Structural, electronic, mechanical and superconducting properties of

Int. Journal of Refractory Metals and Hard Materials 52 (2015) 219–228

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Int. Journal of Refractory Metals and Hard Materials journal homepage: www.elsevier.com/locate/IJRMHM

Structural, electronic, mechanical and superconducting properties of ReC and TcC M. Kavitha a, G. Sudha Priyanga a, R. Rajeswarapalanichamy a,⁎, K. Iyakutti b a b

Department of Physics, N.M.S.S.V.N College, Madurai, Tamilnadu-625019, India Department of Physics and Nanotechnology, SRM University, Chennai, Tamilnadu-603203. India

a r t i c l e

i n f o

Article history: Received 15 January 2015 Received in revised form 28 April 2015 Accepted 18 June 2015 Available online 21 June 2015 Keywords: Ab-initio calculations Structural phase transition Electronic structure Mechanical property Superconducting property

a b s t r a c t The structural, electronic, mechanical and superconducting properties of rhenium carbide (ReC) and technetium carbide (TcC) are investigated using first principles calculations based on density functional theory (DFT). The computed ground state properties like equilibrium lattice constants and cell volume are in good agreement with the available experimental and theoretical data. It is observed that tungsten carbide phase (WC) is the most stable phase for ReC and TcC at normal pressure. Pressure induced structural phase transitions are observed from tungsten carbide to zinc blende phase and then zinc blende to wurtzite phase in rhenium carbide while tungsten carbide to zinc blende phase and then zinc blende to nickel arsenide phase in technetium carbide. Electronic structure reveals that these materials are metallic at ambient condition. The high bulk modulus (B) value of WC phase of both ReC (438 GPa) and TcC (368 GPa) suggests that they are ultra hard and incompressible materials. Also, the superconducting transition temperature is estimated for ReC and TcC at normal pressure. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Transition metal carbides are attracting great attention due to their excellent physical, chemical and mechanical properties such as hardness, high melting temperature and superconducting behavior. Ultra hard materials are of considerable fundamental interest and practical importance because of their excellent properties. Materials that combine mechanical properties (ultra hardness, ultraincompressibility), thermal properties (ultrahigh melting point), and chemical resistance are useful for numerous industrial applications, such as cutting and grinding tools, abrasives, components of gas turbines, coatings for high-speed drill bits, and electronic and semiconductor components. So, the search for new ultra hard materials that combine these useful mechanical, thermal, and chemical properties has intensified. Initially, rhenium metal was thought not to form carbides for many years. Significant process in the synthesis of rhenium carbide is associated with the use of high pressure and high temperature. Popova and Boika [1] have synthesized γ′ MoC type rhenium carbide at a pressure of 6 GPa and a temperature of 800 °C. Later, a phase with cubic NaCl structure was obtained at 16–18 GPa and 1000 °C for rhenium carbide [2]. The formation of hexagonal rhenium carbide phase has been studied in a laser heated diamond anvil cell in the pressure range 20–40 GPa and the bulk modulus of hexagonal Re2C was found to be 405 GPa [3]. Trzebiatowski and Rudzinski [4] first synthesized Tc ⁎ Corresponding author. E-mail address: [email protected] (R. Rajeswarapalanichamy).

http://dx.doi.org/10.1016/j.ijrmhm.2015.06.012 0263-4368/© 2015 Elsevier Ltd. All rights reserved.

carbide and obtained its radiograph that showed that Tc carbides are in hexagonal and face centered cubic structures. They reported that hexagonal structure contains 1 wt.% carbon content, and the lattice constant is expanded from a = 2.740 Å, c = 4.398 Å in the pure state to a = 2.812 Å, c = 4.470 Å for the limiting concentration. German et al. [5] synthesized TcC and their X-ray diffraction study proved the formation of cubic TcC with lattice constant a = 3.98 Å. Gou et al. [6] analyzed the compressibility and the effect of metallic bonding on the hardness of ReC. They found that the incompressibility of ReC exceeds that of diamond under high pressure. Wang [7] has studied the structural, elastic and electronic properties of ReC, TcC, OsC, IrC and PtC. It was found that ReC and TcC with hexagonal WC structure have very large bulk and shear moduli, signifying their incompressible and potentially hard properties. Chen et al. [8] computationally designed ultra incompressible material ReC with very shear modulus. Zhao et al. [9] have reported that Re carbide in cubic phase was identified as a stable Fe4N structured Re4C which was ultra incompressible, hard and superconducting material. Rached et al. [10] have studied the structural stabilities, elastic and electronic properties of 5d transition metal mono nitrides and carbides using the full-potential linear muffin-tin orbital (FP-LMTO) method, based on the density functional theory (DFT) within the local density approximation (LDA). Ivanovskii [11] has estimated the Vickers's micro hardness parameter for rhenium boride, rhenium carbide and rhenium nitride. Giorgi and Szklarz [12] reported the superconducting transition temperature for technetium and technetium carbide. Sun et al. [13] have investigated the properties of TcC with rock salt structure in the pressure range of 0–80 GPa and the temperature up to 2500 K by means of first-principles calculations

