Polarons, vacancies, vacancy associations, and defect states in multiferroic ${\mathrm{BiFeO}}_3$

Oxygen, bismuth, and iron vacancies are studied by density-functional theory calculations in multiferroic bismuth ferrite in oxidative and reducing external conditions. We accurately describe the localized electronic states associated with isolated defects. Oxygen-rich conditions provide fully ionized oxygen vacancies, and ionized or partially ionized cationic vacancies, possibly yielding $p$-type conductivity mediated by oxygen-type holes. Oxygen-poor conditions provide fully ionized vacancies and a much smaller concentration of electronic carriers. Cationic and oxygen vacancies tend to pair (Bi-O, Fe-O) with an associated electric dipole being as close as possible to the polar axis, possibly leading to imprint effects. This clustering also results in an extended stability domain for the ionized defects. Furthermore, O and Fe vacancies are respectively associated with Fe-$3d$ and O-$2p$ type moderately deep defect states spatially localized close to the defect. The charge-neutral Fe vacancy possesses an electric dipole, despite being a point defect. Bi vacancies may be found in several metastable electronic states, with in the most stable one, a mid-gap band of defect states with Fe-$3d$ character, in addition to the three O-$2p$ defect states associated to the acceptor nature of this defect. Finally, electrons released in the lattice by oxygen vacancies, if not captured by acceptor defects, tend to localize as small polarons, while holes released by cationic vacancies, if not captured by donor defects, are associated with delocalized Bloch-like band states.

Figure 1

Electronic density of states of BFO projected on Fe $d$ and O $p$ states, for one of the two spin channels, as obtained by LDA+$U$ ($U=3.87$ eV on Fe $d$ orbitals). The two Fe atoms are antiferromagnetically coupled. The Fermi energy is set at 0 eV.

Figure 2

Formation energies of the isolated vacancies calculated with LDA+$U$, in oxygen-rich conditions (upper panels, $\mathrm{\Delta }{\mu }_O=-0.5$ eV) and oxygen-poor conditions (lower panels, $\mathrm{\Delta }{\mu }_O=-2.0$ eV). Dashed brown vertical line: LDA+$U$ band gap.

Figure 3

(a) The four subsets of oxygen atoms around the Bi atom in $R3c$ BFO. The Bi-O divacancy is simulated in four configurations, each one corresponding to the removal of one oxygen in a given subset. (b) The two subsets of oxygen atoms around the Fe atom in $R3c$ BFO. The Fe-O divacancy is simulated in two configurations, each one corresponding to the removal of one oxygen in a given subset. Bi, Fe, and O atoms are respectively in grey, yellow, and red.

Figure 4

Formation energies of the ${\left[\text{Bi-O}\right]}_1$ and ${\left[\text{Bi-O}\right]}_2$ divacancies, in oxygen-rich (left) and oxygen-poor (right) conditions, for charge states from $q=-3$ to +2. Dashed brown vertical line: LDA+$U$ band gap.

Figure 5

Formation energies of the ${\left[\text{Fe-O}\right]}_1$ and ${\left[\text{Fe-O}\right]}_2$ divacancies, in oxygen-rich (left) and oxygen-poor (right) conditions, for charge states from $q=-3$ to +2. Dashed brown vertical line: LDA+$U$ band gap.

Figure 6

Association energies of the Bi-O vacancy pair (top) and Fe-O vacancy pair (bottom), as a function of the total charge of the bidefect, and for the different configurations studied.

Figure 7

e-DOS for the three charge states of the O vacancy, as obtained from GGA and LDA+$U$. The zero of energy is placed approximately at mid-distance between the highest occupied and the lowest unoccupied state (dashed green vertical line). The arrows indicate where the defect states lie. The neutral O vacancy (double donor) generates two occupied states with opposite spin lying approximately in the middle of the band gap. Upon charging positively the vacancy, these states become unoccupied and their energy raises up to the bottom of the conduction band. Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$. Small violet (respectively, magenta) arrows: occupied (respectively, unoccupied) KS defect states.

Figure 8

Isosurfaces of the density of probability of the two defect states associated to the charge-neutral oxygen vacancy (LDA+$U$). The position of the O vacancy is shown with a red circle. The two states are occupied. The straight grey lines are the edges of the 80-atom supercell.

Figure 9

e-DOS for the four charge states of the Fe vacancy. The zero of energy is placed approximately at mid-distance between the highest occupied and the lowest unoccupied state (dashed green vertical line). The arrows indicate where the defect states lie. The charge-neutral Fe vacancy (triple acceptor) generates three empty states of same spin lying in the band gap. Upon charging negatively the vacancy, these states become occupied and their energy decreases down to the top of the valence band. Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$. Small violet (respectively, magenta) arrows: occupied (respectively, unoccupied) KS defect states.

