J. Phys. Chem. C 2009, 113, 4997–5003
4997
First-Principles Investigations of the Physical Properties of Magnesium Nitridoboride P. Hermet,*,† S. Goumri-Said,† M. B. Kanoun,‡ and L. Henrard† Laboratoire de Physique du Solide and Laboratoire de Chimie The´orique Applique´e, Faculte´s UniVersitaires Notre-Dame de la Paix, 5000 Namur, Belgium ReceiVed: October 15, 2008; ReVised Manuscript ReceiVed: January 17, 2009
In this article, we investigate the electronic, dielectric, dynamic, and elastic properties of new magnesium nitridoboride (MgNB9) using a first-principles approach based on density functional theory. No available experimental or theoretical investigations of these physical properties have been previously reported in the literature. Our work shows that MgNB9 is a semiconducting positive uniaxial trigonal material with an indirect band gap (Z-L) of 1.76 eV and mixed ionic-covalent character. In addition, its electronic dielectric tensor is nearly isotropic, and the magnitude of its components is similar to those reported for ferroelectric materials. By contrast, its static dielectric tensor is strongly anisotropic in the plane orthogonal to its optical axis. This anisotropy is mainly governed by a highly polar low-frequency mode assigned to localized Mg motions. Furthermore, this material is mechanically stable, and its bulk and shear moduli are larger than those reported on III-V semiconductors. These results suggest that MgNB9 could be a promising potential material for applications in optoelectronics. I. Introduction Boron compounds and their derivatives are chemically and physically exciting materials.1 In particular, boron carbides and boron nitrides are widely used in science and industry because of their mechanically, thermally, or physically outstanding properties.2,3 Among these compounds, magnesium nitridoboride (MgNB9) is a new material involved in the MgB2 single-crystal growth process.4 MgNB9 is made of structural building blocks that are characteristic of various modifications of boron and, more generally, borides. It consists of B6 octahedra connecting two layers of B12 icosahedra. Within the layer, the icosahedra are joined through the N atoms, with each of them bonded to three B12 units. Although the discovery of superconductivity in MgB2 below 39 K has stimulated detailed studies on borides,5-8 there are, to the best of our knowledge, no available experimental or theoretical data reported in the literature on MgNB9 except with regard to its crystallographic structure.9 In this paper, we perform a comprehensive study of the structural, electronic, dielectric, dynamic, and elastic properties of MgNB9 using first-principles methods based on density functional theory (DFT). Our study therefore covers all of the linear couplings among the applied static homogeneous electric field, strain, and periodic atomic displacements. The main objective of this work is the investigation of the microscopic physical properties of this new boride material for its potential industrial applications. In addition, this work can provide benchmark theoretical results for the understanding of the important family of boron nitrides or information about the MgB2 synthesis. This paper is organized as follows. In the next section, we describe the theoretical framework of our calculations and the DFT-based codes used in the present study. In section III, we report the crystallographic and electronic structure of MgNB9 * Corresponding author: E-mail:
[email protected] † Laboratoire de Physique du Solide. ‡ Laboratoire de Chimie The´orique Applique´e.
for the understanding of its bonding properties. In particular, we show that MgNB9 is a semiconducting material with an indirect band gap of 1.76 eV and mixed ionic-covalent character. Section IV is devoted to the calculation of the Born effective charges and electronic dielectric tensors. In section V, we focus on the MgNB9 zone-centered phonon modes, and we investigate their contributions to its static dielectric tensor and its infrared reflectivity spectrum. We predict that the static dielectric tensor of this material is strongly anisotropic in the plane orthogonal to its optical axis. This anisotropy is mainly governed by a very polar low-frequency mode assigned to localized Mg motions. In section VI, we report the elastic constants of MgNB9, and we compare its bulk and shear moduli with MgB2 and III-V semiconductors. Finally, section VII concludes with the most important results of this work. II. Computational Details DFT calculations were performed according to the Hohenberg, Kohn, and Sham formalism10,11 as implemented in the ABINIT package,12 a plane-waves pseudopotential DFT code, which in addition to usual ground-state calculations allows linear-response computations of the phonon frequencies, Born effective charges, dielectric tensors, and elastic constants. Furthermore, we have conjointly used the CASTEP package,13 which is also a plane-waves pseudopotential DFT code, for the investigation of the bonding character of MgNB9 from the computation of the Mulliken bond population. This quantity is not presently implemented in ABINIT. These two DFT-based codes are therefore complementary and allow us to perform a complete investigation of MgNB9. The exchange-correlation energy functional was evaluated within the generalized gradient approximation (GGA) as proposed by Perdew, Burke, and Ernzerhof (PBE).14 The all-electron potentials were replaced in ABINIT by norm-conserving pseudopotentials generated according to the Troullier-Martins scheme,15 whereas Vanderbilt ultrasoft pseudopotentials16 were used in CASTEP. Boron (2s22p1), nitrogen (2s22p3), and magnesium (3s2) electrons were considered to be valence states in the construction of the
10.1021/jp8091286 CCC: $40.75 2009 American Chemical Society Published on Web 03/05/2009
4998 J. Phys. Chem. C, Vol. 113, No. 12, 2009
Hermet et al. TABLE 1: Calculated Structural Parameters of the MgNB9 Rhombohedral Unit Cell Provided by ABINIT at the Experimental and Optimized Lattice Parameters, Together with the Experimental Onesa
Figure 1. Structure of MgNB9 in the hexagonal unit cell highlighting the NB6 layer and MgB3 groups (a) and in the rhombohedral unit cell (b).
