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Structural, mechanical and electronic properties study on group 5 transition metals ternary mononitrides from ﬁrst-principles calculations Lei Chen a, b, Junlian Xu d, Meiguang Zhang b, Zengrun Wen c, Zhenyi Jiang a, * a

Shaanxi Key Laboratory for Theoretical Physics Frontiers, Institute of Modern Physics, Northwest University, Xi'an, 710069, China College of Physics and Optoelectronic Technology, Baoji University of Arts and Sciences, Baoji, 721016, China Institute of Photonics and Photon-technology, Northwest University, Xi'an, 710069, China d College of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721016, China b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 June 2019 Received in revised form 9 September 2019 Accepted 10 September 2019 Available online 11 September 2019

Various possible structures of ternary mononitrides of group 5 transition metals (V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N) are systematically explored in 0e300 GPa pressure range through CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) method. Three possible phases with Pmm2, Cmc21, and Fm-3m space group have been uncovered. Their dynamical, mechanical and thermodynamic stabilities are checked through calculations of phonon spectra, elastic constants and formation enthalpy. The elastic anisotropies are fully investigated by the dependence of linear bulk, Young's and shear moduli on the crystal orientations. To study the strength of these three compounds, the compressive, tensile and indentation shear strengths are systematically studied from the ﬁrst-principles calculations. The mononitrides with Pmm2 and Cmc21 structures exhibit much better mechanical property relative to VN, NbN and TaN with traditional B1 (NaCl-like) structure. Our results reveal that the Pmm2 and Cmc21 phases of these three compounds possess the strength values up to 30 GPa under the Vickers indentation simulation calculations, which are comparable to well-known WB3, WB4 and ReB2 reported as potential superhard transition metal borides. Additionally, the electronic structures of these mononitrides are also studied to explore the intrinsic chemical bonding nature, and a novel transition of semiconductor-toconductor from V0.5Ta0.5N, V0.5Nb0.5N to Nb0.5Ta0.5N is exposed for the Fm-3m phases of these compounds. © 2019 Elsevier B.V. All rights reserved.

Keywords: Ternary mononitrides Structure searching First-principles Mechanical property Electronic structure

1. Introduction Owing to excellent properties of high hardness, high-melting points, good thermal and chemical stability, high strength, superior wear and friction properties, transition metal nitrides, borides and carbides are widely studied [1e15]. These properties have made them widely applied in industrial production, such as cutting tools, coating materials, abrasives, etc. [16e22]. Among these compounds, mononitrides have been proved to possess high incompressibility and large hardness values, such as VN, TiN, NbN, TaN, ZrN, HfN, WN, ScN, etc [23]. The N element intercalated in these compounds can form strong covalent interactions with the transition metal elements, which is believed to be the main

* Corresponding author. E-mail address: [email protected] (Z. Jiang). https://doi.org/10.1016/j.jallcom.2019.152246 0925-8388/© 2019 Elsevier B.V. All rights reserved.

driving force of the enhancement of the strength and hardness [24e27]. To improve the mechanical property, some researchers have focused on tuning the VEC (valence electron concentration) which has been reported to be a signiﬁcant indicator in various materials to inﬂuence the crystal structure, thermodynamic and mechanical properties [28e39]. Changing the components and proportion to tune the VEC is believed to be an effective method for designing new potential engineering materials possessing high hardness and toughness [40,41]. H. Kindlund et al. reported the enhanced nanoindentation hardness for the NaCl-structure V1xWxN ternary systems through alloying VN with WN, in which the third element W plays an important role on enhancing the mechanical properties [42]. In these ternary systems, optimized element compositions and VEC promote structural stability and bond strength [43]. Structures and bond strength are two key factors inﬂuencing the

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L. Chen et al. / Journal of Alloys and Compounds 813 (2020) 152246

