Computational Materials Science 88 (2014) 86–91

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Computational Materials Science

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Structural and elastic properties of TiCxN1�x, TiCxO1�x, TiOxN1�x solidsolutions from first-principles calculations

http://dx.doi.org/10.1016/j.commatsci.2014.02.0420927-0256/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +86 10 62334775; fax: +86 10 62334204.E-mail address: hzhu@metall.ustb.edu.cn (H. Zhu).

Jiusan Xiao, Bo Jiang, Kai Huang, Hongmin Zhu ⇑State Key Lab of Advanced Metallurgy, School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e i n f o

Article history:Received 7 November 2013Received in revised form 21 February 2014Accepted 25 February 2014

Keywords:Titanium carbonitridesTitanium oxycarbidesTitanium oxynitridesElastic propertiesFirst-principles calculations

a b s t r a c t

The structural, electronic and elastic properties of TiCxN1�x, TiCxO1�x and TiOxN1�x alloys have been inves-tigated by using the plane-wave pseudopotential method within the density function theory. The presentlattice parameters and bulk modulus of TiCxN1�x alloys generally follow the Vegard’s law. The elasticproperties of TiCxN1�x alloys presented as a function of concentration reveal the anisotropy and ductility.The prediction of elastic properties for TiOxN1�x alloys has been presented due to the consistencybetween the calculated results and the experimental results for TiCxN1�x and TiCxO1�x alloys. The partialdensity of states (PDOS) and total density of states (DOS) for the binary and ternary compounds havebeen obtained, and the metallic behavior of these alloys has been confirmed by the analysis of DOS.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Titanium oxides, carbides and nitrides compounds, with a face-centered cubic (FCC) NaCl type structure, have attracted consider-able interest for their similar and special properties such as highmelting point, hardness, corrosion resistance and metallic conduc-tivity [1–3]. Due to these combined properties, they could be usedas hard coating thin films [4] and strengthening phases [5] to opti-mize the mechanic performance of other materials. Besides, thesecompounds could form a series of continuous solid solution (TiCx-

N1�x, TiCxO1�x and TiOxN1�x, 0 6 x 6 1), when Ti atoms occupy the4a (000) sites and C/N/O atoms occupy the 4b (0.50.50.5) sitesrandomly. It is worth mentioning that titanium oxycarbides espe-cially the TiC0.5O0.5 could be utilized as the consuming anode forthe titanium electrolysis process in molten salt [6,7]; high puritytitanium deposits in the cathode through the electrolysis whilethe carbon and oxygen component in the solid solution evolve intothe CO gas. It describes great potential and bright prospect fordecreasing the cost of titanium production.

The relationship between the mechanic features and the elec-tronic properties has attracted lots of interest, and since first-principles calculation has been proved to be an efficient methodto predict material performance [8,9], a large amount of theoreticalcalculations and experimental studies have been presented for thebinary compounds (TiC, TiN and TiO) in order to understand and

take advantage of the mechanism [10–14]. Ahuja and Erikssonhave studied the elastic properties as well as high-pressure proper-ties of TiC, TiN and TiO [15], and the results obtained from LDA andGGA calculation show a good agreement with experiments. More-over, these compounds are found to undergo structural transitionat elevated pressure. Meanwhile, similar investigations have alsobeen conducted for the ternary alloys (TiCxN1�x, TiCxO1�x and TiOx-

N1�x, 0 6 x 6 1). Jhi et al. have investigated the variation of theshear modulus c44 for transition-metal carbonitrides includingTiCxN1�x with the valence-electron concentration (VEC) throughthe ab initio pseudopotential method [16]; it turns out the highestvalue exists as the VEC = 8.4 and similar results have been providedby Ivashchenko et al. [17]. Fullpotential method based on densityfunction theory has been employed by Zaoui et al. to study thestructural, elastic and electronic properties of TiCxN1�x, the com-puted data have been compared with previous experimental valuesand the results suggest that chemical bonds have an effect on theelastic properties [18]. The elastic constants and microhardnessof TiCxN1�x powders prepared by hot-pressing have been exploredby Yang et al. and the impact of porosity and VEC has been dis-cussed [19]. Unfortunately first-principles calculations on TiCxO1�x

and TiOxN1�x alloys are relatively rare [20,21].In this paper, theoretical investigations of TiCxN1�x, TiCxO1�x

and TiOxN1�x alloys as well as their binary compounds are per-formed by using the first-principles methods based on densityfunction theory. The structural and elastic properties are discussed,and the total density of states (DOS) and partial density of states(PDOS) for binary and ternary compounds are also studied.

