Crystal Structures, Electronic Properties and Hardness

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C: Physical Processes in Nanomaterials and Nanostructures

Prediction of Molybdenum Nitride from First-principle Calculations: Crystal Structures, Electronic Properties and Hardness Li-Ping Ding, Peng Shao, Fang-Hui Zhang, Cheng Lu, and Xiao-Fen Huang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04779 • Publication Date (Web): 15 Aug 2018 Downloaded from http://pubs.acs.org on August 16, 2018

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The Journal of Physical Chemistry

Prediction of Molybdenum Nitride from First-principle Calculations: Crystal Structures, Electronic Properties and Hardness Li-Ping Ding,†,∗ Peng Shao,†,∗ Fang-Hui Zhang,† Cheng Lu,‡,§ Xiao-Fen Huang¶ †

Department of Optoelectronic Science & Technology, College of Elecrical & Information

Engineering, Shaanxi University of Science & Technology, Xi’an 710021, China ‡

Department of Physics, Nanyang Normal University, Nanyang, 473061, China

§

Department of Physics and High Pressure Science and Engineering Center, University of Nevada,

Las Vegas, Nevada 89154, United States. ¶

Physics Department, Sichuan Normal University, Chengdu 610068, China

∗

Correspondence to: Li-Ping Ding, Shaanxi University of Science & Technology, Xi’an 710021, China. Tel./fax: +86 29 86168320. E-mail address: [email protected] (Li-Ping Ding), [email protected] (Peng Shao).

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ABSTRACT Transition metal (TM) nitrides has been widely used in many scientific and technical areas because of their unique physical and mechanical properties. However, most of the well known transition metal nitrides are nitrogen deficient. The reports on nitrogen-rich TM nitrides are rather lack and sometimes even discrepancy in their crystal structures. Herein, the microstructure, stability, electronic property and hardness of nitrogen-rich molybdenum nitride MoN2 compound have been investigated systematically by using an unbiased structure search method CALYPSO combined with first-principle calculations. Our study demonstrates a stable configuration orthorhombic Cmc21 (No.36) for MoN2 crystal, which even lower in energy than the experimental synthesized structure rhombohedral R3m-MoN2 at ambient pressure condition. The formation enthalpies with respect to the reactant Mo+N2, mechanical stabilities and phonon dispersions further confirm the stability of Cmc21-MoN2 phase at the whole ambient condition, indicating that it can be synthesized in experiment. According to the density of states, it is seen that all the considered MoN2 exhibit metallic behavior and there are two types of bonds (covalent and ionic) exist in MoN2. The Vicker hardness of Cmc21-MoN2 is calculated as 11.987 GPa, and the strength and number of covalent bonds may dominate its hardness.

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1. INTRODUCTION In recent years, the transition metal (TM) nitrides have attracted much attention because of their unique physical and mechanical properties, e.g., high hardness, low compressibility, high melting point, high temperature ceramics, strong magnetism, superconductivity and catalysts.1-3 The most important feature of TM nitrides is that the maximum oxidation state of the metal is difficult to reach as high as in homologous fluorides and oxides. The oxidation state of the metal in TM oxides often exceeds +3, while most of the well known transition metal nitrides are nitrogen deficient with x < 1 in TMNx. In this case, the early experimental and theoretical studies4-6 mainly focus on the synthesis and the above mentioned properties of nitrogen deficient transition metal nitrides. Compared with the nitrogen-deficient transition metal nitrides, the reports on the nitrogen-rich transition metal nitrides are rather lack and sometimes even discrepancy in their crystal structures. The reason may be that the incorporation of nitrogen into the crystal lattice of transition metals is always thermodynamically unstable at ambient pressure. On the other hand, the syntheses of nitrogen-rich transition metal nitrides at low temperature condition is a considerable challenge. However, nitrogen-rich transition metal nitrides are promising candidates for the next generation catalysts. Thus, despite the synthesized challenge, the nitrogen-rich transition metal nitrides still have been synthesized by several methods such as methathesis, high temperature ammonlysis, vapor deposition and high pressure techniques.4-8 For example, the nitrogen-rich transition metal nitrides have been addressed the possibility to form and synthesize on experimental side using the high-pressure techniques9 including previously unknown noble metal nitrides PtN2, IrN2, OsN2 and PdN2,10-13 as well as some new polymorphs such as Zr3N4,9 Hf3N4,9 Ta3N514,15 and Ta2N316. Recently, Wang et al.17,18 have successfully synthesized novel nitrogen-rich tungsten nitrides (e.g., CrN and W3N4) by using the same method. While for the molybdenum atom which has the same outermost valence electrons as tungsten, there is still no nitrogen-rich molybdenum nitrides to be synthesized in the binary Mo-N system to date only except for the nitrogen-rich Mo5N619. We should bear in mind that 3



