DFT+$U$ study of self-trapping, trapping, and mobility of oxygen-type hole polarons in barium stannate

The charge-transfer insulating perovskite oxides currently used as fuel cell electrolytes undergo, at high temperature, an oxidation reaction $\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)+{\mathrm{V}}_{\mathrm{O}}^{••}\to {\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}+2{\mathrm{h}}^{•}$, that produces oxygen-type holes. Understanding the nature and mobility of these oxygen-type holes is an important step to improve the performance of devices, but presents a theoretical challenge since, in their localized form, they cannot be captured by standard density functional theory. Here, we employ the DFT+$U$ formalism with a Hubbard correction on the $p$ orbitals of oxygen to investigate several properties of these holes, in the particular case of ${\mathrm{BaSnO}}_3$. We describe the small oxygen-type hole polarons, the self-trapping at their origin, and their trapping by trivalent dopants (Ga, Sc, In, Lu, Y, Gd, La). Strong similarities with protonic defects are observed concerning the evolution of the trapping energy with ionic radius of the dopant. Moreover, we show that long-range diffusion of holes is a complex phenomenon, that proceeds by a succession of several mechanisms. However, the standard implementation of DFT+$U$ within the projector augmented-wave (PAW) formalism leads to use very large, unphysical values of $U$ for the O-$p$ orbital. We propose here a slightly modified DFT+$U$ scheme, that takes into account the fact that the O-$p$ is truncated in usual DFT+$U$ implementation in PAW. This scheme yields more physical values of $U$ than the ones traditionally used in the literature, and describes well the properties of the hole polaron.

Figure 1

Comparison of the radial part of an atomic orbital ${\varphi }_0$, a truncated atomic orbital ${\varphi }_0^t$, and a renormalized atomic orbital $\overline{{\varphi }_0^t}$ for an oxygen-$p$ orbital.

Figure 2

Energy change with respect to piecewise linearity, as a function of fractional charge (number of electrons) in the polaron, and DFT+$U$ contribution to the total energy, due to the orbital on which the hole is located, namely, $U/2(n_i-n_i^2)$. The linearity is subtracted so that the concavity or convexity appears clearly. Left panels: standard DFT+$U$ scheme $\left({\mathrm{o}}_{\mathrm{A}}\right)$, with $U=6$, 7, and 8 eV; right panels: scheme ${\mathrm{o}}_{\mathrm{C}}$ $(U=4$, 5, 6, and 7 eV). The atomic geometry (positions, lattice constant) is fixed to that of the STHP at $U=6$ eV (with scheme ${\mathrm{o}}_{\mathrm{A}})$.

Figure 3

The oxygen-type hole polaron, in its self-trapped state (a), (b), bound to the yttrium dopant in first-neighbor (c) and second-neighbor (d) position, and to a Ga dopant, in first-neighbor (e) and second-neighbor (f) position. Panels (a), (c), (d), (e), and (f) are isosurfaces of the hole state, while (b) illustrates the lattice distortions associated to the self-trapped hole. Panels (a) and (b) show the self-trapped hole polaron in the lattice (the hole occupies a $2p_x$ orbital), with a self-trapping distortion confined to the $\left(x,z\right)$ plane. The insets in (a) and (b) show the quasiabsence of distortion in the $\left(y,z\right)$ plane, and thus the anisotropy of the self-trapping distortion. The orange arrows schematically depict the atomic distortions, and the numbers correspond to the variations of the distance between the atom and ${\mathrm{O}}^{-}$. In second neighbor of the dopant [(d), (f)], the hole has a nonzero probability to be found on the oxygen first neighbor. Figure 3 has been done with the xcrysden software [45].

Figure 4

The three configurations of the hole polaron close to the trivalent dopant. Gray, green, and red circles are, respectively, Sn, dopant, and O atoms.

