Point defects, impurities, and small hole polarons in ${\mathrm{GdTiO}}_3$

The electronic structure of native defects and impurities in ${\mathrm{GdTiO}}_3$, a rare-earth titanate Mott insulator, is studied using density functional theory with a hybrid functional. Among native defects, the cation vacancies have the lowest formation energies in oxygen-rich conditions and oxygen vacancies have the lowest formation energy in oxygen-poor conditions. Among the impurities, ${\mathrm{Sr}}_{\mathrm{Gd}},{\mathrm{H}}_i$, and ${\mathrm{C}}_{\mathrm{O}}$ have low formation energies. A common feature of the native defects and impurities is that they lead to the formation of small hole polarons, which explains the frequent observation of $p$-type hopping conductivity in the rare-earth titanates. These small hole polarons also lead to optical absorption and act as electron traps in devices.

Figure 1

Schematic band structure of ${\mathrm{GdTiO}}_3$. The zero of the Fermi level is referenced to the top of the lower Hubbard band (LHB). The gap is between the lower and upper Hubbard band (UHB).

Figure 2

Allowed values of O and Ti chemical potentials (gray shaded region) defining the stability of ${\mathrm{GdTiO}}_3$. The chemical potentials ${\mu }_{\mathrm{Ti}},{\mu }_{\mathrm{O}}$, and ${\mu }_{\mathrm{Gd}}$ are limited by the formation of secondary phases ${\mathrm{TiO}}_2$ (rutile), ${\mathrm{Gd}}_2{\mathrm{O}}_3$, and ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. The filled black circles correspond to ${\mu }_{\mathrm{O}}=-3.61$ eV and ${\mu }_{\mathrm{O}}=-5.25$ eV, spanning the range of possible values of ${\mu }_{\mathrm{O}}$.

Figure 3

Formation energies as a function of Fermi level for native defects in GTO under (a) oxygen-rich and (b) oxygen-poor conditions. The slopes of the lines indicate the charge state of the defect, and the kinks in the lines correspond to the position of the charge-state transition levels in the gap [Eq. (2)]. The dotted lines indicate charge states corresponding to hole polarons bound to the defect center.

Figure 4

Charge density for the Ti-Ti bonding state in an oxygen vacancy $\left(V_{\mathrm{O}}\right)$, with isosurface set to 10% of the maximum.

Figure 5

Formation of a single small hole polaron for $V_{\mathrm{Gd}}^{-2}$. The charge-density isosurface illustrating the wave function of the polaron state is set to 10% of the maximum value.

Figure 6

Formation energies as a function of Fermi level for impurities in GTO under (a) oxygen-rich and (b) oxygen-poor conditions. The dotted lines indicate charge states corresponding to hole polarons bound to the impurity.

Figure 7

Charge density of the small hole polaron states for ${\mathrm{C}}_{\mathrm{O}}^0$, with the isosurface set to 10% of the maximum value.

Figure 8

Configuration coordinate diagrams (as calculated in Ref. [29]) for (a) the optical excitation of a hole from a localized to a delocalized state and (b) the recombination of an electron with a localized hole (small polaron). $E_{\mathrm{a}}$ is the absorption energy, $E_{\mathrm{G}}$ is the band-gap energy, $E_{\mathrm{ST}}$ the polaron self-trapping energy, $E_{\mathrm{e}}$ the optical emission energy, and $E_{\mathrm{S}}$ is the lattice energy cost (strain energy).

Figure 9

Band alignment between STO and GTO, with positions of charge-state transition levels for native defects and impurities shown within the GTO gap. The zero of energy is set to the top of the GTO valence band (LHB), and the conduction-band minimum (CBM) of STO is indicated.

Figure 10

Thermodynamic transition levels and charge-state switching levels for trapping/detrapping of an electron, for (a) the (+1/0) transition for a polaron in bulk GTO and (b) the $(0/-1)$ transition for ${\mathrm{C}}_{\mathrm{O}}$. The arrows indicate a transition in which the atomic configuration is kept fixed to that of the initial state (unlike the thermodynamic transition levels, for which the atomic configuration of the final state is relaxed). $E_{\mathrm{S}(1,2)}$ are the relaxation energies between the two charge states.