Fe3 O4 [27], double-perovskite Sr2 CrReO6 [28], perovskite structure La0.7 Sr0.3 MnO3 [29]. In 2000, Akinaga et al. [30] first grew a thin film of the zinc-blende CrAs on top of GaAs, it was shown that the total spin magnetic moment is 3µB (Bohr magneton), which is consistent with the half-metallic property predicted by the electronic structure calculations published in the same paper. The sample also shows the desirable high Curie temperature TC , exceeding 400 K. This discovery intrigued the interest from both experimental and theoretical sides.
L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
Afterwards, Zhao et al. [31] and Deng et al. [32] have grown zinc-blende CrSb films by molecular beam epitaxy, and found the ferromagnetic transition temperature (beyond 400 K) to be similar to zinc-blende CrAs. However, it is difficult to grow the zinc-blende half-metallic films to be thick enough for the device fabrication [23,32]. Although many half-metallic ferromagnets have been theoretically predicted and some of them have been confirmed in experiments, it is important to explore new half-metallic ferromagnetic materials, which are compatible with the common semiconductors, and at the same time the halfmetallic gap is large enough. An important aspect on the applications of the half-metallic ferromagnets is the stability of the ferromagnetism and the half-metallic gap against the variation of the lattice parameter. In this work, we have performed the full potential first-principle calculations based on density functional theory for the manganese IV-group compound (MnSn). Based on band structure calculations, the half-metallic ferromagnetism of zinc-blende MnSn is predicted, the influences of lattice parameter on the electronic and magnetic properties are discussed. 2. Computational details We have calculated the structural, electronic, and magnetic properties of MnSn in several structural forms such as NiAs, zinc-blende, NaCl, and CsCl structures, using the full potential augmented plane wave plus local orbitals method (APW+lo) within density functional theory, as implemented in the Wien2k code [33]. In the APW+lo method, the Kohn–Sham orbitals are expanded into atom-like orbitals inside the non-overlapping muffin-tin (MT) spheres and into plane waves in the interstitial region [34–37]. For all four structures, we chose the muffintin sphere radii RMT = 2.1 bohr for Mn atom, and RMT = 2.3 bohr for Sn atom, respectively. The cutoff parameter RMT ∗ K MAX is equal to 7.5, where RMT is the smallest atomic sphere radius (i.e. Mn sphere radius) and K MAX (3.57/bohr) is the largest reciprocal lattice vector used in the plane wave expansion. The basis sets inside each MT sphere are split into core and valence states. For Mn atom 3 p, 3d, 4s states, and for Sn atom 4d, 5s, 5 p states are considered as valence states, respectively, others as core states. The core states are treated in a fully relativistic manner, for the valence states the relativistic effects are included at the scalar-relativistic level. For low-lying Mn 3 p and Sn 4d valence states (usually called semi-core states), additional local orbitals (LOs) [38, 39] are added to the basis sets to improve the variational flexibility. The exchange–correlation effects are treated within the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof [40]. The calculations of Continenza et al. [14] for Mn pnictides (MnSb, MnAs, and MnP) show that the local spin density approximation (LSDA) severely underestimates the equilibrium volume, but GGA obtains accurate equilibrium volume and magnetic moments. For the k-point sampling, we used a 12 × 12 × 12 k-points mesh in the first Brillouin zone for zinc-blende, NaCl, and CsCl structures,
19
a 14×14×9 k-points mesh for NiAs structure, respectively. The self-consistency was achieved by demanding a convergence of and the total energy to be smaller than 5 × 10−5 Ry/MnSn, R at the same time the integrated charge difference ( |ρn (r ) − ρn−1 (r )|dr ) between last 2 iterations to be smaller than 1×10−4 electron. Since our ferromagnetic (FM) calculations revealed only the zinc-blende structure has the half-metallic property amongst four investigated structures, and the NiAs structure has the lowest energy, we also carried out the calculations of antiferromagnetic (AFM) ordering in the zinc-blende and NiAs structures. The zinc-blende structure belongs to the cubic facecentered structure with the space group F43m (No. 216), and has Mn atom at 4a site (0, 0, 0), Sn atom at 4c site (1/4, 1/4, 1/4) in the Wyckoff’s notation, respectively. The AFM configuration chosen by us has the + - + - sequence of ferromagnetically coupled Mn atom layers along z axis, the AFM interaction occurs between the Mn atom