pseudopotentials. The self-consistent cycles were converged within a tolerance of 10-14 Ha on the potential residual. The electronic wave functions were expanded in plane waves up to kinetic energy cutoffs of 50 and 25 Ha in ABINIT and CASTEP, respectively. The Brillouin zone integration was sampled using a 6 × 6 × 6 special k-point grid according to the Monkhorst-Pack scheme,17 except for the calculations of electronic band structure and densities of states (DOS) where a denser 10 × 10 × 10 k-point grid was required to reach convergence. The dynamic matrix, Born effective charges, and dielectric tensors were computed within a variational approach to density functional perturbation theory.18 The total and local DOS were obtained using the tetrahedron method. Local DOS were estimated by projecting the total DOS inside a sphere with a radius of 2 Bohr centered on the atoms. Mulliken charges and bond overlap populations were calculated by projecting the plane waves Kohn-Sham eigenstates onto a localized linear combination of atomic orbitals.19 The charge-spilling parameter, which estimates the reliability of this projection, is 4.6 × 10-3 in our calculation, indicating that an acceptable 0.46% of the valence has been lost in the projection. III. Structural and Electronic Properties A. Structure and Relaxations. The structure of MgNB9 has been recently refined from single-crystal X-ray data.9 Rietveld refinements have assigned MgNB9 as a uniaxial crystal belonging to the R3jm (D53d) trigonal space group with lattice parameters of a ) 5.4960 Å and c ) 20.0873 Å. This structure of 66 atoms consists of two layers, NB6 and MgB3, alternating along the c axis of the crystal as it is shown in Figure 1a. However, to decrease the computational time, calculations were performed with a 22-atom rhombohedral unit cell with lattice parameters of arh ) 7.4096 Å and Rrh ) 43.539° (Figure 1b). In this rhombohedral unit cell, Mg and N atoms occupy the 2c Wyckoff positions, whereas B atoms occupy the 6h positions. All structural relaxations were performed by using the BroydenFletcher-Goldfarb-Shanno algorithm20 until the maximum residual forces on the atoms were less than 6 × 10-6 Ha/Bohr. First, the rhombohedral structure was relaxed at the experimental lattice parameters. The relaxed structural parameters are reported in Table 1 where they are compared to the experimental ones. The relaxed B-N interatomic distances (1.526 Å) are close to the experimental ones (1.524 Å). Although these distances are much longer than the ones reported for boron nitrides (1.34-1.45 Å), they may be as long as 1.50-1.54 Å in some
optimized volume
exp. volume
exp. ref 9
Mulliken bond population
arh (Å) Rrh (deg.) Ω0 (Å3)
7.3618 43.624 172.360
7.4096 43.539 175.156
7.4096 43.539 175.156
7.4096 43.539 175.156
Mg-N Mg-B1 Mg-B2i Mg-B3i Mg-B3ii N-B1 B1-B1iii B1-B2 B1-B2iii B2-B2iv B2-B3 B3-Bv B3-B3vi
2.092 2.843 2.751 2.517 2.799 1.520 1.795 1.794 1.788 1.766 1.693 1.738 1.757
2.099 2.851 2.783 2.538 2.811 1.526 1.802 1.802 1.797 1.774 1.705 1.744 1.767
2.082 2.840 2.793 2.542 2.811 1.524 1.806 1.804 1.801 1.780 1.697 1.749 1.773
0.37 -0.16 0.03 0.03 -0.01 0.85 0.58 0.48 0.56 0.54 1.01 0.52 0.53
a Mulliken bond population analyzes provided by CASTEP at experimental lattice parameters are also reported. Interatomic distances are given in angstroms. Symmetry codes used are (i) x, y, 1 + z; (ii) -y, 1 - x, -z; (iii) -z, 1 - y, -x; (iv) 1 + z, x, -1 + y; (V) z, x, y; and (vi) -x, -z, -y.
