DOI:
10.1039/D2RA06423E
(Paper)
RSC Adv., 2023,
13, 91099118
Theoretical study on structural and mechanical properties of Sicontaining ternary transition metal nitrides M_{0.5}Si_{0.5}N (M = Ti, Zr, Hf)†
Received
12th October 2022
, Accepted 9th March 2023
First published on 20th March 2023
Abstract
Sicontaining transitionmetal nitrides Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N with conventional rock salt B1 structure exhibit superior hardness, strength and oxidation resistance. However, the potential phases of the ternary systems at various pressures remain unexplored. In this work, we firstly studied the potential structures of Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N in pressures of 0–100 GPa. A hexagonal phase with P6_{3}/mmc symmetry was uncovered and verified to be quenchable in the ambient conditions. The structural, mechanical and electronic properties were systematically studied and compared with the wellknown ordered B1 structure. We surprisingly found that Ti_{0.5}Si_{0.5}N within this hexagonal phase displayed much improved ideal indentation shear strength from about 10 GPa for a B1 structure to 30 GPa. The estimated hardness based on the empirical formula is up to 38 GPa, greatly exceeding that of the B1 structure. By the detailed electronic analysis, the underlying atomic mechanism for the outstanding mechanical properties was also studied.
1. Introduction
Binary transition metal (TM) nitrides such as TiN, ZrN, TaN, VN, etc. are widely used as protective coatings for cutting tools.^{1} However, these nitrides have poor oxidation resistance in oxygen or air when the temperature is beyond 500 °C.^{2,3} In cutting applications such as high speed milling, the temperature can be easily improved to exceed 800 °C.^{4} To address this problem, an effective design strategy to enhance the oxidation resistance and mechanical properties is introducing a third element Al or Si into the TM nitrides to form ternary or quaternary systems like TiAlN, TiSiN, TiAlCrN, TiAlCN, etc.^{5–10} These aluminum or silicon nitrides have a distinct improvement of oxidation resistance. Furthermore, the introduction of the third element also induces the formation of a nanocomposite structure containing a nonmiscible nanocrystalline or amorphous structures and results in high hardness.^{11}
As is known, these binary or ternary TM nitrides adopt the metastable NaCltype B1 structure at room temperature thanks to the stabilization of the anharmonic effect.^{12} This B1 structure can't sustain its dynamic stability when the temperature descends to 0 K due to the appearance of unphysical longitudinal acoustic phonon energy.^{12} Most of these ternary B1 nitrides containing two TM elements often have a disordered occupation of TM atoms in the cation locations. As a result of this disordered occupation, the mechanical anisotropy is further improved on the basis of the ordered metastable B1 structure, which is not favorable in engineering application. Additionally, though the cubic B1 structure has relatively high crystal symmetry, the mechanical anisotropy is strong and this has been verified in our previous work for ternary VAlN and quaternary VWCN systems.^{13,14} However, these TM nitrides within hexagonal structure often behave less mechanical anisotropy, which can be ascribed to the better distribution of the bonds between TM and nitrogen atoms. Meanwhile, the electronic structure analysis shows the “pseudogap” separating the bonding and antibonding states is present in the most B1structure compounds and the E_{F} often lies in the antibonding region, which improves the formation enthalpy of the systems and induces the metastability. The occupition of d–d antibonding states also weakens the p–d hybridization between TM and light elements. However, this p–d hybridization is indispensable in determining the antishear properties and macroscopic hardness for this type of compounds. Thus, the hardness of these B1 structure TM nitrides is mostly in the range of 10–30 GPa much less than the threshold of the superhard standard (40 GPa).^{15,16} For the Sicontaining ternary TM nitrides within B1 structure, Si atoms are introduced for replacing partial TM atoms. This replacement can't prominently tune the E_{F} position due to the structural limitation. Thus, it also can't significantly improve the mechanical properties. Therefore, it is greatly meaningful to search new configuration for these Sicontaining nitrides with tuned E_{F} position and enhanced mechanical properties.
In this work, we systematically explored structural, mechanical and electronic properties of the ternary Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems using the firstprinciple and structure searching methods implemented in the CALYPSO code^{17,18} in the pressure of 0–100 GPa. The hexagonal P6_{3}/mmc phase was firstly uncovered. To clarify the structural effects on hardness, strengths, mechanical anisotropies and crystal stability of these three systems within P6_{3}/mmc and B1 structures, we simulated the uniaxial and biaxial stains to obtain the corresponding stress and analyzed the thermal and dynamic stabilities as well as electronic structures.
