Theoretical prediction on electronic structure, mechanical properties and lattice dynamics of YB4 for ultrahigh temperature applications

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Theoretical prediction on electronic structure, mechanical properties and lattice dynamics of YB4 for ultrahigh temperature applications Yanchun Zhou ∗ , Huimin Xiang, Zhihai Feng, Zhongping Li Science and Technology of Advanced Functional Composite Laboratory, Aerospace Research Institute of Materials & Processing Technology, No. 1 South Dahongmen Road, Beijing 100076, China

a r t i c l e

i n f o

Article history: Received 16 June 2015 Received in revised form 1 August 2015 Accepted 12 August 2015 Available online xxx Keywords: YB4 Electronic structure Chemical bonding Anisotropic elastic stiffness Lattice dynamics

a b s t r a c t The electronic structure, mechanical properties and lattice dynamics of YB4 are investigated using firstprinciples calculations. The Y–B bonding is ionic-covalent, the Y–Y bonding is metallic, the B–B bonding within B6 octahedron is weak ␶-type covalent and that connecting the octahedra is strong ␴-type covalent. The chemical bonding anisotropy has also been confirmed by lattice dynamic properties, and is reflected by the anisotropic elastic stiffness. The maximum Young’s modulus (427 GPa) is 1.5 times larger than its minimum (285 GPa). The lowest shear moduli are associated with the [1 1 0](1 0 0) and [1 1 0](1 1 0) systems. The linear compressibility of a (1.70 × 10−3 GPa) is larger than that of c (1.47 × 10−3 GPa). Based on the low c44 , c66 and Pugh’s ratio G/B, YB4 is predicted as a damage to lerant ceramic. YB4 also has a relatively low Young’s modulus (339 GPa), which ensures it good thermal shock resistance. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Transition metal borides (TMBs) have contributed to the success of ultrahigh temperature materials (UHTM) development. Current ZrB2 and HfB2 based ceramics are regarded as the materials of choice for nose and sharp leading edges of hypersonic vehicles, hot structure components for scramjet engines [1–3], matrix and/or surface coatings for ultrahigh temperature ceramic matrix composites [4,5] due to the properties that they display. The unique combination of the properties of ZrB2 and HfB2 based ceramics include high melting point, high strength, good high temperature environmental resistance, and absence of phase transformations in the solid state. However, the intrinsic brittleness, poor thermal shock resistance and difficulty in machining into sharp and/or complex shapes have blocked their real applications in hypersonic vehicles. Previous work demonstrated that the poor thermal shock resistance of ZrB2 and HfB2 was resulted from their high Young’s modulus [1], while the poor oxidation resistance was due to the lack of the formation of stable and protective scales during high temperature oxidation [6–7]. In order to stimulate the applications of transition metal borides for ultrahigh temperature applications, searching for low Young’s modulus TMBs with high melting point and environmental stability becomes emergent.

∗ Corresponding author. Fax: +86 10 68752937. E-mail addresses: [email protected], [email protected] (Y. Zhou).

Materials genome approach has been proved very efficient and powerful in rapid scanning the properties of intrigued materials and accelerating the materials development and application processes [8]. One of the important ways in rapid scanning the properties of materials is through quantum mechanical modeling based on density functional theory (DFT) to obtain microscopic information, which is a key ingredient in deep understanding the macroscopic properties of materials. Thus investigating the electronic structure, chemical bonding and intrinsic properties of TMBs is essential in finding new potential materials and designing new compositions/composites for ultrahigh temperature applications. Previous work on the electronic structure of transition metal diborides (TMB2 ) demonstrated that the bonding in TMB2 is a combination of ionic, covalent and metallic [9–12]. The B–B bonding is covalent of -type formed by overlapping of B sp2 –B sp2 orbitals and -type formed by overlapping of B pz –B pz orbitals; the TM-B bonding is ionic–covalent and the covalent nature increases with increasing occupation of the d-orbitals [9,12]. Among the intrigued TMB2 , YB2 has the lowest Young’s modulus [12]. This result gives us a hint that it is possible to find low Young’s modulus TMBs with better thermal shock resistance and damage tolerance in Y–B system. Very recently, YB6 was predicted as a “ductile” and soft ceramic for ultrahigh temperature applications [13]. The “ductility” is underpinned by chemical bonding anisotropy, i.e., strong  bond connecting the B6 octahedra and weak ␶ bond within the B6 octahedron formed by overlapping the lobes of perpendicular p orbitals. YB4 has higher melting point (2800 ◦ C) than YB6 (2600 ◦ C) [14] and a

