Absence of Anomalous Electron-Phonon Coupling in the Temperature Renormalization of the Gap of CsPbBr$_3$ Nanocrystals

Metal halide perovskites exhibit a fairly linear increase of the bandgap with increasing temperature, when crystallized in a tetragonal or cubic phase. In general, both thermal expansion and electron-phonon interaction effects contribute equally to this variation of the gap with temperature. Herein, we have disentangled both contributions in the case of colloidal CsPbBr$_3$ nanocrystals (NCs) by means of photoluminescence (PL) measurements as a function of temperature (from 80 K to ambient) and hydrostatic pressure (from atmospheric to ca. 1 GPa). At around room temperature, CsPbBr$_3$ NCs also show a linear increase of the bandgap with temperature with a slope similar to that of the archetypal methylammonium lead iodide (MAPbI$_3$) perovskite. This is somehow unexpected in view of the recent observations in mixed-cation Cs$_x$MA$_{1-x}$PbI$_3$ single crystals with low Cs content, for which Cs incorporation caused a reduction by a factor of two in the temperature slope of the gap. This effect was ascribed to an anomalous electron-phonon interaction induced by the coupling with vibrational modes admixed with the Cs translational dynamics inside the cage voids. Thus, no trace of anomalous coupling is found in CsPbBr$_3$ NCs. In fact, we show that the linear temperature renormalization exhibited by the gap of CsPbBr$_3$ NCs is shared with most metal halide perovskites, due to a common bonding/antibonding and atomic orbital character of the electronic band-edge states. In this way, we provide a deeper understanding of the gap temperature dependence in the general case when the A-site cation dynamics is not involved in the electron-phonon interaction.