2. Calculation methods and details
The structure searching process for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems was performed in the pressure of 0–100 GPa, by the method of CALYPSO,^{17,18} which has been fully confirmed as an reliable and effective structural searching approach.^{19–25} We adopted 0, 30, 50 and 100 GPa as the searching pressure points. The crystal cell containing 1–4 formula units (f.u.) was used in the searching process. All the calculations were carried out by the firstprinciple density functional theory (DFT) method implemented in the Vienna ab initio simulation package (VASP).^{26} The generalized gradient approximation (GGA) was used in calculating elastic and structural properties.^{27} The exchange correlation potential within the GGA proposed by Perdew, Burke and Ernzerhoff (PBE) with projector augmented wave (PAW) method^{28,29} was used in all the calculations. The kinetic cutoff energy was set as 600 eV during the optimization process of all the structures. The convergence criteria of the total energy in all the structural relaxations were set as 10^{−5} eV. The 7 × 7 × 7 and 12 × 12 × 3 kpoint meshs in Monkhorst Pack scheme and Gamma centered grid^{30} were used for ordered pseudoB1 and P6_{3}/mmc sturcutes, respectively. The ternary disordered B1 systems were calculated with 4 × 4 × 4 kpoint meshs and 64 atoms supercells using the special quasirandom structure (SQS) method.^{31,32} The SQS structure was constructed for ternary Ti_{0.5}Si_{0.5}N system using unit cells containing 32 cations and 32 anions within fcc structure. The 2 × 2 × 2 supercell containing 64 atoms was generated. The evaluation of the generated model was performed by the objective function provided by the SQS approach. The simulated SQS model as well as the predicted structures were set in the ESI data.† The phonon spectra were calculated by the density functional perturbation theory (DFPT) approach implemented in the PHONOPY code.^{33} The elastic constants C_{ij} were calculated by applying a small strain onto the crystal structures with a finite variation. All the elastic moduli were calculated according to the Voigt–Reuss–Hill approximation.^{34} The 3D and 2D distributions of elastic moduli were investigated by calculating the directional Young's and shear moduli, which were obtained by calculating the Young's and shear moduli along various crystal directions based on the formulas provided by the literature^{35} using the elastic constants. The Chen empirical formula was adopted in assessing the Vickers hardness.^{36} The electron localization function (ELF)^{37} is also calculated to further analyze the bonding nature.
3. Results and discussions
3.1. Structure, property and stability
The CALYPSO method was employed to searching potential structures at the pressure points of 0, 30, 50 and 100 GPa, respectively. A trigonal structure with the space group of R3m (No. 160) was confirmed to be the groundstate phase of these three Sicontaining ternary systems. However, this phase cannot sustain its thermal stability when the pressure ascends up to 30 GPa. At the pressures of 30, 50 and 100 GPa, the results of the structure searches showed the most favorable structure of these systems tuned to be the hexagonal P6_{3}/mmc (No.194) phase, which is displayed in Fig. 1. The TM and Si atoms have an alternate layered distribution. As a comparison, the ordered B1 structure was also exhibited in the figure. It can be seen there are six light element N atoms surrounding the TM and Si atoms, which is consistent with the B1 structure. The difference is the bond angles and lengths of the TMN and Si–N bonds have some variations. Here, we take Ti_{0.5}Si_{0.5}N as an example. It is clearly seen the bond angle of all the adjacent Ti–N and Si–N bonds is 90° in the B1 structure. After full relaxation of both lattice constants and the positions of intra atoms, the obtained bond angles are 77.6°, 87.2° for N–Ti–N and 94.0°, 86.0° for N–Si–N. The bond lengths of Ti–N and Si–N are 2.227 Å and 1.909 Å, respectively, which are all different from those of ordered B1 structure. The Wyckoff sites of Ti, Si and N are 2d (0.667, 0.333, 0.25), 2a (0, 0, 0) and 4f (0.333, 0.667, 0.100) respectively. As listed in Table 1, these three ternary systems within the same structures have approximately identical lattice constants. The lattice constants of hexagonal Ti_{0.5}Si_{0.5}N are a = b = 2.793 Å and c = 10.236 Å. The evaluated density is 3.007 × 10^{3} kg m^{−3}, larger than the ordered B1 structure (2.726 × 10^{3} kg m^{−3}) and the trigonal structure (2.705 × 10^{3} kg m^{−3}) presented in Table 1. Thus, the average TM–N and Si–N bond lengths are smaller than B1 structure and the trigonal structure. For other two Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems, the structural properties are similar to the Ti_{0.5}Si_{0.5}N system.

