Mechanical and metallic properties of tantalum nitrides from first-principles calculations

Abstract

The phase stability, mechanical properties and metallic properties of tantalum nitrides are extensively studied by means of first principles calculations. The relationship between nitrogen concentration and physical properties of tantalum nitrides has been systematically investigated. With the nitrogen concentration increasing, it is found that the feature of covalent bonding enhances and the directionality of the covalent bonding and hardness of tantalum nitrides reduce. While these make the ductility of tantalum nitrides improve with the nitrogen concentration increasing. The intensity of metallic properties of tantalum nitrides can be effectively adjusted by controlling the nitrogen concentration and pressure. When the tantalum: nitrogen ratio reaches Ta : N = 1 : 3, remarkable nitrogen–nitrogen bonds are found in TaN₃. The hardness of TaN₃ abnormally increases with reference to that of the preceding composition Ta₃N₅-II. The potential synthesis routes of tantalum nitrides are suggested.

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1 Introduction

Superhard materials have attracted much attention due to their wide industrial and technological applications. Typical superhard materials come from nature. Diamond is the most important one in typical superhard materials.¹ Recently, to satisfy complicated industrial requirements, some novel superhard materials with unique physical and chemical properties, such as c-BN,² C₃N₄,³ BC₂N,⁴ and BC₅ (ref. 5) etc., have been designed and synthesized by using theoretical and experimental methods. Over the past few decades, it has been proposed that the intercalation of light elements into transition metals might be a good strategy in the search for potential superhard materials. Very recently, four noble metal nitrides PtN,⁶ PtN₂,⁷ IrN₂,⁸ and OsN₂ (ref. 8) have been successfully synthesized by diamond anvil techniques under high pressure and high temperature (HPHT).

The tantalum (Ta) has similar valence electron arrangement and bulk modulus to osmium (Os), iridium (Ir), and platinum (Pt) in the periodic table of elements. So tantalum nitrides are expected to have the same good mechanical properties as osmium, iridium, and platinum nitrides. And tantalum and nitrogen can form semiconductor or weak metallic compounds with different compositions. The tantalum nitrides display rich crystal chemistry. In the Ta–N binary phase diagram, tantalum nitrides have six experimentally known compositions: Ta₂N, TaN, Ta₅N₆, Ta₄N₅, Ta₃N₅, and Ta₂N₃. The α-Ta₂N is an amorphous phase of Ta₂N.⁹ β-Ta₂N adopts the Fe₂N-type structure.¹⁰ The TaN exists in three forms: θ-TaN (WC-type structure), δ-TaN (NaCl-type structure), and ε-TaN (CoSn-type structure).^11,12 The first principle calculations indicate that the presence of Ta vacancies can reduce the density of state (DOS) around the Fermi level and result in a metal-to-insulator transition in δ-TaN.¹³ The Ta₅N₆ (space group: P6₃/mcm) crystallizes with hexagonal symmetry. The Ta₄N₅ (space group: I4/m) persists on the NaCl-like structure, consisting of an ordered arrangement of Ta vacancies, also exhibits a notable low DOS at the Fermi level.¹³ The lower DOS at Fermi level is the key for the formation of superhard materials.¹⁴ The most nitrogen-rich tantalum nitride Ta₃N₅-I (space group: Cmcm) is a semiconductor with a band gap of 1.5 eV.^15–17 The high-pressure polymorph Ta₃N₅-II (space group: Pnma) is a potential hard material with a high bulk modulus of 378 GPa at above 9 GPa.¹⁵ Very recently, an orthorhombic U₂S₃-type Ta₂N₃ (space group: Pbnm) was synthesized under HPHT conditions.¹⁸ It exhibits a high hardness and an extraordinary texture and can be quenched to ambient condition. However, the theoretical study reveals that the experimentally observed U₂S₃-type Ta₂N₃ actually is an oxygen-substituted orthorhombic structure. Oxygen plays a key role for the stabilization of the experimentally observed U₂S₃-type Ta₂N₃ at ambient pressure. The extended enthalpy calculations indicate that tetragonal Ta₂N₃ (space group: P4̄m2) is energetically more favorable to the experimentally observed orthorhombic U₂S₃-type Ta₂N₃ at below 7 GPa.¹⁹ Among the above mentioned tantalum nitrides, δ-TaN is superconducting (T_c = 6.5 K),²⁰ and exhibits the highest hardness of 30–32 GPa in the group of transition metal mononitrides.²¹ Ta₃N₅ has attracted much attention as a visible-light-driven photocatalyst for splitting water.²² However, the current study of tantalum nitrides is still limited to lower nitrogen concentration (the highest nitrogen content reach to x = 1.67 in TaN_x). There is lack of reports on the nitrogen-rich tantalum nitrides and the relationship between physical properties and nitrogen composition. The crystal structures, mechanical properties and metallic properties of nitrogen-rich tantalum nitrides are still far from being clear. It is of fundamental interest to explore nitrogen-rich structures in tantalum nitrides. Moreover, the stability, synthesis routes, and the origin of hardness of tantalum nitrides are least studied.

