The high-pressure stability of Ni2In-type structure of ZrO2 with respect to OII and Fe2P-type phases: A first-principles study

The density-functional theory is used to investigate the stability of the Ni2In-type hexagonal structure with the space group: P63/mmc at high pressures and compared to the orthorhombic OII and Fe2P-type phases of zirconia (ZrO2). The calculations showed that the high-pressure phase transition sequence in ZrO2 is as follows: OII → Fe2P → Ni2In, which is consistent with the recent measurements that are observed in Fe2P as a post-OII phase of ZrO2. We obtained a very small volume change across OII → Fe2P transition, whereas an appreciable volume change is found across the Fe2P → Ni2In transition. The compressibility of Ni2In phase is found to be high as compared to the other two high-pressure phases.


Introduction
The transition-metal dioxide zirconia (ZrO2) is involved in many industrial applications [1][2][3][4][5][6][7] due to its important properties. As a result of the bonding nature of this dioxide, ZrO2 has received an obvious attention regarding its structural and mechanical properties, where many experimental (e.g., Refs. [8][9][10][11][12][13]) and theoretical (e.g., Refs. [8,11,[14][15][16][17]) studies have focused on investigating the high-pressure behavior of ZrO2 including the determination of the EOS of its various polymorphs, phase stability, and mechanical strength. At ambient temperature, the well-known high-pressure phase transition sequence [8,12,14,15,18,19] is as follows: MI Î OI Î OII, where MI has a monoclinic structure (space group: P21/c) and both OI and OII have an orthorhombic structure (space groups: Pbca and Pnma, respectively). Recent measurements and calculations have confirmed the discovery of Fe2P-type structure (Hexagonal, space group: m P 2 6 ) as a post-OII phase for ZrO2 at ultrahigh pressures [11], and thus, the most recent high-pressure sequence becomes as follows: MI Î OI Î OII Î Fe2P (Fig. 1). We should mention that Fe2P-type structure has been also discovered as a post-OII phase in the similar dioxides TiO2 [20] and HfO2 [21]. Recently, after this discovery, another theoretical study [16] has proposed a new hexagonal phase (Ni2In-type structure) of ZrO2 ( Fig. 1) to be the most stable phase at pressures greater than 380 GPa. However, in this work [16], the transition to Ni2In phase is claimed to likely occur from OII phase rather than Fe2P phase, indicating that this study [16] suggests the following transition sequence: MI Î OI Î OII Î Ni2In. In fact, ignoring Fe2P phase from the high-pressure phase transition sequence is not consistent with the recent experimental and theoretical studies [11,20,21] that have confirmed the phase stability of Fe2P as a post-OII phase. Thus, the reasonable expectation is that any new proposed phase transition in ZrO2 should occur from the experimentally observed Fe2P phase.
In our study, we use DFT calculations to test the stability of Ni2In-type structure at high pressures with respect to OII and Fe2P phases. The main goal in this work is to investigate the upper part of highpressure phase transition sequence of ZrO2 in an effort to better understand the high-pressure phase diagram of this dioxide.

Theoretical Methods
To investigate the high-pressure phase transitions and the EOSs within the three ZrO2 phases, we used density-functional theory (DFT) [22] based static first-principles computations. The projectoraugmented wave (PAW) formalism [23,24] was used to treat the interactions between the atoms having a core radii of 2.500 Bohr for hafnium Hf (with the valence configuration of 4s 2 4p 6 5s 2 4d 2 ) and 1.520 Bohr for oxygen O (with the valence configuration of 2s 2 2p 4 ). The generalized gradient approximation (GGA) [25] was used to treat the electronic exchange and correlation effects. We performed our calculations using the VASP software package [26][27][28][29] with an energy cutoff of 600 eV and Γ-centered k-point meshes [30]. For all phases, total energies and pressures were converged to better than ~0.1 meV/atom and ~0.1 GPa, respectively. The Brillouin zone integration was performed using the following k-point meshes for the ZrO2 phases: 4x8x4 for OII, 6x6x10 for Fe2P, and 10x10x8 for Ni2In. For a fixed volume, all internal degrees of freedom and unit-cell parameters of the structure were optimized simultaneously during the geometry optimizations. The ground-state energy for each phase was determined for 13-20 volumes. Up to highest pressure achieved in our calculations, all ZrO2 phases remain insulators.