 Fig. 1 Crystal structures of (a) trigonal R3m, (b) side and top views of hexagonal P6_{3}/mmc as well as the corresponding first Brillouin zone and (c) ordered B1 structures of Ti_{0.5}Si_{0.5}N. The large cyan, blue and small silvery spheres represent Ti, Si and N atoms respectively.  
Table 1 Calculated lattice constants, M–N bond lengths (Å), formation enthalpies (eV/f.u.), and densities (kg m^{−3})

Symmetry 
Source 
a 
c 
M–N 
Si–N 
H_{f} 
Density 
Ti_{0.5}Si_{0.5}N 
R3m 
This work 
10.449 



−2.092 
2.705 × 10^{3} 

P6_{3}/mmc 
This work 
2.793 
10.236 
2.227 
1.909 
−1.918 
3.007 × 10^{3} 

B1 
This work 
4.230 

2.151 
2.105 
−1.172 
2.726 × 10^{3} 
Zr_{0.5}Si_{0.5}N 
R3m 
This work 
10.820 



−1.846 
3.375 × 10^{3} 

P6_{3}/mmc 
This work 
2.898 
10.891 
2.397 
1.953 
−1.666 
3.720 × 10^{3} 

B1 
This work 
4.524 

2.243 
2.281 
−1.050 
3.210 × 10^{3} 
Hf_{0.5}Si_{0.5}N 
R3m 
This work 
10.709 



−1.866 
5.494 × 10^{3} 

P6_{3}/mmc 
This work 
2.878 
10.861 
2.378 
1.946 
−1.596 
6.021 × 10^{3} 

B1 
This work 
4.521 

2.226 
2.226 
−1.104 
5.237 × 10^{3} 
To verify the thermal stability, we calculated the formation enthalpy using the following formula: , (M = Ti, Zr, Hf), where H_{M0.5Si0.5N}, H_{M}, H_{Si}, and H_{N2} are the enthalpies per f.u. of M_{0.5}Si_{0.5}N, pure hcpTi/Zr/Hf, diomandtype Si and αN_{2}. The calculated results are −1.918, −1.666 and −1.596 eV/f.u. for hexagonal Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems respectively. For trigonal R3m structure, the formation enthalpies are −2.092, −1.846 and −1.866 eV/f.u. respectively. As a contrast, the formation enthalpies of them within ordered B1 structure are −1.172, −1.050 and −1.104 eV/f.u., respectively, indicating the trigonal phases are more thermodynamically stable than the hexagonal and ordered B1 phases in principle at ambient condition. Fig. 2 displays the calculated results of enthalpy differences of these three systems within B1 and R3m structures relative to the hexagonal P6_{3}/mmc structure as a function of pressure. It is clearly seen that with the pressure ascending beyond 15 15 and 25 GPa, P6_{3}/mmc structure transforms to be the most energetically favourable phase for them, as it has the lowest total enthalpy. Meanwhile, the enthalpy differences have an increasing tendency with pressure, implying the thermodynamic stability of P6_{3}/mmc structure can be maintained up to 100 GPa.

