pressure induced phase transitions in transition metal nitrides: ab initio study

Phys. Status Solidi B 248, No. 12, 2793–2800 (2011) / DOI 10.1002/pssb.201046589 p s sb

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basic solid state physics

Pressure induced phase transitionsin transition metal nitrides:Ab initio study

Anurag Srivastava*,1, Mamta Chauhan1, and R. K. Singh2

1Advanced Material Research Lab, Indian Institute of Information Technology and Management, Gwalior 474010, India2Department of Physics, ITM University, Gurgaon 122017, India

Received 20 November 2010, revised 24 April 2011, accepted 5 May 2011

Published online 2 June 2011

Keywords bulk modulus, CsCl, density functional theory, high-pressure effects, nitrides, phase transitions

*Corresponding author: e-mail [email protected], Phone: þ91 751 244 9826, Fax: þ91 751 244 9813

We have analyzed the stability of transition metal nitrides

(TMNs)XN (X¼Ti, Zr, Hf, V,Nb, Ta) in their original rocksalt

(B1) and hypothetical CsCl (B2) type phases under high

compression. The ground state total energy calculation

approach of the system has been used through the generalized

gradient approximation (GGA) with the Perdew-Burke-

Ernzerhof (PBE) type parameterization as exchange correlation

functional. In the whole series of nitrides taken into consid-

eration, tantalum nitride is found to be themost stable.We have

observed that under compression the original B1-type phase of

these nitrides transforms to a B2-type phase. We have also

discussed the computation of ground state properties, like the

lattice constant (a), bulk modulus (B0) and first order pressure

derivative of the bulk modulus (B’0) of the TMNs and their host

elements.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The study of transition metal nitrides(TMNs) has been a challenging area relatively to all thetheoretical and experimental researchers due to the unusualcombination of their physical and chemical properties. Thesematerials possess both metallic and non-metallic properties,e.g., very high melting points and high hardness, propertiestypical to covalent materials but they also show lustre,brittleness, and conductivity like metals and in some casessuperconductivity too with a high transition temperature Tc(Tc¼ 14.94K for NbN, 10.0K for ZrN, and 8.83K for HfN)[1]. These unusual properties of nitrides make themtechnologically very important [2, 3]. Owing to theseproperties they have a wide range of applications like incutting tools, resistant coating, magnetic recording, micro-electronics, corrosion, and abrasion resistant layers onoptical and mechanical components. Due to the extremelyhigh melting point, these materials are also called asrefractory materials.

The TMNs have an unusual bonding nature, which isresponsible for their unique properties. They have almost thesame metal–metal distance as in the pure metals but the non-metal atoms fill the interstitial holes of the metal lattice,which categorizes them as interstitial compounds. Blaha

et al. [4] observed some charge transfer in these compoundsand concluded that this charge transfer phenomenon isresponsible for their ionic nature in addition to the covalentand metallic behavior. Advanced theoretical modelingpackages have enabled us to compute and predict the variousproperties of crystals. These possibilities in computersimulations have made possible various theoretical predic-tions about these hard materials [5, 6] leading to a number ofexperimental efforts [7] to synthesize them. The applicationof these super hard materials under extreme temperature andpressure conditions makes this study interesting.

There have been a number of first-principle studies onelectronic structure calculations [8–12] of these TMNsdescribing their electronic band structure and bondingbehavior. In an even earlier work, Neckel et al. [13]calculated the band structures for eight compounds (ScN,ScO, TiC, TiN, TiO, VC, VN, and VO) and reported thecharge analysis with the help of wavefunctions. However,these calculations were still based on the muffin-tinapproximation. The Slater-Koster scheme was also used bythem to yield accurate density of states and partial density ofstates. Haglund et al. [14–16] investigated the cohesiveproperties and enthalpies of formation of 3d, 4d, and 5d


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TMNs and discussed the bonding mechanism by using thelinear muffin-tin orbital (LMTO) method. Wang et al. [17]calculated the mechanical properties of cubic d-NbN andhexagonal d

