- Email: [email protected]
Structure and chemical bond characteristics of LaB6
ARTICLE IN PRESS Physica B 404 (2009) 4086–4089
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Structure and chemical bond characteristics of LaB6 Lina Bai a, Ning Ma b, Fengli Liu c, a b c
School of Physics and Electronic Engineering, Harbin Normal University, Harbin 150025, China Institute of Computer Science and Information Engineering, Harbin Normal University, Harbin 150025, China College of Physical Science and Technology, Heilongjiang University, Harbin 150080, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 April 2009 Received in revised form 22 June 2009 Accepted 21 July 2009
The structure and chemical bond characteristics of LaB6 have been achieved by means of the density functional theory using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method, which are implemented within the EXCITING code. The results show our optimized lattice ˚ parameter z (0.1981) and bulk modulus B (170.4 GPa) are in good agreement with constant a (4.158 A), the corresponding experimental data. Electron localization function (ELF) shows the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest neighbor B atoms (B2 and B3) is a little stronger than that between the nearer neighbor B atoms (B1 and B4). Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved.
PACS: 71.15.Ap 31.15.Ar 31.15.Ew 61.10.Nz Keywords: LaB6 Structure and chemical bond FPLAPW method
1. Introduction The divalent rare earth hexaborides (RB6) have been investigated both experimentally and theoretically. The main interest in these compounds was based on the rare earth 4f electrons, which play a central role in determining the electronic properties. Among the rare earth hexaborides, LaB6 is metallic at room temperature [1] but it will be transformed into superconductor at TC ¼ 0.45 K [2]. Furthermore, it has high electrical conductivity which has recently attracted attention, and it is a thermionic electron emitter with a low work function, high brightness and long life compared with conventional tungsten filaments [3]. At the same time, several powder diffraction reference materials have been developed by the National Institute of Standards and Technology (NIST), including lanthanum hexaboride, LaB6, (NIST Standard Reference Material SRM-660a) [4]. In a word, all of these interesting properties are related with the structure and electronic properties, in particular, chemical bond characteristics of LaB6. Electronic structure of LaB6 has been studied by a few ab initio calculations. One of the earliest band structure calculations was performed by Hasegawa et al. [5], using the symmetrized nonrelativistic self-consistent augmented plane wave (APW) method.
Corresponding author. Tel./fax: +86 045186609817.
E-mail address: [email protected] (F. Liu).
And then electronic structure was estimated by Kubo et al. [6], using the three-dimensional Lock–Crisp–West (LCW) folded momentum densities (3D LCW FMD’s) within local-density approximation (LDA) method. Recently, electronic structure has been studied by Hossain et al. [7], using the Cambridge Serial Total Energy Package (CASTEP) software code. However, differences still exist and the electronic structure of this compound and chemical bond between La and B atom have still no uniform conclusion. Walch et al. have reported that the La–B chemical bond was covalent and was more significant than La–La interaction after they calculated band structure of LaB6 using a discrete variational method based on the Hartree–Fock–Slater model [8]. Nevertheless, Tanaka et al., making use of electrical resistivity measurements, have taken stock of a considerable charge transfer between La and B atom, attributing the ‘‘metallic’’ conduction to polar optical phonon scattering within the ionic structure [9]. Based on the fundamentality as a standard reference material for X-ray powder diffraction and dispute of chemical bond about LaB6, it is necessary to further research the structure and chemical bond of LaB6. In this article, the density functional theory based on the stateof-the-art full-potential linearized augmented plane wave (FPLAPW) method [10], which is implemented within the EXCITING code [11], was used to complete all the calculation for LaB6. This paper is arranged into four sections. In Section 2, crystal structure and computational methodology will be discussed in
0921-4526/$ - see front matter Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.189
ARTICLE IN PRESS L. Bai et al. / Physica B 404 (2009) 4086–4089
detail. In the following section (Section 3), the lattice constant and internal coordinates of B atom will be optimized by using two exchange-correlation energy functions and equations of state in Section 3.1 and then based on the steady structure of LaB6, electronic structure and chemical bond will be discussed in Section 3.2. Finally, conclusions about the structure and chemical bond properties of LaB6 will be presented in Section 4.