cases.21 The relaxed Mg-N interatomic distances (2.099 Å) are consistent with those reported in the literature. Indeed, the experimental distances in nitrides and boron nitrides are found to be in the 1.91-2.18 Å range, but the most reliable data give a 2.03-2.15 Å range for both nitrides (Mg3N2)22 and boron nitrides (Mg3BN3).23 Then, the rhombohedral structure was fully relaxed (both lattice parameters and atomic positions). The calculated equilibrium lattice parameters are also reported in Table 1. We observe that the calculation predicts a rhombohedral angle larger than the experimental value and its underestimate of the lattice parameter, arh, leads to a unit cell volume about 2% smaller than the experimental one. In the hexagonal structure, this implies a slightly underestimate of the lattice parameters with respect to the experimental ones. This unphysical compression of the structure may be related to the lack of van der Waals interactions in the exchange-correlation energy functional. The energy difference between the relaxed configurations at the experimental and theoretical lattice parameters is nevertheless small (17 meV). All of the interatomic distances remain very close by changing the lattice parameters from the experimental values to the theoretical equilibrium ones. Finally, because it is well known that DFT predictions of some physical properties, such as electronic band structure or phonon frequencies, usually are in better agreement with the experimental ones when the experimental volume is considered and to obtain a reliable analysis of the MgNB9 bonding properties conjointly using ABINIT and CASTEP without any problem related to volume effects, the lattice parameters were fixed to the experimental ones in this work. B. Electronic and Bonding Properties. The electronic structure of the valence bands, the energy band gap value (Eg), and the low-energy conduction bands determine the most important properties of a material for electronic devices applications. Figure 2 displays the electronic band structure of MgNB9 along the F-Γ-Z-L-Γ line, together with its total DOS in the -18 to 17 eV range. These calculations have been performed using ABINIT at the MgNB9 experimental lattice
Physical Properties of Magnesium Nitridoboride
Figure 2. Electronic band structure along the F-Γ-Z-L-Γ line and total DOS of MgNB9 in the -18 to 17 eV range. The horizontal dashed line indicates the position of the Fermi level. High-symmetry points of the R3jm Brillouin zone are F(0, 1/2, 1/2), Γ(0, 0, 0), Z(1/2, 1/2, 1/2), and L(1/2, 0, 0).
parameters. The top of the valence band was fixed to 0 eV. The top of the valence band and the bottom of the conduction band are respectively located at the Z and L high-symmetry points of the R3jm trigonal structure. A plane wave PBEpseudopotential DFT approach therefore predicts MgNB9 to be a semiconductor material with an indirect band gap of 1.76 eV. However, because the electronic band gap is usually quite sensitive to the unit cell volume and the used exchange-correlation functional and considering the usual accuracy of band structure calculations, we do not completely wipe out the possibility that the top of the valence band is not located at Z but rather at L. Thus, we have checked that working at the PBE-optimized volume or using the Perdew-Wang (PW) local density approximation (LDA) does not affect the calculated position of the top of the valence band. In addition, Eg is not significantly sensitive when calculations are performed at the PBE-optimized volume (1.73 eV). Similarly, an all-electron approach based on full-potential linearized augmented plane wave method as implemented in Wien2k24 gives Eg ) 1.73 and 1.68 eV when calculations are performed at the experimental lattice parameters using PW (LDA) and PBE (GGA) exchange-correlation functionals, respectively. Nevertheless, because it is well known that DFT-based methods usually underestimate the band gap energy, the experimental band gap should be larger than the predicted values of around 1.74 eV. The analysis of the local DOS shows that the top of the valence band is mainly dominated by the 2p orbitals of the B and N atoms whereas the bottom of the conduction band is dominated by the B 2p orbitals weakly hybridized with the Mg 3s orbitals. The Mulliken bond population is useful for evaluating the bonding character in a material. Indeed, a high value of the bond population indicates a covalent bond, and a low value indicates an ionic nature. Positive and negative values indicate bonding and antibonding states, respectively. The Mulliken bond population reported in Table 1 shows that the Mg atoms have a tendency to decrease the bond population and even to make them negative when they are associated with B or N atoms. By contrast, the Mulliken bond population becomes more covalent when Mg atoms are substituted with B or N atoms. Thus, these results suggest that the chemical bonding in MgNB9 has mixed covalent-ionic character: the bonding between B and N atoms has non-negligible covalent character, and the bonding with Mg atoms is essentially ionic.