Figure 5

Dopant-hole interaction energy as a function of ionic radius of the dopant, as obtained with the two DFT+$U$ schemes used in this work. Note that with scheme ${\mathrm{o}}_{\mathrm{C}}$, we could not obtain the hole polaron as stable in 2nd neighbor position from the Ga dopant (the configuration evolved to a polaron localized as 1st neighbor).

Figure 6

The four possible motions of the self-trapped hole polaron studied in this work: (a) simple hopping, from the $2p_y$ orbital of an O atom onto the $2p_x$ orbital of its first-neighbor oxygen; (b) re- orientation from the $2p_y$ orbital onto the $2p_x$ orbital of the same O atom; (c) parallel hopping, from the $2p_z$ orbital of an O atom onto the $2p_z$ orbital of its first-neighbor oxygen; (d) rotating hopping, from the $2p_y$ orbital of an O atom onto the $2p_z$ orbital of its first-neighbor oxygen. Gray and red circles are, respectively, Sn and O atoms.

Figure 7

Simple hopping of the self-trapped hole polaron. (a) Energy along the path (black: linear interpolation, red: minimum energy path from string method calculation). (b) Magnetic moments $\left({\mu }_B\right)$ of the initial $\left(m_i\right)$ and final $\left(m_f\right)$ oxygen atoms along the minimum energy path. Dashed lines: Gaussian fit.

Figure 8

Reorientation of the hole polaron on a O atom, from a $2p_y$ to a $2p_x$ orbital. (a) Energy along the minimum energy path (black: hole on $2p_y$, red: hole on $2p_x)$. (b) Occupation matrix components of the $p$ orbitals on the O atom along the minimum energy path. Purple: $yy$ component, orange: $xx$ component.

Figure 9

Parallel hopping of the hole polaron, from a $2p_z$ to a $2p_z$ orbital. (a) Energy along the minimum energy path; (b) magnetic moments $\left({\mu }_B\right)$ of the initial $\left(m_i\right)$ and final $\left(m_f\right)$ oxygen atoms along the minimum energy path.

Figure 10

Rotating hopping of the self-trapped hole polaron. (a) Energy along the MEP; (b) magnetic moments $\left({\mu }_B\right)$ of the initial $\left(m_i\right)$ and final $\left(m_f\right)$ oxygen atoms along the minimum energy path; (c) diagonal components of the $p$ orbital occupation matrix on the final atom [see Fig. 6].

Figure 11

Schematic representation of a small polaron in a crystal. Upper panels: schematic view of the potential felt by the excess charge; lower panels: application to the oxygen-type hole polaron in a perovskite oxide (only one plane of the ${\mathrm{SnO}}_2$ sublattice is shown). (a), (b) Undistorted lattice, all the O sites (red circles) are equivalent, the hole is delocalized. (c), (d) Self-trapped configuration: the hole polaron is localized on the yellow O atom. It repels the two nearest-neighbor Sn atoms (purple arrows), which contribute to stabilize its site. (e), (f) Coincidence configuration for hopping: the hole has the same energy whether it is placed on a yellow O or on the other.

Figure 12

Energy change with respect to piecewise linearity, as a function of fractional charge (number of electrons) in the polaron, using an oxygen atomic data set with a radius of 1.4 a.u. The linearity is subtracted so that the concavity or convexity appears clearly. Left panel: standard DFT+$U$ scheme $\left({\mathrm{o}}_A\right)$, with $U=7$, 8, 9, and 10 eV; right panel: scheme ${\mathrm{o}}_{\mathrm{C}}$ $(U=4$, 5, 6, and 7 eV). The atomic geometry (positions, lattice constant) is the same as in Fig. 2.

Figure 13

Hole self-trapping energy (eV) as a function of $U$ (eV), in the two DFT+$U$ schemes, and using two oxygen atomic data sets with two different radii (1.6 and 1.4 a.u.). The arrow indicates, for each scheme, the approximate best value of $U$ as dictated by piecewise linearity of the energy.