 Fig. 2 Enthalpy difference of (a) Ti_{0.5}Si_{0.5}N, (b) Zr_{0.5}Si_{0.5}N and (c) Hf_{0.5}Si_{0.5}N within P6_{3}/mmc structure relative to R3m and B1 structures as a function of pressure.  
To further check the stability and the possibility of artificial synthesis of these systems with new uncovered hexagonal structure, the study of dynamic properties are necessary. Therefore, using the DFPT method which is implemented in the PHONOPY code, the phonon spectra as well as the corresponding projected density of states (PDOS) are calculated by simulating the 3 × 3 × 1 supercells for hexagonal Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems. As is plotted in Fig. 3, there are no imaginary modes in the whole Brillouin zone for all of them, confirming their dynamic stability at ambient condition. For the primitive cell of uncovered hexagonal structure contains 8 atoms, the corresponding phonon spectra have 24 modes for each wave vector q. There are 3 acoustic and 21 optical branches in the spectra. It is obviously seen that the frequency distributions of these three systems are different with each other. The highest frequency of the atomic vibrations is up to near 25 THz for Ti_{0.5}Si_{0.5}N, exceeding those of Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N. Thus, it can be concluded the atomic interactions of Ti_{0.5}Si_{0.5}N are the strongest in them, implying this type of Ti_{0.5}Si_{0.5}N has better mechanical property and stability. The PDOS plots provide the information of frequency distributions of each element contained in the systems. One can see for these three Sicontaining systems, the vibrations of Si and TM atoms dominate the lower frequency regions and the N atoms mainly supplies the higher frequency ranges.

 Fig. 3 Phonon spectra of Ti_{0.5}Si_{0.5}N (a), Zr_{0.5}Si_{0.5}N (b) and Hf_{0.5}Si_{0.5}N (c) within P6_{3}/mmc structure at ambient pressure.  
3.2. Mechanical property
Let's focus our attention on the mechanical properties of these three ternary Sicontaining systems in this section. As is known, ideal strength is an important mechanical property in definiting the macro physical natures of the materials such as hardness, toughness, brittleness and ductility etc. By applying tensile/compressive strains along certain directions while relaxing the crystal along the rest four orientations, the tensile/compressive stress–strain relations can be obtained. In stress–strain curves, the corresponding stress value of the yield point is defined as the ideal strength in this direction. Due to the hexagonal crystal symmetry, stress–strain relations in three directions ([0001], and ) are necessary and plotted in Fig. 4a–c. Here, the tensile/compressive stress is denoted as σ_{zz}. It can be seen the nonlinear tensile/compressive stress occurs at the strains of about 0.04 and all these three Sicontaining systems have the largest/smallest tensile strengths along [0001]/ directions, reflecting their strongest and weakest stress response are along [0001] and directions as shown in Fig. S1,† respectively. The ideal tensile strengths along [0001]/ directions are 71.7/34.5 GPa, 61.3/23.4 GPa and 63.5/25.9 GPa on the strains of 0.212/0.111, 0.194/0.089 and 0.206/0.089 for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems respectively. The compressive strengths are all beyond 75 GPa for them in all these three directions, confirming they all have ultraincompressibility. The fracture is present near the yield points upon tensile strains and the plastic deformation is absent for all of them, implying they all behave brittleness. Because of the longest tensile curves for Ti_{0.5}Si_{0.5}N, that means it has the largest toughness nature relative to the rest two. By comparing the largest and the smallest strengths (denoted as σ_{max} and σ_{min}), the strength anisotropies can also be assessed by the ratios of σ_{max} and σ_{min}. The calculated tensile σ_{max}/σ_{min} values are 2.1, 2.6 and 2.5 for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems respectively, suggesting Ti_{0.5}Si_{0.5}N has the smallest strength anisotropy.