J. Phys. Chem. C, Vol. 113, No. 12, 2009 4999 IV. Dielectric Properties A. Electronic Dielectric Tensor. The electronic dielectric permittivity tensor, ε∞, is related to a second derivative of the electronic energy with respect to an electric field and has been computed using a linear response technique.18 No scissor correction has been included in the calculation. Because of the symmetry properties of the MgNB9 crystal structure, this tensor ∞ ) is diagonal with two independent components, parallel (εzz ∞ ∞ ∞ ∞ ε| ) and perpendicular (εxx ) εyy ) ε⊥ ) to the trigonal axis. The computed values of these two independent components, ε⊥∞ ) 6.96 and ε∞| ) 7.51, suggest that MgNB9 is a positive uniaxial trigonal crystal with a quite isotropic electronic response to a homogeneous electric field. Unfortunately, there are no available experimental or theoretical data reported in the literature on the dielectric properties of MgNB9 for comparison purposes. Nevertheless, the magnitudes of the ε∞ components are similar to those reported in the ferroelectric phases of BaTiO325,26 and in III-V semiconductors.27,28 It is well known that DFT usually overestimates the absolute values of ε∞ with respect to the experimental ones. This problem has been previously discussed in the literature29,30 and has been related to the lack of polarization dependence of local (LDA) and quasi-local (GGA) exchange-correlation functionals. In spite of this error in the absolute value, the evolutions of the optical dielectric permittivity tensor are in general qualitatively well described by LDA or GGA calculations. B. Born Effective Charges. Born effective charges (Z*), also called transverse charges, play a fundamental role in the dynamics of crystal lattices. They govern the amplitude of the long-range Coulomb interactions between nuclei and the splitting between longitudinal optic (LO) and transverse optic (TO) phonon modes. Z*Rβ(κ) is defined as the proportionality coefficient relating, in linear order, the macroscopic polarization (PR) per unit cell, created along the direction R, to the displacement (τ) along the direction β of the atoms belonging to the sublattice κ under the condition of zero electric field (E)31
* (κ) ) Ω ZRβ 0
∂PR ∂τβ(κ)
|
(1) E)0
where Ω0 is the unit cell volume. Thus, Z* is a dynamic quantity that is strongly influenced by dynamic changes in orbital hybridization induced by atomic displacements. The components of these tensors therefore reflect the effects of covalency or ionicity with respect to some reference ionic value. As a consequence, their amplitude is not directly related to that of the static charges and can take anomalous values. Z* related to the atoms constituting the MgNB9 asymmetric unit are reported in Table 2, together with their Mulliken charges provided by CASTEP. These tensors are given in Cartesian coordinates and using the same set of axes as for ε∞ previously calculated. Z* for the other atoms can be obtained from those given in Table 2 by applying the symmetry operations expected by the R3jm trigonal space group. Because of the finite number of plane waves and the k-point sampling, the acoustic sum rule, ∑κZ*Rβ(κ) ) 0, is fulfilled in our calculations within a slight deviation from charge neutrality, which is less than 0.008e. The form of Z* results directly from the site symmetry of the atoms. Thus, Z*(Mg) and Z*(N) are diagonal with two independent components (parallel and perpendicular to the trigonal axis), whereas Z*(B) has nine nonzero components. We observe that Z*(Mg) is nearly isotropic. Its diagonal elements are positive and close to the nominal charge of the Mg atoms
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* TABLE 2: Born Effective Charge Tensors (Zrβ ), Average of Their Eigenvalues (λ ) 1/3 Tr(Z*)), and Mulliken Charges (QM) of the Different Atoms Constituting the MgNB9 Asymmetric Unit (in |e|)a
atom
(
2.71 0.00 0.00 0.00 2.71 0.00 0.00 0.00 2.59
Mg
N
(
B3
)
-2.54 0.00 0.00 0.00 -2.54 0.00 0.00 0.00 -1.16
)
(
0.35 0.71 -0.03 0.71 1.16 -0.05 -0.10 -0.18 0.18
B1
B2
* ZRβ
( (
-0.36 0.19 0.09 0.19 -0.14 0.15 0.07 0.13 -0.45
-0.54 -0.04 0.19 -0.04 -0.58 0.33 0.23 0.41 -0.20
) )
)
λ
QM
2.67
1.89
-2.08
-0.74
0.56
0.15
Figure 3. Representation of the A2g(TO1) librational mode (a) and the Eu(TO1) mode (b). Green, pink, and blue atoms denote B, Mg, and N atoms, respectively. The optical axis (z axis) is along the Mg-N bond. Some bonds in the icosahedral B12 cage have been deleted in the structures for clarity.
