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Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12

from First-Principles Calculations

Article  in  Science of Advanced Materials · December 2013

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Science of Advanced MaterialsVol. 5, pp. 1916–1921, 2013

(www.aspbs.com/sam)

Mechanics, Lattice Dynamics, and ChemicalBonding in ZrB2 and ZrB12 fromFirst-Principles CalculationsBao-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

ABSTRACT

We have calculated the electronic, mechanical, and vibrational properties of ZrB2 and ZrB12 from density-functional theory. Results show that the strong covalent bonding of B layers or clusters is responsible for thegood mechanical and dynamical stabilities. The electronic density of states at the Fermi level N(EF) for ZrB12 isprominently larger than that for ZrB2, especially for Zr 4dyzdxz states. The low frequency vibration of Zr atomsmakes electron–phonon interaction considerable. Through analyzing electronic structures, bonding pictures andionicity for the two borides have been indicated quantitatively.

KEYWORDS: Zirconium Borides, Elastic Constant, Phonon, First-Principles.

1. INTRODUCTION

Zirconium diboride ZrB2 is widely used as refractorymaterials, cutting tools, protection materials, and elec-trodes due to its mechanical strength, stiffness, high melt-ing point, hardness, chemical stability, high thermal andelectronic conductivity. Using ZrB2 as a lattice-matchedbuffer layer, high quality GaN films were successfullygrown on Si.1 For zirconium dodecaboride ZrB12, its valu-able properties of high melting points, hardness, thermaland chemical stability also make it rather promising intechnical applications.At ambient pressure, ZrB2 crystallizes in the AlB2-

type hexagonal crystal structure with space group P6/mmm(No. 191), in which B atoms form two-dimensional hon-eycomb layers and Zr atoms sit above the centers of thehexagons in between the B layers; ZrB12 crystallizes in aface-centered cubic (fcc) structure with space group Fm-3m (No. 225) with Zr in 4a(0, 0, 0) and B in 48e(0.5, y, y)Wyckoff positions, in which the Zr atoms and cuboctahe-dral B12 cluster are arranged in an NaCl type structure.For these two borides, the mechanical properties are gov-erned mainly by B network and the electronic transportproperties are controlled principally by Zr sublattice.Although ZrB2 possesses the same structure as that of

the famous superconductor MgB2, its superconductivitywas doubted by point-contact spectroscopy investigation of

∗Author to whom correspondence should be addressed.Email: wbt11129@sxu.edu.cnReceived: 26 March 2013Accepted: 23 May 2013

the electron–phonon interaction.2 However, among dode-caborides ZrB12 is found to exhibit the highest supercon-ducting transition temperature Tc ∼ 6 K.3 Considering themechanical, electronic, and especially lattice dynamicalinformation on these two borides in current literature arelimited,4–11 we will comparatively investigate those prop-erties from first-principles calculations.

2. COMPUTATIONAL METHODS

First-principles density functional theory (DFT) calcula-tions on the basis of the projected augmented wave (PAW)method of Blöchl12 were performed within the Vienna abinitio simulation package (VASP),13 where the Perdew,Burke, and Ernzerhof (PBE)14 form of the generalizedgradient approximation (GGA) was employed to describeelectron exchange and correlation. For the plane-wave set,a cutoff energy of 500 eV was used. The � -centered kpoint-meshes in the full wedge of the Brillouin zone (BZ)were sampled by 18×18×16 and 6×6×6 grids accord-ing to the Monkhorst-Pack (MP)15 for ZrB2 (three-atomscell) and ZrB12 (52-atoms cell), respectively. All atomswere fully relaxed until the Hellmann-Feynman forcesbecame lower than 0.001 eV/Å. The Zr 4s24p64d35s1 andthe B 2s22p1 orbitals were explicitly included as valenceelectrons.The theoretical equilibrium volume, bulk modulus B,

and pressure derivative of the bulk modulus B′ wereobtained by fitting the energy-volume data in the third-order Birch–Murnaghan equation of state (EOS).16 Elas-tic constants were calculated by applying stress tensors

1916 Sci. Adv. Mater. 2013, Vol. 5, No. 12 1947-2935/2013/5/1916/006 doi:10.1166/sam.2013.1657

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Wang et al. Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations

ARTIC

LEwith various small strains onto the equilibrium structures.The strain amplitude � was varied in steps of 0.006 from� = −0�036 to 0.036. After obtaining elastic constants,the polycrystalline bulk modulus B and shear modulus Gwere calculated from the Voigt–Reuss–Hill (VRH) approx-imations.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 Debyetemperature were derived from polycrystalline bulk andshear modulus. Detailed calculation scheme for mechani-cal properties can be found in Ref. [18].Phonon frequency calculations were performed by using

the supercell approach within the FROPHO code.19 Toreach high accuracy, 3× 3× 3 hexagonal supercell con-taining 81 atoms and 3× 3× 3 rhombohedral supercellcontaining 104 atoms were used for ZrB2 and ZrB12,respectively; 5× 5× 5 and 3× 3× 3 MP k-point meshesare utilized in the BZ integration for ZrB2 and ZrB12,respectively.

