Structural stability of TiO and TiN under high pressure

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1 Cent. Eur. J. Phys. 6(2) DOI: /s z Central European Journal of Physics Structural stability of TiO and TiN under high pressure Research Article Raja Chauhan 1, Sadhna Singh 2, Ram Kripal Singh 3 1 Truba Institute of Engineering and Information Technology, Bhopal , India 2 Physics Dept., Barkatullah University,Bhopal , India 3 Institute of Professional and Scientific Studies and Research, Choudhary Devi Lal University, Sirsa , India Received 6 August 2007; accepted 5 November 2007 Abstract: The high pressure phase transition and elastic behavior of Transition Metal Compounds (TiO and TiN) which crystallize in NaCl-structure have been investigated using the three body potential model (TBPM) approach. These interactions arise due to the electron-shell deformation of the overlapping ions in crystals. The TBP model consists of a long range Coulomb, three body interactions, and the short-range overlap repulsive forces operative up to the second neighboring ions. The authors of this paper estimated the values of the phase transition pressures, associated volume collapses, and elastic constants, all of which were found to be closer to available experimental data than other calculations. Thus, the TBPM approach promises to predict the phase transition pressure and pressure variations of elastic constants of Transition Metal compounds. PACS (2008): p, 62.20dq, Lt., f, Keywords: phase transition transition pressure three body interaction (TBI) elastic constants Gibbs free energy Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction The Transition Metal Compounds (TMC), which crystallize in an NaCl (B 1 )-structure at ambient pressure, have been a topic of great interest because of their optical, magnetic and electrical properties. The high-pressure studies on various materials are significantly important both from basic and applied point of view. The transition metal compounds play an important role in solid-state technology, as they have many scientific, industrial, and technological raja283_74@yahoo.co.in applications 1, 2]. We have studied the behaviour of TiO and TiN under high pressure. The transition metal compounds MX (M denotes a transition metal element and X denotes one of the non-metallic elements C, N or O) have recently attracted much attention because of their high hardness, high melting point, wear, and corrosion resistance 3]. The electronic and structural phase transition of Transition Metal Compounds at high pressure is described by Zhukov et al. 4]. A theoretical study using a FPLMTO approach of the TMC has been carried out by Ahuja et al. 1]. They successfully employed the local density approach (LDA) for TiO and TiN. A linear muffin tin orbital atomic sphere approx- 277