TABLE 3: MgNB9 Zone-Centered Phonon Frequencies (in cm-1), Including LO and TO Characters of Infrared Modes, with Their Corresponding Irreducible Representations Eg
-0.32
-0.21
-0.44
-0.33
a The labels of the atoms correspond to those defined in Table 1. Z* and QM are provided by ABINIT and CASTEP, respectively.
(+2), indicating an ionic-type response for motions involving Mg atoms. By contrast, the strongly anisotropic components of Z*(N) suggest that the charge transfer is significantly different along and perpendicular to the trigonal axis for motions involving N atoms. In spite of the fact that B and N atoms are close in the periodic table, we observe that their effective charges are very different. Indeed, according to their Mulliken charges, the three different boron atoms do not have the same charge character: the B1 atoms, with a positive charge, are rather electron donors whereas the B2 and B3 atoms with negative charges are electron acceptors. Whereas the ionic value of the Mg atoms is clear (+2), the reference values of the N and B atoms are less obvious. However, the fact that Z*(B) and Z*(N) are strongly altered (smaller) from their reference ionic values (configuration considering closed shell model) indicates substantial covalent character in the bonding. These results are consistent with the Mulliken bond population analyses discussed in section III.B. V. Zone-Centered Phonon Modes A. Normal Modes. Zone-centered phonon modes can be classified according to the irreducible representations of the D3d point group. The group theory predicts the following symmetries of the modes:
Γ ) 8A1g x 3A2g x 11Eg x 3A1u x 8A2u x 11Eu The A2u and Eu modes polarized along and perpendicular to the trigonal axis, respectively, are infrared-active. The A1g and Eg modes are Raman-active, whereas the A2g and A1u modes
163 280 516 571 619 737 778 813 849 1064 1077
Eu
A1g
TO1 127 LO1 200 232 TO2 406 LO2 413 530 TO3 547 LO3 549 677 TO4 634 LO4 634 758 TO5 657 LO5 658 789 TO6 777 LO6 779 890 TO7 785 LO7 786 1023 TO8 852 LO8 852 1167 TO9 1052 LO9 1079 TO10 1094 LO10 1097
A2u
A1u A2g
TO1 281 LO1 310 427 307 TO2 486 LO2 492 638 530 TO3 661 LO3 664 775 751 TO4 794 LO4 798 TO5 911 LO5 911 TO6 1008 LO6 1008 TO7 1113 LO7 1114
are silent. One A2u and one Eu mode are uniform translational modes. At the Γ point, the macroscopic electric field splits the infrared-actives modes (A2u and Eu) into TO and LO modes. This splitting can be calculated from the knowledge of Z* and ε∞ as described in ref 18. The calculated phonon frequencies, including the LO and TO character of the polar modes, are reported in Table 3 with their corresponding irreducible representations. These frequencies are calculated for the equilibrium geometry at the experimental lattice parameters. The phonon frequencies calculated at optimized lattice parameters differ at most by 5% with respect to those reported in Table 3. We highlight that no experimental or theoretical phonon investigations on MgNB9 material are presently available in the literature. Analyzing the phonon eigendisplacement vectors of each normal mode, we observe that Mg atoms are mainly involved in low-frequency modes up to 250 cm-1, whereas B atoms are mainly involved for modes centered at 677, 737, and 849 cm-1 and in the 530-658, 890-1030, and 1090-1167 cm-1 ranges. For modes with A2g and A1u symmetry, only B atoms are involved without contributions from the Mg or N atoms. These three A1u and A2g modes are respectively assigned to in-phase and out-of-phase librational modes of the icosahedral B12 cages around the Cartesian axis of the MgNB9 trigonal structure. A representation of the lowest low-frequency librational mode, A2g(TO1), is displayed in Figure 3. Both B and N atoms strongly dominate in the other normal modes. B. Static Dielectric Tensor and Infrared Reflectivity Spectra. In section IV A, we have reported the electronic dielectric permittivity tensor, ε∞. This quantity describes the response of the electron gas to a homogeneous electric field if
Physical Properties of Magnesium Nitridoboride
J. Phys. Chem. C, Vol. 113, No. 12, 2009 5001
TABLE 4: Mode Oscillator Strengths (× 10-5 a.u.) for a MgNB9 Monocrystal (A⊥, A|) or a Powder (〈A〉)a Eu modes ωm
A⊥
A2u modes
〈A〉
ε0⊥
ωm
A|
〈A〉
ε0|
∞
ε 6.96 7.51 TO1 127 36.533 24.356 11.62 281 26.791 8.930 1.74 TO2 406 6.942 4.628 0.22 486 7.947 2.649 0.17 TO3 547 1.693 1.129 0.03 661 5.816 1.938 0.07 TO4 634 0.464 0.309 0.00 794 9.626 3.209 0.08 TO5 657 1.311 0.874 0.02 911 0.015 0.005 0.00 TO6 777 5.629 3.753 0.05 1008 0.390 0.130 0.00 TO7 785 1.932 1.288 0.02 1113 4.228 1.409 0.02 TO8 852 0.708 0.472 0.00 TO9 1052 80.151 53.434 0.37 TO10 1094 3.316 2.211 0.01 total 19.31 9.59 a Ionic contributions of the infrared TO phonon modes of MgNB9 to its static dielectric constant (ε0) are also reported. Frequencies (ωm) are in cm-1.