3. RESULTS AND DISCUSSION

The optimized lattice constants a or c, bulk modulus B,pressure derivative of the bulk modulus B′, obtained byfitting the EOS for ZrB2 and ZrB12, are presented inTable I. For comparison, previous DFT-PBE results10�11

and experimental data4�20,21 are also included in Table I.Our calculated lattice constants and bulk modulus B arewell consistent with previous DFT-PBE results by Lawsonet al.10 and corresponding experimental values, which sup-ply the safeguard for our following study of mechanicalproperties and electronic structure of these two borides.With respect to the experimental values, the small overes-timation of lattice constants and underestimation of bulkmodulus B are due to the using of GGA. For ZrB12, theboron position parameter y is optimized to be 0.1695,which is well consistent with previous experimental andcalculating values of 0.1693 and 0.1619, respectively.7

Our calculated elastic constants for both ZrB2 andZrB12 are also in good agreement with recent DFT-PBEresults10�11 and experiments.4�21 With respect to experi-ments, we deduce closer values of elastic constants thanthose obtained by recent DFT studies.5� 22 The mechanicalstability for both ZrB2 and ZrB12 at their corresponding

Table I. Calculated lattice parameters (a or c�, bulk modulus (B�, pressure derivative of the bulk modulus (B′�, and elastic constants of ZrB2 andZrB12. For comparison, previous DFT-PBE results and experimental values are also listed.

Compound Method a (Å) c (Å) c/a B (GPa) B′ C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa)

ZrB2 This work 3.179 3.552 1.118 236.1 3.97 560.3 56.0 124.0 428.5 247.6DFT-PRB10 3.17 3.56 1.123 563 64 133 446 253DFT-PBE11 557.5 68.1 138.3 437.9 245.9

Expt. 3.17020 3.53220 1.11420 24521 56821 5721 12121 43621 24821

ZrB12 This work 7.410 233.2 3.60 467.6 118.3 269.5Expt.4 7.4077 234 443 129

equilibrium volumes can be predicted from the elasticconstants. While the elastic constants of ZrB2 satisfy themechanical stability criteria23 of the hexagonal structure:C44 > 0, C11 > �C12�, (C11+ 2C12)C12 > 2C13,

2 the elasticconstants of ZrB12 satisfy the mechanical stability crite-ria of the cubic structure: C11 > 0, C44 > 0, C11 > �C12�,�C11 + 2C12� > 0. Compared with -Zr,18 ZrB2 possesseslager values of C11, C33, and C44, but smaller C12. This factmakes ZrB2 more stable than -Zr under condition of com-pression or tension. Actually, the main bonding strengthfor ZrB2 originates from B B covalent bonds (see fol-lowing electronic structure analysis).Based upon structural parameters and elastic con-

stants, the elastic moduli, Poisson’s ratio (�), density (�,transverse sound velocity (�t), longitudinal sound velocity(�l), average sound velocities (�m�, and Debye tempera-ture of ZrB2 and ZrB12 are calculated and tabulated inTable II. Each derived bulk modulus B turns out to bevery close to that obtained by EOS fitting. Our calculatedbulk modulus B of about 239 GPa for ZrB2 is somewhatsmaller than about 248 and 249 GPa reported in a recentDFT-PBE study.10�11 This is due to the fact that our cal-culated volume of about 31.08 Å3 is slightly larger thanthe volume of about 30.98 Å3 derived by previous DFT-PBE calculation.10 On the other hand, our calculated elas-tic moduli, elastic wave velocities, and Debye temperaturesfor both ZrB2 and ZrB12 are wholly consistent with corre-sponding results deduced from previous DFT-PBE10�11 andalso the experimental4�21 structural parameters and elasticconstants by using our scheme. Therefore, our calculationsare consistent and reliable.Phonon spectrum has tight relation with dynamical sta-

bility, phase transition, thermoelectric effect, and super-conductivity. The calculated phonon curves along typicalhigh-symmetry directions and the phonon density of states(PhDOS) for ZrB2 and ZrB12 are displayed in Figure 1. ForZrB2, we note that our results are wholly consistent with arecent theoretical work6 by using SIESTA and PHONONcodes. The large slope values of acoustics branches againinsure the stability of this material. The large gap in thephonon spectrum and PhDOS is due to the mass differencebetween Zr and B atoms. The main contribution to theacoustics branches is from Zr atoms and the main contribu-tion to the optical branches is from B atoms. For ZrB12, thepresent results are in good agreement with experimental

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Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations Wang et al.

