Theoretical study on structural, mechanical and electronic properties of ternary mononitride Ti0.5W0.5N from first-principles calculations

Theoretical study on structural, mechanical and electronic properties of ternary mononitride Ti0.5W0.5N from first-principles calculations

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Journal Pre-proof Theoretical study on structural, mechanical and electronic properties of ternary mononitride Ti0.5W0.5N from first-principles calculations Lei Chen, Junlian Xu, Meiguang Zhang, Yun Zhang PII:




MAC 122476

To appear in:

Materials Chemistry and Physics

Received Date: 27 August 2019 Revised Date:

19 November 2019

Accepted Date: 21 November 2019

Please cite this article as: L. Chen, J. Xu, M. Zhang, Y. Zhang, Theoretical study on structural, mechanical and electronic properties of ternary mononitride Ti0.5W0.5N from firstprinciples calculations, Materials Chemistry and Physics (2019), doi: j.matchemphys.2019.122476. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Theoretical study on structural, mechanical and electronic properties of ternary mononitride Ti0.5W0.5N from first-principles calculations

Lei Chen1, 3*, Junlian Xu2, Meiguang Zhang1, Yun Zhang1 1

College of Physics and Optoelectronic Technology, Baoji University of Arts and Sciences, Baoji 721016, China

2 3

College of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721016, China

Advanced Titanium Alloys and Functional Coatings Cooperative Innovation Center, Baoji University of Arts and Sciences, Baoji 721016, China

Corresponding author: Dr. Lei Chen. E-mail: [email protected]

ABSTRACT In the present work, systematic structure searching is carried out using CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) method combined with first-principles calculations, in the pressure range of 0~300 GPa for ternary mononitride Ti0.5W0.5N. A new possible structure with Cmc21 space group has been uncovered to be the ground-state phase which is much more energetically favorable than the well known B1 structure at 0 GPa and 0 K conditions. There is an excellent enhancement of mechanical properties for the Cmc21-Ti0.5W0.5N relative to B1-Ti0.5W0.5N and its parent compounds B1-TiN and NbO-type WN. The dynamical, mechanical and thermodynamic stabilities are verified through calculating the phonon spectra, elastic constants and formation enthalpy. The elastic properties are fully investigated and the results show that its Vickers hardness value is up to 34 GPa evaluated from bulk modulus (335 GPa) and shear modulus (255 GPa), wihch is much more than that of B1-TiN and NbO-type WN. Meanwhile, the elastic anisotropy is also studied through the dependence of Young’s, linear bulk and shear moduli along the crystal orientations. According with the elastic properties, the calculated ideal strengths also confirm that there is a great improvement for the Cmc21-Ti0.5W0.5N relative to that of the above binary mononitrides. Further analysis of the electronic density of states and the chemical bonding nature reveals that the component of Ti plays an important role through tuning the valence-electron concentration (VEC) to enhance the stability and the mechanical properties. Keywords: Ternary mononitride; First-principles; Elastic properties; Ideal strengths; Electronic structures


1. Introduction Due to outstanding properties of high melting point, corrosion resistance, wear resistance, high strength and hardness, etc, hybrid compounds of transition metals and light elements (TM-LE) are attractive for wide applications in industrial production, such as abrasion resistant coatings, cutting tools, abrasives etc [1-3]. Plenty of theoretical and experimental efforts have been devoted to these compounds which are also considered as a range of important candidates of superhard materials [4], owing to the dramatic enhancement of mechanical properties relative to the pure transition metals. The covalent bonding nature between TM and LE is confirmed to be a key factor of the hardening effect [5-7]. Among these TM-LE compounds, a series of TM mononitrides such as WN, ScN, VN, TiN, TaN, ZrN, HfN, etc have been confirmed to have ultra incompressibility and high hardness[8-12].The introduction of the N element can form bonds with the TM elements and lots of further theoretical investigations have proved the bonding nature can be viewed as a type of mixed bonds containing metallic, ionic and covalent composition [13-16].The strong covalent σ bonding between TM elements and the LE (B, C, N, O) results from the hybridization of the TM-d and the LE-p orbitals, which can greatly improve the resistance against the shear deformation. Moreover, thanks to the high density of valence electron, the TM mononitrides almost all have ultra incompressibility, which is another necessary condition to achieve high hardness. However, many experimental and theoretical researches on this series of compounds have exhibited that most of their Vickers hardness values are limited below 30 GPa, such as TiN (22.5 GPa), GaN (18.1 GPa), VN (14.9 GPa), NbN (13.6 GPa) [9, 17], etc. The hardness of hexagonal δ-MoN with P63mc structure was reported to be about 30 GPa, which is the hardest mononitride with superconducting property [18]. Among these TM mononitrides, molybdenum and tungsten nitrides have been reported to have the largest hardness values [19, 20]. The metastable B1-WN has been successfully synthesized using W3N4 as precursor at high temperature and high pressure conditions and its hardness is about 29 GPa obtained under an applied indenter load of 0.49 N [21]. The ground-state phase of WN has been confirmed to be the NbO-type structure with the theoretical hardness of 26 GPa [22] and it is considered as the most stable phase in the pressure range of 0 ~ 5 GPa [23, 24]. The hardness of NiAs-type structure (P63/mmc, No. 194) has been predicted to be 24.7 GPa [23, 25] which is comparable to the NbO-type WN. There are various physical and chemical deposition methods for fabricating B1-TiN [26] and the hardness of B1-TiN has been reported to be below 25 GPa [9]. An effective approach to enhance or change the mechanical properties of TM mononitrides is introducing additional elements into them to tune the valence-electron concentration (VEC), which is viewed as the valence electrons number [27]. Additionally, VEC is suggested as a key factor to influence the structural and mechanical 2

