## Introduction

Light-induced functions such as photocatalytic and photovoltaic (PV) effects rely on the formation of electron (e’)-hole (h) pairs under illumination. As a representative photocatalyst, titanium dioxide1, absorbs only ultraviolet light owing to its large bandgap (Eg), several approaches for controlling the electronic structure have been used to activate a visible-light response, often with the aid of sacrificial reagents, dyes or precious metal co-catalysts: introducing donor levels into the bandgap via doping2,3, raising the valence band to a more negative level than that arising from O-2p by a composition modification4,5, and narrowing the bandgap by forming a solid solution6 or a two-dimensional structure7.

For strontium titanate, Reunchan et al.8 have demonstrated that the Fermi-level lifting by a co-doping promotes a photocatalytic response under visible-light, which is achieved by stabilizing Cr3+ in the presence of La3+. Defect levels inside the bandgap, called gap states, of Cr3+ are located above the valence band maximum (VBM), and the donor states play a crucial role in visible-light activation9. Wang et al.10 have shown that La- and Rh-codoped SrTiO3 functions as a photocatalyst, where the occupied states of Rh3+ narrows the bandgap.

Ferrophotovoltaics have attracted considerable interest because their bulk PV effect11,12,13 leads to high voltages that are beyond the bandgap limit of semiconductor p–n junctions11,12,14,15. The ferroelectric PV effect has been extensively studied for wide-bandgap oxides such as LiNbO311,16,17, BaTiO318,19,20,21,22, PbTiO3-based perovskites23,24,25, and BiFeO314,15,26. The poor absorption of visible-light is partially overcome by narrowing the bandgap27,28,29,30,31. This is realized by incorporating non-polar components into the widegap ferroelectrics, which is, in principle, accompanied by a substantial decrease in spontaneous polarization (Ps). As the intense photoresponse originates from the large magnitude of Ps13,32, these material modifications are inevitably associated with a marked suppression of photovoltage. Gap-state engineering that utilizes mid-gap states for BiFeO333 can enhance photocurrents without sacrificing photovoltages, where the photon energy of at least half of Eg are required.

Here, we report successive-redox-mediated ferrophotovoltaics, where two gap states realize e’-h pair formation over a wide photon-energy (hv) range. Our approach is based on acceptor and donor states that act as scaffolds for generating photoinduced carriers. A single transition-metal dopant with two different valence states introduces these gap states, thereby providing successive redox cycles under illumination. In principle, it can induce PV currents at small hv without being restricted by the material bandgap. Our experimental and theoretical study on a prototypical ferroelectric BaTiO3 demonstrates that the 3d orbitals of iron derive donor and acceptor levels in the Fe2+-Fe3+ coexisting state and that e’-h pairs injected by two sequential Fe3+/Fe2+ reactions deliver a robust PV response not only under visible-light but also at hv greater than Eg.

## Results

### Strategy for visible-light activation

Because of the wide Eg of BaTiO3 (3.2 to 3.3 eV)34,35, the PV effect occurs under ultraviolet light. To activate a visible-light response, we introduce electronic states derived from Fe2+ and Fe3+ into the bandgap and utilize these gap states as scaffolds for carrier generation. The most striking feature is that e’-h pairs are, in principle, formed at photon energies much smaller than that of the mid-gap-state engineering33, as described later.

In Oh symmetry (Fig. 1a, b), the electronic configurations of iron in the high-spin state are expressed as Fe3+ (d5) with t2g3(up) eg2(up) t2g0(down) eg0(down) and Fe2+ (d6) with t2g3(up) eg2(up) t2g1(down) eg0(down)36. In the BaTiO3 lattice, the states of t2g3(up) eg2(up) are located near the bottom of the valence band37, while it is probable that the states of t2g0(down) of Fe3+ (Fig. 1c) and t2g1(down) of Fe2+ (Fig. 1d) are present inside the bandgap by tuning their local structures. We, therefore, consider the following strategy for generating e’-h pairs under visible-light: Fe3+ plays the role of an electron acceptor that results in hole injection into the valence band38,39, and Fe2+ acts as an electron donor that leads to electron injection into the conduction band40,41,42. Provided that Fe3+ and Fe2+ coexist and also that their acceptor and donor states are present at moderate levels near the VBM (the top of the O-2p band) and the conduction band minimum (CBM: the bottom of the Ti-3d band), respectively, e’-h pairs can be created at hv much smaller than Eg. As displayed in Supplementary Fig. 1, the hybridization between Fe-3d and its adjacent orbitals provides the bonding states (t2g and eg) and the antibonding states (t2g* and eg*), as for the majority spin components. Hereafter, the Fe-3d derived states without an asterisk, such as t2g and dxy, are the bonding states, while those with asterisk (*), e.g., t2g* and dxy*, are the antibonding ones.