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M. Kavitha et al. / Int. Journal of Refractory Metals and Hard Materials 52 (2015) 219–228

based on density functional theory. Zou et al. [14] have investigated the phase transition, elastic and thermo dynamical properties of TcC under high pressure in WC, NiAs, NaCl, CsCl and ZnS structures and it was found that WC and CsCl phases are thermodynamically stable. Liang et al. [15] investigated the phase stability and mechanical properties of TcC and TcN and it was found that TcC and TcN are elastically stable, hard and ultra stiff materials in two hexagonal phases (WC and NiAs). They have reported that the bulk modulus of TcC exceeds cubic BN and even rival to those of diamond, suggesting that they might be ultra stiff and hard materials. Bulk properties and stability of 4d transition metal carbides and nitrides were analyzed by Korir et al. [16]. Cohesive properties of 4dtransition-metal carbides and nitrides in the NaC1 structure were investigated by Guillermet et al. [17]. The elastic, mechanical and thermo dynamical properties of transition metal mono carbides at high pressure and temperature were analyzed [18]. In the present investigation, the electronic structure, structural phase transition and mechanical stability of ReC and TcC are investigated using firstprinciples calculations based on density functional theory. The superconducting transition temperature is estimated. Also, the high

pressure effect on electronic structure and elastic constants is also analyzed. 2. Computational details The ab-initio total energy calculations are performed using firstprinciples calculations based on density functional theory as implemented in Vienna ab-initio simulation package (VASP). Both the generalized gradient approximation (GGA) [19–21] and local density approximation (LDA) [22] are used for the exchange and correlation potentials [23–25]. Ground-state geometries are determined by minimizing stresses and Hellman–Feynman forces using the conjugategradient algorithm with force convergence less than 10−3 eV/Å. The wave function of the valence electron is expanded by a plane wave basis with an energy cutoff of 600 eV. Brillouin zone integrations are performed using the Monkhorst–Pack scheme [26] with a grid size of 12 × 12 × 12 for structural optimization and total energy calculation. Iterative relaxation of atomic positions is stopped when the change in total energy between successive steps is less than 1 meV/cell. The valence electron configurations are 4f14 5d5 6s2 for Re, 4d6 5s1 for Tc

Fig. 1. Crystal structure of different phases of rhenium carbide and technetium carbide.


221

Table 1 Calculated lattice parameters a, c (Å), cohesive energy Ecoh (eV), equilibrium volume V0 (Å3), valence electron density ρ (electrons/Å3), bulk modulus B0 and its derivative B0′ for ReC with possible considered structures. NaCl

a

CsCl

ZB

NiAs

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

4.364 4.34a 4.005c

4.308

2.7

2.54 2.67f

4.657 4.64a

4.597

2.521 2.84c 2.853e

2.482

2.706

2.68 3.04f

2.889

4.62

f

c

6.71 20.78

8.25 19.99

5.7 19.68

6.42 19.24

6.9 25.25

7.21 24.29

ρ B0

0.52 324 369a 4.3

0.55 332

0.55 521

0.57 542

0.45 258

4.1 4.5f

4.7

4.8 4.59f

0.43 243 278a 4.0

b c d e f

3.9 4.2f

LDA 2.854 2.84b

e

2.850 2.96f 8.601

8.742 9.85c 5.593e

Ecoh V0

a

WC

LDA

4.328f

B0 ′

WZ

GGA

4.432

4.391 6.72f

2.81f 2.789 2.77b

2.821 2.786e

2.98 48.2 39.42e 0.45 338 4.2

5.24f 4.51 44.2

4.2 26.94

6.5 25.59

0.47 349

0.42 319

0.43 339

4.1 4.4f

4.0

4.2 4.3f

2.87g 9.21 19.68

8.1 20.41 19.60e 0.53 451

0.55 458 464d 4.2 4.49f

4.3

Ref-[9]-GGA. Ref-[6]-LDA. Ref-[4]-expt. Ref-[7]-LDA. Ref-[8]-GGA. Ref-[10]-LDA.

and 2s2 2p2 for C. The crystal structure of the six considered structures of ReC and TcC is shown in Fig. 1. The tight binding linear muffin tin orbital method [27–31] is used for the estimation of electron–phonon coupling constant and electron– electron interaction parameter. This method treats one electron potential in a relativistic form. The exchange correlation potential within the local density approximation is calculated using the parameterization scheme of Von Barth and Hedin. The tetrahedron method [32] of Brillouin zone integration is used to calculate the total density of states.

