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First-principles study of electronic structure for the Laves-phase compounds HfFe2 and HfV2
Journal of Alloys and Compounds 448 (2008) 53–58
First-principles study of electronic structure for the Laves-phase compounds HfFe2 and HfV2 Changwen Zhang a,∗ , Zhong Zhang a , Shaoqing Wang a , Hua Li b , Jianmin Dong b , Naisheng Xing b , Yongquan Guo c , Wei Li c a School of Science, Jinan University, Jinan 250022, PR China School of Physics and Microelectronics, Shandong University, Jinan 250100, PR China c Institute of Functional materials, Central Iron and Steel Research Institute, Beijing 100081, PR China b
Received 27 September 2006; received in revised form 9 November 2006; accepted 9 November 2006 Available online 12 December 2006
Abstract Ab initio calculations have been performed to calculate the density of states (DOS), energy bands and charge density distributions of paramagnetic cubic Laves-phase HfFe2 and HfV2 based on the method of augmented plane waves plus local orbital to reveal its electronic structure. The results revealed the overlap of charge densities between the neighboring Fe–Fe (V–V) atoms. This showed a rather strong covalent bonding between Fe–Fe (V–V) atoms and a metallic bonding between Hf–Fe (Hf–V) atoms. The Fe-d states are also more spatially confined than those of V-d, therefore, the hybridization of the valence electrons between Hf–V are stronger than that between Hf–Fe, leading to a stronger covalent bonding in HfV2 compound. These results are helpful to understanding the physical and chemical properties for both compounds. © 2006 Elsevier B.V. All rights reserved. Keywords: Electronic structure; Laves-phase compound; First principles; Density of states; Bonding
1. Introduction Because of the growing interest in intermetallic compounds as potential high-temperature structural materials, there have been many investigations carried out on various compounds, mostly structures that are ordered forms of simple fcc, bcc, and hcp metals [1]. If new intermetallic-based alloys are to be selected on the basis of low density and high-melting temperature, then three groups of materials emerge as promising candidates: aluminides [2], topologically close-packed compounds [3], and silicide-based compounds [4]. In the group of topologically close-packed compounds their structures generally are complex, in that the unit cell contains many atoms, even though the crystal structures may have high symmetry [5]. A number of these compounds have quite high-melting temperatures, low densities, and high-oxidation resistance, the properties necessary for high-temperature structural applications. However, all three groups of materials are often brittle
∗
Corresponding author. E-mail address: [email protected] (C. Zhang).
0925-8388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2006.11.039
at low temperatures up to temperatures of hundreds of degrees Celsius. This low-temperature brittleness adversely affects the fabrication and use of these materials. Consequently, it is highly desirable to find ways to improve the low-temperature ductility without compromising much on the attractive high-temperature properties. Topologically close-packed Laves phase are the most abundant among intermetallic compounds, with over 360 binary Laves phases reported, in which a promising approach to enhance ductility at low-temperatures is the addition of a third metal to form a ternary alloy. However, since there are many Laves-phase materials, the possibilities of forming ternary alloys for this purpose are countless. A good understanding of the basic properties of binary Laves-phase materials will help materials scientists focus on a limited number of systems instead of manufacturing and testing every possible candidate. The experiments have been proved that among the three Laves-phase structures, C15 Laves phases are expected to show better deformability than the other two Laves phases because of their fcc-based structure. Nakamichi et al. [6] confirmed that the stoichiometric composition of HfFe2 shows mainly the cubic MgCu2 -type structure containing a small amount of hexagonal MgZn2 -type
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C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
Fig. 1. The energy bands for HfM2 compound.