2 Structural stability of TiO and TiN under high pressure imation (LMTO-ASA) approach has been employed to accurately study the theoretical high pressure behavior of TMC. As TMC s have interesting properties, and the fact that no study has been conducted using the three body interactions, we thought it pertinent to apply a three body interaction potential approach 5] which has a realistic potential. The importance of inclusion of three body interactions in a potential model to improve results has also been emphasized by Sims et al. 6] and W. Cochran 7]. In view of earlier studies 8 11], we decided to employ our three body interaction potential (TBIP) approach 10, 11] to study the high pressure behavior of TiO and TiN. This TBIP includes the long range Coulomb, three body interactions and the short range overlap repulsive interaction operative up to the second neighboring ions within the Hafemeister and Flygare framework 12]. 2. TBP model and method of computations The application of pressure on the crystals causes the decrease in their volume, which in turn leads to an increased charge transfer (or three-body interaction effects) due to the existence of the deformed (or exchange) charge between the overlapping electron shells of the adjacent ions. This overlapping leads to the transfer of charges which when interacts with another distant charge which gives rise to many body interactions (MBI). The dominant part of MBI is three body interactions (TBI) 10]. To understand this mechanism, let us designate A, B, and C ions with positions (lk), (l k ) and (l k ) in an ionic crystal having an ionic charge ± Ze with l and k as the cell and basis indices as shown in Fig. 1. Also, C is the nearest neigh- (dq k ) takes place between them dq k = ±zef k r(lk, l k ) = ±zef k (r). (1) The occurrence of the above transferred charge leads to a modified charge of A (or C) as z m e = z k e + nef k r(lk, l k ) = z k e1 + (2n/z)f k r(lk, l k )] 1/2. (2) Here, n is the number of the nearest neighbor (nn) ions, e is the electronic charge, f k (r) is the interionic potential force and 1+(2n/z)f k r(lk, l k )] 1/2 has been approximated as 1+(2n/z)f(r)]. The expression for the modified Coulombic energy due to the three body potential (TBI) is Φ m (r) = Φ c + Φ T, (3) Φ m (r) = α Mz 2 e nz ] r f(r), (4) where α M s the Madelung constant, which is ( ) for NaCl (CsCl) structure, r is the equilibrium nn ion separation and n is the number of nn ions, f(r) is the TBI parameter and is dependent on the nearest neighboring ion distance as f(r) = f 0 exp r. (5) These TBP effects have been incorporated into Gibbs free energy (G = U+PV T S). Here, U is the internal energy which at T = 0 K is equivalent to the lattice energy, S is the vibrational entropy at absolute temperature T. At T = 0 K and pressure P, the Gibbs free energy for the rock salt (B 1, real) and CsCl (B 2, hypothetical) structures are given by G B1 (r) = U B1 (r) + PV B1, (6) G B2 (r ) = U B2 (r ) + PV B2, (7) 278 Figure 1. Schematic representation of three body interactions model showing the three ions A, B and C with positions (lk, l k, l k ). bor (nn) of ion A and separated by distance r(lk, l k ) ; B is the nearest neighbor separation (r = r(lk, l k ) ) away from the ion A. During lattice vibrations, the electron shells of ions A and C overlap and a transfer of charge with V B1 (= 2.00r 3 ) and V B2 (= 1.54r 3 ) as the unit cell volumes for B 1 and B 2 phases respectively. Here r(r ) is the interionic separation for NaCl (CsCl)- structures respectively (awkward phrasing). The first term in the Eqns. (6) and (7) are the lattice energies for B 1 and B 2 structures and they are expressed as:

3 Raja Chauhan, Sadhna Singh, Ram Kripal Singh U B1 (r) = α Mz 2 e 2 12α Mze 2 f(r) ri + r j r + 6bβ ij exp r r 2ri 1.41r 2rj 1.41r + 6bβ ii exp + 6bβ jj exp, (8) U B2 (r ) = α M z2 e 2 16α M ze2 f(r ) ri + r j r + 8bβ r r ij exp 2ri 1.154r 2rj 1.154r + 3bβ ii exp + 3bβ jj exp. (9) Here r i (r j ),, b and β ij are the ionic radii, range parameter, hardness parameter and Pauling coefficients of the compounds respectively. These lattice energies consist of the long-range Coulomb energy (first term in Eqns. (8) and (9)), and the three body interaction energy term which are expressed by the second term in Eqns. (8) and (9). The energy due to the overlap repulsion extended up to the second neighboring ions are based on Hafemeister and Flygare(HF) type potential (the third, fourth and fifth terms in Eqns. (8) and (9)). To understand the elastic properties of these TMC s, we have calculated the second order elastic constants (SOEC), (C 11 C 12 and C 44 ) and their pressure derivatives. Since these elastic constants are functions of first and second order derivatives of short range potential, their calculations will provide knowledge about the effect of short range forces on these materials. Following the procedure adopted by Singh 10] and Sharma 13], we can obtain the expressions of the SOEC as: C 11 = e z { z + 12f(r) } + A 4r A ] 2 + B 2, 2 (10) C 12 = e z { z + 12f(r) } B 4r A ] 2 5B 2, (11) 4 C 44 = e z { z + 12f(r) } + B 4r A ] 2 + 3B 2, (12) 4 where, { } A 1 = 8r3 0 b e 2 exp ri + r j r, (13) 2 r=r 0 { } B 1 = 8r2 0 b e 2 r exp ri + r j r, (14) r=r 0 { A 2 = 8r3 0 3b e 2 exp 2 { B 2 = 8r2 0 3b e 2 r exp 2ri r 2ri r ] + b } exp 2rj r, 2 r= 2r 0 (15) ] + b } r exp 2rj r. r= 2r 0 (16) The first and second terms in Eqns. (10)-(12) are the contributions from the long-range Coulomb and TBP and the remaining contributions from the short-range overlap repulsion expressed as short range parameters (A 1,B 1 ) and (A 2,B 2 ) due to the nearest neighbour and next nearest neighbour (nnn) interactions. There values are obtained from the expressions defined by Singh and coworkers 10, 13] which use the values of b and s and whose determination procedure is describe below. Here r 0 is the inter-ionic separation at zero pressure. 3. Results and discussion The input data on crystal properties of transition metal compounds are presented in Table 1. The values of the model parameters, s, b, f(r)] namely the range, hardness and TBI parameter are evaluated from the knowledge of r 0, and the first and second space derivatives of internal energy U for the NaCl-stucture du/dr ] r=ro = 0 and d 2 U/dr 2] = 9kr 0 B T. (17) 279

