A new hard phase and physical properties of Tc₂C predicted from first principles

Abstract

Using the first principles particle swarm optimization algorithm for crystal structural prediction, we have predicted a hexagonal P6₃/mmc structure of Tc₂C. The new phase is mechanically and dynamically stable, as verified by its elastic constants and phonon dispersion. The formation enthalpy–pressure curves show that the predicted P6₃/mmc-Tc₂C is more energetically favorable than the previously proposed Mo₂C-, anti-MoS₂-, Re₂P-, and Fe₂N-type structures in the considered pressure range. The calculated mechanical properties show that it is an ultra-incompressible and hard material. Meanwhile, the directional dependences of the Young's modulus, bulk modulus, and shear modulus for Tc₂C are systematically investigated. The analyses of the density of states and electronic localization function reveal the presence of strong covalent bonding between Tc and C atoms, which is of crucial importance in forming a hard material.

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1. Introduction

Designing and searching for ultra-incompressible and superhard materials are of great scientific interest due to their various industrial applications, such as cutting and polishing tools, abrasives, oil exploitations, and coatings. Previously, it was generally accepted that superhard materials are strong covalently bonded compounds formed by light atoms B, C, N, and O, such as diamond,¹ c-BN,² B₆O,³ etc. These materials easily form strong three-dimensional covalent bonding networks. Though diamond is the hardest known material with a measured hardness of 60–120 GPa, it is unstable in the presence of oxygen at a moderate temperature and reacts easily with iron-based materials. The second-hardest material is cubic boron nitride (c-BN), but it can be synthesized only under high temperature and high temperature conditions, which results in a great cost of synthesis. Thus, intense effort has been devoted for hunting the new hard and ultra-incompressible materials. Recently, it has been reported that the incorporation of light atoms (B, C, N and O) into heavy transition metals (TMs) with high valence electron densities provides a good candidate for designing new hard materials. The compounds formed by transition metal and light atoms usually possess high valence electron density and directional covalent bonds, and these covalent bonds are strong enough to inhibit creation and movement of dislocations, which significantly improve their mechanical properties and create high hardness. Following this design criterion, recent design of the new intrinsically potential superhard materials has focused on light element TM compounds, e.g. ReB₂,⁴ WC,⁵ IrN₂,⁶ RuO₂ ⁷ and so on, the obtained results show that these materials usually possess the large bulk and shear moduli. Thereby, these pioneering studies open up a new route for pursuing new superhard materials.

Carbides, especially TM carbides, are widely used for industrial application because of their high melting temperature, extreme hardness, and chemical stability, which makes them useful in cutting tools, dental drills, rock drills in mining, and abrasives. Experimentally, some monocarbides, such as TiC,⁸ ZrC,⁹ PtC,¹⁰ and WC⁵ have been synthesized, the obtained results show that they all hold very bulk modulus. A high bulk modulus is a good indicator of a superhard material, thus TM carbides are potential candidates for superhard materials. Re₂C belongs to the family of TM carbides, and some experimental and theoretical studies of Re₂C have been investigated.^11–15 Meanwhile, these theoretical calculations and experimental data reveal that Re₂C is ultra-incompressible and has a high bulk module of about 400 GPa.

Technetium (Tc) lies directly above Re in the periodic table and has the same valence electron number, so it is worth studying the mechanical properties of its carbides. Experimentally, TcC has been first synthesized in 1962 and was presumed to be hexagonal and faced-centred cubic structures.¹⁶ Later, Giorgi and Szklarz interpreted the cubic compound on the authority of a body-centred cubic phase.¹⁷ Recently, an experimental synthesis has verified the existence of TcC,¹⁸ which also provides a useful information for understanding the crystal structure of TcC. Theoretically, the crystal structures and the related physical properties of TcC have been considerably investigated.^19–25 These calculated results indicated that TcC with considered structures is an ultra-incompressible and hard material. However, the research works of the compounds with lower carbon contents are seldom reported so far.