J. Xiao et al. / Computational Materials Science 88 (2014) 86–91 87

2. Computational details

The calculation procedures were performed by the plane-wavepseudopotential method based on the density function theory [22]as implemented in the CASTEP package [23]. Ultrasoft pseudopo-tentials were employed to describe the ion–electron interactions[24]. The exchange–correlation potential was treated by the gener-alized gradient approximation (GGA) in the scheme of Perdew–Burke–Ernzerhof (PBE). Brillouin–zone (BZ) integrations wereachieved by discrete summation over a special set of k-pointsdue to the Monkhorst–Pack scheme [25]. In order to guaranteethe convergence of the calculations, the criteria for structure opti-mization and energy calculation were set to fine quality with afixed kinetic energy cutoff of 500 eV and k-point meshes of6 � 6 � 6 and 6 � 3 � 3 were used to sample the Brillouin–zone(BZ) for 8-atom structure and 32-atom structure, respectively.The self-consistent convergence accuracy was set at1.0 � 10�5 eV/atom, and the convergence criterion for the maximalforce between atoms was 0.03 eV/Å. The maximum displacementwas 1.0 � 10�3 Å, and the maximum stress was 0.05 GPa. Thesecomputational parameters were obtained through strict conver-gence test. For simulation, a series of unit cells containing 4 tita-nium atoms and 4 nonmetal (C/N/O) atoms were utilized, asdepicted in Fig. 1(a)–(c). The variety of different composition forthe ternary alloys was achieved by replacing the nonmetal atomsone at a time. As examples, the crystal structures of TiC0.5N0.5,TiC0.5O0.5 and TiO0.5N0.5 are displayed in Fig. 1(d)–(f), respectively.

3. Results and discussion

3.1. Structural properties

First, the structural properties of binary compounds (TiC, TiNand TiO) are analyzed using the GGA calculations and the calcu-lated lattice constants a, bulk modulus B are shown in Table 1, with

Fig. 1. Atomic arrangement of (a) Ti4C4, (b) Ti4O4, (c) Ti4N4, (d) Ti4C2N2, (e) Ti4C2O2,(f) Ti4O2N2, (g) Ti15C7O8, and (h) Ti15O8N7.

the available previous experimental and theoretical data includedfor comparison. It reveals that our calculated a and B values for bin-ary compounds are consistent with the results obtained throughexperiment, except for TiO whose calculated lattice constant isabout 2% higher than the experimental value. This overestimatemay be attributed to the phenomenon that vacancies appear atboth Ti and O sites [31], thus the actual structure of TiO couldnot be simulated by perfect FCC structure and vacancies need tak-ing into account. In this work, vacancies were achieved throughtaking away Ti atoms and O atoms without changing the over-allstructure. Results are found to fit with the experimental valueswhen vacancy concentration is set to 16.7%, indicating that thereare about 16.7% vacancies at both Ti and O sites in the actual TiOstructure. The computed a and B of TiCxN1�x are plotted inFig. 2(a) and (b) as a function of concentration, respectively. Previ-ous theoretical and experimental data are also presented and highconsistency could be observed through comparison. What’s more,it can be easily observed that there is a linear relationship betweenlattice constants of TiCxN1�x and concentration, which indicatesthat Vegard’s law is valid for this series of continuous solid solu-tion. However, for TiCxO1�x and TiOxN1�x alloys, as shown inFigs. 3(a) and 4(a), deviation of the calculated lattice parametersfrom experimental values exists and becomes larger as the concen-tration of O in the ternary compounds rises, due to the vacanciesbrought in by TiO, as mentioned before. Therefore, two kinds ofmodels (including Ti15C7O8 and Ti15O8N7) which present a vacancyconcentration of 6.25% have been introduced for calculation, asshown in Fig. 1(g) and (h), respectively. It can be easily observedin Figs. 3 and 4 that after taking vacancy into account great consis-tency between our calculated structural properties and previousexperimental data has been achieved; except for TiOxN1�x alloysno experimental values of B have been published so far. Thus,our results obtained through first-principles calculation provide areasonable prediction for the bulk modulus of TiOxN1�x continuoussolid solution.