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LETable II. Calculated elastic moduli, Poisson’s ratio (��, density (�, transverse (�t�, longitudinal (�l�, and average (�m� sound velocities derived frompolycrystalline bulk and shear modulus, and Debye temperature of ZrB2 and ZrB12. For comparison, previous DFT-PBE results and experimental valuesare also listed.

Compound Method B (GPa) G (GPa) E (GPa) � (g/cm3� �t (km/s) �I (km/s) �m (km/s) �D (K)

ZrB2 This work 239.2 229.0 520.8 0.137 6.030 6.162 9.502 6.760 922.9DFT-PRBa 247.7 230.5 527.8 0.145 6.051 6.172 9.578 6.776 926.1DFT-PBEa 248.8 224.3 517.4 0.154Expt.a 240.6 232.0 526.8 0.135 6.089 6.169 9.497 6.766 927.1

ZrB12 This work 234.7 226.5 514.1 0.135 3.607 7.924 12.198 8.690 1302.9Expt.a 234 215 493 0.148 3.612 7.712 11.999 8.468 1270

Note: aThese data are derived from elastic constants of previous theoretical calculations10�11 and experiments.4�21

phonon spectrums7 as well as previous theoretical phononfrequencies at the � point.24 The PhDOS is well consistentwith the DFT results in recent lattice dynamics study onZrB12.

7 The PhDOS can be viewed as being composed oftwo parts. One is the part below 5 THz where the maincontribution comes from the Zr sublattice, while the otherpart above 5 THz is dominated by the dynamics of the lightB atoms. The vibrational frequency of Zr atoms in ZrB12

is wholly lower than that in ZrB2. We believe that the lowfrequency vibration of Zr is responsible for a considerableelectron–phonon interaction and the superconductivity.Basically, all the macroscopical properties of materi-

als, such as hardness, elasticity, and conductivity, origi-nate from their electronic structure properties as well aschemical bonding nature. The calculated total and partialdensity of states (DOSs) of ZrB2 and ZrB12 are shown inFigure 2. For both ZrB2 and ZrB12, the main features of

Fig. 1. Phonon dispersion of (a) ZrB2 and (b) ZrB12 at their corre-sponding equilibrium volumes. For comparison, experimental phononspectrums from Ref. [7] as well as theoretical and experimental phononfrequencies at the � point from Ref. [24] are also presented.

the orbital occupation are consistent with the experimentalX-ray photoemission spectra measurements.7 While the Zrp and B p states for ZrB12 do not exhibit significant differ-ence along x/y/z directions, the p states for ZrB2 emergesome differences. This is due to the different structures thetwo compounds crystallizing in. In fact, the structure ofZrB12 is isotropic, but the structure of ZrB2 is anisotropic.For ZrB2, the conductivity is mainly contributed by Zr

4d and B pz orbitals. This observation is consistent withprevious DFT study performed by Zhang et al.9 where itis stated that the conductivity of ZrB2 is not only in the cdirection but also in the a–b plane. Along the c direction,the Zr B bonds possess metallic property coming mainlyfrom the Zr 4d and B pz orbitals. In a–b plane, the Zr Zrbonds contribute to the conductivity. The B layers, exhibit-ing insulating nature, contain mainly sp2 hybridization.9

Theses features make ZrB2 different from the isostruc-tural superconductor MgB2, where the conductivity arisesmainly from the strong coupling between the in-plane E2g

phonon mode and the � bonding states (spxpy hybridiza-tion within the B layers).25 For ZrB12, the conductivitymainly originates from Zr 4dyzdxz and B pz orbitals. TheDOS at the Fermi level, N�EF �, of ZrB12 is prominentlylarger than that of ZrB2, especially for Zr 4d states. Sincethe increase of the N�EF �, a weak mixture of B p andZr d states is believed crucial for the superconductivity inZrB12.

8�18

To study the bonding nature of ZrB2 and ZrB12, we plotthe isosurfaces of charge density and difference chargedensity of ZrB2 in Figure 3 and the valence charge den-sity and difference charge density of ZrB12 in the {001}-planes crossing the (0, 0, 0) and (0, 0, 0.3304) pointsin Figure 4. The difference charge density is obtained bysubtracting the densities of noninteracting component sys-tems, (Zr)+(Bx� (x = 2 or 12), from the density of theZrBx system, (ZrBx�, while maintaining the positions ofthe component systems at the same location as in ZrBx.We also plot the line charge density distribution along thenearest B B, Zr B, and Zr Zr bonds (not shown) andperform the Bader analysis.26 The Bader charges, Badervolumes, bond lengths, and line charge density at the cor-responding bond points (CDb� are tabulated in Table III.