properties of many ternary or binary compounds [28], and this parameter has been reported to have significant effect on the phase stability of high entropy alloys [29]. Tuning the VEC to 8.4 per formula unit has been verified to achieve maximum mechanical strength and structural stability for ternary TiC1-xNx systems [30, 31]. Moreover, Zhao et al have investigated mechanical properties of VSiN coating with various Si content [32]. They found the intercalated third element Si can greatly improve the hardness values of VSiN systems from 19.8 GPa up to 27 GPa. G. Soto et al have studied on the role of VEC in influencing the properties of YBxN1-x, YCxN1-x, and YNxO1-x , they have confirmed the bulk modulus is an increasing function of VEC up to 4.1 per atom [33]. D. G. Sangiovanni et al have reported their systematic study on cubic Ti0.5W0.5N systems with ordered and disordered atomic arrangement [34, 35]. In their ternary systems, the VEC is 10 per formula unit and the ductility is considerably enhanced, while the hardness is comparable to B1-TiN, which is attributed to the high VEC. Furthermore, the value of VEC also seems to be a significant parameter, if one wants to design a superhard material using LEs, due to the fact that many superhard LE compounds possess the optimal VEC (8 per two atoms), such as diamond, cubic-BN, cubic-BC2N etc [36]. For these compounds, besides the diamond-like structure they have, the optimal VEC plays an important role to achieve superhard properties. All the valence electrons contribute to the strong covalent bonds between LEs. However, for the TM-LE compounds, the covalent component in the bonds between TMs and LEs mainly stems from the hybridization of d-orbitals (TM) and the p-orbitals (LE), while the outermost s-orbitals supply little contributions to the chemical bonding. Thus, the optimal VEC of these TM-LE compounds is greatly different from that of LE compounds. In this paper, we focus on the Ti0.5W0.5N compound, because the third intercalated element Ti decreases its VEC to 10 per two atoms and especially, this value is changed to 8 (the optimal value) per two atoms when the outermost s-electrons are not taken into consideration. In this paper, we first carry out systematic study on the structural, mechanical and electronic properties of Ti0.5W0.5N compound based on the first-principles DFT calculations. Through the Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) method [37, 38], the structures searching is performed in the pressure range of 0 ~ 300 GPa and two structures with Cmc21 and Fm-3m space group respectively have been uncovered. Their thermal, mechanical and dynamical stability are checked through the formation enthalpy, elastic constants and phonon spectra calculations. The orthorhombic sturcture (Cmc21) is the ground-state phase because of the lowest formation enthalpy at 0 K and 0 GPa conditions relative to the B1 and the predicted Fm-3m structures. The calculated mechanical properties showed the Vickers hardness of the Cmc21 structure is up to 34.1 GPa and this


value exceeds all the tungsten and titanium mononitrides. The electronic structures are also calculated to explore the hardening mechanisms.