Figure 10

Isosurface of the density of probability of the lowest-energy defect state associated to the iron vacancy (dn1 on Fig. 9), in the neutral case ($q=0$). The two other defect states have similar density of probability, localized on the same three oxygen atoms and the same $2p$ orbitals. The position of the Fe vacancy is shown with a green circle. This defect state (here empty) may be seen as associated to an oxygen-type hole localized on three O atoms first-neighbor of the missing Fe (this is also the case for the two other defect states). The three O atoms are the ones made the closest from Fe by the FE distortion [see Fig. 3]. The defect carries therefore as its own an electric dipole, schematically shown by a green arrow. The straight grey lines are the edges of the 80-atom supercell.

Figure 11

e-DOS of the 80-atom BFO supercell with one Bi vacancy, as obtained in LDA+$U$. Left (respectively, right) Charge state $1-$ (respectively, $2-$). (Top) The e-DOS for the configuration optimized with the electron gas in the most stable state that we found. (Bottom) The e-DOS for the configuration optimized with the electron gas in a metastable state. The green dashed line separates occupied from unoccupied states. Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$.

Figure 12

e-DOS for the four charge states of the Bi vacancy. The zero of energy is placed approximately at mid distance between the highest occupied and the lowest unoccupied state (dashed green vertical line). The arrows indicate where the defect states lie. Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$. Small violet (respectively, magenta) arrows: occupied (respectively, unoccupied) KS defect states. Small orange arrows: unoccupied Fe-$3d$ KS states in the band gap.

Figure 13

e-DOS for (a) the partially ionized Bi vacancy ($q=-1$), and (b) the fully ionized Bi vacancy ($q=-3$). Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$. An isosurface of the associated defect states is shown in both cases. The zero of energy is set approximately at mid-distance between the highest-occupied and the lowest-unoccupied KS levels. The straight grey lines are the edges of the 80-atom supercell. Note that, due to periodic boundary conditions, periodic images of the Bi vacancy lie at all the corners of the supercell.

Figure 14

(Effective) elastic dipole tensor components (eV) vs the inverse of the number of atoms $N_{\mathrm{at}}$ in the perfect supercell, for the charge-neutral Bi, Fe, and O vacancies. $z$ corresponds to the polar axis of the crystal ([111] direction in terms of pseudocubic axis).

Figure 15

(a) Isosurface of the KS state occupied by the self-trapped electron polaron in BFO (as obtained from LDA+$U$ with $U=3.87$ eV). The electron is localized on a single Fe atom (which is formally reduced to the ${\mathrm{Fe}}^{2+}$ state), and occupies a state with marked Fe $3d$ character. The figure also displays the atomic geometry around it, with the self-trapping distortion visible (the distance of Fe to nearest oxygens is enhanced to 2.00 and 2.13 Å vs 1.94 and 2.07 Å in bulk). The blue arrow indicates the direction of polarization. (b) e-DOS of the BFO supercell with the self-trapped electron polaron ($U=3.87$ eV). The KS state occupied by the electron polaron is spotted with a violet arrow.

Figure 16

Hole polaron self-trapping energy (eV) as a function of Hubbard parameter $U_p$ applied on oxygen $p$ orbitals (with $U=3.87$ eV on Fe $d$ orbitals). The arrow indicates the value of $U_p$ for which the energy as a function of fractional polaronic charge is close to piecewise linearity (see Appendix pp3). (Inset) Part of the hole charge which is localized on a single oxygen atom.

Figure 17

e-DOS for the charge-neutral Bi vacancy as obtained with LDA+$U+U_p$, i.e., Hubbard term applied on oxygen $p$ ($U_p=8.5$ eV), and on Fe $d$ ($U=3.87$ eV). The zero of energy (0 eV) is placed approximately at mid-distance between the highest occupied and the lowest unoccupied KS state. Positive/red curves: spin $\uparrow$, negative/black curves: spin $\downarrow$. An isosurface of the density of probability of the KS states associated with the defect (in the band gap) is shown. The picture in the top left corner schematically represents the six oxygen atoms and the six $2p$ orbitals that receive the holes released by the Bi vacancy (which is located one unit cell further along the [111] direction). The straight grey lines are the edges of the 80-atom supercell.

Figure 18

Schematic evolution of the KS quantum eigenstates associated with the defect in the case of the O, Fe, and Bi vacancies, from the neutral to the completely ionized defect.

Figure 19

e-DOS obtained for the O and Bi vacancies in a $3\times 3\times 3$ supercell.

Figure 20

Energy change with respect to piecewise linearity as a function of the fractional number of electrons in a supercell containing a self-trapped electronic polaron (left) or a self-trapped hole polaron (right) in BFO. In the first case, different $U$ are applied on Fe $d$ orbitals (within LDA+$U$), without $U_p$ applied on oxygen $p$. In the second case, $U=3.87$ eV is fixed on Fe $d$ orbitals, and different $U_p$ are applied on oxygen $p$ orbitals (LDA+$U+U_p$).