3. Results and discussion 3.1. Structural properties The ab initio total energy calculations are performed for rhenium carbide (ReC) and technetium carbide (TcC) in various structures, namely, NaCl, zinc blende (ZB), cesium chloride (CsCl), tungsten carbide (WC), nickel arsenide (NiAs) and wurtzite (WZ). The optimized lattice parameters of ReC and TcC are determined by computing the total energies for various volumes for all the considered phases. The volume

Table 2 Calculated lattice parameters a, c (Å), cohesive energy Ecoh (eV), equilibrium volume V0 (Å3), valence electron density ρ (electrons/Å3), bulk modulus B0 (GPa) and its derivative B0′ for TcC with possible considered structures. NaCl

a

CsCl

ZB

NiAs

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

4.34

4.28 4.26a 4.25c

2.69 2.62c

2.58

4.62

4.56 4.54a 4.52c

2.71

2.67 2.81a 2.81c

2.70

2.61

2.87

2.83 2.81a 2.79c

5.33

5.24 5.53a 5.53c 5.11 16.25 38.03c

4.43

4.28

2.797

3.32 13.88

5.86 12.62

7.51 19.98

2.75 2.745a 2.72c 9.38 19.20 18.5c

2.71b 4.64d

c

Ecoh V0

6.4 20.48

ρ B0

20.44d 0.53 358

B0 ′

335d 4.2

b c d e

WC

GGA

4.34d 4.18e

a

WZ

Ref-[15]-LDA. Ref-[7]-LDA. Ref-[14]-LSDA. Ref-[16]-GGA. Ref-[17]-Expt.

8.63 19.60 19.29c 0.56 372 380a 378c 4.3 4.39a 4.33c

5.47 19.25 20.01b 0.5 385

4.1 4.2c

6.24 18.9 18.2c 0.58 392 412c 418b

6.32 24.80 24.9d 0.44 271

245d 4.4 4.1

7.91 23.77 23.49c

3.67 17.05

0.46 279 285a 289c

0.32 389

0.34 398 405a 405c

0.40 309

0.45 319

0.55 392

4.1 4.39b 4.51c

4.2

4.1 4.33a 4.38c

3.9

4.0

4.3

0.57 398 416a 426b 414c 4.2 4.25a 4.28c

222


Fig. 2. Energy versus pressure curve for rhenium carbide using GGA and LDA.

corresponding to the minimum energy is the equilibrium volume V0. The bulk modulus and its pressure derivative have been computed by minimizing the total energy for different values of the lattice constants by means of the Birch–Murnaghan equation of state [33]. The calculated ground state properties like lattice constants a, c (Å), cell volume V0 (Å3), valence electron density ρ (electrons/ Å3), bulk modulus (B0) and its first derivative (B0′) for considered phases of ReC and TcC using both LDA and GGA are listed in Tables 1–2 respectively along with the available experimental [4] and theoretical results [7–11, 15–18]. From the calculated values, In ReC the calculated lattice constant is in good agreement with Zhao et al.[9] and Rached et al. [10] in NaCl phase. For CsCl and ZB phase the lattice constant is in close agreement with Rached et al. [10] and Zhao et al.[9] respectively. But for WC phase the lattice parameters are closer with the results of Chen et al. [8]. In NiAs and wurtzite phases there is a slight deviation from the other results. In TcC the calculated lattice constant is in good agreement with Korir et al. [16] and Liang et al. [15] in NaCl phase and ZB phase. For CsCl phase the lattice constant values are comparable with the results of Wang [7], but for WC phase the lattice parameters are closer with the results of Chen et al. [8]. In NiAs phase there is slight deviation from the other results.

The high bulk modulus (B) value of WC phase for both ReC (438 GPa) and TcC (368 GPa) suggests that they are ultra hard and incompressible materials compared with c-BN of bulk modulus 402 GPa and diamond whose bulk modulus is 469 GPa [34]. 3.2. Structural phase transition The structural stability of ReC and TcC is analyzed among six different structures namely NaCl, ZB, CsCl, WC, NiAs, and WZ under normal and high pressure. The total energy of rhenium carbide and technetium carbide is plotted for various pressures and are given in Figs. 2 and 3. It is observed that WC phase is the most stable phase for ReC and TcC. As the pressure is increased, structural phase transition is noted in both the carbides. In rhenium carbide, the phase transition from tungsten carbide to zinc blende phase occurs at a pressure of 206 GPa and zinc blende to wurtzite phase transition occurs at 232 GPa (Fig. 2(a)). Similarly in technetium carbide, tungsten carbide to zinc blende phase transition occurs at a pressure of 185 GPa and zinc blende to nickel arsenide phase transition occurs at 208 GPa (Fig. 3(a)) respectively. The transition pressure values obtained with LDA functional are almost same (Figs. 2(b) and 3(b)).