0-NbN with density functional theory (DFT) and

found that the calculated ideal strength of d0-NbNwas higher

than that of d-NbN. Puska et al. [18] have predicted the first-principle electronic structure and positron-state calculationsfor carbides as well as nitrides of transition metals anddetermined their positron affinities and lifetimes. Kobayashi[19] has analyzed the electronic and structural properties ofTMN surfaces. In a recent report by Hernandez et al. [20] onthe study of electronic and structural properties of TiN, thestability of TiN has been analyzed in its various phases likeNaCl (B1), CsCl (B2), zinc blende (B3), and wurtzite (B4)type. Ahuja et al. [21] have discussed the stability of TiC,TiN, and TiO in NaCl (B1) and CsCl (B2) type phases byusing FPLMTO method and also investigated the structuralphase transition in them. In an another study, TiN,MoN,VN,NbN, HfN, and ZrN have been analyzed under highcompression by Ojha et al. [22] through model calculationswhere predictions were made on a B1 to B2 type structuralphase transition. However, there is anomaly among theresults of model calculation and FPLMTO calculation,specially in case of TiN.

Recently, our group has done the stability analysis oftransition metal carbides (TMCs) [23] as well as othermaterials [24] using the DFT approach and discussed thestructural phase transitions in them. Not many groups haveattempted to predict the phase transition in TMNs under highcompression; particularly in TaN, the structural phasetransition pressure has not yet been discussed, but only itsenergetic stability in different structures has been discussedby Cao et al. [25] and Zhao et al. [26]. Zhao et al. haveobserved a structural transition of TaN from TaN typestructure (P62m, No. 189) to WC type structure (P6m2,No.187) at a very low pressure of about 5GPa and have notgone beyond 100GPa. The technological applications ofthesematerials and success of ab initiomethodsmotivated usto analyze the stability and structural changes in TiN, ZrN,HfN, VN, NbN, and TaN super hard materials. Apart fromthe structural phase transitions in these materials, the presentstudy also includes the discussions on the computation ofground state properties, like lattice parameter (a), bulkmodulus (B0), first order pressure derivative of bulkmodulus(B’0) and cohesive energies of TMNs in B1 and B2 typephases and their constituent elements in their stablestructures.

2 Computational methodology The present com-putation of structural stability and phase transition in TMNshas been done by using the DFT-based SIESTA code [27]which is appropriate for electronic structure calculations andab initio molecular dynamics simulations of molecules andsolids. Within the framework of DFT [28], the first-principlepseudopotential approach has been used to perform thestability analysis of the original rocksalt (B1) and hypothe-tical CsCl (B2) type phases of TMNs. The effects of


exchange–correlation interactions are handled by thegeneralized gradient approximation (GGA) with thePerdew-Burke-Ernzerhof (PBE) [29] type parameterizationin the self consistent run for all the above mentioned superhard materials. The pseudopotentials used in the presentcalculation are norm conserving and non-relativistic. In thepresent computation, the nitrides are assumed to be withoutany defect and the stoichiometric composition for all thenitrides is taken as 1:1 ratio ofmetal atom and nitrogen atom.The electronic wave functions are expanded in a plane wavebasis set with an energy cut-off of 200 Ry. The atomic orbitalbasis set employed in our calculations is double-z withpolarization for Ti 4s, Zr 5s, Hf 6s, V 4s, Nb 5s, Ta 6s, and N2p states whereas only double-z for Ti 3d, Zr 4B, Hf 5B, V3d, Nb 4B, Ta 5B, andN 2s states. For the k-point sampling, agrid of 7� 7� 7 (343) k-points is used for HfN and TaN inthe whole irreducible Brillouin zone, whereas for TiN, ZrN,VN, NbN, V, Nb, and Ta, a grid of 8� 8� 8 (512) points andfor Ti, Zr, and Hf grid of 8� 8� 5 (320) points is used, as aresult of convergence tests for these compounds.

3 Results and discussion The structural propertiesare very imperative in analyzing the behavior of materials.To determine the structural ground state properties, the totalenergy is calculated at different unit cell volumes around theequilibrium volume, for which the total energy of the systembecomes lowest, by the self-consistent run of Kohn Shamequations [30]. The calculated total energies and theircorresponding unit cell volumes are used in Murnaghan’sequation of state [31] for the computation of the equilibriumground state properties like lattice parameter, bulk modulusand first pressure derivative of the bulk modulus. Both theLDA and GGA schemes have been used in the presentanalysis of TMNs in their original B1-type phase, as in ourearlier work on TMCs [23] the present observation withGGA has given better results than LDA. The discussionsregarding stability analysis, ground state properties andhigh-pressure phase transition in a series of the above TMNsalong with the ground state properties of constituent metalsare given in following sub sections.