In this study, we report a systematic computational study on the crystal structures, stability, mechanical properties, metallic properties and synthesis routes of tantalum nitrides (TaN_x, x ≥ 1). The crystal structures of nitrogen-rich tantalum nitrides are explored by ab initio evolutionary crystal structure prediction USPEX method.^23–25 The relationship between mechanical and metallic properties and nitrogen concentration are systematically studied. The potential synthesis routes of tantalum nitrides are suggested. We uncover a synthesizable new composition TaN₃ with strong covalent nitrogen–nitrogen bonds chains along the crystallographic b axis. Our study could be extremely helpful for future experiments.

2 Computational details

In this study, the experimentally observed compositions of TaN, Ta₅N₆, Ta₄N₅, Ta₂N₃, and Ta₃N₅ were adopted for the calculation. For nitrogen-rich tantalum nitrides, since no experimental structure parameters were found to date, we used the evolutionary algorithm (USPEX code),^23–25 designed to search for the structure possessing the lowest free energy at given P/T conditions, to predict the possible structures. The evolutionary variable-cell structure predictions were performed at 0, 10, 30, 50, and 80 GPa with systems containing one, two, three, and four formula units (f.u.) in the simulation cell. In contrast to the early calculations and experiments, the underlying structure relaxations were performed using density functional theory (DFT) within the generalized gradient approximation (GGA),²⁶ as implemented in the Vienna ab initio simulation package (VASP).^27,28 The electron interaction was described by means of the projector-augmented wave method.²⁹ The 2s² 2p³ and 6s² 5d³ were treated as valence electrons for N and Ta, respectively. Convergence tests gave a kinetic energy cutoff as 900 eV for all phases. Monkhorst–Pack³⁰ k point meshes with a grid of 0.035 Å⁻¹ for Brillouin zone sampling were chosen to ensure the total energies converged to be better than 1 meV per f.u. Gamma-point-centered k-mesh were used for the hexagonal structures. The phonon spectra were calculated using the direct supercell method, performed by PHONON software.³¹ The elastic constants were calculated by strain-stress method, and the bulk modulus, shear modulus, Young's modulus, and Poisson's ratio were thus derived from the Voigt–Reuss–Hill approximation.^32–34 The theoretical Vickers hardness was estimated by using the Chen's model.¹⁴ Formation enthalpy was calculated by the equation of ΔH = E(Ta_xN_y)−xE(Ta)−yE(N), in which bcc phase of Ta and α-N₂ were adopted for the calculations. The details of convergence tests have been described elsewhere.^35–37