4 Structural stability of TiO and TiN under high pressure Table 1. Input and model parameters of TiO and TiN. Compounds Input data Model parameters r i Å] r j Å] r 0 Å] b J] Å] f(r) TiO ] ] ] TiN ] ] ] Using the above equations, we have calculated the model parameters, bf(r)] and listed them in Table 1. We have computed the values of G B1 and G B2 at T = 0K using model parameters and minimization technique at different pressures. We have plotted the Gibbs free energy differences G = (G B1 G B2 ) against pressure (P) for TiO and TiN as shown in Fig. 2. We can see from Fig. 2 that the G Figure 3. Variation of relative volume change with pressure for TiO and TiN. Figure 2. Variation of Gibbs free energy differences G KJ/mol] against pressure PGPa] for TiO and TiN. is positive at zero pressure, which is a required criterion for a structure to be stable. As the pressure is increased, G decreases and approaches zero. The corresponding pressure at which G approaches zero is the phase transition pressure (P t ). From Fig. 2, it is evident that the phase transition pressure for TiO is 100 GPa and TiN is 310 GPa respectively. We find that these values are comparable with estimated value of P t from other theoretical work listed in Table 2. Furthermore, we have estimated a relative volume change V (P)/V (0)] and have plotted it with various high pressures for TiO and TiN depicted in Fig. 3. We find a sudden volume collapse in volume at phase transition pressure, which is a characteristic of first order phase transition. The magnitude of relative volume change at the transition pressure for TiO and TiN lies at 9%, and 8%, respectively, and they are close to the value of 10% reported by others. The second order elastic constants (SOEC) and their combination, C L = (C 11 + C C 44 )/2 and C s = (C 11 C 12 )/2 are calculated and listed in Table 3. The present results are aligned with the first-order character of the transition for these compounds and they are similar to the earlier reported results for calcium chalcogenides which belongs to the same family of compounds of NaCl (B1) structure. Also, they show a similar type of phase transition from NaCl (B1) to CsCl (B2) structure14]. According to the Vukcevich 15], the stable phase of a crystal is one in which the shear elastic constant C 44 is nonzero (for mechanical stability) and one of which has the lowest potential energy among the mechanically stable lattices. We have followed the Born criterion for a lattice to be mechanically stable which states that the elastic energy density must be a positive definite function of strain. This requires that the principal minors (the eigen values) of the elastic constant matrix should all be positive. Thus using the above stability, criterion for NaCl-stucture in 280

