First-principles studies of the electronic and elastic properties of metal nitrides XN (X = Sc, Ti, V, Cr, Zr, Nb)

Computational Materials Science 51 (2012) 380–388

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First-principles studies of the electronic and elastic properties of metal nitrides XN (X = Sc, Ti, V, Cr, Zr, Nb) M.G. Brik ⇑, C.-G. Ma Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia

a r t i c l e

i n f o

Article history: Received 2 July 2011 Received in revised form 1 August 2011 Accepted 2 August 2011 Available online 31 August 2011 Keywords: Electronic band structure Optical properties Elastic properties

a b s t r a c t Electronic and elastic properties of a series of the transition metal ion mononitrides (ScN, TiN, VN, CrN, ZrN, NbN) have been modeled in the framework of ab initio plane wave spin-polarized calculations using the generalized gradient and local density approximations. The calculated band structures are typical for metallic compounds, except for ScN, whose band structure is that one of the gapless semiconductor. Strongly delocalized d states of transition metal ions are spread over a wide region of about 10–12 eV and are strongly hybridized with the nitrogen 2p states. Among the considered nitrides, only CrN exhibits a clear difference between the spin-up and spin-down states, which would manifest itself in magnetic properties. The overall appearance of the calculated cross-sections of the electron density difference changes drastically when going from Sc to Nb in the considered series of compounds. For the first time the calculated tensors of the elastic constants and elastic compliance constants were used for the analysis and visualization of the directional dependence of the Young’s moduli. It was shown that ScN and VN can be characterized as more or less elastically isotropic materials, whereas in TiN, CrN, ZrN, and NbN the Young’s moduli vary significantly in different directions. The maximal values of the Young’s moduli are along the crystallographic axes, the minimal values are along the bisector direction in the coordinate planes; the difference between them in the case of CrN exceeds one order of magnitude. In addition, pressure dependence of the ‘‘metal – nitrogen’’ distance was modeled. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The physical properties of the transition metal nitrides XN (X = Sc, Ti, V, Cr, Zr, Nb) represent a unique combination of high hardness, high melting point and metallic conductivity [1], which makes them suitable for many important technological applications, e.g. coating, corrosion resistance on machine tooling etc. In addition, relatively high superconducting temperature has been reported for a number of compounds from this group [2,3]. In view of these factors, thorough theoretical modeling of physical properties of these nitrides (preferably, in the framework of reliable ab initio methods, to avoid employing any fitting parameters) becomes an important problem, which has attracted so far considerable interest of various research groups. Early calculations of the band structure for ScN, TiN, VN (based on the Xa method) were described in Ref. [4]. Further refined calculations for some nitrides can be also found in the literature. In particular, electronic structure of ScN and pressure effects were considered in Ref. [5]. Experimental studies of the CrN electronic structure and vibrational modes have been reported in Ref. [6], whereas the electronic structure of the

⇑ Corresponding author. Tel.: +372 7374751; fax: +372 738 3033. E-mail address: brik@fi.tartu.ee (M.G. Brik). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.08.008

same CrN was calculated in Ref. [7]. Elastic properties of a number of transition metal mononitrides were the subject of study in Ref. [8]; the band structures and elastic properties of mononitrides were calculated in Ref. [9] using the full-potential linearized augmented plane wave method. The X-ray absorption spectra of TiN and VN were compared to the calculated band structures in Ref. [10]. However, quite a number of phenomena remained out of consideration in the above-mentioned publications. Thus, variation of the structural (lattice constant, interionic distances) and electronic (band structure, density of states) properties with pressure was not modeled yet. Anisotropy of the elastic properties, revealed in dependence of the Young’s moduli on the direction in the crystal lattice, was not considered at all. So, in the present work we report on a consistent and systematic study of the structural, electronic, elastic properties of ScN, TiN, VN, CrN, ZrN and NbN in their dependence on pressure. The paper organization is as follows: in the next section the calculation details are described along with a short summary of the literature structural properties of these compounds. Then we proceed with a separate description (in each section) of the electronic and elastic properties. All obtained results are compared to the available experimental and/or theoretical data. The paper is concluded with a short summary.

381

M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388 Table 1 Structural properties of metal nitrides: lattice constant a (in Å). Experiment

a b c d e f g

This work

Other calculations

GGA

LDA

ScN

4.44a

4.516016

4.444841

TiN VN CrN ZrN NbN

4.235a 4.128a 4.148b 4.56847c 4.702a

4.250492 4.119166 4.06304 4.598469 4.415868

4.184637 4.0565 4.032732 4.528893 4.355801

Ref. Ref. Ref. Ref. Ref. Ref. Ref.