In this paper, we have extensively investigated the ground state structure of Tc₂C by using the first principles particle swarm optimization algorithm (PSO) on crystal structural prediction.²⁶ This method has been successfully applied to various systems,^27–29 unbiased by any known information. A novel hexagonal P6₃/mmc structure is uncovered for Tc₂C, which is more energetically favorable than the earlier proposed Mo₂C-, anti-MoS₂-, Re₂P-, and Fe₂N-type structures in the pressure range of 0–120 GPa. First principles calculations are then performed to study the total energy, lattice parameters, phase stability, elastic properties, elastic anisotropy, and density of states for this novel hexagonal phase.

2. Computational methods

The PSO technique for crystal structural prediction has been implemented in the crystal structure analysis by the particle swarm optimization (CALYPSO) code³⁰ at 0 GPa with 1–4 formula units (f.u.) each simulation cell. The underlying ab initio calculations are performed using density functional theory as implemented in the Vienna ab initio simulation package (VASP).³¹ The generalized gradient approximation (GGA) is used for the exchange and correlation potentials. The all-electron projector augmented wave (PAW) method³² is employed with 2s²2p² and 4p⁶4d⁵5s² treated as the valence electrons for C and Tc, respectively. Geometry optimization is performed by using the conjugate gradient algorithm method with a plane-wave cutoff energy of 520 eV. The calculations are conducted with 14 × 14 × 4, 14 × 14 × 4, 8 × 6 × 7, 14 × 14 × 4, 7 × 12 × 4, and 14 × 14 × 4 for the predicted P6₃/mmc-Tc₂C structure and our considered anti-ReB₂-type (space group P6₃/mmc), Mo₂C-type (space group Pbcn), anti-MoS₂-type (space group P6₃/mmc), Re₂P-type (space group Pnma), and Fe₂N-type (space group P-3m1) structures, respectively. For hexagonal structures, Γ centered k mesh is adopted, and for other structures, Monkhorst–Pack k meshes³³ are used to ensure that all the total energy calculations are well converged to better than 1 meV per atom. The phonon calculation is carried out by using a supercell approach as implemented in the PHONOPY code.³⁴ Single crystal elastic constants are calculated by a strain-energy approach, i.e., applying a small strain to the equilibrium lattice and fitting the dependence of the resulting change in energy on the strain. The bulk modulus, shear modulus, Young's modulus, and Poisson's ratio are derived from the Voigt–Reuss–Hill approximation.³⁵

3. Results and discussion

Using the PSO technique, we perform variable-cell structure prediction simulation for Tc₂C containing 1–4 f.u. in the simulation cell at 0 GPa, a hexagonal P6₃/mmc structure is found, as shown in Fig. 1. The P6₃/mmc structure contains two Tc₂C f.u. in a unit cell. At ambient pressure, the equilibrium lattice parameters are a = b = 2.838 Å and c = 9.774 Å, in which the Tc and C atoms occupy Wyckoff 4f (1/3, 2/3, 0.89092) and 2c (1/3, 2/3, 1/4) sites, respectively. From Fig. 1(a), each C atom is coordinated by 6 neighboring Tc atoms, forming edge-shared Tc₆C trigonal prisms with Tc–C bond distance of 2.141 Å, which is shorter than the Re–C (2.16 Å) and Os–C distance (2.32 Å) in hard materials Re₂C¹¹ and Os₂C,³⁶ respectively. To check the structural stability of the new phase of Tc₂C, the phonon dispersion curves are calculated at 0 GPa and 120 GPa, respectively, and the results are given in Fig. 2. Clearly, the P6₃/mmc structure is dynamically stable at ambient pressure and high pressure because of the inexistence of the imaginary phonon frequency in the whole Brillouin zone. It is well known that the thermodynamic stability of a compound can be described by the knowledge of its ground-state energy. Therefore, we calculate the total energy per f.u. as a function of the volume for the predicted P6₃/mmc structure. For comparison, the previously known five structures of anti-ReB₂, Mo₂C, anti-MoS₂, Re₂P, and Fe₂N are also considered for Tc₂C. The calculated results are shown in Fig. 3. As seen from this figure, the predicted P6₃/mmc structure for Tc₂C has the lower energy minimum than the Mo₂C-, anti-MoS₂-, Re₂P-, and Fe₂N-type structures but the same energy–volume curve as the anti-ReB₂-type structure. On the one hand, this indicates that the predicted P6₃/mmc structure is the ground-state phase at 0 GPa. On the other hand, this confirms that the P6₃/mmc-Tc₂C has the same structure with anti-ReB₂.