3.1 Stability and ground state properties Foranalyzing the stability of 4B and 5B transition metals andtheir nitrides, the DFT-based SIESTA code is used forcalculating their total energies. The mechanical strength of asolid is determined with the help of its cohesive energy,which tells the strongness of binding between the constituentatoms in a solid. The cohesive energy of a solid is thedifference between the total energy per atom of the bulkmaterial at equilibrium and the atomic energies of the atomsbelonging to the unit cell of the material. Cohesive energiesare taken to be positive and plotted in Fig. 1(a) and 1(b) fortransition elements and their nitrides respectively. Ourcalculated cohesive energies are in close agreement withthe experimental observations [14, 15, 32]. However, thecalculated values for the transition metals are about 15–18%higher (much more for vanadium) than the experimental

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Phys. Status Solidi B 248, No. 12 (2011) 2795

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Figure 1 (online color at: www.pss-b.com) (a) Cohesive energiesof transitionmetals in their stable structures. (b)Cohesiveenergiesoftransition metal nitrides in B1 structures.

Table 1 Lattice constant (a), bulk modulus (B0), and pressure deri

element (structure) a (A)

Ti (hcp) present 2.887experimental 2.95 [40]TB method 2.97 [40]

Zr (hcp) present 3.215experimental 3.23 [40]TB method 2.99 [40]

Hf (hcp) present 3.228experimental 3.19 [40]TB method 3.07 [40]

V (bcc) present 3.077experimental 3.03 [40]TB method 2.94 [40]APW-LDA 2.927 [41]

Nb (bcc) present 3.378experimental 3.30 [40]TB method 3.25 [40]APW-LDA 3.260 [41]

Ta (bcc) present 3.380experimental 3.30 [40]TB method 3.30 [40]APW-LDA 3.294 [41]

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values, whereas for TMNs this variation is from 35 to 40%with respect to the corresponding experimental values [32].Plots show that the cohesive energies of nitrides are higherthan that of their constituent metals. Thus, it can beconcluded that the bonding strength of nitrides is higherthan that of the host elements. Here, it is also observed thatZrN and TaN with the highest cohesive energies can beconsidered as the most stable in their respective 4B and 5Bseries, whereas TaN owing to its highest cohesive energy isthemost stable among all theTMNs taken into consideration.However, experimentaly [15] in the 4B series HfN isreported as the most stable, whereas the cohesive energy inthe 4B series is highest for ZrN as reported by Stample et al.[10] very similar to our observations. The present stabilitytrends of transition metals as well as their nitrides are verysimilar in nature except that in the 4B series, where Hf isfound to be the most stable one, which is well supported byexperimental results [32]. The present calculated bindingenergy (�8.658 eV) and bond length (1.1 A) of a nitrogenmolecule are in close agreement with experimental obser-vations particularly for the binding energy [33]. Thecalculated ground state properties, like lattice parameters,bulk moduli, and pressure derivatives are obtained by a fit toMurnaghan’s equation of state for the host transitionelements and are listed in Table 1. The ground stateproperties for both the B1 and B2 type structures have alsobeen computed and are listed in Table 2 along with the otherreported experimental [34–38] and/or theoretical [8, 10–12,20–22, 25, 26, 39] counterparts.

In the B1 type phase of TMNs, our calculated values oflattice constants are found to be 0.4–1.5%higher and the bulkmoduli about 8–15% smaller in comparison to their availableexperimental counterparts. The lattice constants of host

vative (B00) of transition metals in their stable structures.

c (A) B0 (GPa) B00

4.684 155.87 2.734.68 [40] 105 [40]4.80 [40] 122 [40]5.148 116.51 2.055.15 [40] 83 [40]5.57 [40] 108 [40]5.051 107.08 2.585.05 [40] 109 [40]5.08 [40] 111 [40]

132.06 2.49162 [40]211 [40]223.0 [41]133.91 2.53170 [40]187 [40]194.8 [41]161.25 2.67200 [40]185 [40]200.8 [41]



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Table 2 Lattice constant (a), bulk modulus (B0), pressure derivative (B00), and phase transition pressure (PT) of TMNs.

crystal a (A) B0 (GPa) B00 PT (GPa)