4.52d, 4.51e 4.42–4.54f 4.18–4.32f 4.06–4.23f 4.02g 4.53–4.57f 4.36–4.42f

[11]. [12]. [13]. [5]. [14]. [9]; different methods. [15].

2. Input structural data and details of calculations Under normal conditions, all considered compounds crystallize in the rock-salt structure (space group Fm3m, No. 225) [11]. There

are four formula units per one unit cell. Each ion has six nearest neighbors forming an octahedron. Table 1 below shows a summary of the literature data (both experimental and theoretical) on the crystal lattice constants of the considered crystals, in comparison with the results obtained in the present work. All calculations reported in this paper have been performed using the CASTEP module [16] of Materials Studio package, in two independent runs, with either the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof functional [17] or the Ceperley–Alder–Perdew–Zunger parameterization [18,19] in the local density approximation (LDA) to treat the exchange–correlation effects. The ultrasoft pseudopotentials were employed for a description of interaction between the ionic cores and valence electrons. The following electronic configurations were taken: Sc 3s23p63d14s2; Ti 3s23p63d24s2; V 3s23p63d34s2; Cr 3s23p63d54s1; Zr 4s24p64d25s2; Nb 4s24p64d45s1; N 2s22p3. The convergence parameters were set as follows: energy tolerance 105 eV/atom, force tolerance 0.03 eV/Å; stress tolerance 0.05 GPa, and maximum displacement 103 Å. The Monkhorst–Pack k-points grid sampling was set as 10  10  10. The plane-waves cut off (in eV) was 280 for ZrN and VN, 290 for ScN, 310 for TiN and CrN, and 320 for NbN.

LDA GGA

ScN 6

LDA GGA

TiN 4 2

2

Energy, eV

Energy, eV

4

0 -2

W

G

L

X

W

-6

K

LDA GGA

VN 6

Energy, eV

Energy, eV

X

W

K

spin-up spin-down

4

0 -2 -4

2 0 -2 -4

W

L

G

X

W

-6

K

LDA GGA

ZrN 6

W

L

G

X

W K

G

X

W

LDA GGA

NbN 6 4

Energy, eV

4

Energy, eV

G

L

CrN

2

2 0 -2 -4 -6

W

6

4

-6

-2 -4

-4 -6

0

2 0 -2 -4

W

L

G

X

W

K

-6

W

L

K

Fig. 1. Calculated band structures for ScN, TiN, VN, CrN, ZrN and NbN. The LDA- and GGA-calculated states are shown by the solid lines and filled circles, respectively. In the case of CrN the LDA-calculated bands are distinguished between the spin-up and spin-down states (the solid and dashed lines, correspondingly); the GGA results for CrN are not shown.

M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

Density of states, electrons/eV

4

Sc

t2g

4s states 4p states 3d states

2

4 eg

0

N

2

2s states 2p states

1 0 4

ScN Total DOS

2

Densitry of states, electrons/eV

382

t2g

Ti

2 0 2

N 2s states 2p states

1 0 4

TiN Total DOS

2 0

0 -20

-15

-10

-5

0

5

10

15

-20

20

-15

-10

-5

t2g

V 2

4s states 4p states 3d states

eg

0

N 2s states 2p states

1 0 4

VN Total DOS

2

-20

10

15

t2g

20

4s states 4p states 3d states

Cr

2

eg

0

N

2s states 2p states

1 0 4

CrN Total DOS

2

-15

-10

-5

0

5

10

15

-20

20

-15

-10

eg

N 2s states 2p states

0 4

ZrN Total DOS 2

Density of states, electrons/eV

5s states 5p states 4d states

t2g

Zr

-5

0

5

10

15

20

Energy, eV

Energy, eV

Density of states, electrons/eV

5

0

0

0 2

0

Energy, eV Density of states, electrons/eV

Density of states, electrons/eV

Energy, eV

2

4s states 4p states 3d states

eg

2

t2g

Nb

1 0 2

5s states 5p states 4d states

eg

N 2s states 2p states

1 0 4

NbN Total DOS

2 0

0 -20

-15

-10

-5

0

5

10

15

20

-20

-15

-10

-5

0

5

10

15

20

Energy, eV

Energy, eV

The results of the geometry optimization with these parameters are shown in Table 1. The structural data from that table allow to conclude that the chosen parameters ensured good agreement between our calculated lattice constants and results of other research groups.