the ions are taken as fixed at their equilibrium positions. To include the response of the crystal lattice to the electric field, one can use a model that assimilates the solid to a system of undamped harmonic oscillator. The static dielectric tensor, ε0, can be therefore decomposed into an electronic and an ionic term such as18 0 ∞ εRβ ) εRβ +
4π Ω0
∑ m
ARβ(m)
(2)
ωm2
The infrared oscillator strengths, A, is a second-order tensor given by
ARβ(m) )
[∑ γ,κ
* (κ) ZRγ
√Mκ
] [∑ *
eγ(κ, m)
γ,κ
* (κ) Zβγ
√Mκ
Figure 4. Infrared reflectivity spectra of a MgNB9 monocrystal: (A) A2u modes, (B) Eu modes, and (C) powder.
Furthermore, the three modes cited above will dominate the infrared absorption spectrum of MgNB9 with intense bands as a result of their dominant oscillator strengths. We have also computed the infrared reflectivity spectra to motivate complementary experimental investigations on MgNB9 monocrystals and to obtain an estimate of the LO/TO splitting strength in this material. The reflectivity of optical waves normal to the surface, with their electric field along an optical axis of the crystal q, is given by32
]
R(ω) )
eγ(κ, m) (3)
(
ε1/2 qˆ (ω) - 1 ε1/2 qˆ (ω) + 1
)
2
(4)
where the dielectric tensor is obtained as where the sums run over all atoms κ and space directions γ, Mκ is the mass of the κth atom, and eγ(κ, m) and ωm are, respectively, the γκ component of the eigenvector and the frequency of the mth mode obtained from the diagonalization of the analytical part of the dynamic matrix. The computed main values of the mode oscillator strengths are reported in Table 4 for a MgNB9 monocrystal and powder. According to the symmetry of the MgNB9 structure, ε0 is a diagonal tensor with two different components: ε0⊥ ) 19.31 and ε0| ) 9.59. By contrast to ε∞, the inclusion of the ionic contribution results in a strongly anisotropic ε0 with ε0| being twice as small as ε⊥0 . The dielectric response of MgNB9 is therefore mainly ionic in the plane orthogonal to the optical axis and electronic along this axis. From the decomposition of ε0, we observe that the Eu(TO1) and Eu(TO9) modes, respectively, calculated to be 127 and 1052 cm-1, represent the most important contributions of the oscillator strength (Table 4). However, because of their different frequencies, they do not contribute equally to ε⊥0 . The main contribution (60%) comes from the Eu(TO1) mode, but only 2% originates from the Eu(TO9) mode. The analyses of the eigendisplacement vectors assign the Eu(TO1) mode to a normal mode of vibration where the Mg atoms are mainly involved (Figure 3). Thus, we may increase the ε⊥0 component by substituting Mg atoms with a heavier alkaline earth atom. By contrast, we observe that the A2u(TO1) mode gives the most important contribution to ε0| .
