Prediction of stability of tenfold-coordinated silica up to 200 TPa pressure based on ab initio calculations with all-electron pseudopotentials

The ultrahigh-pressure structural evolution and phase diagram of silica (${\mathrm{SiO}}_2$) from 10 TPa (${10}^{12}\mathrm{Pa}$) to 200 TPa are studied. Using a combination of ab initio simulations and a structure search algorithm, we reveal the phase diagram of ${\mathrm{SiO}}_2$ above 10 TPa and confirm that $I4/mmm{\mathrm{SiO}}_2$ with the coordination number of 10 is the most stable phase of ${\mathrm{SiO}}_2$ at ultrahigh pressures. The phase transition pressures from ${\mathrm{Fe}}_2\mathrm{P}$-type to $I4/mmm$ phase are obtained for different temperatures. To correctly model the ultrahigh-pressure structures, the inner-shell electron interactions are necessary to be considered. The pseudopotentials with different valence electron configurations are tested and the results show that neglecting the inner-shell electrons can predict incorrect phase transition sequence and thermodynamic stability. Based on simulations with all-electron pseudopotentials, it is found that the $I4/mmm{\mathrm{SiO}}_2$ is the most stable ten-coordinated structure up to 200 TPa. The thermodynamic stability of $I4/mmm{\mathrm{SiO}}_2$ up to a temperature of 8 kK is inferred from the Gibbs free energy calculations and the dynamic stability of $I4/mmm{\mathrm{SiO}}_2$ at 180 TPa is demonstrated by phonon calculations.

Figure 1

(a) Static enthalpies of ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$ (red line) and Cot-type ${\mathrm{SiO}}_2$ (blue line) with respect to the pyrite-type ${\mathrm{SiO}}_2$ (horizontal black line), as calculated using GGA (PBEsol). The transition pressure is about 0.67–0.68 TPa, which is almost consistent with previous studies [5, 15]. (b) The static enthalpies of $I4/mmm$ (red line) and $C_2/m{\mathrm{SiO}}_2$ (blue line) with respect to ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$ are calculated using GGA (PBEsol), and the results of the LDA (blue dashed line) and PBE calculations (red dotted line) are plotted in (b) and the latter in the insert figure. (c) The pressure dependencies of $△E_{\mathrm{tot}}$ (blue dot line) and $△\left(PV\right)$ (red dot line) between $I4/mmm$ and ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$.

Figure 2

Crystalline structures of (a) ${\mathrm{Fe}}_2\mathrm{P}$-type, (b) $I4/mmm$, (c) mixed-type, and (d) $C_2/m{\mathrm{SiO}}_2$ at 14 TPa. The possible mixed structure has characteristics of both ${\mathrm{Fe}}_2\mathrm{P}$-type and $I4/mmm$, but the deformation is large. It contains two kinds of ${\mathrm{SiO}}_2$ polyhedra with the CoNs of 9 and 10, which correspond to ${\mathrm{Fe}}_2\mathrm{P}$-type and $I4/mmm{\mathrm{SiO}}_2$, respectively. The charge densities are calculated using GGA (PBEsol) for (e) $I4/mmm{\mathrm{SiO}}_2$ on the (1 1 0) plane and for (f) ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$ on the (0 0 1) plane at 14 TPa. The unit cell is indicated by black dashed lines.

Figure 3

An ultrahigh-pressure $P\text{−}T$ diagram of ${\mathrm{SiO}}_2$. Dashed boundaries indicate $P$, $T$ conditions outside of the QHA validity range. The core-mantle boundary (CMB) pressures for a mega-Earth with 45 ${\mathrm{M}}_{\oplus }$ are plotted by vertical dashed lines. The Gibbs free energies of pyrite-type, cotunnite-type, and ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$ marked in orange are from Ref. [15], and the Gibbs free energies of pyrite-type, $R\overline{3}$, cotunnite-type, and ${\mathrm{Fe}}_2\mathrm{P}$-type ${\mathrm{SiO}}_2$ marked in cyan are from Ref. [18]. Additionally, the estimated and partial $P$, $T$ ranges of mega-Earth's mantle are plotted. The pink and purple ellipses represent K2-66b [10], with more than 20 ${\mathrm{M}}_{\oplus }$ and Kepler-277c with more than 60 ${\mathrm{M}}_{\oplus }$, respectively.

Figure 4

Phonon-dispersion relations obtained by the PBEsol calculation. The results of (a) ${\mathrm{Fe}}_2\mathrm{P}$-type and (b) $I4/mmm{\mathrm{SiO}}_2$ at 14.5 TPa along the highly symmetric path.