the oxidation state of molybdenum atom is up to +6 in some chemical systems (e.g., MoO3). Early studies only reported three different phases with nitrogen-deficient of the binary Mo-N system (tetragonal Mo2N,20 cubic Mo3N221 and hexagonal MoN19). Most recently, Wang’s group18 synthesized nitrogen-rich molybdenum nitride MoN2 under the high-pressure condition. On the theoretical side, Zhao et al.22 have investigated the rhenium nitrides. The results showed that the 3D polyhedral stacking with strong covalent bonding N-N can stabilize transition nitrides to form nitrogen-rich phases and improve the mechanical performance. More importantly, they have also proved that the N content is the key factor affecting the metallicity and hardness of rhenium nitrides. The real group-state structures of MoN2 is explored by Frapper and co-workers based on ab initio evolutionary crystal structure prediction method (USPEX)23. The result show that a pernitride phase (P63/mmc) is the ground-state structure. Zhang’s group24 predicted that the 2D Tetra-MoN2 sheet is much more stable than the H phase proposed previously. Thus, the deep theoretical investigation on nitrogen-rich molybdenum nitrides is rather necessary. In this work, we systematically studied the structure, high pressure behavior and phase transition of the nitrogen-rich molybdenum nitride MoN2 with various different structures including experiment synthesized rhombohedral R3m type18 and our theoretical predicated structures. The aim of the present work is to give a comprehensive understanding of the structural characterization and mechanical property of MoN2. We hope that this investigation can provide help for the synthesis of novel nitride compound in the future. 2. COMPUTATIONAL DETAILS The structure searches of molybdenum nitride (MoN2) are performed using the CALYPSO (Crystal structure Analysis by Particle Swarm Optimization) structure prediction method,25-29 which has been validated with various known systems, including clusters, crystals and two-dimension materials.30-35 The first generation of structures is generated randomly and subsequently optimized. Each generation contain 50 structures, 60% of which are generated via Particle Swarm Optimization and the others are new and will be produced randomly. We usually follow 50 generations to 4


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achieve convergence. All the candidate lowest-energy structures of the global minimum are optimized using the first-principle method within CASTEP code.36 The generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA-PBE)37 is used to describe the exchange correlation function. The ultrasoft pseudopotentials treat 2s22p3 and 4d55s1 as valence electrons for N and Mo, respectively. A cutoff energy of 550 eV for the expansion of the wave function into plane waves and fine Monkhorst-Pack k meshes38 are chosen. These ensure that the total energy is well converged to be within 1×10-6 eV. Besides, the tolerances of geometry optimization are set as the difference in total energy being within 5×10-6 eV/atom, the maximum stress within 0.02 GPa, the maximum ionic Hellmann-Feynman force within 0.01 eV/Å, and the maximum ionic displacement within 5×10-4 Å, respectively. The phonon dispersion calculations are performed by using a supercell approach as implemented in PHONOPY code.39, 40 3. RESULTS AND DISCUSSION 3.1 Crystal structure Based on structure searching, the orthorhombic (space group Cmc21), rhombohedral (space group R3-mh), rhombohedral (space group R3mh), monoclinic (space group P121/c1), hexagonal (space group P63/mmc) and hexagonal (space group P6/mmm) phases are predicted (denoted as Cmc21-, R3-mh-, R3mh-, P121/c1-, P63/mmc, and P6/mmm-MoN2,

respectively,

hereinafter).