structure. They also observed the temperature dependence of the magnetization. Belosevic-Cavor [8] and Yamada and Shimizu [7] investigated the relation of electronic structure and magnetic properties of HfFe2 compound. The pseudobinary system (Hf, Zr)V2 have been investigated as superconducting materials [9,10]. Because of the high-melting point, high strength at elevated temperatures and reasonably good oxidation resistance, HfV2 have been studied recently for high-temperature structural applications [11–17]. In order to further understand the physical and chemical properties of these compound materials, an investigation of the electronic structure and bonding characteristic of Laves-phase compounds has both fundamental scientific and technological significance. Therefore Laves-phase compounds HfFe2 and HfV2 have been chosen as representative examples. 2. Details of calculation HfM2 (M = Fe and V) compound crystallizes in the cubic Fd3m structure (space group no. 227) and therefore it belongs to the large class of Laves-phase compounds. The unit cell consists of two formula units (i.e. six atoms). The calculations of the electronic structure and total energy of HfFe2 and HfV2 compounds was done using the method of APW + lo, which was implemented in the software package WIEN2K [18]. Core states are treated fully relativistically while semi-core and valence states are treated within a scalar relativistic approximation in the calculation. The APW + lo method converged practically the identical results as the linearized augmented plane wave (LAPW) method, but using smaller basis sets in the APW + lo method leads to significantly reduce the computation time by up to an order of magnitude [19]. Potential and charge densities inside the atomic spheres are expanded in lattice harmonics up to L = 8 and using GGA to treat exchange and correlation effects within the density functional theory. Muffin-tin (Mt) radius (Rmt ) of 2.2 a.u. was chosen for Hf, and 2.3 and 2.5 a.u. for Fe and V atoms, respectively, and a plane wave cut-off (Rmt Kmax ) of 10.0 was used. The calculation was performed with a 84-k point mesh in
the irreducible wedge of the Brillouin zone. When the energy difference was less than 0.1 mRy, convergence was assumed. The Brillouin zone integration was carried out according to a modified tetrahedron method [20]. 3. Results and discussion In this section, the ground-state properties are obtained by a minimization of the total energy with respect to the volume with GGA and LDA approach in the Laves-phase compounds HfFe2 and HfV2 . By analyzing more carefully these results, we observe that the lattice constant with GGA approach is evaluated at 2.16% and 5.17%, respectively, smaller than the available experimental data, in better agreement with the experiments (than the LDA approach). So, the experimental lattice constant is used in our calculations. Figs. 1–3 show the energy bands, the calculated total and partial density of states (DOS) and charge densities for HfFe2 and HfV2 , respectively. The Fermi level of HfFe2 crosses the density of states curve quite close to a small local maximum, whereas that of HfV2 crosses at a relatively high-local minimum. The density of states at the Fermi energy, n(EF ), is found to be 194 and 216 states/eV for HfFe2 and HfV2 , respectively. Ormeci et al. obtained the density of states to be 197.9 states/eV for HfV2 with LMTO method, which is similar with us. In both cases, the density of states near the Fermi level is dominated by the d states. In the case of HfV2 the contribution of the V-d states is 120 states/eV compared to 25 states/eV for Hf-d per atom. For HfFe2 on the other hand, Fe-d and Hf-d contributions are 158 and 12 states/eV, respectively. We also notice one difference between HfV2 and HfFe2 regarding the contributions of the B-p states to n(EF ), where the V-p contribution to n(EF ) is very significant compared with the Fe-p states. It also can be seen from energy band structures in both cases that the peaks arise from the almost dispersionless bands near the Fermi level. Hence, a precise location of EF is very important for the accurate calculation of those physical properties (such as magnetism and
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
55
Fig. 2. The calculated density of states for HfFe2 .
Fig. 3. The calculated density of states for HfV2 .
superconductivity), which depends critically on the value of the DOS at EF . On the other hand, the dominant contribution to the DOS at lower energies comes from bonding Hf–M s-like band beginning at −7.92 and −6.63 eV for HfFe2 and HfV2 , respectively. The bandwidths, as measured by the 1 − X1 energy difference, are nearly the same for both compounds (3 eV). It is seen that in the region −3 to 2 eV near EF the M-d states are dominant, with important contributions from the Hf-d states and others as well. The Fe-d states appear to be confined to this energy range, while the V-d states are seen to have significant additional contributions in the range of 7 eV near EF . The most important feature of these DOS for both compounds is the overlapping of d states
from Hf and M in the construction of electronic energy band, which implies hybridization between Hf-d and M-d states on formation of the HfM2 compound. Moreover, the hybridization between the Hf–V states is stronger than that between Hf–Fe states in the region near EF , where the Fermi levels fall, while the corresponding states in HfFe2 compound are more diffuse, which the Hf-d states show peaks near −2 eV. Hf–Fe hybridizations result in a bigger shift of empty states to the Fermi level for the Hf-d states upon formation of HfFe2 compared to HfV2 . As seen from Figs. 2 and 3, there are high densities of unoccupied states spreading far above the Fermi level for d states for HfFe2 , and the unoccupied states are fewer and nearer the Fermi level for the Fe d states compared with those for the V-d states.
