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Electronic band structure, stability, structural, and elastic properties of IrTi alloys
Physica B 407 (2012) 2744–2748
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Electronic band structure, stability, structural, and elastic properties of IrTi alloys Wen-Zhou Chen a, Qian Li a, Zhen-Yi Jiang a,n, Xiao-dong Zhang a, Liang Si a, Li-Sha Li b, Rui Wu a a b
Institute of Modern Physics, Northwest University, Xi’an 710069, PR China Department of Physics, Northwest University, Xi’an 710069, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2012 Received in revised form 4 April 2012 Accepted 7 April 2012 Available online 12 April 2012
The structural properties and mechanical stabilities of B2-IrTi have been investigated using firstprinciple calculations. The elastic constants calculations indicate that the B2-IrTi is unstable to external strain and the softening of C11 C12 triggers the B2-IrTi (cubic) to L10-IrTi (tetragonal) phase transformation. Detailed electronic structure analysis revealed a Jahn–Teller-type band split that could be responsible for elastic softening and structure phase transition. The cubic–tetragonal transition is accompanied by a reduction in the density of states (DOS) at the Fermi level and the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi phase. & 2012 Elsevier B.V. All rights reserved.
Keywords: Alloys Density functional theory Electronic structure Elastic properties
1. Introduction Elucidating the structural stability trends and the underlying mechanisms in elemental metals is a fundamental topic in condensed matter physics. IrTi is an important Ti-based alloy owing to its high melting temperature and good high-temperature strengths [1]. With an abundance of possible applications such as medical devices and materials for aerospace, industrial and commercial use, this material has been studied for many years. The structures of the IrTi alloys used to be a subject of controversy [2–5]. The IrTi alloy exhibits an ordered cubic CsCltype for temperatures above 2130 1C. At room temperature, the crystal transforms into a monoclinic structure [2]. In disagreement with this observation, Raman and Schubert [3] reported that IrTi (annealed at 820 1C) has an end-centered orthorhombic lattice. Subsequently, Chen and Franzen [4] and Okamoto [5] reported that contrary to Ref. [2], the high temperature phase of IrTi is the tetragonal AuCu-type rather than the B2 cubic structure, and the low temperature phase is the orthorhombic NbRutype rather than the monoclinic lattice. Considering that we have obtained the low-temperature phase which is an orthorhombic cell with Cmmm space group [6], this paper will discuss the experimental debate regarding the high-temperature phase as to whether it is cubic or tetragonal structure and explore the
n
Corresponding author. Tel.: þ86 29 88303491; fax: þ86 29 88302331. E-mail addresses: [email protected], [email protected] (Z.-Y. Jiang). 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.04.017
underlying phase transformation mechanism further. It is well known that many properties, such as the phase stability and the transformation behavior are closely related to their electronic and elastic properties [7–12]. To the best of our knowledge, there are no theoretical results for these properties. In the present study, we have performed the first-principle total energy calculations on B2-IrTi under small strains to determine its elastic constants and identify the optimal tetragonal structure. Detailed electronic structure analysis associated with the tetragonal strain revealed a Jahn–Teller-type band split that could be responsible for elastic softening and structural phase transition. We also found that the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi cubic phase.
2. Calculatuions Our calculations were performed within the density functional theory, as implemented in the MedeA-Vasp package [13,14] with the projector augmented wave method (PAW) [15]. For the GGA [16] exchange-correlation function, the Perdew–Burke–Ernzerhof [17] form was employed. For comparison, the LDA [18,19] exchange-correlation function is also used for the calculation of the equilibrium structures and the elastic constants. The pseudopotentials represented 6s15d8 and 3d34s1 electron configurations for Ir and Ti, respectively. The Brillouin-zone intergrations were carried out by the Hermite–Gaussian smearing technique [20] with the smearing parameter of 0.01 eV. As the elastic constants calculations require a very high degree of precision, an energy
W.-Z. Chen et al. / Physica B 407 (2012) 2744–2748
cutoff of 600 eV was chosen. The Brillouin-zone (BZ) integration was carried out using the special k-point sample of the Monkhorst–Pack [21] type. The dense k-point grids 30 30 30 were used for all the cubic and tetragonal structures. The chosen plane-wave cutoff and the number of k-points were carefully checked to ensure that the total energy converged to better than 0.1 meV per formula unit (f.u.). The internal atomic positions and the external cell shape and volume were optimized simulta˚ neously until the force on each atom is less than 0.1 meV/A.
Table 2 Calculated values of elastic constants (GPa) of the cubic and tetragonal structures in IrTi alloys. Phase
C11
C12
C13
C33
C44
C66
C’
Cubic(LDA) Cubic(PBE) Ta(LDA) Ta(PBE) T.b(LDA) Tb(PBE)
54 48 212 192 419 367
354 304 365 309 172 143
– – 186 155 205 172
– – 436 382 381 338
92 82 88 79 77 71
– – 92 83 68 63
300 256 153 117 247 224
a
3. Results and discussions
2745
b
Tetragonal structure with c/a ¼0.910. Tetragonal structure with c/a¼ 1.189.
