Copyright © 2013 by American Scientific Publishers All rights reserved. Printed in the United States of America
Science of Advanced Materials Vol. 5, pp. 1916–1921, 2013 (www.aspbs.com/sam)
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations Bao-Tian Wang∗ , Wenxue Zhang, and Wei-Dong Li Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, People’s Republic of China
ARTICLE
ABSTRACT We have calculated the electronic, mechanical, and vibrational properties of ZrB2 and ZrB12 from densityfunctional theory. Results show that the strong covalent bonding of B layers or clusters is responsible for the good mechanical and dynamical stabilities. The electronic density of states at the Fermi level N(EF ) for ZrB12 is prominently larger than that for ZrB2 , especially for Zr 4dyz dxz states. The low frequency vibration of Zr atoms makes electron–phonon interaction considerable. Through analyzing electronic structures, bonding pictures and ionicity for the two borides have been indicated quantitatively. KEYWORDS: Zirconium Borides, Elastic Constant, Phonon, First-Principles. the electron–phonon interaction.2 However, among dodeDelivered by Publishing Technology Baotian Wang caboridesto:ZrB 12 is found to exhibit the highest superconIP: 166.111.120.71 On: Thu, 05 Dec 2013 12:42:41 Zirconium diboride ZrB2 is widely used as refractory ducting transition temperature Tc ∼ 6 K.3 Considering the Copyright:and American Publishers materials, cutting tools, protection materials, elec- Scientific mechanical, electronic, and especially lattice dynamical trodes due to its mechanical strength, stiffness, high meltinformation on these two borides in current literature are ing point, hardness, chemical stability, high thermal and limited,4–11 we will comparatively investigate those propelectronic conductivity. Using ZrB2 as a lattice-matched erties from first-principles calculations. buffer layer, high quality GaN films were successfully grown on Si.1 For zirconium dodecaboride ZrB12 , its valu2. COMPUTATIONAL METHODS able properties of high melting points, hardness, thermal and chemical stability also make it rather promising in First-principles density functional theory (DFT) calculatechnical applications. tions on the basis of the projected augmented wave (PAW) At ambient pressure, ZrB2 crystallizes in the AlB2 method of Blöchl12 were performed within the Vienna ab type hexagonal crystal structure with space group P6/mmm initio simulation package (VASP),13 where the Perdew, (No. 191), in which B atoms form two-dimensional honBurke, and Ernzerhof (PBE)14 form of the generalized eycomb layers and Zr atoms sit above the centers of the gradient approximation (GGA) was employed to describe hexagons in between the B layers; ZrB12 crystallizes in a electron exchange and correlation. For the plane-wave set, face-centered cubic (fcc) structure with space group Fma cutoff energy of 500 eV was used. The -centered k 3m (No. 225) with Zr in 4a(0, 0, 0) and B in 48e(0.5, y, y) point-meshes in the full wedge of the Brillouin zone (BZ) Wyckoff positions, in which the Zr atoms and cuboctahewere sampled by 18 × 18 × 16 and 6 × 6 × 6 grids accorddral B12 cluster are arranged in an NaCl type structure. ing to the Monkhorst-Pack (MP)15 for ZrB2 (three-atoms For these two borides, the mechanical properties are govcell) and ZrB12 (52-atoms cell), respectively. All atoms erned mainly by B network and the electronic transport were fully relaxed until the Hellmann-Feynman forces properties are controlled principally by Zr sublattice. became lower than 0.001 eV/Å. The Zr 4s 2 4p6 4d 3 5s 1 and Although ZrB2 possesses the same structure as that of the B 2s 2 2p1 orbitals were explicitly included as valence the famous superconductor MgB2 , its superconductivity electrons. was doubted by point-contact spectroscopy investigation of The theoretical equilibrium volume, bulk modulus B, and pressure derivative of the bulk modulus B were ∗ Author to whom correspondence should be addressed. obtained by fitting the energy-volume data in the thirdEmail:
[email protected] order Birch–Murnaghan equation of state (EOS).16 ElasReceived: 26 March 2013 Accepted: 23 May 2013 tic constants were calculated by applying stress tensors
1. INTRODUCTION
1916
Sci. Adv. Mater. 2013, Vol. 5, No. 12
1947-2935/2013/5/1916/006
doi:10.1166/sam.2013.1657
Wang et al.