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C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
˚ −3 with 3 Einstein A ˚ −3 intervals and Fig. 4. Contour plot of charge density in the (1 1 0) plane for HfFe2 . The contour lines are drawn (a) from 1.0 to 100 Einstein A ˚ −3 intervals. ˚ −3 with 0.02 Einstein A (b) from 0 to 1 Einstein A
The Fe-d states are also more spatially confined than those of V. Table 1 presents DOS values at EF together with the charges Q in the MT spheres. This can be seen by noting from Table 1 that Fe MT charges are QV d = 2.8529 and Qd = 5.9816, respectively, V Fe realizing that Qd exceed Qd by more than the three extra electrons in Fe compared to V, which is consistent with DOS above. Belosevic-Cavor [8] investigated the spin-polarized electronic structure of HfFe2 compound with LMTO method. The results showed that the spin-polarized total charges are nearly equal to total MT charges we calculated in LAPW method. This is probably because the Fe MT radii are smaller than V due to the smaller lattice constants of the former. Ohba et al. [21] obtained the structure factors of MgZn2 and MyCu2 by the single crystal X-ray diffraction method. They showed that the charge transfers occurred between the constituent atoms by the population
analysis and localized electrons were seen in the center of the tetrahedral formed by the small atoms in the difference Fourier maps. A similar phenomena is observed from Table 1, in which in both cases some electrons are transferred from Hf atom to Fe or V atoms when one account for the different MT radii. In order to exploring the bonding characteristics in the HfM2 compound, the charge density maps of the (1 1 0) plane for both cases are shown in Figs. 4 and 5. As can be seen from Figs. 4 and 5, the (1 1 0) plane is bisecting the B tetrahedral, containing the [1 1 1] close-packing direction. Figs. 4 and 5(a) shows the higher density region of the plane, which corresponds to the core electron distribution of Hf and M atoms. Fe or V atoms have almost spherical distribution. In comparison, however, Cu atoms are slightly deformed in Laves-phase MgCu2 [22]. Figs. 4 and 5(b) shows the lower density region of the
˚ −3 with 3 Einstein A ˚ −3 intervals and Fig. 5. Contour plot of charge density in the (1 1 0) plane for HfV2 . The contour lines are drawn (a) from 1.0 to 100 Einstein A ˚ −3 with 0.02 Einstein A ˚ −3 intervals. (b) from 0 to 1 Einstein A
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58 Table 1 The total DOS n(EF ) (states/eV) and MT charges (electrons/atom) for HfM2 compounds
57
Material
Atom
n(EF )
Qs
Qp
Qd
Qf
HfFe2
Hf Fe
194
2.1774 0.3601
5.9019 6.3021
1.0997 5.9816
13.9506 0.0160
nificant in HfV2 than that in HfFe2 . In face, the d electrons from the bonding did not contribute to magnetic moment at all, and the magnetic properties of C15 compounds come mainly from the other d electrons. The strength of covalent bonding for HfV2 is stronger than that of HfFe2 , which are helpful to increasing low-temperature ductility and Young’s modulus of compound.
HfV2
Hf V
216
2.1847 2.2497
5.8875 6.1166
1.1521 2.8529
13.9440 0.0117
4. Conclusions
same plane. The contour lines of the higher density regions are omitted. Figs. 4 and 5(b) shows that Hf atoms are almost spherical, but the contour line of low-charge density around Hf ˚ −3 for Hf–Fe and atoms, for example 0.39 and 0.43 Einstein/A Hf–V compounds, moves obviously towards Fe and V, respectively. In contrast, M atoms are anisotropic. Our calculation results, which are very similar with the contour plot obtained by Ormeci et al. [16], clearly indicate the directional d bonding in HfV2 , much stronger than that in Fe compound. The reason for this behavior is probably the transfer of electrons form the Hf to the Fe (V) atom, to fill up the incompletely filled inner shell. On the other hand, the overlap of electron densities between M–M atoms implies a covalent bonding between M–M atoms due to hybridization of the d orbitals. The height of the charge density at the bond midpoint between Fe–Fe was found to ˚ −3 , smaller than 0.64 Einstein/A ˚ −3 between be 0.57 Einstein/A V–V, whereas the height of the charge density at the bond mid˚ −3 . It point between Hf–M is almost the same 0.38 Einstein/A could be assumed that the bonding between Hf–M is much weaker than that between M–M in the HfFe2 , while the strength of bonding between V–V is rather stronger that corresponding that between Fe–Fe. This could be explained from the point of view of the C15 Laves-phase structure. The nearest neighboring sites of Fe are occupied by 6 Fe atoms in the C15 Lavesphase HfFe2 . The atomic distance between the nearest Fe–Fe