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of states were also given by these authors [21]. However, detailed information on the chemical bonding and mechanical properties of YB4 is still lacking. In this work, the electronic structure, chemical bonding, anisotropic elastic stiffness and lattice-dynamic properties of YB4 were investigated using first-principles calculations. Detailed chemical bonding nature was disclosed and its relation to the mechanical properties was established. In addition, hardness, damage tolerance and shear deformation resistance were discussed based on the calculated mechanical properties. 2. Calculation methods

Fig. 1. Crystal structure of YB4 .

low theoretical density of 4.33 g/cm3 . The possible oxidation product of YB4 , i.e., Y2 O3 , also has a high-melting point of 2415 ◦ C. In addition, the crystal structure of YB4 is strongly related to both YB2 and YB6 (Fig. 1). The unique properties and structure features of YB4 render it as a promising UHTM for ultrahigh temperature applications [15]. Although growth of YB4 single crystals [16,17], synthesis of YB4 powders [15,18] and preparation of bulk materials [15] have been reported, the structure and property relationships of this material have not been established. Waskowska et al. [19] determined the lattice parameters and bulk modulus of YB4 using a combination of high-pressure X-ray diffraction and DFT calculation. The bulk modulus of YB4 was determined 185.4 ± 3.5 GPa based on the least square fit of the pressure–volume curve. Jäger et al. [20] investigated the bonding properties of YB4 and YB6 by analyzing the band structure, density of states and valence-electron densities. Very recently, the elastic properties of YB4 were calculated by Fu et al. [21] and the bulk modulus (182 GPa) and Debye temperature (874 K) were predicted. The phonon dispersion and density

To elucidate the chemical bonding and mechanical properties of YB4 , first-principles calculations based on density functional theory (DFT) were performed using the Cambridge Serial Total Energy Package (CASTEP) code [22], wherein the Vanderbilt-type ultrasoft pseudopotential [23] and generalized gradient approximation (GGA) based on the Wu–Cohen (WC) scheme [24] for the exchange–correlation function were used. This approach has been approved quite reliable when investigating the equilibrium structure, electronic structure and mechanical properties of carbides and borides [12,13,25,26]. The plane-wave basis set cutoff was 600 eV for all calculations. The special points sampling integration over the Brillourin zone was employed by using the Monkhorst Pack method with 4 × 4 × 6 special k-points mesh [27]. Lattice parameters were modified to minimize the enthalpy and interatomic forces. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme [28] was used in geometry optimization. The tolerances for geometry optimization were difference on total energy within 5 × 10−6 eV/atom, maximum ionic Hellmann–Feynman force within 0.01 eV/Å, maximum ionic displacement within 5 × 10−4 Å and maximum stress within 0.02 GPa. Calculations of projected density of states were performed using a projection of the planewave electron states onto a localized linear combination of atomic orbitals (LCAO) basis set. The elastic coefficients of single crystals, cij , were determined from a first-principles calculation by applying a set of given homogeneous deformations with a finite value and calculating the resulting stress with respect to optimizing the internal degrees

Table 1 Experimental and geometry optimized lattice parameters and bond lengths of YB4 . Compound Space group Z formula units Density (g/cm3 ) HHH Experimental lattice parameters Lattice parameters

YB4 P4/mbm (127) 4 4.33

7.1035 (Å) Ref. [32] 4.0206 (Å) 90 (◦ ) (0.19, 0.69, 0.5) (0.042, 0.174, 0) (0.586, 0.086, 0) (0, 0, 0.286)

7.111 (Å) Ref. [33] 4.017 (Å) 90 (◦ ) (0.1821, 0.69, 0.5) (0.0389, 0.173, 0) (0.586, 0.086, 0) (0, 0, 0.30)

