Higher-order effects in the temperature dependence of the energy levels in Methylammonium Lead Iodide Perovskite

We determine temperature effects on the bandgap and band edges of CH$_3$NH$_3$PbI$_3$ from first principles by accounting for electron-phonon coupling and thermal expansion. From $290$ to $380$K, the computed bandgap change of $40$ meV coincides with the experimental change of $30$-$40$meV. We show that the calculation of electron-phonon coupling in CH$_3$NH$_3$PbI$_3$ is particularly intricate, as the commonly used Allen-Heine-Cardona (AHC) theory significantly overestimates the bandgap change, and excellent agreement with experiment is only obtained when high-order terms in the electron-phonon coupling are included in conjunction with spin-orbit corrections. In contrast, we find that the inclusion of nonlocal correlations using hybrid functionals has little effect. We show that the low-energy phonon modes contribute the most to the energy levels renormalization due to temperature, which suggests that the AHC failure is likely common to metal-halide perovskites and not only to hybrid ones. We have verified this on CsPbI$_3$. Our results unambiguously confirm for the first time the importance of higher-order terms in the electron-phonon coupling by directly comparing with experimental results