Phase stability and transformations in the halide perovskite ${\mathrm{CsSnI}}_3$

We employ the quasiharmonic approximation to study the temperature-dependent lattice dynamics of the four different phases of cesium tin iodide $\left({\mathrm{CsSnI}}_3\right)$. Within this framework, we obtain the temperature dependence of a number of structural properties, including the cell volume, bulk modulus, and Grüneisen parameter. The Gibbs free energy of each phase is compared against the temperature-dependent Helmholtz energy obtained from the equilibrium structure within the harmonic approximation. We find that the black tetragonal perovskite phase is not dynamically stable up to at least 500 K, with the phonon dispersion displaying negative optic modes, which pass through all of the high-symmetry wave vectors in the Brillouin zone. The main contributions to the negative modes are found to be motions of the Cs atom inside the perovskite cage. The black cubic perovskite structure shows a zone-boundary instability, indicated by soft modes at the special $\mathbf{q}$ points $M$ and $R$. These modes are present in calculations at the equilibrium (0 K) lattice constant, while at finite temperature additional negative modes develop at the zone center, indicating a ferroelectric instability. The yellow crystal, composed of one-dimensional $\left({\mathrm{SnI}}_6\right){}_n$ double chains, has the same heat of formation as the orthorhombic perovskite phase at 0 K, but becomes less energetically favorable at higher temperatures, due to its higher free energy.

Figure 1

The four polymorphs of ${\mathrm{CsSnI}}_3$. On the top row, the three black perovskite structures are shown: (a) cubic $\mathrm{B}\alpha$, (b) tetragonal $\mathrm{B}\beta$, and (c) orthorhombic $\mathrm{B}\gamma$. The bottom row shows the yellow crystal where the tin halide octahedral networks are fragmented into one-dimensional chains (d). In all four images, the green spheres represent the Cs atom, while the purple polyhedra represent the octahedral perovskite cage formed by the bonding of the Sn (steel blue) and I (dark purple) atoms.

Figure 2

Free energy as a function of temperature for the three black ${\mathrm{CsSnI}}_3$ polymorphs relative to the Y phase, which is the lowest-energy structure at 0 K. Plots (a) and (b) show the Helmoltz and Gibbs free energies, respectively. Energies are given per ${\mathrm{CsSnI}}_3$ formula unit.

Figure 3

Structural properties of the four phases of ${\mathrm{CsSnI}}_3$ as a function of temperature within the quasiharmonic approximation. (a) Grüneisen parameter, $\gamma$(T); (b) volumetric-expansion coefficient, ${\alpha }_{\mathrm{V}}$(T); (c) bulk modulus, $B\left(T\right)$; and (d) volume per formula unit, $V\left(T\right)$.

Figure 4

Phonon dispersions and densities of states of the four different polymorphs of ${\mathrm{CsSnI}}_3$, plotted for various lattice temperatures obtained from the quasiharmonic approximation (the corresponding structural parameters are listed in Table 5): (a) $\mathrm{B}\alpha$, (b) $\mathrm{B}\beta$, (c) $\mathrm{B}\gamma$, and (d) Y. The unit cell of the $\mathrm{B}\alpha$ phase contains five atoms $\left(Z=1\right)$, whereas the $\mathrm{B}\beta$ unit cell is twice as large (10 atoms; $Z=2$), and the the two orthorhombic structures both contain 20 atoms in the primitive cell $(Z=4$).

Figure 5

Partial phonon density of states of the $\mathrm{B}\alpha$ (a) and $\mathrm{B}\beta$ (b) phases of ${\mathrm{CsSnI}}_3$, calculated at the 0 K equilibrium lattice parameters. It can be seen that distortions of the Sn-I cage are the major contributor to the negative part of the DOS in the $\mathrm{B}\alpha$ phase, whereas the imaginary modes in the $\mathrm{B}\beta$ phase are predominantly due to the motion of the caged Cs ion.

Figure 6

Phonon band dispersion (a) and partial density of states (b) of the $\mathrm{B}\alpha$ polymorph of ${\mathrm{CsSnI}}_3$, at low temperatures. An imaginary optical mode emerges at the $\Gamma$ point as the temperature increases, which appears to be due predominantly to the “rattling” of the Cs atom within the perovskite cage.

Figure 7

Phonon band dispersions of the $\mathrm{B}\beta$ phase of ${\mathrm{CsSnI}}_3$ calculated with plane-wave cutoffs of 500 eV (electronically converged value; top) and 800 eV (bottom).

Figure 8

Helmoltz free-energy equation-of-state fits for the four ${\mathrm{CsSnI}}_3$ polymorphs at different lattice temperatures: (a) $\mathrm{B}\alpha$, (b) $\mathrm{B}\beta$, (c) $\mathrm{B}\gamma$, and (d) Y. Note that the free energies are given per unit cell, and not per formula unit.