Efficacy of the DFT + $U$ formalism for modeling hole polarons in perovskite oxides

We investigate the formation of self-trapped holes (STH) in three prototypical perovskites (${\mathrm{SrTiO}}_3$, ${\mathrm{BaTiO}}_3$, ${\mathrm{PbTiO}}_3$) using a combination of density functional theory (DFT) calculations with local potentials and hybrid functionals. First we construct a local correction potential for polaronic configurations in ${\mathrm{SrTiO}}_3$ that is applied via the DFT + $U$ method and matches the forces from hybrid calculations. We then use the DFT + $U$ potential to search the configuration space and locate the lowest energy STH configuration. It is demonstrated that both the DFT + $U$ potential and the hybrid functional yield a piecewise linear dependence of the total energy on the occupation of the STH level, suggesting that self-interaction effects have been properly removed. The DFT + $U$ model is found to be transferable to ${\mathrm{BaTiO}}_3$ and ${\mathrm{PbTiO}}_3$, and STH formation energies from DFT + $U$ and hybrid calculations are in close agreement for all three materials. STH formation is found to be energetically favorable in ${\mathrm{SrTiO}}_3$ and ${\mathrm{BaTiO}}_3$ but not in ${\mathrm{PbTiO}}_3$, which can be rationalized by considering the alignment of the valence band edges on an absolute energy scale. In the case of ${\mathrm{PbTiO}}_3$ the strong coupling between Pb $6s$ and O $2p$ states lifts the valence band minimum (VBM) compared to ${\mathrm{SrTiO}}_3$ and ${\mathrm{BaTiO}}_3$. This reduces the separation between VBM and STH level and renders the STH configuration metastable with respect to delocalization (band hole state). We expect that the present approach can be adapted to study STH formation also in oxides with different crystal structures and chemical compositions.

Figure 1

(a) Force matching between LDA + $U$ models with $U$ applied to O $2p$ states and the HSE06 hybrid functional for cubic ${\mathrm{SrTiO}}_3$ based on $l_1$ and $l_2$ norm as well as maximum deviation of forces $\Delta f_{max}$ (40-atom cell, one hole). Comparison of force components from (b) LDA ($U=0\mathrm{eV}$) and (c) LDA + $U$ with $U=8\mathrm{eV}$ with HSE06 calculations.

Figure 2

(a) Geometric structure of STH in cubic ${\mathrm{SrTiO}}_3$ and (b) hole density corresponding to STH level projected onto (100) plane. (c) Band structure for $3\times 3\times 3$ supercell showing the STH level in the band gap (LDA + $U$). $\mathbf{k}$-point labels are equivalent to the primitive cell.

Figure 3

Variation in total energy with excess charge calculated using the HSE06 hybrid functional as well as several LDA + $U$ models. Ionic positions are fixed at the relaxed STH configuration. Both HSE06 and $U=7\mathrm{eV}$ exhibit only small deviations from linear dependence.

Figure 4

(a) Convergence of formation energy of STH in ${\mathrm{SrTiO}}_3$ computed via Eqs. (5) and (4) with $\mathbf{k}$-point grid (LDA + $U$ with $U\left(\mathrm{O}2p\right)=8\mathrm{eV}$). The difference between the open and closed squares illustrates the effect of the potential alignment correction term $q\Delta v_{\mathrm{PA}}$. (b) Electrostatic potential shift averaged around ionic cores between neutral ideal and charged STH supercells as a function of the distance to the oxygen ion at which the STH is centered.

Figure 5

(a) Band lineup for ${\mathrm{BaTiO}}_3$, ${\mathrm{PbTiO}}_3$, and ${\mathrm{SrTiO}}_3$ according to different XC functionals. [(b)–(d)] Band structures and (e) densities of states for ${\mathrm{SrTiO}}_3$, ${\mathrm{PbTiO}}_3$, and ${\mathrm{BaTiO}}_3$ according to LDA + $U$ calculations with scissor corrections. The significantly higher VBM in ${\mathrm{PbTiO}}_3$ originates from the coupling between O $2p$ and Pb $6s$ states. The latter appear just below the O $2p$ derived topmost valence band and are visible in both panels (c) and (e) between approximately $-4$ and $-8\mathrm{eV}$. The bands in panels (b)–(d) have been colored according to the dominant character of the entire band manifold to indicate the assignments qualitatively rather than quantitatively. The arrow in panel (e) indicates the position of the VBM in ${\mathrm{PbTiO}}_3$.

Figure 6

(a) STH formation energies and (b)–(d) maximum atomic displacements per atom type for cubic ${\mathrm{SrTiO}}_3$, ${\mathrm{PbTiO}}_3$, and ${\mathrm{BaTiO}}_3$ as a function of lattice parameter. Filled symbols indicate data corresponding to the equilibrium lattice constant.