3. Summary of the electronic properties After having optimized the crystal structure, we proceed with calculations and analysis of the electronic and elastic properties. The calculated band structures (in the vicinity of the Fermi’s level) of all six considered nitrides are shown in Fig. 1. Closeness of the GGA (filled circles) and LDA (solid lines) results stresses out their consistency. The first glance at the calculated bands suggests that all crystals are metals, since there is no distinct clearly resolved band gap (in all diagrams the Fermi level is set at 0 eV). However, a more detailed consideration allows to classify ScN as a gapless

Density of states, electrons/eV

Fig. 2. Calculated partial and total density of states (DOS) diagrams for ScN, TiN, VN, CrN, ZrN and NbN.

CrN 1

spin-up

0

spin-down -1

-10

-8

-6

-4

-2

0

2

4

6

Energy, eV Fig. 3. Difference between the spin-up and spin-down 3d states of chromium in CrN.

M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

383

Fig. 4. Electron density difference distribution for ScN, TiN, VN, CrN, ZrN and NbN.

semiconductor. The minimal energy of a direct band-to-band transition in ScN is about 0.7 eV at the X point. All electronic bands exhibit well-pronounced dispersion, indicating high mobility of electrons as one additional feature of metallic behavior of these materials. Although all calculations were spin-polarized, only in one case – that one of chromium mononitride – there was a clear and obvious difference between the spin-up and spin-down states. To emphasize such a difference, the electronic band structure for CrN is plotted only for the LDA results (for the sake of brevity), but for the spin-resolved states.

The composition of the calculated energetic bands can be understood with the help of the density of states (DOS) diagrams, depicted in Fig. 2 (GGA data). The lowest in energy states shown in these diagrams are those of N – the 2s and 2p. It is interesting to note that in ScN the nitrogen 2s states are located at about 12.5 eV, and then they move deeper in other crystals – to 15 and even to 17 eV (in CrN). However, the energy separation between these 2s and 2p N states remains practically the same – about 7 eV, which implies that the N 2p states are also deeper in energy in all crystals, except for ScN. The widths of the N 2p states distribution is very large – they

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M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

Table 2 Elastic constants of ScN, TiN, VN, CrN, ZrN and NbN (all in GPa, except for non-dimensional B0 = dB/dP). Other literature data C11

C12 a

C44 a

B a

C12

C44

B

B

C11

C12

C44

B

B0

3.78 3.89c

354.06

100.20

170.00

184.82 216.97

3.15

418.72

101.79

173.48

207.43 220.96

3.64

5.4g

534.67

117.70

175.42

256.69 268.78 305.98 325.37 310.41 324.81 226.14 263.42 299.52 331.66

3.94

648.96

129.01

193.96

3.90

3.81

763.16

148.70

158.27

3.99

518.26

220.24

9.54

3.47

605.99

99.25

128.06

3.52

829.77

112.15

100.84

302.33 312.80 353.52 372.95 319.58 275.72 268.16 280.04 351.36 358.15

a

380.90

TiN

625e

165e

163e

196.67 195b 197c 288f

VN

645d

349d

124d

282d

–

628.70

144.63

147.41

CrN

546h

184h

20h

327h

4.46h

502.77

214.23

4.05

–

491.55

93.44

117.32

–

696.61

100.99

94.32

ZrN NbN

471 304j 608i

i

167.18

C11

B a

This work, LDA

ScN

i

104.56

This work, GGA 0

i

88 114j 134i

f

138 511j 117i

215 246j 292f,i

0

3.80 4.67 3.86 3.86

The B and B0 values obtained from the Murnaghan equation of state (3) are given in italic. a Calculations; Ref. [27]. b Calculations; Ref. [5]. c Calculations; Ref. [14]. d Calculations; Ref. [28]. e Experiment; Ref. [29]. f Experiment; Ref. [8]. g Experiment; Ref. [8,30]. h Calculations; Ref. [31]. i Experiment; Ref. [32]. j Calculations; Ref. [33].

contribute to the conduction band (although, of course, their contribution is a minor one) due to the hybridization effects. The transition metal d states (3d states of Sc, Ti, V, Cr, and 4d states of Zr and Nb) are also spread over a wide region, from about 5–7 eV (this is due to the above-mentioned hybridization effects with the nitrogen 2p states) to about 5 eV. The structure of the d states distributions is due to the crystal field splitting of the d orbi-

