Empirical approaches to Fröhlich excitonic polarons in polar semiconductors

The interaction between charge carriers with opposite charge, i.e. excitonic correlation between electron and hole, and the charge carriers' interactions with underlying crystal lattice govern the optoelectronic response of semiconductors. The latter is dominated in ionic crystals by the Fröhlich interaction corresponding to the long-range electric field generated by polar phonons at the zone centre. Theoretical description of complex interplay arising from competition and interference between these two interactions, excitonic and polaronic, has been a formidable challenge. At the same time, the emergence of optoelectronics based on ionic semiconductors such as metal-halide perovskites has made it highly relevant to revisit theoretical models developed in the past, and push them further for application to real ionic semiconductors of interest. To this end, the present paper reviews the physics of Fröhlich excitonic polarons from the point of view of empirical approaches and supplements it with a few original developments to the state-of-the-art. At first, we review the models for excitonic polarons in ionic semiconductors, initially built by Toyozawa and Hermanson in analogy with Lee-Low-Pines (LLP) model for free polarons, and later extended by Pollman and Büttner (PB). These models have been applied in past to the simpler cases of weakly interacting polarons (e.g. GaAs). We further consider their applicability to ionic solids such as TlCl or 3D lead halide perovskites, where the 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 expression of the approximated version of the 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 theoretical developments carried out in this paper also include extension of Kane and PB's semi-empirical models to incorporate Fröhlich-like interaction with multiple polar phonons, essential for reconciliation of experimental observations in the case of multi-atom systems. In the end, we discuss the relation of the empirical approaches with emerging ab initio methods and provide an outlook of their application to lower-dimensional systems.