Fig. 3. Energy versus pressure curve for technetium carbide using GGA and LDA.


223

3.3. Electronic properties of ReC and TcC

Fig. 4. Electronic band structure of rhenium carbide and technetium carbide at normal pressure.

The energy band structure computed for ReC and TcC along the symmetry directions Γ–K–M–Γ–A–L–H–A is depicted in Fig. 4. The valence band minimum is located at the Γ (Gamma) point. It is observed that ReC and TcC have 6 valence bands corresponding to 11 valence electrons which come from 5d5 6s2 of Re atom (4d6 5s1 of Tc atom) and 2s2 2p2 electrons of C atom. The low lying band is mainly due to the non-metal C-2s state and this band is separated from the series of bands formed by Re (Tc)-s state and p state electrons by an energy gap of 9 eV. However, the main hybridization is due to Re-5d (Tc-4d) bands and C-2p bands. It is seen that there are several bands crossing the Fermi level which indicates the well metallic feature of both the carbides. The valence states are separated by a wide gap from the occupied states, indicating the presence of covalent bonding in hexagonal WC phase ReC and TcC. Above the Fermi level, the empty conduction bands are present with a mixed s, p, and d characters. The density of states (DOS) and partial density of states (states/eV.cell) histogram corresponding to the stable phase of ReC and TcC for normal pressure is given in Fig. 5. The level arising from C2s state gives the highest spike with narrow width below the Fermi level (EF) represented by vertical dotted line in figure. The short spike near the origin is due to the s state electrons of metal. The highest spike with a broad width above EF is due to the d state electrons of metal. The charge-density contour of hexagonal WC-ReC and WC-TcC is displayed in Fig. 6. It can be seen that there is an increase in electron density around the carbon atoms while a decrease in electron density in the interstitial region between metal atoms is observed. Furthermore, strong directional charge redistribution is observed from metal to carbon atoms. These charge rearrangements reflect the electronegative

Fig. 5. Density of states and partial density of states of rhenium carbide and technetium carbide at normal pressure.

224


Fig. 6. Charge density distribution of rhenium carbide and technetium carbide in the WC structure.

nature of carbon atoms. It reveals an ionic contribution to the bonding in addition to the metallic character. Also, in density of states histogram, covalent character is observed due to the strong hybridization between metal and carbon states. Thus, the bonding is a mixture of metallic, covalent, and ionic attribution in both ReC and TcC.

3.4. Mechanical properties Elastic constants are the measure of the resistance of a crystal to an externally applied stress. The elastic constants provide important information about the stability, stiffness and hardness of materials.

Table 3 Calculated elastic constants C11, C12, C44, C13, and C33 (GPa), Young's modulus E (GPa), shear modulus G (GPa), B/G ratio, Poisson's ratio ν, Zener isotropy (A), Lame constants (λ, μ), Kleinman parameter (ζ),Vicker's hardness parameter (Hυ), Cauchy pressure (C.P.),Linear compressibility coefficient (Kc/Ka) and melting temperature (Tm) for ReC with possible considered structures. NaCl