mechanical properties of TM-LE (transition-metal light-elements, herein, these light elements mainly contain B, C, N, O) compounds, such as borides, carbides, nitrides and oxides [26,35]. Thanks to the high density of valence electron, these compounds generally possess high incompressibility. The introduction of light element can form short covalent interactions between TM and LE atoms which can greatly improve the resistance to the shearing deformation. For TM-LE compounds, exploring the best structures with high bond strength is an important approach to design potential superhard materials. In recent years, the development of high-pressure technology has greatly expanded the boundary of the structure searching space. As an approach of breaking the energy barrier in synthesizing materials, high-pressure is an effective method to form new phases of TM-LE compounds. Some materials with unique properties (high hardness, interesting magnetic or optoelectronic features) have been uncovered, such as novel metal nitrides, oxonitrides as well as the new class of nitride-diazenide compounds, all formed under high-pressure conditions [44]. Traditionally, the ground-state phase has the minimum formation enthalpy and the maximum orbital hybridization which are conducive to high hardness, moreover, the optimal structure determining the spatial distribution of the chemical bonds is also necessary for high hardness. Take the graphite and diamond for example, as the ground-state phase of C element, the graphite has the smaller CeC distance at ambient pressure and stronger sp2 orbital hybridization between C atoms than the sp3 orbital hybridization of diamond. However, due to the layered structure with strong intra-layer bonds and weak (van der Waals) inter-layer interactions, the macroscopic hardness of graphite is much lower than the diamond. Based on the above two conditions for high hardness, ternary or multivariate nitrides, carbides and borides are becoming research hotspots in materials science. Some researchers have focused on adding one or more transition metal elements with comparable atom radiuses and different valence electrons numbers into TM-LE mononitrides or monoborides by tuning the VEC to enhance the interactions between TM elements and light elements. Ternary mononitrides formed with the same group TM elements are studied to clarify the relationship between the component proportion and the mechanical properties or phase stability in some literatures [45,46]. In the natural conditions, nearly all the same group TM elements exist in the form of associated minerals, such as Nb and Ta, Zr and Hf, etc. Thus, exploring the structural and mechanical properties of the same group TM-LE ternary compounds has great signiﬁcance in synthesis of mononitrides or monoborides. Based on the ﬁrst-principles calculations, systematic studies on the structural, mechanical properties and electronic structures of the group VB elements mononitrides (V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N) are carried out in this work. Using CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) method [47,48], the structures of these three compounds are explored under different pressures up to 300 GPa and three different structures with Pmm2, Cmc21, and Fm-3m space group are conﬁrmed quenchable to the ambient conditions. The thermal and dynamical stability are veriﬁed through the formation enthalpy and phonon calculations. The mechanical properties are systematically studied through calculating the elastic constants, ideal compressive and tensile strength. As a reliable method to evaluate the Vickers hardness, Vickers indentation shear strength is also calculated to check the strength. The results show that the Pmm2 and Cmc21 structures exhibit the comparable hardness values in the range of 25 GPae30 GPa for these three compounds. Their electronic structures are calculated to further explore the nature of chemical bonds.

2. Methods and calculations details The CALYPSO method implemented in the CALYPSO code is fully performed in the structures predictions for V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N under the pressure of 0 GPa, 50 GPa, 100 GPa, 200 GPa, 300 GPa with the simulation cell sizes of 1e4 formula units (f.u.). The effectiveness of this structure searching method has been conﬁrmed by the successes in predicting new structures of different systems. It mainly contains several techniques (e.g. particle-swarm optimization algorithm, symmetry constraints on structural generation, bond characterization matrix on elimination of similar structures, partial random structures per generation on enhancing structural diversity, and penalty function, etc.). Here, the population size, which is the structural number in each searching generation is set as 50 and the generation number is set as 30, in the structural searching process. In the output ﬁle, CALYPSO sorts the structures by enthalpy and the optimal structure can be obtained. The Vienna ab initio simulation package (VASP) based on the density functional theory (DFT) [49] is employed to carry out all the ﬁrst-principles calculations. The projector-augmented wave (PAW) method [50] and the generalized gradient approximation (GGA) using the parameterization of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation potential are implemented in the calculations [51]. The kinetic cutoff energy is set up to 640 eV for the accuracy of calculations. The Monkhorst Pack k-point mesh [52] is set as 11116, 667, 666 for the orthorhombic (No. 25, Pmm2 and No. 36, Cmc21), cubic (No. 225 Fm-3m) structures respectively, to ensure all the energy calculations are accurately converged to less than 1meV/atom. In consideration of underestimating the band gap of semiconductors for the traditional DFT, Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [53] is used to obtain the accurate band alignments of Fm-3m structures for these compounds. The PHONOPY code [54] is used to calculate the phonon dispersion through the density functional perturbation theory (DFPT) approach [55,56]. The elastic moduli of these ternary mononitrides, such as bulk modulus, shear modulus, Young's modulus, and Poisson’ ratios are achieved using the Voigt-Reuss-Hill approximation [57]. Their theoretical Vickers hardness values are estimated through the empirical relations proposed by Chen et al. [58]. Their ideal compressive and tensile strength are obtained by continuously compressing or stretching the crystal cell along the direction of the applied strain and relaxing the cell in other directions simultaneously. The indentation shear strength is achieved through calculations in a biaxial stress ﬁeld which includes two stress components (shear szx and compressive szz) and these two stress components obey the relation: szx ¼ sxx tan 4, where the 4is the centerline-to-face angle of the indenter [1]. Chemical bonding is analyzed by performing the crystal orbital Hamilton population (COHP) [59] method implemented in the LOBSTER code [60]. To further study the bonding nature, the electron localization function (ELF) [61] is also calculated. 3. Results and discussion 3.1. Structural properties and stability To verify the potential structures existing in the pressure range of 0e300 GPa, the CALYPSO code is performed with global optimization in the structural searching under 0 GPa, 50 GPa, 100 GPa, 200 GPa and 300 GPa pressures respectively. The results show that these three mononitrides (V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N) adopt the identical structures in the analogous pressure range. As shown in Fig. 1, two orthorhombic structures with Pmm2, Cmc21 space group and a cubic phase with Fm-3m symmetry respectively are predicted. The coordination numbers of the TM elements are six