layers at z = 0 and z = 1/2. A tetragonal structure was chosen for the zincblende AFM calculation, this structure has the space group P4m2 (No. 115), with two Mn atom sites at 1a (0, 0, 0) and 1c (1/2, 1/2, 1/2), and one Sn atom site at 2g (0, 1/2, 1/4), the magnetic moments of two Mn atoms are antiparallel. The lattice of this tetragonal structure are a 0 = b0 = √ constants 0 a/ 2, c = a, where a is the lattice constant of zinc-blende. The hexagonal NiAs structure has the space group P63 /mmc (No. 194), with Mn atom at 2a site (0, 0, 0), Sn atom at 2c site (1/3, 2/3, 1/4). In the AFM-NiAs configure chosen by us, the AFM interaction occurs between the Mn atom layers at z = 0 and z = 1/2. We used a hexagonal structure for AFM-NiAs calculation, which has the space group P3m1 (No. 164), with two Mn atom sites at 1a (0, 0, 0) and 1b (0, 0, 1/2), and one Sn atom site at 2d (1/3, 2/3, 1/4), the magnetic moments of two Mn atoms are antiparallel. For the AFM zinc-blende (P4m2) and NiAs (P3m1) calculations, 14 × 14 × 10 and 14 × 14 × 9 kpoints meshes were used, respectively, but other calculation parameters are the same as the FM calculations. In order to check the numerical precision, we also calculated the FM state of the tetragonal structure (P4m2, two Mn atoms have parallel moments). By comparison of the total energies between AFM and FM, we try to find the magnetic ordering of ground state. 3. Results and discussions The volumes of the zinc-blende, NaCl, CsCl, and NiAs structures and the c/a ratio of the hexagonal NiAs structure were optimized by the total energy minimization. Fig. 1 depicts the total energy–volume and magnetic moment–volume curves of MnSn in the different structures. It is clear that the ground state phase of MnSn is in the AFM-NiAs structure, the FM-NiAs structure has a little (0.04 eV/MnSn) higher energy than the AFM-NiAs structure, this property also was found in the similar compound CrSb (FM state has 0.066 eV higher energy [17]). The zinc-blende structure has the highest energy amongst four structures, so the zinc-blende MnSn should not exist as bulk crystal. The zinc-blende MnSn can be achieved only when the material is grown on top of a
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L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
Fig. 2. The spin-dependent band structures of zinc-blende MnSn at the predicted equilibrium lattice constant. The deeper Sn 5s valence bands are not shown in the figure. The Fermi level is set to 0 eV.
Fig. 1. The total energy and magnetic moment per formula unit, as a function of volume. E tot is the total energy for different structures, E eq (AFM-NiAs) is the equilibrium total energy of AFM-NiAs structure. The perpendicular dashed lines indicate the equilibrium volumes for corresponding structures. Table 1 The calculated equilibrium lattice constants, magnetic moments M, and the relative energies Er , per formula unit
Zinc-blende NaCl CsCl NiAs
˚ Lattice constant (A)
M (µB )
Er (eV)
FM: a = 6.145 FM: a = 5.499 FM: a = 3.333 FM: a = 4.081, c/a = 1.36 AFM: a = 4.178, c/a = 1.27
3.00 3.16 3.08 2.79 0
1.41 0.43 0.37 0.04 0
compatible semiconductor substrate. Table 1 lists the calculated equilibrium lattice constants, magnetic moments, and the total energies relative to the AFM-NiAs structure for four investigated structures. It can be seen from Fig. 1 that when increasing the volume from the equilibrium volume of NiAs to that of zinc-blende, the difference in total energies between NiAs and zinc-blende decreases. The calculated equilibrium lattice constant of FM zinc-blende MnSn is quite close to that of the zinc-blende ˚ [19], 6.14 A ˚ [17]) half-metallic ferromagnets CrSb (6.15 A ˚ and MnSb (6.19 A [19]). The total energy–volume curves (the lower panel of Fig. 1) reveal that for the zinc-blende structure the FM state is energetically preferable compared to the AFM one in a large range of volume, the AFM ordering makes its equilibrium total energy per formula unit increase by 0.27 eV. The discrepancy of equilibrium total energy between FM zinc-blende and FM tetragonal structures is about 2 × 10−4 Ry/MnSn, which demonstrates the validity of our self-consistent calculations. The magnetic moment–volume relations (the upper panel of Fig. 1) show the moments in all four structures increase with volume. However, the moments of zinc-blende structure remain 3.00µB between −16% and at least 29% in the relative volume, which implies that the
Fig. 3. The partial density of states (DOS) of zinc-blende MnSn at the predicted equilibrium lattice constant. The positive and negative values are corresponding to the spin-up and spin-down states, respectively. The inset shows the hybridization between Sn p and Mn d-t2g above Fermi level.