Geometry optimized lattice parameters a=b Lattice parameters c ˛=ˇ= Y Atomic positions B(1) B(2) B(3)

7.1343 (Å) 4.0480(Å) 90 (◦ ) (0.1804, 0.6804, 0.5) (0.0391, 0.1754, 0) (0.5877, 0.0877, 0) (0, 0, 0.2964)

7.1091 (Å) Ref. [20] 4.028 (Å) 90 (◦ ) (0.1818, 0.69, 0.5) (0.0386, 0.1760, 0) (0.586, 0.0870, 0) (0, 0, 0.30)

Bond lengths B(3)–B(3) B(1)–B(2) B(1)–B(3) B(2)–B(2) B(1)–B(1)

B(1)-Y B(1)-Y B(2)-Y B(3)-Y –

2.7582 (Å) 2.8466 (Å) 2.7620 Å 2.7449 Å –

Atomic positions

a=b c ˛=ˇ= Y at 4 h B(1) at 8i B(2) at 4g B(3) at 4e

1.6480 (Å) inter-octahedral 1.7256 (Å) graphic like B ring 1.7559 (Å) intra-octahedral 1.7691 (Å) inter graphic like B ring 1.8130 (Å) intra-octahedral in ab plane

7.0977 (Å) Ref. [21] 4.0177 (Å) 90 (◦ ) (0.8177,0.3177, 0.5) (0.0386, 0.1761, 0) (0.5870, 0.0870,0) (0, 0, 0.30)

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Fig. 2. Electron density difference maps on (0 0 1) and (−1 5 0) planes that are across the B atoms ((a) and (b)), and on (3 5 0) plane that is across both Y and B atoms (c). The unit is electron·A˚ −3 .

of freedoms, as implemented by Milman and Warren [29] and Ravindran et al. [30]. The criteria for convergence in optimizing atomic internal freedoms were selected as follows: difference on total energy within 5 × 10−6 eV/atom, ionic Hellmann–Feynman forces within 0.002 eV/Å and maximum ionic displacement within 5 × 10−4 Å. The polycrystalline bulk modulus B, shear modulus G, Young’s modulus E were estimated according to the Voigt, Reuss, and Hill approximation [31–33]. 3. Results and discussion 3.1. Lattice parameters and bond lengths YB4 crystallizes in a UB4 -type structure with the space group of P4/mbm (No. 127), which can be described as a quasilayer structure with alternating sheets of Y atoms and B6 octahedra linked together laterally by B2 units. In this structure, Y atoms are located at 4 h (0.19, 0.69, 1/2) Wyckoff positions; B(1) atoms are located at 8i (0.042, 0.174, 0), B(2) atoms are located at 4 g (0.586, 0.086, 0), and B(3) atoms are located at 4e (0, 0, 0.286) Wyckoff positions. Before performing calculations to obtain the elastic constants and electronic structure, we have first optimized the structural parameters of YB4 . In Table 1 we lists the experimental and geometry optimized lattice parameters a and c, atomic positions and B–B and B–Y bond lengths of YB4 . It is obvious that the theoretical lattice parameters are agree well with the experimental [34,35] and the previous theoretical data calculated using VASP

(GGA) code [20] and CASTEP code [21]. In the structure of YB4 , the B6 octahedra are linked together through B(3) atoms forming B(3)2 units. The octahedron is made of B(1) squares lying in the a–b plane bicapped by B(3) atoms. The B(2) atoms lie in the same a–b plane as the B(1) atoms to which they are connected. Thus, YB4 can be reformulated as Y2 B6 B2 . In Y2 B6 B2 , there are five different nearest-neighbor B–B distances. The shortest bond length is B(3)–B(3) = 1.6480 Å, which links the B6 octahedra. The B(1)–B(1) bond length, which constructs the square of B6 octahedron on a–b plane is 1.8130 Å. The bond length between B(1) in the square of the octahedron and the caped B(3) atom is B(1)–B(3) = 1.7559 Å. B(2) atom is tri-coordinated, i.e., it bonds to both B(1) and another B(2) [36]. The bond lengths are B(1)–B(2) = 1.7256 Å and B(2)–B(2) = 1.7691 Å. B(1) and B(2) atoms form a two-dimensional planar consisting of network of B(1) squares and heptagons of –B(2)–B(1)–B(1)–B(2)–B(2)–B(1)–B(1)–B(2)–. Y atoms also form a two dimensional planar network on the (002) plane. The B-Y bond lengths are B(1)–Y = 2.7582 Å, B(1)–Y = 2.8466 Å, B(2)–Y = 2.7620 Å, and B(3)–Y = 2.7449 Å. The significant difference in bond lengths can be considered as indication of anisotropic in chemical bonding, which will be illustrated in later sections. 3.2. Electronic structure and bonding properties In Section 3.1, different B–B bond lengths in the geometry optimized crystal structure of YB4 are given. It is well accepted that these differences are underpinned by the chemical bonding

