^{1,a)}and Karsten Albe

^{1}

### Abstract

The thermodynamic and kinetic properties of mono- and di-vacancy defects in cubic (para-electric) bariumtitanate are studied by means of density-functional theory calculations. It is determined which vacancy types prevail for given thermodynamic boundary conditions. The calculations confirm the established picture that vacancies occur in their nominal charge states almost over the entire band gap. For the dominating range of the band gap the di-vacancy binding energies are constant and negative. The system, therefore, strives to achieve a state in which, under metal-rich (oxygen-rich) conditions, all metal (oxygen) vacancies are bound in di-vacancy clusters. The migration barriers are calculated for mono-vacancies in different charge states. As oxygen vacancies are found to readily migrate at typical growth temperatures, di-vacancies can be formed at ease. The key results of the present study with respect to the thermodynamic behavior of mono- and di-vacancies influence the initial defect distribution in the ferroelectric phases and therefore the conditions for aging.

This project was funded by the *Sonderforschungsbereich 595* “Fatigue in functional materials” of the Deutsche Forschungsgemeinschaft.

I. INTRODUCTION

II. METHODOLOGY

A. Computational setup

B. Defect calculations

1. Formation energies

2. Transition energies

3. Migration energies

III. RESULTS

A. Band structure

B. Chemical potentials and stability diagram

C. Defect formation energies

D. Migration energies

IV. DISCUSSION

V. CONCLUSIONS

## Figures

Stability diagram for cubic barium titanate as determined from density-functional theory calculations. The area confined between points A, B, C, and D is the chemical stability range of . The line through points C and D corresponds to maximally oxygen-rich conditions and an oxygen chemical potential of . Along lines parallel to C–D the oxygen chemical potential is constant. The most negative value of is obtained in the upper right corner of the diagram.

Stability diagram for cubic barium titanate as determined from density-functional theory calculations. The area confined between points A, B, C, and D is the chemical stability range of . The line through points C and D corresponds to maximally oxygen-rich conditions and an oxygen chemical potential of . Along lines parallel to C–D the oxygen chemical potential is constant. The most negative value of is obtained in the upper right corner of the diagram.

(Color online) Variation of defect formation energies with Fermi level for representative thermodynamic conditions indicated in Fig. 1. The numbers indicate the charge states. Parallel lines correspond to identical charge states. The solid and dashed lines correspond to mono- and di-vacancies, respectively. The arrows indicate the position of the Fermi level pinning energy under different conditions.

(Color online) Variation of defect formation energies with Fermi level for representative thermodynamic conditions indicated in Fig. 1. The numbers indicate the charge states. Parallel lines correspond to identical charge states. The solid and dashed lines correspond to mono- and di-vacancies, respectively. The arrows indicate the position of the Fermi level pinning energy under different conditions.

Transition levels for mono- and di-vacancies in . Only the band edges are shown. The dashed transition levels are positioned inside the valence or the conduction bands (indicated by the light gray shaded areas) and are only included for illustration. The dark gray shaded areas indicate the sum of the extrapolation errors for each transition.

Transition levels for mono- and di-vacancies in . Only the band edges are shown. The dashed transition levels are positioned inside the valence or the conduction bands (indicated by the light gray shaded areas) and are only included for illustration. The dark gray shaded areas indicate the sum of the extrapolation errors for each transition.

(Color online) Binding energies for and di-vacancies as a function of Fermi level. The kinks correspond to charge transition points of the isolated defects (compare Fig. 2 and Fig. 3).

(Color online) Binding energies for and di-vacancies as a function of Fermi level. The kinks correspond to charge transition points of the isolated defects (compare Fig. 2 and Fig. 3).

## Tables

Bulk properties of cubic barium titanate as obtained from experiment and first-principles calculations. U.S.-PP: ultrasoft pseudopotentials; FP-LAPW: full potential-linearized augmented plane waves; TB-LMTO: tight-binding linear muffin-tin orbitals; ASA: atomic sphere approximation; LDA: local-density approximation; GGA: generalized-gradient approximation; PBE: Perdew-Burke-Ernzerhof parameterization of the GGA; : cohesive energy ; : lattice constant ; : equilibrium volume ; , : bulk modulus (GPa) and its pressure derivative; : direct band gap at -point (eV); : indirect band gap measured between points and ; , : effective electron (hole) mass at the -point along in units of the electron mass.

Bulk properties of cubic barium titanate as obtained from experiment and first-principles calculations. U.S.-PP: ultrasoft pseudopotentials; FP-LAPW: full potential-linearized augmented plane waves; TB-LMTO: tight-binding linear muffin-tin orbitals; ASA: atomic sphere approximation; LDA: local-density approximation; GGA: generalized-gradient approximation; PBE: Perdew-Burke-Ernzerhof parameterization of the GGA; : cohesive energy ; : lattice constant ; : equilibrium volume ; , : bulk modulus (GPa) and its pressure derivative; : direct band gap at -point (eV); : indirect band gap measured between points and ; , : effective electron (hole) mass at the -point along in units of the electron mass.

Bulk properties of Ba, Ti and O and their compounds in their respective ground-states. Experimental data from Refs. 64–66. : cohesive energy ; : axial ratio; : dimer bond length ; : enthalpy of formation ; other symbols as in Table I.

Bulk properties of Ba, Ti and O and their compounds in their respective ground-states. Experimental data from Refs. 64–66. : cohesive energy ; : axial ratio; : dimer bond length ; : enthalpy of formation ; other symbols as in Table I.

Formation energies of mono- and di-vacancies under the chemical conditions indicated in Fig. 1. Note that if and are given is uniquely determined by Eq. (4). The charge state, , of the defect which determines the Fermi level dependence of the formation energies via Eq. (1) is given in the second column. The number of electrons occupying conduction band states and holes occupying valence band states are relevant for the band gap correction via Eq. (2) and are given in the third column where positive and negative values indicate and , respectively. All energies are given in units of eV. The finite-size scaling extrapolation error is given in the last column.

Formation energies of mono- and di-vacancies under the chemical conditions indicated in Fig. 1. Note that if and are given is uniquely determined by Eq. (4). The charge state, , of the defect which determines the Fermi level dependence of the formation energies via Eq. (1) is given in the second column. The number of electrons occupying conduction band states and holes occupying valence band states are relevant for the band gap correction via Eq. (2) and are given in the third column where positive and negative values indicate and , respectively. All energies are given in units of eV. The finite-size scaling extrapolation error is given in the last column.

Calculated migration energies of mono-vacancies in units of electron volts. The temperature ranges above which the defects become mobile are given in the last column. The negative charge states of the titanium vacancy were not considered as already the neutral charge state displays a huge barrier and, following the trends for the barium and oxygen vacancies, the addition of electrons can only be expected to further increase this value.

Calculated migration energies of mono-vacancies in units of electron volts. The temperature ranges above which the defects become mobile are given in the last column. The negative charge states of the titanium vacancy were not considered as already the neutral charge state displays a huge barrier and, following the trends for the barium and oxygen vacancies, the addition of electrons can only be expected to further increase this value.

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