3.1. Elastic properties and their related stability As mentioned in the introduction section, a previous experimental study [2] has proposed that the high-temperature structure of B2-IrTi has a Pm-3m symmetry. The Ir atom is located at 1a position (0.0, 0.0, 0.0), and Ti at 1b position (0.5, 0.5, 0.5). The optimized lattice constants of B2-IrTi are compared in Table 1 with the experimental results. In the GGA–PBE calculation, the equilibrium lattice constant is predicted to be 3.1226 A˚ and its ˚ [2] is deviation with respect to the experiment value (3.106 A) 0.53%, while the LDA calculation predicted that the equilibrium ˚ which is about 1.34% smaller than the lattice constant is 3.0645 A, experimental value. This indicates that the present GGA–PBE yielded results that are in good agreement to the available experimental data. The stability against small deformations can be studied by considering the elastic constants. According to Hook’s law, the elastic energy of a solid can be written as a quadratic function of the strain components; hence, the elastic constants can be derived from the second-order derivatives of the energy–strain relations. A cubic crystal (B2-IrTi) has three independent elastic constants: C11, C12, and C44. So three equations are needed to determine all the three elastic constants. For this, we employed following three sets of calculations: the first set determines tetragonal shear constant [C0 ¼(C11 C12)] through the energy– tetragonal strain relationship, the second set determines bulk modulus [B ¼(C11 þ2C12)/3] through the energy–volume relationship and the third set determines C44/2 through the energy– orthorhombic strain relationship. For each set, its (B2-IrTi) unit cell was deformed by three different strain modes, whose nonzero strains are as follows: e1 ¼(d,d,(1 þ d) 2 1,0,0,0), e2 ¼ (d,d,d,0,0,0) and e3 ¼(0,0,d2(4 d2) 1,0,0,d) to calculate 3(C11 C12), 3(C11 þ2C12)/2, and C44/2, respectively. For each type of lattice deformation in our calculations d varies from 0.012 to 0.012 in steps of 0.003. The calculated elastic constants are listed Table 1 ˚ for the B2 and tetragonal structures of IrTi Equilibrium lattice constants (A) determined from GGA–PBE, compared with LDA and experimental results given in Refs. [2,4] and energies (DE ¼E EB2 eV/f.u.) relative to B2 in Tb and Tc. Structure
a
c
DE
B2(LDA) B2(GGA–PBE) B2a Tb(LDA) Tb(GGA–PBE) Tc(LDA) Tc(GGA–PBE) L10d
3.0645 3.1226 3.1060 3.1630 3.2224 2.8879 2.9416 2.9484
– – – 2.8778 2.9312 3.4352 3.4974 3.4986
0.00 0.00 – 0.0659 0.0640 0.1759 0.1633 –
a
Ref. [2]. Tetragonal structure with c/a¼ 0.910. c Tetragonal structure with c/a¼ 1.189. d Ref. [4]. b
Fig. 1. Total energies of deformed B2-IrTi as a function of the strain magnitude.
in Table 2. It is to be noticed that there are no other theoretical or experimental results for comparing with the present work. So we have used rather large k-meshes and energy cutoffs with two different exchange correlation potentials to reduce deviations. The discrepancy of the elastic constants between the two different exchange correlation potentials is due to the well-known tendency that LDA overestimates the elastic constants while GGA underestimates the elastic constants. For a cubic crystal there are four generally accepted elastic stability criteria [22,23]: (C11 þ2C12)4 0, C44 40, C11 40 and (C11 C12)40. The numerical values of our computation of all the elastic constants of B2-IrTi satisfy all the stability conditions except the condition (C11 C12)40. In the present calculations, the softening of C11 C12 indicates that the B2-IrTi is mechanically unstable under the tetragonal distortion. In order to study the crystal behavior at the point of instability, we do more calculations for the B2-IrTi under the tetragonal strain e1 ¼(d,d,(1þ d) 2 1,0,0,0). The total energies of deformed B2-IrTi as a function of the d (strain magnitude) are shown in Fig. 1. It clearly shows two local minima of energy, at d ¼ 0.055 and d ¼0.030. Furthermore, we perform geometrical optimization and similar symmetry analysis to determine the crystal structure of these resulting phases. It is found that the d ¼ 0.055 phase has P4/mmm symmetry with c/a¼1.189 and atomic positions: Ti (0.5, 0.5, 0.5), Ir (0.0, 0.0, 0.0). The d ¼ 0.030 structure has the same P4/mmm space group with c/a¼0.910 and atomic positions Ti (0.5, 0.5, 0.5), Ir (0.0, 0.0, 0.0). The detailed lattice constants of these two phases are also listed in Table 1. The calculated volume of the phase with c/a¼1.189 is 30.26 A˚ 3, while that of c/a¼0.910 is 30.44 A˚ 3. The equilibrium volume of the B2-IrTi is 30.45 A˚ 3. This is understandable because the martensitic transformation always involves slight change in volume. So these two resulting structures can be the candidate structures for B2-IrTi under deformation. One