CsCl

ZB

NiAs

WZ

WC

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

C11

650.2 670.7a

668.1

1082

1093

274.2 259.4a

280.9

584.5 726c

630.5

552.3

580.9

760.1

791.1 812b

C12

166.2 219.2a

178.9

259

284

243.9 287.4a

263.1

236.9 229c

250.2

207.9

230.5

796c 232.5

−51.2 −72.9a

−84.3

35.52

42.8

161.9 373.1a

199.1

105.2 284c

112.1

196.8

203.2

242c 234.1

C44

242.5 258b 255.2 205b

c

C13

–

–

–

–

–

–

110.8 225c

121.2

78

94

195 211

C33

–

–

–

–

–

–

585.2 993c

601.5

445.8

470.2

192c 1049.5

226.8 250b 1081.1 1063b

c

B0

328 369.7a

342

533

553

254 278.1a

269

354 422c

368

322

347

1123 431

G

–

–

185.9

188

101.6

123

243 273c

251

187

192

440c 246

446 464b 263 283b

c

E λ μ ζ ν B/G A Hν Hν = 0.147G Hν = 0.060E C.P. Kc/Ka Tm a b c

Ref-[9]-GGA. Ref-[7]-LDA. Ref-[8]-GGA

–

–

– – – – – – – – – – – –

– – – – – – – – – – –

515.8 158 216.7 0.38 0.19

508 141.1 211.6 0.40 0.20

2.8 0.08 11.7 27.42 31.30 11.22 223 –

2.9 0.105 10.9 27.73 30.83 241.2 – –

268.9 2180 90.84 0.92 0.48 0.47a 2.5 1.21 9.11 14.98 16.32 82 – –

320.1 5261 326.6 0.97 0.49

593.2 294.9 231.7 0.54 0.28

613 304.7 239.4 0.53 0.28

470 217.2 185.04 0.51 0.27

486 241.6 189.8 0.53 0.28

223 620 214.6 252.03 0.45 0.23

2.1 0.71 11.4 18.14 19.43 64 – –

1.45 0.60 29.30 35.84 36.007 5.6 1.24 2985.3

1.46 0.69 29.77 37.02 37.20 9.1 1.32 3147.7

1.72 1.14 20.13 27.58 28.52 −118.8 1.64 2679.6

1.80 1.15 19.4 28.32 29.50 −109.2 1.65 2802

1.78 0.88 23.9 36.28 37.63 −23 0.71 4208.5

620.2 189.7 255.6 0.47 0.213 0.25b 1.79 0.93 26.08 38.79 37.64 −28.4 0.67 4348.9


225

unstable because of its negative C44 value. From the calculated elastic constants Cij, the bulk modulus and shear modulus of cubic and hexagonal crystals are calculated using the Voigt–Reuss–Hill (VRH) averaging scheme [39–41]. The calculated elastic constants C11, C12, C44, C13, and C33 (GPa), bulk modulus B (GPa),Young's modulus E (GPa), shear modulus G (GPa), B/G ratio, Poisson's ratio ν, Zener isotropy (A), Lame constants (λ, μ), Kleinman parameter (ζ), Vicker's hardness (H ν ), Cauchy pressure (C.P.), Linear compressibility coefficient (K c /Ka ) and melting temperature (T m ) for ReC and TcC are presented in Tables 3 and 4. It is found that the calculated elastic constants are in agreement with the available theoretical data [7-9,14,15]. For the NaCl and ZB phases of ReC, the calculated elastic constants are in good agreement with Zhao et al. [9]. The elastic constants for WC phase of ReC are in close agreement with Wang et al. [7]. The elastic constants for CsCl phase of TcC is in agreement with Zou et al. [14].

The elastic constants Cij are computed using the total energy method [35–37] where the unit cell is subjected to a number of finite strains along several directions. For cubic systems, there are three independent elastic constants (C11, C12, C44) and five (C11, C12, C44, C13, C33) for hexagonal crystals [35]. For a stable hexagonal structure, the five independent elastic constants Cij (C11, C12, C33, C13, and C44) should satisfy the well known Born–Huang [38] stability criteria: C12 N 0, C33 N 0, C11 N C12, C44 N 0, (C11 + C12) C33 N 2C213 while for a cubic crystal, the three independent elastic constants Cij (C11, C12, C44) should satisfy the Born–Huang [38] stability criteria: C44 N 0, C11 N |C12|, (C11 + 2C12) N 0. Clearly, the calculated elastic constants for hexagonal ReC and TcC satisfy Born–Huang criteria, suggesting that they are mechanically stable. At normal pressure, NaCl phase of ReC and TcC are mechanically

Table 4 Calculated elastic constants C11,C12, C44, C13, and C33 (GPa), Young's modulus E (GPa), shear modulus G (GPa), B/G ratio, Poisson's ratio ν, Zener isotropy (A), Lame constants (λ, μ), Kleinman parameter (ζ), Vicker's hardness parameter (Hυ), Cauchy pressure (C.P.),Linear compressibility coefficient (Kc/Ka) and melting temperature (Tm) for TcC with possible considered structures. NaCl