3 Results and discussion

The optimized crystal structures of tantalum nitrides are shown in Fig. 1. Each Ta atom has six nearest neighboring N atoms in WC-type TaN, NaCl-type TaN, CoSn-type TaN, Ta₅N₆, and Ta₄N₅ structures. Each Ta atom has seven nearest neighboring N atoms in P4̄m2-Ta₂N₃. The nearest neighbor number of Ta atoms is five or six in Ta₃N₅-I. The TaN₃ has seven N atoms nearest neighboring Ta atom at 50 GPa. The calculated lattice parameters, cell volumes, and formation enthalpy of tantalum nitrides with various nitrogen concentrations in the paramagnetic and ferromagnetic states are summarized in Table 1. For tantalum mononitrides, the WC-type structure is energetically more stable than NaCl-type and CoSn-type structures at ambient pressure. The calculated lattice parameters of tantalum nitrides are in good agreement with the previous experimental results.^11,38–40 So the parameters of the calculations used in the study are sufficient to predict Ta–N systems. The bond length of Ta–N bond is 2.235 Å in WC-type TaN, which is larger than the sum of bonding radii (2.09 Å) of Ta (r = 1.34 Å) and N (r = 0.75 Å) atoms. The Ta₅N₆ persists on the hexagonal structure with 22 atoms in the unit cell. The shortest bond length of Ta–N bond is 2.193 Å in Ta₅N₆, which is still larger than the sum of the bonding radii (2.09 Å), indicating the obvious weaker covalent bonding feature. The NaCl-like Ta₄N₅ has intrinsic Ta vacancies in the center of the structure as shown in Fig. 1. The shortest bond length of Ta–N bond is 2.086 Å in Ta₄N₅, which is very close to the sum of bonding radii (2.09 Å), indicating strong covalent bonding feature. The P4̄m2-Ta₂N₃ is energetically more favorable compared to the U₂S₃-type Ta₂N₃ at ambient pressure as summarized in Table 1. The shortest Ta–N bond length of P4̄m2-Ta₂N₃ is 2.117 Å which is still larger than the sum of bonding radii (2.09 Å). For Ta₃N₅, Ta₃N₅-I has a lower formation enthalpy than that of Ta₃N₅-II, suggesting that the Ta₃N₅-I is thermodynamically stable at ambient condition. The calculated lattice constants of Ta₃N₅ are very close to the experimental values, indicating the reliability of our calculations. Obvious covalent bonding feature is found in Ta₃N₅-I because of the presence of the shortest Ta–N bond length (1.963 Å) among the above mentioned tantalum nitrides. We find that the bond length of Ta–N bond of tantalum nitrides decrease with the nitrogen concentration increasing (TaN_x, x < 1.67). The degree of covalent bonding, which is more important for the hardness of materials, can be enhanced by increasing nitrogen concentration in tantalum nitrides. The shortest nitrogen–nitrogen distance (2.42 Å) of the above mentioned tantalum nitrides is still larger than the bond length of the single N–N bond (1.45 Å for molecule N₂H₄). So no nitrogen–nitrogen covalent bonds are found in these phases (TaN_x, x < 1.67). However, the distinct nitrogen–nitrogen bonds chain is found in nitrogen-rich TaN₃ (space group: P2₁/m). The shortest bond length of nitrogen–nitrogen bond (1.396 Å) is obviously smaller than the single nitrogen–nitrogen bond length of the hydrazine molecule N₂H₄ (1.45 Å), indicating the presence of strong covalent bonding feature in nitrogen-rich TaN₃. The hardness of nitrogen-rich TaN₃ could be expected to be enhanced because of the presence of strong covalent nitrogen–nitrogen bonds.

Fig. 1 The crystal structures of the Ta–N system (TaN_x, x ≥ 1): (a) the WC structure of TaN. (b) The NaCl structure of TaN. (c) The CoSn structure of TaN. (d) The Ta₅N₆ structure with space group P6₃/mcm. (e) The Ta₄N₅ structure with space group I4/m. (f) The Ta₂N₃ structure with space group P4̄m2. (g) The U₂S₃ structure of Ta₂N₃. (h) The Ta₃N₅-I with space group Cmcm. (i) The Ta₃N₅-II with space group Pnma. (j) The TaN₃ is a monoclinic structure (space group P2₁/m) with Ta at 2e (0.681, 0.25, 0.344), N1 at 4f site (0.022, 0.439, 0.186), and N2 at 2e (0.497, 0.25, 0.756). The large and small spheres represent Ta and N atoms, respectively.