Fig. 1 Crystal structure of the P6₃/mmc-Tc₂C structure. Large and small spheres represent Tc and C atoms, respectively.

Fig. 2 Phonon dispersion curves of the P6₃/mmc-Tc₂C at 0 GPa (a) and 120 GPa (b).

Fig. 3 Total energy versus f.u. volume for Tc₂C with five different structures.

To explore the thermodynamic stability for further experimental synthesis, the formation enthalpy of the Tc₂C with respect to the separate phases is examined by the reaction route ΔH = H_Tc2C − 2H_Tc − H_c, here ΔH is the formation enthalpy, the hexagonal Tc (space group: P6₃/mmc) and the graphite C are chosen as the reference phases. The calculated formation enthalpy under pressure is shown in Fig. 4. As shown in this figure, the stabilities of Tc₂C with different structures are gradually enhanced with increasing the pressure, therefore, the pressure is helpful to their stabilities. At ambient condition, the negative values of the formation enthalpies for the predicted P6₃/mmc-Tc₂C and anti-ReB₂-type structures indicate that they are thermodynamically stable, whereas the positive formation enthalpies for other structures show that they are not thermodynamically stable. Moreover, it is can be seen that the enthalpy–pressure curves of the P6₃/mmc- and anti-ReB₂-type structures merge together in our considered pressure range, which further proves that they belong to the same structure. The present calculated results suggest that the predicted P6₃/mmc-Tc₂C can be synthesized at ambient condition, thus further experimental synthesis is highly desirable.

Fig. 4 The calculated formation enthalpy–pressure curves for Tc₂C with five different structures.