C11

C44 C13 C33 B0

CsCl

ZB

NiAs

WC

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

GGA

LDA

649

678 689b 661c

848

883 989c

262

270 299b 289c

716

720 738b 749c

560.8

584.5

720

185.5

204 230b 239c

123.1

138 126c

256

262 273b 288c

206

210 187b 189c

198.1

202.8

212

−49.2

−80.1 −61b −69c

62.5

89 85c

143.5

144.8 130b 116c

265

288.3 288b 283c

180.8

187.7

160

–

–

–

–

–

–

219

228 227b 219c

91

118

182

–

–

–

–

–

–

842

894 966b 976c

424.9

432.2

721.8

340

362 382b 378c

364.7

386.3 412c

257

268 285b 289c

376

380 405b 405c

319

330

368

–

–

180.7

202.4 175c

86.7

90.5 56b 36c

261

275 290b 290c

181

189

268.7

–

–

465.2

515.4 460c

242.9

635

479

649

– – – –

65.58 207.6 0.12

0.13

654 288b 702c 213.8 272.1 0.43 0.26 0.214b

456

– – – –

233 158b 104c 1889 78.7 0.95 0.41 0.407b

196.08 180.9 0.49 0.26

191.6 185.2 0.49 0.25

224.7 263.8 0.44 0.23

– – – – – – –

–

2.01

1.9 2.35c

1.76

1.74

1.37

0.99 19.15 26.69 27.67 −89.8 1.72 2673.7

0.98 19.92 27.87 29.07 −69.7 1.75 2755.8

0.62 33.69 39.63 39.39 22 1.05 3596.7

735 749a 752b 759c 217.5 244a 228b 232c 182 199a 198b 188c 190 199a 199b 188c 743.1 1031a 1030b 1044c 390 426a 416b 414c 272 270a 252b 249c 601 629b 702c 208.11 244.3 0.44 0.23 0.24a 0.246b 1.61 1.39c 0.70 32.32 35.69 36.48 35.5 1.03 3673.6

G

E

λ μ ζ ν

WZ

GGA

B/G A Hν Hν = 0.1475G Hν = 0.0607E C.P Kc/Ka Tm

3993 81.5 0.99 0.42 2.9

a b c

Ref-[7]-LDA. Ref-[15]-LDA. Ref-[14]-LSDA.

– – – – –

0.17 16.36 26.65 28.23 52.6 – –

0.23 18.94 29.85 31.28 – –

35.5 6.29 12.78 14.74 112.5 – –

315.6 248.08 0.43 0.28 1.44

2.9 8.03c 36.4 6.5 13.34 14.14 117.2 – –

1.03 31.22 38.49 38.49 −46 0.77 3765

1.38 1.39c 0.82 33.98 40.56 39.69 −60.3 0.71 3855

226


From Tables 1–4, it is seen that the bulk modulus values computed using elastic constants are close to the bulk modulus values calculated using EOS fit. The bulk modulus of ReC and TcC is greater than the bulk modulus of super hard materials Hf3N4 (275.9 GPa) [42,43], Zr3N4 (250 GPa) [43,44], and Si3N4 of bulk modulus 233 GPa [45]. Thus both ReC and TcC are super hard materials. Young's modulus (E) is the longitudinal elasticity modulus, which represents the ability of resistance to deformation and a large value of E implies the stiffness of the material. The WC phase of ReC and TcC has large Young's modulus values, 620 GPa and 649 GPa respectively and ZB phase of ReC and TcC has small Young's modulus values, 268 GPa and 242 GPa respectively. This signifies that the WC phase is more effective than all the other phases for deformation resistance. The obtained Young's modulus values for ReC and TcC is greater than the Young's modulus of cubic Hf3N4 (395 GPa), which indicates that ReC and TcC are stiff materials. Poisson's ratio reflects the stability of the crystal against shear. The ratio can formally takes the values between −1 and 0.5, which corresponds to the lower bound where the material does not change its shape or to the upper bound when the volume remains unchanged. It has been proved that ν = 0.25 is the lower limit for central force solids and 0.5 is the upper limit which corresponds to infinite elastic anisotropy [46]. The obtained Poisson's ratio values for WC-ReC and WC-TcC are 0.235 and 0.231 respectively, which indicates that both has central inter atomic forces and is relatively against shear. To estimate the brittleness and ductility of ReC and TcC, Pugh's criterion (B/G ratio) [47] is used. If B/G N 1.75, the materials behave in a ductile manner. On the contrary if B/G b 1.75, the materials behave in brittle manner. The calculated values of B/G predict that ReC is brittle in nature and TcC is ductile in nature with stable WC phase. The Cauchy pressure ((C13-C44) for hexagonal crystals and (C12-C44) for cubic crystals) is also calculated to analyze the ductility and brittleness of these materials. Generally, a positive Cauchy pressure reveals damage tolerance and ductility of a crystal, while a negative Cauchy pressure demonstrates brittleness which also reveals that ReC is brittle and TcC is ductile in nature with stable WC phase. The elastic anisotropy of a crystal has an important application in engineering science since it is highly correlated with the possibility to induce micro cracks in materials [48]. The anisotropy factor A = (2C44 + C12) / C11 has been evaluated. For isotropic crystal, A = 1, while values smaller or larger than 1 indicates that the crystal is anisotropic. The deviation of A from 1 is the measure of the degree of elastic anisotropy possessed by the crystal. From Tables 3 and 4, it is seen that ReC and TcC are elastically anisotropic at ambient pressure in stable phase. The Lame constants (λ, μ) are often measured for polycrystalline materials when investigating their hardness. They are two constants which relate stress to strain in an isotropic, elastic material. Physically, the first Lame constants (λ) represent the compressibility of the material, while the second Lame constant (μ) reflects its shear stiffness [49]. Usually, the second Lame's constant μ is positive, the Lame's first parameter λ can be negative, in principle; however, for most materials it is also positive. The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media. From Tables 3 and 4 it is observed that both Lame's constants (λ, μ) are positive. Another important parameter is the Kleinman parameter that describes the relative positions of the cation anion sub lattices under volume conserving strain distortions for which the positions are not fixed by symmetry [32]. The Kleinman parameter would quantify the balance between bond bending and bond stretching forces. From Tables 3 and 4 it is observed that the Kleinman parameter (ζ) is for ReC and TcC is low. A low value of ζ implies a large resistance against bond bending or bond-angle distortion while the reverse is true for high value [50,51]. The investigation of the stiffness can be completed by providing the Vicker's hardness parameter (Hυ). Chen et al. [52] derived the formula for Vicker's hardness (Hυ) using B and G as, Hυ = 2(k2G)0.375 − 3.