2. Crystal structure and computational methodology The crystal structure of LaB6 is simple cubic (CsCl structure type, P/m-3 m) as shown in Fig. 1 [12]. La atom occupies at corners of the unit cell while B6-octahedra sites in body-centered positions. La atom occupies the 1a (0, 0, 0) Wyckoff site and B atoms occupy the 6f (0.5, 0.5, z) Wyckoff sites [13], where z is an internal parameter. It determines the ratio between inter- and intra-octahedron B–B distances. On purpose to keep the numerical analysis as accurate as possible, all calculations are performed using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method [10], which is implemented within the EXCITING code [11]. The single electron potential is calculated exactly without any shape approximation, the lattice space is divided into muffin-tin (MT) regions, where atomic orbitals are used as a basis and interstitial region, where plane waves are used as basis. Likewise, the magnetization and current densities and their conjugate fields are all treated as unconstrained vector fields throughout space. The deep-lying core states (6 Ry below the Fermi level) are treated as Dirac spinors and valence states as Pauli spinors. To obtain the Pauli spinors states, the Hamiltonian containing only the scalar fields is diagonalized in the LAPW basis: this is the firstvariational step. The scalar states thus obtained are then used as a basis to set up a second-variational Hamiltonian with spinor degrees of freedom, which consists of the first-variational eigenvalues along the diagonal, the matrix elements are obtained from the external and effective vector fields. This is more efficient than simple using spinor LAPW functions, but care must be taken to ensure there are a sufficient number of first-variational eigenstates for convergence of the second-variational problem.
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Spin polarization is also included at these calculations. The main parameters are as follows. The muffin tin sphere size of La and B atoms are 2.2 and 1.45 a.u., respectively. Rmin max (|G+k|) ¼ 7.0, which sets the maximum length for the G+k vectors and the Rmin is the smallest muffin-tin radius. The other parameters are as follows: lmaxinr ¼ 2, lmaxmat ¼ 5 and lmaxvr ¼ 7, which represents angular momentum cut-off for the muffin-tin density and potential on the inner part of the muffin-tin, for the outer-most loop in the Hamiltonian and overlap matrix setup and for the muffin-tin density and potential, respectively. The optimizing k-point mesh sizes is 5 5 5 in the irreducible Brillouin zone (IBZ). The relaxation of internal coordinates are obtained by minimizing the Hellman–Feynman forces and the force components on each atom are relaxed to 0.001 eV/A. The energy accuracy is thus less than 0.0001 eV.
3. Results and discussion 3.1. Optimized structure The optimized internal coordinate of B atom and equilibrium lattice constant a, bulk modulus B and total energy Et are listed in Table 1. The exchange-correlation energy function is local spin density approximation (LSDA) [14] and general gradient approximation (GGA) [15]. The equation of state (EOS) is Universal EOS [16] and Birch–Murnaghan (B–M) third-order EOS [17]. The fitting accuracy of total energy Et, lattice constant a and bulk modulus B are 0.005 eV, 0.001 A˚ and 0.2 GPa, respectively. Additionally, the previous reported internal coordinate of B atom [18], lattice constant a [13,19] and bulk modulus B [12,20] are also presented in Table 1. It is clearly presented in Table 1 that the fitted lattice constant a is a little bigger by GGA than by LSDA method. Similarly, the parameter z (internal coordinate of B atom) is a few longer, but the bulk modulus B has no same trend. Based on the LSDA, the fitted lattice constant a is a little smaller by Universal EOS than by B–M EOS while the fitted bulk modulus B is on the contrary. Based on the GGA, the comparative result is the same. These may indicate that there is no regular conclusion. However, it is obvious that based on the GGA method, the lattice constant a and bulk modulus B are in better agreement with the corresponding experimental data by fitting the B–M EOS than others, furthermore, the total energy Et also is the lowest one among the four energies. Anyhow, for the steady structure the lattice constant is Table 1 ˚ bulk modulus (B; GPa) The internal coordinate of B atom, lattice constant (a; A), and total energy (Et; eV) of LaB6.