2. Methods and calculation details Under the pressure of 0 ~ 300 GPa, the CALYPSO method [37, 38] implemented in the CALYPSO code is performed in the structure searching process using the simulated cell sizes of 1-4 f.u. The CALYPSO method 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.) and it contains four main steps: (i) generating random structures with the constraint of symmetry; (ii) local structural optimization; (iii) post-processing for the identification of unique local minima by bond characterization matrix; (iv) generating new structures by particle swarm optimization for iterations. After sufficient cycles, CALYPSO code sorts the structures by enthalpy and the optimal structure can be obtained. All of the first-principles calculations are performed using the VASP code (Vienna ab initio simulation package) based on the DFT [39]. The PAW (projector-augmented wave) method [40] is adopted in the potentials and the exchange-correlation is used as GGA-PBE (the generalized gradient approximation of Perdew-Burke-Ernzerhof) [41]. The 3p63d24s2, 5p65d46s2 and 2s22p3 electrons are explicitly considered as valence for Ti, W and N respectively. The cutoff energy is set as 640 eV in the plane-wave expansions. The Monkhorst Pack k-points mesh [42] is set as 6×6×7, 6×6×6 and 6×6×6 for the Cmc21, Fm-3m and B1 structures respectively, to ensure all the energy calculations are accurately converged to less than 1 meV/atom. The PHONOPY code [43] is employed for the calculations of the phonon spectra and the projected phonon density of states (PDOS) using the density functional perturbation theory (DFPT) approach [44, 45]. The elastic moduli of all the compounds are obtained through the Voigt-Reuss-Hill approximation [46]. The theoretical Vickers hardness is evaluated using the empirical correlation suggested by Chen et al [47] based on the elastic bulk (B) and shear (G) moduli. The ideal compressive and tensile strength are achieved through incremental deformation applied to the crystal cells along various directions. The Vickers indentation shear strength is produced by calculating the stress in a biaxial stress field including two stress components (shear σzx and compressive σzz ) and the relation of these two stress components obeys the correlation:

σ zx = σ xx tan φ , where the φ (68°) represents the centerline-to-face angle of the indenter [48]. The crystal orbital Hamilton population (COHP) [49] method implemented in the LOBSTER code [50] is used in analyzing the


chemical bonds. To further explore the mechanisms of the indentation shear instability, the electron localization function (ELF) [51] is also calculated with the process of the incremental deformation.

3. Results and discussion 3.1 Structures and stability WN was reported to adopt the hexagonal structure at 0 GPa pressure [25]. Furthermore, B1-WN was synthesized at high temperature and high pressure conditions [21], but it is confirmed to be mechanically metastable. Further theoretical research clarified the fact that the NbO-type structure has the lowest formation enthalpy [24] and the transition from the unstable B1 structure to the stable NbO-type is attributed to the high vacancy concentration (Cv=25%) which induced a smaller electron rearrangement of the t2g orbitals than that of the B1 structure[22]. In the present work, we intercalate another TM (Ti) into W-N system and employ the CALYPSO code to carry out the structure searching systematically in the pressure range of 0 ~ 300 GPa. As shown in Fig. 1 (a, b, c), an orthorhombic structure of Ti0.5W0.5N with Cmc21 space group is found to be the most energetically favorable at 0 GPa and 0 K conditions and it is significantly different from the phases of WN (NbO-type) and TiN (B1) with the space group of Pm-3m and Fm-3m respectively, which are confirmed as the ground-state phases of WN and TiN (Mehl et al [24] found that NbO-type WN has the lowest energy at 0 GPa and 0 K conditions and B1-TiN has been verified to be the most energetically stable by Yang et al [52]). It can be seen the TM coordination numbers atoms are four for NbO-type WN, six for B1-TiN, and six for our Cmc21 structure. Due to the high vacancy concentration (Cv=25%), NbO-type WN has the lowest atomic density. In our Cmc21 structure, the N atoms occupy the Wyckoff 8b (0.752, 0.127, 0.571) sites and the Wyckoff sites of Ti, W are 4a (0, 0.624, 0.906) and 4a (0, 0.876, 0.402) respectively. After the full relaxations of the lattice constants and the atomic coordination at 0 GPa pressure, the obtained ground-state lattice parameters are a=5.597 Å, b=5.737 Å and c=4.960 Å. Additionally, when the pressure increases above 243.6 GPa, another cubic structure with Fm-3m space group becomes the most energetically favorable phase. Its coordination number of the TM atoms is eight and the lattice parameters are a=b=c=5.325 Å. The Wyckoff sites of Ti, W and N are 4a (0, 0, 0), 4b (0.5, 0.5, 0.5), and 8c (0.25, 0.25, 0.75) respectively. All the optimized equilibrium structural parameters of Ti0.5W0.5N, WN and TiN are listed in Table 1, which shows that our calculated lattice parameters of WN and TiN agree well with the previous results, suggesting the high reliability of our results. The thermodynamic stability is checked by the calculations of the formation enthalpy ∆Hf of Ti0.5W0.5N, WN and TiN based on the relations: ∆H f = H Ti0.5W0.5N − 0.5H Ti − 0.5H W − 0.5H N2 , where pure TMs of bcc-Ti 5