The

experimental

synthesized

rhombohedral R3m-MoN218 is also certainly observed, which provides important support for the reliability of the present structure searches. Table 1 summarizes the optimized lattice parameters (a, b and c), unit cell volumes (V) and formation enthalpies ∆H of various MoN2 structures. The formation enthalpies are calculated by formulas ∆H1 = E(MoN2) - E(Mo) - E(N2) and ∆H2 = E(MoN2) - E(MoN) -

1 E(N2), 2

where E(MoN2) and E(MoN) are the total energies per formula unit of the MoN2 and MoN compounds, E(Mo) and E(N2) are the energies of pure Mo and solid molecular N2 at the equilibrium structure (using their bcc phase, hexagonal α phase with space group P63mc, respectively). It is generally known that a compound is stable if it meets 5



the following three criteria: (1) mechanical stable if the elastic constants meet the mechanical stability criteria41; (2) thermodynamically stable if its formation enthalpy is negative; (3) dynamic stable if its phonon dispersion spectra possesses no imaginary frequency. According to the calculated total energies, it can be found that the energy order of various different MoN2 structures is Cmc21-MoN2 < P121/c1-MoN2 < R3-mh-MoN2 < R3m-MoN2 < R3mh-MoN2 < P6/mmm-MoN2 < P63/mmc-MoN2, indicating that orthorhombic Cmc21 phase is the most stable structure at ambient condition. The calculated equilibrium lattice parameters, volumes and formation enthalpies are listed in Table 1. For the experimental synthesized R3m-MoN2 crystal , the calculated lattice parameters are very close to the experimental results18 with the deviations less than 2%. The first four stable structures are depicted in Figure 1. From Figure 1, it is clearly found that various MoN2 phases have intriguing polyhedral stacking structures, which is accord with the result on RexNy obtained by Zhao’s group22. Zhao et al. point out that polyhedral stacking configuration can be an effective way to stable RexNy and it often forms in dense N-rich phase. For Cmc21-MoN2 structures (Figure 1a), the stacking consists of corner and edge shared MoN7 decahedrons and bonds with N-N connections too. The unit cell has 12 atoms. Mo atoms take the Wyckoff 4a (0, 0.0925, 0) position, and N atoms occupy the Wyckoff 4a (0, 0.2981, 0.8015) and 4a (0, 0.4299, 0.1188) positions in this structure. The shortest bond length of Mo-N is 2.101 Å. The stacking structure for P121/c1-MoN2 (see Figure 1b) consists of MoN6 octahedrons. As to the R3-mh-MoN2 (Figure 1c) and experimental observed R3m-MoN2 structures (Figure 1d), the linkage between close-packed layers of Mo atoms is connected via the single N-N bond of the pernitride unit. Mo atoms exhibit 6-fold coordinates in nitrides and the oxidation state of Mo is +4. 3.2 Stability Mechanical stability and elastic property The mechanical stability criteria41 of the considered crystal structures are introduced as following. The elastic constants are calculated using the strain-stress method. The 6


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bulk modulus B and shear modulus G are determined by the Voigt-Reuss-Hill (VRH) approximations42. The Young’s modulus E and Poission’s ratio ν are estimated via the following formulas41: E = 9BG/(3B+G),

ν = (3B - 2G)/[2(3B + G)] .

The mechanical stability criteria of orthorhombic phase are given by C11 > 0,

C22 > 0,

C33 > 0,

C44 > 0,

C55 > 0,

C66 > 0,

[C11 + C22 + C33 + 2(C12 + C13 + C23)] > 0, (C11 + C22 - 2C12) > 0, (C11 + C33 -2C13) > 0,

(C22 + C33 - 2C23) >0

The criteria of mechanical stability for rhombohedral is that C11 > |C12|, (C11 + C12) C33 - 2C132 > 0, (C11 - C12) C44 - 2C142 > 0 Monoclinic phase (C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35 and C46) C11 > 0,

C22 > 0,

C33 > 0,

C44 > 0,

C55 > 0,

C66 > 0,

[C11 + C22 + C33 + 2(C12 + C13 + C23)] > 0, (C33C55 - C352) > 0,

(C44C66 - C462) > 0,

(C22 + C33 -2C23) > 0,

[C22 (C33C55 - C352) + 2C23C25C35 - C232C55 - C252C33] > 0, {2[C15C25 (C33C12 - C13C23) + C15 (C22C13 - C12C23) + C25C35 (C11C23 - C12C13)] - [C152 (C22C33 - C232) + C252 (C11C33 - C132) + C352 (C11C22 - C122) + C55g] > 0 where g = C11C22C33 - C11C232 - C22C132 - C33C122 + 2C12C13C23 Hexagonal phase (C11, C33, C44, C12 and C13) C44 > 0,