Fig. 1. Crystal structure of LaB6.
Exchange-correlation function
Universal EOS [16]
B–M EOS [17]
LSDA [14] B atom a B Et
(0.5, 0.5, 0.1961) 4.141 175.9 235194.5689
(0.5, 0.5, 0.1964) 4.144 173.8 235194.8537
GGA [15] B atom a B Et B atom (calc.) [18] a (exp.) [13] a (calc.) [19] B (exp.) [20] B (calc.) [12]
(0.5, 0.5, 0.1984) 4.161 178.2 235194.9581 (0.5, 0.5, 0.1970) 4.156 4.154 163.0 142.0
(0.5, 0.5, 0.1981) 4.158 170.4 235195.9176
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L. Bai et al. / Physica B 404 (2009) 4086–4089
4.158 A˚ and the parameter z is 0.1981. The following electronic structure and chemical bond calculation are based on this steady structure and GGA method.
The chemical bond between La and B atom in LaB6 has been studied using Mulliken population analyses [7] and wave function method [8]. In present work, the chemical bond characteristics was explored by the electron localization function (ELF) [23,24].
3.2. Electronic structure and chemical bond The calculated densities of states (DOS) of LaB6 and projected densities of states (PDOS) of La f, d, p states and B p, s states are shown in Figs. 2–7, respectively. The DOS of LaB6 is in accordance with the result reported by Singh et al. [21] but this is a little different from the figure calculated by Walch et al. [8]. This may be cause that the different methods are employed in calculation of LaB6. It is also found in Fig. 2 that the DOS curve passes through Fermi energy (EF) level, which indicates that LaB6 is conductor. This has been testified by experimental data [22]. Compared with Figs. 3–7, it is clearly found that the La-f state mainly occupies the conduction band and the La-d state occupies the valence and conduction band but there is very small DOS. The majority of La-p state occupies the valence band from 17.5 to 14.5 eV while the B-s and -p state synchronously occupy the valence and conduction band, which indicates there may be the covalence bond between B atoms and ionic bond between La atoms. This will be discussed in the following paragraph. Fig. 4. The partial densities of states of La-d.
Fig. 2. The total densities of states of LaB6.
Fig. 3. The partial densities of states of La-f.
Fig. 5. The partial densities of states of La-p.
Fig. 6. The partial densities of states of B-p.
ARTICLE IN PRESS L. Bai et al. / Physica B 404 (2009) 4086–4089
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Fig. 9. Electron localization function among B atoms in LaB6. Fig. 7. The partial densities of states of B-s.
4. Conclusions The structure and chemical bond calculations of LaB6 have been achieved by means of the density functional theory using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method, which is implemented within the EXCITING code. Important conclusion was as follows: ˚ (1) The optimized lattice constant a (4.158 A), parameter z (0.1981) and fitted bulk modulus B (170.4 GPa) are in well agreement with corresponding experimental data. (2) The ELF figures show the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest-neighbor B atoms (B2 and B3) is a little stronger than that between the nearer-neighbor B atoms (B1 and B4). References
Fig. 8. Electron localization function among La and B atoms in LaB6.
ELF varies between 0 and 1, where 1 is corresponding to perfect localization and 1/2 is corresponding to electron gas like behavior. Thus ELF quantifies the chemical bond between La and B atom; it is an idea for studying the nature of chemical bond. Hence the ELF of La (0, 0, 0), La (0, 1, 0) and B (0.5, 0.5, 0.1981) is shown in Fig. 8 while that of B1 (0.5, 0.8019, 0.5), B2 (0.5, 0.1981, 0.5), B3 (0.5, 0.5, 0.8019) and B4 (0.5, 0.5, 1.1981) is also shown in Fig. 9. It is clear seen from Fig. 8 that there is almost no localization charge between the La atoms while a few localization of charge between the La and B atoms, which implies that La–La bond mainly is ionic bond and the ionic bond of La–La bonding is stronger than that of La–B bonding. It is also found in Fig. 9 that the covalent bond has been formed between the B2 (0.5, 0.1981, 0.5) and B3 (0.5, 0.5, 0.8019) atom because of a strong localization of charge between them and the bonding character among B1 (0.5, 0.8019, 0.5) and B2 (0.5, 0.1981, 0.5), B3 (0.5, 0.5, 0.8019) and B4 (0.5, 0.5, 1.1981) is between the ionic and covalent bond character due to localization of a few charge between them. The above result is in good agreement with the conclusion reported by Hossain et al. [7]. It can be concluded that the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest-neighbor B atoms (B2 and B3) is a little stronger than that between the nearer-neighbor B atoms (B1 and B4).