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations
with various small strains onto the equilibrium structures. The strain amplitude was varied in steps of 0.006 from = −0036 to 0.036. After obtaining elastic constants, the polycrystalline bulk modulus B and shear modulus G were calculated from the Voigt–Reuss–Hill (VRH) approximations.17 The Young’s modulus E and Poisson’s ratio were calculated through E = 9BG/(3B + G) and = 3B − 2G/[2(3B + G)]. The transverse (t , longitudinal (l , and average (m sound velocities as well as the Debye temperature were derived from polycrystalline bulk and shear modulus. Detailed calculation scheme for mechanical properties can be found in Ref. [18]. Phonon frequency calculations were performed by using the supercell approach within the FROPHO code.19 To reach high accuracy, 3 × 3 × 3 hexagonal supercell containing 81 atoms and 3 × 3 × 3 rhombohedral supercell containing 104 atoms were used for ZrB2 and ZrB12 , respectively; 5 × 5 × 5 and 3 × 3 × 3 MP k-point meshes are utilized in the BZ integration for ZrB2 and ZrB12 , respectively.
Table I. Calculated lattice parameters (a or c, bulk modulus (B, pressure derivative of the bulk modulus (B , and elastic constants of ZrB2 and ZrB12 . For comparison, previous DFT-PBE results and experimental values are also listed. Compound ZrB2
ZrB12
Method
a (Å)
c (Å)
c/a
B (GPa)
B
C11 (GPa)
C12 (GPa)
C13 (GPa)
C33 (GPa)
C44 (GPa)
This work DFT-PRB10 DFT-PBE11 Expt. This work Expt.4
3.179 3.17
3.552 3.56
1.118 1.123
236.1
3.97
3.17020 7.410 7.4077
3.53220
1.11420
24521 233.2 234
56.0 64 68.1 5721 118.3 129
124.0 133 138.3 12121
428.5 446 437.9 43621
3.60
560.3 563 557.5 56821 467.6 443
247.6 253 245.9 24821 269.5
Sci. Adv. Mater., 5, 1916–1921, 2013
1917
ARTICLE
equilibrium volumes can be predicted from the elastic constants. While the elastic constants of ZrB2 satisfy the mechanical stability criteria23 of the hexagonal structure: C44 > 0, C11 > C12 , (C11 + 2C12 )C12 > 2C13 ,2 the elastic constants of ZrB12 satisfy the mechanical stability criteria of the cubic structure: C11 > 0, C44 > 0, C11 > C12 , C11 + 2C12 > 0. Compared with -Zr,18 ZrB2 possesses lager values of C11 , C33 , and C44 , but smaller C12 . This fact makes ZrB2 more stable than -Zr under condition of compression or tension. Actually, the main bonding strength for ZrB2 originates from B B covalent bonds (see following electronic structure analysis). Based upon structural parameters and elastic constants, the elastic moduli, Poisson’s ratio (), density ( , transverse sound velocity (t ), longitudinal sound velocity (l ), average sound velocities (m , and Debye temperature of ZrB2 and ZrB12 are calculated and tabulated in Table II. Each derived bulk modulus B turns out to be very close to that obtained by EOS fitting. Our calculated bulk modulus B of about 239 GPa for ZrB2 is somewhat smaller than about 248 and 249 GPa reported in a recent DFT-PBE study.10 11 This is due to the fact that our cal3. RESULTS AND DISCUSSION culated volume of about 31.08 Å3 is slightly larger than the volume of about 30.98 Å3 derived by previous DFTThe optimized lattice constants a or c, bulk modulus B, PBE calculation.10 On the other hand, our calculated elaspressure derivative of the bulk modulus B , obtained by tic moduli, wave velocities, and Debye temperatures ZrB12 , are fitting the EOS for ZrB2 andDelivered by presented PublishinginTechnology to: elastic Baotian Wang 10 11 and ZrB Table I. For comparison, previous DFT-PBE resultsOn: IP: 166.111.120.71 Thu,for05both DecZrB 2013 12:42:41 2 12 are wholly consistent with correCopyright: spondingPublishers results deduced from previous DFT-PBE10 11 and and experimental data420, 21 are also included in American Table I. Scientific also the experimental4 21 structural parameters and elastic Our calculated lattice constants and bulk modulus B are well consistent with previous DFT-PBE results by Lawson constants by using our scheme. Therefore, our calculations et al.10 and corresponding experimental