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tribution around Y is not perfect sphere, i.e., the Y-B bonding is ionic–covalent. Details of atomic bonding characteristics of YB4 can be seen from the total density of states (TDOS) and projected density of states (PDOS), as well as the decomposed distribution of charge density from different covalent bonding peaks in the TDOS. Fig. 3 shows the total and projected electronic density of states of Y, B(1), B(2) and B(3) atoms in YB4 . Two features are obvious in the density states of YB4 . First, there are finite states at Fermi level. Second, there are strong hybridization between B 2s and B 2p orbitals. The finite states at the Fermi level indicates that there is metallic bond in YB4 and it is electrically conductive with contributions from Y 4d states, B(1) 2p, B(2) 2p and B(3) 2p states. Strong hybridization is evident because the B 2s orbitals are extended and overlapped in a large energy range with B 2p orbitals. The density states of both B(1) and B(3) atoms show characters of hybridization between one B 2s and one B 2p states (sp hybridization); while those of B(2) show characters of hybridization between one B 2s and two B 2p states (sp2 hybridization). The lowest lying states from −15.4 to −14.9 eV are quasi-core sp band of B(1) and B(3) atoms and very less contribution from p band of B(2) atom and they are not involved in chemical bonding. The states from −10.9 eV to −7.6 eV are mainly from overlapping of B(1) sp-B(2) sp2 , B(2) sp2 -B(2) sp2 , and B(3) sp-B(3) sp orbitals, as shown in the distribution of charge density in Fig. 4(a) and (b). In the energy range from −7.2 eV to −3.0 eV, beside the overlapping of B(1) sp–B(2) sp2 , B(3) sp–B(3) sp, and B(2) sp2 –B(2) sp2 orbitals, overlappings of B(1) 2px  –B(1) py  (in Fig. 4(c)) and B(1) 2pz –B(3) 2px orbitals (in Fig. 4(d)) are also seen. Here x’ and y’ are used because the edges of B(1)6 octahedron cut by (0 0 1) plane is inclined to x and y axis. The states from −3.2 eV and −2.2 eV are dominated by B(2) sp2 –B(1) 2pz , B(1) 2pz –B(3) 2px , B(3) sp–B(3) sp, and B(2) sp2 –B(2) sp2 orbitals with some contribution from B(2) sp2 –B(1) sp interactions, as shown in Fig. 4(e) and (f), respectively. The charge density near Fermi level shows B(1) 2px  like, B(1) 2py  like, B(1) 2sp like and B(2) 2sp2 like orbitals (in Fig. 4(g)), B(1) 2pz like, B(2) 2pz like, B(3) 2px like orbitals (in Fig. 4(h)). Fig. 3. Total density of states (TDOS) and projected electron density of states (PDOS) of Y, B(1), B(2) and B(3) atoms in YB4 .