 Fig. 4 Tensile/compressive (a–c) and indentation shear (d–f) stress–strain curves for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N.  
By simulating the real experimental measuring process of Vickers hardness, we also evaluated the values of hardness for these three systems. According to the measuring process, a biaxial strain field containing two stress components (shear σ_{zx} and compressive σ_{zz}) forms in the tested materials. The relation of the two stress components obeys the formula: σ_{zx} = σ_{zz} tanφ, where φ (=68°) denotes the facetocenterline angle of the diamond indenter.^{38} Using this method, the Vickers hardness of them should be well assessed. The calculated results of indentation shear stress–strain relations are plotted in Fig. 4d–f. In accord with the tensile strength, the weakest stress response occurs in the (0001) for all of these three systems, suggesting the (0001) plane can be viewed as their first cleavage plane. To obtain the indentation shear strength along as many crystal directions as possible, three typical planes along six slip directions were adopted to calculate the indentation shear strength. The largest/smallest strengths are 38.5/30.0, 25.0/17.4 and 25.6/19.6 on the strains of 0.116/0.139, 0.113/0.122 and 0.116/0.141 for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems respectively. Correspondingly, the indentation shear σ_{max}/σ_{min} values are about 1.28, 1.44 and 1.31 respectively, meaning Ti_{0.5}Si_{0.5}N has the best shear strength anisotropy again. To be noted that this ideal strength calculations don't take the defect and impurity inside the real crystals into consideration. However, it is effective to assess the Vickers hardness for the ideal crystalline materials. Thus, it can be concluded that this hexagonal P6_{3}/mmc Ti_{0.5}Si_{0.5}N possesses high Vickers hardness up to 38.5 GPa under real experimental conditions in principle, very close to the threshold of superhard bound (40 GPa). Meanwhile, Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems all have larger tensile/compressive strengths in various directions relative to their shear strengths, indicating the failure modes of them are mainly induced by the shear loadings in the future engineering applications.
In comparison, the tensile strengths of Ti_{0.5}Si_{0.5}N in ordered B1 and disordered SQSB1 structures are also studied. As is plotted in Fig. S2,† it can be obviously seen both of them exhibit much weaker tensile and compressive strengths relative to those of hexagonal phase. The largest/smallest strengths are 13.6/6.5 GPa and 12.4/6.4 GPa for ordered and disordered structures respectively, which are very close to each other, implying there is little effect on the mechanical strengths for the occupations of the TM and Si atoms. However, most experimental hardness values of this B1–Ti_{0.5}Si_{0.5}N are up to beyond 20 GPa, which is mainly ascribed to the hardening effect of superlattice formed as nanocomposite.