C11 > |C12|,

(C11 + 2C12) C33 > 2C132

The elastic stiffiness constants Cij for MoN2 with various space groups (SG) are calculated using strain-stress method and tabulated in Table 2. According to Table 2, it can be seen that R3mh-MoN2 and P63/mmc-MoN2 phases do not satisfy the mechanical stability criteria, indicating they are mechanical instable. Thus, these two crystal structures are ruled out firstly. The Cmc21-MoN2 and P6/mmm-MoN2 phases possess the larger values of elastic constants C11 (580 and 328 GPa) and C33 (613 and 758 GPa), respectively. This suggests that they are extremely difficult to compress along the a axis and c axis. Moreover, their extremely large C33 (613 and 758 GPa) are comparable to that of c-BN (773 GPa)43,44, revealing the extremely high incompressibility along the c axis. 7



Based on VRH approximations42, the bulk modulus B, shear modulus G, Young's modulus E and Poission's ratio ν are estimated and collected in Table 3. It can be seen from Table 3 that Cmc21-MoN2 phase has the largest bulk modulus (324 GPa), shear modulus (187 GPa) and Young's modulus (470 GPa) among the considered stable structures. The bulk modulus of Cmc21-MoN2 (324 GPa) is higher than those of hard materials SiC (212 GPa)45 and Al2O3 (252 GPa)46, which further suggests that it is hard to compress. The large Young's modulus shows that it has strong capability to resist tension and pressure in the range of elastic deformation. The shear modulus is a better indicator of potential hardness and can show the resistance to shear deformation. Compared with B and E, the smaller shear modulus (187 GPa) suggests that Cmc21-MoN2 phase has weak resistance to the change of shape. In addition, the Poission's ratio ν is an important parameter to describe the directional degree of covalent bonding. According to the ν collected in Table 3, we find that there are strong covalent bonds exist in these mechanically stable MoN2 phases. Thermodynamic and dynamical stability The thermodynamic stability of a compound is very importance for its application. Therefore, the formation enthalpies of MoN2 with various different structures are explored under the high pressure up to 100 GPa. The curves of calculated formation enthalpies of the first four stable structures (Cmc21-MoN2, P121/c1-MoN2, R3-mh-MoN2 and R3m-MoN2), with respect to the reactant of Mo + N2, are shown in Figure 2a. From Figure 2a, it can be clearly seen that the Cmc21-MoN2 and P121/c1-MoN2 crystals are thermodynamically stable at ambient pressure condition. With the increasing of pressure, the formation enthalpies of the considered structures decrease gradually. Above 3.9 GPa, the R3-mh-MoN2 structure transforms to thermodynamically stable structure. The R3m-MoN2 structure, which can be synthesized experimentally at a moderate pressure of 3.5 GPa18, is thermodynamically stable when the pressure is up to 13.8 GPa. These all indicate that the high pressure is beneficial to their thermodynamic stabilities. The difference between our theoretical predication and the experiment result of Wang et al18 should be further confirmed. At ambient pressure, our theoretical predicted Cmc21-MoN2 is the most stable phase. 8


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When the pressure is up to 25.8 GPa, the P121/c1-MoN2 phase is more stable than the Cmc21-MoN2 phase. And the difference between the formation enthalpies of Cmc21-MoN2 and P121/c1-MoN2 increase with the increasing of pressure. For the reactant of

