Dynamic stabilization of cubic $\mathrm{Ca}\mathrm{Si}{\mathrm{O}}_3$ perovskite at high temperatures and pressures from ab initio molecular dynamics

The stability of cubic $\mathrm{Ca}\mathrm{Si}{\mathrm{O}}_3$ perovskite (CaPv) at high temperatures and pressures is investigated by vibrational normal-mode analysis. We compute power spectra of mode autocorrelation functions using a recently developed hybrid approach combining ab initio molecular dynamics with lattice dynamics. These power spectra, together with the probability distributions of atomic displacements, indicate that cubic CaPv is stabilized at $T\sim 600$ K and $P\sim$ 26 GPa. We then utilize the concept of phonon quasiparticles to characterize the vibrational properties of cubic CaPv at high temperature and obtain anharmonic phonon dispersions through the whole Brillouin zone. Such temperature-dependent phonon dispersions pave the way for more accurate calculations of free-energy, thermodynamic, and thermoelastic properties of cubic CaPv at Earth's lower mantle conditions.

Figure 1

CaPv in a $2\times 2\times 2$ cubic supercell. (a) The ideal cubic structure ($a^0{a}^0{a}^0$) being dynamically unstable at 0 K. (b) The $a^{+}{b}^{-}{b}^{-}$ structure with the atomic degrees of freedom fully relaxed. It is 5.5 meV/atom more stable than $a^0{a}^0{a}^0$. Its tilting angles in the $x$, $y$, and $z$ directions are 1.4, 4.5, and $4.5^{\circ }$, respectively. Harmonic phonon calculations indicate that all of its vibrational frequencies are positive.

Figure 2

(a) Probability distributions of atomic displacements with respect to the ideal cubic structure ($a^0{a}^0{a}^0$ in Glazer's notation [42]) at $T=1000$ K. For Ca and Si atoms, the distributions of atomic displacements in the three Cartesian directions are the same, while for O atoms displacements in the $z$ direction (octahedral stretching) are more restricted than displacements in the $x$ or $y$ direction (octahedral tilting). (b) Distributions of the O displacements in the $x$ or $y$ direction (octahedral tilting) with respect to the ideal cubic structure ($a^0{a}^0{a}^0$) at various temperatures.

Figure 3

Correlation of an off-diagonal component of the stress tensor ${\sigma }_{yz}$ at various temperatures.

Figure 4

(a) Power spectrum of the velocity autocorrelation functions projected on different wave vectors. Dotted lines correspond to the total ${\sum }_{ls}{\langle {\mathbf{v}}_{ls}·{\mathbf{v}}_{ls}\rangle }_{\omega }$. (b) Power spectrum of individual mode autocorrelation functions with $\mathbf{q}$ vector $\mathbf{R}$ $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ . Dotted lines correspond to the ${\sum }_s{\langle {\mathbf{v}}_{\mathbf{q}s}^{*}·{\mathbf{v}}_{\mathbf{q}s}\rangle }_{\omega }$ at $\mathbf{R}$. The temperature is 1000 K.

Figure 5

Power spectra of the mode autocorrelation function of $R_{25}$ with $\mathbf{q}$ vector $\mathbf{R}(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ at low temperatures.

Figure 6

Temperature dependence of renormalized phonon frequencies at constant volume. For each inequivalent $\mathbf{q}$ vector, Two modes with the largest positive and one mode with the largest negative temperature dependence are shown. Each mode is labeled by its symmetry and frequency at 600 K (shown in parentheses in units of cm${}^{-1}$). The plotted frequency and error bar correspond to the mean value and variance of the frequencies of all symmetrically equivalent modes. Solid lines show the $\sqrt{T}$ temperature dependence.

Figure 7

Phonon dispersions at 1000 K. The circles represent the frequencies extracted from the normal-mode correlation functions. Fourier interpolating these frequencies yields the renormalized phonon dispersions, shown in solid black line. Dashed lines are the phonon dispersions determined by fitting an effective harmonic force constant matrix [28]. Harmonic phonon dispersions calculated with DFPT are shown in orange (light) line.

Figure 8

(a) Vibrational entropy predicted by PGM in the $2\times 2\times 2$ supercell. The line labeled as “$\tilde{\omega }\left(600\mathrm{K}\right)$” denotes the entropy calculated with phonon frequencies obtained at 600 K. Open circles with label “$\tilde{\omega }\left(T\right)$” are the entropies calculated with phonon frequencies at the corresponding temperatures. (b) Anharmonic entropy of cubic CaPv determined by TI (line) vs the one from PGM using Eq. (8) (circles) in the $2\times 2\times 2$ supercell. The system at 600 K is chosen as the reference state.