anisotropy. To visually illustrate the chemical bonding nature, the electron density difference maps of YB4 are shown first. The electron density difference produces a density difference field which shows the changes in the electron distribution due to the formation of all the bonds in a system. The electron density difference map is appropriate to illustrate the charge redistribution due to chemical bonding in solids. Fig. 2 shows the electron density difference maps on (0 0 1) and (−1 5 0) planes that are across the B atoms (Fig. 2(a) and (b)), and on (3 5 0) plane that is across both Y and B atoms (Fig. 2(c)). In Fig. 2(a), the B6 octahedron is cut by (0 0 1) plane into a square with its axis inclined 58.5◦ and 32.5◦ to the x and y axis, respectively. The B(1)–B(1) bonds are formed by overlapping of perpendicular B-2p orbitals of adjacent B(1) atoms, which is similar to the ␶-bond in B6 octahedron of YB6 [13]. The B(1)–B(2) bond shows characteristics of B(1) spB(2) sp2 overlapping. The B(2)–B(2) bond is formed through the overlapping of B(2) sp2 –B(2) sp2 orbitals. The above bonding properties can also be seen from the electron density difference map on (−150) plane of YB4 shown in Fig. 2(b). In this figure, the B(1)–B(3) bonds in B6 octahedron are formed by overlapping of perpendicular B(1)-2pz and B(3)-2px orbitals. The B(1)–B(2) bonds are formed by overlapping of B(1) sp–B(2) sp2 orbitals and the high density region is close to B(1) atom. The B(3)–B(3) bonding is obviously through the overlapping of hybridized B(3) sp-B(3) sp orbitals in c direction similar to the inter-octahedra bonding in YB6 [13]. The bonding between Y and B is mainly ionic, however, the dis-

3.3. Elastic properties In the last two sections, the anisotropic chemical bonding nature in YB4 has been disclosed. It is well established that the bonding properties can be reflected by the elastic modulus [37]. To disclose the chemical bonding dictated elastic stiffness, the second order elastic constants of single crystals and elastic stiffness of polycrystalline YB4 were calculated. For tetragonal YB4 , there are six independent constants c11 , c12 , c13 , c33 , c44 and c66 . The anisotropic Young’s modulus was calculated from the compliance matrix: Exi =

1 , sii

i = 1, 2, 3

(1)

The polycrystalline bulk modulus B, shear modulus G, Young’s modulus E can be estimated from elastic constants, cij , according to the Voigt, Reuss, and Hill approximation [31–33]. The computed second order elastic constants cij , anisotropic Young’s modulus Ex and Ez , bulk modulus B, shear modulus G, Poisson’s ratio  of YB4 are given in Table 2. Born and Huang [38] systematically studied the lattice mechanical stability and formulated the stability criteria in terms of the elastic constants cij by expanding the internal crystal energy in a range of strains and imposing the convexity of the energy. The Born criterion for a lattice to be mechanically stable is that the elastic energy density must be a positive definite quadratic function of strain. For a tetragonal crystal, the mechanical stability criteria are given by: c 11 > 0,c 33 > 0,c 44 > 0,c 66 > 0,

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Fig. 4. Decomposed charge density on (0 0 1) and (−1 5 0) planes in the energy range from −10.9 eV to −7.6 eV ((a) and (b)), −7.2 eV to −3.0 eV ((c) and (d)), −3.2 eV and −2.2 eV ((e) and (f)), and near Fermi level ((g) and (h)). The unit for all distributions is electron·A˚ −3 . Table 2 Second order elastic constants cij , anisotropic Young’s modulus Ex and Ez , bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio of YB4 . Second order elastic constants (cij ) and anisotropic Young’s modulus of single crystal c33 c44 c66 c11 446 436 110 129 Elastic moduli of polycrystalline YB4 B G 141 189

c12 90 E 339

c13 48

Ex 425

Ez 427

 0.201

Solidity index: s = 3G/4B = 0.559, G/B = 0.746.

(c 11 − c 12 ) > 0, (c 11 + c 33 − 2c 13 ) > 0 (2(c 11 + c 12 ) + c 33 + 4c 13 ) > 0

The stability conditions are clearly satisfied for YB4 , suggesting the stability of this quasi-layered compound under elastic strain perturbations. Analysis of the second order elastic constants listed in Table 2, one can see that the elastic constants c11 and c33 of YB4 , which represent stiffness against principal strains, are relative higher; while the elastic constant c44 and c66 , which represents shear

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Fig. 5. Surface contour of directional dependent Young’s modulus (a) and planar projections on (1 0 0), (0 0 1) and (1 1 0) plane (b) of YB4 .

