High Pressure/Temperature Phase Transformations of Ilmenite Materials

Introduction

Under atmospheric pressure, ZnSnO₃ exists in the ilmenite crystal structure which is known to exhibit excellent pyroelectric, dielectric, piezoelectric, and photostrictive properties¹. Ilmenite materials belong to the trigonal crystal system with -3m point group and R3 space group. ZnSnO₃ has an observed band gap of 3.7eV². Ferroelectric properties arise from cation sequence along the c-axis to be Sn-Zn-Vac-Sn-Zn-Vac³, as seen in Figure 1. The intrinsic vacancy concentration allows cations to shift positions relative to the applied field. Both cations sit on octahedral sites. To date, only a limited number of studies exist and there is much to be learned about ZnSnO₃.

Inaguma et al³ have synthesized ZnSnO₃ under high pressure conditions (7 GPa @ 1000C), yielding a lithium niobate [LN] type crystal structure. It’s well known that materials with LN crystal structure are noncentrosymmetric space group R3c exhibiting many ferroelectric properties as well³. Polarization is calculated to be up to 59uC/cm².

It's known that ilmenite materials may also transform to the perovskite and postperovskite structures when subjected to extreme pressures and or temperatures. Given the environmentally friendly elements and promising potential for ferroelectric applications, it's worth theoretically investigating the high pressure and temperature properties and phases of ZnSnO₃.

Figure 1: Crystal structure of ZnSnO₃ with representation of SnO₆ and ZnO₆ octahedra³. C-axis cation sequence: Sn-Zn-Vac-Sn-Zn-Vac

To provide further insight into high pressure/temperature phases of ZnSnO₃, it’s important to understand mechanisms behind the phases transformation related to pressure and temperature⁵. Since studies of ZnSnO₃ are limited, insight from other ilmenite material will be analyzed.

Literature Review

First, it will be of benefit to discuss DFT methods used to generate the theoretical electronic structure of LN-ZnSnO₃ as Wang et al⁴ have preformed. In their calculations, they used the full-potential linearized augmented plane-wave method (FP-LAPW). Exchange and correlation effects we treated under the guidelines of the generalized gradient approximation (GGA). Optical constants, such as energy dependence of absorption coefficient, refractive index and reflectivity were derived from both the imaginary and real components of the dielectric constant. The imaginary component of the dielectric constant was solved from the momentum matrix elements between the unoccupied and occupied wave functions given by the equation:

(1)

\begin{align} \varepsilon_{2}(\omega)=\frac{Ve^{2}}{2\pi \hbar m^{2}\omega^{2}}\int d^{3}k\sum \mid\langle kn\mid p\mid kn'\rangle\mid^{2}f(kn)(1-f(kn'))\delta (E_{kn}-E_{kn'}-\hbar\omega) \end{align}

The real component of the dielectric constant, $\varepsilon_{1}(\omega)$, was evaluated from the above equation using the Kramer-Kronig transformation. A 10X10x10 mesh was used for k-point sampling in the irreducible wedge of the Brillouin zone, neglecting spin-orbit coupling.

Figure 2: Theoretical absorption, refractive, excitation, energy loss and reflectivity as a function of energy for the LN-ZnSnO₃ Solid line corresponds to xx direction and dotted line corresponds to zz direction ⁴

ZnSnO₃ is estimated to have ~1eV direct band gap and valence and conduction bands are dictated by the O 2p and Sn 5s electron states, respectively. Optical/electronic information found in figure 2. The optical spectra mostly coincide along the two directions (xx and zz) with minor deviations. Phonon energy absorption steadily increases up 17eV, where it then begins to decline. Reflectivity in the range of 0-6eV was found to be less than 25% suggesting ZnSnO₃ is transparent to visible light. There are no experimental studies which can confirm this data. DFT is notorious for underestimating band gap and therefore the predicted band gap may be up to 2.5eV larger than predicted.

Since no theoretical study exists to provide insight into the mechanisms and properties of high pressure phase transformations ZnSnO₃ exists, a close look into high pressure phase transformations of other ilmenite structures is necessary. Theory of the high pressure phase transformations of FeTiO₃, MeGeO₃ and NaSbO₃ using DFT are available.

FeTiO3

Ilmenite FeTiO₃ is the most stable phase under normal ambient condition⁵. To study the effect of phase change and related electronic properties, first principles calculations are performed. Electronic exchange and correlation was approximated with Becke’s three-parameter hybrid functional. Within this approximation the exchange-correlation energy is represented by:

(2)

\begin{equation} E_{xc}=(1-a_{o})E_{X}^{LDA}+a_{o}E_{X}^{HF}+a_{x}E_{X}^{B88}+a_{c}E_{C}^{LYP}+(1-a_{c})E_{c}^{VWN} \end{equation}

Where:
$a_{x}\Delta E_{X}^{B88}$ is Becke’s gradient correction to the exchange functional
$E_{x}^{HF}$ is the Fock exchange energy
$E_{c}$ is the correlation energy given by: [Lee, Yang, and Parr]⁶ & [Vosko, Wilk, and Nussair]⁷

“Unlike the local density approximation (LDA) or GGA, [Becke’s three-parameter hybrid functional gives] a qualitatively correct ground state for a wide range of transition metal oxides”, generally in close agreement to experiment⁵.