atoms is 4.688 a.u., which is 2.79% shorter than that in pure Fe (4.802 a.u.) with an hcp structure. Compared with the atomic distance between the next nearest Hf–Fe atoms (5.497 a.u.), the Fe–Fe covalent bonding is thus stronger than that between Hf–Fe atoms, which is a mainly metallic bonding in HfFe2 . The hybridization of Fe–Fe therefore is stronger than Hf–Fe hybridization in HfFe2 compound and even stronger than Hf–Hf hybridization in Hf metal. In the interatomic region, the electrons distribute evenly like a metallic bond whose nature is well explained by the nearly free electron model. For HfV2 , on the other hand, the atomic distance between the nearest V–V atoms is 4.8643 a.u., which is 11.12% shorter than that in pure V metal. The atomic distance between the next nearest Hf–V atoms is 5.7040 a.u. Thus the hybridization of V–V in HfV2 is stronger than Fe–Fe hybridization in HfFe2 . A further analysis of the decomposed d DOS showed the bonding between the Hf–Fe atoms is mainly contributed by hybridization between the d2g orbital on the Hf sites and the orbital made up of dx2–y2 and dxy on the Fe sites. In comparison, the bonding between the H–V atoms is mainly contributed by hybridization between the d2g orbital on the Hf sites and the orbital dxy on the V sites. Therefore, it can be said that the hybridization of Hf-d and V-d states is more sig-
The electronic structure and bonding character of C15 type Laves-phase HfFe2 were investigated in the paper. Ab initio calculations have been performed to calculate the density of states (DOS), charge density distribution and band structure of the intermetallic compound HfFe2 based on the method of augmented plane waves plus local orbitals (APW + lo). The results revealed the overlap of charge densities between the neighboring Fe–Fe (V–V) atoms. This showed a rather strong covalent bonding between Fe–Fe (V–V) atoms and a metallic bonding between Hf–Fe (Hf–V) atoms. Furthermore, the Fe-d states are also more spatially confined than those of V-d, the hybridization of the valence electrons between Hf–V are stronger than that between Hf–Fe, leading to a stronger covalent bonding in HfV2 compound. On the other hand, there is virtually no covalency between Hf–Fe (or V) atoms, although HfFe2 (HfV2 ) are called intermetallic compounds. In the interatomic region, the electrons are distributed evenly like a metallic bond. In fact, the d electrons from the bonding did not contributed to magnetic moment, and the magnetic properties come mainly from the other d electrons. These results are helpful to understanding the physical and chemical properties for both compounds. References [1] J.K. Tien, T. Caulfield, Superalloys, Supercomposites and Superceramics, Academic Press, New York, 1989. [2] J. Horton, I. Baker, S. Hanada, R.D. Noebe, MRS Symposia Proceedings No. 364, Materials Research Society, Pittsburgh, 1995. [3] D.P. Pope, F. Chu, Structural Intermetallics, TMS Publications, Warrendale, PA, 1993. [4] T.E. Mitchell, R.G. Castro, J.J. Petrovic, S.A. Maloy, O. Unal, M.M. Chadwick, Mater. Sci. Eng. A 155 (1992) 241. [5] J.H. Westbrook, Intermetallic Compounds, John Wiley and Sons Inc., New York, 1967. [6] T. Nakamichi, K. Kai, Y. Aoki, K. Ikeda, M. Yamamoto, J. Phys. Soc. Jpn. 29 (1970) 794. [7] H. Yamada, M. Shimizu, J. Phys. F: Met. Phys. 16 (1986) 1039. [8] J. Belosevic-Cavor, V. Koteski, N. Novakovic, G. Concas, F. Congiu, G. Spano, Eur. Phys. J. B: Condens. Matter Phys. 50 (2006) 425. [9] K. Inoue, K. Tachikawa, IEEE Trans. Magn. 15 (1979) 635. [10] Z. Wu, N.L. Saini, S. Agrestini, D.D. Castro, A. Bianconi, A. Marcelli, M. Battisti, D. Gozzi, G. Balducci, J. Phys.: Condens. Matter 12 (2000) 6971. [11] F. Chu, D.J. Thoma, P.G. Kotula, S. Gerstl, T.E. Mitchell, I.M. Anderson, J. Bentley, Philos. Mag. A 77 (1998) 941. [12] F. Chu, D.J. Thoma, T.E. Mitchell, M. Sob, C.L. Lin, Philos. Mag. B 77 (1998) 121. [13] C.T. Liu, J.H. Zhu, M.P. Brady, C.G. McKamey, L.M. Pike, Intermetallics 8 (2000) 1119. [14] F. Chu, D.P. Pope, Mater. Sci. Eng. A 170 (1993) 39. [15] J.D. Livingston, E.L. Hall, J. Mater. Res. 5 (1990) 59. [16] A. Ormeci, F. Chu, M.W. John, T.E. Mitchell, R.C. Albers, S.P. Chen, Phys. Rev. B: Condens. Matter 54 (1996) 12753.
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[17] T. Jarlborg, A.J. Freeman, Phys. Rev. B 22 (1980) 2332. [18] K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. [19] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
[20] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [21] T. Ohba, Y. Kitano, Y. Komuar, Acta Crystallogr. C 40 (1984) 1. [22] Y. Kubota, M. Takata, M. Sakata, T. Ohba, K. Kifune, T. Tadaki, J. Phys.: Condens. Matter 12 (2000) 1253.