Figure 1e plots the defect concentrations at 25 °C as a function of Po2900 °C in Fe (0.3%)-doped BaTiO3. These calculations are based on the assumption that the oxygen vacancy concentration, [VO••], equilibrated in a high-temperature state at 900 °C remains unchanged at 25 °C (Po2900 °C is the oxygen partial pressure at 900 °C). The defect feature is divided into four regions. In the regions I and II, the majority of iron is trivalent, namely, the concentration of Fe3+, [Fe3+], is several orders of magnitude higher than others; the minority is Fe4+ at a higher Po2 (I) and Fe2+ at a lower Po2 (II). We note that the region III is regarded as the coexisting state of Fe3+ and Fe2+, the concentrations of which have the same order. At the higher Po2900 °C side (I, II, and III), the charge neutrality is approximated as 2[Fe2+] + [Fe3+] ≈ 2[VO••]. The region IV is semiconducting with a high-electron concentration (n), along with Fe2+ as the majority, where the charge neutrality is expressed as n ≈ 2[VO••] except near the transition Po2900 °C. According to the strategy described above, we expect that the coexisting state of Fe3+ and Fe2+ in the region III is capable of inducing a robust PV response under visible-light. We therefore adopt an Po2900 °C of 10−23 atm as an annealing condition, where the concentrations are calculated to be [Fe3+] ≈ 3.5 × 1019 cm−3 and [Fe2+] ≈ 4.3 × 1019 cm−3.

### Formation of defect associates

Figure 2 displays the DFT energy versus n of VOn••, where VOn•• denotes the oxygen vacancy on the n-th nearest-neighbor site with respect to Fe (Supplementary Fig. 2). The cells with n = 1 to 3 exhibit small energies compared with those with n > 4 regardless of the Fe valence. Considering the energy difference of 0.2 to 0.8 eV, we posit that an attractive interaction is formed between iron and VO••. If VO•• is mobile below the Curie temperature (TC), we consider that VO•• is trapped by Fe and is eventually positioned on the nearest-neighbor O1 site (Fig. 3b, c) after a certain period of time. These results suggest that VO•• exists as a defect associate of Fe3+-VO•• and/or Fe2+-VO••.

The formation of the defect associates can be explained in terms of the energy levels of the Fe-derived states. The orbital interactions provide bonding and antibonding states in both the majority (↑) and minority (↓) spin bands. In the ↑ band, the bonding states appear near the bottom of the valence band (Supplementary Fig. 3), and the antibonding states (marked with asterisk) arise near or inside the bandgap (Fig. 4 and Supplementary Fig. 4). For the Fe3+-VO1•• and Fe3+-VO4•• cells, the t2g* and eg* states are similar and their energy levels are close, while the t2g and eg states exhibit different features (Supplementary Fig. 3): these bonding states of the Fe3+-VO1•• cell are lower in energy by ≈0.5 eV than those of the Fe3+-VO4•• cell. The attractive interaction between Fe3+ and VO•• originates from these low-lying bonding states. As for Fe2+, the Fe2+-VO4•• cell (Supplementary Fig. 4c, d) has electron-occupying gap states: the Fe-3dxz state (↓) and the Fe-3$$d_{z^2}{\,\!}^*$$, -3$$d_{x^2 - y^2}{\,\!}^*$$ and -3$$d_{yz} {\,\!}^*$$ states in the ↑ band. In contrast, the Fe2+-VO1•• cell (Fig. 4e, f,) exhibits only two gap states of Fe-3$$d_{z^2}$$ (↓) and -3$$d_{x^2 - y^2} {\,\!}^*$$ (↑). The remaining ↑ states, namely, Fe-3$$d_{xz}$$, -3$$d_{yz} {\,\!}^*$$, -3$$d_{xy} {\,\!}^*$$, and -3$$d_{z^2}$$, are lower in energy and are near the bottom of the valence band. The stabilization of VO•• on the O1 site, i.e., the formation of the Fe2-VO•• associate, results from these lower-lying Fe-3d states.