[1] E. Kauer, Phys. Lett. 7 (1963) 171. [2] J.M. Vanderberg, B.T. Matthias, E. Corenzwit, H. Borz, Mater. Res. Bull. 10 (1975) 889. [3] S. Otani, Y. Ishizawa, Progr. Crystal Growth Charact. 23 (1991) 153. [4] National Institute of Standards and Technology Certificate, Standard Reference Materials 660a, lanthanum hexaboride powder, line position and line shape standard for powder diffraction /http://www.nist.gov/S. [5] A. Hasegawa, A. Yanase, J. Phys. F Met. Phys. 7 (1977) 1245. [6] Y. Kubo, S. Asano, Phys. Rev. B 39 (1989) 8822. [7] F.M. Hossain, P. Daniel Riley, G.E. Murch, Phys. Rev. B 72 (2005) 235101. [8] P.F. Walch, D.E. Ellis, F.M. Mueller, Phys. Rev. B 15 (1977) 1859. [9] T. Tanaka, T. Akahane, E. Bannai, S. Kawai, N. Tsuda, Y. Ishizawa, J. Phys. C 9 (1976) 1235. [10] D.J. Singh, in: Planewaves, Pseudopotentials and the LAPW Method, Kluwer, Dordrecht, 1994. [11] J.K. Dewhurst, S. Sharma, C. Ambrosch-Draxl, /http://exciting.sourceforge. net/S. [12] P. Teredesai, D.V.S. Muthu, N. Chandrabhas, S. Meenakshi, V. Vijayakumar, P. Modak, R.S. Rao, S.K. Sikka, A.K. Sood, Solid State Commun. 129 (2004) 791. [13] R.W.G. Wyckoff, Crystal Structures, vol. 2, University of Arizona, Tucson, AZ, 1965. [14] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [15] Y. Zhang, W. Yang, Phys. Rev. Lett. 80 (1998) 890. [16] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, J. Phys. Condens. Matter 1 (1989) 1941. [17] F. Birch, Phys. Rev. 71 (1947) 809. [18] Y.Q. Cai, Chinese J. Low Temp. Phys. 5 (2005) 515. [19] Z. Yahia, S. Turrell, G. Turrell, J.P. Mercurio, J. Mol. Struct. 224 (1990) 303. [20] T. Tanaka, J. Yoshimoto, M. Ishii, E. Bannai, S. Kawai, Solid State Commun. 22 (1977) 203. [21] N. Singh, S.M. Saini, T. Nautiyal, S. Auluck, J. Phys. Condens. Matter 19 (2007) 346226. ˇ Moloka cˇ, A. Belling, J. Bischof, E. Konovalova, [22] K. Flachbart, M. Reiffers, S. Y. Paderno, Physica B 263–264 (1999) 749. [23] A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. [24] T. Burnus, M.A.L. Marques, E.K.U. Gross, Phys. Rev. A 71 (2005) 010501.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Structure and chemical bond characteristics of LaB6 Lina Bai a, Ning Ma b, Fengli Liu c, a b c
School of Physics and Electronic Engineering, Harbin Normal University, Harbin 150025, China Institute of Computer Science and Information Engineering, Harbin Normal University, Harbin 150025, China College of Physical Science and Technology, Heilongjiang University, Harbin 150080, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 April 2009 Received in revised form 22 June 2009 Accepted 21 July 2009
The structure and chemical bond characteristics of LaB6 have been achieved by means of the density functional theory using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method, which are implemented within the EXCITING code. The results show our optimized lattice ˚ parameter z (0.1981) and bulk modulus B (170.4 GPa) are in good agreement with constant a (4.158 A), the corresponding experimental data. Electron localization function (ELF) shows the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest neighbor B atoms (B2 and B3) is a little stronger than that between the nearer neighbor B atoms (B1 and B4). Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved.