and hcp-W, as well as the solid-state α-phase N2 are adopted in the calculations. The calculated formation entalpy of Ti0.5W0.5N with three structures are listed in Table. 1. One can see all the formation enthalpies are negative, suggesting their thermodynamic stability. As the predicted ground-state phase, the Cmc21 structure has the lowest formation enthalpy, implying it is the most thermodynamically stabile in these three structures. The dependences of the total energy on the volume of f.u. for Ti0.5W0.5N with various structures are shown in Fig. 2 (a), from which it can be seen our predicted Cmc21 structure is more energetically stable at equilibrium volume than the B1 structure and the high pressure phase (Fm-3m structure). From the data of E-V curves, the bulk moduli at ambient pressure are obtained through fitting the third-order Birch-Murnaghan (EOS) [53] and the results are listed in Table 1. One can see they are all above 300 GPa, suggesting the property of high incompressibility. Through the formula H = E + PV , the relations of relative enthalpies (∆H) to the Cmc21 structure and the pressure in the range of 0 ~ 300 GPa are plotted in Fig. 2 (b). It can be seen the Cmc21 structure is the most stable phase under the pressure of 0 ~ 243.6 GPa and when the pressure increased above 243.6 GPa, it turns into Fm-3m structure. In the whole pressure range, the formation enthalpy of B1-Ti0.5W0.5N is always above the Cmc21 structure, indicating it is a metastable phase. To further check the resistance of Cmc21-Ti0.5W0.5N against decomposing to its parent compounds like Cmc21-TiN & Cmc21-WN or the ground-state structures B1-TiN & NbO-type Pm-3m-WN as well as B1-TiN & P63/mmc-WN, their relative enthalpies are also calculated through the formula:

∆H = 0.5H TiN − 0.5H WN − H Ti0.5W0.5N and are plotted in Fig. 2 (b), in which it can be seen all the calculated ∆H are positive in the whole referred pressure range, suggesting our Cmc21-Ti0.5W0.5N is energetically stable. As shown in Fig. 2 (c), we also present the formation enthalpy versus composition curve (convex hull),from which one can see the Cmc21-Ti0.5W0.5N is stable and can be synthesized in principle. The dynamical stability of Cmc21-Ti0.5W0.5N at 0 GPa pressure is also studied through the linear response method with DFPT (density functional perturbation theory) [44, 45] implemented in the PHONOPY code. The 2×2×2 supercell is modeled in the calculations and the calculated phonon spectra and projected density of states (PDOS) are plotted in Fig. 3, in which it can be seen there are no modes with imaginary frequencies in the whole Brillouin zone suggesting the dynamical stability of our Cmc21-Ti0.5W0.5N. The PDOS indicate the lower frequencies of the total DOS are dominated by the lattice dynamics of TMs (Ti, W) and the higher frequencies by lignt N atoms implying the TM sublattice dynamics plays a crucial role for the dynamical stability of these TM-LE compounds.

3.2 Elastic properties 6

For a potential engineering material, elastic properties are important mechanical properties. To get the elastic constants of a compound from first-principles method, one can apply a small strain onto the crystal, and then the change of energy or stress can be obtained. The calculated elastic constants are listed in Table 2 and they are also used to evaluate the mechanical stability of Cmc21-Ti0.5W0.5N. The Born criterion is given here as following:

C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, [C11 + C22 + C33 + 2(C12 + C13 + C23 )] > 0,


(C11 + C12 − 2C 12 ) > 0, (C11 + C33 − 2C13 ) > 0, (C22 + C33 − 2C23 ) > 0, It can be concluded that our calculated elastic contants can well satisfy the stability criteria, suggesting its mechanical stability at ambient pressure. The bulk, Young’s and shear moduli are determined from the elastic constants through the Voigt-Reuss-Hill approximations [46]. To verify the accurancy of our calculations, we compare our results of B1-TiN and NbO-type WN to the reported theoretical results also listed in Table. 2 and there are little relative differences less than 5%, indicating the high reliability of our results. Additionally, the calculated bulk moduli are all comparable to B0 obtained through fitting the third-order Birch-Murnaghan equation of state (EOS), which is listed in Table. 1, suggesting the high accurancy of our calculations again. It is well known that the resistances to volume and shear deformations are the two main indicators of hardness. It also can be seen that Cmc21-Ti0.5W0.5N possesses the largest shear modulus in all the related compounds and its bulk modulus is comparable to the arithmetic average of that of B1-TiN and WN with various structures, implying its excellent mechanical properties. Based on the bulk and shear moduli, we also evaluate the Vickers hardness of these compounds using Chen’s empirical model [47] as following:

H v = 2(k 2G ) 0.585 − 3,


Where k is the ratio of shear modulus (G) to bulk modulus (B). Due to the largest shear modulus, one can see Cmc21-Ti0.5W0.5N has the highest theoretical hardness up to 34.1 GPa which is much more than its parental compounds (TiN and WN). Another information about ductility can be provided by the values of G/B through comparing to the critical G/B ratio of 0.57, in which the bulk and shear modulus are simply considered as the resistance against the fracture and plastic deformation respectively. If G/B > 0.57, the material exhibits brittle behavior, otherwise the material exhibits ductile behavior [54]. As listed in Table 1, it can be seen Cmc21-Ti0.5W0.5N exhibits the brittle behavior with the G/B value of 0.77. Mechanical anisotropy is important for an engineering material, especially for the crystalline ceramics containing directional covalent bonds. Because in various engineering applications, the lower limit of the 7

mechanical properties for these crystalline materials is determined by the weakest crystalline orientations [55]. Because of the crystal anisotropy, atoms arranged in different crystalline directions with different periodicities and densities lead to distinct orientation-dependent physical and chemical properties. It is well known that elastic anisotropy plays an indispensable role in engineering science and crystal physics. Therefore, in this work, the elastic anisotropy is also studied through calculating the directional linear bulk, Young’s and shear moduli of B1-TiN, p63/mmc-WN and Cmc21-Ti0.5W0.5N and the results are plotted as three-dimensional surface representations and their two-dimensional slices in xy, yz and xz planes, as shown in Fig. 4 ~ Fig. 6. To be noted that, in consideration of the analogous structures of B1-TiN and NbO-type WN and NbO-type WN has the similar shape of three-dimensional surface representations, thus it is not shown here. The linear bulk and Young’s moduli on random directions for cubic, hexagonal and orthorhombic symmetries are provided as followings [56]: For cubic symmetry (B1 structure):



= s11 + 2s12


E −1 = s11 − (2 s11 − 2 s12 − s44 )(l12l22 + l22l32 + l12l32 )


For hexagonal symmetry (p63/mmc): −1

= ( s11 + s12 + s13 ) − ( s11 + s12 − s13 − s33 )l32


E −1 =(1 - l32)2 s11 + l34 s33 + l32 (1 − l32 )(2 s13 + s44 )



For orthorhombic symmetry (Cmc21):



= ( s11 + s12 + s13 )l12 + ( s12 + s 22 + s23 )l22 + ( s13 + s 23 + s33 )l32

E −1 = s11l14 + s22l24 + s33l34 + (2 s12 + s66 )l12l22 + (2 s23 + s44 )l22l32 + (2 s13 + s55 )l12l32

(7) (8)

Where l1, l2, l3 are the direction cosines, and sij (=Cij-1) are independent elastic compliance constants determined from the calculated elastic constants Cij. Because of the equivalent C11, C22 and C33 for B1-TiN and NbO-type WN, their directional linear bulk moduli are constant values in random directions which leads to an isotropic compressive property within the elastic limit. The caculated values of 909 GPa and 1110 GPa of isotropic linear bulk moduli for B1-TiN and NbO-type WN indicate their high incompressibility in various directions. However, in contrary to these cubic structures, the hexagonal p63/mmc-WN and orthorhombic Cmc21-Ti0.5W0.5N exhibit little anisotropy and they are shown here two spheroids in the Fig. 4 (b, c). Their largest (least) linear bulk moduli (Bl) are 1400 GPa (1000 GPa) along z direction (in xy plane) and 1200 GPa (900 GPa) along x direction (in xy plane) for p63/mmc-WN and Cmc21-Ti0.5W0.5N respectively. Here, the anisotropic 8