As a short-range VO•• motion with a distance of a few unit cells occurs even near room temperature43,44, it is reasonable to assume that all oxygen vacancies are present as the defect associates of Fe3+-VO•• and/or Fe2+-VO•• in our samples, as reported for SrTiO345,46 and PbTiO347,48 and BaTiO349,50. Considering the charge neutrality and the strong attractive interaction between iron and VO•• (Fig. 2), we can say that the Fe2+-VO•• associate51 is the majority, i.e., the concentration of [Fe2+-VO••] is several orders or magnitude higher than that of an isolated Fe2+ (Fig. 3c). Similarly, the Fe3+-VO•• associate51,52 (Fig. 3b) and an isolated Fe3+53 are present with approximately the same concentration (Fig. 3a).

### Gap states derived from Fe-3d

Figure 4 displays the electronic structures of the Fe3+, Fe3+-VO••, and Fe2+-VO•• cells, the crystal structures of which are shown in Fig. 3. In Fig. 5, the defect levels for the Fe2+-VO1•• and Fe3+-VO1•• cells are illustrated along with the corresponding partial charges of the Fe-3d derived gaps states. As reported in the literature54,55, the valence band is mainly formed by O-2p, while the conduction band is primarily composed of Ti-3d, where an orbital hybridization between them occurs.

Here, we focus on the electronic configuration and the defect levels derived from Fe-3d. In the Fe3+ cell (Fig. 4a, b), the empty t2g (↓) state appears in the bandgap at a depth of 1.9–2.3 eV from the VBM. For the Fe3+-VO•• cell (Fig. 4c, d), four empty bonding (↓) states arise near the middle of the bandgap; at the Γ point, two are low-lying degenerate states of Fe-3dxz and -3dyz (Fig. 5e), and the remaining two are higher-lying Fe-3$$d_{z^2}$$ (Fig. 5f) and -3dxy (Fig. 5g) states. The energy difference between these gap states and the VBM is 1.8–2.1 eV. Both for the single isolated Fe3+ and the Fe3+-VO•• associate (Fig. 5d), we consider that visible-light at hv above 1.8 to 1.9 eV can pump electrons from the VBM to the empty states, namely, visible-light drives h injection from Fe3+ to the valence band. This is achieved by an electron transfer to Fe3+ from the adjacent atoms, which is accompanied by a reduction from Fe3+ to Fe2+.

Figure 4e, f displays the electronic structure of the Fe2+-VO1•• cell, and a schematic representation is shown in Fig. 5a. There are several Fe-3d derived states in the bandgap: the low-lying two states are electron-occupied, while three states near the CBM are empty. The filled Fe-3$$d_{z^2}$$ (↓) state (Fig. 5b) emerges in the middle of the bandgap, which is positioned at a depth of 1.5–1.9 eV from the CBM. This energy difference in the vicinity of the k-points of ZARZ remains unchanged at ≈1.5 eV. Therefore, this occupied state can deliver an onset of electron injection to the conduction band at this photon energy. As shown in Fig. 5c, the occupied Fe-3$$d_{x^2 - y^2} {\,\!}^*$$ (↓) state with significant dispersion is present. As this state is located at a depth of 2.3–2.6 eV from the CBM, an additional onset of electron injection appears at ≈2.3 eV. These electron pumping processes can be regarded as an electron transfer from Fe2+ to titanium along with an oxidation from Fe2+ to Fe3+.

In the Fe2+ and Fe3+ coexisting state under illumination, the first onset is at ≈1.5 eV, above which the electron in the Fe2+-3$$d_{z^2}$$ (↓) state is pumped to the conduction band (Fig. 5a). This serves as a PV response originating from electron conduction through a trapping-detrapping process16,17,56. The second onset is at 1.8–1.9 eV, above which holes are injected into the valence band from the empty gap states (↓) of Fe3+ (Fig. 5d). At hv greater than the second onset energy, the photoinduced carrier is regarded as e’-h pair. The third onset is at 2.3 eV, above which the electron in the Fe2+-3$$d_{x^2 - y^2} {\,\!}^*$$ (↑) state is additionally injected into the conduction band (Fig. 5a). We note that e’-h pairs are effectively generated above the third onset, because visible-light can pump electrons from the valence band to the conduction band mediated through these occupied and empty gap states, namely, electrons are excited from the two occupied states of Fe2+ (resulting in an oxidation to Fe3+), and holes are injected into the valence band from the empty gap states of Fe3 (leading to a reduction to Fe2+). This sequential electron pumping is accompanied by two successive redox reactions of Fe3+/Fe2+. The PV response mediated through the occupied (donor) and empty (acceptor) states is termed the successive redox-mediated ferrophotovoltaics.