MoN +

1 N 2 (Figure 2b), the Cmc21-MoN2 and 2

P121/c1-MoN2 phases become thermodynamic stable above 17.5 and 20 GPa, respectively. This further verify the importance of the pressure for their thermodynamic stabilities. It is worth pointing out that the phase transformation between Cmc21-MoN2 and P121/c1-MoN2 occurs at the same pressure as that of Mo + N2 reactant. In the whole range of pressure (0-100 GPa), the R3-mh-MoN2 and R3m-MoN2 phases are thermodynamic unstable toward dissociation into the corresponding production MoN and solid molecular N2, suggesting that the MoN compound may further dissociate into Mo and N2. We hope that the present results could provide a theoretical prerequisite for the experimental synthesis and practical applications of MoN2. In order to estimate the dynamic stabilities of the considered crystals, we calculate their phonon dispersion within the finite displacement theory. The calculated results are presented in Figure 3. According to Figure 3a, it can be clearly seen that the phonon dispersion of Cmc21-MoN2 crystal structure has no imaginary frequency in the whole Brillouin zone, indicating that this structure is dynamically stable and can be synthesized in experiment under ambient condition. However, the P121/c1-MoN2, R3-mh-MoN2 and R3m-MoN2 crystals are dynamically unstable due to their imaginary frequencies of phonon dispersion (see Figures 3b, 3c and 3d). In addition, we also explored the phonon dispersion at the high pressure (0-100GPa) to detect the effect of pressure on dynamical stability. The results shown that the Cmc21-MoN2 structure is dynamically stable in the entire range of pressure, while the other three phases are dynamically unstable. 3.3 Electronic density of states The electronic density of states is very important to understand the origin of physic properties of crystal structure. Thus, we calculated the total and partial density of

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states (TDOS and PDOS) of the selected four structures and listed them in Figures. 4a, 4b, 4c and 4d respectively, in which the dotted lines indicate Fermi level EF. From Figure 4, it is clearly found that MoN2 with these structures have the similar metallic bonding character because of the finite DOS at the Fermi level (EF), which originates mostly from 4d electrons of Mo. This metallic character can also be confirmed by the overlapping of the valence bands and conduction bands around the Fermi level. In addition, we find that the monoclinic P121/c1-MoN2 phase has the smallest DOS value at the EF. The PDOS can be decomposed into three energy regions: (a) the lowing-energy region of -22 eV to -12.5 eV is mainly originated from N-2s states with full area (see Figure 4); (b) the region of valence-band complex stems mainly from N-2p states mixed with Mo-4d states; (c) the energy region above EF is dominated mainly by unoccupied 4d states of transition metal Mo and N-2p states. It can be seen that there is a strong hybridization between Mo-4d states and N-2p states due to their orbitals energy degenerate. Thus, the strong covalent bonding is formed in MoN2 crystals, which may benefit for their large bulk modulus and shear modulus. The total density of states around the Fermi level EF for Cmc21-MoN2 and P121/c1-MoN2 lies in a dip, confirming the two phases are stable. 3.4 Hardness The high hardness of materials may be related to their high bulk modulus, shear modulus, low Poisson’s ratios, and strong covalent bonds. Therefore, we estimate the Vicker hardness of various MoN2 by our group’s hardness formula, in which the metallicity of bonds is taken into crystal structures. The expression is written as follows: H v (Gpa ) = 699 Pvb−5/3 exp ( −3005 f m1.553 )

(1)

3 3 where νb is the volume of bond, it can be given by vbµ = ( d µ ) / ∑ ( d v ) N bv  , P is  

Mulliken population, and fm is a factor of metallicity f m =

0.026 DF . This method has ne

been successfully used to estimate transition-metal carbides and borides47-49. 10


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For the complex compounds which possess at least two types of chemical bonds in the unit cell, the total hardness can be expressed as the average of hardness of all binary systems. 1/ ∑ n µ µ µ µ n  H v (Gpa) =  ∏ ( H v )   

(2)

where nµ presents the number of µ-type bonds. Using the above formula (2) of hardness, we have estimated the hardness of MoN2 with four different structures Cmc21-MoN2, P121/c1-MoN2, R3-mh-MoN2 and R3m-MoN2. The calculated bond parameter and hardness are collected in Table 4. In addition, we also present the calculated hardness by using the method of Chen50 and list them in Table 4 as comparison. The Chen’s model is given as the following formula: Hν (GPa ) = 2(k 2G ) 0.585 − 3

(3)