Table 1 The calculated formation enthalpy per atom ΔH, optimized equilibrium lattice parameters a, b, and c (Å), and cell volume per formula unit (V in ų) of tantalum nitrides in the paramagnetic (PM) and ferromagnetic (FM) states^a

	Structure	Mag	ΔH	a		b		c		V
a Experiment: a (ref. 11), b (ref. 38), d (ref. 39), c (ref. 52), g (ref. 18) and i (ref. 16). Theory: e (ref. 13), f (ref. 19) and h (ref. 15).
TaN	WC	PM	−1.228	2.948	2.913^a	2.948	2.913^a	2.897	2.862^a	21.799
					2.93^b		2.93^b		2.86^b
		FM	−1.230	2.947		2.947		2.896		21.7763
	NaCl	PM	−0.932	4.415	4.427^c					86.041
					4.413^a
		FM	−0.930	4.413						85.935
	CoSn	PM	−0.785	5.269	5.18^a	5.269	5.18^a	2.920	2.90^a	70.202
					5.196^d		5.196^d		2.911^d
					5.221^a		5.221^a		2.921^a
		FM	−0.753	5.232		5.232		2.929		69.4287
Ta4N5	I4/m	PM	−1.176	6.895	6.831^e	6.895	6.831^e	4.279	4.269^e	203.447
		FM	−1.135	6.848		6.848		4.339		203.505
Ta2N3	P4̄m2	PM	−1.138	2.984	2.99^f	2.984	2.99^f	5.813	5.82^f	51.749
		FM	−1.126	2.984		2.984		5.810		51.726
	U2S3	PM	−1.119	8.227	8.19^g	8.180	8.18^g	2.996	2.98^g	201.585
					8.19^f		8.24^f		3.00^f
		FM	−1.117	8.243		8.160		2.994		201.397
Ta3N5	Cmcm	PM	−1.116	3.909	3.905^h	10.307	10.321^h	10.329	10.349^h	416.196
					3.89^i		10.22^i		10.28^i
		FM	−1.088	3.569		10.752		10.795		414.219
	Pnma	PM	−0.968	10.978	10.998^h	2.960	2.968^h	9.594	9.607^h	311.704
		FM	−0.953	10.973		2.968		9.579		311.902
TaN3	P21/m	PM	−0.366	5.483		3.798		3.794		74.123
		FM	−0.363	5.482		3.801		3.789		74.093

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To understand the thermodynamic stability and synthesis routes of tantalum nitrides, we performed convex hull calculation for tantalum nitrides at 0 and 50 GPa. The convex hull is defined as the formation enthalpy versus composition plot. Any structure whose formation enthalpy lies on the convex hull is deemed stable and synthesizable in principle.^41–45 As shown in Fig. 2(a), the convex hull is composed of five structures: Ta₂N, TaN-WC, Ta₅N₆, Ta₄N₅, Ta₂N₃-P4̄m2, and Ta₃N₅-I at ambient pressure. Above mentioned six structures have been observed in the experiments. For TaN, the WC-type TaN has the lowest formation enthalpy on the convex hull, indicating that the WC-type TaN is easier to be synthesized than NaCl-type and CoSn-type structures. In addition, the formation enthalpy of P4̄m2-Ta₂N₃ lies on the convex hull, indicating the P4̄m2-Ta₂N₃ structure is thermodynamically stable with respect to U₂S₃-type Ta₂N₃ at ambient condition. For Ta₃N₅, the Ta₃N₅-I is thermodynamically stable with reference to Ta₃N₅-II at ambient condition. However, the high-pressure convex hull at 50 GPa is different from the convex hull at ambient pressure. Fig. 2(b) shows that the Fe₂N-type Ta₂N, WC-type TaN, U₂S₃-type Ta₂N₃, Ta₃N₅-II, and P2₁/m-TaN₃ lie on the convex hull at 50 GPa. The Ta₅N₆ and Ta₄N₅ lie above the convex hull, indicating they are not thermodynamically stable at 50 GPa. The U₂S₃-type Ta₂N₃ and Ta₃N₅-II are energetically more stable than P4̄m2-Ta₂N₃ and Ta₃N₅-I at high pressure, respectively, which is in good agreement with previous study.^15,19 At 50 GPa, a novel nitrogen-rich TaN₃ structure with distinct N–N bonds chain has been found. It lies on the convex hull indicating that it could be experimentally synthesized by high-pressure technique. The calculations of the phonon dispersion and elastic constants suggest that the P2₁/m-TaN₃ is dynamically and mechanically stable at 50 GPa. The calculations of convex hull can also suggest potential synthesis routes in tantalum nitrides. It can be found that the Ta₃N₅-II could decompose into 3/2P4̄m2-Ta₂N₃ + 1/4N₂, 3/2U₂S₃-type Ta₂N₃ + 1/4N₂, or 3WC-type TaN + N₂ at ambient pressure. At high pressure, the Ta₃N₅-I could decompose into the 3/2U₂S₃-type Ta₂N₃ + 1/4N₂, 3/2P4̄m2-Ta₂N₃ + 1/4N₂, or 3WC-type TaN + N₂. The Ta₄N₅ (Ta₅N₆) could decompose into the U₂S₃-type Ta₂N₃ + 2WC-type TaN (U₂S₃-type Ta₂N₃ + 3WC-type TaN) or 4WC-type TaN + 1/2N₂ (5WC-type TaN + 1/2N₂). It is noteworthy that the synthesis routes of Ta₃N₅ → 3/2Ta₂N₃ + 1/4N₂ and Ta₃N₅ → 3TaN + N₂ have been confirmed by previous experiments.^46,47 Two potential synthesis routes (Ta₃N₅ + 2N₂ → 3TaN₃ and Ta₄N₅ + 3.5N₂ → 4TaN₃) are also be found to synthesize nitrogen-rich TaN₃. For the further study, we only consider the thermodynamically stable structures at 0 and 50 GPa, respectively.