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W.-Z. Chen et al. / Physica B 407 (2012) 2744–2748
should note that the methodology for calculating the elastic constants fails when the distortion is large. The results here are useful for predicting the phase transition, but are not suitable for calculating elastic constants. We now identify a better candidate structure with optimal c/a by considering its elastic constants. For a tetragonal structure, its independent elastic constants are C11, C12, C13, C33, C44 and C66. To determine the elastic constants of tetragonal structures by means of the curvature of the internal energy vs. strain curves, six strain modes [24] are adopted: e1 ¼(d,0,0,0,0,0), e2 ¼(0,0,d,0,0,0), e3 ¼(0,0,0,2d,0,0), e4 ¼(2d, d, d,0,0,0), e5 ¼ ( d, d,2d,0,0,0) and e6 ¼(d,d, 2d,0,0,2d) for C11/2, C33/2, 2C44, (5C11–4C12–2C13 þ C33)/2, (C11 þC12–4C13 þ2C33), and (C11 þC12 4C13 þ2C33 þ2C66), respectively. The available calculated elastic constants are listed in Table 2. The mechanical stability criterion can be formulated in terms of the elastic constants for tetragonal structures as [22,23]: (C11 C12)40, (C11 þC33–2C13)40, (2C11 þC33 þ2C12 þ4C13)40, C11 40, C33 40, C44 40 and C66 40. The elastic constants of the tetragonal phase (c/a¼1.189) fulfill all the stability criteria, implying its mechanical stability. It is to be noted that the tetragonal phase (c/a¼ 1.189) is stable only against long-wavelength oscillations, and there could be some phonon instabilities at shorter wavelengths [6]. The absolute value of C11 C12 for the c/a¼0.910 case is decreased when compared with the cubic cell (c/a¼ 1); however, it still remains negative. This suggests that the structure with the c/a¼1.189 value is more stable against tetragonal deformation than the structure with c/a¼0.910 value. In addition, the local minimum at c/a¼1.189 is lower in energy relative to the c/a¼0.910 minimum. Thus, we can conclude that the B2-IrTi is more likely to transform into the
tetragonal structure with c/a¼1.189, which is very close to the experimentally observed super-high temperature (tetragonal structure) value 1.187 [5]. The calculated and experimental lattice constants of the B2 phase are listed in Table 1; the good agreement indicates that the B2 phase indeed exists in IrTi alloys; moreover, the B2 phase is higher in energy than the L10 tetragonal structure. Thus, we theoretically prove that the high temperature phase of the IrTi alloys is a cubic structure with Pm-3m symmetry.
3.2. Electronic structure of IrTi In this section, we study in more detail the effects of the tetragonal distortions in the electronic structure. We decomposed the total DOS into site and angular momentum contributions. Fig. 2 depicts the total density of states (DOS) and projected DOS for d-orbital of Ti and Ir as a function of c/a. It is to be noted that the total DOS is mainly due to d partial DOS. The p and s states contribute very little to the total DOS and have been neglected. The Fermi level (EF) is set at 0 eV and denoted by the solid line. For both Ir and Ti sites, the DOS is dominated by contributions from the d states, as shown in Fig. 2. The total DOS below EF comes mainly from the Ti d states, while the DOS above EF is mainly due to the Ir d states. Within a rigid band model, the total DOS at the Fermi level (N(EF)) is an important indication of the stability of alloys [25–27]. Higher stability corresponds to a lower N(EF). As shown in Table 3, the N(EF) of the B2-IrTi, the tetragonal structure (c/a ¼0.910) and L10-IrTi (c/a¼1.189) are 3.41 states/ eV f.u., 2.30 states/eV f.u. and 2.28 states/eV f.u. respectively.
Fig. 2. Total and partial DOS of IrTi as a function of c/a: (a) 0.910, (b) 1.00, (c) 1.10 and (d) 1.189.