deformation resistance, are relative lower. Bulk modulus B measures the resistance of a material against a volume change and provides an estimation of response to a hydrostatic pressure. The calculated bulk modulus is close to the experimental determined 185.4 ± 3.5 GPa [19], but slightly larger than the theoretical predicted 182 GPa by Fu et al. [21] and 180.7 GPa by Wa´skowaska et al. [19]. The main reason for the discrepancy in theoretical predicted values may be different DFT codes were used. Shear modulus G describes the resistance of a material against a shape change. The calculated shear modulus G is 141 GPa and the Poisson’s ratio is 0.201. Low shear modulus and high Poisson’s ratio indicate low shear deformation resistance and relatively weak directional bonding of atoms. Pugh’s shear to bulk modulus ratio G/B [39] and the solidity index s = 3G/4B proposed by Gilman [40] are often used as criteria to distinguish ductile or damage tolerant materials from brittle ones. Low G/B and solidity index indicate intrinsic ductile or damage tolerant. The theoretical estimated Pugh’s shear to bulk modulus ratio G/B and the solidity index s = 3G/4B are 0.76 and 0.57, respectively. Although not intrinsic ductile, these values are close to those of damage tolerant ceramics like Ti3 SiC2 [41], Ti3 AlC2 [42], Cr2 AlC [43] and Y2 Si2 O7 [44,45], indicating that YB4 is a possible damage tolerant ceramic. Using the known elastic constants, the intrinsic hardness can also be estimated although it is a highly complex property. Recently, Chen et al. [46] proposed an empirical model, which considered both shear and bulk modulus. In this model, the Vickers hardness is estimated using the following equation:



Hv = 2 k2 G

0.585

−3

(2)

where k is the Pugh’s modulus ratio, namely, k = G/B. This model well predicted the hardness of a large family of materials including metals, intermetallics and ceramics. Using this model, the hardness of YB4 is predicted 22.7 GPa, which is lower but close to the experimental microhardness (Hv ) of 27.40 GPa determined under 1 kg load [15]. 3.4. Elastic anisotropy The significant difference in cij indicates anisotropic in mechanical properties, although small elastic anisotropy was concluded by Fu et al. [21]. In ceramics, microcracks are induced due to the anisotropy of thermal expansion coefficient and elastic constants [47], thus it is useful to calculate the elastic anisotropic factor. To describe the elastic anisotropy of a crystal, the space surface con-

struction of elastic moduli as a function of direction is practical. The directional dependent Young’s modulus (E) of a tetragonal crystal is defined as [48]:

  1 = s11 l14 + l24 + (2s12 + s66 ) l12 l22 + s33 l34 + (2s13 + s44 ) l32 (l − l32 )(3) E where sij is the elastic compliance constant and l1 , l2 , and l3 are the directional cosines of angles with the three principle directions, respectively. The overall surface contour of Young’s modulus of YB4 is shown in Fig. 5(a) and the planar projections on (1 0 0), (0 0 1) and (1 1 0) planes are shown in Fig. 5(b). For different crystallographic planes, A and B represent different crystallographic directions. A represents [1 0 0], and B represents [0 1 0] for (0 0 1) plane; in the case of (1 0 0) plane, they are [0 1 0] and [0 0 1]; while for (1 1 0) plane, they are [1 1¯ 0] and [0 0 1] directions. The anisotropy of Young’s moduli for YB4 is noticeable. The anisotropy on the (1 1 0) plane is much stronger than the other two planes. The minimum Young’s modulus is 40.40◦ inclined to the [1 1¯ 0] direction (C direction as indicated in Fig. 5(b)), which is only 285 GPa. Low Young’s modulus is associated with low fracture energy, thus the low Young’s modulus direction (C direction in Fig. 5(b)) is the weakest direction under uniaxial tension for YB4 . The maximum Young’s modulus of YB4 is 427 GPa, which is 1.5 times larger than the minimum, demonstrating strong anisotropy in Young’s modulus. To get a clear view on the shear behavior of YB4 , it is practical to discuss the directionality of shear modulus. However, it is impossible to construct a spatial surface for shear modulus since there are two determining directions while a surface presents only one radius vector. It can only be presented by the polar plots superposed on a particular shear plane or direction. In this work, we choose three most likely shear planes ((1 0 0), (0 0 1) and (1 1 0)) and shear directions ([1 0 0], [0 0 1] and [1 1 0]) to examine the shear anisotropy of YB4 . The shear modulus is related to the compliances as follow: Gij =