Calculated elastic constants C_{ij}, corresponding elastic moduli and the assessed Vickers hardness are given in Table 2. The hexagonal P6_{3}/mmc structure has five independent elastic constants (C_{11}, C_{33}, C_{44}, C_{12} and C_{13}) and the cubic B1 has three (C_{11}, C_{12} and C_{44}). However, this ternary systems have three elements with different radii, thus, the ordered B1 structure have different lattice constants of a and b due to the layered and alternated distribution of TM and Si atoms. Thus, the cubic B1 structure degenerates to be the tetragonal phase, correlating to the six independent elastic constants. For the firstly uncovered P6_{3}/mmc structure, the mechanical stability was checked using the criteria:^{34} C_{44} > 0, C_{11} > C_{12}, (C_{11} + 2C_{12})C_{33} > 2C^{2}_{13}. All these three Sicontaining systems well satisfy the criteria, suggesting their mechanical stabilities. As is seen in Table 2, the present calculated three elastic constants for binary TiN are excellently agreement with the previous theoretical and experimental results, confirming the high reliability of our calculations. For the hexagonal ternary systems, the firstly calculated elastic constants can be used as good references for future theoretical or experimental study. Notably, with the addition of Si element, there is a decreasing trend of bulk modulus, which is attributed to the reduction of electron concentrations. However, due to the enhanced p–d hybridization and weakened d–d metallic states between TM and N atoms, the shear moduli are maintained or improved at large values, leading to larger G/B values for them. Thus, these three ternary systems all behave brittle nature (G/B > 0.57).^{39,40} Meanwhile, based on the elastic moduli, the Vickers hardness is calculated using the Chen formula.^{35} Their hardness values are 38.1, 27.3 and 30.2 GPa for Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N respectively. Remarkably, despite the fact that these hardness values are evaluated in the elastic limit of the crystals, they are highly consistent with our results using the indentation shear stress–strain method above. Such high hardness supports they can be viewed as potential superhard materials. Also, it can be seen the hardness of B1 structure of Ti_{0.5}Si_{0.5}N is much smaller, though it was reported to possess the high hardness up to 30 GPa. The reason of the deviation can be mainly ascribed to the superlattice effect in the experimental condition. Additionally, it can be seen the mechanical properties of these three ternary systems such as the G/B values, hardness as well as strengths are not linear while going from Ti_{0.5}Si_{0.5}N to Zr_{0.5}Si_{0.5}N to Hf_{0.5}Si_{0.5}N. The hardness of the hexagonal Hf_{0.5}Si_{0.5}N is slightly higher than Zr_{0.5}Si_{0.5}N while Ti_{0.5}Si_{0.5}N has the largest value. Similar results can be seen in the “indentation shear” strengths. This seeming irregularity may be ascribed to the nonlinear distribution of the atomic radii from Ti to Zr to Hf. Zr has the largest atomic radius resulting in the largest lengths of Zr–N bonds which leads to the weaker bond strength. Thus, Zr_{0.5}Si_{0.5}N has the smallest hardness and lowest strengths.
Table 2 Calculated elastic constants (C_{ij}, GPa), bulk modulus (B, GPa), shear modulus (G, GPa), Young's modulus (E, GPa), Poisson's ratio ν, G/B ratio and the Vickers hardness (H_{v}, GPa)