W.-Z. Chen et al. / Physica B 407 (2012) 2744–2748
Thus, the most stable phase is the tetragonal structure with c/a ¼1.189 followed by the c/a ¼0.910 phase and finally the B2-IrTi phase, which are in agreement with the results obtained from the elastic constants calculations. During the cubic– tetragonal transformation, DOS at the EF declines by 33%, as shown in Table 3. At the same time, the contribution from the Ti d states to N(EF) is 65% (from 1.30 states/eV f.u. in the B2 phase down to 0.45 states/eV f.u. in the L10 phase), while the contribution from the Ir d states is 13%. These results indicates that the reduction of N(EF) is governed mostly by Ti d-states while the contribution of Ir d-states to the N(EF) is relatively small. We may safely conclude that the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi phase. This result is in agreement with the results of other titanium compounds that the Ti d states play a crucial part in the phase stability [28]. The change of the total DOS at the Fermi level can also be traceable to the behavior of band structures in the vicinity of the Fermi energy. To compare the band structures of these phases, we fold the B2-IrTi band structure to a L10-IrTi Brillouin zone. In Fig. 3 the band structure for cubic (c/a¼1) distortions is compared to the band structures for the three different tetragonal distortions. All the band structures are mainly dominated by contributions from the d bands. The Fermi level is set at 0 eV and denoted by the solid line. The strain involved in the cubic–tetragonal transition means that the Fermi surfaces are quite different (L10-IrTi having the lowest DOS at EF). The band structure in the cubic structure (Fig. 3(b))
Table 3 Density of d electron states at the Fermi level (states/eV cell). Structure
N(EF) Total
N(EF) Ti
N(EF) Ir
c/a ¼0.910 c/a ¼1.000 c/a ¼1.100 c/a ¼1.189
2.30 3.41 2.55 2.28
0.66 1.30 0.65 0.45
1.15 1.50 1.35 1.31
2747
clearly show two bands near the Fermi level along the Z–A degenerate at the endpoints. Compared with the cubic phase, the band structures of the tetragonal structures are very similar. The small difference lies in the splitting of the degenerate bands at the A point near the EF which increased the stability. For the c/ a¼0.910 (Fig. 3(a)) and 1.1 (Fig. 3(c)) cases, one of the split bands is below and the other is just at the Fermi level. The distortion to c/a¼ 1.189 (Fig. 3(d)) is just enough to complete the tetragonal phase transition that eliminates the electron pocket and splits the double degenerate bands, thereby finally stabilizing the structure. The reduction of the total DOS value at the Fermi level is caused by the splitting of bands in this region. So we could conclude that the transition from the B2-IrTi to L10-IrTi can be regarded as a Jahn–Teller-type band split which leads to a lowering of the total energy.
4. Conclusions The structural properties and elastic constants of B2-IrTi and L10-IrTi are calculated within the density functional theory. The cubic–tetragonal transition is monitored with respect to total energies, elastic constants, band structures and density of states. We theoretically confirm that the high temperature phase of IrTi alloys does indeed have a cubic CsCl structure (Pm-3m). Our calculations indicated that the B2-IrTi cubic structure is unstable to the C11 C12 tetragonal shear. This tetragonal distortion leads to two energy minimal structures with c/a 41 and c/ ao1. We identified the resulting structures by considering their elastic constants and found that the B2-IrTi is more likely to transform into the tetragonal structure with c/a ¼1.189, which is very close to the experimental value (1.187). Detailed electronic structure analysis revealed a Jahn–Teller-type band split that could be responsible for elastic softening and phase transition. By decomposing the total DOS into site and angular momentum contributions, we found that the Ti d states play a crucial role in destabilizing the cubic phase. The present investigation deals
Fig. 3. Band structures along GMXGZARZ for different values of c/a. All band structures are set in the tetragonal Brillouin zone as a function of c/a: (a) 0.910, (b) 1.00, (c) 1.10 and (d) 1.189.
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W.-Z. Chen et al. / Physica B 407 (2012) 2744–2748
with the tetragonal distortion along the Bain path and provided a step towards a better understanding of the electronic structure involved in this transition.
Acknowledgments The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant nos. 10647008 and 50971099), the Research Fund for the Doctoral Program of Higher Education (no. 20096101110017), Key Project of Natural Science Foundation of Shaanxi Province of China (No. 2010JZ002), and Graduate’s Innovation Fund of Northwest University of China (Nos. 10YSY14 and 10YZZ38). References [1] [2] [3] [4]
Zhao-Yi Zeng, et al., Physica B 405 (2010) 3665. J.L. Murray, J. Phase Equilib. 3 (1982) 205. A. Raman, K. Schubert, Z. Metallkd. 55 (1964) 704. Chen Baihao, Hugo F. Franzen, J. Less-Common Metals 158 (1990) 11.
