High-pressure phases of group-II difluorides: Polymorphism and superionicity

We investigate the high-pressure behavior of beryllium, magnesium, and calcium difluorides using ab initio random structure searching and density functional theory (DFT) calculations, over the pressure range $0–70$ GPa. Beryllium fluoride exhibits extensive polymorphism at low pressures, and we find two new phases for this compound—the silica moganite and ${\mathrm{CaCl}}_2$ structures—which are stable over the wide pressure range $12–57$ GPa. For magnesium fluoride, our searching results show that the orthorhombic “O-I” ${\mathrm{TiO}}_2$ structure $(Pbca,Z=8)$ is stable for this compound between 40 and 44 GPa. Our searches find no new phases at the static-lattice level for calcium difluoride between 0 and 70 GPa; however, a phase with $P\overline{6}2m$ symmetry is close to stability over this pressure range, and our calculations predict that this phase is stabilized at high temperature. The $P\overline{6}2m$ structure exhibits an unstable phonon mode at large volumes which may signal a transition to a superionic state at high temperatures. The group-II difluorides are isoelectronic to a number of other ${\mathrm{AB}}_2$-type compounds such as ${\mathrm{SiO}}_2$ and ${\mathrm{TiO}}_2$, and we discuss our results in light of these similarities.

Figure 1

Static lattice enthalpies for ${\mathrm{BeF}}_2$ calculated with the PBE functional over the pressure ranges (a) $0–8$ GPa, (b) $5–35$ GPa, and (c) $25–70$ GPa. In (a) and (b), enthalpies are shown relative to the $\alpha$-quartz phase, while in (c) they are shown relative to the rutile phase.

Figure 2

(a) ${\mathrm{BeF}}_2$ in the $C2/c$ moganite phase at 20 GPa with fourfold coordinated Be atoms, viewed down the $b$ (and in this case, monoclinic) axis. The lattice $a$ and $c$ axes point horizontal and almost vertical in the page, respectively. (b) ${\mathrm{BeF}}_2$ in the $Pnnm{\mathrm{CaCl}}_2$ phase at 50 GPa, with sixfold coordinated Be atoms. Beryllium atoms in yellow, fluorine atoms in green. Lattice parameters and atomic positions for these structures are given as Supplemental Material [18].

Figure 3

Optical band gaps in Be-, Mg-, and ${\mathrm{CaF}}_2$ as calculated using the HSE06 functional, over the pressure range $0–70$ GPa. For visibility, the band gaps in ${\mathrm{CaF}}_2$ have been shifted down by 3 eV. Discontinuities in the solid curves are due to phase transitions between different structures, while the shaded regions serve to guide the eye. The dashed curve in ${\mathrm{CaF}}_2$ corresponds to the $P\overline{6}2m$ phase, which is not stable at the static lattice level but which we predict is stabilized by temperature.

Figure 4

Static-lattice enthalpies and results from structure searches on ${\mathrm{MgF}}_2$ over the pressure range $0–70$ GPa. The inset plot shows the small enthalpy differences between a few phases in the vicinity of 10 GPa. Enthalpies are shown relative to the pyrite (main figure) and $\alpha –{\mathrm{PbO}}_2$ (inset) phases.

Figure 5

$2\times 1\times 1$ slab of ${\mathrm{MgF}}_2$ in the $Pbca$ “orthorhombic-I” structure $(Z=8)$, which we predict to be stable between 39.6 and 44.1 GPa. This view looks down the $b$ axis. Magnesium atoms in blue; fluorine atoms in green.

Figure 6

Results from structure searches on ${\mathrm{CaF}}_2$ over the pressure range $0–70$ GPa: static-lattice enthalpies relative to the $Pnma$ $(\gamma )$ ${\mathrm{CaF}}_2$ phase.

Figure 7

$2\times 2\times 2$ slabs of (a) our predicted $P6\overline{2}m$ structure, and (b) the $Pnma$ structure (phase $\gamma )$ of ${\mathrm{CaF}}_2$. Calcium atoms are in red; fluorine atoms in green. In both (a) and (b), the right-hand view is obtained from the left-hand view by rotating the structure by ${90}^{\circ }$ about an axis running vertically up the page.

Figure 8

Calculated quasiharmonic phase diagram of ${\mathrm{CaF}}_2$. We find the known $Fm\overline{3}m$ $(\alpha )$ and $Pnma$ $(\gamma )$ phases at low temperature, but our calculations indicate that a previously unreported phase of $P\overline{6}2m$ symmetry becomes stable at high temperature and pressure. “S/I” indicates the superionic region of the phase diagram, and dashed lines show boundaries due to unstable phonons at the quasiharmonic level.

Figure 9

Softening of phonon modes in the fluorite $Fm\overline{3}m–{\mathrm{CaF}}_2$ structure. (a) Phonon dispersion curves of this structure at a static-lattice pressure of 0 GPa (cell volume 41.79 $\AA {}^3$/f.u.). (b) The blue-colored mode in (a) as a function of decreasing static pressure, from 0 to $-10$ GPa. The mode softens and first develops imaginary phonon frequencies at $X$ (shown as negative frequencies).

Figure 10

Phonon dispersion curves for the proposed $P\overline{6}2m–{\mathrm{CaF}}_2$ structure. This phase has stable phonons at a static pressure of 10 GPa (cell volume 34.45 $\AA {}^3$/f.u.). The mode colored blue softens and becomes unstable at the Brillouin zone $K$ point with decreasing pressure, as shown by the blue dashed curve. Only the portion from $\mathrm{\Gamma }$ to $A$ is shown at 0 GPa.

Figure 11

Illustration of the unstable phonon mode in $P\overline{6}2m–{\mathrm{CaF}}_2$ at the Brillouin zone $K$ point, shown here at 0 GPa. A $3\times 3\times 1$ slab of the structure is depicted, viewed along the $c$ axis as in the left-hand panel of Fig. 7. Arrows indicate the direction of movement of atoms in this mode, and are color-coded according to their relative amplitudes: yellow for largest amplitude through to dark red for smallest amplitude. Calcium atoms are in red; fluorine atoms in green.