Thermal Stabilization of Defect Charge States and Finite-Temperature Charge Transition Levels

Point defects introduce localized electronic states that critically affect carrier trapping, recombination, and transport in functional materials. The associated charge transition levels (CTLs) can depend on temperature, requiring accurate treatment of vibrational and electronic free-energy contributions. In this work, we use machine-learned interatomic potentials to efficiently compute temperature-dependent CTLs for vacancies in MgO, LiF, and CsSnBr3. Using thermodynamic integration, we quantify free-energy differences between charge states and calculate the vibrational entropy contributions at finite temperatures. We find that CTLs shift with temperature in MgO, LiF and CsSnBr3 from both entropy and electronic contributions. Notably, in CsSnBr3 a neutral charge state becomes thermodynamically stable above 60 K, introducing a temperature-dependent Fermi-level window absent at 0 K. We show that the widely used static, zero-kelvin defect formalism can miss both quantitative CTL shifts and the qualitative emergence of new stable charge states.

💡 Research Summary

This paper introduces a comprehensive framework for calculating temperature‑dependent defect charge transition levels (CTLs) by combining density‑functional‑theory‑trained neural‑network interatomic potentials (NEPs) with thermodynamic integration (TI). The authors focus on vacancy defects in three representative materials—MgO, LiF, and the halide perovskite CsSnBr₃—covering a range of ionic, covalent, and soft‑lattice chemistries. For each system, NEP models are trained on DFT data (PBEsol for MgO and LiF, r2SCAN for CsSnBr₃) that include both pristine and defect configurations across relevant charge states. Crucially, atoms neighboring the vacancy are labeled according to the defect’s charge, allowing a single structural configuration to represent multiple charge states within the same potential.

Two TI routes are employed. The traditional Frenkel–Ladd path connects the NEP‑described system to an Einstein crystal, yielding absolute Gibbs free energies for each charge state. The authors also devise a direct charge‑state TI path that interpolates between two NEPs differing only by the charge‑state labeling. This latter approach requires only one TI calculation per CTL and converges substantially faster because the energy differences along the path are small. Both routes produce consistent CTL values, validating the methodology.

Free‑energy differences are decomposed into vibrational entropy contributions and temperature‑dependent shifts of the valence‑band maximum (VBM), which are obtained from MD snapshots via deep core‑level alignment. For MgO, the ε⁺²/0 transition moves downward by ~0.785 eV over a 1000 K range; roughly half of this shift originates from the VBM rise (~0.414 eV) and the other half from vibrational entropy (≈‑0.371 eV K⁻¹). LiF shows even larger shifts for ε⁺¹/0 and ε⁰/‑1 (≈‑0.994 eV and ‑1.035 eV over 500 K). The most striking result is for CsSnBr₃: the ε⁺¹/0, ε⁰/‑1, and ε⁺¹/‑1 levels shift by about 0.111 eV, 0.324 eV, and 0.218 eV respectively across 300 K, and a neutral charge state becomes thermodynamically favored above ~60 K. This creates a temperature‑dependent Fermi‑level window that does not exist at 0 K, demonstrating that new stable charge states can emerge solely due to finite‑temperature effects.

Optical transition energies are also evaluated. Zero‑K vertical transitions, obtained from configurational coordinate diagrams, agree with prior GW₀ and Bethe‑Salpeter calculations. Finite‑temperature optical levels are extracted from MD trajectories by monitoring the instantaneous energy difference between two charge states; these follow the same temperature trends as the CTLs, confirming that vibrational and band‑edge effects dominate both thermodynamic and optical defect properties.

Overall, the study provides (i) a scalable, ML‑accelerated workflow for accurate finite‑temperature defect thermodynamics, (ii) quantitative insight into how vibrational entropy and band‑edge shifts jointly dictate CTL movement, and (iii) concrete evidence that temperature can stabilize charge states absent at 0 K. These findings imply that defect modeling for semiconductors, photovoltaics, and solid‑state ionic conductors must go beyond static 0 K calculations and incorporate full free‑energy contributions to reliably predict material performance under realistic operating conditions.