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H. Okamoto, J. Phase Equilib. 13 (1992) 329. W.-Z. Chen, et al., Solid State Commun. 151 (2011) 1433. I.R. Shein, D.V. Suetin, A.L. Ivanovskii, Physica B 406 (2011) 676. I.R. Shein, K.I. Shein, N.I. Medvedeva, A.L. Ivanovskii, Physica B 389 (2007) 296. I.R. Shein, K.I. Shein, A.L. Ivanovskii, Physica B 387 (2007) 184. I.R. Shein, N.I. Medvedeva, A.L. Ivanovskii, Physica B 371 (2006) 126. I.R. Shein, A.L. Ivanovskii, Physica B 405 (2010) 3213. I.R. Shein, A.L. Ivanovskii, Phys. Rev. B 83 (2011) 104501. G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1993) 11169. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. J.P. Perdew, et al., Phys. Rev. B 46 (1992) 6671. John P. Perdew, Kieron Burke, Matthias Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. M. Methfessel, A.T. Paxton, Phys. Rev. B 40 (1989) 3616. H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. J.F. Nye, Physical Properties of Crystals, Oxford University press, Oxford, 1985. I.R. Shein, Physica B 406 (2011) 3525. S.K.R. Patil, S.V. Khare, et al., Phys. Rev. B 73 (2006) 104118. I.R. Shein, et al., Phys. Rev. B 66 (2002) 024520. I.R. Shein, V.L. Kozhevnikov, A.L. Ivanovskii, Phys. Rev. B 78 (2008) 104519. I.R. Shein, A.L. Ivanovskii, Phys. Rev. B 84 (2011) 184509. H.J. Liu, Y.Y. Ye, Solid State Commun. 106 (1998) 197.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Electronic band structure, stability, structural, and elastic properties of IrTi alloys Wen-Zhou Chen a, Qian Li a, Zhen-Yi Jiang a,n, Xiao-dong Zhang a, Liang Si a, Li-Sha Li b, Rui Wu a a b
Institute of Modern Physics, Northwest University, Xi’an 710069, PR China Department of Physics, Northwest University, Xi’an 710069, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2012 Received in revised form 4 April 2012 Accepted 7 April 2012 Available online 12 April 2012
The structural properties and mechanical stabilities of B2-IrTi have been investigated using firstprinciple calculations. The elastic constants calculations indicate that the B2-IrTi is unstable to external strain and the softening of C11 C12 triggers the B2-IrTi (cubic) to L10-IrTi (tetragonal) phase transformation. Detailed electronic structure analysis revealed a Jahn–Teller-type band split that could be responsible for elastic softening and structure phase transition. The cubic–tetragonal transition is accompanied by a reduction in the density of states (DOS) at the Fermi level and the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi phase. & 2012 Elsevier B.V. All rights reserved.
Keywords: Alloys Density functional theory Electronic structure Elastic properties
1. Introduction Elucidating the structural stability trends and the underlying mechanisms in elemental metals is a fundamental topic in condensed matter physics. IrTi is an important Ti-based alloy owing to its high melting temperature and good high-temperature strengths [1]. With an abundance of possible applications such as medical devices and materials for aerospace, industrial and commercial use, this material has been studied for many years. The structures of the IrTi alloys used to be a subject of controversy [2–5]. The IrTi alloy exhibits an ordered cubic CsCltype for temperatures above 2130 1C. At room temperature, the crystal transforms into a monoclinic structure [2]. In disagreement with this observation, Raman and Schubert [3] reported that IrTi (annealed at 820 1C) has an end-centered orthorhombic lattice. Subsequently, Chen and Franzen [4] and Okamoto [5] reported that contrary to Ref. [2], the high temperature phase of IrTi is the tetragonal AuCu-type rather than the B2 cubic structure, and the low temperature phase is the orthorhombic NbRutype rather than the monoclinic lattice. Considering that we have obtained the low-temperature phase which is an orthorhombic cell with Cmmm space group [6], this paper will discuss the experimental debate regarding the high-temperature phase as to whether it is cubic or tetragonal structure and explore the
n
Corresponding author. Tel.: þ86 29 88303491; fax: þ86 29 88302331. E-mail addresses: [email protected], [email protected] (Z.-Y. Jiang). 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.04.017
underlying phase transformation mechanism further. It is well known that many properties, such as the phase stability and the transformation behavior are closely related to their electronic and elastic properties [7–12]. To the best of our knowledge, there are no theoretical results for these properties. In the present study, we have performed the first-principle total energy calculations on B2-IrTi under small strains to determine its elastic constants and identify the optimal tetragonal structure. Detailed electronic structure analysis associated with the tetragonal strain revealed a Jahn–Teller-type band split that could be responsible for elastic softening and structural phase transition. We also found that the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi cubic phase.