Table 5 Density ρ (g/cm3), longitudinal velocity v1 (m/s), transverse velocity vt (m/s), average velocity vm (m/s), and Debye temperature (K) for ReC and TcC with possible considered structures. ρ

νl

νt

νm

θD

GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA

– – 15.70 15.81 11.84 12.36 12.87 13.49 12.51 13.22 15.14

– – 7050.971 7129.503 5727.801 5918.616 7255.655 7215.203 5082.044 5255.144 7079.434

– – 3440.331 3448.262 2919.846 3154.487 4343.750 4312.317 2911.833 3127.121 4030.373

– – 3864.778 3875.531 3271.695 3524.214 4806.280 4772.271 3234.829 3642.149 4479.888

– – 537.2 539.8 418.4 457.2 625.2 630.6 504.3 520.5 615.1

LDA

15.70

7122.083

4092.103

4544.927

GGA LDA GGA LDA

– – 8.442 8.599

– – 8469.531 8735.285

– – 4626.297 4851.491

– – 5159.293 5403.496

ZB

GGA LDA

6.55 6.83

7540.260 7490.192

3637.261 3560.928

4088.528 4005.757

NiAs

GGA LDA

9.532 9.930

8715.063 8671.360

5232.648 5262.477

5788.191 5815.012

WZ

GGA LDA GGA LDA

11.625 11.78 8.076 8.404

6942.488 6746.652 9482.939 9208.545

3945.765 3844.658 5768.046 5366.060

4386.465 4273.109 6372.210 5952.439

631.7 634a – – 722 761 725b 707c 526 522 348b 843 861 1314b 684 688 881 834 831a 864b

Compound ReC

NaCl CsCl ZB NiAs WZ WC

TcC

NaCl CsCl

WC

a b c

Ref-[7]-LDA. Ref-[14]-LSDA. Ref-[15]-LDA.

Tian et al. [53] modified it to obtain positive values of Hυ always as: Hυ = 0.92 k1.137 G0.708. Liu et al. [54,55] computed the Vickers hardness of NbO-structured 3d, 4d and 5d transition metal nitrides and pyrite type transition metal per nitrides using Tian et al. [53] formulation. The Vickers hardness of period IV, V and VI transition metals was computed by Zhou et al. [56]. In this paper, Tian et al. [53] formulation is used to calculate the Vicker's hardness (Hυ) of ReC and TcC and the calculated values are reported in Tables 5 and 6 respectively. The high Hυ value of ReC and TcC again confirms that they are super hard materials. Vicker's micro hardness (H) can be calculated using G and E as, H = 0.1475G and H = 0.0607E [54]. It is also seen that Kc/Ka b 1 [57] demonstrates that the compressibility for TcC along the c-axis is larger than along the a-axis. The melting temperature [58] of ReC and TcC is high which also confirms the hardness of ReC and TcC in stable phase. The relationship between the elastic constants (C11, C12, C44, C13, and C33), bulk modulus (B), shear modulus and Young's modulus (E) and pressure for WC phase of ReC and TcC is shown in Fig. 7. It is seen that the elastic constants C44 and C12 increase monotonically with increase in pressure. It is worth to notice that C11 increase rapidly with pressure.