B1 B2 B1 B2 B1 B2

TiNpresent 4.214 2.66 238.65 194.32 3.48 3.16 474experimental 4.26 [35], 4.235 [37] 288 [34]PP-LDA [11] 4.32 304PP-GGA [39] 4.27 292FPLAPW-GGA [10] 4.26 286FPLAPW-LDA [10] 4.18 322FPLMTO-LDA [21] 370interionic potential theory [22] 126FPLAPW-GGA [20] 4.249 2.64 272.2 244.5 4.34 4.25

ZrNpresent 4.561 2.88 231.74 193.54 3.23 2.79 333experimental 4.61 [37], 4.537 [38]PP-LDA [12] 4.52 285FPLAPW-GGA [10] 4.57 264FPLAPW-LDA [10] 4.53 292interionic potential theory [22] 94

HfNpresent 4.539 2.86 243.82 198.50 3.27 3.03 396experimental 4.52 [38]PP-LDA [12] 4.45 305FPLAPW-GGA [10] 4.54 278FPLAPW-LDA [10] 4.48 320interionic potential theory [22] 130

VNpresent 4.148 2.61 241.41 224.13 3.63 3.21 278experimental 4.13 [35] 268 [36]PP-LDA [11] 4.19 338PP-GGA [39] 4.128 320FPLAPW-GGA [10] 4.12 333FPLAPW-LDA [10] 4.06 376APW [8] 4.14interionic potential theory [22] 102

NbNpresent 4.440 2.78 266.96 236.04 3.45 3.04 207experimental 4.392 [38]PP-LDA [12] 4.35 351FPLAPW-GGA [10] 4.42 317FPLAPW-LDA [10] 4.36 354APW [8] 4.38interionic potential theory [22] 127

TaNpresent 4.448 2.79 294.64 282.09 3.48 2.99 300experimental 4.385 [38]PP-LDA [12] 4.33 374FPLAPW-GGA [10] 4.42 338FPLAPW-LDA [10] 4.37 378APW [8] 4.40US-PP-GGA [25] 4.413 2.75 327 307 4.35 4.28PAW-GGA [26] 4.415 2.74 347 334

transition elements show 0.4–2.4% variation and bulkmodulus 2–40% variation from the corresponding exper-imental values. One reason for these deviations in ourcalculated lattice constants and bulk moduli from the


experimental counterparts may be caused by using GGA asexchange correlation functional in the present calculationand another could be the temperature effect since theexperimental work is done at room temperature and our

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Figure 2 (online color at:www.pss-b.com) (a) Lattice constants inpresent B1 andB2 phaseswith experimental results available for B1phase and (b) bulk modulus in present B1 and B2 phases for TMNs.

Figure 3 Enthalpy as a function of pressure for TiN.

Figure 4 Enthalpy as a function of pressure for ZrN.

Figure 5 Enthalpy as a function of pressure for HfN.

theoretical work corresponds to 0K. The trends of latticeconstants and bulk moduli for 4B and 5B TMNs in both theNaCl and CsCl type phases are presented in Fig. 2(a)and 2(b), where it is seen that our theoretical trend for latticeconstants in the B1 phase (shown as solid line joining thecircular dots) almost retraces the experimental trend (shownby dotted lines joining the triangular dots). In the same graphthe lattice constants of the B2 type phase are also shown by asolid line joining the square dots, which shows that thechange in lattice constant from material to material is thesame in both the B1 and B2 type phases in the respective 4Band 5B series. The bulk moduli in both the B1 and B2 typephases also have a similar trend for thesematerials. The plotsof lattice constants and bulk moduli also show that as wemove from left to right in the periodic table, the latticeconstant decreases while the bulk modulus increases, whichis in agreement with the observation reported by Stampflet al. [10] who have reported results only for the B1 typephase. The present paper reports a similar trend in both theB1 as well as B2 type phase for these nitrides. In absence ofany experimental or theoretical data on lattice parameters,bulkmoduli and pressure derivatives in the B2 type phase forthese materials except for TiN [20] and TaN [25, 26] ourcomputed results may probably be the first.

3.2 High pressure phase transition Under ambi-ent conditions all the TMNs taken into consideration

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crystallize in the NaCl type phase and are found to undergoa structural transformation to the CsCl type phase underhigh compression. In order to investigate this pressureinduced phase transition in the TMNs, we have calculatedthe enthalpies (H¼Eþ PV) as a function of pressurefor the different nitrides and plotted them in Figs. 3–8.The transition pressure (PT) is obtained as the pressureat which the enthalpies of the two competitive phasesare identical. In terms of stability of phases, it is foundthat below the transition pressure, the B1 type phase of



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Figure 7 Enthalpy as a function of pressure for NbN.