2. Calculatuions Our calculations were performed within the density functional theory, as implemented in the MedeA-Vasp package [13,14] with the projector augmented wave method (PAW) [15]. For the GGA [16] exchange-correlation function, the Perdew–Burke–Ernzerhof [17] form was employed. For comparison, the LDA [18,19] exchange-correlation function is also used for the calculation of the equilibrium structures and the elastic constants. The pseudopotentials represented 6s15d8 and 3d34s1 electron configurations for Ir and Ti, respectively. The Brillouin-zone intergrations were carried out by the Hermite–Gaussian smearing technique [20] with the smearing parameter of 0.01 eV. As the elastic constants calculations require a very high degree of precision, an energy
W.-Z. Chen et al. / Physica B 407 (2012) 2744–2748
cutoff of 600 eV was chosen. The Brillouin-zone (BZ) integration was carried out using the special k-point sample of the Monkhorst–Pack [21] type. The dense k-point grids 30 30 30 were used for all the cubic and tetragonal structures. The chosen plane-wave cutoff and the number of k-points were carefully checked to ensure that the total energy converged to better than 0.1 meV per formula unit (f.u.). The internal atomic positions and the external cell shape and volume were optimized simulta˚ neously until the force on each atom is less than 0.1 meV/A.
Table 2 Calculated values of elastic constants (GPa) of the cubic and tetragonal structures in IrTi alloys. Phase
C11
C12
C13
C33
C44
C66
C’
Cubic(LDA) Cubic(PBE) Ta(LDA) Ta(PBE) T.b(LDA) Tb(PBE)
54 48 212 192 419 367
354 304 365 309 172 143
– – 186 155 205 172
– – 436 382 381 338
92 82 88 79 77 71
– – 92 83 68 63
300 256 153 117 247 224
a
3. Results and discussions
2745
b
Tetragonal structure with c/a ¼0.910. Tetragonal structure with c/a¼ 1.189.
3.1. Elastic properties and their related stability As mentioned in the introduction section, a previous experimental study [2] has proposed that the high-temperature structure of B2-IrTi has a Pm-3m symmetry. The Ir atom is located at 1a position (0.0, 0.0, 0.0), and Ti at 1b position (0.5, 0.5, 0.5). The optimized lattice constants of B2-IrTi are compared in Table 1 with the experimental results. In the GGA–PBE calculation, the equilibrium lattice constant is predicted to be 3.1226 A˚ and its ˚ [2] is deviation with respect to the experiment value (3.106 A) 0.53%, while the LDA calculation predicted that the equilibrium ˚ which is about 1.34% smaller than the lattice constant is 3.0645 A, experimental value. This indicates that the present GGA–PBE yielded results that are in good agreement to the available experimental data. The stability against small deformations can be studied by considering the elastic constants. According to Hook’s law, the elastic energy of a solid can be written as a quadratic function of the strain components; hence, the elastic constants can be derived from the second-order derivatives of the energy–strain relations. A cubic crystal (B2-IrTi) has three independent elastic constants: C11, C12, and C44. So three equations are needed to determine all the three elastic constants. For this, we employed following three sets of calculations: the first set determines tetragonal shear constant [C0 ¼(C11 C12)] through the energy– tetragonal strain relationship, the second set determines bulk modulus [B ¼(C11 þ2C12)/3] through the energy–volume relationship and the third set determines C44/2 through the energy– orthorhombic strain relationship. For each set, its (B2-IrTi) unit cell was deformed by three different strain modes, whose nonzero strains are as follows: e1 ¼(d,d,(1 þ d) 2 1,0,0,0), e2 ¼ (d,d,d,0,0,0) and e3 ¼(0,0,d2(4 d2) 1,0,0,d) to calculate 3(C11 C12), 3(C11 þ2C12)/2, and C44/2, respectively. For each type of lattice deformation in our calculations d varies from 0.012 to 0.012 in steps of 0.003. The calculated elastic constants are listed Table 1 ˚ for the B2 and tetragonal structures of IrTi Equilibrium lattice constants (A) determined from GGA–PBE, compared with LDA and experimental results given in Refs. [2,4] and energies (DE ¼E EB2 eV/f.u.) relative to B2 in Tb and Tc. Structure
a
c
DE
B2(LDA) B2(GGA–PBE) B2a Tb(LDA) Tb(GGA–PBE) Tc(LDA) Tc(GGA–PBE) L10d
3.0645 3.1226 3.1060 3.1630 3.2224 2.8879 2.9416 2.9484
– – – 2.8778 2.9312 3.4352 3.4974 3.4986
0.00 0.00 – 0.0659 0.0640 0.1759 0.1633 –
a
Ref. [2]. Tetragonal structure with c/a¼ 0.910. c Tetragonal structure with c/a¼ 1.189. d Ref. [4]. b
Fig. 1. Total energies of deformed B2-IrTi as a function of the strain magnitude.