L. Chen et al. / Journal of Alloys and Compounds 813 (2020) 152246

3

Fig. 1. Crystal structures of V0.5Ta0.5 N: Orthorhombic structures with spaceg group Pmm2 (a), Cmc21 (b) and Cubic structure Fm-3m (c), respectively. The large red, brown spheres and small gray spheres represent V, Ta and N atoms respectively. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.)

for Pmm2, Cmc21 structures and eight for Fm-3m structure. For Pmm2-V0.5Ta0.5N, V and Ta atoms arrange with the parallel permutation and they are in the staggered arrangement for Fm-3mV0.5Ta0.5N. Their lattice constants and atomic coordination are all fully relaxed at 0 GPa pressure and the parameters of equilibrated structure are listed in Table 1. Because of the difference of the atomic radius of the TM elements, there is a little discrepancy for the bond length of TM-N. Here, we take V0.5Ta0.5N for example, for the Pmm2 structure, V and Ta atoms occupy Wyckoff 1c (0.5, 0, 0.035) and 1d (0.5, 0.5, 0.53) sites, and the N atoms occupy Wyckoff 1a (0, 0, 0.707) and 1b (0, 0.5, 0.192) sites. The obtained equilibrium lattice parameters are a ¼ 2.793 Å, b ¼ 2.875 Å and c ¼ 4.929 Å. For the Cmc21 structure, the Wyckoff sites of V, Ta and N atoms are 4a (0.5, 0.124, 0.353), 4a (0.5, 0.371, 0.85), and 8b (0.748, 0.123, 0.684), the equilibrium lattice parameters are a ¼ 5.557 Å, b ¼ 5.730 Å and c ¼ 4.973 Å. For the Fm-3m structure, the Wyckoff sites of V, Ta and N are 4a (0, 0, 0), 4b (0.5, 0.5, 0.5), and 8c (0.25, 0.25, 0.75) respectively, the equilibrium lattice parameters are a ¼ b ¼ c ¼ 5.308 Å. As the atomic radii of Nb and Ta are larger than V atoms, the volume of one Nb0.5Ta0.5N f.u. is larger than that of V0.5Ta0.5N and V0.5Nb0.5N. Due to the largest coordination number of the TM atoms, the TM-N bond length of the Fm-3m structure is the largest one for all these three compounds, implying the weaker interaction between TM and N atoms. Owing to the fact that these three new structures are theoretically predicted in the present work, there are no previous experimental and theoretical results can be used to compare. However, the B1 structure of VN, NbN and TaN has been presented in some literatures, thus, as listed in Table 1,

it can be seen that the calculated lattice parameters of VN, NbN and TaN with B1 structure agree well with the previous results [62,63], which suggests the high reliability of our calculations. The thermodynamic stability is veriﬁed to evaluate the possibility of the synthesis through calculating the formation enthalpy DHf of these three compounds based on the reaction route: DHf ¼ HA0:5 B0:5 N 0:5HA 0:5HB 0:5HN2 , where A and B represent two pure TMs of body-centered cubic phase (bcc-V, Nb and Ta) with the space group of Im-3m [64], and the solid-state a-phase N2 [65] with the space group of Pa-3 are chosen as the reference phase, for bcc-V, Nb and Ta and solid-state a-phase N2 are the ground-state phases at 0 GPa pressure and 0 K conditions. The calculated results of these three compounds for all the predicted structures are listed in Table 1. It is clear that all the formation enthalpy are negative, indicating the thermodynamic stability of these three compounds. For the Pmm2 and Cmc21 structures, the values of DHf are comparable, and they are all greatly smaller than that of the Fm-3m structure, suggesting much more thermodynamic stability. For V0.5Ta0.5N and V0.5Nb0.5N, the Cmc21 structures possess the smaller DHf than that of the Pmm2 structures and it is opposite for Nb0.5Ta0.5N. However, the difference of DHf between Cmc21 and Pmm2 is getting smaller and smaller from V0.5Ta0.5N, V0.5Nb0.5N to Nb0.5Ta0.5N with the difference of the two TM atomic radius getting smaller and smaller, indicating the TM atoms tends to occupy the same Wyckoff sites randomly. Additionally, the DHf of VN, NbN and TaN with B1 structure are also calculated and the higher values of DHf indicate they are energetically metastable at 0 K and 0 pressure conditions. In

Table 1 Calculated the crystal lattices, bond length (measured in Å), volume (measured in Å3/f.u.), formation enthalpy (DHf, measured in eV/f.u.) and obtained B0 (measured in GPa). Compounds