First-principles study of electronic structure for the Laves-phase compounds HfFe2 and HfV2 Changwen Zhang a,∗ , Zhong Zhang a , Shaoqing Wang a , Hua Li b , Jianmin Dong b , Naisheng Xing b , Yongquan Guo c , Wei Li c a School of Science, Jinan University, Jinan 250022, PR China School of Physics and Microelectronics, Shandong University, Jinan 250100, PR China c Institute of Functional materials, Central Iron and Steel Research Institute, Beijing 100081, PR China b
Received 27 September 2006; received in revised form 9 November 2006; accepted 9 November 2006 Available online 12 December 2006
Abstract Ab initio calculations have been performed to calculate the density of states (DOS), energy bands and charge density distributions of paramagnetic cubic Laves-phase HfFe2 and HfV2 based on the method of augmented plane waves plus local orbital to reveal its electronic structure. The results revealed the overlap of charge densities between the neighboring Fe–Fe (V–V) atoms. This showed a rather strong covalent bonding between Fe–Fe (V–V) atoms and a metallic bonding between Hf–Fe (Hf–V) atoms. The Fe-d states are also more spatially confined than those of V-d, therefore, the hybridization of the valence electrons between Hf–V are stronger than that between Hf–Fe, leading to a stronger covalent bonding in HfV2 compound. These results are helpful to understanding the physical and chemical properties for both compounds. © 2006 Elsevier B.V. All rights reserved. Keywords: Electronic structure; Laves-phase compound; First principles; Density of states; Bonding
1. Introduction Because of the growing interest in intermetallic compounds as potential high-temperature structural materials, there have been many investigations carried out on various compounds, mostly structures that are ordered forms of simple fcc, bcc, and hcp metals [1]. If new intermetallic-based alloys are to be selected on the basis of low density and high-melting temperature, then three groups of materials emerge as promising candidates: aluminides [2], topologically close-packed compounds [3], and silicide-based compounds [4]. In the group of topologically close-packed compounds their structures generally are complex, in that the unit cell contains many atoms, even though the crystal structures may have high symmetry [5]. A number of these compounds have quite high-melting temperatures, low densities, and high-oxidation resistance, the properties necessary for high-temperature structural applications. However, all three groups of materials are often brittle
∗
Corresponding author. E-mail address: [email protected] (C. Zhang).
0925-8388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2006.11.039
at low temperatures up to temperatures of hundreds of degrees Celsius. This low-temperature brittleness adversely affects the fabrication and use of these materials. Consequently, it is highly desirable to find ways to improve the low-temperature ductility without compromising much on the attractive high-temperature properties. Topologically close-packed Laves phase are the most abundant among intermetallic compounds, with over 360 binary Laves phases reported, in which a promising approach to enhance ductility at low-temperatures is the addition of a third metal to form a ternary alloy. However, since there are many Laves-phase materials, the possibilities of forming ternary alloys for this purpose are countless. A good understanding of the basic properties of binary Laves-phase materials will help materials scientists focus on a limited number of systems instead of manufacturing and testing every possible candidate. The experiments have been proved that among the three Laves-phase structures, C15 Laves phases are expected to show better deformability than the other two Laves phases because of their fcc-based structure. Nakamichi et al. [6] confirmed that the stoichiometric composition of HfFe2 shows mainly the cubic MgCu2 -type structure containing a small amount of hexagonal MgZn2 -type
54
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
Fig. 1. The energy bands for HfM2 compound.