The bielectronic Coulomb and exchange series were converged to within 0.1mHa. Sampling of the Brillouin zone using the Monkhorst-Pack method was defined with a 14x14x14 k-point mesh⁵. Lattice parameters were optimized to a tolerance of 0.001Å and 10^-7Ha per formula unit of total energy. Pressures were simulated by applying hydrostatic pressure while minimizing the enthalpy with respect to cell parameters⁵.

The LN structure is found to be ~0.1eV per formula unit higher in energy than the ilmenite phase with no predicted transition pressure. Mulliken population analysis results are consistent with Fe ions being in a high-spin d⁶ state, which is different than the low-spin Fe d⁶ state found in the ilmenite structure. The high spin Fe²⁺ is responsible for the LN geometry which is not as stable as ilmenite. With regard to LN Fe³⁺-Ti³⁺ bond state, an electron localizes on the Ti dz² orbital, confirmed with Mulliken analysis. Fe²⁺ ion state is the most stable with the Fe³⁺ being 2.14eV higher in energy⁵ Bulk modulus was found to be in close agreement with experiment as seen in table 2.

Method	Lithium Niobate	Perovskite	Postperovskite
B3LYP	191	222	207
Experiment	182 +/-7	211 +/-14	None

Table 2: Theoretical and experiment bulk moduli for FeTiO₃ in Gpa.

The perovskite phase transition pressure was found to be at 23Gpa, close to the experimental value of 25 +/-4Gpa. Calculations were carried out using the Fe³⁺ and Fe²⁺ states in both small (octahedral coordination) and large cation sites (12-coordination), and it was found that Fe²⁺ in the large cation site is the most energetically favorable. Fe²⁺ ions are found in the high spin d⁶ electron configuration⁵. Ti is found to have larger radius than Fe and thus Ti sits on the large cation site. Bulk modulus can be seen in table 1.

The phase transition pressure is predicted to be at 44Gpa for the postperovskite phase. There is no experimental data for this phase of FeTiO₃. The Fe²⁺ low-spin phase is stable and Fe³⁺ to be unstable. The high-spin state of Fe²⁺ is found to be 3.5eV more favorable than low-spin Fe²⁺.

Enthalpy/unit volume is plotted in figure 3. The plot shows phase stability of the discussed structures within a range of spin and charge states (valence and high/low spin) for a given set of pressures.

Figure 3: Relative enthalpy as a function of pressure for various crystal structures associated with FeTiO₃⁵

MgGeO3

At 0Gpa MgGeO₃ exists in the ilmenite crystal structure. With increasing pressure, it undergoes phase change to perovskite then postperovskite. The lithium niobate (LN) phase is found to be meta-stable. Based on DFT and density functional perturbation theory (DFPT). Tsuchiya et al¹¹ have performed an in-depth study of the temperature/pressure relation of the various phases.

Their computations have been carried out using both the LDA and GGA. Ultrasoft pseudopotentials have been utilized for the core electron configurations. Plane-wave cutoff was taken at 50 Ry and Brillouin zone was sampled on 2x2x2 (lithium niobate), 4x4x2 (pervoskite and postpervoskite) leading to energy convergence within 0.01eV/atom¹¹.

From the VDOS calculations for perovskite and postperovskite phases can be found in figure 4. Eigenvector analysis at 0GPa show that the VDOS of ilmenite, perovskite and postperovskite around 250-400cm^-1 is attributed to Mg atoms displacement and 650-800 cm^-1 is attributed to GeO₆ octahedra deformation. VDOS of perovskite around 100-250cm^-1^^ are explained by octahedra rotation. It’s found that all mentioned phases have no vibrational instability up to 150Gpa. Therefore, each phase is mechanically stable and phase transition are simply caused by thermodynamic energy balance¹¹.

Figure 4: Vibrational density of states for MgGeO₃ at 0Gpa (blue) and 100Gpa (pink) for ilmenite (top) perovskite (middle) and postperovskite (bottom)¹¹.

Determination of high temperature phase boundaries, starts with calculation of Helmholtz free energy, F(V,T), calculated within the quasiharmonic approximation (QHA). From there, Gibbs free energy, G(P,T) = F(V,T) + P(V,T)V, is calculated and used to define the phase boundaries. Free energy differences, relative to perovskite, ΔG(P,T), is calculated within LDA and plotted in figure 5 at several temperatures as a function of pressure¹¹ along with a theoretical pressure/temperature diagram in figure 6.