PACS: 71.15.Ap 31.15.Ar 31.15.Ew 61.10.Nz Keywords: LaB6 Structure and chemical bond FPLAPW method
1. Introduction The divalent rare earth hexaborides (RB6) have been investigated both experimentally and theoretically. The main interest in these compounds was based on the rare earth 4f electrons, which play a central role in determining the electronic properties. Among the rare earth hexaborides, LaB6 is metallic at room temperature [1] but it will be transformed into superconductor at TC ¼ 0.45 K [2]. Furthermore, it has high electrical conductivity which has recently attracted attention, and it is a thermionic electron emitter with a low work function, high brightness and long life compared with conventional tungsten filaments [3]. At the same time, several powder diffraction reference materials have been developed by the National Institute of Standards and Technology (NIST), including lanthanum hexaboride, LaB6, (NIST Standard Reference Material SRM-660a) [4]. In a word, all of these interesting properties are related with the structure and electronic properties, in particular, chemical bond characteristics of LaB6. Electronic structure of LaB6 has been studied by a few ab initio calculations. One of the earliest band structure calculations was performed by Hasegawa et al. [5], using the symmetrized nonrelativistic self-consistent augmented plane wave (APW) method.
Corresponding author. Tel./fax: +86 045186609817.
E-mail address: [email protected] (F. Liu).
And then electronic structure was estimated by Kubo et al. [6], using the three-dimensional Lock–Crisp–West (LCW) folded momentum densities (3D LCW FMD’s) within local-density approximation (LDA) method. Recently, electronic structure has been studied by Hossain et al. [7], using the Cambridge Serial Total Energy Package (CASTEP) software code. However, differences still exist and the electronic structure of this compound and chemical bond between La and B atom have still no uniform conclusion. Walch et al. have reported that the La–B chemical bond was covalent and was more significant than La–La interaction after they calculated band structure of LaB6 using a discrete variational method based on the Hartree–Fock–Slater model [8]. Nevertheless, Tanaka et al., making use of electrical resistivity measurements, have taken stock of a considerable charge transfer between La and B atom, attributing the ‘‘metallic’’ conduction to polar optical phonon scattering within the ionic structure [9]. Based on the fundamentality as a standard reference material for X-ray powder diffraction and dispute of chemical bond about LaB6, it is necessary to further research the structure and chemical bond of LaB6. In this article, the density functional theory based on the stateof-the-art full-potential linearized augmented plane wave (FPLAPW) method [10], which is implemented within the EXCITING code [11], was used to complete all the calculation for LaB6. This paper is arranged into four sections. In Section 2, crystal structure and computational methodology will be discussed in
0921-4526/$ - see front matter Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.189
ARTICLE IN PRESS L. Bai et al. / Physica B 404 (2009) 4086–4089
detail. In the following section (Section 3), the lattice constant and internal coordinates of B atom will be optimized by using two exchange-correlation energy functions and equations of state in Section 3.1 and then based on the steady structure of LaB6, electronic structure and chemical bond will be discussed in Section 3.2. Finally, conclusions about the structure and chemical bond properties of LaB6 will be presented in Section 4.