structure. They also observed the temperature dependence of the magnetization. Belosevic-Cavor [8] and Yamada and Shimizu [7] investigated the relation of electronic structure and magnetic properties of HfFe2 compound. The pseudobinary system (Hf, Zr)V2 have been investigated as superconducting materials [9,10]. Because of the high-melting point, high strength at elevated temperatures and reasonably good oxidation resistance, HfV2 have been studied recently for high-temperature structural applications [11–17]. In order to further understand the physical and chemical properties of these compound materials, an investigation of the electronic structure and bonding characteristic of Laves-phase compounds has both fundamental scientific and technological significance. Therefore Laves-phase compounds HfFe2 and HfV2 have been chosen as representative examples. 2. Details of calculation HfM2 (M = Fe and V) compound crystallizes in the cubic Fd3m structure (space group no. 227) and therefore it belongs to the large class of Laves-phase compounds. The unit cell consists of two formula units (i.e. six atoms). The calculations of the electronic structure and total energy of HfFe2 and HfV2 compounds was done using the method of APW + lo, which was implemented in the software package WIEN2K [18]. Core states are treated fully relativistically while semi-core and valence states are treated within a scalar relativistic approximation in the calculation. The APW + lo method converged practically the identical results as the linearized augmented plane wave (LAPW) method, but using smaller basis sets in the APW + lo method leads to significantly reduce the computation time by up to an order of magnitude [19]. Potential and charge densities inside the atomic spheres are expanded in lattice harmonics up to L = 8 and using GGA to treat exchange and correlation effects within the density functional theory. Muffin-tin (Mt) radius (Rmt ) of 2.2 a.u. was chosen for Hf, and 2.3 and 2.5 a.u. for Fe and V atoms, respectively, and a plane wave cut-off (Rmt Kmax ) of 10.0 was used. The calculation was performed with a 84-k point mesh in
the irreducible wedge of the Brillouin zone. When the energy difference was less than 0.1 mRy, convergence was assumed. The Brillouin zone integration was carried out according to a modified tetrahedron method [20]. 3. Results and discussion In this section, the ground-state properties are obtained by a minimization of the total energy with respect to the volume with GGA and LDA approach in the Laves-phase compounds HfFe2 and HfV2 . By analyzing more carefully these results, we observe that the lattice constant with GGA approach is evaluated at 2.16% and 5.17%, respectively, smaller than the available experimental data, in better agreement with the experiments (than the LDA approach). So, the experimental lattice constant is used in our calculations. Figs. 1–3 show the energy bands, the calculated total and partial density of states (DOS) and charge densities for HfFe2 and HfV2 , respectively. The Fermi level of HfFe2 crosses the density of states curve quite close to a small local maximum, whereas that of HfV2 crosses at a relatively high-local minimum. The density of states at the Fermi energy, n(EF ), is found to be 194 and 216 states/eV for HfFe2 and HfV2 , respectively. Ormeci et al. obtained the density of states to be 197.9 states/eV for HfV2 with LMTO method, which is similar with us. In both cases, the density of states near the Fermi level is dominated by the d states. In the case of HfV2 the contribution of the V-d states is 120 states/eV compared to 25 states/eV for Hf-d per atom. For HfFe2 on the other hand, Fe-d and Hf-d contributions are 158 and 12 states/eV, respectively. We also notice one difference between HfV2 and HfFe2 regarding the contributions of the B-p states to n(EF ), where the V-p contribution to n(EF ) is very significant compared with the Fe-p states. It also can be seen from energy band structures in both cases that the peaks arise from the almost dispersionless bands near the Fermi level. Hence, a precise location of EF is very important for the accurate calculation of those physical properties (such as magnetism and
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
55
Fig. 2. The calculated density of states for HfFe2 .
Fig. 3. The calculated density of states for HfV2 .
superconductivity), which depends critically on the value of the DOS at EF . On the other hand, the dominant contribution to the DOS at lower energies comes from bonding Hf–M s-like band beginning at −7.92 and −6.63 eV for HfFe2 and HfV2 , respectively. The bandwidths, as measured by the 1 − X1 energy difference, are nearly the same for both compounds (3 eV). It is seen that in the region −3 to 2 eV near EF the M-d states are dominant, with important contributions from the Hf-d states and others as well. The Fe-d states appear to be confined to this energy range, while the V-d states are seen to have significant additional contributions in the range of 7 eV near EF . The most important feature of these DOS for both compounds is the overlapping of d states
from Hf and M in the construction of electronic energy band, which implies hybridization between Hf-d and M-d states on formation of the HfM2 compound. Moreover, the hybridization between the Hf–V states is stronger than that between Hf–Fe states in the region near EF , where the Fermi levels fall, while the corresponding states in HfFe2 compound are more diffuse, which the Hf-d states show peaks near −2 eV. Hf–Fe hybridizations result in a bigger shift of empty states to the Fermi level for the Hf-d states upon formation of HfFe2 compared to HfV2 . As seen from Figs. 2 and 3, there are high densities of unoccupied states spreading far above the Fermi level for d states for HfFe2 , and the unoccupied states are fewer and nearer the Fermi level for the Fe d states compared with those for the V-d states.