Fig. 2 Convex hull of the Ta–N system at a pressure of (a) 0 and (b) 50 GPa. The solid line denotes the convex hull at different pressures.

In order to understand the mechanical properties, the elastic constants are calculated as summarized in Table 2. It can be found that the obtained elastic constants of tantalum nitrides all satisfy the Born–Huang criterion.⁴⁸ Interestingly, the P4̄m2-Ta₂N₃ has the largest C₁₁ and C₂₂ (689 GPa) among tantalum nitrides which make the P4̄m2-Ta₂N₃ have high incompressibility along the crystallographic a and b axis. The WC-type TaN has the largest C₃₃ (798 GPa) value among all the tantalum nitrides which make it have high incompressibility along the crystallographic c axis. It is well known that superhard materials should have high bulk modulus and high shear modulus to resist the volume change and shape change, respectively.

Table 2 The calculated elastic constants of the Ta–N system at 0 and 50 GPa

	Type	P	C11	C12	C13	C15	C22	C23	C25	C33	C35	C44	C55	C66
TaN	WC	0	622.5	213.2	140.2		614.0			798.0		233.0
Ta5N6	P63/mcm	0	528	169.2	168.5		534.5			541.7		163.3
Ta4N5	I4/m	0	536.3	168.6	122.5		536.3			751.5		138.0		167.5
Ta2N3	P4̄m2	0	689.0	146.8	178.5		689.0			607.0		144.3		1683.3
Ta3N5	Cmcm	0	393.5	206.2	166.3		391.0	127.3		385.0		114.0	120.3	58.3
TaN	WC	50	910.8	366.2	295.7		898.7			1126.5		344.3
Ta2N3	U2S3	50	728.3	415.5	381.2		962.2			899.7		249.3		186.0
Ta3N5	Pnma	50	901.5	321.8	373.8		1025.2	313.3		935.7		230.8	271.1	238.0
TaN3	P21/m	50	620.0	177.2	197.5	0	401.8	193.2	0	612.0	0	123.0	145.7	146.2