2. Crystal structure and computational methodology The crystal structure of LaB6 is simple cubic (CsCl structure type, P/m-3 m) as shown in Fig. 1 [12]. La atom occupies at corners of the unit cell while B6-octahedra sites in body-centered positions. La atom occupies the 1a (0, 0, 0) Wyckoff site and B atoms occupy the 6f (0.5, 0.5, z) Wyckoff sites [13], where z is an internal parameter. It determines the ratio between inter- and intra-octahedron B–B distances. On purpose to keep the numerical analysis as accurate as possible, all calculations are performed using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method [10], which is implemented within the EXCITING code [11]. The single electron potential is calculated exactly without any shape approximation, the lattice space is divided into muffin-tin (MT) regions, where atomic orbitals are used as a basis and interstitial region, where plane waves are used as basis. Likewise, the magnetization and current densities and their conjugate fields are all treated as unconstrained vector fields throughout space. The deep-lying core states (6 Ry below the Fermi level) are treated as Dirac spinors and valence states as Pauli spinors. To obtain the Pauli spinors states, the Hamiltonian containing only the scalar fields is diagonalized in the LAPW basis: this is the firstvariational step. The scalar states thus obtained are then used as a basis to set up a second-variational Hamiltonian with spinor degrees of freedom, which consists of the first-variational eigenvalues along the diagonal, the matrix elements are obtained from the external and effective vector fields. This is more efficient than simple using spinor LAPW functions, but care must be taken to ensure there are a sufficient number of first-variational eigenstates for convergence of the second-variational problem.
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Spin polarization is also included at these calculations. The main parameters are as follows. The muffin tin sphere size of La and B atoms are 2.2 and 1.45 a.u., respectively. Rmin max (|G+k|) ¼ 7.0, which sets the maximum length for the G+k vectors and the Rmin is the smallest muffin-tin radius. The other parameters are as follows: lmaxinr ¼ 2, lmaxmat ¼ 5 and lmaxvr ¼ 7, which represents angular momentum cut-off for the muffin-tin density and potential on the inner part of the muffin-tin, for the outer-most loop in the Hamiltonian and overlap matrix setup and for the muffin-tin density and potential, respectively. The optimizing k-point mesh sizes is 5 5 5 in the irreducible Brillouin zone (IBZ). The relaxation of internal coordinates are obtained by minimizing the Hellman–Feynman forces and the force components on each atom are relaxed to 0.001 eV/A. The energy accuracy is thus less than 0.0001 eV.
3. Results and discussion 3.1. Optimized structure The optimized internal coordinate of B atom and equilibrium lattice constant a, bulk modulus B and total energy Et are listed in Table 1. The exchange-correlation energy function is local spin density approximation (LSDA) [14] and general gradient approximation (GGA) [15]. The equation of state (EOS) is Universal EOS [16] and Birch–Murnaghan (B–M) third-order EOS [17]. The fitting accuracy of total energy Et, lattice constant a and bulk modulus B are 0.005 eV, 0.001 A˚ and 0.2 GPa, respectively. Additionally, the previous reported internal coordinate of B atom [18], lattice constant a [13,19] and bulk modulus B [12,20] are also presented in Table 1. It is clearly presented in Table 1 that the fitted lattice constant a is a little bigger by GGA than by LSDA method. Similarly, the parameter z (internal coordinate of B atom) is a few longer, but the bulk modulus B has no same trend. Based on the LSDA, the fitted lattice constant a is a little smaller by Universal EOS than by B–M EOS while the fitted bulk modulus B is on the contrary. Based on the GGA, the comparative result is the same. These may indicate that there is no regular conclusion. However, it is obvious that based on the GGA method, the lattice constant a and bulk modulus B are in better agreement with the corresponding experimental data by fitting the B–M EOS than others, furthermore, the total energy Et also is the lowest one among the four energies. Anyhow, for the steady structure the lattice constant is Table 1 ˚ bulk modulus (B; GPa) The internal coordinate of B atom, lattice constant (a; A), and total energy (Et; eV) of LaB6.
Fig. 1. Crystal structure of LaB6.
Exchange-correlation function
Universal EOS [16]
B–M EOS [17]
LSDA [14] B atom a B Et
(0.5, 0.5, 0.1961) 4.141 175.9 235194.5689
(0.5, 0.5, 0.1964) 4.144 173.8 235194.8537
GGA [15] B atom a B Et B atom (calc.) [18] a (exp.) [13] a (calc.) [19] B (exp.) [20] B (calc.) [12]
(0.5, 0.5, 0.1984) 4.161 178.2 235194.9581 (0.5, 0.5, 0.1970) 4.156 4.154 163.0 142.0
(0.5, 0.5, 0.1981) 4.158 170.4 235195.9176
ARTICLE IN PRESS 4088
L. Bai et al. / Physica B 404 (2009) 4086–4089
4.158 A˚ and the parameter z is 0.1981. The following electronic structure and chemical bond calculation are based on this steady structure and GGA method.