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C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
˚ −3 with 3 Einstein A ˚ −3 intervals and Fig. 4. Contour plot of charge density in the (1 1 0) plane for HfFe2 . The contour lines are drawn (a) from 1.0 to 100 Einstein A ˚ −3 intervals. ˚ −3 with 0.02 Einstein A (b) from 0 to 1 Einstein A
The Fe-d states are also more spatially confined than those of V. Table 1 presents DOS values at EF together with the charges Q in the MT spheres. This can be seen by noting from Table 1 that Fe MT charges are QV d = 2.8529 and Qd = 5.9816, respectively, V Fe realizing that Qd exceed Qd by more than the three extra electrons in Fe compared to V, which is consistent with DOS above. Belosevic-Cavor [8] investigated the spin-polarized electronic structure of HfFe2 compound with LMTO method. The results showed that the spin-polarized total charges are nearly equal to total MT charges we calculated in LAPW method. This is probably because the Fe MT radii are smaller than V due to the smaller lattice constants of the former. Ohba et al. [21] obtained the structure factors of MgZn2 and MyCu2 by the single crystal X-ray diffraction method. They showed that the charge transfers occurred between the constituent atoms by the population
analysis and localized electrons were seen in the center of the tetrahedral formed by the small atoms in the difference Fourier maps. A similar phenomena is observed from Table 1, in which in both cases some electrons are transferred from Hf atom to Fe or V atoms when one account for the different MT radii. In order to exploring the bonding characteristics in the HfM2 compound, the charge density maps of the (1 1 0) plane for both cases are shown in Figs. 4 and 5. As can be seen from Figs. 4 and 5, the (1 1 0) plane is bisecting the B tetrahedral, containing the [1 1 1] close-packing direction. Figs. 4 and 5(a) shows the higher density region of the plane, which corresponds to the core electron distribution of Hf and M atoms. Fe or V atoms have almost spherical distribution. In comparison, however, Cu atoms are slightly deformed in Laves-phase MgCu2 [22]. Figs. 4 and 5(b) shows the lower density region of the
˚ −3 with 3 Einstein A ˚ −3 intervals and Fig. 5. Contour plot of charge density in the (1 1 0) plane for HfV2 . The contour lines are drawn (a) from 1.0 to 100 Einstein A ˚ −3 with 0.02 Einstein A ˚ −3 intervals. (b) from 0 to 1 Einstein A
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58 Table 1 The total DOS n(EF ) (states/eV) and MT charges (electrons/atom) for HfM2 compounds
57
Material
Atom
n(EF )
Qs
Qp
Qd
Qf
HfFe2
Hf Fe
194
2.1774 0.3601
5.9019 6.3021
1.0997 5.9816
13.9506 0.0160
nificant in HfV2 than that in HfFe2 . In face, the d electrons from the bonding did not contribute to magnetic moment at all, and the magnetic properties of C15 compounds come mainly from the other d electrons. The strength of covalent bonding for HfV2 is stronger than that of HfFe2 , which are helpful to increasing low-temperature ductility and Young’s modulus of compound.
HfV2
Hf V
216
2.1847 2.2497
5.8875 6.1166
1.1521 2.8529
13.9440 0.0117
4. Conclusions
same plane. The contour lines of the higher density regions are omitted. Figs. 4 and 5(b) shows that Hf atoms are almost spherical, but the contour line of low-charge density around Hf ˚ −3 for Hf–Fe and atoms, for example 0.39 and 0.43 Einstein/A Hf–V compounds, moves obviously towards Fe and V, respectively. In contrast, M atoms are anisotropic. Our calculation results, which are very similar with the contour plot obtained by Ormeci et al. [16], clearly indicate the directional d bonding in HfV2 , much stronger than that in Fe compound. The reason for this behavior is probably the transfer of electrons form the Hf to the Fe (V) atom, to fill up the incompletely filled inner shell. On the other hand, the overlap of electron densities between M–M atoms implies a covalent bonding between M–M atoms due to hybridization of the d orbitals. The height of the charge density at the bond midpoint between Fe–Fe was found to ˚ −3 , smaller than 0.64 Einstein/A ˚ −3 between be 0.57 Einstein/A V–V, whereas the height of the charge density at the bond mid˚ −3 . It point between Hf–M is almost the same 0.38 Einstein/A could be assumed that the bonding between Hf–M is much weaker than that between M–M in the HfFe2 , while the strength of bonding between V–V is rather stronger that corresponding that between Fe–Fe. This could be explained from the point of view of the C15 Laves-phase structure. The nearest neighboring sites of Fe are occupied by 6 Fe atoms in the C15 Lavesphase HfFe2 . The atomic distance between the nearest Fe–Fe atoms is 