Incipient geometric lattice instability of cubic fluoroperovskites

Inorganic metal halide perovskites are promising materials for next-generation technologies due to a plethora of unique physical properties, many of which cannot be observed in the oxide perovskites. On the other hand, the search for ferroelectricity and multiferroicity in lead-free inorganic halide perovskites remains a challenging research topic. Here, we experimentally show that cubic fluoroperovskites exhibit proximity to incipient ferroelectrics, which manifested in the softening of the low-frequency polar phonons in the Brillouin zone center at cooling. Furthermore, we reveal the coupling between harmonic and anharmonic force constants of the softening phonons and their correlation with the perovskite tolerance factor. Next, using first-principles calculations, we examine the lattice dynamics of the cubic fluoroperovskites and disclose the incipient lattice instability at which the harmonic force constants of low-lying phonons tend to decrease with a reduction of tolerance factor at all high-symmetry points of the Brillouin zone. The correlations with the tolerance factor indicate the geometric origin of observed incipient lattice instability in the cubic fluoroperovskites caused by the steric effect due to the volume filling of the unit cell by different radius ions. These results provide insights into the lattice dynamics and potential ferroelectric properties of inorganic lead-free metal halide perovskites, relevant to further design and synthesis of new multifunctional materials.

Figure 1

The crystal structure of cubic fluoroperovskites $AM{\mathrm{F}}_3$ with the tolerance factor (a) $t<1$ and (b) $t>1$. (c) The Brillouin zone of a face-centered cubic lattice indicating the high-symmetry $\mathrm{\Gamma }-M-R-\mathrm{\Gamma }-X$ path used in the lattice dynamic calculations. The $k_x, k_y$, and $k_z$ are the primitive reciprocal lattice vectors.

Figure 2

(a)–(d) Room temperature and (e)–(h) temperature color map of far-infrared reflectivity spectra of the cubic fluoroperovskites ${\mathrm{KZnF}}_3, {\mathrm{RbMnF}}_3, {\mathrm{KNiF}}_3$, and ${\mathrm{KMgF}}_3$, respectively. The solid black lines are results of fits based on the generalized oscillator model according to Eq. (2). Color vertical dashed lines indicate ${\omega }_{\text{TO}}$ and ${\omega }_{\text{LO}}$ phonon frequencies, with ${\omega }_{\text{TO}}<{\omega }_{\text{LO}}$. Horizontal black dashed lines indicate antiferromagnetic phase transition temperatures $T_N$.

Figure 3

Temperature dependences of frequencies ${\omega }_j$ of the $j=1–3$ polar phonons for the cubic fluoroperovskites (a) ${\mathrm{KZnF}}_3$, (b) ${\mathrm{RbMnF}}_3$, (c) ${\mathrm{KNiF}}_3$, and (d) ${\mathrm{KMgF}}_3$. The color circles represent the experimental data. The black lines correspond to the fit under assumption of anharmonic temperature behavior in the absence of magnetic ordering according to Eq. (3). The green lines are fits of the shift due to spin-phonon coupling according to Eq. (4). Values of the spin-phonon coupling $\mathrm{\Delta }{\omega }^{\text{SP}}$ (${\mathrm{cm}}^{-1}$) are given. Differences between nonmagnetic and magnetic fit functions are shown by green filled areas. The paramagnetic and antiferromagnetic phases are shown in blue and red color filled backgrounds, respectively.

Figure 4

(a) Temperature and (b) tolerance factor $t$ (at room temperature) dependences of the differences of squared phonon frequencies ${\omega }_{1\text{TO}}^2\left(T\right)-{\omega }_{1\text{TO}}^2\left(5\text{K}\right)$ which represent the ratio of anharmonic and harmonic force constants $\mathrm{\Delta }{k}_{\text{ah}}/k_0$ in the cubic fluoroperovskites. The data for ${\mathrm{KCoF}}_3$ and ${\mathrm{RbCoF}}_3$ have been adapted from Ref. [56]. (c) Dependence of the squared phonon frequency ${\omega }_{1\text{TO}}^2\left(5\text{K}\right)$ at low temperature which represent the reduced harmonic force constant $k_0\left(T\right)/\mu$ on the tolerance factor $t$ in two groups of the cubic fluoroperovskites: ${\mathrm{KMnF}}_3$ [52, 53] (at room temperature), ${\mathrm{KCoF}}_3$ [56], ${\mathrm{KNiF}}_3$ and ${\mathrm{RbMnF}}_3, {\mathrm{RbFeF}}_3$ [59] (at room temperature), ${\mathrm{RbCoF}}_3$ [56] with insignificant difference of the reduced mass $\mu$. Temperature dependences of the ${\omega }_{1\text{TO}}^2\left(T\right)-{\omega }_{1\text{TO}}^2\left(5\text{K}\right)$ which represent the reduced anharmonic force constant $k_{\text{ah}}\left(T\right)/\mu$ in groups of the cubic fluoroperovskites (d) ${\mathrm{KCoF}}_3, {\mathrm{KNiF}}_3$ and (e) ${\mathrm{RbMnF}}_3, {\mathrm{RbCoF}}_3$ in which the difference in the value of $\mu$ is neglected.