5 Raja Chauhan, Sadhna Singh, Ram Kripal Singh Table 2. Calculated transition pressures in GPa], associated volume collapses, elastic constants and their combination in GPa]. Compounds Transition Transition Pressure Volume collapes GPa] Present Others Present Others TiO B 1 B ] 9% 10% TiN B 1 B ] 8% 10% Table 3. The calculated values of the elastic constants and their combination GPa] of TiO and TiN. Compounds BT LDA GGA Exp. Present C S = 1 / 2(C 11 - C 12 ) C L = 1 / 2(C 11 +C 12 +2C 44 ) TiO 280 1] 230 1] 270 1] TiN 310 1] 270 1] 320 1] terms of the elastic constants is as follows C L = (C C 12 )/3 > 0, C 44 > 0, C S = (C 11 C 12 )/2 > 0. (18) The C 44 and C S are the tetragonal and shear modulus of a Cubic crystal. The value of C 44 /B T and C S /B T at the phase transition pressure shown in Table 2 satisfy the condition given by Eq. (18). Also, the estimated tetragonal moduli of these compounds are C 44 =145 GPa and C 44 =185 GPa respectively and the shear moduli for TiO and TiN are C S =289 GPa and C S =295 GPa respectively. Fig. 4 shows theoretical value while the value of C 12 nearly agrees with the theoretical data and better matching with experimental value. Present value of C 44 is larger than the theoretical data but it is in comparable range with theoretical and experimental value. As there are no experimental data for elastic constants for TiO, we could compare our results with theoretical results of Ahuja et al. In case of TiN our values are matching with experimental and also they are comparable with GGA and LDA values reported by Ahuja et al. 1] The values of C 11 and C 44 are lower then the LDA but C 12 are larger than the LDA. If we see GGA the present value of C 11 C 12 and C 44 are found to be larger then these value. The computed values of bulk modulus B T for TiO and TiN are compared with LDA and GGA value and experimental data in Fig. 5. The present values of B T are close to these values. Finally, we may conclude that during the crystallographic transition from NaCl to CsCl, the volume discontinuity in the pressure-volume phase diagram, identifies the same trends as was exhibited by the experimental and other theoretical technique. We have also checked the stability criterion for these compounds in terms of the elastic constants. On the basis of an overall achievement, we may claim that the TBP approach is appropriately suitable for the description of the phase transition and elastic behavior under pressure in Transition Metal Compounds. Figure 4. Variation of combination of elastic constants for TiO and TiN. References that for TiO the present value of C 11 is lower than the 1] R. Ahuja, O. Eriksson, Phys. Rev. B 53, 3072 (1996) 281

6 Structural stability of TiO and TiN under high pressure Table 4. Elastic constants of TiO and TiN (GPa). Compounds C 11 C 12 C 44 LDA GGA Exp. Present LDA GGA Exp. Present LDA GGA Exp. Present TiO 693 1] ] ] 145 TiN 735 1] 610 1] 625 1] ] 100 1] 165 1] ] 168 1] 163 1] 185 Figure 5. Variation of bulk modulus for TiO and TiN. 2] X. Li, T. Kobayashi, T. Sekine, Solid State Commun. 130, 79 ( 2004) 3] Y. Zhang, J. Li, Solid State Commun. 121, 411 (2002) 4] V.P. Zhukov, N.I. Medvedeva, V.A. Gubanov, Phys. Status Solidi B 21, 411 (2001) 5] D.W. Hafemeister, W.H. Flygare, J. Chem. Phys. 43, 795 (1965) 6] C.E. Sims, G.D. Barrera, N.L. Allan, Phys. Rev. B 57, (1998) 7] W. Cochran, Crit. Rev. Solid State 2, 1 (1971) 8] K.N. Jog, R.K. Singh, S.P. Sanyal, Phys. Rev. B 35, 5235 (1987) 9] R.K. Singh, D.C. Gupta, Phys. Rev. B 43, (1991) 10] R.K. Singh, Phys. Rep. 85, 259 (1982) 11] R.K. Singh, S. Singh, Phys. Rev. B 45, 1019 (1992) 12] D.W. Hafemeister, W.H. Flygare, J. Chem. Phys. 43, 795 (1965) 13] U.C. Sharma, PhD Thesis, Agra University (Agra, India, 1985) 14] S. Singh, J. Phys. Soc. Jpn. 71, 2477 (2002) 15] M.R. Vukecevich, Phys. Status Solidi B 54, 435 (1972) 282

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