3.2. Elastic properties

The elastic properties such as bulk modulus B, shear modulus G,Young’s modulus E and Poisson’s ratio t are important parametersto reveal the relationship between the mechanic and dynamicbehaviors of crystals and provide important information on thestability and stiffness of materials concerning the nature of theforces exerting on solids. The calculated elastic properties are ob-tained from the elastic constants c11, c12 and c44 using the equa-tions below:

B ¼ c11 þ 2c12

3ð1Þ

G ¼ 12

c11 � c12 þ 3c44

5þ 5c44ðc11 � c12Þ

4c44 þ 3ðc11 � c12Þ

� �ð2Þ

E ¼ 9BG3Bþ G

ð3Þ

m ¼ E� 2G2G

ð4Þ

The computed elastic properties of the binary compounds are sum-marized in Table 2. Our calculated B, G, E and t values for the binarycompounds agree well with the results acquired via theoreticalcalculation performed by Yang et al. [2] and Marques et al. [20],but when compared with the experimental data large deviationhas been found for TiO while for TiC and TiN only slight differenceexists. This problem has been well solved by introducing andcontrolling the vacancies, as mentioned before. Elastic properties

Fig. 2. Calculated lattice constants a and elastic properties (bulk modulus B, shear modulus G and Poisson’s ratio t) for TiCxN1�x alloys, including previous theoretical andexperimental results for comparison. (See above-mentioned references for further information.)

Table 1Calculated equilibrium lattice constants a, bulk modulus B of the binary compounds together with available previous experimental and theoretical data for comparison.

Compounds Parameters Present work Experiment Other theoretical works

TiC a (Å) 4.332 4.33 [26] 4.27 [18]4.3311 [2]

B (GPa) 249 240 [26] 281 [18]249 [2]

TiN a (Å) 4.245 4.24 [26] 4.18 [18]4.246 [2]

B (GPa) 278 288, 320 [26] 318 [18]279 [2]

TiO a (Å) 4.280 4.185 [27] 4.271 [28]4.181 [29] 4.264 [20]

4.185 (vacancy)B (GPa) 222 177 [30] 270.18 [28]

220.0 [20]176 (vacancy)

88 J. Xiao et al. / Computational Materials Science 88 (2014) 86–91

of TiO including 16.7% vacancies are found to be accordant with theexperimental values, thus justifying the vacancies in the actual TiOstructure.

The elastic properties of TiCxN1�x, TiCxO1�x and TiOxN1�x alloysare depicted as a function of concentration in Figs. 2–4, respec-tively. The computed shear modulus G of TiCxN1�x, TiCxO1�x andTiOxN1�x alloys, as shown in Figs. 2(c), 3(c) and 4(c), are achievedby taking the average of Voigt’s shear modulus and Reuss’s shearmodulus as the best estimate of the theoretical polycrystallineshear modulus [40]. Since there is no vacancy in TiC and TiNstructure, our computed elastic properties for TiCxN1�x show goodagreement with both the experimental and theoretical results ob-tained by others, as shown in Fig. 2. Particularly, Fig. 2(b) indicatesa linear behavior of bulk modulus with the concentration of theTiCxN1�x alloys, which could also be found in first-principles calcu-lation performed by others [18]. For TiCxO1�x and TiOxN1�x alloys,vacancy needs to be considered to reduce the difference between