Symmetry 
Source 
C_{11} 
C_{22} 
C_{33} 
C_{44} 
C_{55} 
C_{66} 
C_{12} 
C_{13} 
C_{23} 
B 
G 
E 
ν 
G/B 
H_{v} 
Ref. 41. Ref. 42. Ref. 43. Ref. 44. 
Ti_{0.5}Si_{0.5}N 
P6_{3}/mmc 
This work 
640 

570 
223 

251 
137 
101 

280 
240 
559 
0.17 
0.86 
38.1 

B1 
This work 
316 
399 

156 
47 

129 

101 
203 
89 
232 
0.31 
0.44 
7.5 

B1 
Exp^{a} 











260 


25 

B1 
Exp^{b} 











334 


26 
Zr_{0.5}Si_{0.5}N 
P6_{3}/mmc 
This work 
528 

469 
168 

197 
134 
107 

246 
185 
443 
0.20 
0.75 
27.3 
Hf_{0.5}Si_{0.5}N 
P6_{3}/mmc 
This work 
568 

477 
174 

216 
136 
96 

251 
197 
468 
0.19 
0.79 
30.2 
TiN 
B1 
This work 
598 


155 


144 


295 
181 
451 
0.25 
0.61 
20.6 

B1 
Theory^{c} 
590 


169 


145 


294 
189 
466 
0.24 
0.51 
22.6 

B1 
Exp^{d} 
625 


163 


165 


320 




23 
As is known, the elastic properties are achieved by estimating the stress response of the crystals under micro strains. Normally, there is elastic anisotropy depending on the crystal symmetry and the bond distributions in the materials. Here, we adopt Ti_{0.5}Si_{0.5}N as the representative to further compare the elastic anisotropies of hexagonal P6_{3}/mmc, B1 and trigonal R3m structures through calculating the Young's (E) and shear (G) moduli in various directions based on the formulas provided by the literature.^{35} The results are displayed in Fig. 5 (Young's modulus E) and Fig. 6 (shear modulus G) using the threedimensional (3D) surface representations and twodimensional (2D) projections on xy, xz and yz crystal planes. Here, we use the distance from the coordinate origin to the 3D surface denote the directional modulus E/G. Thus, if a material behaves isotropic elastic properties, the 3D plots would be a perfect sphere. The deviation from the sphere can reflect the elastic anisotropy. It is obviously deduced the P6_{3}/mmc structure possesses the best isotropic elastic properties thanks to the smallest deviation from the perfect sphere. This is in accordance with the calculated tensile/compressive stress–strain curves, in which the curve slopes of those three typical crystal directions near the coordinate origin are close with each other. The largest/smallest Young's moduli (E_{max}/E_{min}) for P6_{3}/mmc, B1 and R3m structures are 599/528 GPa, 348/152 GPa and 500/277 GPa, respectively. The quantitative elastic anisotropies can be compared by the ratios of E_{max} and E_{min}. The E_{max}/E_{min} values of them are 1.13, 2.29 and 1.81, meaning the P6_{3}/mmc structure has the best elastic isotropy. Also, 3D and 2D directional dependence of the shear modulus (G) are plotted in Fig. 6. Notably, the shear modulus is different from the Young's modulus, because the G value varies with the slipping direction for a given crystal plane. Therefore, each crystal plane has minimum and maximum G values (G_{min} and G_{max}). Here, the 3D directional dependence of G_{min} are plotted in Fig. 6a–c and the 2D projections of G_{min} and G_{max} on xy, xz and yz planes are plotted in Fig. 6d–f. Obviously, P6_{3}/mmc structure exhibits the best shear anisotropy due to the fact that its 3D plot is close to a perfect sphere. The ratios of G_{max}/G_{min} are 251 GPa/223 GPa ≈ 1.13 (P6_{3}/mmc), 156 GPa/47 GPa ≈ 3.32 (B1) and 216 GPa/105 GPa ≈ 2.06 (R3m). It can be seen the ordered B1 structure has large shear anisotropy. For the Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems within these three structures, the results of the strength and elastic anisotropies are similar to the Ti_{0.5}Si_{0.5}N. Hence, they are not repetitively discussed here.

 Fig. 5 Threedimensional (3D) surface representations of Young's moduli for Ti_{0.5}Si_{0.5}N within P6_{3}/mmc (a), B1 (b) and R3m (c) structures and twodimensional (2D) projections (d–f) of the Young's moduli on the xy, yz and xz planes.  

