Ferroelectricity in $d^0$ double perovskite fluoroscandates

Ferroelectricity in strain-free and strained double perovskite fluorides, ${\mathrm{Na}}_3{\mathrm{ScF}}_6$ and ${\mathrm{K}}_2{\mathrm{NaScF}}_6$, is investigated using first-principles density functional theory. Although the experimental room temperature crystal structures of these fluoroscandates are centrosymmetric, i.e., ${\mathrm{Na}}_3{\mathrm{ScF}}_6$ ($P{2}_1/n$) and ${\mathrm{K}}_2{\mathrm{NaScF}}_6$ ($Fm\overline{3}m$), lattice dynamical calculations reveal that soft polar instabilities exist in each prototypical cubic phase and that the modes harden as the tolerance factor approaches unity. Thus the double fluoroperovskites bear some similarities to $AB{\mathrm{O}}_3$ perovskite oxides; however, in contrast, these fluorides exhibit large acentric displacements of alkali metal cations (Na, K) rather than polar displacements of the transition metal cations. Biaxial strain investigations of the centrosymmetric and polar ${\text{Na}}_3{\text{ScF}}_6$ and ${\text{K}}_2{\text{NaScF}}_6$ phases reveal that the paraelectric structures are favored under compressive strain, whereas polar structures with in-plane electric polarizations ($\sim 5–18\mu \mathrm{C}{\mathrm{cm}}^{-2}$) are realized at sufficiently large tensile strains. The electric polarization and stability of the polar structures for both chemistries are found to be further enhanced and stabilized by a coexisting single octahedral tilt system. Our results suggest that polar double perovskite fluorides may be realized by suppression of octahedral rotations about more than one Cartesian axis; structures exhibiting in- or out-of-phase octahedral rotations about the $c$ axis are more susceptible to polar symmetries.

Figure 1

Common structure types adopted by double perovskite fluorides. Elpasolites (a) have stoichiometry $A_{2}BM{\mathrm{F}}_6$ and are typically found in cubic $Fm\overline{3}m$ symmetry. Cryolites (b) have stoichiometry $A_{3}M{\mathrm{F}}_6$ and often crystallize in a rotationally distorted monoclinic $P{2}_1/n$ symmetry. In cryolites the 12-fold coordinated $A$ cation is chemically identical to the octahedrally coordinated $B$-site cation ($A=B$). The structure stability range for the each family within the given symmetry is specified by the tolerance factor, $t$, obtained using Shannon radii [29, 30].

Figure 2

Unstable lattice distortions at $\mathrm{\Gamma }$ and $X$ for ${\text{Na}}_3{\text{ScF}}_6$. (a) ${\mathrm{\Gamma }}_4^{-}$ is a polar displacement of $A$-site Na atoms and F atoms in the [001] direction indicated by the direction of the red arrows; the length indicates the relative magnitude of the displacement for the ions. (b) $X_3^{+}$ is an in-phase octahedral rotation about the [001] axis. The rotation angle is measured as $(90-\theta )/2$. (c) ${\mathrm{\Gamma }}_4^{+}$ is an out-of-phase rotation of octahedra about the [110] direction. The tilt angle is measured as $(180-\varphi )/2$. These modes may be visualized using the resources in the Supplemental Material [41].

Figure 3

(a) Stability of the monoclinic $P{2}_1/n$ ground state (circles) and polar $I4mm$ (squares) structure with epitaxial strain. The inset shows a schematic of the epitaxial relationship between ${\text{Na}}_3{\text{ScF}}_6$ and the cubic substrate. (b) The energy difference between the structures is obtained as $\mathrm{\Delta }E=E\left(I4mm\right)-E\left(P{2}_1/n\right)$.

Figure 4

(a) Total energy evolution with biaxial strain for low-energy ${\text{Na}}_3{\text{ScF}}_6$ polymorphs. The evolution in the (b) in-phase rotation mode amplitude and (c) electric polarization for the $Pm$ phase with strain. The gray line in (c) indicates the region where the polarization is inaccessible at 0 K.