values, which supare consistent and reliable. ply the safeguard for our following study of mechanical Phonon spectrum has tight relation with dynamical staproperties and electronic structure of these two borides. bility, phase transition, thermoelectric effect, and superWith respect to the experimental values, the small overesconductivity. The calculated phonon curves along typical timation of lattice constants and underestimation of bulk high-symmetry directions and the phonon density of states modulus B are due to the using of GGA. For ZrB12 , the (PhDOS) for ZrB2 and ZrB12 are displayed in Figure 1. For boron position parameter y is optimized to be 0.1695, ZrB2 , we note that our results are wholly consistent with a which is well consistent with previous experimental and recent theoretical work6 by using SIESTA and PHONON calculating values of 0.1693 and 0.1619, respectively.7 codes. The large slope values of acoustics branches again Our calculated elastic constants for both ZrB2 and insure the stability of this material. The large gap in the ZrB12 are also in good agreement with recent DFT-PBE phonon spectrum and PhDOS is due to the mass difference results10 11 and experiments.4 21 With respect to experibetween Zr and B atoms. The main contribution to the ments, we deduce closer values of elastic constants than acoustics branches is from Zr atoms and the main contributhose obtained by recent DFT studies.5 22 The mechanical tion to the optical branches is from B atoms. For ZrB12 , the stability for both ZrB2 and ZrB12 at their corresponding present results are in good agreement with experimental
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations
Wang et al.
Table II. Calculated elastic moduli, Poisson’s ratio (, density ( , transverse (t , longitudinal (l , and average (m sound velocities derived from polycrystalline bulk and shear modulus, and Debye temperature of ZrB2 and ZrB12 . For comparison, previous DFT-PBE results and experimental values are also listed. Compound ZrB2
ZrB12
Method
B (GPa)
G (GPa)
E (GPa)
(g/cm3
t (km/s)
I (km/s)
m (km/s)
D (K)
This work DFT-PRBa DFT-PBEa Expt.a This work Expt.a
239.2 247.7 248.8 240.6 234.7 234
229.0 230.5 224.3 232.0 226.5 215
520.8 527.8 517.4 526.8 514.1 493
0.137 0.145 0.154 0.135 0.135 0.148
6.030 6.051
6.162 6.172
9.502 9.578
6.760 6.776
922.9 926.1
6.089 3.607 3.612
6.169 7.924 7.712
9.497 12.198 11.999
6.766 8.690 8.468
927.1 1302.9 1270
ARTICLE
Note: a These data are derived from elastic constants of previous theoretical calculations10 11 and experiments.4 21
phonon spectrums7 as well as previous theoretical phonon the orbital occupation are consistent with the experimental frequencies at the point.24 The PhDOS is well consistent X-ray photoemission spectra measurements.7 While the Zr p and B p states for ZrB12 do not exhibit significant differwith the DFT results in recent lattice dynamics study on ence along x/y/z directions, the p states for ZrB2 emerge ZrB12 .7 The PhDOS can be viewed as being composed of some differences. This is due to the different structures the two parts. One is the part below 5 THz where the main two compounds crystallizing in. In fact, the structure of contribution comes from the Zr sublattice, while the other ZrB12 is isotropic, but the structure of ZrB2 is anisotropic. part above 5 THz is dominated by the dynamics of the light For ZrB2 , the conductivity is mainly contributed by Zr B atoms. The vibrational frequency of Zr atoms in ZrB12 4d and B pz orbitals. This observation is consistent with is wholly lower than that in ZrB2 . We believe that the low previous DFT study performed by Zhang et al.9 where it frequency vibration of Zr is responsible for a considerable is stated that the conductivity of ZrB2 is not only in the c electron–phonon interaction and the superconductivity. direction but also in the a–b plane. Along the c direction, Basically, all the macroscopical properties of materiB bonds possess metallic property coming mainly the Zr to: als, such as hardness, elasticity, and