Table 6 Superconducting transition temperature Tc (K), electron–phonon coupling constant λ, electron–electron interaction parameter μ* and ωlog for ReC and TcC. Element

λ

μ*

ωlog

Tc (K)

ReC TcC

0.2241 0.3569

0.1452 0.1173

331 412

4.08 4.2 3.85a

a

Ref-[12]-expt.


227

Fig. 7. Pressure dependence of elastic constants of rhenium carbide and technetium carbide.

The elastic constant C11 represents the elasticity in length. A longitudinal strain produces a change in C11. The elastic constants C12 and C44 are related to the elasticity in shape, which is a shear constant. A transverse strain causes change in shape without change in volume. Therefore, C12 and C44 are less sensitive to pressure as compared to C11. It is also observed that the pressure has an important influence on Young's modulus, bulk modulus and shear modulus. The Debye temperature (θD) is the important parameter closely related to many physical properties of materials, such as specific heat, elastic constants and melting temperature. The Debye temperature is calculated from the elastic constants data using average sound velocity vm [59]. The calculated values of longitudinal, transverse, average sound velocities and Debye temperature for ReC and TcC using GGA and LDA are listed in Table 5 along with available results (7, 14 and 15). The high value of the Debye temperature for TcC implies that its thermal conductivity is more when compared with ReC. 3.5. Superconductivity The continuous promotion of s electron to d shell in solids is one of the factors which will induce superconductivity. The calculated Tc values depend more sensitively on λ rather than θD and μ*. ReC is found to have superconducting nature at ambient condition. For normal pressure, the superconducting transition temperature is estimated by using the McMillan equation modified by Allen and Dynes [60],

Tc ¼

ωlog −1:04ð1 þ λÞ exp λ−μ ð1 þ 0:62λÞ 1:2

ð1Þ

where λ is the electron–phonon coupling constant, μ* is the electron– electron interaction parameter and ωlog is the average phonon frequency. The average of the phonon frequency square is, bω2log N ¼ 0:5θ2D :

ð2Þ

The above expression gives a good estimate of the Tc value. The electron–phonon coupling constant λ can be written as [61]

λ¼

NðE F ÞbI2 N Mbω2 N

ð4Þ

where N(EF) is the density of states at the Fermi energy. M is the atomic mass.

bI2N is the square of the electron–phonon matrix element averaged over the Fermi energy. bI2N (in Rydbergs) can be written as [62], ( ) X ðl þ 1Þ Nl ðE F ÞNlþ1 ðE F Þ M2i jþ1 bI2 N ¼ 2 ð2l þ 1Þð2l þ 3Þ NðE F Þ2 l

ð5Þ

where Ml, l + 1 are the electron–phonon matrix elements which can be expressed in terms of the logarithmic derivatives. Dl ¼

d lnφl d ln r r¼s

ð6Þ

is evaluated at the sphere boundary, h i Ml;lþ1 ¼ −φl φlþ1 ðDl ðE F Þ−1ÞðDlþ1 ðE F Þ þ l þ 2Þ þ ðE F −VðSÞÞS2

ð7Þ

where φl is the radial wave function at the muffin-tin sphere radius corresponding to the Fermi energy. The electron–electron interaction parameter μ* is estimated using the relation [63], μ ¼

0:26NðE F Þ : ð1 þ NðE F ÞÞ

ð8Þ

The TC, μ* and λ values are computed for ReC and TcC at normal pressure using the results obtained from the electronic structure calculated using TB-LMTO and are tabulated in Table 6. The superconducting transition temperature (TC) is calculated as 4.08 K and 4.2 K for ReC and TcC respectively. The superconducting transition temperature of TcC is in close agreement with the value reported by Giorgi and Szklarz [12] as 3.85 K. 4. Conclusion The structural, electronic and mechanical properties of ReC and TcC are investigated. The calculated ground state properties are in good agreement with the experimental and other available theoretical results. Our results suggest that ReC and TcC are stable in WC structure at ambient pressure. A pressure-induced structural phase transition occurs from WC to ZB phase in ReC at 206 GPa, TcC at 190 GPa and ZB to WZ phase transition occurs at 232.5 GPa for ReC and ZB-NiAs phase transition occurs at 200 GPa. The electronic band structure and density of states of ReC and TcC confirm their metallic nature. The computed elastic constants agree well with the available results. Moreover, the pressure dependence of elastic properties, bulk modulus, Young's modulus and shear modulus are also investigated. It is found that the pressure has an important influence on these physical properties. The

228


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