Figure 5

Energetic dependence of various mode distortions at 4% biaxial strain for ${\text{Na}}_3{\text{ScF}}_6$. Coupling between ${\mathrm{\Gamma }}_4^{-}$ and fixed amplitudes of (a) the in-phase rotation, $X_3^{+}$, (b) the antipolar distortion, $X_5^{-}$, and (c) the linear combination of the two modes, $X_3^{+}\oplus X_5^{-}$. (d) The polar displacements of ions in the ${\mathrm{\Gamma }}_4^{-}$ mode. The normalized energy gains are obtained by increasing the amplitudes of the in-phase and antipolar distortions at fixed percentages found in the equilibrium structure at 4% tensile strain.

Figure 6

Evolution of the (a) relative total energy and (b) energy differences ($\mathrm{\Delta }E$) with biaxial strain for low-energy polar phases of ${\text{K}}_2{\text{NaScF}}_6$ with respect to the equilibrium centrosymmetric ($I4/mmm$) structure. In (a) the area shaded gray represents the biaxial strain region over which the centrosymmetric phase is stable.

Figure 7

Evolution of the (a) out-of-phase rotation angle and (b) in-plane electric polarization in $Cm {\text{K}}_2{\text{NaScF}}_6$ as a function of tensile strain. The gray line represents the region where the polar state is inaccessible.

Figure 8

Evolution in the energy for ${\text{K}}_2{\text{NaScF}}_6$ as a function of interacting lattice modes at 5% strain. (a) The coupling between the polar ${\mathrm{\Gamma }}_4^{-}$ and the out-of-phase rotation ${\mathrm{\Gamma }}_4^{+}$ modes, (b) the polar ${\mathrm{\Gamma }}_5^{-}$ and the out-of-phase rotation ${\mathrm{\Gamma }}_4^{+}$ modes, (c) the two polar modes, i.e., ${\mathrm{\Gamma }}_4^{-}\oplus {\mathrm{\Gamma }}_5^{-}$ (no rotations), and (d) the linear combination of all three modes, ${\mathrm{\Gamma }}_4^{-}\oplus {\mathrm{\Gamma }}_5^{-}\oplus {\mathrm{\Gamma }}_4^{+}$. Here the symmetry equivalent minima are obtained by computing the energy for positive and negative amplitudes of the polar and rotational distortion modes. The normalized energy gains are obtained by increasing the amplitudes of the distortions at fixed percentages found in the equilibrium structure at 5% tensile strain.

Figure 9

Layer-resolved polarizations for polar ${\text{Na}}_3{\text{ScF}}_6$ ($Pm$) and ${\text{K}}_2{\text{NaScF}}_6$ ($Cm$) at 10% tensile strain. (a) The cation displacements and (b) the layer-resolved polarization for ${\text{Na}}_3{\text{ScF}}_6$ along the [100] direction for ${\text{Na}}_3{\text{ScF}}_6$. (c) The cation displacements and (d) the layer-resolved polarization for ${\text{Na}}_3{\text{ScF}}_6$ along the [010] direction. Similarly, panels (e),(f) and (g),(h) depict the same information for ${\text{K}}_2{\text{NaScF}}_6$ along the [100] and [010] directions, respectively. Berry phase calculations result in the total electric polarization for ${\text{Na}}_3{\text{ScF}}_6$ at 10% to be $P=12.7\widehat{x}+12.8\widehat{y}$, which is close to that obtained by summing the layer dipoles, i.e., $\mathrm{\Sigma }p=13.0\widehat{x}+13.1\widehat{y}$. Similarly, for ${\text{K}}_2{\text{NaScF}}_6$ at 10%, $P=1.4\widehat{x}+17.9\widehat{y}$ and $\mathrm{\Sigma }p=0.9\widehat{x}+16.6\widehat{y}$.