Origin of the tetragonal-to-hexagonal phase transitions in Fe-doped BaTiO$_3$

Abstract

Based on detailed first-principles calculations, we investigate the tetragonal-to-hexagonal phase transition in Fe-doped BaTiO$_3$. Total energy calculations confirm a crossover from the tetragonal to hexagonal phases around 4\% Fe, in agreement with experimental observations, where comparative calculations show that neither CaTiO$_3$ nor SrTiO$_3$ exhibits similar behavior under equivalent substitution. Furthermore, three possible mechanisms are quantified: oxygen vacancies shift the crossover concentration from $\sim$4\% to $\sim$2\% through charge compensation, Jahn-Teller distortions impose a larger elastic penalty, both favoring tetragonal-to-hexagonal phase transitions; whereas the tolerance factor is reduced in comparison with that of pristine BaTiO$_3$ for reasonable Fe valence states, disfavoring the occurrence of the hexagonal phases. Detailed analysis on the electronic structure reveals that the charge redistribution induced by oxygen vacancy is strongly orbital dependent due to the local crystal structure distortions.

Figures (5)

• Figure 1: Concentration-dependent total energy analysis. (a) Stoichiometric BaTiO$_3$: The inset shows the relative stability ($E_{\text{tetra}} - E_{\text{hexa}}$), with a crossover at $\sim 4\%$. (b) CaTiO$_3$ and (c) SrTiO$_3$: Neither host stabilizes the hexagonal phase, acting as control groups. (d) BaTiO$_3$ with oxygen vacancies ($V_{\text{O}}$): The presence of $V_{\text{O}}$ shifts the hexagonal crossover from $\sim 4\%$ down to $\sim 2\%$, favoring the hexagonal phase significantly.
• Figure 2: Analysis of Jahn-Teller (JT) distortions as a driving force for the phase transition. (a) JT stabilization energy per Fe from constrained geometry. (b) Concentration dependence of JT distortion index ($\lambda$).
• Figure 3: Unfolded band structures and Fe $3d$ projected density of states (PDOS) for 2% Fe-substituted tetragonal BaTiO$_3$ (a) without and (b) with a compensating oxygen vacancy ($V_\text{O}$). Black lines indicate the band structure of pristine tetragonal BaTiO$_3$ for reference. The PDOS panels show the Fe $3d$ states (red). Horizontal solid and dashed lines indicate the Fermi level, which is set to zero. The PDOS is plotted in units of states/eV/u.c. with a range of 0--40, where u.c. refers to the 270-atom supercell.
• Figure 4: Charge-density difference (CDD) isosurfaces ($\pm 0.02~e/\text{\AA}^3$) under frozen-ion conditions. (a) Effect of Fe substitution: $\Delta\rho = \rho(\text{Fe}_\text{Ti}) - \rho(\text{pristine})$. (b) Effect of oxygen vacancy: $\Delta\rho = \rho(\text{Fe}_\text{Ti}{+}V_\text{O}) - \rho(\text{Fe}_\text{Ti})$. Yellow and cyan isosurfaces indicate electron accumulation and depletion, respectively. Atom colors: Ba (green), Ti (blue), Fe (brown), O (red). The vacancy site is circled in (b).
• Figure 5: Spin-resolved Fe $3d$ projected density of states for (a) tBTO 2% and (b) tBTO 2% $+ V_\text{O}$ (relaxed structures). Solid and dashed lines denote spin-up and spin-down channels, respectively. The Fermi level is set to zero.