Figure 9 (online color at: www.pss-b.com) Transition pressuresfor TMNs.

Figure 10 Relative volume as a function of pressure for TiN.

Figure 6 Enthalpy as a function of pressure for VN.

all the TMNs is preferred in comparison to the hypotheticalB2 type phase, whereas above this pressure the B2 typephase becomes more stable than the B1 type phase. Thecalculated transition pressures are tabulated in Table 2 andcompared with other reported results. In reference to TiN,our calculated transition pressure and the reported data byAhuja et al. [21] using a full potential method show aclose match. However, in comparison to a recent report ontransition pressures for TiN, ZrN, HfN, VN, and NbN [22]using model calculation our results are found to be onthe higher side. This deviation might be due to the type ofinteractions taken into consideration in two approaches.Recently, a similar analysis on the structural transitions

Figure 8 Enthalpy as a function of pressure for TaN.


in TMCs has been reported by our group [23], where wehave compared our findings with the FPLMTO and a modelcalculation results. The variation of transition pressures fordifferent nitrides is plotted in Fig. 9 where it can be clearlyseen that in the 4B series TiN has the highest transitionpressure and ZrN has the lowest one while in the 5B seriesthe transition pressure is highest for TaN and lowest forNbN.

For a better understanding of hardness, the amount ofvolume collapse at the transition pressure has also beencomputed. The relative change in volume as a functionof pressure is plotted in Figs. 10–15. In these materials,

Figure 11 Relative volume as a function of pressure for ZrN.

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Figure 12 Relative volume as a function of pressure for HfN.

Figure 13 Relative volume as a function of pressure for VN.

Figure 15 Relative volume as a function of pressure for TaN.

the small change of about 2–3% in volume during thetransition from the NaCl to CsCl type phase indicates theextreme hardness of these materials. As mentioned inthe introduction Zhao et al. [26] and Cao et al. [25] haveonly discussed the stability of TaN in its various phasesand did not report any transition from the B1 to B2 typephase; hence our results on TaN under high compressionmay be the first one.

Figure 14 Relative volume as a function of pressure for NbN.

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4 Conclusion All calculations in the present studyhave been done using the ab initio pseudopotential approachwithin the framework ofDFT as implemented in the SIESTAcode. The ground state properties of TMNs and the hostelements are in reasonable agreement with other reportedtheoretical and/or experimental results. Similar trends havebeen found for the stability, lattice constants as well as bulkmoduli of TMNs in their NaCl and CsCl type phases. Allthese nitrides undergo a structural B1!B2 phase transitionunder high compression in the range of 200–500GPa. Such ahigh phase transition pressure reveals the extreme hardnessof these materials, which is consistent with their high bulkmoduli and small values of volume collapses at theirphase transition pressures. Calculated cohesive energiesand bulk moduli of TMNs are found to be higher incomparison to that of the pure transition elements due tothe stronger bonding and mechanical strength of TMNs thantheir host elements.

Acknowledgements Authors gratefully acknowledge theDRDO for providing financial support for the research. One of usMamta Chauhan, is thankful to DRDO for providing JRF.

References

[1] X.-J. Chen, V. V. Struzhkin, S. Kung, H.-K. Mao, and R. J.Hemley, Phys. Rev. B 70, 014501 (2004).

[2] H. J. Goldschmidt, Interstitial Alloys, Vol. 1 (Butterworths,London, 1967).

[3] L. E. Toth, Transition Metal Carbides and Nitrides (Aca-demic Press, New York/London, 1971).

[4] P. Blaha, J. Redinger, and K. Schwarz, Phys. Rev. B 31, 2316(1985).

[5] M. L. Cohen, Phys. Rev. B 32, 7988 (1985).[6] A. Y. Liu and M. L. Cohen, Science 245, 841 (1989).[7] J. V. Badding and D. C. Nesting, Chem. Mater. 8, 535

(1996).[8] D. A. Papaconstantopoulos, W. E. Pickett, B. M. Klein, and

L. L. Boyer, Phys. Rev. B 31, 752 (1985).[9] S.-H. Jhi, J. Ihm, S. G. Louie, and M. L. Cohen, Nature

(London) 399, 132 (1999).[10] C. Stampfl,W.Mannstadt, R. Asahi, and A. J. Freeman, Phys.