indicator can be described as Bl(max)/Bl(min) and it can be concluded Cmc21-Ti0.5W0.5N possess less anisotropy than p63/mmc-WN. Contrary to the linear bulk modulus, Young’s modulus is defined as response of the crystals applied the stretching stress within the elastic limit and it is also anisotropic for the crystalline materials. As shown in Fig. 5, B1-TiN exhibits the largest Young’s modulus along <100> directions, which are parallel to the covalent TM-N bonds. The calculated E(max) (E(min)) are 544 GPa (395 PGa), 765 GPa (448 GPa), 659 GPa (459 GPa) and 789 GPa (570 GPa) as well as the values of E(max)/E(min) are 1.38, 1.71, 1.44 and 1.38 for B1-TiN, NbO-type WN, p63/mmc-WN and Cmc21-Ti0.5W0.5N respectively. It is obvious that Cmc21-Ti0.5W0.5N possess the largest E(min) and not that prominent anisotropy in these four related structures. It is well known that the resistant of materials against shear deformation is closely related to their engineering applications due to its important role in determining the hardness. Thus, the anisotropy of shear modulus is also studied in this work. Fig. 6 shows the calculated three-dimensional surface representations of minimum shear modulus in various directions and their two-dimensional slices of minimum and maximum shear moduli in xy, yz and xz planes. We use G(min-max) and G(min-min) denote the maximum and minimum values of the G(min) which is the minimum shear modulus in certain directions. The calculated ratios of G(min-max)/G(min-min) are 224 GPa / 174 GPa≈1.29, 190 GPa / 155 GPa≈1.23, 295 GPa / 174 GPa≈1.70 and 250 GPa / 230 GPa≈1.08 for B1-TiN, NbO-type WN, p63/mmc-WN and Cmc21-Ti0.5W0.5N respectively, suggesting Cmc21-Ti0.5W0.5N has the smallest anisotropy which can be confirmed through the slight deviation from a standard sphere as shown in Fig. 6 (c). The value of G(max)/G(min) is the ratio of maximum and minimum shear moduli, which is another indicator of shear anisotropy. The calculated results are 337 GPa / 174 GPa≈1.94 (B1-TiN), 227 GPa / 155 GPa≈1.47 (NbO-type WN), 295 GPa / 174 GPa≈1.70 (p63/mmc-WN), and 295 GPa / 230 GPa≈1.29 (Cmc21-Ti0.5W0.5N). It is clear that the smallest shear anisotropy of Cmc21-Ti0.5W0.5N is confirmed again.

3.3 Ideal strength and structural deformation mechanism The calculations of stress-strain relations containing ideal compressive, tensile and indentation shear strength are conducted to further explore the response of Cmc21-Ti0.5W0.5N to the continuously increasing strains in various crystal orientations. The acquired compressive and tensile strengths are calculated through applying continuous uniaxial strains along the given crystal orientations and relaxing the crystal cell as well as inner atoms in the other directions to ensure the stress is zero along the transverse directions. Additionally, through simulating two stress components (shear σzx and compressive σxx ) in a biaxial stress field the indentation shear stress-strain relations are calculated to describe the plastic deformation under Vickers indenter and the shear σzx and 9

compressive σxx obey the relation: σzx = σxx tan φ, where the φ is the centerline-to-face angle of the indenter. The results are plotted in Fig. 7. For comparisons with Cmc21-Ti0.5W0.5N, the ideal strengths of B1-TiN and Pm-3m-WN (NbO-type) are also calculated. From Fig. 7 (a, b, c), it can be seen the compressive strength of Cmc21-Ti0.5W0.5N i.e. the peak value of 79.4 GPa along [101] direction is slightly higher than that of B1-TiN (67 GPa) and NbO-type WN (75 GPa) along [111] direction, suggesting its high incompressibility. The anisotropic ratio of its compressive strength of σ[100] / σ[101]=235 GPa / 79.4 GPa≈2.96 is less than that of σ[100] / σ[111]=348 GPa / 67 GPa≈5.19 and σ[100] / σ[111]=237 GPa / 75 GPa≈3.16 for B1-TiN and NbO-type WN respectively. Fig. 7 (d, e, f) shows the results of calculated tensile strengths. The smallest peak values of σ[100] =32.5 GPa (B1-TiN), σ[100] =49.8 GPa (NbO-type WN) and σ[101] =37.1 GPa (Cmc21-Ti0.5W0.5N) suggest NbO-type WN has the largest tensile strength, and the anisotropic ratio of σ[100] / σ[101]=73.3 GPa / 37.1 GPa≈1.98 for Cmc21-Ti0.5W0.5N is much smaller than that of B1-TiN with the value of σ[111] / σ[100]=99.7 GPa / 32.5 GPa≈3.07, but a little higher than that of NbO-type WN with the value of σ[111] / σ[100]=76.0 GPa / 49.8 GPa≈1.53. However, unlike the compressive and tensile strengths, indentation shear strength can provide the response of the crystals in a biaxial stress field and the corresponding stresses containing shear σzx and compressive σzz. The relation of these two stresses is given by σ zx = σ xx tan 68° . The indentation shear strength calculation mainly simulates the crystal deformations under Vickers indenter and it can provide a reliable theoretical Vikers hardness value [57]. In Fig. 7 (g, h, i), it obviously shows Cmc21-Ti0.5W0.5N possesses the larger indentation shear strength with the peak value of 30.0 GPa along (101)[101] direction than those of B1-TiN and NbO-type WN with the peak values of 22.4 GPa and 21.0 GPa respectively, indicating its highest Vickers hardness in these three compounds which is consistent with our theoretical hardness based on elastic property. The reason for the weakest shear direction appearing in (101) plane is ascribed to the fact that the weakest tensile and compressive directions are all present in [101] direction, thus the corresponding (101) plane can be confirmed as cleavage plane which has the smallest resistance against shear deformations. The anisotropic ratio of σ(101)[101] / σ(100)[010]=45.6 GPa / 30.0 GPa≈1.52 for Cmc21-Ti0.5W0.5N is much less than those of σ(110)[001] / σ(111)[112]=48.6 GPa / 22.4 GPa≈2.17 for B1-TiN and σ(110)[001] / σ(111)[112]=39.2 GPa / 21.0 GPa≈1.87 for NbO-type WN, indicating its less anisotropy of indentation shear strength than the related two parent compounds, which is in agreement with anisotropy of shear modulus shown in Fig 6. To further explore the deformation mechanism under indentation shear strain for Cmc21-Ti0.5W0.5N along the weakest (101)[101] direction, the relations of the applied strains and electronic bonding nature as well as the changes of bond length are presented in Fig. 8. Note that there are six types of Ti-N bonds (A1=2.183, A2=2.169 Å 10