As is well known, the mechanical stability is a necessary condition for the stabilization of a crystal to exist, and the mechanical properties, such as elastic constants and elastic moduli, are important for potential technological and industrial applications. Thus we perform further the studies on the mechanical properties of the P6₃/mmc-Tc₂C phase. By using a strain-energy method, we obtain the zero-pressure elastic constants (C_ij) of the P6₃/mmc-Tc₂C and P-6m2-TcC, respectively. The calculated elastic constants C_ij are listed in Table 1, along with the theoretical values and available experimental data of other TM carbides (WC,⁵ PtC,¹⁰ Re₂C,^11,12,14,15 Ru₂C,^37,38 and Os₂C^36,39) for comparison. For a stable hexagonal crystal, C_ij should satisfy the following criteria:⁴⁰ C₃₃ > 0, C₄₄ > 0, C₁₂ > 0, C₁₁ > |C₁₂|, (C₁₁ + 2C₁₂)C₃₃ > 2C₁₃². As shown in Table 1, the elastic constants of the predicted P6₃/mmc-Tc₂C satisfy completely the elastic stability criteria for a hexagonal crystal, thus it is mechanically stable at ambient condition. The large values of C₁₁ and C₃₃ for the compound suggest that it is extremely difficult to be compressed along a-axis (or b-axis) and c-axis, respectively. Moreover, the C₃₃ value of the P6₃/mmc phase (781 GPa) is close to those for WC,⁵ TcC,^19–21 and Re₂C¹¹ but much larger than Ru₂C³⁸ and Os₂C,³⁹ suggesting its high linear incompressibility along the c-axis. Based on the obtained elastic constants C_ij, the polycrystalline bulk modulus B and shear modulus G are thus determined by the Voigt–Reuss–Hill approximation. The Young's modulus E and Poisson's ratio v can be derived by the following equations E = 9BG/(3B + G) and v = (3B − 2G)/(6B + 2G). The calculated bulk modulus, shear modulus, Young's modulus, and Poisson's ratio of the P6₃/mmc phase together with the reference materials mentioned above are listed in Table 1. As shown in Table 1, the bulk modulus of the P6₃/mmc phase is 343 GPa, which is comparable to the experimental data of WC (439 GPa),⁵ PtC (301(±15)),¹⁰ and Re₂C (405(30) and 386(10))^14,15 but much more higher than that of Ru₂C (178(4) GPa),³⁷ indicating its ultra-incompressible structure nature. Moreover, the calculated bulk modulus (B = 343 GPa) for the P6₃/mmc-Tc₂C agrees well with that directly obtained from the fitting results (B₀ = 344 GPa) of the third-order Birch–Murnaghan equation of states, further verifying the reliability of the present elastic calculations. In order to further compare the incompressibility of the P6₃/mmc-Tc₂C, WC, PtC, Re₂C, and Ru₂C under pressure, the volume compressions as a function of the pressure are presented in Fig. 5. It can be seen that the volume incompressibility of the P6₃/mmc-Tc₂C is comparable to those of WC and Re₂C but exceeds those of PtC and Ru₂C. Compared with the bulk modulus, the shear modulus of a material quantifies its resistance to the shear deformation and acts as a better indicator of the potential hardness. Obviously, the predicted P6₃/mmc phase has a large shear modulus of 216 GPa, thus it is expected to withstand the shear strain to a large extent. In addition, Young's modulus can also provide a good measure of the stiffness of materials except the bulk modulus and shear modulus. The larger Young's modulus the material has, the harder it is to deform. The calculated Young's modulus of the P6₃/mmc-Tc₂C is 536 GPa, indicating that it is a hard material. Poisson's ratio v is a crucial parameter to describe the degree of directionality of the covalent bonding. Generally speaking, the typical v value is 0.1 for covalent materials and 0.33 for metal materials, respectively. From Table 1, it can be seen that the v value of the P6₃/mmc-Tc₂C phase is below 0.33 and at the same time the P6₃/mmc-Tc₂C has the smallest Poisson's ratio 0.239 in contrast to other considered TM carbides, such as Ru₂C,³⁸ Os₂C,³⁹ and TcC.^19–21 Consequently, the P6₃/mmc-Tc₂C has a strong degree of covalent boding. Furthermore, the B/G ratio is commonly used to predict the brittle or ductile behavior of materials. According to the Pugh criterion,⁴¹ if B/G > 1.75, the material behaves in a ductile way, otherwise, the material behaves in a brittle manner. For the case of the P6₃/mmc-Tc₂C, our predicted value of B/G is 1.59, suggesting that it has a little brittle nature. Using the empirical formula for the hardness prediction proposed by Chen et al.,⁴² the estimated Vickers hardness for the P6₃/mmc-Tc₂C is 24.11 GPa, which is comparable to the known hard materials of SiO₂ (30.6 GPa)⁴³ and B₄C (32.8 GPa).⁴⁴ All of these excellent mechanical properties strongly support that the P6₃/mmc-Tc₂C is an ultra-incompressible and hard material.

Table 1 Calculated elastic constants C_ij, bulk modulus B (GPa), EOS fitted bulk modulus B₀ (GPa), shear modulus G (GPa), Young's modulus E (GPa), B/G, and Poisson's ratio v, Debye temperature Θ (K) for P6₃/mmc-Tc₂C and P-6m2-TcC, together with other theoretical results and available experimental data

Structure		C11	C12	C13	C33	C44	C66	B	B0	G	E	B/G	v	Θ
a Our calculated result from ref. 39 on the basis of v = (3B − 2G)/(6B + 2G).
P63/mmc-Tc2C	This work (GGA)	612	201	172	781	199	206	343	344	216	536	1.59	0.239	675
P63/mmc-Re2C	Theor. (GGA)^11	748	201	219	939	252	273	410	400	273	671		0.227	564
	Theor.^12							388.9
	Expt.^14							405(30)
	Expt.^15							386(10)
P-3m1-Ru2C	Expt.^37							178(4)
P-31m-Ru2C	Theor. (GGA)^38	500	198	239	481	151		315		139		2.27	0.307^a
P63/mmc-Os2C	Theor. (GGA)^36								418.48
	Theor. (GGA)^39	639	142	329	579	111		381		152	402	2.51	0.32
P-6m2-TcC	This work (GGA)	682	200	166	940	169	241	371	372	225	562	1.65	0.247	756
	Theor. (LDA)^19	749	244	211	1031	199	252	426		270		1.59	0.24	831
	Theor. (LDA)^20	752	228	199	1030	198			416	252	629		0.249
	Theor. (LDA)^21	759	232	188	1044	188		414	416	249	623	1.67	0.249	864
P-6m2-WC	Expt.^5	720	254	267	972	328		439
Fm-3m-PtC	Expt.^10							301(±15)

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Fig. 5 The calculated volume compression as a function of pressure for the P6₃/mmc-Tc₂C compared with WC, PtC, Re₂C, and Ru₂C.