calculated results and experimental values, as have been previ-ously mentioned. Likewise, the elastic properties of Ti15C7O8 modeland Ti15O8N7 model which were formerly described in Fig. 1(g) and(h) have been investigated. Apparently our computed bulk modu-lus B for TiCxO1�x alloys are in good agreement with the experi-mental data provided by Ivanov et al. [30] while shear modulusand Poisson’s ratio are slightly underestimated. Unfortunately,most of the published data do not report the experimental elasticconstants of TiOxN1�x alloys, therefore our calculated results couldbe regarded as a rational prediction for the elastic properties ofTiOxN1�x alloys considering the consistency between calculatedand experimental results we have obtained from TiCxN1�x and TiCx-

O1�x alloys.It is known that a high B/G value is correlated with ductility and

the threshold which separates brittle and ductile is 1.75 [41].According to Table 2, the calculated B/G for binary compoundsTiC, TiN, TiO are 1.31, 1.42 and 1.96, respectively. This indicates

Fig. 3. Calculated lattice constants a and elastic properties (bulk modulus B, shear modulus G and Poisson’s ratio t) for TiCxO1�x alloys, including previous theoretical andexperimental results for comparison, results with vacancy involved are also presented. (See above-mentioned references for further information.)

Fig. 4. Calculated lattice constants a and elastic properties (bulk modulus B, shear modulus G and Poisson’s ratio t) for TiOxN1�x alloys, including previous experimentalresults for comparison, results with vacancy involved are also presented. (See above-mentioned references for further information.)

J. Xiao et al. / Computational Materials Science 88 (2014) 86–91 89

that TiC and TiN crystal as well as the TiCxN1�x alloys behave in abrittle manner and TiO crystal is ductile.

Poisson’s ratio can provide more information about the proper-ties of the bonding force than other elastic constants [42]. It is re-lated with the volume change during the elastic deformation andthe lower the Poisson’s ratio is, the larger volume change occurs.

As displayed in Figs. 2(d), 3(d) and 4(d), the Poisson’s ratios of allthe ternary alloys are below 0.25, which demonstrates the anisot-ropy hardness, since it has been proved that 0.25 is the lower limitfor central-force solids and 0.5 is the upper limit [43]. Therefore,the TiCxN1�x, TiCxO1�x and TiOxN1�x alloys are all found to beanisotropic.

Table 2Calculated elastic properties (bulk modulus B, Young’s modulus E, shear modulus G and Poisson’s ratio t) of the binary compounds, other calculations and experimentalmeasurements are included for comparison.

Compounds Methods B (GPa) E (GPa) G (GPa) t

TiC Present work 249 454 190 0.196Other theoretical work [2] 249 455 190 0.196Experiment [32] 242 437 182 0.199

TiN Present work 278 477 196 0.214Other theoretical work [2] 279 477 197 0.21Experiment [33] 318 469 187 0.25

TiO Present work 222 514 83 0.114Other theoretical work [20] 220.0 499.2 222.5 0.122a

Present work (with vacancy) 176 267 90 0.246Experiment [30] 177 270 112 0.26

a The value is deduced through the Eq. (4).

Fig. 5. Total density of states and partial density of states for TiC0.5N0.5.

Fig. 6. Total density of states and partial density of states for TiC0.5O0.5.

Fig. 7. Total density of states and partial density of states for TiO0.5N0.5.

90 J. Xiao et al. / Computational Materials Science 88 (2014) 86–91

3.3. Electronic properties

The calculated total density of states (DOS) and partial densityof states (DOS) around the Fermi level for ternary alloys of TiC0.5-

N0.5, TiC0.5O0.5 and TiO0.5N0.5 are shown in Figs. 5–7, respectively.