conductivity, origi-Technology Delivered by Publishing Baotian Wang the Zr 4d and B pz orbitals. In a–b plane, the Zr Zr nate from their electronic structure properties as wellOn: as Thu,from IP: 166.111.120.71 05 Dec 2013 12:42:41 Copyright: Publishers bonds contribute to the conductivity. The B layers, exhibitchemical bonding nature. The calculated total andAmerican partial Scientific ing insulating nature, contain mainly sp2 hybridization.9 density of states (DOSs) of ZrB2 and ZrB12 are shown in Theses features make ZrB2 different from the isostrucFigure 2. For both ZrB2 and ZrB12 , the main features of tural superconductor MgB2 , where the conductivity arises mainly from the strong coupling between the in-plane E2g phonon mode and the bonding states (spx py hybridization within the B layers).25 For ZrB12 , the conductivity mainly originates from Zr 4dyz dxz and B pz orbitals. The DOS at the Fermi level, N EF , of ZrB12 is prominently larger than that of ZrB2 , especially for Zr 4d states. Since the increase of the N EF , a weak mixture of B p and Zr d states is believed crucial for the superconductivity in ZrB12 .8 18 To study the bonding nature of ZrB2 and ZrB12 , we plot the isosurfaces of charge density and difference charge density of ZrB2 in Figure 3 and the valence charge density and difference charge density of ZrB12 in the {001}planes crossing the (0, 0, 0) and (0, 0, 0.3304) points in Figure 4. The difference charge density is obtained by subtracting the densities of noninteracting component systems, (Zr) + (Bx (x = 2 or 12), from the density of the ZrBx system, ( ZrBx , while maintaining the positions of the component systems at the same location as in ZrBx . We also plot the line charge density distribution along the nearest B B, Zr B, and Zr Zr bonds (not shown) and Fig. 1. Phonon dispersion of (a) ZrB2 and (b) ZrB12 at their correperform the Bader analysis.26 The Bader charges, Bader sponding equilibrium volumes. For comparison, experimental phonon volumes, bond lengths, and line charge density at the corspectrums from Ref. [7] as well as theoretical and experimental phonon frequencies at the point from Ref. [24] are also presented. responding bond points (CDb are tabulated in Table III. 1918
Sci. Adv. Mater., 5, 1916–1921, 2013
Wang et al.
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations
ARTICLE
Delivered by Publishing Technology to: Baotian Wang IP: 166.111.120.71 On: Thu, 05 Dec 2013 12:42:41 Copyright: American Scientific Publishers
Fig. 2.
Total and partial densities of states for (a) ZrB2 and (b) ZrB12 . The Fermi energy level is set at zero.
From Figure 3 and Table III, one can deduce the following features for ZrB2 : (i) the B layers are strongly bonded by B B covalent bonds with CDb (B B) = 0107 e/au3 . This value is a slightly larger than 0.104 e/au3 found for the Si covalent bond;27 (ii) above the B hexagons, Zr atoms are connected to each other with relatively weak metal bonds; (iii) the adjacent Zr and B layers are bonded by Zr B bonds with mixed features of ionic and covalent. The CDb value for Zr B bonds of 0.044 e/au3 is prominently higher than 0.007 e/au3 found for the Na Cl bond in the typical ionic crystal NaCl.27 The isosurfaces of the difference charge density of ZrB2 indicate that the charge is accumulated from each B atoms towards the vertical direction of the Zr layer, i.e., to the triangular regions between groups of three Zr atoms. The charge depleted from the B layer is more evident than from the Zr layer. Thus, the charge is dragged principally from the atomic Sci. Adv. Mater., 5, 1916–1921, 2013
Fig. 3. (a)–(b) Side and (c) (d) top views of the isosurfaces of the charge density (left one) and the isosurfaces of the difference charge density (right one) of ZrB2 . While the isosurfaces of the charge density are drawn at 0.08 and 0.1 e/au3 , the isosurfaces of the difference charge density are drawn at −0.01 and 0.01 e/au3 .
1919
ARTICLE
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations
Wang et al.