 Fig. 6 Threedimensional (3D) surface representations of shear moduli for Ti_{0.5}Si_{0.5}N within P6_{3}/mmc (a), B1 (b) and R3m (c) structures and twodimensional (2D) projections (d–f) of the shear moduli on the xy, yz and xz planes.  
3.3. Electronic properties
We now turn to discuss the electronic properties of these Sicontaining ternary systems. Due to the same valence electrons for Ti, Zr and Hf elements, the electronic structures of Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems are similar with each other in the same structures. Fig. 7 shows the calculated projected band structures and partial density of states (PDOS) of Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N within P6_{3}/mmc structure. It can be seen their band structures are analogous with each other. The band gaps are absent for all of them, indicating their metallic nature. In particular, it can be seen the conduction bands are mainly dominated by the M–d orbitals, the valence bands by partial M–d and N–p orbitals. The p–d hybridization can be obviously observed near the E_{F}. Meanwhile, one can see the difference of crystal structure enormously impact the band structures. As shown in Fig. S3(a),† for Ti_{0.5}Si_{0.5}N within B1 phase, the E_{F} lies at the antibonding regions, in contrast, it is modified effectively at the overlap of valence band top and conduction band bottom for P6_{3}/mmc structure, leading to the enhanced electronic stability and mechanical property. In comparison, TiN and SiN within hexagonal P6_{3}/mmc structure are also simulated to study the difference of the band structure from the ternary system. As seen in Fig. S3(b and c),† it can be seen some conduction bands descend below the E_{F} for TiN, as a contrast, the valence bands spread through the E_{F} for SiN. That means the ternary hexagonal phase possesses the best electronic stability and the maximized covalent p–d hybridization, inducing the enhanced antishear property. The PDOS plots show that the pseudogaps near the E_{F} separating the bonding and antibonding states can be obviously seen for Ti_{0.5}Si_{0.5}N within P6_{3}/mmc and B1 structures. However, thanks to the introduction of Si element, the E_{F} is exactly modulated at the valley of the pseudogap for P6_{3}/mmc structure. It suggests the bonding states are fully occupied, whereas the antibonding states are not. The p–d hybridization can be seen in the energy range of 0 to −5 eV for Ti_{0.5}Si_{0.5}N. B1 structure exhibits the obvious occupation of d–d states near the E_{F}, resulting in the improved total formation enthalpy and the metallic nature, which is not favorable for the mechanical properties of compounds. Though the Si element induces the modulated electronic structure, it has weaker interactions with N element due to its small contributions to the bonding states for both sorbitals and porbitals in the related energy range. The plot of calculated ELF is presented in Fig. 8. To clearly exhibit the Ti–N and Si–N bonding nature, herein, we selected a crystal plane holding the Ti, Si and N atoms within a common plane. The ELF value is dimensionless and it has a range of 0–1. ELF = 1/ELF = 0 have the meaning of highly localized/lack of electrons. Hence, the bonding nature can be evaluated by discerning the color of the region among the atoms in the plot. It can be seen the electrons are highly localized between Si and N atoms, suggesting the covalent Si–N bonding. Thanks to the highly p–d coupling, the Ti–N bonds also exhibit strong covalent feature, though comprising some of metallic component.

 Fig. 7 Projected band structures and projected density of states (PDOS) for (a) Ti_{0.5}Si_{0.5}N, (b) Zr_{0.5}Si_{0.5}N and (c) Hf_{0.5}Si_{0.5}N within P6_{3}/mmc structure.  

 Fig. 8 Calculated ELF slice of the selected plane of P6_{3}/mmcTi_{0.5}Si_{0.5}N.  
4. Conclusions
In summary, the structure diversity of Ti_{0.5}Si_{0.5}N, Zr_{0.5}Si_{0.5}N and Hf_{0.5}Si_{0.5}N systems under the pressure of 0–100 GPa are firstly explored using the developed crystal structure searching method combined with the firstprinciple calculations. Theoretical explorations on their structural stability, electronic and mechanical properties were systematically performed by firstprinciple calculations. Our results confirmed that the P6_{3}/mmc structure of these three Sicontaining ternary systems has surprisingly enhanced ideal strengths and larger hardness up to 38.1 GPa, as well as better mechanical isotropy relative to the ordered B1 structure. Electronic analysis in depth was also carried out to further study the mechanism of the hardening inducement. For this new hexagonal P6_{3}/mmc structure, the addition of Si element well tuned the electronic structure to make the E_{F} position optimized and maximized the p–d hybridization and minimized the occupation of antibonding d–d states. Meanwhile, the structural stability was also improved by this electronic adjust. All these findings would be expected to stimulate further experimental and theoretical works on these Sicontaining systems and provide effective strategy in designing new type of hard materials with composite combination of beneficial properties.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work was supported by State Key Laboratory of Precision Blasting and Hubei Key Laboratory of Blasting Engineering, Jianghan University (No. PBSKL2022D03), the Natural Science Foundation of China (Grant Nos. 51808554), the Natural Science Basic Research plan in Shaanxi Province of China (Grant No. 2020JQ892).
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