It’s found that the transition pressure of ilmenite to perovskite is found to be at ~24GPa at 300K and the transition pressure decreases with increasing temperature. This is close to experimental phase change at ~23Gpa. Transition pressure from perovskite to postperovskite is found at 51Gpa @ 300K and 63GPa at 2000K indicating phase transition pressure increases with increasing temperature, which agrees with experiment¹¹.

Figure 5: ΔG(kJ/mol) as a function of pressure at fixed temperature¹¹

Figure 6: Theoretical high pressure/temperature diagram¹¹

It can be seen that ΔG for the LN phase is always larger than the ilmenite and therefore it no thermodynamic stability. The reason can be attributed to the large lattice energy associated with the LN phase of MgGeO₃. As a result, the LN phase doesn’t appear on the P-T diagram. The ilmenite-perovskite phase transition at 28Gpa, close to the 23Gpa predicted by experiment¹¹. In contrast, GGA yields a transition at 38Gpa. Postperovskite transition is predicted at 51Gpa predicted by LDA, also close to experiment.

Compression information is plotted in figure 7. Calculated compression curves using LDA are close to experimental values for all three phases. GGA was found to produce volumes 2.5%-3% larger than experiment and thus, LDA is closer to experiment in this case. Change from ilmenite to perovskite, results in volume reduction of -4.5% (compared to -5.5% experimentally) due to Mg coordination change from six fold to eight fold during phase change¹¹. Volume change is -1.8% for perovskite - postperovskite phase change which is 0.3% larger than experiment. The small volume change is associated with change in the GeO₆ octahedra connectivity, changing from corner sharing to edge sharing configuration. Table 2 contains calculated thermodynamic data.

Regarding the ilmenite to perovskite phase transition, the Mg-O bong distance was found to increase by 9% due to the increase in the Mg coordination number from 6 to 8 fold, and the average Ge-O bond shortens¹¹. Because there is no changes in coordination number within the perovskite to postperovskite phase change, only small changes in bond distance are founds.

Figure 7: Specific volume as a function of pressure for theory (solid lines) and experimental data (labeled dots)¹¹.

Table 2: Thermodynamic Properties of MgGeO₃ for ilmenite, perovskite and postperovskite phases¹¹.

Change in entropy for ilmenite to perovskite transition is found to be 7.2, 10.4, and 12.2 J/mol K @ 300, 1000, and 2000K respectively and therefore has strong temperature dependence. Change in entropy for perovskite to postperovskite is -2.9 J/mol K with weak temperature dependence¹¹. Regarding the Goldschmidt tolerance, which describes the cubic nature of perovskite structures where a value of 1=perfectly cubic and anything <1 is further away from being perfectly cubic, it’s found that ‘t’ decreases with increasing pressure for both perovskite and postperovskite. It’s found that perovskite is stable when t>0.83 and postperovskite is stable when t<0.83¹¹.

NaSbO3 andNaBiO3

Liu et al¹² have investigated the high pressure phase transformations of NaSbO₃ and NaBiO₃ using first principal calculations. Exchange correlation energy was handled under the guidelines of the GGA. Core electrons were represented with pseudopotentials. Cutoff energy , E_cut, was set to a value of 370eV.

Pseudowave functions and potentials were represented with fast-Fourier-transform grids with size 30x30x45 deemed sufficient for the E_cut¹². Valence electrons were described by Vaderbilt-type non-local ultrasoft potentials with O-2s²2p⁴, Na-2s² 2p⁶3s¹ Sb-5s²5p³, and Bi-6s²6p³. The Brillouin zone was sampled with a 5x5x2 Monkhorst-Pack mesh. Their calculation parameters and convergence criteria were left as the default values of the CASTEP DFT code¹².

Over of pressure range from 0-15Gpa, it was found that both NaBio₃ and NaSbO₃ undergo a change from the ilmenite structure to an orthorhombic structure (space group Pnma) at 10Gpa and 12Gpa respectively.

Based on the theory from the Wilson et al⁸ and Tsuchiya et al¹¹, it can be assumed both NaBiO₃ and NaSbO₃ may undergo phase transformations into both the LN and post-perovskite structures. Unfortunately their work didn’t investigate pressures high enough to observe these phases and would benefit the quality of the paper to do so.

Open Issues for Further Investigation

There is no DFT data documenting the thermodynamic phase relationship of ZnSnO₃ to either temperature or pressure. Experimental data, so far, has identified decomposition into Zn₂SnO₄ around 800C, but fails to accurately identify the temperature. An in-depth theoretical study, similar to the work of Wilson et al⁵, would help to clarify the high temperature and pressure phases. In addition, DFT study of ferroelectric properties of each of the high pressure species will help to identify favorable properties.

In addition, the intrinsic vacancy concentration within ZnSnO₃ may allow ample room for added dopants. A theoretical study relating alterations to the electronic properties of of LN-type ZnSnO₃ due to different dopants will be adventitious. This will provide insight as the to feasibility alterations to the electronic structure of ZnSnO₃. Corresponding experimental data will prove necessary if the theory supports concepts.

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