Structure

a

b

c

VeN bond

V0.5Ta0.5N

4.929 4.973

2.882 5.566

4.955 4.986

2.136 2.146 2.298 2.144 2.142 2.302

2.967 5.802

5.136 5.135

NbN

P-6m2 B1

TaN

P-6m2 B1

2.793 5.730 5.308 2.787 5.758 5.317 2.900 5.929 5.508 4.124 4.1231 2.740 4.453 4.4291 2.976 4.420 4.4272 2.952

2.875 5.557

VN

Pmm2 Cmc21 Fm-3m Pmm2 Cmc21 Fm-3m Pmm2 Cmc21 Fm-3m B1

V0.5Nb0.5N

Nb0.5Ta0.5N

P-6m2 1 [62]. 2 [63].

NbeN bond

2.197 2.184 2.302 2.254 2.238 2.385

TaeN bond

Volume

DHf

B0

2.175 2.174 2.298

19.785 19.794 18.69 19.895 19.970 18.793 22.095 22.083 20.889 17.535

2.301 2.370 1.048 2.154 2.182 0.947 2.260 2.256 0.823 1.934

314 324 313 304 306 300 319 320 304

17.197 22.068

2.332 1.738

2.210

22.242 21.585

2.149 1.706

2.238

21.885

2.311

2.216 2.235 2.385

2.062 2.646

2.062 2.226

2.899

2.900

2.248

4

L. Chen et al. / Journal of Alloys and Compounds 813 (2020) 152246

consideration of the facts that VN, NbN and TaN with B1 sturcture are all dynamically unstable at 0 K [66,67] and owing to the anharmonic vibrations, they are dynamically stabilized with the temperature increasing above 250 K [68], the structures searching through CALYPSO method is carried out to verify the ground-state of these binary mononitrides. Consistent with previous results reported by R. Weinberger et al. [69] and Meenaatci et al. [70], our results showed that the WC-type structure with the space group of P-6m2 symmetry of these mononitrides is the most energetically favorable conﬁguration. Thus, the DHf of the WC-type structure deﬁned as the ground-state phase for VN, NbN and TaN are also calculated and the results are listed in Table 1. Through the equation DH ¼ HA0:5 B0:5 N 0:5HAN 0:5HBN , where AN and BN represent binary mononitrides of V, Nb and Ta with the WC-type structure, we calculate the DH of these three compounds with the predicted structures. For Pmm2 and Cmc21 structures, the values of DH are in the range of 0.1 eV/f.u. ~0.1 eV/f.u. suggesting these ternary mononitrides are energetically comparable with the binary ground-state phases. We take V0.5Ta0.5N as representative compounds for discussion, and a series of lattice parameters of the crystal cells are set to calculate their total energy and the relations of energy and the volume per f.u. are plotted in Fig. 2 (a), from which it can be seen the obvious higher total energy for the Fm-3m structure than the Pmm2 and Cmc21 structures at the equilibrium volume, and its equilibrium volume is much smaller than the other two, indicating the Fm-3m structure has higher density and is energetically metastabile at ambient pressure. The E-V curves of the Pmm2 and Cmc21 structures nearly coincide in the whole volume range, indicating the high structural accordance. Meanwhile, from the data of E-V curve, the bulk moduli at 0 pressure can be obtained through ﬁtting the third-order Birch-Murnaghan EOS [71], and the results are listed in Table 1. Using the fomula H ¼ Eþ PV the enthalpy differences relative to the Cmc21 structure as a function of pressure in 0e300 GPa pressure range are plotted in Fig. 2 (b), from which, it is clear that the Cmc21 structure is the most energetically favorable in the pressure range of 0 GPae220.7 GPa and the Fm-3m structure becomes the most stable structure with the pressure increased. Using the linear response method with density functional perturbation theory (DFPT) [55,56], as implemented in the PHONOPY code, the dynamical stability is examined at 0 GPa pressure through the calculations of phonon spectra. We have relaxed all the atomic positions for the related structures by setting the stricter break condition in the ionic relaxation loop before the phonon

calculations. As shown in Fig. 3 (a, b, c), there are no imaginary modes in the whole Brillouin zone for V0.5Ta0.5N with these three structures, conﬁrming that they are all dynamically stable. The phonon spectra for V0.5Nb0.5N and Nb0.5Ta0.5N are also calculated and the results are accord with V0.5Ta0.5N. From the phonon projected density of states, it can be seen that the lower frequency part is mainly dominated by the lattice dynamics of TM and the higher frequency by the light N atoms. For these three mononitrides with three predicted structures, the values of calculated elastic constants are listed in Table 2 and the mechanical stability is examined through Born criteria [57] as followings: For Pmm2 and Cmc21 structures:

C11 >0;C22 >0;C33 >0;C44 >0;C55 >0;C66 >0; ½C11 þ C22 þ C33 þ 2ðC12 þ C13 þ C23 Þ>0; ðC11 þ C12 2C 12 Þ>0;ðC11 þ C33 2C13 Þ>0;ðC22 þ C33 2C23 Þ>0: (1) For Fm-3m structure:

C11 > 0; C44 > 0; C11 > jC12 j; C11 þ 2C12 > 0:

(2)

It can be concluded that the calculated elastic constants can well satisfy the above conditions for these three mononitrides, suggesting they are all mechanically stable at ambient pressure. 3.2. Elastic properties Besides used for checking the mechanical stability, the elastic property is important for the potential superhard materials, because they can provide the information concerning fundamental solid state properties, such as bonding characteristics, interatomic potentials, speciﬁc heat, equation of states, etc [72]. As the results of the response of external forces, the elastic constants are closely related to bulk modulus, Young's modulus, shear modulus, and Poisson's ratio, and thus play an important role in determining the mechanical strength of materials. The bulk modulus B, Young's modulus E and shear modulus G are achieved by the Voigt-ReussHill approximations [57]. As shown in Table 2, it can be seen that all these three ternary mononitrides possess the larger values of bulk, Young's and shear moduli than the traditional binary mononitrides with B1 structure, implying the enhancement of the mechanical property and the potential superhard property. Additionally, the calculated bulk moduli are all comparable to B0

Fig. 2. Total energy of V0.5Ta0.5 N with three structures as a function of volume of formula unit (a), enthalpy difference of V0.5Ta0.5 N with Pmm2 and Fm-3m structures relative to the Cmc21 structure as a function of pressure (b).

L. Chen et al. / Journal of Alloys and Compounds 813 (2020) 152246

5

Fig. 3. Phonon dispersion curves at ambient pressure for V0.5Ta0.5 N with Pmm2 (a), Cmc21 (b) and Fm-3m (c) structures respectively.

Table 2 Calculated elastic constants (Cij, GPa), bulk modulus (B, GPa), shear modulus (G, GPa), Young's modulus (E, GPa), Poisson's ratio n, G/B ratio and the Vickers hardness (Hv, GPa). Symmetry

Source

C11

C22

C33

C44

C55

C66

C12

C13

C23

B

G

E

n

G/B

Hv

229 234

185 138

591 568

224 220

226 218

139 134

197 201

600 601

597 602

221 223

222 222

138 140

194 195

B1

320 329 320 321 316 309 324 325 309 314 313 320 304 309 328 332

236 233 213 226 217 198 228 227 205 152 153 163 124 137 114 130

570 566 524 549 529 489 553 553 504 393 393 418 327 358 306 347

0.20 0.21 0.23 0.21 0.22 0.24 0.21 0.22 0.23 0.29 0.29 0.28 0.32 0.31 0.34 0.33

0.74 0.71 0.67 0.71 0.69 0.64 0.70 0.70 0.66 0.48 0.49 0.51 0.41 0.44 0.35 0.39

31.3 29.5 25.6 28.7 26.9 23.2 28.6 28.5 24.9 13.3

TaN

119 136 12.8 125 128 8.8 142 141 0.5 168 178 162 150 179 166 165

126 197

582 566

210 231 120 194 178 106 205 202 110 119 126 126 80 107 65 82

225 198

B1

823 603 934 804 795 909 780 779 928 605 585 636 610 568 651 667

604 605

NbN

This work This work This work This work This work This work This work This work This work This work Exp1 Theory2 This work Theory3 This work Theory3

605 827

VN

Pmm2 Ccm21 Fm-3m Pmm2 Ccm21 Fm-3m Pmm2 Ccm21 Fm-3m B1

V0.5Ta0.5N

V0.5Nb0.5N

Nb0.5Ta0.5N

10.6 6.2

1 [77]. 2 [78]. 3 [62].

obtained through ﬁtting the third-order Birch-Murnaghan equation of state (EOS), suggesting the high accuracy of our calculations. The moduli of Pmm2 and Cmc21 structures for each of these three mononitrides are all comparable and for the Fm-3m structure, there is obvious reduction, implying the weaker mechanical property than that of the other two structures. The ratio of shear modulus G to bulk modulus B is used to assess the ductile VS. brittle behavior of the material through comparing to the critical value (0.57) which divides ductile (G/B < 0.57) from brittle (G/B > 0.57) property [73]. The results showed these three compounds all exhibit brittle property. The Vickers hardness Hv are also estimated by Chen's empirical model [58] as following:

length of an applied line when the crystal is subjected to unit hydrostatic pressure [74]. The variation of linear bulk, Young's and shear moduli along a random direction for orthorhombic symmetry (Pmm2 and Cmc21 structures) are given by: 2 2 2 B1 l ¼ ðs11 þ s12 þ s13 Þl1 þ ðs12 þ s22 þ s23 Þl2 þ ðs13 þ s23 þ s33 Þl3