For engineering applications that make use of single crystals, it is necessary to know the values of the Young's modulus, bulk modulus, and shear modulus as a function of crystal orientation. For the hexagonal P6₃/mmc-Tc₂C, the Young's modulus and bulk modulus are expressed by:

E⁻¹ = s₁₁(α² + β²)² + s₃₃γ⁴ + (2s₁₃ + s₄₄)(β²γ² + α²γ²), (1)

B⁻¹ = (s₁₁ + s₁₂ + s₁₃) − (s₁₁ + s₁₂ − s₁₃ − s₃₃)γ², (2)

where α, β, and γ are the direction cosines of [uvw] direction and s_ij are the elastic compliance constants which are obtained by Nye.⁴⁵ The shear modulus on the (hkl) shear plane with the shear stress applied alone the [uvw] direction is written as

G⁻¹ = 4s₁₁(α₁²α₂² + β₁²β₂²) + 4s₃₃γ₁²γ₂² + 8s₁₂α₁α₂β₁β₂ + 8s₁₃(α₁α₂ + β₁β₂)γ₁γ₂ + s₄₄[(β₁γ₂ + β₂γ₁)² + (α₁γ₂ + α₂γ₁)²] + s₆₆(α₁β₂ + α₂β₁)², (3)

where α₁, β₁, γ₁, α₂, β₂, γ₂ are the direction cosines of the [uvw] and [hkl] directions in the coordinate systems, and the [hkl] direction shows the vector normal to the (hkl) shear plane.