1 4sijij

(4)

where sijij is the compliances under certain coordination system. The shear modulus on every considered plane can be expressed as: G(100)  = G001  =

1 s44 + (s11 + s33 − 2s13 − s44 ) sin2 2 1

s66 + (2s11 − 2s12 − s66 ) sin2 

(5)

(6)

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Fig. 6. Shear modulus of YB4 on different planes (a) and along different orientations (b).

G(110)  =

s44 +

1

s

11

2

+

s12 2

+ s33 − 2s13 +

s66 4



− s44 sin2 

(7) Fig. 7. Linear compressibility of YB4 .

And the shear modulus along every considered direction is as follows: G[100]  =

1 s66 + (s44 − s66 )sin2 

G[100]  =

1 s44

G[110]  =

1 s44 + (2s11 − 2s12 − s44 ) cos2 

(8) (9)

3.5. Lattice dynamics properties (10)

The deduction procedure has been discussed thoroughly in previous reported works [49]. The shear modulus on different planes and along different orientations are plotted in Fig. 6, which shows that YB4 exhibits obvious shear anisotropy. To provide a measurement of the degree of anisotropy, we define a shear anisotropic factor: A=

Gmax Gmin

(11)

For (1 0 0) shear plane, A(100) = s44 /(s11 + s33 − 2s13 ), for (0 0 1) shear plane, A(001) = s66 /(2 (s11 − s12 )), while for (1 1 0) plane, A(110) = s44 /((s11 /2) + (s12 /2) + s33 + (s66 /4) − 2s13 ). For an isotropic crystal, the factor A(100) , A(001) and A(110) must be unity, while any value that unequal to unity is a measure of the degree of elastic anisotropy. The anisotropic factors for (1 0 0), (0 0 1) and (1 1 0) shear planes are 1.78, 1.38 and 1.61, respectively, demonstrating high degree of shear anisotropy. High degree of elastic anisotropy is due to the anisotropy in chemical bonding. The lowest shear moduli, which are associated with the [1 1 0](1 0 0) and [1 1 0](1 1 0) systems, suggest that [1 1 0](1 0 0) and [1 1 0](1 1 0) systems are the most likely slip systems in YB4 due to the presence of weak ␶-bond formed by overlapping of perpendicular B 2p orbitals. This is consistent with the electron density difference maps in Fig. 2. The linear compressibility of a crystal is the relative decrease in length of a line when the crystal is subjected to unit hydrostatic pressure, which varies with direction. For a tetragonal crystal, the directionality of linear compressibility is expressed as: ˇ = (s11 + s12 + s13 ) − (s11 + s12 − s13 − s33 ) l32

surface contour on (1 0 0) and (0 0 1) planes. Fig. 7 shows the compressibility of YB4 . The linear compressibility of a (1.70 × 10−3 GPa) is larger than that of c (1.47 × 10−3 GPa) for YB4 .

(12)

The compressibility of YB4 is rotationally symmetrical about the four fold axis (c axis), thus it is sufficient to describe the characteristic of the directionality of linear compressibility by projecting the

The anisotropic chemical bonding nature is not only reflected by the anisotropic elastic properties but also by the lattice dynamic properties of the crystal, which involve the vibration of the lattice, i.e., the vibration of the chemical bonds. To reveal the anisotropy of the lattice vibration, the lattice dynamics of YB4 were examined. The calculated phonon dispersion relations and the phonon density of states of YB4 are illustrated in Fig. 8. As shown in Fig. 8, there is no imaginary phonon frequency throughout the examined highsymmetry direction in the Brillouin zone, which indicates that the structure of YB4 is dynamical stable against mechanical perturbations. The zone-center (-point) phonons can be classified by the irreducible representation of the point group D4 h . As predicted from the group theory, the symmetries of the optical phonon modes could be expressed as:  = 4A2u + 10Eu + 3B2u + 6Eg + 5A1g + 4B2g + 4B1g + 5A2g + 2A1u + Bu

