Anharmonic stabilization and band gap renormalization in the perovskite ${\mathrm{CsSnI}}_3$

Amongst the $X(\text{Sn},\text{Pb})Y_3$ perovskites currently under scrutiny for their photovoltaic applications, the cubic $\mathrm{B}-\alpha$ phase of ${\text{CsSnI}}_3$ is arguably the best characterized experimentally. Yet, according to the standard harmonic theory of phonons, this deceptively simple phase should not exist at all due to rotational instabilities of the ${\mathrm{SnI}}_6$ octahedra. Here, employing self-consistent phonon theory, we show that these soft modes are stabilized at experimental conditions through anharmonic phonon-phonon interactions between the Cs ions and their iodine cages. We further calculate the renormalization of the electronic energies due to vibrations and find an unusual opening of the band gap, estimated as 0.24 and 0.11 eV at 500 and 300 K, which we attribute to the stretching of Sn-I bonds. Our work demonstrates the important role of temperature in accurately describing these materials.

Figure 1

Phonon band structures obtained for the $\mathrm{B}-\alpha$ phase of ${\text{CsSnI}}_3$ calculated under the harmonic approximation or with self-consistent phonon frequencies ($\mathrm{SC}\omega$). Imaginary frequencies are shown as negative. The supercell calculations do not include the nonanalytic correction accounting for the long-wavelength splitting of polar modes [23].

Figure 2

(a) $\mathrm{B}-\alpha$ phase of ${\text{CsSnI}}_3$, with blue/yellow/gray atoms = Cs/Sn/I. (b) $\mathrm{B}-\alpha$ structure distorted along modes at the $M$ point corresponding to ${\mathrm{SnI}}_6$ octahedral rotation (orange arrows) and Cs vibration (pink arrows in/out of page). The respective amplitudes along each mode are $x_{\mathrm{rot}}$ and $x_{\mathrm{Cs}}$. (c) Energy $V$ vs deviation in Sn-I-Sn bond angle $\theta$ for constant values of $x_{\mathrm{Cs}}$ of (in direction of arrow) 0.0, 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, and 2.5, in units of ${\sqrt{\langle x_{\mathrm{Cs}}^2\rangle }}_{T=500\text{K}}$.

Figure 3

GLLB-SC band gaps calculated with/without (green squares/gray circles) derivative discontinuity contribution for an ensemble of 300 configurations vs average Sn-I bond length $D_{\mathrm{SnI}}$. A linear fit to the data is shown. $E_g$ is taken as the difference between the highest occupied state and the average of the three lowest unoccupied states at $R$ [35]. We also show the gaps of the unperturbed $\mathrm{B}-\alpha , \mathrm{B}-\beta$, and $\mathrm{B}-\gamma$ structures (orange stars). Note that these calculations were performed in a $2\times 2\times 2$ supercell and are subject to the finite size effects discussed in the text.

Figure 4

Gap spectral functions $g^{2}F\left(\omega \right)$ calculated using $\partial E_g/\partial n_{\nu }=l_{\nu }^2{\partial }^2{E}_g/\partial x_{\nu }^2$ (red dotted line) and Eq. (5) (blue solid line). The expected gap renormalization at temperature $T$ is obtained as $\int d\omega g^{2}F\left(\omega \right)[n_B(\omega ,T)+1/2]$.