Lattice instability, anharmonicity and Raman spectra of BaO under high pressure: A first principles study

Alkaline-earth metal oxides, in particular MgO and CaO dominate Earths lower mantle, therefore, exploring high pressure behavior of this class of compounds is of significant geophysical research interest. Among all these compounds, BaO exhibits rich polymorphism in the pressure range of 0-1.5 Mbar. Static enthalpy calculations revealed that BaO undergoes a pressure induced structural phase transition from NaCl-type (B1) $\rightarrow$ NiAs-type (B8) $\rightarrow$ distorted CsCl-type (d-B2) $\rightarrow$ CsCl-type (B2) at 5.1, 19.5, 120 GPa respectively. B1 $\rightarrow$ B8 $\&$ B8 $\rightarrow$ d-B2 transitions are found to be first order in nature whereas d-B2$\rightarrow$ B2 is a second order or weak first order phase transition. Interestingly, d-B2 phase shows stability over a wide pressure range, $\sim$19.5-113 GPa. Mechanical and dynamical stabilities of ambient and high pressure phases are demonstrated through computed elastic constants and phonon dispersion curves, respectively. Under high pressure, significant phonon softening and soft phonon mode along M-direction are observed for B8, d-B2 and B2 phases, respectively. Pressure dependent Raman spectra suggest a phase transition from d-B2 to Raman inactive phase under pressure. Overall, the present study provides a comprehensive understanding of underlying mechanisms behind pressure-induced structural phase transitions in BaO.


Introduction
Alkaline-earth metal oxides, MO (M = Mg, Ca, Sr, Ba) have gained a considerable attention due to their technological applications in diverse fields, 1-5 ranging from catalysis [6][7][8] to electrode coating 9 and interfaces. 10 These metal oxides are suitable candidates for CO 2 sorbent applications due to their high absorption capacity at moderate temperatures. 3 Moreover, these mono-oxides are prime candidates for the Earth's interior, especially light metal oxides such as MgO and CaO are abundant in the Earth's lower mantle, in which the pressure varies from 24 to 136 GPa from top of the upper mantle to the lower mantle. 11-15 Therefore, exploring high pressure and/or temperature behavior of these metal oxides provides an insight on the deep interior of the Earth. These metal oxides crystallize in the rocksalt NaCl-type (B1) structure with space group F m3m at ambient conditions. Upon compression, these metal oxides usually undergo a body centred cubic CsCl-type (B2) structure and it is well established for MgO, CaO and SrO. Amongst all these compounds, BaO has interesting structural properties and it exhibits a rich polymorphism in the pressure range of 0-1.5 Mbar over the remaining alkaline earth oxides, CaO, 16-24 SrO. [16][17][18][19][25][26][27][28][29] Therefore, understanding the high pressure structural and lattice dynamical behavior of BaO is crucial as this can be a possible candidate of the Earth's mantle. 12 Several high pressure X-ray diffraction (HP-XRD) measurements 23,[29][30][31][32][33][34][35] and theoretical studies using first principles calculations were focused on exploring the structural phase transitions, 18,21,25,27,[36][37][38][39] 30 revealed that BaO undergoes a structural transition from B1 to distorted B2 (d-B2) at 9.2 GPa and remains in the same structure till 29 GPa. Energy dispersive X-ray diffraction study 31 disclosed that BaO transforms from B1 to an intermediate hexagonal NiAs-type (B8) structure at around 10 GPa, later to a tetragonal PH 4 I-type (d-B2) structure with pseudo 8-fold coordination at around 15 GPa and this phase approaches to B2 structure with increasing pressure above 1 Mbar. First principles studies 17,53,54 reported that BaO transforms from B1 to B8 phase at 11.3 GPa, 53