Figure 5

Phonon dispersion curves along the $\mathrm{\Gamma }-M-R-\mathrm{\Gamma }-X$ high-symmetry path of the Brillouin zone of the cubic fluoroperovskites with different tolerance factors $t$. The lines represent the calculated data. Imaginary frequencies are signified by negative numbers and correspond to unstable phonons. The circles denote the adapted experimental data from the neutron scattering for ${\mathrm{KCoF}}_3$ [68], ${\mathrm{KZnF}}_3$ [69], and ${\mathrm{KMgF}}_3$ [70]. The squares represent the polar phonon frequencies in the Brillouin zone center from our infrared spectroscopy experiments for ${\mathrm{KZnF}}_3, {\mathrm{RbMnF}}_3, {\mathrm{KNiF}}_3$, and ${\mathrm{KMgF}}_3$, and adapted from the literature for ${\mathrm{RbCaF}}_3$ [75], ${\mathrm{KMnF}}_3$ [52], ${\mathrm{KCoF}}_3$ [56], and ${\mathrm{RbCoF}}_3$ [56].

Figure 6

Calculated force constants $k$ of the low-lying $T_{1u}, X_5, M_2$, and $R_{{15}^{\prime }}$ phonons at the $\mathrm{\Gamma }, X, M$, and $R$ high-symmetry points of the Brillouin zone, respectively, in the cubic fluoroperovskites $AM{\mathrm{F}}_3$ with different tolerance factors $t$.

Figure 7

(a) Calculated frequencies $\omega$ of the polar phonons in cubic fluoroperovskite ${\mathrm{KNiF}}_3$ in (left frames) G-AFM and (right frames) FM states as a function of lattice parameter $a$. The equilibrium value of $a$ is in the center. Values of the obtained Grüneisen parameters $\gamma$ for phonons are given. (b) Phonon frequency ${\omega }_{1\text{TO}}$ for ${\mathrm{KNiF}}_3$ in G-AFM state as a function of biaxial epitaxial strain $\eta$. Negative and positive signs of $\eta$ correspond to compression and expansion, respectively. For $\eta \ne 0$ the symmetry is tetragonal with the space group $P4/mmm$.

Figure 8

(a) Sketch of the ion displacements for polar phonons in the cubic fluoroperovskites according to the DFT simulations. (b) The $M-\mathrm{F}-M$ superexchange bond angle ${\theta }_0$ and length $r_0$ which are dynamically changed due to ion displacements of polar phonons. ${\mathrm{F}}_{\perp }$ and ${\mathrm{F}}_{\parallel }$ indicate the displacements perpendicular and parallel to the $M-\mathrm{F}-M$ bond, respectively. (c) Computed absolute values of ion displacements related to polar phonons in ${\mathrm{RbMnF}}_3$ and ${\mathrm{KNiF}}_3$. (d) Relationship between the calculated dynamical changes of the bond angle $\mathrm{\Delta }\theta /{\theta }_0$ and length $\mathrm{\Delta }r/r_0$ and the computed frequency shift $\mathrm{\Delta }{\omega }^{\text{SP}}/{\omega }^{\text{NM}}$ due to the spin-phonon coupling in ${\mathrm{RbMnF}}_3$ and ${\mathrm{KNiF}}_3$. Picture was prepared using the vesta software [104].

Figure 9

Temperature dependences of (bottom frames) the dielectric strength $\mathrm{\Delta }{\varepsilon }_j$ for polar phonons $j=1–3$, (middle frames) the static dielectric permittivity ${\varepsilon }_0$, and (upper frames) the low-frequency dielectric permittivity ${\varepsilon }_0^{\text{lf}}$ at $f=100\mathrm{kHz}$ for the cubic fluoroperovskites (a) ${\mathrm{KZnF}}_3$, (b) ${\mathrm{RbMnF}}_3$, (c) ${\mathrm{KNiF}}_3$, and (d) ${\mathrm{KMgF}}_3$. The temperature dependence of the ${\varepsilon }_0^{\text{lf}}$ for ${\mathrm{RbMnF}}_3$ has been adapted from Ref. [91]. The color circles correspond to the experimental data. The black and green lines are fits assuming the anharmonic and spontaneous magnetodielectric effects, respectively. Values of the spontaneous magnetodielectric effect $\mathrm{\Delta }{\varepsilon }^{\text{MD}}$ are given. The paramagnetic and antiferromagnetic phases are shown in blue and red color filled backgrounds, respectively.