Fig. 6(a) and (c) are the three-dimensional surface representations which show the variation of the Young's modulus and bulk modulus, respectively. For an isotropic crystal, one would see a spherical shape, while a deviation from a spherical shape can directly reflect the degree of the elastic anisotropy in the crystal. From these two figures, one can see that the Young's modulus exhibits a high degree of the elastic anisotropy due to a large deviation from a spherical shape, whereas the bulk modulus is nearly isotropic because of a small deviation from a spherical shape. The projections of the Young's modulus and bulk modulus on the ab and bc planes are also plotted in Fig. 6(b) and (d) for comparison. As can be seen, in-plane anisotropy in the ab plane is not existent, whereas in-plane anisotropy in the bc plane is directly revealed. A deeper insight of the change of the Young's modulus alone different directions can be obtained by investigating the directional dependences of the Young's modulus alone tensile axis in the (0001), (101̄0), (12̄12), and (1̄012) planes, respectively, and the obtained results are given in Fig. 7(a). For the (0001) plane, since the direction cosines are α = cos θ, β = sin θ, and γ = 0 (θ is the angle between tensile stress and [0001]), one obtains E⁻¹ = s₁₁ from eqn (1). This implies that the Young's modulus on the basal plane is independent of the tensile stress direction for the P6₃/mmc-Tc₂C, which is the reason of the elastic isotropy in the basal plane for hexagonal crystal. For the directional dependences of the Young's modulus from [0001] to [1̄21̄0] in the (101̄0) plane, the P6₃/mmc-Tc₂C possesses a maximum of E_[0001] = 708.2 GPa and a minimum of E_[1̄21̄0] = 527.8 GPa. For the (12̄12) plane, the Young's modulus alone the [101̄0] direction has the minimal value (E_min = 527.8 GPa) and the Young's modulus alone the [1̄21̄1] direction has the maximal one (E_max = 653.6 GPa). For the change of the Young's modulus in the (1̄012) plane for the quadrant of directions between [1̄21̄0] and [101̄1], the P6₃/mmc-Tc₂C exhibits a maximum of E_[101̄1] = 591.7 GPa and a minimum of E_[1̄21̄0] = 527.8 GPa. Consequently, the order of the Young's modulus as a function of the principal crystal tensile [uvw] for the P6₃/mmc-Tc₂C is: E_[0001] > E_[1̄21̄1] > E_[101̄1] > E_[101̄0]. To understand the plastic deformation for the predicted P6₃/mmc-Tc₂C phase, we investigate the dependence of the shear modulus on the stress direction, and the corresponding result is presented in Fig. 7(b). For the (0001) shear plane with the shear stress direction varied from [101̄0] to [1̄21̄0], since the direction cosines are α₁ = β₁ = 0, γ₁ = 0, α₂ = cos θ, β₂ = sin θ, and γ₂ = 0 (θ is the angle between the shear direction and [101̄0]), one can obtain G⁻¹ = s₄₄ from eqn (3). This implies that the shear modulus is isotropic in the basal plane, as shown by the green curve in Fig. 7(b). For the shear plane (101̄0) with the shear stress direction changed from [0001] to [1̄21̄0], α₁ = 1, β₁ = γ₁ = 0, α₂ = 0, β₂ = sin θ, and γ₂ = cos θ (θ is the angle between the shear direction and [0001]), so the shear module can be reduced to G⁻¹ = s₆₆ + (s₄₄ − s₆₆)cos² θ. Since s₄₄ > s₆₆ in our calculations, the shear module on the (101̄0) plane is the highest when θ = 90° and the lowest when θ = 0, corresponding to G_max = 205.5 GPa for the [1̄21̄0] direction and G_min = 199 GPa for the [0001] direction. For the shear plane (12̄12) with the shear stress direction rotated from [101̄0] to [1̄21̄1], the maximum of the shear modulus is 212.6 GPa for the [1̄21̄1] direction and the minimum of the shear modulus is 204.9 GPa for the [101̄0] direction. When the pyramidal plane (1̄012) is the shear plane and the shear stress direction is changed from [1̄21̄0] to [101̄1], the shear modulus is the largest alone the [101̄1] direction (G_max = 232.4 GPa) and the shear modulus is the smallest along the [1̄21̄0] direction (G_min = 204.2 GPa). Through the above analysis, we draw conclusions: (1) the Young's modulus and shear modulus are constant on the basal plane and orientation-dependent on both the pyramidal and prismatic planes; (2) the bulk modulus is almost isotropic.

Fig. 6 Three-dimension surface representations of the Young's modulus (a) and bulk modulus (c) in the P6₃/mmc-Tc₂C. The plane projections of the directional dependence of the Young's modulus (b) and bulk modulus (d) in the P6₃/mmc-Tc₂C.

Fig. 7 Directional dependences of Young's modulus (a) and the shear modulus (b) in the (0001), (101̄0), (1̄012), and (12̄12) planes.

The elastic anisotropy of crystals can exert great effects on the properties of the physical mechanism, such as anisotropic plastic deformation, crack behavior, and elastic instability. Therefore, it is important and necessary to investigate the elastic anisotropy to improve their mechanical durability. The compression and the shear anisotropic factors provide the measures of the degrees of anisotropy in atomic bonding in different crystallographic planes. For hexagonal crystal, the anisotropies in compressibility and shear are defined as:

For an isotropic crystal, the factors A_comp and A_shear must be one, while any departure from one is a measure of the degree of elastic anisotropy possessed by the crystal. According to the above definitions, the values of A_comp and A_shear are calculated to be 0.77 and 0.968, respectively. This indicates that the P6₃/mmc-Tc₂C phase has a certain degree elastic anisotropy.

Debye temperature is related to many physical properties of materials, such as specific heat, elastic constants, and melting temperature.⁴⁶ It is used to differentiate between high and low temperature regions for a solid. When the temperature T > Θ_D, all modes are expected to have the energy of k_BT; when T < Θ_D, the high-frequency modes are expected to be frozen, namely the vibrational excitations origin only from the acoustic vibrations. The Debye temperature Θ_D for the studied P6₃/mmc-Tc₂C is estimated from the average sound velocity (v_m) by the following equation:

where h is Planck's constant, k is Boltzmann's constant, N_A is Avogadro's number, n is the number of atoms per formula unit, M is the molecular mass per f.u., and ρ is the density. The average sound velocity v_m is given by:where v_t and v_l are the transverse and longitudinal elastic wave velocities of the polycrystalline material, which can be determined by Navier's equation.⁴⁷ The obtained Debye temperatures of the P6₃/mmc-Tc₂C under pressure are presented in Fig. 8. At zero pressure and zero temperature, the Debye temperature for the predicted P6₃/mmc-Tc₂C phase is 675 K. Generally, the higher Debye temperature the materials possess, the larger microhardness they have. From Fig. 8, one can see clearly that the Debye temperature increases with the increase of the pressure. This means that the pressure is better for the enhancement of the hardness for the P6₃/mmc-Tc₂C.

Fig. 8 Debye temperature of the P6₃/mmc-Tc₂C as a function of pressure.

The electronic structure is the key to understand the mechanical properties of the P6₃/mmc-Tc₂C, the total and partial densities of states (DOS) are calculated and given in Fig. 9, where the vertical dashed line denotes the Fermi level. Distinctly, the P6₃/mmc-Tc₂C exhibits a good metallic behavior due to the adequately large total DOS at the Fermi level. This metallicity might make it a better candidate for hard conductors. From partial DOS, it can be clearly seen that the peaks from −13 eV to −10.8 eV are mainly contributed by C-2s and Tc-4d states with small contributions from the 4p and 5s electrons of Tc. The states from −7.3 eV to the Fermi energy (0 eV) mainly originate from the Tc-4d and C-2p orbitals with small contributions of Tc-4p and Tc-5s. In addition, the partial DOS profiles of Tc-4d and C-2p are very similar in the energy region from −7.3 eV to 0 eV, which reflects that Tc-4d orbital has a significant hybridization with C-2p orbital. This fact also shows the existence of a strong covalent bonding between the Tc and C atoms. For the total DOS, the typical feature is the presence of so-called pseudogap, which is regarded as the borderline between the bonding states and antibonding states. Since the Fermi energy is located below the pseudogap for the P6₃/mmc-Tc₂C phase, its bonding states are partially occupied and full antibonding states are unoccupied. This property also increases the structural stability of Tc₂C. Fig. 10 presents the calculated electronic localization function (ELF) of the predicted P6₃/mmc-Tc₂C phase on the (110) plane. From this figure, one can see that the large ELF value between the Tc and C atoms indicates the partially Tc–C covalent bonding interaction in the P6₃/mmc-Tc₂C phase. This also explains the reason of the high bulk modulus and large shear modulus of Tc₂C.

Fig. 9 Total (a) and partial (b) densities of states of the P6₃/mmc-Tc₂C. The vertical dashed line denotes the Fermi level E_F.

Fig. 10 Contours of ELF of the P6₃/mmc-Tc₂C on the (110) plane at 0 GPa.

4. Conclusions

In conclusion, a new hexagonal P6₃/mmc structure is unraveled to be the ground-state structure for Tc₂C using the PSO algorithm, and it is energetically much superior to the previously known Mo₂C-, anti-MoS₂-, Re₂P-, and Fe₂N-type structures in the considered pressure range. Based on the calculated phonon dispersion and formation enthalpy curves, we demonstrate that the predicted P6₃/mmc phase is dynamically stable and synthesizable under the ambient condition. The high bulk modulus, large shear modulus, small Poisson's ratio, and considerable hardness indicate that the P6₃/mmc-Tc₂C is an ultra-incompressible and hard material. Moreover, the Young's modulus, bulk modulus, and shear modulus as a function of crystal orientation for Tc₂C are also investigated systematically. The presence of strong covalent Tc–C bonds, which is of crucial importance in forming a hard material, is confirmed by the detailed analyses of the electronic structure and electronic localization function. We expect that our theoretical results will stimulate further experimental research on this material in the future.

Acknowledgements

This work was supported by the Science Foundation of Baoji University of Arts and Sciences of China (Grant No. ZK16069 and ZK11135), the Natural Science Foundation of the Education Committee of Shaanxi Province of China (Grant No. 2013JK0637), the Natural Science Basic Research plan in Shaanxi Province of China (Grant No. 2016JM1012), and the Natural Science Foundation of China (Grant No. 11404008).

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