Similar DOS structures are observed from these ternary alloysand all can be divided into three main regions, including the firstregion mainly containing the nonmetal (C, N, O) 2s states, thesecond region originating from the intense hybridization of Ti 3dstates and nonmetal 2p states which represents the metal–non-metal covalent interaction and the third region dominated by Ti3d states which contributes to the metal–metal bond. Accordingto Jhi’s work [16], the highly directional coupling between metald and non-metal 2p electrons results in a shear resistive covalentbonding and such bonding gives a positive contribution to thehardness. Thus, the second region which consists of hybridized Ti3d states and nonmetal 2p states gives a rise to the hardness ofthese alloys and the metal–metal bond formed by Ti 3d statesreduces the hardness on the contrary. Furthermore, nonzero DOSvalues presented at the Fermi level demonstrate the metallic fea-ture of these alloys and Ti 3d states dominating the conductionband improve the electrical conductivity.

4. Conclusion

The structural, elastic and electronic properties of TiCxN1�x,TiCxO1�x and TiOxN1�x alloys as well as their binary compoundshave been investigated by means of first-principles calculationbased on density function theory. The computed equilibriumlattice constants a, bulk modulus B, shear modulus G and Poisson’sratio t for TiCxN1�x alloys are found to be consistent with the

J. Xiao et al. / Computational Materials Science 88 (2014) 86–91 91

experimental results, while for TiCxO1�x similar results have beenachieved after taking vacancy into account. Based on the consis-tency between calculated results and experimental values we haveobtained in not only TiCxN1�x and TiCxO1�x alloys but also the lat-tice parameters of TiOxN1�x alloys, rational prediction of the elasticproperties for TiOxN1�x alloys has been made. Moreover, for TiCx-

N1�x alloys bulk modulus are found to be linear with the concen-tration and the elastic properties presented as a function ofconcentration reveal the anisotropy and ductility of this kind ofcontinuous solid solution. The results of total density of statesand partial density of states for these alloys confirm the hybridiza-tion of Ti 3d states and nonmetal 2p states which produces a strongcovalent bond and gives a rise to the hardness of the alloys.

Acknowledgements

This work was supported by National Science Foundation ofChina (Nos. 50934001, 21071014, 51102015 and 51004008), theFundamental Research Funds for the Central Universities (Nos.FRF-AS-11-002A, FRF-TP-12-023A, FRF-MP-09-006B), ResearchFund for the Doctoral Program of Higher Education of China (No.20090006110005), National High Technology Research and Devel-opment Program of China (863 Program, No. 2012AA062302), Pro-gram of the Co-Construction with Beijing Municipal Commission ofEducation of China (Nos. 00012047 and 00012085) and the Pro-gram for New Century Excellent Talents in University (NCET-11-0577).

References

[1] S.-H. Jhi, J. Ihm, Phys. Rev. B 56 (1997) 13826–13829.[2] Y. Yang, H. Lu, C. Yu, J.M. Chen, J. Alloys Compd. 485 (2009) 542–547.[3] S.P. Denker, J. Less-Common Metall. 14 (1967) 1.[4] Y.T. Pei, C.Q. Chen, K.P. Shaha, J.Th.M. De Hosson, J.W. Bradley, S.A. Voronin, M.

Cada, Acta Mater. 56 (2008) 696–709.[5] F. Akhtar, J. Alloys Compd. 459 (2008) 491–497.[6] S.Q. Jiao, H. Zhu, J. Mater. Res. 21 (2006) 2172–2175.[7] S.Q. Jiao, H. Zhu, J. Alloys Compd. 438 (2007) 243–246.[8] M. Marlo, V. Milman, Phys. Rev. B 62 (2000) 2899–2907.[9] K.B. Panda, K.S. Ravi, Acta Mater. 54 (2006) 1641–1657.

[10] L. Tsetseris, S.T. Pantelides, Acta Mater. 56 (2008) 2864–2871.

[11] J. Wang, Z. Chen, C. Li, F. Wang, Y. Zhong, Adv. Mater. Res. 415–417 (2012)1451–1456.