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Wang et al. Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations

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LE

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 fol-lowing features for ZrB2:(i) the B layers are strongly bonded by B B cova-

lent bonds with CDb(B B)= 0�107 e/au3. This value is aslightly larger than 0.104 e/au3 found for the Si covalentbond;27

(ii) above the B hexagons, Zr atoms are connected toeach 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 promi-

nently higher than 0.007 e/au3 found for the Na Cl bondin the typical ionic crystal NaCl.27 The isosurfaces of thedifference charge density of ZrB2 indicate that the chargeis accumulated from each B atoms towards the verticaldirection of the Zr layer, i.e., to the triangular regionsbetween groups of three Zr atoms. The charge depletedfrom the B layer is more evident than from the Zr layer.Thus, the charge is dragged principally from the atomic

Fig. 3. (a)–(b) Side and (c) (d) top views of the isosurfaces of thecharge density (left one) and the isosurfaces of the difference chargedensity (right one) of ZrB2. While the isosurfaces of the charge densityare drawn at 0.08 and 0.1 e/au3, the isosurfaces of the difference chargedensity are drawn at −0.01 and 0.01 e/au3.

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Mechanics, Lattice Dynamics, and Chemical Bonding in ZrB2 and ZrB12 from First-Principles Calculations Wang et al.

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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 to0.12 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 localiza-tion function (ELF), Lawson et al.10 have clearly found thetriangular accumulation regions within the Zr layer. Webelieve that it is this kind of charge accumulation behav-ior which is responsible for the covalent bonding of theZr B bonds.For ZrB12, the B atoms within B12 cluster are bonded

by strong B B covalent bonds with CDb (B B) =0�114 e/au3. The B12 clusters are connected by strongerB B covalent bonds with CDb (B B) = 0�146 e/au3.The Zr atoms are isolated to each other by B12 clus-ters. The Zr atoms and B12 cluster are bonded by Zr Bbonds with mixed features of ionic and covalent. The dif-ference charge density of ZrB12 illustrates that the main

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 oneindicates the bonding within the B12 clusters. The latter one is indicated by data within parentheses.

QB (Zr) QB (B) VB (Zr) VB (B) B B Zr B Zr Zr CDb CDb CDb

Compound (e) (e) (Å3� (Å3� (Å) (Å) (Å) (B B) (e/au3� (Zr B) (e/au3� (Zr Zr) (e/au3�

ZrB2 10.46 3.77 12.23 9.43 1.835 2.554 3.179 0.107 0.044 0.027ZrB12 10.08 3.16 12.20 7.45 1.686 (1.776) 2.752 5.239 0.146 (0.114) 0.033

contribution to the charge accumulation is from B atoms,not Zr. The charge of noninteracting component systemB12 is crowded on the B B bonds, which results in anexternal pressure of about 0.5 GPa. After interacting withZr, 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 intersti-tial region of B plane is also due to the dragging force fromZr atoms [see Fig. 4(c)]. Compared with the B B bondsin ZrB2, the B B bonds in ZrB12 are more covalentlybonded. But the B12 clusters network weakens the bond-ing strength of Zr B bonds. In analyzing the ionicity, theionic charges of ZrB12 also have a little change comparedwith ZrB2. While each Zr atom losses 1.54 electrons in

1920 Sci. Adv. Mater., 5, 1916–1921, 2013

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LEZrB2, it transfers 1.92 electrons to B atoms in ZrB12. Theionic charge for ZrB2 and ZrB12 can be represented asZr1�54+B0�77−

2 and Zr1�92+B0�16−12 , respectively.

4. CONCLUSION

In summary, we have calculated structural parameters,elastic constants, elastic moduli, elastic wave velocities,Debye temperature, and the phonon modes for ZrB2 andZrB12, which are good agreement with experimental data.Superconductivity for ZrB12 is found mainly contributedby Zr 4dyzdxz states as well as the low frequency vibrationof Zr atoms. For ZrB2, the B layers are strongly bonded byB B covalent bonds; Zr atoms are connected with weakmetal bonds; the adjacent Zr and B layers are bonded byZr B bonds with mixed ionic and covalent properties. ForZrB12, the covalently bonded B12 clusters are connected bystronger B B covalent bonds in fcc form; the Zr atomsare isolated to each other by B12 clusters and are bonded toB12 clusters by Zr B bonds with mixed features of ionicand covalent. For both compounds, their good mechanicalproperties are governed mainly by B network.

Acknowledgments: This work was supported by NSFCunder Grant Nos. 11104170 and 11074155.

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