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As listed in Table 3, tantalum nitrides have high bulk moduli, which indicate that they are difficult to be compressed. At ambient condition, the bulk moduli follow the descending order WC-type TaN > P4̄m2-Ta₂N₃ > Ta₄N₅ > Ta₅N₆ > Ta₃N₅-I. The WC-type TaN has the largest bulk modulus (337.9 GPa) among all the tantalum nitrides. Generally, bulk modulus might have a direct correlation with valence electron densities.^49,50 So we investigate the average valence electron density (VED) for tantalum nitrides. It is found that the VED values of WC-type TaN (0.73 electrons per ų), P4̄m2-Ta₂N₃ (0.71 electrons per ų), Ta₅N₆ (0.688 electrons per ų), Ta₄N₅ (0.68 electrons per ų), and Ta₃N₅-I (0.56 electrons per ų) are following the similar tendency as the variation of bulk moduli. In contrast, at 50 GPa, the bulk moduli of tantalum nitrides follow the descending order WC-type TaN > U₂S₃-type Ta₂N₃ > P2₁/m-TaN₃ > Ta₃N₅. The WC-type TaN structure still has the largest bulk modulus (336.7 GPa) among the stable high-pressure structures. The Ta₃N₅-II has the smallest bulk modulus (241.0 GPa). The VED values of tantalum nitrides follow the descending order Ta₃N₅-II (0.832 electrons per ų) > U₂S₃-type Ta₂N₃ (0.823 electrons per ų) > WC-type TaN (0.818 electrons per ų) > TaN₃ (0.798 electrons per ų) at high pressure. It is noteworthy that the variation of the valence electron density does not follow the same tendency as the variation of bulk moduli at high pressure. It is different from the result at ambient pressure. The WC-type TaN also has the largest shear modulus (239.7 GPa and 244.5 GPa at 0 and 50 GPa, respectively) among the tantalum nitrides. The P4̄m2-Ta₂N₃ has the second larger shear modulus (190.2 GPa) at ambient condition. The Ta₃N₅-I has the highest nitrogen concentration among the experimentally observed tantalum nitrides. However, the shear modulus of Ta₃N₅-I is the smallest one among all the relevant structures. The Poisson's ratio and B/G ratio are also two important parameters to describe the mechanical properties except the bulk modulus and shear modulus. The Poisson's ratio is indicator of the degree of directionality of the covalent bonding. Lower Poisson's ratio (about 0.2) indicates that the directionality of the covalent bonding is good in materials.⁵¹ The B/G value is associated with the ductility (brittleness) of materials, and the critical value is about 1.75. The Poisson's ratio v was obtained by the following formula:

Table 3 Bulk modulus B, Shear modulus G, Young's modulus Y, Poisson's ratio v, and Vickers hardness of the Ta–N system at 0 and 50 GPa

	Type	P	B	G	B/G	Y	v	Hv
TaN	WC	0	337.9	239.7	1.41	581.7	0.21	30.0
Ta5N6	P63/mcm	0	290.4	175.2	1.65	437.5	0.25	19.7
Ta4N5	I4/m	0	294.6	182.7	1.61	454.3	0.24	21.1
Ta2N3	P4̄m2	0	332.5	190.2	1.75	479.2	0.26	19.4
Ta3N5	Cmcm	0	241.0	103.2	2.32	270.9	0.31	8.2
TaN	WC	50	336.7	244.5	1.37	590.4	0.21	31.3
Ta2N3	U2S3	50	332.5	190.1	1.75	479.1	0.26	19.4
Ta3N5	Pnma	50	241.0	103.2	2.32	270.8	0.31	8.2
TaN3	P21/m	50	307.7	154.0	2	396.0	0.29	14

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Table 3 reflects that the ascending order of Poisson's ratio is WC-type TaN < Ta₄N₅ (Ta₅N₆) < P4̄m2-Ta₂N₃ < Ta₃N₅-I at ambient condition. It is found that the degree of directionality of the covalent bonding can reduce with the nitrogen content increasing in tantalum nitrides. The WC-type TaN, Ta₄N₅, Ta₅N₆ and P4̄m2-Ta₂N₃ have smaller Poisson's ratios (∼0.25), implying higher degree of directionality of covalent bonding in those structures. The variation of Poisson's ratio at 50 GPa is similar to that of Poisson's ratio at ambient pressure. However, when the nitrogen content reaches to x = 3 in TaN_x, the Poisson's ratio of TaN₃ decreases larger than that of the preceding composition Ta₃N₅-II. The relative directionality of covalent bonds has an important effect on the hardness of materials. So we can estimate that the hardness of tantalum nitrides decreases with the nitrogen contents increasing. The following hardness calculations confirm this point. The Vickers hardness H_v of tantalum nitrides is estimated by recently proposed empirical model H_v = 2.0(k²G)^0.585–3.0 (H_v in GPa),¹⁴ which has better results for the anisotropic structures. The variation of hardness has the opposite tendency to that of Poisson's ratio in tantalum nitrides as listed in the Table 3. The WC-type TaN has the highest hardness (30 GPa). The hardness of Ta₃N₅-I is only 8.2 GPa at 0 GPa. At high pressure, when the nitrogen composition reaches to x = 3 in TaN_x, the Poisson's ratio decreases while the hardness of TaN₃ (14 GPa) abnormally increases with respect to that of the preceding composition Ta₃N₅-II (8.2 GPa) because of the presence of the strong covalent nitrogen–nitrogen bonds chain along the crystallographic b axis in TaN₃. The higher nitrogen concentration is the key for the formation and directionality of strong covalent nitrogen–nitrogen bonds, which make the tantalum nitrides have good mechanical properties. With the nitrogen content increasing, the variation of B/G has the same tendency as the variation of Poisson's ratio. The B/G value of Ta₂N₃ lies on the critical value (1.75). The WC-type TaN, Ta₅N₆ and Ta₄N₅ have small B/G values (1.41, 1.65 and 1.61, respectively), which indicates they are brittle. The B/G value of Ta₃N₅-I (2.32) is much larger than the critical value, reflecting it is a good ductile material. It is found that the ductility of tantalum nitrides can also be effectively improved by increasing the nitrogen content.

