Doping-induced phase transitions in ferroelectric BaTiO${}_3$ from first-principles calculations

Carrier-electron-induced phase transition from tetragonal to cubic phases in BaTiO${}_3$ is studied using first-principles calculation. Our results show that the disappearance of the ferroelectric phase is an intrinsic effect resulting from carrier electron doping in BaTiO${}_3$. We further clarify that the lattice disorder induced by donor dopants such as oxygen vacancies and substitutionally doped Nb${}^{5+}$ at Ti${}^{4+}$ sites accelerates the disappearance of the tetragonal phase in BaTiO${}_3$.

Figure 1

(a) BaTiO${}_3$ tetragonal crystal structure and doped-electron distribution of 0.1 $e$/(unit cell). (b) Shifts of $c$-axis atomic positions of Ba, Ti, O${}_{\text{I}}$, and O${}_{\text{II}}$ in the BaTiO${}_3$ unit cell induced by electron doping. (c) The $c/a$ ratio and unit cell volume of BaTiO${}_3$ as a function of carrier electron doping.

Figure 2

Partial density of states of (a) Ti-3$d$ and O-2$p$ in $T$ and $C$ phases of BaTiO${}_3$ and (b) $C$ phase SrTiO${}_3$ in charge neutral condition. The lattice constant and inner-atomic coordinates of crystal structures are optimized. In both figures, O-$2p$ states are the sum of contributions from all oxygen atoms in the unit cell.

Figure 3

(a) Total energy difference $\Delta E_{\text{total}\text{(Neutral)}}$ (the deformation potential) as a function of soft-mode distortion in neutral BaTiO${}_3$. Kinetic and Coulomb ($\Delta E_{\text{kinetic}}$ and $\Delta E_{\text{Coulomb}}$, respectively) energy contributions to $\Delta E_{\text{total}\text{(Neutral)}}$. $\Delta E_{\text{total}\text{(Charged)}}$ is the result for BaTiO${}_3$ doped at 0.15 $e$/(unit cell). All values are set to zero at $\text{Ti}\text{position}=0.50$. (b) Contribution of kinetic ($\delta E_{\text{kinetic}}$) and Coulomb ($\delta E_{\text{Coulomb}}$) energy terms to the change in the deformation potential between charged and neutral states ($\delta E_{\text{total}}$).

Figure 4

Supercell models composed of $n\times n\times n$ BaTiO${}_3$ unit cells used the defect calculations. The notations bc (body center) and fc (face center) denote the defect positions within the supercells. Primitive cells are adopted in the calculations with fc and bc supercells.

Figure 5

(a) The $c/a$ ratio of Nb-doped supercells. (b) Average volume obtained by dividing supercell volume by the number of unit cells within the supercell.

Figure 6

$c/a$ ratios and unit cell volumes at local positions $A$, $B$, $C$, and $D$ in the cross-sectional view of the relaxed supercell that contains Nb${}^{5+}$. The supercell size is 444 (bc), which corresponds to a Nb${}^{5+}$ concentration of 0.031 per unit cell.

Figure 7

(a) The $c/a$ ratio and (b) average volume calculated for $V_{\text{O}}^{2+}$-doped BaTiO${}_3$ supercells.

Figure 8

$c/a$ ratios and unit cell volumes at different local positions $E$, $F$, $G$, and $H$ in the relaxed structure of 444 (bc) supercell with $V_{\text{O}}^{2+}$ type I vacancy.