(4) E1 ¼ s11 l41 þ s22 l42 þ s33 l43 þ ð2s12 þ s66 Þl21 l22 þ ð2s23 þ s44 Þl22 l23 þ ð2s13 þ s55 Þl21 l23 (5)

2

Hv ¼ 2ðk GÞ

0:585

3;

(3)

where k represents the ratio of G to B. As shown in Table 2, there is a great increase from TMNs with B1 structure to these ternary mononitrides with three predicted structures. Accord with the moduli values, the Pmm2 and Cmc21 structures possess the comparable values for each of the compounds which are larger than that of the Fm-3m structures, implying the stronger interactions between TM and N elements in the ground-state structures. The hardness value of 31.3 GPa indicating Pmm2-V0.5Ta0.5N is a potential superhard material for engineering applications. For an engineering material, the elastic anisotropy is an important factor greatly inﬂuencing its applications. In this work, the elastic anisotropy is studied in linear bulk, Young's and shear moduli. Here, the linear bulk modulus (Bl) is deﬁned as the inverse of the linear compressibility, which is the relative decrease in

For cubic symmetry (Fm-3m structure):

B1 l ¼ s11 þ 2s12

(6)

E1 ¼ s11 ð2s11 2s12 s44 Þðl21 l22 þ l22 l23 þ l21 l23 Þ

(7)

where l1, l2, l3 are the direction cosines on the compressive stress direction, and sij (¼C1 ij ) are independent elastic compliance constants determined from the calculated elastic constants Cij. The directional dependence of the linear bulk moduli for V0.5Ta0.5N with three predicted structures and the according slices of xy, yz, xz planes are plotted in Fig. 4 (a ~ f). The cubic phase of V0.5Ta0.5N with Fm-3m symmetry exhibits the isotropic linear bulk modulus with the value of 1191 GPa, implying the high incompressibility in all the directions. For the other two orthorhombic structures, it can be

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L. Chen et al. / Journal of Alloys and Compounds 813 (2020) 152246

Fig. 4. Three-dimensional and two-dimensional surface representations of linear bulk modulus for the three structures of V0.5Ta0.5 N: Pmm2 (a, d), Cmc21 (b, e), and Fm-3m (c, f).

seen a spheroid due to the elastic anisotropy. The difference of orientation of these two spheroids is caused by choosing the direction of coordinate axis differently. The maximum values of linear bulk moduli are 1137 GPa and 1185 GPa along x and y directions for Pmm2 and Cmc21 structures respectively, meanwhile, the minimum values are 878 GPa and 905 GPa along z directions. The values of Bl(max)/Bl(min) are 1.29 and 1.31 for Pmm2 and Cmc21 structures respectively, exhibiting the comparable anisotropy of incompressibility. The directional dependence of the Young's moduli for V0.5Ta0.5N are plotted in Fig. 5(a, b, c) and the slices in xy, yz, xz planes are shown in Fig. 5 (d, e, f), from which, it can be seen that the anisotropy of Young's moduli for Pmm2 and Cmc21 structures is accord with the results of linear bulk modulus. The maximum (and minimum) values are 784 (530) GPa and 779 (519) GPa for Pmm2 and Cmc21 structures respectively, and the values of E(max)/E(min) are 1.48 and 1.5, indicating the comparable anisotropy of Young's modulus. For the Fm-3m structure, it exhibits great difference from the results of directional linear bulk modulus, that the value of E(max)/E(min) is 2.45 indicating much more anisotropy than the above two structures. To further study the elastic anisotropy, the directional shear moduli are also calculated and three-dimensional surface representations of minimum shear moduli in random direction are plotted in Fig. 6 (a, b, c). The slices of the minimum and maximum shear moduli in xy, yz, xz planes are shown in Fig. 6 (d, e, f). For Pmm2 and Cmc21 structures, the minimum shear moduli in random direction exhibit almost a perfect sphere with a slight anisotropy and the values of G(max)/G(min) are 1.15 and 1.05. Accord with the results of Young's modulus anisotropy, the value of G(max)/ G(min) is 1.83 for the Fm-3m structure, indicating a large anisotropic character. Accordingly, the directional linear bulk moduli, Young's moduli and shear moduli of V0.5Nb0.5N and Nb0.5Ta0.5N are also calculated but not shown here due to the consistent results with V0.5Ta0.5N.