The electronic structure is crucial to understand the origin of physical properties of these nitrides. The total and partial density of states (PDOS) of the structures lied on the convex hull at different pressures are shown in Fig. 3. Fig. 3(a) shows that WC-type TaN, Ta₅N₆, Ta₄N₅, and P4̄m2-Ta₂N₃ are metallic because of their finite electron DOS at the Fermi level at 0 GPa. The Ta₃N₅-I is a semiconductor with band gap of 1.2 eV. The whole DOS curves are composed of three parts in the Ta–N system. For WC-type TaN, three parts are in range of −8.8–−3.3 eV, −3.3–0 eV and 0–7.5 eV. It can be found that the second part is close to the Fermi level and is moving to the high energy range with the nitrogen content increasing. The intensity of the second part is decreasing with the nitrogen content increasing as shown in the zone surrounded by the green dot line. The second part disappears while the Ta : N ratio reach to 1 : 1.67. The tantalum nitrides transform from metal to semiconductor with the nitrogen content increasing. At 50 GPa, the result shows that the variation of the second part follow the same trend as that at 0 GPa as shown in Fig. 3(b). It is decreasing with the nitrogen content increasing. The pressure makes the Ta₃N₅-II has metallic properties due to finite electron DOS at the Fermi level. However, when the nitrogen composition reaches to x = 3 in TaN_x, the second part of TaN₃ abnormally increases with reference to that of the preceding composition Ta₃N₅-II because of the unpaired d electron of tantalum atoms in TaN₃. Generally, the DFT underestimates band gap. However, this does not change the trend of variation. So the metallic properties of tantalum nitrides can be effectively adjusted by controlling the nitrogen content and pressure in tantalum nitrides.

Fig. 3 The density of states of the TaN_x (x ≥ 1) at 0 (a) and 50 GPa (b). All the structures are simplified to one Ta atom and x N atoms. The black dot line denotes the fermi level.

4 Conclusions

The phase stability, mechanical properties and metallic properties of tantalum nitrides (TaN_x, x ≥ 1) are investigated by first principles calculations. We uncover a synthesizable new composition TaN₃ with strong covalent N–N bonds chains along the crystallographic b axis. The established enthalpy versus composition phase diagram of tantalum nitrides is of fundamental interest and important for future experimental synthesis. With the nitrogen concentration increasing, the feature of covalent bonding of tantalum nitrides can be enhanced while the degree of directionality of the covalent bonding and hardness reduce. These make the ductility of the tantalum nitrides improved with the nitrogen composition increasing. However, the hardness of the nitrogen-rich TaN₃ increases obviously with respect to that of the preceding composition Ta₃N₅-I because of the presence of remarkable nitrogen–nitrogen bonds in TaN₃. Moreover, the intensity of metallic properties of tantalum nitrides can be effectively adjusted by controlling nitrogen composition and pressure. We hope that these calculations will stimulate extensive experimental work on these technologically important tantalum nitrides.

Acknowledgements

This work was supported by the National Basic Research Program of China (No. 2011CB808200), Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1132), National Natural Science Foundation of China (no. 51032001, 11074090, 10979001, 51025206), and National Found for Fostering Talents of basic Science (no. J1103202). Parts of calculations were performed in the High Performance Computing Center (HPCC) of Jilin University.

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