Theoretical approaches to Fröhlich excitonic polarons in polar semiconductors

TL;DR

This article surveys a cohesive, empirical framework for Fröhlich excitonic polarons in polar semiconductors, aligning the classic LLP theory for free polarons with the Pollmann–Büttner–Kane (PBK) treatment of excitonic polarons. It develops and compares single- and multimode phonon models (PB, PBK, Iad) and derives new analytical expressions for polaron energies, effective masses, virtual phonon populations, and lattice-mediated electron–hole interactions; extends the formalism to multiple polar phonon branches (MPB/MPBK/MIad) and to nanostructures. The work also connects empirical models with first-principles approaches (DFT/GW/DFPT/BSE), illustrating parameter extraction and validating predictions against materials such as TlCl and halide perovskites, while highlighting limitations at finite temperature and in disordered or strongly anharmonic lattices. It shows that PB and PBK provide accurate descriptions in weak-to-intermediate coupling, while ABS offers efficient approximations for excited states; multimode treatments are essential to capture the full phonon landscape of perovskites. Overall, the framework offers practical, tunable tools to interpret excitonic polaron effects in 3D perovskites and nanoscale structures, guiding materials design for next-generation optoelectronic devices and informing where first-principles methods can provide complementary insight.

Abstract

Short abstract: The paper reviews the physics of Fröhlich excitonic polarons from the viewpoint of empirical approaches with some original developments. Models for excitonic polarons in ionic semiconductors in the spirit of the Lee Low and Pines (LLP) model for free polarons were initiated by Toyozawa and Hermanson and extended by Pollman and Buttner (PB). The dominant electron-hole interaction with the lattice introduced by Frohlich is represented by a long-range effective interaction with a single longitudinal optical polar mode. The properties of the excitonic polarons are characterized by various physical quantities such as effective dielectric constants, effective masses, virtual phonon populations, carrier self-energies and binding energies, and effective electron-hole interactions mediated by the lattice. In 3D perovskites, the excitonic polarons deviate from the simplified picture of weakly interacting (almost free) polarons, with sizeable effects of electron-hole correlations on all the physical properties.

Figures (21)

• Figure 1: (left) Artist view of the polar lattice distortion for a negative Fröhlich polaron. (center) The electron and the hole are represented by large shallow spheres, while lattice anions or cations are represented by smaller negatively or positively charged spheres. (right) Artist view of the polar lattice distortion for a positive Fröhlich polaron.
• Figure 2: Various levels of theory analyzed in this work, with corresponding material parameters and physical observables. Continuum-based model parameters used for uncorrelated charge carriers and the 1S exciton ground state are further introduced in Tab. \ref{['tab:table2']}.
• Figure 3: Schematic representation of (a) the zinc-blende structure of GaAs (b) the CsCl structure of TlCl (c) the perovskite structure of CsPbI3.
• Figure 4: Illustration of the regime where the e-h correlations weakly influence free polaronic distortions. The two charges are separated by a distance greater than the free polaron radius, with here the same radius $R_{e} = R_{h} = R_{\text{pol}}$ for the electron and the hole. $\ {\widetilde{R}}_{\text{pol}}$ denotes the reduced free polaron radius ${\widetilde{R}}_{\text{pol}} = \frac{R_{\text{pol}}}{a_{B}}\ \ll 1$. $a_{B}$ is the 1S exciton Bohr radius. (Top right) Reduced radial probability densities $\widetilde{\rho}\left( \widetilde{r} \right) = \rho\left( r \right)*a_{B}\ $of hydrogenic 1S and 2S Wannier exciton wavefunctions as a function of the reduced e-h distance $\widetilde{r} = \frac{r}{a_{B}}\ $ (the same value of $a_{B}$ is considered here for both 1S and 2S excitons) (Right) Artist view of two charges separated by a short distance, where the polaronic distortion differs from the simple superposition of the distortions related to free polarons. (Bottom left) Variation of the free polaron radius (blue line), Bohr radius within the PBK model (green line) and the PB model (red line) as a function of $\alpha$ taking materials parameters for TlCl (Tab. \ref{['tab:Table1']}). The variation of $\alpha$ is obtained by varying the effective phonon frequency $\hbar \omega_{\text{LO}}$.(Bottom right) Variation of the reduced free polaron radius as obtained within the PBK model (blue straight line) and the PB model (red dashed line) as a function of $\alpha$ taking materials parameters for MAPbI3 (Tab. \ref{['tab:Table1']}). The variation of $\alpha$ is obtained by varying the effective phonon frequency $\hbar \omega_{\text{LO}}$. The differences between the two models are coming from small differences between the computed Bohr radii $a_{B}$. To match the experimentally determined exciton binding energy of 16 meV, Miyata_2015 the effective longitudinal optical phonon energy derived within the PBK model amounts to 8.2 meV ($\alpha = 3.18$). The dashed horizontal line indicates that the crossover from the weak to intermediate exciton-polaron coupling regime occurs for ${\widetilde{R}}_{\text{pol}} = 1/4$, that is ${4R}_{\text{pol}} = a_{B}$.
• Figure 5: Illustration of the calculations of 1S exciton binding energies for TlCl. The variations of exciton energies are represented by straight lines as a function of the Bohr radius, for a Wannier exciton model (a) with $\varepsilon_{\text{eff}} = \varepsilon_{\infty}$ and a PB excitonic polaron model (b). The Bohr radii are scaled by the value $a_{B,\infty}$ at the minimum energy. Energies are scaled by the minimum energy $E_{b}^{\infty}$. The binding energies represented by arrows are computed at the energy minima by subtracting the energies for free carriers (a) or free polarons (b). The minimum energy for free carriers (band gap) is set as a reference to 0 to define the vertical axis. The minimum energy for free polarons is lowered by self-energy terms computed within LLP theory.