Association of oxygen vacancies with impurity metal ions in lead titanate

Thermodynamic, structural, and electronic properties of isolated copper and iron atoms as well as their complexes with oxygen vacancies in tetragonal lead titanate are investigated by means of first principles calculations. Both dopants exhibit a strong chemical driving force for the formation of $M_{\mathrm{Ti}}-V_{\mathrm{O}}$ $(M=\mathrm{Cu},\mathrm{Fe})$ defect associates. The most stable configurations correspond to a local dipole aligned along the tetragonal axis parallel to the spontaneous polarization. Local spin moments are obtained and the calculated spin densities are discussed. The calculations provide a simple and consistent explanation for the experimental findings. The results are discussed in the context of models for degradation of ferroelectric materials.

Figure 1

(Color online) Finite-size scaling of formation enthalpies to obtain formation energies in the limit of infinite dilution for three of the most important defects. The small filled and large open circles show the data before and after application of the monopole-monopole correction term.

Figure 2

(Color online) Variation of the formation energy of unbound oxygen vacancies with the Fermi level for extreme metal-rich conditions $(\Delta {\mu }_{\mathrm{Pb}}=\Delta {\mu }_{\mathrm{Ti}}=0\mathrm{eV})$. The white area corresponds to the calculated band gap. The numbers indicate the defect charge state.

Figure 3

(Color online) Fermi level dependence of formation energies of free and complexed Cu impurities. Parallel lines imply equal charge states as indicated in the plot. The white stripe indicates the (calculated) band gap while the gray stripes indicate the valence and conduction bands, respectively. Extrapolated formation energies from Table were used except for ${\mathrm{Cu}}_{\mathrm{Ti}}-V_c^{\left(4\right)}$ for which the data obtained with 80-atom cells are shown.

Figure 4

(Color online) Spin-decomposed total and partial densities of states for neutral uncomplexed Cu impurities in lead titanate. The partial densities of states for the Cu atom have been scaled by a factor of 20 and are shown separately in the upper part of the figure. The density of states for the ideal crystal is shown below. In the latter case, the density of states is independent of the spin orientation. A Gaussian filter with $\sigma =0.05\mathrm{eV}$ was applied to optimize the visualization.

Figure 5

(Color) [(a)–(d)] Isolated copper impurity in charge state $q=-1$ and [(e)–(h)] copper impurity-oxygen-vacancy complex in charge state $q=0$. [(a) and (e)] Position of atoms in the (100) plane containing the defect; numbers given in brackets give the equivalent distances in the defect-free crystal. [(b) and (f)] Total charge density; a logarithmic scale has been chosen for the charge density in order to enhance the features. [(c) and (g)] Difference between total charge density of defective cell and defect-free cell. [(d) and (h)] Spin densities.

Figure 6

(Color online) Schematic of different conceivable vacancy-impurity atom configurations. The tetragonal perovskite lattice is shown along the [100] axis and Pb atoms have been omitted for clarity. The different gray shaded areas indicate the size of the different supercells employed in the present work.

Figure 7

(Color online) Binding energies for Cu impurities in tetragonal lead titanate obtained from the extrapolated formation energies given in Tables . Negative energies imply a chemical driving force for association.

Figure 8

(Color online) Fermi level dependence of formation energies of free and complexed Fe impurities. Parallel lines imply equal charge states as indicated in the plot. The white stripe indicates the (calculated) band gap, while the gray stripes indicate the valence and conduction bands, respectively. Extrapolated formation energies from Table were used except for ${\mathrm{Fe}}_{\mathrm{Ti}}-V_c^{\left(4\right)}$ for which the data obtained with 80-atom cells are shown.

Figure 9

(Color online) Spin-decomposed total and partial densities of states for neutral uncomplexed Fe impurities in lead titanate. The partial densities of states for the $\mathrm{Fe}d$ orbitals are shown separately in the upper part of the figure. The $\mathrm{Fe}s$ and $\mathrm{Fe}p$ states have been omitted since in the energy window shown they are practically zero. A Gaussian filter with $\sigma =0.05\mathrm{eV}$ was employed to smoothen the data.

Figure 10

(Color) [(a)–(d)] Isolated iron impurity in charge state $q=0$ and [(e)–(h)] iron impurity-oxygen-vacancy complex in charge state $q=+1$. [(a) and (e)] Position of atoms in the (100) plane containing the defect; numbers given in brackets give the equivalent distances in the defect-free crystal. [(b) and (f)] Total charge density; a logarithmic scale has been chosen for the charge density in order to enhance the features. [(c) and (g)] Difference between total charge density of defective cell and defect-free cell. [(d) and (h)] Spin densities.

Figure 11

(Color online) Binding energies as defined by Eq. (2) for Fe impurities in tetragonal lead titanate obtained from the extrapolated formation energies given in Tables . Negative energies imply a chemical driving force for association.

Figure 12

(Color online) Summary of key results for the most important charge states of nearest neighbor $\mathrm{Cu}∕\mathrm{Fe}$-vacancy complexes (${[{\mathrm{Cu}}_{\mathrm{Ti}}-V_{\mathrm{O}}]}^{\times }$, ${[{\mathrm{Fe}}_{\mathrm{Ti}}-V_{\mathrm{O}}]}^{●}$). The large arrow on the far left indicates the direction of the spontaneous polarization with respect to the ideal unit cell. The short arrows indicate the direction of the defect induced dipole moments. Distances are given between the impurity atom and the position of the vacancy (as defined by the positions of the neighboring oxygen planes). Values in brackets refer to the ideal (undistorted) lattice. The lower panel visualizes the formation energy differences.