A3=2.156 Å, A4= A1, A5= A2, A6= A3) and W-N bonds (B1=2.153, B2=2.166 Å, B3=2.175 Å, B4= B1, B5= B2, B6= B3) in equilibrium structure. With the applied indentation shear strain increasing to the critical value (0.13), A1, A5, A6 and B1, B5, B6 decrease, in contrast, A2, A3, A4 and B2, B3, B4 increase. As shown in Fig. 8 (d), when the strain exceeds the critical value, A2, A3 and B2, B3 increase sharply over 3 Å, i.e. they are stretched to be fractured, on the contrary, A4 and B4 retract with the break of A2, A3 and B2, B3, because they don’t achieve their critical bond length under the shear strain. This changing process can be clearly seen from the electronic localization function (ELF) [51] in the selected crystal plane as shown in Fig. 8. The definite values of ELF between TM and N atoms indicate there are high electronic localizations in these regions, suggesting the strong covalent bonds nature. When the indentation shear strain increases up to 0.14, the values of ELF between TM and N (corresponding the bonds of A2, A3 and B2, B3) appears obvious reducing down to 0, indicating the break of A2, A3 and B2, B3. It can be conclude that the structural instability induced by indentation shear strain is mainly attributed to the break of TM-N bonds.

3.4 Electronic structures and chemical bonds In order to have a deep understanding of the enhancement of the mechanical properties for Cmc21-Ti0.5W0.5N, we calculate the total electronic density of states (t-DOS), projected DOS (p-DOS) and the crystal orbital Hamilton population (COHP) [49] for B1-TiN, NbO-type WN and Cmc21-Ti0.5W0.5N which are plotted in Fig. 9 (a, b, c). As it is shown, they all exhibit the metallic behavior thanks to the definite t-DOS at Fermi level (EF) mainly provided by TM-d orbitals. In the regions below the EF (-8 ~ 0eV), the typical strong hybridizations of N-p and TM-d orbitals for all these three compounds are present, indicating the covalent bonds nature between TM and N atoms which is confirmed by the –COHP curves. To be noted that the “pseudogaps” appear near the EF and separate the DOS into the bonding states and the antibonding states. For B1-TiN and NbO-type WN, the “pseudogaps” fall below the EF, i.e. all the bonding states are filled and the antibonding states are partly occupied, in contrast, for Cmc21-Ti0.5W0.5N, the “pseudogap” falls at the EF, indicating the bonding states are filled and the antibonding states are not occupied. Thus, it can be confirmed from band filling theory [58~60] that the electronic stability of Cmc21-Ti0.5W0.5N is enhanced relative to B1-TiN and NbO-type WN. Meanwhile, from the bond-breaking process discussed above it can be seen that the covalent TM-N bonds play an important role in determining the resistance to the applied indentation shear strain. The introduction of the third element (Ti) tunes the VEC per two atoms to 10 (or 8 when the outermost s-electrons are not taken into consideration), which can greatly enhance the covalent TM-N bonds and leads to the maximum stability and strongest mechanical property. In Fig.12, we also compare the projected band structures of Cmc21-Ti0.5W0.5N with (a) and without (b) spin-orbit 11

splitting (SOC) effects. It can be seen there are no band gap nearby the EF, indicating the metallic property of Cmc21-Ti0.5W0.5N, which is in accordance with DOS. Meanwhile, there are several degenerate points e.g. the X, Y, G and U points which are split with including the SOC effects, but the change nearby the EF is little and the band gap is still not opened up.