Bulk modulus determination
To study the compressibility of the three phases, we used Birch-Murnaghan (BM-) EOS to determine the bulk modulus for each phase. In the third-order BM-EOS [31], the pressure P is given by where V is the volume at pressure P, V0 is the zero-pressure volume, K0 is the zero-pressure bulk modulus, and K0' is the first pressure derivative of the bulk modulus at zero pressure. From the thermodynamic relationship: Where E0 is the total energy at zero pressure. One can obtain the second-order BM-EOS by substituting K0' = 4 in Eqns. 1 and 2. The EOS parameters for each ZrO2 phase were obtained by fitting the total energy as a function of volume to the second-and third-order BM-EOS [31] (Table I). We determined the EOSs for the three phases and summarized them in Table I. We note that our bulk modulus (K0) for both OII and Fe2P phases is in excellent agreement with previous studies [8,11,16]. On the other hand, our computed K0 for Ni2In phase is ~16% less than previous results [16]. However, we should note that BM-EOS is sensitive to both V0 and K0' [31], where K0 decreases with increasing V0 and/or K0'. Although our K0 value (200 GPa) of Ni2In is less than the previously reported one (239 GPa) [16], V0 (31.81 Å 3 ) and K0' (4) are greater than the reported values [16] of V0 (29.21 Å 3 ) and K0' (3.86).
The EOS calculations show that the change in K0 across the phases is as follows: OII Î Fe2P (K0 increases) and Fe2P Î Ni2In (K0 decreases), in agreement with previous studies [8,11,16]. In details, using the second-order BM-EOS, K0 increases by ~5.8% across OII Î Fe2P and decreases by ~26% across Fe2P Î Ni2In. This clearly indicates that Ni2In-ZrO2 phase is noticeably more compressible than OII and Fe2P phases. Table I: Theoretical EOSs of various ZrO2 phases using the second-and third-order BM-EOS [31].
To explain the noticeable compressibility in Ni2In phase, we have investigated the change in the lattice parameters (a and c) of this phase with increasing pressure. Figure 2 clearly shows that up to a few tens Ref. [8] Ref. [11] Ref. [11] Ref. [ of GPa, the c/a ratio sharply decreases as pressure increases. This indicates that at relatively low pressure the lattice parameter a is much more incompressible than the parameter c, which likely explains the low K0 value of Ni2In phase. Figure 2: c/a ratio for Ni2In-ZrO2 phase as a function of pressure. The lattice parameter a is more incompressible than the parameter c, especially at low pressures. Figure 3 shows the change in enthalpy of Fe2P and Ni2In phases with respect to OII phase. Our calculated transition pressures are as follows: across OII Î Fe2P is 94 GPa (99 GPa) using the second-(third-) order BM-EOS [31], and across Fe2P Î Ni2In is 317 GPa (311 GPa) using the second-(third-) order BM-EOS [31]. Although our calculated transition pressure across OII Î Fe2P of ~26-30% is less than previously reported results [11,16], this transition is consistent with the recent measurements and calculations, where Fe2P is observed to be the post-OII phase in ZrO2 [11] and in the similar dioxide TiO2 [20], and thus, any high-pressure phase transition to a new structure should occur from Fe2P phase. However, a recent theoretical study [16] has concluded that the transition to Ni2In phase is likely to occur from OII phase and not from Fe2P phase which is not consistent with the recent experimental and theoretical results [20,11]. These studies have obviously shown that Fe2P phase is more stable at high pressures compared to OII phase. On the other hand, our high-pressure phase transition sequence for ZrO2 (OII Î Fe2P Î Ni2In) is in reasonable agreement with previous experimental studies [20,11] as we propose the transition to Ni2In to occur from Fe2P rather than from OII. Even though Ni2In-ZrO2 phase has not been experimentally observed yet, likely due to the extreme pressure-temperature conditions required to stabilize it, our calculations show that it is the most stable phase of ZrO2 at ultrahigh pressures in agreement with previous calculations [16].

Volume change across transitions
In this section we discuss the volume decrease across each transition of the phase sequence OII Î Fe2P Î Ni2In. Figure 4 shows the volume of each ZrO2 phase as a function of pressure as well as the volume decrease across each transition. Our calculations predict a small volume decrease of ~0.4% across OII Î Fe2P, while we predict a large volume change of ~3.6% across Fe2P Î Ni2In. However, we should note that the coordination number (CN) across the transition OII Î Fe2P remains unchanged, which explains the small volume decrease across this transition. On the other hand, the Fe2P Î Ni2In transition is associated with a CN increase from 9 to 11, and therefore, a large volume decrease is found as expected. This large volume decrease is consistent with previous studies on ZrO2 [8] and similar dioxides TiO2 [32] and HfO2 [33] that have shown a large volume collapse when the transition across the phases is associated with a CN increase. across Fe2P Î Ni2In, lower inset: across OII Î Fe2P).

Conclusions
In summary, using DFT calculations, we investigated the stability of Ni2In phase at high pressures with respect to OII and Fe2P phases for ZrO2. Our calculations showed that the predicted high-pressure phase sequence across these phases is as follows: OII Î Fe2P Î Ni2In, where Ni2In phase is stable at pressures greater than ~317 GPa. Furthermore, we computed BM-EOS for the three phases, and found that the compressibility of Ni2In phase is obviously high when compared to the other two high-pressure phases. Finally, we have predicted a very small volume change across OII Î Fe2P and a noticeably large volume change across Fe2P Î Ni2In transition.