in Table 2. It is to be noticed that there are no other theoretical or experimental results for comparing with the present work. So we have used rather large k-meshes and energy cutoffs with two different exchange correlation potentials to reduce deviations. The discrepancy of the elastic constants between the two different exchange correlation potentials is due to the well-known tendency that LDA overestimates the elastic constants while GGA underestimates the elastic constants. For a cubic crystal there are four generally accepted elastic stability criteria [22,23]: (C11 þ2C12)4 0, C44 40, C11 40 and (C11 C12)40. The numerical values of our computation of all the elastic constants of B2-IrTi satisfy all the stability conditions except the condition (C11 C12)40. In the present calculations, the softening of C11 C12 indicates that the B2-IrTi is mechanically unstable under the tetragonal distortion. In order to study the crystal behavior at the point of instability, we do more calculations for the B2-IrTi under the tetragonal strain e1 ¼(d,d,(1þ d) 2 1,0,0,0). The total energies of deformed B2-IrTi as a function of the d (strain magnitude) are shown in Fig. 1. It clearly shows two local minima of energy, at d ¼ 0.055 and d ¼0.030. Furthermore, we perform geometrical optimization and similar symmetry analysis to determine the crystal structure of these resulting phases. It is found that the d ¼ 0.055 phase has P4/mmm symmetry with c/a¼1.189 and atomic positions: Ti (0.5, 0.5, 0.5), Ir (0.0, 0.0, 0.0). The d ¼ 0.030 structure has the same P4/mmm space group with c/a¼0.910 and atomic positions Ti (0.5, 0.5, 0.5), Ir (0.0, 0.0, 0.0). The detailed lattice constants of these two phases are also listed in Table 1. The calculated volume of the phase with c/a¼1.189 is 30.26 A˚ 3, while that of c/a¼0.910 is 30.44 A˚ 3. The equilibrium volume of the B2-IrTi is 30.45 A˚ 3. This is understandable because the martensitic transformation always involves slight change in volume. So these two resulting structures can be the candidate structures for B2-IrTi under deformation. One
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should note that the methodology for calculating the elastic constants fails when the distortion is large. The results here are useful for predicting the phase transition, but are not suitable for calculating elastic constants. We now identify a better candidate structure with optimal c/a by considering its elastic constants. For a tetragonal structure, its independent elastic constants are C11, C12, C13, C33, C44 and C66. To determine the elastic constants of tetragonal structures by means of the curvature of the internal energy vs. strain curves, six strain modes [24] are adopted: e1 ¼(d,0,0,0,0,0), e2 ¼(0,0,d,0,0,0), e3 ¼(0,0,0,2d,0,0), e4 ¼(2d, d, d,0,0,0), e5 ¼ ( d, d,2d,0,0,0) and e6 ¼(d,d, 2d,0,0,2d) for C11/2, C33/2, 2C44, (5C11–4C12–2C13 þ C33)/2, (C11 þC12–4C13 þ2C33), and (C11 þC12 4C13 þ2C33 þ2C66), respectively. The available calculated elastic constants are listed in Table 2. The mechanical stability criterion can be formulated in terms of the elastic constants for tetragonal structures as [22,23]: (C11 C12)40, (C11 þC33–2C13)40, (2C11 þC33 þ2C12 þ4C13)40, C11 40, C33 40, C44 40 and C66 40. The elastic constants of the tetragonal phase (c/a¼1.189) fulfill all the stability criteria, implying its mechanical stability. It is to be noted that the tetragonal phase (c/a¼ 1.189) is stable only against long-wavelength oscillations, and there could be some phonon instabilities at shorter wavelengths [6]. The absolute value of C11 C12 for the c/a¼0.910 case is decreased when compared with the cubic cell (c/a¼ 1); however, it still remains negative. This suggests that the structure with the c/a¼1.189 value is more stable against tetragonal deformation than the structure with c/a¼0.910 value. In addition, the local minimum at c/a¼1.189 is lower in energy relative to the c/a¼0.910 minimum. Thus, we can conclude that the B2-IrTi is more likely to transform into the
tetragonal structure with c/a¼1.189, which is very close to the experimentally observed super-high temperature (tetragonal structure) value 1.187 [5]. The calculated and experimental lattice constants of the B2 phase are listed in Table 1; the good agreement indicates that the B2 phase indeed exists in IrTi alloys; moreover, the B2 phase is higher in energy than the L10 tetragonal structure. Thus, we theoretically prove that the high temperature phase of the IrTi alloys is a cubic structure with Pm-3m symmetry.
3.2. Electronic structure of IrTi In this section, we study in more detail the effects of the tetragonal distortions in the electronic structure. We decomposed the total DOS into site and angular momentum contributions. Fig. 2 depicts the total density of states (DOS) and projected DOS for d-orbital of Ti and Ir as a function of c/a. It is to be noted that the total DOS is mainly due to d partial DOS. The p and s states contribute very little to the total DOS and have been neglected. The Fermi level (EF) is set at 0 eV and denoted by the solid line. For both Ir and Ti sites, the DOS is dominated by contributions from the d states, as shown in Fig. 2. The total DOS below EF comes mainly from the Ti d states, while the DOS above EF is mainly due to the Ir d states. Within a rigid band model, the total DOS at the Fermi level (N(EF)) is an important indication of the stability of alloys [25–27]. Higher stability corresponds to a lower N(EF). As shown in Table 3, the N(EF) of the B2-IrTi, the tetragonal structure (c/a ¼0.910) and L10-IrTi (c/a¼1.189) are 3.41 states/ eV f.u., 2.30 states/eV f.u. and 2.28 states/eV f.u. respectively.