4.Conclusions To conclude, using the CALYPSO method, we successfully predict the orthorhombic structure of Cmc21 symmetry is the ground-state phase of the ternary mononitride Ti0.5W0.5N. Its dynamical, thermodynamic and mechanical stabilities are confirmed by the calculations of phonon spectra, formation enthalpy and elastic constants through first-principles DFT method. Through comparing the mechanical properties of Cmc21-Ti0.5W0.5N with its parent binary materials B1-TiN and NbO-type WN, we evidence that the Cmc21-Ti0.5W0.5N possesses the greatly improved hardness (34 GPa) and ideal strengths. Mechanical anisotropy analysis reveals that the Cmc21-Ti0.5W0.5N has smaller elastic anisotropy than B1-TiN and WN with various structures. The indentation shear strength of Cmc21-Ti0.5W0.5N is up to 30 GPa which is much more than those of B1-TiN (22.4 GPa) and NbO-type WN (21 GPa). Through the calculations of electronic structure, it can be inferred that the enhanced stability and mechanical properties for Cmc21-Ti0.5W0.5N are mainly ascribed to the introduction of the third element Ti. The hardness value of 34 GPa indicates it can be used as potential superhard material. The present findings will stimulate further experimental and theoretical works on these technologically important materials.

Acknowledgments: The authors thank the support from National Natural Science Foundation of China (No. 11704007), 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. 2019JM-353).


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Table1. Calculated lattice constants a(Å), b(Å), c(Å), equilibrium volume of per formula unit (Å3/f.u.), formation enthalpy (∆Hf , eV/f.u.) and obtained B0 from the third-order Birch-Murnaghan (EOS) (GPa). Compounds











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1. [63] 2. [64] 3. [65] 4. [61] 5. [62] 6. [15] 7. [66]


Table2. Elastic constants (Cij, GPa), elastic moduli (B, G, E, GPa), Poisson’s ratio ν, G/B ratio and the Vickers hardness (Hv, GPa).






















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1. [61] 2. [9] 3. [23]






Fig.1 Crystal structures of B1-TiN (a), NbO-type WN (b) and Cmc21-Ti0.5W0.5N (c). The large blue, dark gray spheres and small silvery spheres represent Ti, W and N atoms respectively.


Fig.2 Calculated E-V curves (total energy versus volume of per f.u. for Ti0.5W0.5N with three different str uctures) (a), relative enthalpies of B1-Ti0.5W0.5N, Fm-3m-Ti0.5W0.5N as well as the related TiN and WN with various structures to the ground-state phase of Cmc21-Ti0.5W0.5N as a function of pressure (b) (Cmc 21-TiN & Cmc21-WN, B1-TiN & P63/mmc-WN and B1-TiN & Pm-3m-WN represent the enthalpies of 0. 5*TiN and 0.5*WN) and convex hull of Ti0.5W0.5N at the pressure of 0 GPa (c).


Fig. 3 Calculated phonon spectra of Cmc21-Ti0.5W0.5N at 0 GPa.


Fig. 4 Three-dimension surface representations of linear bulk modulus (Bl) and their two-dimension slices in xy, yz and xz planes for B1-TiN (a, d), P63/mmc-WN (b, e) and Cmc21-Ti0.5W0.5N (c, f).


Fig. 5 Three-dimension surface representations of Young’s modulus (E) and their two-dimension slices in xy, yz and xz planes for B1-TiN (a, d), P63/mmc-WN (b, e) and Cmc21-Ti0.5W0.5N (c, f).


Fig. 6 Three-dimension surface representations of minimum shear modulus in random directions and the two-dimension slices of the minimum and maximum shear moduli in random directions for B1-TiN (a), P63/mmc-WN (b) as well as Cmc21-Ti0.5W0.5N (c).


Fig.7 Calculated compressive, tensile and indentation shear stress-strain relations for B1-TiN (a, d, g), P63/mmc-WN (b, e, h) as well as Cmc21-Ti0.5W0.5N (c, f, i).


Fig.8 The changes of bond lengths and ELF in the selected crystal plane under indentation shear strain for Cmc21-Ti0.5W0.5N along the weakest (101)[101] direction.


Fig.9 Calculated total and projected density of states as well as COHP of B1-TiN (a), P63/mmc-WN (b) and Cmc21-Ti0.5W0.5N (c) respectively. The vertical dashed lines denote the Fermi level EF.


Fig.10 Calculated projected band structures with (a) and without SOC effect (b) for Cmc21-Ti0.5W0.5N. The horizon dashed lines denote the Fermi level EF. The red, blue, and cyan bubbles represent V, Ta and N respectively.


Highlights: : 

The ground-state phase of Ti0.5W0.5N is predicted through CALYPSO method.

Mechanical properties study reveal Cmc21-Ti0.5W0.5N has a large hardness of 34 GPa.

The origin of enhanced mechanical properties is ascribed to the introduction of Ti.


Declaration of interests ■ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.