4.688 a.u., which is 2.79% shorter than that in pure Fe (4.802 a.u.) with an hcp structure. Compared with the atomic distance between the next nearest Hf–Fe atoms (5.497 a.u.), the Fe–Fe covalent bonding is thus stronger than that between Hf–Fe atoms, which is a mainly metallic bonding in HfFe2 . The hybridization of Fe–Fe therefore is stronger than Hf–Fe hybridization in HfFe2 compound and even stronger than Hf–Hf hybridization in Hf metal. In the interatomic region, the electrons distribute evenly like a metallic bond whose nature is well explained by the nearly free electron model. For HfV2 , on the other hand, the atomic distance between the nearest V–V atoms is 4.8643 a.u., which is 11.12% shorter than that in pure V metal. The atomic distance between the next nearest Hf–V atoms is 5.7040 a.u. Thus the hybridization of V–V in HfV2 is stronger than Fe–Fe hybridization in HfFe2 . A further analysis of the decomposed d DOS showed the bonding between the Hf–Fe atoms is mainly contributed by hybridization between the d2g orbital on the Hf sites and the orbital made up of dx2–y2 and dxy on the Fe sites. In comparison, the bonding between the H–V atoms is mainly contributed by hybridization between the d2g orbital on the Hf sites and the orbital dxy on the V sites. Therefore, it can be said that the hybridization of Hf-d and V-d states is more sig-
The electronic structure and bonding character of C15 type Laves-phase HfFe2 were investigated in the paper. Ab initio calculations have been performed to calculate the density of states (DOS), charge density distribution and band structure of the intermetallic compound HfFe2 based on the method of augmented plane waves plus local orbitals (APW + lo). The results revealed the overlap of charge densities between the neighboring Fe–Fe (V–V) atoms. This showed a rather strong covalent bonding between Fe–Fe (V–V) atoms and a metallic bonding between Hf–Fe (Hf–V) atoms. Furthermore, the Fe-d states are also more spatially confined than those of V-d, the hybridization of the valence electrons between Hf–V are stronger than that between Hf–Fe, leading to a stronger covalent bonding in HfV2 compound. On the other hand, there is virtually no covalency between Hf–Fe (or V) atoms, although HfFe2 (HfV2 ) are called intermetallic compounds. In the interatomic region, the electrons are distributed evenly like a metallic bond. In fact, the d electrons from the bonding did not contributed to magnetic moment, and the magnetic properties come mainly from the other d electrons. These results are helpful to understanding the physical and chemical properties for both compounds. References [1] J.K. Tien, T. Caulfield, Superalloys, Supercomposites and Superceramics, Academic Press, New York, 1989. [2] J. Horton, I. Baker, S. Hanada, R.D. Noebe, MRS Symposia Proceedings No. 364, Materials Research Society, Pittsburgh, 1995. [3] D.P. Pope, F. Chu, Structural Intermetallics, TMS Publications, Warrendale, PA, 1993. [4] T.E. Mitchell, R.G. Castro, J.J. Petrovic, S.A. Maloy, O. Unal, M.M. Chadwick, Mater. Sci. Eng. A 155 (1992) 241. [5] J.H. Westbrook, Intermetallic Compounds, John Wiley and Sons Inc., New York, 1967. [6] T. Nakamichi, K. Kai, Y. Aoki, K. Ikeda, M. Yamamoto, J. Phys. Soc. Jpn. 29 (1970) 794. [7] H. Yamada, M. Shimizu, J. Phys. F: Met. Phys. 16 (1986) 1039. [8] J. Belosevic-Cavor, V. Koteski, N. Novakovic, G. Concas, F. Congiu, G. Spano, Eur. Phys. J. B: Condens. Matter Phys. 50 (2006) 425. [9] K. Inoue, K. Tachikawa, IEEE Trans. Magn. 15 (1979) 635. [10] Z. Wu, N.L. Saini, S. Agrestini, D.D. Castro, A. Bianconi, A. Marcelli, M. Battisti, D. Gozzi, G. Balducci, J. Phys.: Condens. Matter 12 (2000) 6971. [11] F. Chu, D.J. Thoma, P.G. Kotula, S. Gerstl, T.E. Mitchell, I.M. Anderson, J. Bentley, Philos. Mag. A 77 (1998) 941. [12] F. Chu, D.J. Thoma, T.E. Mitchell, M. Sob, C.L. Lin, Philos. Mag. B 77 (1998) 121. [13] C.T. Liu, J.H. Zhu, M.P. Brady, C.G. McKamey, L.M. Pike, Intermetallics 8 (2000) 1119. [14] F. Chu, D.P. Pope, Mater. Sci. Eng. A 170 (1993) 39. [15] J.D. Livingston, E.L. Hall, J. Mater. Res. 5 (1990) 59. [16] A. Ormeci, F. Chu, M.W. John, T.E. Mitchell, R.C. Albers, S.P. Chen, Phys. Rev. B: Condens. Matter 54 (1996) 12753.
58
C. Zhang et al. / Journal of Alloys and Compounds 448 (2008) 53–58
[17] T. Jarlborg, A.J. Freeman, Phys. Rev. B 22 (1980) 2332. [18] K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. [19] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
[20] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [21] T. Ohba, Y. Kitano, Y. Komuar, Acta Crystallogr. C 40 (1984) 1. [22] Y. Kubota, M. Takata, M. Sakata, T. Ohba, K. Kifune, T. Tadaki, J. Phys.: Condens. Matter 12 (2000) 1253.