3.3. Ideal strength calculations Traditionally, the stress-strain relations calculated from ﬁrsprinciples can provide an accurate description of deformation and strength of materials [75]. Through setting the continuously increasing strains in the assigned direction and relaxing the crystal cell and inner atoms in the other directions, the ideal compressive, tensile and indentation shear strength are calculated to further study the mechanical properties of these three compounds with predicted structures. The results of V0.5Ta0.5N are plotted in Fig. 7, from which one can see these three structures display different anisotropy of compressive strength. The smallest compressive strengths are 72.3 GPa along [101] direction, 78.4 GPa along [111] direction and 29.8 GPa along [110] direction for Pmm2, Cmc21 and Fm-3m structures respectively, suggesting the two orthorhombic structures possess the comparable ideal compressive strength, which is accord with the elastic property. The anisotropy ratios of ideal compressive strength of V0.5Ta0.5N with these three structures are 3.7, 3.2 and 8.1 for Pmm2, Cmc21 and Fm-3m structures respectively. The cubic structure exhibits the largest anisotropy of compressive strength and the smallest strength values. As shown in Fig. 7 (d, e, f), the smallest values of the ideal tensile strength are 31.1 GPa ([101] direction), 33.2 GPa ([111] direction) and 10.5 GPa ([111] direction), and the anisotropy ratios are 3.2, 2.4 and 8.7 for Pmm2, Cmc21 and Fm-3m structures respectively. To accurate describe the plastic deformation under Vickers indenter, the Vickers indentation shear strengths are obtained through calculations in a biaxial stress ﬁeld which includes two stress components (shear szx and compressive szz) and these two stress components obey the relation: szx ¼ sxx tan 4, where the 4is the centerline-toface angle of the indenter. The calculated results are plotted in Fig. 7 (g, h, i). The weakest strength with the values of 24.5 GPa and 29.4 GPa are found in (011)[011] direction for Pmm2, and Cmc21 and

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Fig. 5. Three-dimensional and two-dimensional surface representations of Young's modulus for the three structures of V0.5Ta0.5 N: Pmm2 (a, d), Cmc21 (b, e), and Fm-3m (c, f).

Fig. 6. Three-dimensional surface representations of minimum shear modulus in random directions for the three structures of V0.5Ta0.5 N: Pmm2 (a), Cmc21 (b), and Fm-3m (c), and the slices of the minimum and maximum shear moduli in random directions: Pmm2 (d), Cmc21 (e), and Fm-3m (f).

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Fig. 7. Calculated compressive, tensile and Vickers indentation shear stress-strain relations for V0.5Ta0.5 N: Pmm2 (a, d, g), Cmc21 (b, e, h), and Fm-3m (c, f, i).

8.6 GPa in (110)[001] direction for Fm-3m structures, suggesting V0.5Ta0.5N with these two orthorhombic structures are potential superhard materials and the value of 29.4 GPa is comparable to the reported ReB2 (28 GPa), OsB2 (<20 GPa) [76]. The calculated ideal strength for V0.5Nb0.5N and Nb0.5Ta0.5N are listed in Table 3. Because of the shortest TM-N bonding length which implying the strongest interaction between TM and N in these compounds, Cmc21V0.5Ta0.5N possesses the highest Vickers indentation shear strength. 3.4. Electronic structure and chemical bonds To explore the origin of the enhanced strength in fundamental level, the electronic structure and chemical bonds are studied using V0.5Ta0.5N as a representative material. The calculated total (t-DOS) DOS and projected DOS (p-DOS) and the crystal orbital Hamilton population (COHP) [56] for V0.5Ta0.5N as well as the projected band structures with these three predicted structures are plotted in

Figs. 8 and 9 respectively. The dash lines represent the Fermi level (EF). From the t-DOS curves and the band structures, one can see the Pmm2 and Cmc21 structures exhibit a conductive property conﬁrmed by the ﬁnite electronic DOS and the absence of the band gap at the EF, and the DOS are mainly supplied by N-p and TMd electrons. The N-s electrons which are mostly localized far away from EF play a dispensable role in bonding between TM and N atoms. For the Pmm2 and Cmc21 structures, it can be seen the obvious overlap of the peaks of N-p and TM-d orbitals from 8 eV to 3 eV which is in accord with the projected band structures, implying there is strong hybridization viewed as covalent bonds nature between TM and N atoms. On the contrary, for the Fm-3m structure, it can be seen in the energy range of 9 eV ~ 3 eV, the DOS are mainly dominated by N-p states, and from 3 eV to the EF, the TM-d states become the main suppliers. The allopatric distribution of N-p and TM-d is the origin of the weaker mechanical property relative to the two orthorhombic structures. It is notable

Table 3 Calculated ideal strength (GPa). Compounds

V0.5Ta0.5N V0.5Nb0.5N Nb0.5Ta0.5N

Pmm2

Cmc21

Fm-3m

Compressive

Tensile

Shear

Compressive

Tensile

Shear

Compressive

Tensile

Shear

72.3 63.7 77.8

31.1 31.4 32.1

24.5 21.5 26.1

78.4 59.8 70.7

33.2 30.5 32.8

29.4 19.1 23.6

29.8 23.9 25.9

10.5 7.8 6.9

8.6 7.0 7.3

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Fig. 8. Calculated total, projected DOS and COHP for V0.5Ta0.5 N: Pmm2 (a), Cmc21 (b), and Fm-3m (c), vertical dash lines represent the Fermi level (EF).