Fig. 2. Total and partial DOS of IrTi as a function of c/a: (a) 0.910, (b) 1.00, (c) 1.10 and (d) 1.189.
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Thus, the most stable phase is the tetragonal structure with c/a ¼1.189 followed by the c/a ¼0.910 phase and finally the B2-IrTi phase, which are in agreement with the results obtained from the elastic constants calculations. During the cubic– tetragonal transformation, DOS at the EF declines by 33%, as shown in Table 3. At the same time, the contribution from the Ti d states to N(EF) is 65% (from 1.30 states/eV f.u. in the B2 phase down to 0.45 states/eV f.u. in the L10 phase), while the contribution from the Ir d states is 13%. These results indicates that the reduction of N(EF) is governed mostly by Ti d-states while the contribution of Ir d-states to the N(EF) is relatively small. We may safely conclude that the d-DOS of Ti at Fermi level plays a decisive role in destabilizing the B2-IrTi phase. This result is in agreement with the results of other titanium compounds that the Ti d states play a crucial part in the phase stability [28]. The change of the total DOS at the Fermi level can also be traceable to the behavior of band structures in the vicinity of the Fermi energy. To compare the band structures of these phases, we fold the B2-IrTi band structure to a L10-IrTi Brillouin zone. In Fig. 3 the band structure for cubic (c/a¼1) distortions is compared to the band structures for the three different tetragonal distortions. All the band structures are mainly dominated by contributions from the d bands. The Fermi level is set at 0 eV and denoted by the solid line. The strain involved in the cubic–tetragonal transition means that the Fermi surfaces are quite different (L10-IrTi having the lowest DOS at EF). The band structure in the cubic structure (Fig. 3(b))
Table 3 Density of d electron states at the Fermi level (states/eV cell). Structure
N(EF) Total
N(EF) Ti
N(EF) Ir
c/a ¼0.910 c/a ¼1.000 c/a ¼1.100 c/a ¼1.189
2.30 3.41 2.55 2.28
0.66 1.30 0.65 0.45
1.15 1.50 1.35 1.31
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clearly show two bands near the Fermi level along the Z–A degenerate at the endpoints. Compared with the cubic phase, the band structures of the tetragonal structures are very similar. The small difference lies in the splitting of the degenerate bands at the A point near the EF which increased the stability. For the c/ a¼0.910 (Fig. 3(a)) and 1.1 (Fig. 3(c)) cases, one of the split bands is below and the other is just at the Fermi level. The distortion to c/a¼ 1.189 (Fig. 3(d)) is just enough to complete the tetragonal phase transition that eliminates the electron pocket and splits the double degenerate bands, thereby finally stabilizing the structure. The reduction of the total DOS value at the Fermi level is caused by the splitting of bands in this region. So we could conclude that the transition from the B2-IrTi to L10-IrTi can be regarded as a Jahn–Teller-type band split which leads to a lowering of the total energy.
4. Conclusions The structural properties and elastic constants of B2-IrTi and L10-IrTi are calculated within the density functional theory. The cubic–tetragonal transition is monitored with respect to total energies, elastic constants, band structures and density of states. We theoretically confirm that the high temperature phase of IrTi alloys does indeed have a cubic CsCl structure (Pm-3m). Our calculations indicated that the B2-IrTi cubic structure is unstable to the C11 C12 tetragonal shear. This tetragonal distortion leads to two energy minimal structures with c/a 41 and c/ ao1. We identified the resulting structures by considering their elastic constants and found that the B2-IrTi is more likely to transform into the tetragonal structure with c/a ¼1.189, which is very close to the experimental value (1.187). Detailed electronic structure analysis revealed a Jahn–Teller-type band split that could be responsible for elastic softening and phase transition. By decomposing the total DOS into site and angular momentum contributions, we found that the Ti d states play a crucial role in destabilizing the cubic phase. The present investigation deals
Fig. 3. Band structures along GMXGZARZ for different values of c/a. All band structures are set in the tetragonal Brillouin zone as a function of c/a: (a) 0.910, (b) 1.00, (c) 1.10 and (d) 1.189.
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with the tetragonal distortion along the Bain path and provided a step towards a better understanding of the electronic structure involved in this transition.
Acknowledgments The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant nos. 10647008 and 50971099), the Research Fund for the Doctoral Program of Higher Education (no. 20096101110017), Key Project of Natural Science Foundation of Shaanxi Province of China (No. 2010JZ002), and Graduate’s Innovation Fund of Northwest University of China (Nos. 10YSY14 and 10YZZ38). References [1] [2] [3] [4]
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