Figure 5

The static enthalpies based on $M\text{−}L$ pseudopotential with (a) GGA-PBEsol functional and (b) LDA-PZ functional of ${\mathrm{Fe}}_2\mathrm{P}$-type, $C2/m$, $P{6}_{2}222$, $P{6}_{4}222$, $P\overline{6}m2$ and ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ differences relative to the $I4/mmm{\mathrm{SiO}}_2$ (horizontal black line) plotted as a function of pressure, where the $P{6}_{2}222$ and $P{6}_{4}222$ phases overlap in the figure because the enthalpies are almost identical. The calculated phase transition pressure from $I4/mmm$ phase to ${\mathrm{Ni}}_2\mathrm{In}$-type phase is indicated by the red arrow.

Figure 6

The static enthalpies based on $M\text{−}L$ pseudopotential with (a) GGA-PBEsol functional, (b) meta-GGA-TPSS functional and (c) meta-GGA-SCAN functional of ${\mathrm{Fe}}_2\mathrm{P}$-type, $C2/m$, and ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ differences relative to the $I4/mmm{\mathrm{SiO}}_2$ (horizontal black line) plotted as a function of pressure. The calculated phase transition pressure from ${\mathrm{Fe}}_2\mathrm{P}$-type to $I4/mmm$ phase to ${\mathrm{Ni}}_2\mathrm{In}$-type phase is indicated by the blue and red arrows, respectively.

Figure 7

The static enthalpies based on (a) GGA-PBE functional, (b) GGA-PBEsol functional and (c) LDA-PZ functional of ${\mathrm{Fe}}_2\mathrm{P}$-type, $C2/m$, and ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ differences relative to the $I4/mmm{\mathrm{SiO}}_2$ (horizontal black line) plotted as a function of pressure. Dotted, dashdot, dashed and solid lines indicate the results of $M\text{−}L$, $L\text{−}L$, $L\text{−}K$, and $K\text{−}K$ pseudopotentials, respectively. (d) The transition pressure from the ${\mathrm{Fe}}_2\mathrm{P}$-type phase to the $I4/mmm$ phase calculated by the $K\text{−}K$ pseudopotential, while the result of the $M\text{−}L$ pseudopotential is marked by blue pentagram.

Figure 8

Unit crystal structures of (a) $I4/mmm$ and (b) ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ at 180 TPa. The lattice parameters for the $I4/mmm$ phase are $(a,b,c,\alpha ,\beta ,\gamma )=(0.96\AA ,0.96\AA ,2.59\AA ,{90}^{\circ }$, ${90}^{\circ }$, ${90}^{\circ }$), and for ${\mathrm{Ni}}_2\mathrm{In}$-type phase are $(a,b,c,\alpha ,\beta ,\gamma )=(1.34\AA ,1.34\AA ,1.52\AA ,{90}^{\circ }$, ${90}^{\circ }$, ${120}^{\circ }$). Si atoms are represented by the color blue, and the different O atoms are represented by the colors red, yellow and cyan, respectively. The crystal structure is rendered by vesta [49]. The lattice parameters and volume for (c) $I4/mmm$ and (d) ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ plotted as a function of pressure. The red and blue colors indicate the results of the $K\text{−}K$ pseudopotential and the $M\text{−}L$ pseudopotential, respectively. Lattice constants $a$, $b$, $c,$ and the volume are indicated by upper triangular, lower triangular, circular and square dots, respectively.

Figure 9

The Gibbs free energy based on $K\text{−}K$ pseudopotential with GGA-PBEsol functional of ${\mathrm{Ni}}_2\mathrm{In}$-type ${\mathrm{SiO}}_2$ differences relative to the $I4/mmm{\mathrm{SiO}}_2$ (horizontal black line) plotted as a function of temperature at pressures ranging from 80 to 180 TPa.

Figure 10

Phonon-dispersion relations (left) and PDOS curves (right) of the $I4/mmm$ phase at 180 TPa based on $K\text{−}K$ pseudopotential with GGA-PBEsol functional. In the PDOS plot, the projections onto Si and O atoms are shown as overlaid blue and Red shaded curves, and the total phonon density of states is represented by the black curve.

Figure 11

Calculated (a) band gap and (b) ZPE of pyrite-type, ${\mathrm{Fe}}_2\mathrm{P}$-type and $I4/mmm{\mathrm{SiO}}_2$ with GGA-PBEsol functional as a function of pressure. The band gaps and ZPE for 0.5–18 TPa is calculated by $M\text{−}L$ pseudopotential, and the band gaps and ZPE for 20–200 TPa is calculated by $K\text{−}K$ pseudopotential. The band gaps for 0.5–3 TPa marked by blue squares are from Ref. [16].