zinc-blende MnSn has the half-metallic ferromagnetism. The band structure calculations show the NiAs (both AFM and FM states), NaCl, and CsCl structures of MnSn are metallic. The band structures of zinc-blende phase are shown in Fig. 2. It is clear that the spin-up bands (majority spin electrons) are metallic, but there is a gap of 0.74 eV around the Fermi level for spin-down bands (minority spin electrons), where the bottom of conduction bands is at the X -point (0.50 eV) of the Brillouin zone, the top of valence bands is at the X - or K -point (−0.24 eV, the energy difference in these two points is smaller than 0.001 eV). In the spin-down bands, the minimal gaps for creating hole and electron are 0.24 and 0.50 eV, respectively, so the minimal energy gap for a spin excitation (half-metallic gap) is 0.24 eV, the smaller one of these two gaps. From the band structures (Fig. 2) and partial density of states (as shown in Fig. 3) it can be seen that the two quite narrow bands are mainly formed by the Mn d-eg orbitals, around −2 eV for spin-up bands and +1.3 eV for spin-down bands, reflecting the large exchange splitting. Their hybridization with the states of Sn neighbors is weak, especially, at Γ -point (doubly degenerate Γ12 ) even zero, due to the tetrahedral coordination environment. If the exchange splitting is expressed as 1E x ≈ IM [41], we obtain I ≈ 1.1 eV for the exchange integral, where M = 3 for the spin moment. These flat bands can accommodate two electrons per spin. On the other hand, the
L.H. Yu et al. / Solid State Communications 144 (2007) 18–22
tetrahedral environment allows the Mn d-t2g states to hybridize with the p states of the four Sn neighbors. Below the eg bands, there are three symmetry-induced p–d hybridization bonding-bands (at k = 0, triply degenerate Γ15 ), with a large dispersion due to the p–d interaction. These full-filled bands do not contribute to the total moment. Above the eg bands, the substantially wider antibonding p–d hybridization bands appear, crossing the Fermi level for spin-up bands and above the Fermi level for spin-down bands. Their large bandwidths also can be attributed to the strong p–d hybridization. An evident bonding–antibonding splitting can be seen from Fig. 2. The bonding–antibonding splitting and large exchange splitting together drive the half-metallic ferromagnetism to appear, this physical mechanism involved is similar to the V-, Cr-, Mnpnictides and chalcogenides [20,19,15]. The Mn and Sn atoms have seven and four valence electrons, respectively. In Fig. 2 five spin-up bands and three spin-down bands are fully filled. The s bands of Sn are very low in energy and omitted from the Fig. 2, where two Sn 5s electrons are accommodated (one per spin). The hybridization bondingbands (below the d-eg bands) can accommodate six electrons (three per spin). After eight of the valence electrons fully fill these four bands, the remaining electrons will fully fill the spinup d-eg bands (creating the magnetic moment of 2µB ), two partly filled spin-up bands together contribute 1µB to the total moment. Consequently, an integer total moment will be M = (Z tot − 8)µB = 3.0µ B /MnSn, where Z tot is the total number of valence electrons in the primitive cell (per formula unit). This similar Slater–Pauling behavior was previously found in the half-metallic full-Heusler [8] and half-Heusler [7] alloys (as the rule: M = (Z tot − 24)µB or M = (Z tot − 18)µB , respectively). The Sn atom has an induced local magnetic moment, which is antiparallel with the moment of the Mn atom, the magnetic moments in the Mn, Sn MT spheres used by us, and the interstitial regions are 3.35, −0.24, and −0.11µB , respectively. This property was also found in the other zinc-blende halfmetallic compounds [14,20]. We plot the total density of states at the calculated equilibrium volume and the compressed, expanded volumes in the Fig. 4. The volume compression causes a stronger hybridization, and increases the bandwidths, so some states of the spin-up bands near the Fermi level will be shifted to higher energies, the Fermi level must move upwards to conserve the total number of electrons (accommodate the integer electrons). When the volume is expanded, an opposite movement of the Fermi level occurs. Under the large volume compression or expansion, the Fermi level can move into the conduction or valence bands of spin-down bands, the half-metallicity will be lost. In summary, we have applied the full potential APW+lo method to investigate the electronic and magnetic properties of the transition-metal compound MnSn, and found that the zincblende phase is a half-metallic ferromagnet with a magnetic moment of 3.00µB per formula unit. Its half-metallic gap reaches 0.24 eV, and keep non-zero over a large volume range. Amongst four investigated structure: zinc-blende, NaCl, CsCl, and NiAs, only the zinc-blende phase exhibits the half-
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Fig. 4. The total spin-dependent DOS for the zinc-blende MnSn at the compressed (a), expanded (b), and the equilibrium (c) volumes.
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