Delivered by Publishing Technology to: Baotian Wang IP: 166.111.120.71 On: Thu, 05 Dec 2013 12:42:41 Copyright: American Scientific Publishers
Fig. 4. Valence charge density (upper panels) and difference charge density (lower panels) of ZrB12 in the {001}-planes cross the (a) (c) (0, 0, 0.3304) and (b) (d) (0, 0, 0) points. Here, the coordinates are indicated in fraction format. Contour lines for the valence charge density are drawn from 0.00 to 0.12 at 0.01 e/au3 intervals and for the difference charge density are drawn from 0.00 to 0.01 at 0.002 e/au3 intervals.
2p state of the B layer. By drawing the electron localization function (ELF), Lawson et al.10 have clearly found the triangular accumulation regions within the Zr layer. We believe that it is this kind of charge accumulation behavior which is responsible for the covalent bonding of the Zr B bonds. For ZrB12 , the B atoms within B12 cluster are bonded by strong B B covalent bonds with CDb (B B) = 0114 e/au3 . The B12 clusters are connected by stronger B B covalent bonds with CDb (B B) = 0146 e/au3 . The Zr atoms are isolated to each other by B12 clusters. The Zr atoms and B12 cluster are bonded by Zr B bonds with mixed features of ionic and covalent. The difference charge density of ZrB12 illustrates that the main
contribution to the charge accumulation is from B atoms, not Zr. The charge of noninteracting component system B12 is crowded on the B B bonds, which results in an external pressure of about 0.5 GPa. After interacting with Zr, charge is dragged from B B bonds to Zr [see Fig. 4(d)], which, leads to a release of the pressure for the B12 fcc cell. Besides, the accumulation of charge at the interstitial region of B plane is also due to the dragging force from Zr atoms [see Fig. 4(c)]. Compared with the B B bonds in ZrB2 , the B B bonds in ZrB12 are more covalently bonded. But the B12 clusters network weakens the bonding strength of Zr B bonds. In analyzing the ionicity, the ionic charges of ZrB12 also have a little change compared with ZrB2 . While each Zr atom losses 1.54 electrons in
Table III. Calculated charge and volumes according to Bader partitioning as well as the bond lengths and charge density values at bond points (CDb for ZrB2 and ZrB12 . For ZrB12 , there are two typical B B bonds. One of them stands for the connecting between the B12 clusters and another one indicates the bonding within the B12 clusters. The latter one is indicated by data within parentheses. Compound ZrB2 ZrB12
1920
QB (Zr) (e)
QB (B) (e)
VB (Zr) (Å3
VB (B) (Å3
B B (Å)
Zr B (Å)
Zr Zr (Å)
10.46 10.08
3.77 3.16
12.23 12.20
9.43 7.45
1.835 1.686 (1.776)
2.554 2.752
3.179 5.239
(B
CDb B) (e/au3
0.107 0.146 (0.114)
(Zr
CDb B) (e/au3 0.044 0.033
(Zr
CDb Zr) (e/au3 0.027
Sci. Adv. Mater., 5, 1916–1921, 2013
Wang et al.
Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations
5. R. Escamilla, M. Romero, and F. Morales, Solid State Commun. 152, 249 (2012). 6. E. Deligoz, K. Colakoglu, and Y. O. Ciftci, Solid State Commun. 150, 405 (2010). 7. A. V. Rybina, K. S. Nemkovski, P. A. Alekseev, J. M. Mignot, E. S. Clementyev, M. Johnson, L. Capogna, A. V. Dukhnenko, A. B. 4. CONCLUSION Lyashenko, and V. B. Filippov Phys. Rev. B 82, 024302 (2010). 8. L. Huerta, A. Duran, R. Falconi, M. Flores, and R. Escamilla, In summary, we have calculated structural parameters, Physca C 470, 456 (2010). elastic constants, elastic moduli, elastic wave velocities, 9. X. Zhang, X. Luo, J. Han, J. Li, and W. Han, Comput. Mater. Sci. Debye temperature, and the phonon modes for ZrB2 and 44, 411 (2008). ZrB12 , which are good agreement with experimental data. 10. J. W. Lawson, C. W. Bauschlicher Jr, and M. S. Daw, J. Am. Ceram. Superconductivity for ZrB12 is found mainly contributed Soc. 94, 3493 (2011). 11. R. Kumar, M. C. Mishra, B. K. Sharma, V. Sharma, J. E. Lowther, by Zr 4dyz dxz states as well as the low frequency vibration V. Vyas, and G. Sharma, Comput. Mater. Sci. 61, 150 (2012). of Zr atoms. For ZrB2 , the B layers are strongly bonded by 12. P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). B B covalent bonds; Zr atoms are connected with weak 13. G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). metal bonds; the adjacent Zr and B layers are bonded by 14. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 Zr B bonds with mixed ionic and covalent properties. For (1996). 15. H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1972). ZrB12 , the covalently bonded B12 clusters are connected by 16. F. Birch, Phys. Rev. 71, 809 (1947). stronger B B covalent bonds in fcc form; the Zr atoms 17. R. Hill, Phys. Phys. Soc. London 65, 349 (1952). are isolated to each other by B12 clusters and are bonded to 18. B. T. Wang, P. Zhang, H. Y. Liu, W. D. Li, and P. Zhang, J. Appl. B12 clusters by Zr B bonds with mixed features of ionic Phys. 109, 063514 (2011). and covalent. For both compounds, their good mechanical 19. A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008). properties are governed mainly by B network. 20. V. A. Gasparov, N. S. Sidorov, I. I. Zve´rkova, and M. P. Kulakov, JETP Lett. 73, 532 (2001). Acknowledgments: This work was supported by NSFC 21. N. L. Okamoto, M. Kusakari, K. Tanaka, H. Inui, M. Yamaguchi, under Grant Nos. 11104170 and 11074155. and S. Otani, J. Appl. Phys. 93, 88 (2003). 22. H. Fu,to: Y. Lu, W. Liu,Wang and T. Gao, J. Mater. Sci. 44, 5618 (2009). Delivered by Publishing Technology Baotian F. Nye, Physical Properties of Crystals Oxford University Press, 2013 12:42:41 References and Notes IP: 166.111.120.71 On: Thu,23.05J. Dec OxfordPublishers (1985). Copyright: American Scientific 24. H. Werheit, V. Filipov, K. Shirai, H. Dekura, N. Shitsevalova, 1. Y. Yamada-Takamura, Z. T. Wang, Y. Fujikawa, T. Sakurai, Q. K. U. Schwarz, and M. Armbr\"{u}ster, J. Phys.: Condens. Mater Xue, J. Tolle, P.-L. Liu, A. V. G. Chizmeshya, J. Kouvetakis, and 23, 065403 (2011). I. S. T. Tsong, Phys. Rev. Lett. 95, 266105 (2005). 25. M. Iavarone, G. Karapetrov, A. E. Koshelev, W. K. Kwok, G. W. 2. Y. G. Naidyuk, O. E. Kvitnitskaya, I. K. Yanson, S.-L. Drechsler, Crabtree, D. G. Hinks, W. N. Kang, E.-M. Choi, H. J. Kim, H.-J. G. Behr, and S. Otani, Phys. Rev. B 66, 140301 (2002). Kim, and S. I. Lee, Phys. Rev. Lett. 89, 187002 (2002). 3. B. T. Matthias, T. H. Geballe, K. Andres, E. Corenzwit, G. W. Hull, 26. W. Tang, E. Sanville, and G. Henkelman, J. Phys.: Condens. Matter and J. P. Maita, Science 159, 530 (1968). 4. G. E. Grechnev, A. E. Baranovskiy, V. D. Fil, T. V. Ignatova, I. G. 21, 084204 (2009). Kolobov, A. V. Logosha, N. Yu. Shisevalova, V. B. Filippov, and 27. B. T. Wang, P. Zhang, H. L. Shi, B. Sun, and W. D. Li, Eur. O. Eriksson, Low Temp. Phys. 34, 921 (2008). Phys. J. B 74, 303 (2010).
ZrB2 , it transfers 1.92 electrons to B atoms in ZrB12 . The ionic charge for ZrB2 and ZrB12 can be represented as and Zr192+ B016− , respectively. Zr154+ B077− 2 12
1921
ARTICLE
Sci. Adv. Mater., 5, 1916–1921, 2013