where k=G/B is Pugh’s modulus ratio. During calculation of hardness, the antibonding bonds with negative Mulliken population are not considered. From Table 4, it is found that the experimental synthesized R3m-MoN2 (space group R3m, No. 160) contains 6 N-Mo bonding bonds. Half of them possess bond lengths of 1.980 Å, and the lengths of the other N-Mo bonds are 2.133 Å. The Mulliken overlap population of two types of N-Mo bonds are 1.22e and 1.19e, respectively. Based on our semiempirical model, the predicted Vicker hardness value is 2.341 GPa, and the result of Chen’s model is 4.304 GPa. A similar situation can also be found in rhombohedral R3-mh type and monoclinic P121/c1 type MoN2, in which only bonding bonds N-Mo exist. As for our predicted phase Cmc21-MoN2, there are 4 N-N bonds and 16 N-Mo bonds. The calculated Vicker hardness value is 11.987 GPa (19.419 GPa), which is comparable to those of MoN (11.7 GPa51, 25.4 GPa52 and 28.0 GPa53). The Mulliken overlap population of N-N bonds is 0.60e, which is significant large among all of the bonds. Thus, we inferred that the strong covalent N-N bond dominates the hardness of Cmc21-MoN2 phase, and the strength and number of covalent bonds are important to hardness.

4. CONCLUSION 11



In summary, we have systematically investigated the structures, stabilities, electronic properties and hardness of MoN2 with various different structures based on the first principle method. The obtained results are summarized as follows. (1) Based on the structural search, formation enthalpy and phonon dispersion, we found that the orthorhombic Cmc21-MoN2 is the most stable phase for MoN2 compound at ambient pressure condition. The experimental synthesized rhombohedral R3m type MoN2 is thermodynamically stable structure above 13.8 GPa. (2) According to the formation enthalpy, it is concluded that the high pressure is beneficial to the thermodynamic stabilities of materials. Meanwhile, a phase transformation is observed between Cmc21-MoN2 and P121/c1-MoN2 structure when the pressure is up to 25.8 GPa. (3) The electronic density of states show that all the considered MoN2 structures have the similar metallic bonding nature. The DOS near the Fermi level (EF) originates mainly from Mo-4d electrons. By analyzing PDOS, we find that there is at least two types of bonds (covalent and ionic bonds) exist in MoN2. (4) The predicted Vicker hardness of MoN2 by using our group’s semiempirical method indicates that the Vicker hardness of theoretical obtained MoP2-MoN2 phase is 11.987 GPa, which is comparable with that of MoN. Moreover, it is found that the strength and number of covalent bonds dominate the hardness.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

Notes The authors declare no competing financial interest.

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ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (Nos. 11604194,

11304167 and 21671114),

2014CB660804),

The

973 Program of China (No.

Natural Science Foundations

of Shaanxi Province (Nos.

2016JQ1028 and 2016JQ1003), the Shaanxi University of Science & Technology Key Research Grant (Nos. 2016BJ-01 and BJ15-07), and the Program for Science & Technology

Innovation

Talents

in

Universities

of

Henan

Province

(No.

15HASTIT020

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17



Page 18 of 22

Table 1. Calculated lattice parameters a, b and c (Å), unit cell volume V (Å3), and formation enthalpies ∆H1, ∆H2 (eV/atom) of MoN2 with various different structures phase

a

b

c

V

∆H1

∆H2

R3m-MoN2 Exp.a Cmc21-MoN2 R3-mh-MoN2 R3mh-MoN2 P121/c1-MoN2 P63/mmc-MoN2 P6/mmm-MoN2 a Reference 18

2.86 2.85 2.87 3.06 3.07 5.94 4.63 3.43

2.86 2.85 9.11 3.06 3.07 4.82 4.63 3.43

15.67 15.94 4.17 24.25 12.48 5.98 17.16 2.61

111.30 112.43 108.74 196.11 101.77 146.01 327.62 26.64

0.37

1.66

-1.02 0.11 3.57 -0.88 3.77 0.38

0.27 1.39 4.85 5.05 0.40 1.67

Table 2. Elastic stiffiness constants Cij for MoN2 from various space groups (SG) SG

C11

C22

R3m Cmc21 R3-mh R3mh P121/c1 P63/mmc P6/mmm

160 580 371 124 155 -660 328

675

342

C33

C44

55 613 74 -104 91 280 758

28 223 9 27 64 -450 10

C55

C66

215

99

C12

C13

244 165 165 122 13 -868 430

82 170 22 -154 56

C15

C23

C25

C35

C46

0.4

0.4

0.2

194

2

0.2

70

Table 3. Calculated bulk modulus B (GPa), shear modulus G (GPa), young’s modulus E (GPa), poission’s ratio ν, and density ρ (g/cm3) of MoN2 system at different structures phase

