Electron-phonon couplings in polymorphous crystals

Abstract

Positional polymorphism in solids refers to locally disordered unit cells that, on average, reproduce the high-symmetry structures observed in diffraction experiments. Standard theories of electron-phonon interactions fail to describe the temperature-dependent electronic structure of such polymorphous systems. Hybrid halide perovskites are a prime example, where configurational entropy from both polymorphism and molecular disorder plays a central role. Here we generalize the special displacement method to polymorphous crystals, providing an efficient ab initio framework for electron-phonon couplings without resorting to molecular dynamics. We resolve long-standing discrepancies in hybrid halide perovskite physics, including temperature-dependent anharmonic phonons and band gaps. Our approach provides a practical route to link local disorder, configurational entropy, and electron-phonon interactions, with applicability across diverse material classes, from optoelectronics and ferroelectrics to thermoelectrics.

Figures (3)

• Figure 1: (a-f) Ball-stick models of cubic FAPbI$_3$ and MASnI$_3$ representing monomorphous (a,b), reference (c,d), and polymorphous (e,f) structures. Structures in (a,b) refer to unit cells (12 atoms), in (c,d,e,f) to 2$\times$2$\times$2 supercells. (g,h) Calculated temperature-dependent band gaps (coloured discs) of Pb (g) and Sn (h) cubic hybrid halide perovskites including electron-phonon coupling (EPC) and thermal expansion (TE) effects; EPC is incorporated via the ASDM using ZG polymorphous structures and TE contribution is evaluated by allowing lattice expansion according to the experimental expansion coefficients Zacharias2026b. Experimental gaps (triangles) are from Refs. [Milot2015Kontos2018Parrott2016Mannino2020Chen2020Kahmann2020]. ASDM calculations refer to 4$\times$4$\times$4 supercells (768 atoms) and the DFT-PBEsol including SOC. Data are shifted close to PBE0 corrections (see Table I of Ref. [Zacharias2026b]) to match experiment. PBE0 corrections to electron-phonon renormalized band gaps of halide perovskites are negligible Zacharias2023npj.
• Figure 2: (a,b) Electron spectral functions (color maps) of the polymorphous cubic FAPbI$_3$ (a) and MASnI$_3$ (b). Black dispersions represent the band structure of the monomorphous networks. Calculations are at the PBEsol+SOC level and thus underestimate the band gap. Combining hybrid functionals with SOC and polymorphous structures reproduce experiments well Zacharias2026bGarba2025. (c,d) Average band gap increase due to positional polymorphism $\Delta E_g$ (c) and tolerance factor $t$ (d) versus the average metal-halide-metal angle decrease $\Delta \theta_{\rm B-X-B}$ calculated for 10 distinct polymorphous configurations. Solid lines are fits obtained by linear regression and shaded regions represent three standard deviations on either side of the lines. The error bars represent the standard deviation across 10 configurations used for each material.
• Figure 3: (a,b,c) Temperature-dependent band gap of cubic MAPbBr$_3$ calculated considering (a) electron-phonon coupling (EPC), (b) lattice thermal expansion (TE), and (c) both effects combined. Calculated data are shifted by 1.38 eV (orthorhombic), 1.79 eV (monomorphous cubic) and 1.44 eV (polymorphous cubic) to match the experiment Mannino2020 (grey triangles) at $T=275$ K. (d) Eliashberg function showing the one-phonon energy-resolved band gap renormalization of MAPbBr$_3$ at $T=300$ K. (e) Fan-Migdal (FM) and Debye-Waller (DW) self-energy corrections to the VBM of the reference and polymorphous cubic FAPbI$_3$ due to a bending mode. (f,g) Linear variations in the charge density at the VBM obtained for a single motif of the reference (f) and polymorphous (g) networks. Arrows representing the bending mode are scaled by 50. The isosurface value $u$ is indicated.