The chemical bond between La and B atom in LaB6 has been studied using Mulliken population analyses [7] and wave function method [8]. In present work, the chemical bond characteristics was explored by the electron localization function (ELF) [23,24].
3.2. Electronic structure and chemical bond The calculated densities of states (DOS) of LaB6 and projected densities of states (PDOS) of La f, d, p states and B p, s states are shown in Figs. 2–7, respectively. The DOS of LaB6 is in accordance with the result reported by Singh et al. [21] but this is a little different from the figure calculated by Walch et al. [8]. This may be cause that the different methods are employed in calculation of LaB6. It is also found in Fig. 2 that the DOS curve passes through Fermi energy (EF) level, which indicates that LaB6 is conductor. This has been testified by experimental data [22]. Compared with Figs. 3–7, it is clearly found that the La-f state mainly occupies the conduction band and the La-d state occupies the valence and conduction band but there is very small DOS. The majority of La-p state occupies the valence band from 17.5 to 14.5 eV while the B-s and -p state synchronously occupy the valence and conduction band, which indicates there may be the covalence bond between B atoms and ionic bond between La atoms. This will be discussed in the following paragraph. Fig. 4. The partial densities of states of La-d.
Fig. 2. The total densities of states of LaB6.
Fig. 3. The partial densities of states of La-f.
Fig. 5. The partial densities of states of La-p.
Fig. 6. The partial densities of states of B-p.
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Fig. 9. Electron localization function among B atoms in LaB6. Fig. 7. The partial densities of states of B-s.
4. Conclusions The structure and chemical bond calculations of LaB6 have been achieved by means of the density functional theory using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method, which is implemented within the EXCITING code. Important conclusion was as follows: ˚ (1) The optimized lattice constant a (4.158 A), parameter z (0.1981) and fitted bulk modulus B (170.4 GPa) are in well agreement with corresponding experimental data. (2) The ELF figures show the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest-neighbor B atoms (B2 and B3) is a little stronger than that between the nearer-neighbor B atoms (B1 and B4). References
Fig. 8. Electron localization function among La and B atoms in LaB6.
ELF varies between 0 and 1, where 1 is corresponding to perfect localization and 1/2 is corresponding to electron gas like behavior. Thus ELF quantifies the chemical bond between La and B atom; it is an idea for studying the nature of chemical bond. Hence the ELF of La (0, 0, 0), La (0, 1, 0) and B (0.5, 0.5, 0.1981) is shown in Fig. 8 while that of B1 (0.5, 0.8019, 0.5), B2 (0.5, 0.1981, 0.5), B3 (0.5, 0.5, 0.8019) and B4 (0.5, 0.5, 1.1981) is also shown in Fig. 9. It is clear seen from Fig. 8 that there is almost no localization charge between the La atoms while a few localization of charge between the La and B atoms, which implies that La–La bond mainly is ionic bond and the ionic bond of La–La bonding is stronger than that of La–B bonding. It is also found in Fig. 9 that the covalent bond has been formed between the B2 (0.5, 0.1981, 0.5) and B3 (0.5, 0.5, 0.8019) atom because of a strong localization of charge between them and the bonding character among B1 (0.5, 0.8019, 0.5) and B2 (0.5, 0.1981, 0.5), B3 (0.5, 0.5, 0.8019) and B4 (0.5, 0.5, 1.1981) is between the ionic and covalent bond character due to localization of a few charge between them. The above result is in good agreement with the conclusion reported by Hossain et al. [7]. It can be concluded that the La–La bond mainly is ionic bond, La–B bond is between ionic and covalent bond while the covalent bond between the nearest-neighbor B atoms (B2 and B3) is a little stronger than that between the nearer-neighbor B atoms (B1 and B4).
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