GGA LDA

1.00 B=220.96 ± 1.06 GPa B'=3.64 ± 0.03

0.95 0.90 0.85

Relative volume change, V/V0

0.70

B=216.97 ± 2.14 GPa B'=3.22 ± 0.06

B = 312.80 ± 1.34 GPa B' = 3.90 ± 0.04

0.95 0.85 0.80 0.75

0.75

B = 275.72 ± 7.69 GPa B' = 4.67 ± 0.23

1.00

B=280.04 ± 1.26 GPa B' = 3.86 ± 0.04

0.95

ZrN

0.80 B = 268.78 ± 1.28 GPa B' = 3.94 ± 0.04 B = 372.95 ± 1.62 GPa B' = 3.80 ± 0.04

0.75 0.70

B=263.42 ± 2.69 GPa B' = 3.47 ± 0.07

1.00

B = 358.15 ± 1.26 GPa B' = 3.86 ± 0.03

0.95

VN

0.90

0.85

0.75

0.80

0.85

0.95

0.80

CrN

0.90

TiN

1.00

0.90

B = 324.81 ± 1.51 GPa B' = 3.99 ± 0.04

0.95

0.85

1.00

0.90

1.00

0.90

ScN

0.80 0.75

tals into the t2g (lower) and eg (higher) states in the octahedral crystal field. Although direct comparison between the crystal field data for 3d ions as impurities in crystals and present results is not possible, we note in passing that the difference between their barycenters is about 2 eV, which is close to a typical value of the crystal field strength 10Dq for the sixfold coordinated transition metal ions. In addition, the conduction band of all these nitrides contains also con-

NbN

0.85 B = 325.37 ± 1.51 GPa B' = 3.81 ± 0.04

0.80 0.75

0

20

40

60

80 100 120 140 160

Pressure, GPa

B = 331.66 ± 2.70 GPa B' = 3.52 ± 0.07

0

20

40

60

80 100 120 140 160

Pressure, GPa

Fig. 5. Dependence of the relative volume change V/V0 on pressure for ScN, TiN, VN, CrN, ZrN and NbN. The calculated values are shown by symbols (GGA – squares; LDA – open circles); fitting to the Murnaghan equation of state are shown by the solid (GGA) and dashed (LDA) lines, respectively. Parameters of fits are given in each figure.

M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

385

Fig. 6. Directional dependence of the Young’s moduli for ScN, TiN, VN, CrN, ZrN and NbN. The GGA results were used for plotting all surfaces and cross-sections. The axes units are GPa.

tributions from the 4s, 4p states of the metal ions (ScN, TiN, VN, CrN) and 5s, 5p states (ZrN, NbN), which extend up to 30 eV. The density of states at the Fermi level for ScN is only 0.07 – an additional indication of its ‘‘semiconductor-like’’ properties. The same value for ZrN, NbN, and TiN is 0.67, 0.936, and 0.98, correspondingly. It is considerably greater for VN (1.70), and, finally, reaches its maximum value among the considered materials for CrN (2.75), highlighting metallic properties of these compounds. As it was already mentioned, in all crystals but CrN the d states distribution showed absolutely no difference between the spin-up and spin-down states. The Cr 3d spin-up and down states are depicted in Fig. 3; there is a clear asymmetry in their distribution, which would manifest itself in magnetic properties of this com-

pound. Further studies of the magnetic properties of CrN are beyond the scope of the present paper. Changes of the transition metal ion in the considered series result in a drastic change of the electron density distribution in the space between the crystal lattice ions. Fig. 4 shows the electron density difference distribution (GGA data) for all six considered nitrides. All diagrams are plotted in one scale, for easier comparison of different bonding properties, which are illustrated by the density distribution in the space surrounding the transition metal and nitrogen ions. The weakest interaction between the metal ions and nitrogen is in ScN and ZrN, which also correlates with the largest interionic separation between these ions among studied crystals; the strongest overlap between the transition metal ions

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M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

pressure; the obtained results are collected in Table 2. A common feature of the calculated results is that the LDA elastic constants Cij are greater than the GGA ones. Comparison with other literature data (both experimental and theoretical, also shown in Table 2) yields good agreement and consistency. Mechanical stability of these compounds can be checked by applying the following criterion for a cubic crystal [20]:

Table 3 Parameters of the a þ b1 P þ b2 P 2 fit for the ‘‘metal – nitrogen’’ distance in ScN, TiN, VN, CrN, ZrN and NbN. The GGA and LDA results are in the upper and lower rows for each crystal. Metal – nitrogen distance

ScN TiN VN CrN ZrN NbN

a (Å)

b1 (Å/GPa)

b2 (Å/GPa2) (in 106)

2.25470 2.21794 2.12199 2.08988 2.05735 2.02653 2.05934 2.01431 2.29686 2.26127 2.20649 2.17595

0.00280 0.00260 0.00211 0.00185 0.00177 0.00156 0.00253 0.00194 0.00242 0.00219 0.00192 0.00173

7.68512 7.18065 5.56057 4.59182 4.28884 3.56473 8.69821 5.55253 6.33631 5.64583 4.60908 4.05372

C11 þ 2C12 > 0;

C44 > 0;

C11  C12 > 0:

ð1Þ

These conditions hold true for all six nitrides, which indicates their structural and mechanical stability. The values of the elastic constants can also shed some light upon the nature of interaction and chemical bonding between atoms. Analysis of the deviations from the Cauchy condition C12 ¼ C44 [21] for the cubic crystals suggests [22–24] that the CrN and NbN can be categorized as compounds with mostly ionic bonds, whereas the four remaining nitrides exhibit more pronounced covalent behavior. The bulk moduli B were calculated in the present work by two methods. The first one is a direct calculation of the elastic constants available in CASTEP, when the bulk modulus is estimated as [25]

B ¼ ðC11 þ 2C12 Þ=3:

ð2Þ

The second method is based on the numerical fitting of the ‘‘volume V – pressure P’’ dependence for a unit cell to the Murnaghan equation of state [26]

  1 V P B0 ¼ 1 þ B0 ; V0 B

ð3Þ 0

where V0 is the volume at ambient pressure, B and B are the bulk modulus and its pressure derivative, respectively. For application of Eq. (3) the crystal lattice structures have been optimized at different values of external hydrostatic pressure from 0 to 140 GPa with a step of 20 GPa. Fig. 5 below shows the results of the ‘‘volume – pressure’’ calculations and fitting to the Murnaghan equation for all considered compounds. The fitting lines excellently follow the calculated V/ V0 values as a function of pressure. Indeed, the values of B (between ca 200–370 GPa) obtained by both ways confirm a high hardness of the considered compounds. We also note, in passing, that the calculated values of B in Ref. [9] (using different calculating techniques for each of listed compounds) were as follows: ScN – from 201 to 235; TiN from 286 to 389; VN from 333 to 414; ZrN from 264 to 292; NbN from 317 to 354, which is in agreement with our results. Dependence of the ‘‘metal – nitrogen’’ distances on pressure was also studied. Decrease of these structural parameters with increasing pressure was fitted to the second order polynomial

Fig. 7. Cross-sections (in the ab plane) of the Young’s moduli surfaces for ScN, TiN, VN, CrN, ZrN and NbN. The axes units are GPa.

and nitrogen is observed for VN and CrN, in accordance with the shortest ‘‘metal – nitrogen’’ distance. 4. Summary of the elastic properties and pressure effects on the lattice constants and interionic distances For cubic crystal three independent elastic constants C11 ; C12 ; C44 are needed to describe response of a crystal to any applied stress. These constants have been calculated for the optimized crystal structures in both GGA and LDA at zero-th external

Table 4 The calculated (this work) elastic compliance constants Sij (all in 1/GPa), shear moduli GV, GR (in GPa), the maximum/minimum (Emax/Emin) (in GPa) values of the Young’s moduli (in GPa) and anisotropy factor AG (in %) for ScN, TiN, VN, CrN, ZrN and NbN. The GGA/LDA results are given in the upper/lower rows, respectively.