εqˆ(ω) )
4π ∑ qˆRεRβ∞ (ω)qˆβ + Ω ∑ 0 Rβ
m
∑Rβ qˆRARβ(m)qˆβ ωm2 - ω2
(5)
Panels A and B of Figure 4 display the calculated infrared reflectivity spectra, at normal incidence, associated with the A2u and Eu modes, respectively. The reflectivity related to the A2u modes can be associated with incident light parallel to the trigonal axis of the crystal (i.e., parallel to the c direction, which is perpendicular to the (001) surface). Similarly, the Eu modes can be associated with incident light perpendicular to the trigonal axis (i.e., parallel to the a(b) directions, which are perpendicular to the (100) ((010)) surfaces). We also report in panel C the average reflectivity spectrum assuming a quasi-continuous and random distribution of crystallite orientations. Because our approach neglects the damping of the phonon modes, their reflectivities saturate to 1. As expected from the frequency calculation of the TO and LO modes, we observe that the LO-TO splitting is negligible for all modes in MgNB9 except for those below 350 cm-1 and the Eu modes around 1050 cm-1. VI. Elastic Properties and Mechanical Stability Elastic constants characterize the ability of a material to deform under small stresses. They can be described by a fourth-
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j ) and Relaxed-Ion (C) Elastic TABLE 5: Clamped-Ion (C Tensors (in GPa) and Corresponding Compliance Tensors (Sj and S in TPa-1) in MgNB9a index 11 12 13 14 33 44
j C
C
Sj
S
543.63 63.47 64.01 -9.22 489.93 222.13
470.37 89.80 98.06 -31.02 380.45 173.62
1.89 -0.20 -0.22 0.09 2.10 4.51
2.33 -0.37 -0.51 0.48 2.89 5.93
a The indexes denote the Cartesian directions given in Voigt notation.
rank tensor (C) relating the second-rank stress tensor (σ) to the (also second-rank) strain tensor (η) via the generalized Hooke law
CRβ )
∂σR ∂ηβ
(6)
1 B ) [2(C11 + C12) + C33 + 4C13] 9 G)
(9)
1 (7C11 - 5C12 + 2C33 - 4C13 + 12C44) (10) 30
The corresponding predicted values are B ) 210.34 GPa and G ) 176.52 GPa, which are of the same order of magnitude as those predicted in MgB2.36 In addition, these moduli are larger than those reported in GaN37 or the other III-V semiconducting compounds.38-41 The hardness of MgNB9 is mainly due to the presence of boron polyhedra. The BN compound is the hardest in the nitride borides family. Finally, the typical relations between B and G are G = 1.1B and G = 0.6B for covalent and ionic materials, respectively. In our case, the calculated value of G/B is 0.84, indicating mixed covalent-ionic bonding in MgNB9 according to the previous sections. VII. Conclusions
(8)
In this article, we have investigated the electronic, dielectric, dynamic, and elastic properties of MgNB9 within the DFT framework. Our study covers all of the linear couplings among applied static homogeneous electric field, strain, and periodic atomic displacements, including the Born effective charge, dynamic matrix at the zone center, optical dielectric tensor, and elastic tensors. The band structure of MgNB9 states that this material is a semiconductor with an indirect band gap of 1.76 eV between the Z and L high-symmetry points of the trigonal space group. An inspection of the Mulliken bond population shows that this material has mixed ionic-covalent character. Its electronic dielectric tensor is nearly isotropic, and the magnitude of its components is similar to those reported in ferroelectric materials or III-V semiconductors. By contrast, its static dielectric tensor is strongly anisotropic in the plane orthogonal to the optical axis of this material. This anisotropy is mainly governed by a very polar low-frequency Eu mode assigned to localized Mg motions. The calculation of the complete set of zero-pressure elastic constants has been performed and satisfies the Born mechanical stability criteria for trigonal materials, highlighting the fact that the MgNB9 structure is mechanically stable. In addition, its bulk and shear moduli are larger than those reported on III-V semiconductors. We have also reported and discussed the infrared reflectivity spectra and the zone-centered phonon frequencies of TO and LO modes. These results suggest that the physical properties between MgNB9 and III-V semiconductors are similar, leading to some potential applications of this new boride material in optoelectronic-like short-wavelength ranges and high-temperature, highpower, and high-frequency electronic devices where the III-V semiconductors are currently used. It is noteworthy that the first-principles techniques that are used have such predictive capability. However, we also hope that this work will motivate experimental investigations on this new boride material to confirm our predictions.
