The plane-wave density functional package Quantum Espresso (50) was used to perform the structural relaxation and electronic structure calculation with the PBEsol exchange-correlation functional (51). Norm-conserving, designed nonlocal pseudopotentials were generated with the OPIUM package (52, 53). For the electrostatic potential calculation, the ferroelectric/paraelectric domains were built by stacking two hexagonal supercells (indicating ferroelectric and paraelectric phases, respectively) with around 33% La substitution concentration. For the electronic structure calculations, a 2 × 2 × 2 supercell of the rhombohedral BFO unit cell (80 atoms) was used to model different concentrations of La (0%, 6.25%, 12.5%, and 18.75%). A plane-wave kinetic energy cutoff of 50 Ry and k point sampling on a 4 × 4 × 4 grid were sufficient to yield converged total energies. Because of the well-known deficiency of DFT in overly delocalizing electrons, the DFT + U method was used, with effective Ueff applied on the Fe 3d orbitals for all the calculations. Ueff = 4 eV was shown to improve the structural relaxations significantly (28). In addition, as shown in (21), Ueff = 5 eV was optimal for the electronic structure calculations. Therefore, we used Ueff = 4 eV for the structural relaxation and Ueff =5 eV to compute the electronic band structures. The structures with various La atom positions and doping concentrations were all fully relaxed with a force tolerance of 0.001 Ry/bohr. The band structures of the doped structures with low total energies were computed on a denser uniform k-point grid (16 × 16 × 16) to calculate their recombination rates.

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