(13)

where Eg , A1g , B2g , and B1g modes are Raman active, and Eu and A2u modes are infrared (IR) active. Using the phonon eigenvectors, the symmetries of all optical modes at  are identified. The Raman and IR active phonon modes and their symmetries are listed in Table 3. The phonon density of states shown in Fig. 8(b) exhibits separated bands. In order to examine the contribution of vibration of different bonds to the total density of states, the site projected phonon density of states of Y, B(1), B(2) and B(3) atoms have been calculated. Clearly the vibration of Y atoms dominates the lowest frequency interval at around 120 − 240 cm−1 . This is due to the weak Y–Y and Y–B bonding and small values of force constants. The vibration of B(1) and B(2) atoms covers a wide high frequency range from 300 to 1000 cm−1 , indicating both relative low force constant of weak ␶-bond and high force constants of strong B(1)–B(2) and B(2)–B(2) bonds. The vibration of B(3) atoms locates in a more wider

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Fig. 9. Calculated contribution of lattice vibration to isochoric specific heat Cv of YB4 .

high frequency range from 300 to 1150 cm−1 , demonstrating relative low force constant of weak ␶-bond and higher force constants of much stronger B(3)–B(3) bonding within the structure of YB4 . From the lattice dynamic properties, anisotropic chemical bonding in YB4 is further confirmed. From the phonon density of states, basic thermodynamic functions can be calculated within the quasiharmonic approximation. The contribution of lattice vibration to isochoric specific heat Cv is calculated and given in Fig. 9. As the temperature increases, the calculated Cv increases and approaches a constant of 3 R (R is the gas constant) at high temperature. To the authors’ knowledge, no experimental specific heat is available. This set of data is useful in experimental investigation on the thermal properties of YB4 . 4. Conclusion remarks

Fig. 8. Calculated phonon dispersion (a) and phonon density of states (b) of YB4 .

Table 3 Theoretical frequencies (ω in cm−1 ) and irreducible representations (Irrep.) of zonecenter optical modes of YB4 . Raman active modes

IR active modes

ω

Irrep.

ω

Irrep.

152 192 207 451 579 613 624 637 689 706 714 737 760 762 929 942 967 1151

Eg B2g B1g Eg A1g Eg B2g Eg B1g B2g B1g A1g Eg Eg B1g A1g B2g A1g

181 272 298 396 455 545 591 630 698 900 932 963

Eu A2u Eu Eu A2u Eu Eu Eu Eu A2u Eu Eu

The electronic structure, bonding properties, elastic stiffness, hardness and lattice dynamics of YB4 are investigated using firstprinciples calculations based on density functional theory. The Y–B bonding is ionic-covalent, the Y–Y bonding is metallic, the B–B bonding within the B6 octahedron is weak ␶-type covalent and that connecting the octahedral is strong -type covalent. The anisotropic bonding results in anisotropic elastic stiffness. The maximum Young’s modulus (427 GPa) is 1.5 times larger than the minimum (285 GPa). The lowest shear moduli are associated with the [1 1 0](1 0 0) and [1 1 0](1 1 0) systems, suggesting that these systems are the most likely slip systems in YB4 due to the presence of weak ␶-bond formed by overlapping of perpendicular B 2p orbitals. The compressibility also shows anisotropic feature, i.e., the linear compressibility of a (1.70 × 10−3 GPa) is larger than that of c (1.47 × 10−3 GPa) for YB4 . Based on the low second order elastic resistance c44 , c66 and shear to bulk modulus ratio G/B, YB4 is predicted to be a damage tolerant ceramic. The microhardness of YB4 is predicted to be 22.7 GPa, which is close to the experimental measured value. YB4 also has a relatively low Young’s modulus of 339 GPa, which ensures it a good thermal shock resistance. Acknowledgments This work was supported by the National Outstanding Young Scientist Foundation for Y. C. Zhou under Grant No. 59925208, and the Natural Sciences Foundation of China under Grant No. U1435206.

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