[12] M. Zhang, J. He, Surf. Coat. Technol. 142–144 (2001) 125–131.[13] S. Bartkowski, M. Neumann, Phys. Rev. B 56 (1997) 10656–10667.[14] J. Graciani, A. Marquez, J.F. Sanz, Phys. Rev. B 72 (2005) 054117.[15] R. Ahuja, O. Eriksson, Phys. Rev. B 53 (1996) 3072–3079.[16] S.-H. Jhi, J. Ihm, S.G. Louie, M.L. Cohen, Nature 399 (1999) 132–134.[17] V.I. Ivashchenko, P.E.A. Turchi, A. Gonis, L.A. Ivashchenko, P.L. Skrynskii, Metall.

Mater. Trans. A 37 (2006) 3391–3396.[18] A. Zaoui, B. Bouhafs, P. Ruterana, Mater. Chem. Phys. 91 (2005) 108–115.[19] Q. Yang, W. Lengauer, T. Koch, M. Scheere, I. Smid, J. Alloys Compd. 309 (2000)

L5–L9.[20] L.S. Marques, A.C. Fernandes, F. Vaz, M.M.D. Ramos, Plasma Process. Polym. 4

(2007) S195–S199.[21] H.M. Pinto, J. Coutinho, M.M.D. Ramos, F. Vaz, L. Marques, Mater. Sci. Eng. B

165 (2009) 194–197.[22] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopolos, Rev. Mod. Phys.

64 (1992) 1045.[23] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M.C.

Payne, J. Phys. B: Condens. Matter 14 (2002) 2717–2744.[24] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892–7895.[25] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192.[26] V.P. Zhukov, V.A. Gubanov, O. Jepsen, N.E. Christensen, O.K. Anderson, J. Phys.

Chem. Solids 49 (1988) 841–849.[27] R.E. Loehman, C.N.R. Rao, J.M. Honig, J. Phys. Chem. 73 (1969) 1781–1784.[28] Y. O Ciftci, Y. Ünlü, K. Colakoglu, E. Deligo, Phys. Scr. 80 (2009) 025601.[29] A.D. Pearson, J. Phys. Chem. Solids 5 (1958) 316–327.[30] N.A. Ivanov, L.P. Andreeva, S.I. Alyamovskii, P.V. Gel’d, Inorg. Mater. 12 (1976)

1705–1707.[31] S. Andersson, B. Collen, U. Kuylenstierna, A. Magneli, Acta Chem. Scand. 11

(1957) 1641–1652.[32] J.J. Gilman, B.W. Roberts, J. Appl. Phys. 32 (1961) 1405.[33] J.O. Kim, J.D. Achenbach, P.B. Mirkarimi, M. Shinn, S.A. Barnett, J. Appl. Phys. 72

(1992) 1805–1811.[34] W. Feng, S. Cui, H. Hu, G. Zhang, Z. Lv, Physica B 406 (2011) 3631–3635.[35] T. Mastuda, H. Matsubara, J. Alloys Compd. 562 (2013) 90–94.[36] B. Jiang, N. Hou, S. Huang, G. Zhou, J. Hou, Z. Cao, H. Zhu, J. Solid State Chem.

204 (2013) 1–8.[37] V.A. Zhilyaev, S.I. Alyamovskii, V.D. Lyubimov, G.P. Shveikin, Inorg. Mater. 10

(1974) 1846–1849.[38] M.V. Kuznetsov, Y.F. Zhuravlev, V.A. Zhilyaev, V.A. Gubanov, Russ. J. Inorg.

Chem. 36 (1991) 560–563.[39] G.D. Bogomolov, G.P. Shveikin, S.I. Alyamovskii, Y.G. Zainulin, V.D. Lyubimov,

Inorg. Mater. 7 (1971) 60–64.[40] R. Hill, Proc. Phys. Soc. London Sect. A 65 (1952) 349.[41] S.F. Pugh, Philos. Mag. 45 (1954) 823–843.[42] M.H. Ledbetter, in: R.P. Reed, A.F. Clark (Eds.), Materials at Low Temperatures,

American Society for Metals, Metals Park, OH, 1983, p. 6.[43] W. Koster, H. Franz, Metall. Rev. 6 (1961) 1–56.