Computational details
All the first principles calculations were carried out using projected augmented wave (PAW) method as implemented in Vienna ab-initio simulation package (VASP). 59 Electron-electron interactions are treated within generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) functional while PAW pseudopotentials are used to capture the electron-ion interactions. The plane wave basis orbitals 5s 2 5p 6 6s 2 for Ba (Ba sv GW) and 2s 2 2p 4 for O (O GW) are used as valence electrons. Plane wave cutoff energy and k-spacing are tested against the total energy to ensure the convergence. The plane wave cutoff energy was set to 600 eV and a k-spacing of 2π × 0.024Å −1 with Γ-centered Monkhorst-Pack scheme to compute structural and mechanical properties at ambient as well as at high pressure. To compute phonon dispersion curves, forces are calculated using finite displacement method for ambient and high pressure phases and then force constants are extracted and then postprocessed by using phonopy package. 60 Pressure dependent Raman spectra is calculated using norm conserving pseudo potentials within linear response approach as implemented in CASTEP package. 61 Crystal structures are visualized and analyzed using the VESTA software. 62 To include anharmonic effects, we also have calculated the finite temperature lattice dynamics for B2 phase of BaO using ab-initio molecular dynamics (AIMD) simulations and then post-processed using temperature dependent effective potential method (TDEP). 63

Results and Discussion
Crystal structure and pressure-induced structural phase transitions As shown in Figure 1a, BaO crystallizes in the face centred cubic (SG: F m3m) rocksalt NaCl-type (B1) structure with 6-fold coordination and having Z = 4 formula units (f.u.) per unit cell at ambient conditions. 31 High pressure X-ray diffraction measurements 30,31,64 revealed that BaO exhibits rich polymorphism under high pressure. BaO undergoes to a high pressure NiAs-type (B8) structure (SG: P 6 3 /mmc) from B1 phase at 9.2 GPa without changing the coordination number (CN) by having Z = 2 f.u. per primitive cell. The crystal structure of B8 phase is an hexagonal analog of the B1 structure. Later, the B8 phase transforms to PH 4 -type distorted CsCl-type (d-B2) at 14.0 GPa, 30 18 GPa 30 with pseudo 8-fold coordination (SG: P 4/nmm) with Z = 2 f.u. per primitive cell, and then to a 8-fold coordinated CsCl-type (B2) structure (SG: P m3m) with Z = 1 f.u. per primitive cell upon further compression i.e., above 1 Mbar pressure (see Figure 1 and Table 1). 30,31,64 The crystal structure of d-B2 phase is analogous to PbO-type structure without a lone-pair at Ba atom having two in-equivalent bond lengths as shown in Figure 1c. To get further insights on the observed structural phase transitions and their sequence, as a first step, we have tested with various DFT functionals such as LDA, PBE, PW91 and SCAN to reproduce the B1 to B8 transition pressure (see Figure S1). The transition pressure obtained using PBE, PW91 and SCAN functionals are consistent with each other while LDA poorly describe the phase stability between B1 and B8 phases. Later, we carried out static enthalpy calculations with PBE functional for the polymorphic phases of BaO and plotted enthalpy difference with respect to B2 phase as a function of pressure as shown in Figure 2a. The obtained phase transition sequence is B1 → B8 → d-B2 → B2 and the corresponding transition pressures 5.1, 19.5, 120 GPa are in good agreement with the HP-XRD measurements 30,31 as well as with previous first principles calculations. 52,53 Moreover, TlI-type (B33) phase is found to be meta-stable for CaO 65 and we also have included this B33 phase in the phase diagram of BaO to verify it's relative phase stability w.r.t high pressure phases of BaO and then we found that the B33 phase is also meta-stable for BaO over the maximum studied pressure range (see Figure 2a). As presented in Figure 2b, the enthalphy difference between d-B2 and B2 phases is continuously decreasing with increasing pressure and they become iso-enthalpic at around 120 GPa. As shown in Figure 2c, the calculated lattice constants for the ambient and high pressure phases are monotonically decreasing with pressure. The lattice constant 'c' of d-B2 phase becomes lattice constant 'a' for B2 phase (c d−B2 = a B2 ). The detailed transition mechanism for d-B2 to B2 transition is explained through elastic constants in the following section. As shown in Figure 2d, the calculated volume decreases monotonically with pressure. We observe a volume collapse during the phase transition from B1 → B8 is ∼ 3.65 % at 5.1 GPa and it is ∼ 6.4 % at 19.5 GPa for the transition B8 → d-B2. The obtained volume reduction is consistent with high pressure X-diffraction measurements and previous first principles calculations as presented in Table 2. This clearly demonstrates that B1 to B8 & B8 to d-B2 transitions are first order in nature, while d-B2 to B2 transition is a second order or weak first order in nature due to continuous change in volume during the phase transition (see Figure 2d). We then calculated the equilibrium bulk modulus (B 0 ) and its pressure derivative (B 0 ') by fitting the obtained pressure-volume (P-V) data to 3 rd GPa for high pressure B2 phase disclose that B2 phase is harder than B1 phase (see Table 3).
In addition, the calculated in-equivalent bond lengths are plotted as a function of pressure and the same are presented in Figure 2e. The in-equivalent bond lengths of d-B2 phase are found to converge to became a single bond length for B2 phase above 120 GPa. In addition, in the d-B2 phase, the Ba atoms are located at (0.5, 0.0, 0.5+∆) and (0.0, 0.5, 0.5-∆), where ∆ is the atomic position parameter, ∆ in combination axial ratio (c/a) provides a useful information on d-B2 → B2 transition. If ∆ = 0 & c/a = 1/ √ 2 = 0.7071, then the structure is of undistorted B2 phase. 31 To confirm this, we have calculated c/a ratio and ∆ as a function of pressure for d-B2 phase. As shown in Figure