B

G

E

ν

ρ

R3m-MoN2 Cmc21-MoN2 R3-mh-MoN2 P121/c1-MoN2 P6/mmm-MoN2 a Reference 18

90 324 101 189 285

42 187 41 139 40

109 470 108 335 115

0.29 0.25 0.32 0.21 0.43

6.23; 5.49a 7.57 6.29 5.37 7.73

18


Page 19 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4. The calculated bond parameters and vicker hardness of the various structural MoN2 crystals

Cmc21-MoN2

P121/c1-MoN2

R3-mh-MoN2 R3m-MoN2

bond type

dµ

νbµ

P

fm(×10-3)

Hνµ

N-N

1.429

0.456

0.60

0

154.164

N-Mo

2.101

1.448

0.41

4.566

7.639

N-Mo

2.158

1.569

0.55

4.566

8.964

N-Mo

2.178

1.614

0.71

4.566

11.048

N-Mo

2.220

1.709

0.15

4.566

2.122

N-Mo

1.832

0.983

0.84

1.235

54.680

N-Mo

1.843

1.001

0.77

1.235

48.651

N-Mo

1.981

1.243

0.51

1.235

22.454

N-Mo

2.004

1.287

0.52

1.235

21.612

N-Mo

2.273

1.878

0.15

1.235

3.321

N-Mo

2.576

2.733

0.11

1.235

1.303

N-Mo

2.044

2.623

1.15

8.613

1.257

N-Mo

2.095

2.824

1.22

8.613

1.179

N-Mo

1.980

5.496

1.22

5.866

2.856

N-Mo

2.133

6.871

1.19

5.866

1.919

Hνcalc 11.987a 19.419b

13.320a 22.018b

1.484a 3.999b 2.341a 4.301b

a, b represent two types different methods of our hardness formula and Chen’s model, respectively.

19



Figure 1. Optimized crystal structures: (a) Cmc21-MoN2, (b) P121/c1-MoN2, (c) R3-mh-MoN2 and (d) R3m-MoN2.

(a)

(b)

(d)

(c)

Figure 2. The calculated formation enthalpies with respect to the different reactants (a)

(a)

1

13.8GPa

0

1 N2 . 2

Cmc21-MoN2

P121/c1-MoN2

R3-mh-MoN2

R3m-MoN2

3.9GPa

-1

25.8GPa

-2 -3 -4

Formation enthalpy (eV)

Mo + N 2 , (b) MoN +

Formation enthalpy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 22

-5 0

10

20

30

40

50

60

70

80

90 100

(b)

2.0 1.5

Cmc21-MoN2

P121/c1-MoN2

R3-mh-MoN2

R3m-MoN2

1.0 0.5 0.0

20 GPa 17.5 GPa

-0.5

25 GPa

-1.0 -1.5 -2.0 -2.5

0

10

20

30

40

50

60

70

Pressure (GPa)

Pressure (GPa)

20


80

90 100

Page 21 of 22

Figure 3. Phonon dispersion curves: (a) Cmc21-MoN2, (b) P121/c1-MoN2, (c) R3-mh-MoN2 and (d) R3m-MoN2. 30

(a)

(b)

(c)

(d)

25 20 15 10 5 0

2018/8/15

Figure 4. Total and partial density of states: (a) Cmc21-MoN2, (b) P121/c1-MoN2, (c)

R3-mh-MoN2 and (d) R3m-MoN2. The Fermi level is at zero. 15

(a)

8

s d

s d

(b)

p TDOS

p TDOS

DOS (states/eV)

DOS (states/ev)

12 6

4

2

9 6 3

0 -25

-20

-15

-10

-5

0

5

0 -25

10

-20

-15

Energy (eV)

(c)

s d

-10

-5

0

5

10

0

5

10

Energy (eV) 12

20 p TDOS

(d)

s d

10

p TDOS

DOS(states/eV)

15

DOS (states/eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

5

-20

6 4 2

2018-5-5

0 -25

8

-15

-10

-5

0

5

10

0 -25

-20

-15

Energy (eV)

21


-10

-5

Energy (eV)


TOC Graphic

30

s d

8

p TDOS

DOS (states/ev)

25

Frequency (THZ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 22

20 15 10

6

4

2

5 0 G

Z

T

Y S

X

U

R

0 -25

-20

-15

-10

-5

Energy (eV)

22


0

5

10

Crystal Structures, Electronic Properties and Hardness

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