Oxygen vacancies in strontium titanate: A $\mathrm{DFT}+\mathrm{DMFT}$ study

We address the long-standing question of the nature of oxygen vacancies in strontium titanate, using a combination of density functional theory and dynamical mean-field theory ($\mathrm{DFT}+\mathrm{DMFT}$) to investigate in particular the effect of vacancy-site correlations on the electronic properties. Our approach uses a minimal low-energy electronic subspace including the $\mathrm{Ti}-t_{2g}$ orbitals plus an additional vacancy-centered Wannier function, and it provides an intuitive and physically transparent framework to study the effect of the local electron-electron interactions on the excess charge introduced by the oxygen vacancies. We estimate the strength of the screened interaction parameters using the constrained random phase approximation, and we find a sizable Hubbard $U$ parameter for the vacancy orbital. Our main finding, which reconciles previous experimental and computational results, is that the ground state is either a state with double occupation of the localized defect state or a state with a singly occupied vacancy and one electron transferred to the conduction band. The balance between these two competing states is determined by the strength of the interaction both on the vacancy and the Ti sites, and on the Ti-Ti distance across the vacancy. Finally, we contrast the case of vacancy doping in $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ with doping via La substitution, and we show that the latter is well described by a simple rigid-band picture.

Figure 1

Supercells of the calculated structures for (a) $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ (20-atom unit cell, $Pm\overline{3}m$ symmetry) and (b) $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3-\delta }$ (19-atom unit cell, $P4/mmm$ symmetry). (c) Geometry of the relaxed structure containing an oxygen vacancy (indicated with the orange circle) with the Ti-Ti distances given in Å. For comparison, the calculated Ti-Ti distance for stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ is 3.862 Å. Note that the orientation of part (c) is rotated relative to that of parts (a) and (b).

Figure 2

Calculated DFT band structures for (a) stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ and (b) $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{2.75}$. DFT and MLWF bands are shown as gray and blue solid lines, respectively; the vacancy band in (b) is highlighted in orange (see the main text). (c) MLWF-projected density of states (DOS) for the $t_{2g}$-like and ${\mathrm{O}}_{\mathrm{V}}$ MLWFs. Part (d) shows the real-space representation of a $d_{xz}$-type MLWF in the stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ system, while (e) corresponds to the ${\mathrm{O}}_{\mathrm{V}}$ MLWF.

Figure 3

$\mathrm{DFT}+\mathrm{DMFT}$ results for different settings of U(${\mathrm{O}}_{\mathrm{V}}$)/U(Ti), from 0 (a) to 1 (e). The left column in each row shows the site occupations for the three types of correlated sites: Ti-next, Ti-farther, and ${\mathrm{O}}_{\mathrm{V}}$ (lighter blue, darker blue and orange, respectively). The second to left column shows the corresponding quasiparticle weights $Z$. The following three plots in each row show the spectral functions for the three aforementioned sites for three different $U\left(\mathrm{Ti}\right)$ values.

Figure 4

Averaged values for the partially screened interaction parameters $U\left(\mathrm{Ti}\right)$ and $U\left({\mathrm{O}}_{\mathrm{V}}\right)$ obtained within cRPA. Blue dots represent the values for the Ti sites, while the orange star corresponds to the ${\mathrm{O}}_{\mathrm{V}}$ site. Displayed on the horizontal axis are the different schemes for choosing the target subspace (see the main text for more details).

Figure 5

Total energy as a function of the Ti-${\mathrm{O}}_{\mathrm{V}}$-Ti distance (in units of the Ti-O-Ti distance in stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$) obtained within $\mathrm{LDA}+\mathcal{U}$ for three different choices of the Hubbard $\mathcal{U}$ parameter. The vertical line highlights the distance in cubic stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$

Figure 6

Evolution of the $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3-\delta }$ band structure with respect to the Ti-${\mathrm{O}}_{\mathrm{V}}$-Ti distance, measured in units of the Ti-O-Ti distance in stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$. The central plot corresponds to the Ti-O-Ti distance in stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$. We see that bringing the two Ti ions closer together (left side) pushes the ${\mathrm{O}}_{\mathrm{V}}$ band down in energy, favoring charge localization, while pulling the Ti atoms apart (right side) enhances the entanglement of the ${\mathrm{O}}_{\mathrm{V}}$ band with the Ti-$t_{2g}$ bands, and redistributes some of its charge onto the Ti ions.

Figure 7

$\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3-\delta }$ DMFT occupations for the three different sites as a function of the Ti-${\mathrm{O}}_{\mathrm{V}}$-Ti distance (in units of the Ti-O-Ti distance in stoichiometric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$), for three different values of $U\left({\mathrm{O}}_{\mathrm{V}}\right)/U\left(\mathrm{Ti}\right)$. Lighter blue, darker blue, and orange represent Ti-next, Ti-farther, and ${\mathrm{O}}_{\mathrm{V}}$ sites, respectively. Electron transfer from the ${\mathrm{O}}_{\mathrm{V}}$ to the Ti ions is favored by large Ti-${\mathrm{O}}_{\mathrm{V}}$-Ti distance and by large $U$(${\mathrm{O}}_{\mathrm{V}}$).

Figure 8

LDA Band structure of ${\mathrm{La}}_x{\mathrm{Sr}}_{1-x}\mathrm{Ti}{\mathrm{O}}_3$ for different values of $x$. DFT and MLWF bands are shown in gray and in color (from blue to red), respectively. While the general shape and bandwidth of the Ti-$t_{2g}$ remain basically constant for the whole series, its relative position with respect to $E_F$ moves gradually to lower energies for increasing $x$, marking the expected filling of the $t_{2g}$ bands from $d^0$ for $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ ($x=0$) to $d^1$ in $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ ($x=1$).

Figure 9

(a) Ti-$t_{2g}$ DMFT spectral functions and (b) corresponding quasiparticle weights for three different concentrations $x$ of ${\mathrm{La}}_x{\mathrm{Sr}}_{1-x}\mathrm{Ti}{\mathrm{O}}_3$ for a 40-atom unit cell and two different values of $U$. Spectral functions in (a) are shifted on the $y$-axis for clarity.