Elastic constants under high pressure
To get deeper insight on the structural phase transition(s) and verify mechanical stability of the BaO polymorphs, we have calculated second order elastic constants (C ij ) as a function of pressure for all the four B1, B8, d-B2 and B2 phases. The cubic (F m3m, P m3m), hexagonal (P 6 3 /mmc) and tetragonal (P 4/nmm) crystal symmetries possess 3, 5 and 6 independent elastic constants.
When a finite hydrostatic pressure is applied to a crystal system, the Born stability criteria 68 at ambient conditions must be revised with modified elastic constants at a given hydrostatic pressure conditions. According to Sinko and Smirnov the modified elastic constants at a given pressure (P) are C ii = C ii -P, (for i = 1-6) and C 1j = C 1j + P (for j = 2-3) for cubic and hexagonal/tetragonal systems. Therefore, the modified Born stability criteria under hydrostatic pressure for cubic (equation 1) and hexagonal/tetragonal (equation 2) are given as follows: The calculated pressure dependent C ij for B1, B8, d-B2 and B2 phases satisfy the necessary and sufficient conditions for the revised Born stability criteria under hydrostatic pressure indicating that the polymoprhs are mechanically stable at the corresponding pressures. We shed more light on understanding lattice transformation from d-B2 to B2 phase through the computed pressure dependent elastic constants of d-B2 and B2 phases. The relationships between lattice constants of d-B2 and B2 phases at the phase transition are given as follows: Figure 3a), thus, results in merging of the following elastic constants of d-B2 to B2 phase under pressure are as depicted in Figure 3 as given below:

13
→ C B2 12 , which corroborates with pressure dependent structural and bond parameters at the d-B2 → B2 transition.
Lattice dynamics and Raman spectra under high pressure To compliment the structural stability obtained from total energy calculations and verify dynamical stability of the ambient and high pressure phases of BaO, we calculated phonon dispersion curves including LO-TO splitting at 0 and 9 GPa for B1 phase (see Figure 4a  phase (see Figure 5) and finally at 100, 110, 120 and 140 GPa for B2 phase (see Figure 6). As shown in Figure 4a Figure 7). This result is in very good agreement with the Raman spectroscopic measurements, 70 which show that BaO is Raman inactive until 10 GPa (B1 phase) and above 11 GPa it becomes Raman active (B8), where one Raman active E 2g mode is observed and this Raman mode shows blue-shift with increasing pressure. Later, significant changes are observed in the Raman spectra at around 14 GPa and it indicates a transition from B8 to to d-B2 phase. The similar trends are observed in the present work with deviations in the transition pressures which might be due to Raman measurements were carried out at 300 K, while the calculations are performed at 0 K. Since, B8 and d-B2 phases crystallize in P 6 3 /mmc and P 4/nmm space groups, respectively and they consist of four atoms per primitive cell, the symmetry decomposition of group theory predicts 12 vibrational modes along Γ for both of these phases, which are given as follows: Among the 12 vibrational modes of B8 phase, only E 2g is Raman active mode which is consistent with high pressure Raman measurements, 70 E 1u , E 2u , A 2u and B 1u are IR active modes but B 2g is neither Raman nor IR active (silent) mode; where E 1u , E 2u and E 2g are doubly degenerate modes. Variation of E 2g mode as a function of pressure is shown in Figure   7a, which is mainly due to in-plane vibrational motion of Ba atom and its intensity decreases with pressure.
On the other hand, for Litharge PbO-type d-B2 phase, the zone-center Raman active modes B 2g , A 1g and E g are polarized along c-and a/b-axis, respectively. E u and A 2u are IR active modes; where E u and E g modes are doubly degenerate. Pressure dependence of vibrational frequency modes are presented in Figure 7d. For each vibrational mode, frequency increases with pressure except for A 1g mode. When the pressure is applied, the interatomic distance becomes smaller and hence interaction between neighboring atoms becomes stronger.
Therefore, usually, the vibrational modes increase with pressure as shown in Figure 7b & d except for A 1g mode. As E g mode is in plane vibrational mode, the vibrational fre-quency/wavenumber increases as the interaction of atoms becomes stronger. As shown in Figure 2f; c/a ratio decreases with increasing pressure. Due to the out-of-plane vibrations of heavier Ba atoms in A 1g mode (Figure 7c), the decrease in the wavenumber of Raman-shift is observed with pressure. The red-shift of the A 1g mode with pressure shows a negative pressure coefficient and it might be responsible for introducing dynamical instability at higher pressure in the d-B2 phase. In contrast, this mode was tentatively assigned as E g mode due to in-plane vibration of Ba atoms in the HP-Raman study. 70 However, we thoroughly analyzed and assigned it as A 1g mode rather than E g mode, which actually shows phonon softening with pressure which is associated with displacive nature of Ba atoms along c-axis.
The Intensity of four Raman active modes (E g , B 2g and A 1g ) are gradually diminishing with pressure ( Figure 7b) for d-B2 phase, this strongly suggests that high pressure phase might be Raman inactive. Finally, the high pressure B2 phase can be stabilized above 113 GPa (see Figure 7e) which is consistent with the observations from the HP-XRD measurements. 31 As shown in Figure S3 & 6, the calculated room temperature phonon dispersion curves in combination with AIMD and TDEP disclose that B2 phase is dynamically unstable at 80 GPa and 300 K whereas it becomes dynamically stable at 100 GPa and 300 K, while harmonic lattice dynamical calculations show that B2 phase is dynamically unstable below ∼ 113 GPa.
This clearly shows the importance of anharmonicity to determine the stability of B2 phase above 85 GPa and this is in good accord with the HP-XRD diffraction measurements. 31

Conclusions
To summarize, we have systematically investigated the polymorphism of BaO under hydrostatic compression using first principles calculations based on density functional theory. We    GPa at 0K as well as at 300 K. B2 phase is dynamically stable including anharmonic effects at 100 and 110 GPa at 300 K but it is dynamically unstable within the harmonic approximation at 0 K.     S-4