Contrasting temperature dependence of the band gap in CH 3 NH 3 Pb X 3 ( X = I , Br , Cl ) : Insight from lattice dilation and electron-phonon coupling

Although hybrid halide perovskites $(\text{MAPb}{X}_{3},$ $\mathrm{MA}={\mathrm{CH}}_{3}{\mathrm{NH}}_{3} \mathrm{and} X=\mathrm{I}, \mathrm{Br}, \mathrm{Cl})$ have been ubiquitously explored from the photovoltaic perspective, there are still a few unanswered questions which require a more fundamental understanding. One such unsettled issue is the puzzling behavior of the band gap. Unlike conventional semiconductors, $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$ is found to show a blueshift (increase) in the band gap $({E}_{g})$ with increasing temperature $(T)$, while ${\mathrm{MAPbCl}}_{3}$ shows an unusual redshift (decrease). In order to understand this, we performed a detailed $T$-dependent study of electronic, optical, and structural properties of $\text{MAPb}{X}_{3}$ combined with the state-of-the-art first-principles calculations. With increasing $T$, two dominant mechanisms which come into play are lattice dilation and electron-phonon coupling (EPC). The former (latter) is responsible for an increase (decrease) in ${E}_{g}$. We found that lattice dilation effect dominates in $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$, causing an enhancement in ${E}_{g}$. EPC involves various contributions, of which the interaction of charge carriers with the longitudinal optical phonon mode via Fr\"ohlich interaction is the most dominant one at room temperature. We quantify this contribution using Fr\"ohlich's theory of large polarons and show that the ${E}_{g}$ correction due to this effect in ${\mathrm{MAPbCl}}_{3}$ is almost double as compared to ${\mathrm{MAPbBr}}_{3}$ and thus explain the reduction in ${E}_{g}$ for the former.