Empirical approaches to Fr\"ohlich excitonic polarons in polar semiconductors
The excitonic correlation between electron and hole and these carriers' interactions with underlying crystal lattice govern the opto-electronic response of semiconductors. The latter is dominated in ionic crystals by the Fr\"{o}hlich interaction corresponding to the long-range electric field generated by polar phonons. Theoretical description of complex interplay of these two effects, excitonic and polaronic, has been a formidable challenge. The present paper reviews the physics of Fr\"ohlich excitonic polarons from the point of view of empirical approaches and supplements it with a few original developments. At first, we review the models for excitonic polarons in ionic semiconductors built analogously to Lee-Low-Pines (LLP) model for free polarons, and later extended by Pollman and B\"uttner (PB). These models have been applied in past to the case of weakly interacting polarons (e.g. GaAs). We consider their applicability to ionic solids such as TlCl or 3D lead halide perovskites, where electron-hole correlations are relatively stronger. In these compounds, electrons and holes have almost equal effective masses, which allows us to derive a new analytical expressions of PB effective interaction potential. The refined Kane approach to PB's model is shown to (i) bridge the regime between weakly interacting polarons and excitonic polarons with strong electron-hole correlations and (ii) recover the LLP model for free polarons in the limit of vanishing correlation. Various developments carried out in this paper also include extension of Kane and PB's semi-empirical models to incorporate Fr\"ohlich-like interaction with multiple polar phonons, essential for reconciliation of experimental observations in multi-atom systems. In the end, we discuss the relation of the empirical approaches with ab initio methods and provide an outlook of their application to lower-dimensional systems.