Self-consistent calculations of charge self-trapping energies: A comparative study of polaron formation and migration in ${\mathrm{PbTiO}}_3$

This study presents a systematic assessment of the behavior of self-trapped electrons in ${\mathrm{PbTiO}}_3$, which is a prototypical ferroelectric material with a wide range of technological applications. Since modeling of polarons depends sensitively on the applied method, the goal of this work is to identify the parameters used in density functional theory (DFT), which allow to predict the properties of polarons with high accuracy. The $\mathrm{DFT}+U$ method is employed to benchmark how the choice of $k$-mesh grids, lattice parameters, and pseudopotential (PP) affects the polaron trapping energy. Then, the appropriate parameters were used to study polaron trapping energy and its optical transition using the HSE06 hybrid functional. It is shown that the magnitude of the trapping energy is highly sensitive to the choice of the PP and the applied lattice parameters. A comparison of polaron trapping energies using the two functionals indicates proximity of the $\mathrm{DFT}+U$ result to the HSE06 result. Finally, configuration coordinate diagrams for the polaron-associated absorption and luminescence peaks in ${\mathrm{PbTiO}}_3$ are presented and compared to experiments.

Figure 1

The relaxed configuration of the STel on Ti site in ${\mathrm{PbTiO}}_3$. The blue, pink, and yellow spheres represent Pb, Ti, and O ions, respectively. The charge density isosurface illustrates the polaron wave function.

Figure 2

Calculated trapping energies of STel on Ti atom obtained with $\mathrm{P}\mathrm{B}\mathrm{E}+U$, using different PPs of Ti. $Z$ denotes the number of electrons treated as valence electrons in the PP of Ti. @PBE, @HSE, and @Exp denote $\mathrm{P}\mathrm{B}\mathrm{E}+U$ calculations performed with PBE, HSE, and experimental lattice constants, respectively. Calculations were carried out using $4\times 4\times 4$ $\mathrm{\Gamma }$-centered grids.

Figure 3

Top row: Electronic band structure of polaronic configuration, calculated by treating (a) 4, (b) 10, and (c) 12 outermost electrons of Ti as valence. Middle row: The schematic configuration coordinate diagram depicting the excitation of a small STel to a delocalized electron configuration energy as a functional of lattice distortion, calculated by treating (d) 4, (e) 10, and (f) 12 outermost electrons of Ti as valence. Bottom row: Schematic configuration coordinate diagram depicting the excitation of a small STel to a nearest-neighbor site, calculated by treating (g) 4, (h) 10, and (i) 12 outermost electrons of Ti as valence.

Figure 4

Charge distribution corresponding to the small STel hopping from the (a)initial to the (b) saddle point, and (c) final configurations using $\mathrm{P}\mathrm{B}\mathrm{E}+U$ calculations.

Figure 5

Spin-polarized projected density of states of Ti ions for two configurations: (a) The added electron is delocalized, forming a band electron, and (b) the added electron is localized on a Ti site, forming a STel.

Figure 6

Schematic configuration coordinate diagram showing the optical absorption process, associated with the STel in ${\mathrm{PbTiO}}_3$.