Fig. 9. Calculated projected band structures for V0.5Ta0.5 N: Pmm2 (a), Cmc21 (b), and Fm-3m (c), the red, blue, and cyan bubbles represent V, Ta and N respectively, the horizon dash lines represent the Fermi level (EF). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.)

that “pseudogap” nearby EF are present for Pmm2 and Cmc21 structures, suggesting the existence of the strong hybridizationeffect between TM and N atoms, and the EF separate the bonding and antibonding states which are clearly veriﬁed by the curves of eCOHP. In the whole occupied regions, these ternary structures almost all present bonding states for TM-N combinations. From Fig. 9 (c), it can be seen the obvious band gap with the value of 0.178 eV is present, indicating the semiconductive property for the Fm-3m structure. In band structures calculations, because of the fact that PBE method traditionally underestimates the band gap, Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [53] is used to calculate the band alignments of Fm-3m structure for these three compounds to evaluate the band gap accurately. As shown in Fig. 10, from V0.5Ta0.5N, V0.5Nb0.5N to Nb0.5Ta0.5N the values of band gap are from 0.7531 eV, 0.1767 eVe0 eV, implying the increased metallicity. It is well known that, traditionally, the TM mononitrides exhibit the conductive nature. To compare with V0.5Ta0.5N,

V0.5Nb0.5N and Nb0.5Ta0.5N, we also calculate the band structures of VN, NbN and TaN with the same Fm-3m structure. The results are shown in Fig. 11, it can be seen they all exhibit conductive property. It is to be noted that, on the contrast to NbN and TaN, VN with this predicted cubic structure is dynamically metastable, because it has imaginary frequencies in the calculated phonon spectra as shown in Fig. 12. However, V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N are all dynamically stable, and the reason of this phenomenon is mainly explained by the shift of the conductive band towards the lower energy regions leading to opening the band gap which enhances the electronic stability. The electronic localization function (ELF) implying the localizing degree of electrons in the certain regions [61] is also calculated to further study the nature of chemical bonds of the TM-N combinations. The value of ELF (0e1) can provide useful information of the chemical bonding property by deﬁning the bonding nature to iconicity, covalency or metallicity in qualitative terms. The localizing degree of electrons is gradually

Fig. 10. Calculated band structures by HSE06 method for V0.5Ta0.5N (a), V0.5Nb0.5N (b) and Ta0.5Nb0.5N (c) with Fm-3m structure, the horizon dash lines represent the Fermi level (EF).

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Fig. 11. Calculated band structures by PBE method for VN (a), NbN (b) and TaN (c) with the predicted cubic structure, the horizon dash lines represent the Fermi level (EF).

Fig. 12. Phonon dispersion curves for VN (a), NbN (b) and TaN (c) with the predicted Fm-3m structure.

Fig. 13. Contours of ELF for V0.5Ta0.5N: (100) plane for Pmm2 (a), (100) plane for Cmc21 (b), and (011) plane for Fm-3m (c).

increasing from the value of 0e1 in the related regions. The covalent bonds between atoms present high ELF value up to 1, on the opposite, the ionic bonds present low ELF value down to 0. The results of calculated ELF on the assigned planes are shown in Fig. 13, from which, it can be seen that the deﬁnite ELF value between TM and N atoms, suggesting the covalent interactions which can be considered as the main origin of the enhancement of the mechanical property. 4. Conclusions In summary, by the CALYPSO method, three structures for group VB ternary mononitrides (V0.5Ta0.5N, V0.5Nb0.5N and Nb0.5Ta0.5N) are predicted successfully, and the mechanical properties and electronic structures are systematically studied through ﬁrstprinciples calculations. It can be concluded that the Pmm2 and Cmc21 structures are competitive in the process of synthesis due to the close formation enthalpy. Because of the higher formation enthalpy, the Fm-3m structure is deﬁned as metastable structure which is energetically stable in the pressure range of above

220.7 GPa. The calculations of elastic property reveal the Pmm2 and Cmc21 structures for these three mononitrides possess the similar mechanical properties, and V0.5Ta0.5N with the Pmm2 structure possesses the highest hardness and the smallest elastic anisotropy. In accord with the elastic properties, the further ideal strength calculations reveal the obvious improved strength relative to VN, NbN and TaN with B1 structure. From the calculations of electronic structures, it can be seen the covalent interactions between TM and N atoms which are considered as the origin of the high hardness and strength. Meanwhile, it is followed that the semiconductive property is an unnecessary condition to the high hardness for this group TM nitrides. Acknowledgments This work was supported by the Natural Science Foundation of China under Grants (No.51572219, 51872227, 11447030), the Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (Grant No. 2017KJXX-53) and Natural Science Basic Research plan in Shaanxi Province of China (Grant No.

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