Rev. B 63, 155106 (2001).



hys

ica ssp st

atu

s

solid

i b

[11] J. C. Grossman, A. Mizel, M. Cote, M. L. Cohen, and S. G.Louie, Phys. Rev. B 60, 6343 (1999).

[12] Z. Wu, X.-J. Chen, V. V. Struzhkin, and R. E. Cohen, Phys.Rev. B 71, 214103 (2005).

[13] A. Neckel, P. Rastl, R. Eibler, P. Weinberger, andK. Schwarz, J. Phys. C 9, 579 (1976).

[14] J. Haglund, G. Grimvall, T. Jalborg, and A. F. Guillermet,Phys. Rev. B 43, 14400 (1991).

[15] A. F. Guillermet, J. Haglund, and G. Grimvall, Phys. Rev. B45, 11557 (1992); Phys. Rev. B 48, 11673 (1993).

[16] J. Haglund, A. F. Guillermet, G. Grimvall, and M. Korling,Phys. Rev. B 48, 11685 (1993).

[17] C. Wang, M. Wen, Y. D. Su, L. Xu, C. Q. Qu, Y. J. Zhang,L. Qiao, S. S. Yu, W. T. Zheng, and Q. Jiang, Solid StateCommun. 149, 725 (2009).

[18] M. J. Puska, M. Sob, G. Brauer, and T. Korhonen, Phys. Rev.B 49, 10947 (1994).

[19] K. Kobayashi, Surf. Sci. 493, 665 (2001).[20] R. J. G. Hernandez, W. R. L. Perez, and J. A. R. Martinez,

J. Phys. 39(2), 519 (2007).[21] R. Ahuja, O. Ericsson, J. M. Wills, and B. Johansson, Phys.

Rev. B 53, 3072 (1996).[22] P. Ojha, M. Aynyas, and S. P. Sanyal, J. Phys. Chem. Solids

68, 148 (2007).[23] A. Srivastava, M. Chauhan, and R. K. Singh, Phase Transit.

84, 58 (2011).[24] A. Srivastava, N. Tyagi, U. S. Sharma, and R. K. Singh,

Mater. Chem. Phys. 125, 66 (2011).[25] C. L. Cao, Z. F. Hou, and G. Yuan, Phys. Status Solidi B 245,

1580 (2008).[26] E. Zhao, B. Hong, J. Meng, and Z. Wu, J. Comput. Chem. 30,

2358 (2009).


[27] J. M. Soler, E. Artacho, J. D. Gale, A. Garcia, J. Junquera,P. Ordejon, and D. Sanchez-Portal, J. Phys.: Condens.Matter 14, 2745 (2002).

[28] P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).[29] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.

77, 3865 (1996).[30] W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).[31] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 30, 244 (1944).[32] D. H. Buckley, Surface Effects in Adhesion, Friction, Wear,

and Lubrication (Elsevier, The Netherlands, 1981).[33] W. A. Harrison, Pure Appl. Chem. 61, 2161 (1989).[34] V. A. Guvanov, A. L. Ivanovsky, and V. P. Zhukov,

Electronic Structure of Refractory Carbides and Nitrides(Cambridge University Press, Cambridge, 1994).

[35] W. B. Pearson, A Handbook of Lattice Spacings and Struc-tures of Metals and Alloys, Vol. 2 (Pergamon Press, Oxford,1967).

[36] J. O. Kim, J. D. Achenbach, P. B. Mirkarimi, M. Shinn, andS. A. Barnett, J. Appl. Phys. 72, 1805 (1992).

[37] R. W. G.Wyckoff, Crystal Structures, Vol. 1, 2nd ed. (Wiley,New York, 1963).

[38] Structure reports, edited by W.B. Pearson (InternationalUnion of Crystallography, Oosthoek, Scheltema, and Holk-ema, Utrecht, 1913–1993). ZrN (Vol. 30A, p. 155), NbN(Vol. 30A, p. 155), HfN (Vol. 17, p. 70), and TaN (Vol. 35A,p. 121).

[39] P. Lazar, J. Redinger, and R. Podloucky, Phys. Rev. B 76,174112 (2007).

[40] M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54,4519 (1996).

[41] M. M. Sigalas and D. A. Papaconstantopoulos, Phys. Rev. B50, 7255 (1994).

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pressure induced phase transitions in transition metal nitrides: ab initio study

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