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Temperature Dependence of the Energy Levels of Methylammonium Lead Iodide Perovskite from First Principles

Environmental effects and intrinsic energy-loss processes lead to fluctuations in the operational temperature of solar cells, which can profoundly influence their power conversion efficiency. Here we determine from first principles the effects of temperature on the band gap and band edges of the hybrid pervoskite CH$_3$NH$_3$PbI$_3$ by accounting for electron-phonon coupling and thermal expansion. From $290$ to $380$ K, the computed band gap change of $40$ meV coincides with the experimental change of $30$-$40$ meV. The calculation of electron-phonon coupling in CH$_3$NH$_3$PbI$_3$ is particularly intricate, as the commonly used Allen-Heine-Cardona theory overestimates the band gap change with temperature, and excellent agreement with experiment is only obtained when including high-order terms in the electron-phonon interaction. We also find that spin-orbit coupling enhances the electron-phonon coupling strength, but that the inclusion of nonlocal correlations using hybrid functionals has little effect. We reach similar conclusions in the metal-halide perovskite CsPbI$_3$. Our results unambiguously confirm for the first time the importance of high-order terms in the electron-phonon coupling by direct comparison with experiment.