ScN TiN VN CrN ZrN NbN

S11

S12

S44

GV

GR

AG

Emax

Emin

0.003227 0.002639 0.002032 0.001650 0.001740 0.001399 0.002668 0.002585 0.002166 0.001730 0.001490 0.001245

0.000712 0.000516 0.000367 0.000274 0.000325 0.000228 0.000797 0.000771 0.000346 0.000243 0.000189 0.000148

0.005882 0.005764 0.005701 0.005156 0.006784 0.006318 0.246914 0.104822 0.008524 0.007809 0.010602 0.009917

152.77 167.47 188.65 220.37 185.26 217.85 60.14 65.33 150.01 178.18 175.72 204.03

149.68 167.15 187.30 215.89 174.74 196.35 6.63 15.25 140.38 159.64 129.79 141.55

1.02 0.098 3.57 1.03 2.92 5.19 80.15 62.15 3.32 5.49 15.03 18.08

390.37 407.00 492.13 606.06 574.71 714.80 374.81 386.85 461.68 578.03 671.14 803.21

309.89 378.93 428.63 479.39 381.00 413.17 12.10 28.34 300.06 331.38 256.10 276.09

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M.G. Brik, C.-G. Ma / Computational Materials Science 51 (2012) 380–388

a þ b1 P þ b2 P 2 ; the obtained parameters of this fit are collected in Table 3. Although all six crystals considered in the present work are cubic, this does not mean at all that they should exhibit isotropic elastic properties. For an analysis of the elastic anisotropy the values of the shear moduli are needed. The upper (Voigt, GV) and lower (Reuss, GR) estimates of the shear moduli values are given by the following equations [34,35]

C11 þ C22 þ C33  C12  C13  C23 C44 þ C55 þ C66 GV ¼ þ ; 15 5

ð4Þ

15 : 4ðS11 þ S22 þ S33 Þ  4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ

GR ¼

ð5Þ The Sij values in Eq. (5) are the elastic compliance constants (measured in GPa1), which form the matrix inverse to the matrix of elastic constants Cij. In the case of cubic crystals ðC11 ¼ C22 ¼ C33 ; C12 ¼ C23 ¼ C31 ; C44 ¼ C55 ¼ C66 Þ these Sij constants can be calculated as follows:

S11 ¼

C11 þ C12 C211

S12 ¼ 

þ C11 C12  2C212 C12

;

C211 þ C11 C12  2C212

;

S44 ¼

1 : C44

ð6Þ

The numerical values of Sij for the studied materials are collected in Table 4. Eqs. (4) and (5) in the case of cubic crystals are further simplified to

GV ¼

C11  C12 þ 3C44 ; 5

5 4 3 ¼ þ : GR C11  C12 C 44

ð7Þ

An important non-dimensional quantity, which describes the anisotropy of the elastic properties, is the so called anisotropy factor AG [36]

AG ¼

GV  GR : GV þ GR

ð8Þ

As follows from Table 4, the AG value varies in a very wide range: from several percent for ScN, TiN, VN, ZrN (low anisotropy) to about 15–18% for NbN and 60–80% for CrN, thus showing remarkably high elastic anisotropy for the last two compounds. A useful visualization of the elastic anisotropy can be obtained by plotting three-dimensional picture of dependence of the Young’s modulus E on a direction in a crystal. For cubic materials, it is described by the following equation [37]:

Þ ¼ Eðn

1 ; S11  b1 ðn21 n22 þ n21 n23 þ n22 n23 Þ

¼ 2S11  2S12  S44 ;

where b1 ð9Þ

and n1, n2, n3 are the direction cosines, which determine the angles between the a, b, c axes of a crystal and a given direction. Eq. (9) determines a three-dimensional closed surface, and the distance from the origin of system of coordinate to this surface equals to the Young’s modulus in a given direction. For a perfectly isotropic medium this surface would be a sphere, but, as was stressed out in Ref. [37], this is often not a case even for cubic crystals. Fig. 6 shows six surfaces, which exhibit directional dependence of the Young’s modulus in the six considered nitrides; the coordinate axes in all figures coincide with the crystallographic ones. In all cases the shape of the surface deviates from the spherical one. It is closer to a perfect spherical shape in the case of ScN and TiN, shows well-pronounced anisotropy for VN and ZrN, and is remarkably different from a sphere in the case of NbN and (espe-

cially) CrN. The cross-sections of all these six surfaces in the ab plane are shown in Fig. 7. Analytical equations for determination of the maximum and minimum values of the Young’s moduli are given in Ref. [37]:

Emax ¼

3 ; S11 þ 2S12 þ S44

Emin ¼

1 ; S11

if b1 > 0:

ð10Þ

The ‘‘max’’ and ‘‘min’’ subscripts should be interchanged if b1 < 0 [37]. The estimated in this way Emax and Emin values are given in Table 4, which – along with Figs. 6 and 7 and Table 2 – gives complete information about elastic properties of the studied materials. Considerable difference between Emax and Emin (which reaches one order of magnitude in the case of CrN) can impose certain limitations and restrictions on possible applications of these materials: the stresses applied in different directions can produce quite different response of the medium. As seen from Fig. 7, for TiN, VN, CrN, ZrN, and NbN the maximum value of the Young’s modulus (the highest hardness) is realized along the crystallographic axes, whereas the minimum values of the Young’s modulus occur along the bisector direction in each of the ab, bc, ac coordinate planes. The situation is opposite in the case of ScN. We also note that CrN – quite a hard material if the stress is applied along the crystallographic axes – turns out to be quite soft if the direction of the applied stress makes an angle with any of three axes of a crystal. In addition, such a difference between the Emax and Emin values can sometimes make difficult direct comparison of the theoretical and experimental results, since the latter ones would depend on the sample’s orientation with respect to the applied stress.

5. Conclusions The reported in the present paper first-principles calculations of the structural, electronic, and elastic properties of a group of transition metal mononitrides (ScN, TiN, VN, CrN, ZrN and NbN) revealed certain peculiarities in their behavior. For the sake of consistency, both LDA- and GGA-based calculations (as implemented in the CASTEP module of Materials Studio) were performed. From the analysis of the calculated band structures and estimated values of the density of states at Fermi level, ScN can be classified as a so called gapless semiconductor, whereas the remaining five studied crystals can be described as metals. Strong delocalization of the d states of transition metal ions and 2p states of nitrogen indicates importance of the hybridization effects in formation of the physical properties of these compounds. The calculated cross-sections of the electron density distribution show noticeable difference in the interionic interaction and overlap of the when going from ScN to NbN across the considered series. Elastic properties of all six nitrides were studied in details. The elastic constants, elastic compliance constants, bulk moduli and their pressure derivatives, shear and Young’s moduli all have been calculated. For the first time the analysis of the directional dependence of the Young’s moduli has been performed, which has uncovered interesting results. In spite of the fact that all six considered crystals are cubic, they exhibit remarkable anisotropy of the Young’s moduli values. The difference between the highest (along the crystallographic axes) and lowest (along the bisectors of the coordination planes) values of the Young’s moduli is quite noticeable, and even exceeds one order of magnitude in the case of CrN. Such elastic anisotropy should be taken into account when discussing the areas of applications of these materials. In addition, pressure dependence of the characteristic ‘‘transition metal ion – nitrogen’’ distances has been modeled by the second order polynomial. Such a detailed analysis of the elastic and

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structural properties can be very useful for prediction of behavior of these compounds under extreme conditions. Acknowledgments M.G. Brik’s research was supported by European Social Fund’s Doctoral Studies and Internationalisation Programme DoRa. C.-G. Ma appreciates financial support from European Social Fund Grant No. GLOFY 054MJD. The authors thank Prof. Ü. Lille (Tallinn University of Technology) for giving an opportunity to use the Materials Studio package. References [1] L.E. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York, 1971. [2] A. Nigro, G. Nobile, V. Palmieri, G. Rubino, R. Vaglio, Phys. Scr. 38 (1988) 483. [3] K. Kawaguchi, M. Sohma, Jpn. J. Appl. Phys. 30 (1991) L2088. [4] A. Neckel, P. Rastl, R. Eibler, P. Weinberger, K. Schwarz, J. Phys. C: Solid State Phys. 9 (1976) 579. [5] P.-F. Guan, C.-Y. Wang, T. Yu, Chin. Phys. B 17 (2008) 3040. [6] X.Y. Zhang, D. Gall, Phys. Rev. B 82 (2010) 045116. [7] A. Herwadkar, W.R.L. Lambrecht, Phys. Rev. B 79 (2009) 035125. [8] P. Ojha, M. Aynyas, S.P. Sanyal, J. Phys. Chem. Solids 68 (2007) 148. [9] C. Stampfl, W. Mannstadt, R. Asahi, A.J. Freeman, Phys. Rev. B 63 (2001) 155106. [10] L. Soriano, M. Abbate, H. Pen, P. Prieto, J.M. Sanz, Solid State Commun. 102 (1997) 291. [11] R.W.G. Wyckoff, Cryst. Struct. 1 (1963) 85–237. [12] M. Nasr Eddine, E.F. Bertaut, M. Roubin, J. Paris, Acta Cryst. B 33 (1977) 3010.

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