It is well known that the bulk modulus (B) or the shear modulus (G) can estimate the hardness of a material in an indirect way:34 materials with high B or G are likely to be hard materials. On the basis of the Voigt approximation, we have calculated these quantities using the following relations:35
Acknowledgment. S.G.-S., L.H., and P.H. are respectively supported by the CERUNA project of the University of Namur, the Belgian FRS-FNRS, and the European Commission under the 6 Framework Programme (STREP project BNC Tubes, contract number NMP4-CT-2006-03350). Calculations using CASTEP have been performed at the Centre de Ressources Informatiques de Haute-Normandie (CRIHAN, France). Those
where R, β ) 1, 2,..., 6 denote the Cartesian directions given in Voigt notation. The above equation can be split into two main contributions:
CRβ )
∂σR ∂ηβ
|
+ τ
∂σ
R ∑ ∂τR(κ) κ
∂τR(κ) ∂ηβ
(7)
j ), The first term is the frozen (clamped) ion elastic tensor (C whereas the second term includes contributions from forceresponse internal stress and displacement-response internal strain tensors. The second term also accounts for the ionic relaxations in response to strain perturbations. The addition of the two contributions is the relaxed ion elastic tensor C. Because of the symmetry of the R3jm structure of MgNB9, these tensors have only 6 independent elements to be determined instead of 21. The independent elements of the clamped-ion and relaxed-ion elastic tensors at constant electric field are reported in Table 5, together with their corresponding compliance tensors (Sj and S). The compliance tensors are defined as the inverse of the elastic tensors. These results show that the physical elastic constant (C) is generally smaller than the frozenion ones (at least for diagonal elements) because the additional internal relaxation allows some of the stress to be relieved. Similarly, the diagonal S values are larger than the Sj ones, reflecting the increased compliance allowed by the relaxation of the atomic coordinates. In addition, MgNB9 is mechanically stable because the elastic tensor constants satisfy the Born mechanical stability restrictions for trigonal structure given by the following system of equations:33
{
C11 - |C12 | > 0 (C11 + C12)C33 - 2C213 > 0 (C11 - C12)C44 - 2C214 > 0
Physical Properties of Magnesium Nitridoboride using ABINIT and Wien2k have been performed at the Namur Interuniversity Scientific Computing Facilicity (Namur-ISCF, Belgium). Namur-ISCF is a common project among FNRS, SUN Microsystems, and Les Faculte´s Universitaires NotreDame de la Paix (FUNDP). We acknowledge F. Wautelet (FUNDP) for permanent computer assistance. References and Notes (1) Choi, H. J.; Roundy, D.; Sun, H.; Cohen, M. L.; Louie, S. G. Nature (London) 2002, 418, 758. (2) Tang, C.; Bando, Y.; Liu, C.; Fan, S.; Zhang, J.; Ding, X.; Golberg, D. J. Phys. Chem. B 2006, 110, 10354. (3) An, W.; Wu, X.; Zeng, X. C. J. Phys. Chem. B 2006, 110, 16346. (4) Karpinski, J.; Kazakov, S. M.; Jun, J.; Angst, M.; Puzniak, R.; Wisniewski, A.; Bordet, P. Physica C 2003, 385, 42. (5) Pallasand, A.; Larsson, K. J. Phys. Chem. B 2006, 110, 5367. (6) Johnson, O.; Joyner, D. J.; Hercules, D. M. J. Phys. Chem. 1980, 84, 542. (7) Nagamatsu, J.; Nakagawa, N.; Murakana, T.; Zenitani, Y.; Aksimitsu, J. Nature (London) 2001, 410, 63. (8) Liu, A. Y.; Mazin, I. I.; Kortus, J. Phys. ReV. Lett. 2001, 87, 087005. (9) Mironov, A.; Kazakov, S.; Jun, J.; Karpinski, J. Acta Crystallogr., Sect. C 2002, 58, i95. (10) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (11) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (12) Gonze, X.; Beuken, J.-M.; Caracas, R.; Detraux, F.; Fuchs, M.; Rignanese, G.-M.; Sindic, L.; Verstraete, M.; Zerah, G.; Jollet, F.; Torrent, M.; Roy, A.; Mikami, M.; Ghosez, Ph.; Raty, J.-Y.; Allan, D. C. Comput. Mater. Sci. 2002, 25, 478. (13) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter 2002, 14, 2717. (14) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (15) Troullier, N.; Martins, J.-L. Phys. ReV. B 1991, 43, 1993. (16) Vanderbilt, D. Phys. ReV. B 1990, 41, 7897. (17) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (18) Gonze, X.; Lee, C. Phys. ReV. B 1997, 55, 10355. (19) Sa´nchez-Portal, D.; Artacho, E.; Soler, J. M. J. Phys.: Condens. Matter 1996, 8, 3859.
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