### Visible-light PV properties

Figure 6 shows the current density normalized by optical intensity, Jsc Iopt−1, as a function of hv. Both the samples exhibit a complicated PV response above Eg of 3.2 eV. While this behavior is assumed to be explained by the shift current theory13, we focus on the PV properties at hv below Eg.

The oxidized (Fig. 6a, b) and non-doped (Supplementary Fig. 5) samples exhibit a PV onset at 1.9–2.0 eV, which is in good agreement with the energy difference between the VBM and the empty states of the isolated Fe3+ and the Fe3+-VO•• associate (Fig. 4). As displayed in Fig. 6c, d, the reduced sample indicates a much higher Jsc Iopt−1 in a wide hv range. Considering the strong emission lines of the light source (Xe lamp) in the near-infrared region of >820 nm (>1.5 eV, Supplementary Fig. 6) and the low-hv data (the inset of Fig. 6d), we posit that the first onset is below 1.6 eV. This agrees with the 1.5 eV difference of the CBM to Fe2+-3$$d_{z^2}$$ (↓) (Fig. 5a). The second onset was found at ≈1.9 eV, which is in quantitative agreement with the difference between the VBM and the empty Fe3+ (↓) states (Fig. 5b). The third onset was observed at ≈2.4 eV, which accords with the gap of the CBM to the filled Fe2+-3$$d_{x^2 - y^2} {\,\!}^*$$ (↑) state.

Here, we address the photogenerated carriers under visible-light. The carrier for the oxidized sample is h above the first onset, while that for the reduced sample is e’ in the hv range from the first to the second onset. These single-carrier-type PV effects lead to a |Jsc Iopt−1| of ≈1 × 10−9 V−1 at most. For the reduced sample above the second onset, h is generated in addition to e’. This yields e’-h pairs, thereby leading to an intense PV response. In the hv range from the second to the third onset, the concentration of h is higher than that of e’. This is because the two orbitals of Fe3+ and the three orbitals of Fe3+-VO•• act as an accepter state for h injection whereas the only one orbital of Fe2+-VO•• contributes to a donor state for e’ injection. We therefore consider that e’-h pairs and the surplus h contribute to the PV currents in this energy range. Above the third onset, e’ pumped from the Fe2+-3$$d_{x^2 - y^2} {\,\!}^*$$ (↑) state is superimposed; hence, the e’-h concentration increases markedly, thereby delivering the robust PV effect.

Figure 7a presents the current density (J) versus bias voltage (V) properties under light at hv= 3.1 eV. The oxidized sample exhibits a short-circuit J (Jsc) of −32 nA cm−2 and an open-circuit voltage (Voc) of 5.9 V. The reduced sample shows a large response: compared with the oxidized sample, the reduced sample has a Jsc of −520 nA cm−2 that is more than one order of magnitude and an extrapolated Voc of ≈35 V that is approximately six times.

The current density Ji arising from the bulk-PV effect is expressed by a third-rank tensor βijk. For the tetragonal crystal in 4 mm point group, the non-zero components of the bulk PV tensor are β31, β33, and β15. Figure 7b indicates the data of J3 (Θ) at hv= 3.1 eV and the results of the fitting analysis, where Θ denotes the light-polarization angle. As listed in Supplementary Table 1, the oxidized sample has β33 = −1.77 × 10−9 V−1 and β31 = −2.86 × 10−9 V−1, while the reduced sample features larger components by over an order of magnitude: β33 = −3.25 × 10−8 V−1 and β31 = −5.54 × 10−8 V−1. We confirmed that β15 is much smaller than β31 and β33 for both samples.

### Photo-luminescence analysis

Figure 8 displays the photo-luminescence (PL) spectra of the single-crystal samples at 10 K, where the samples have a rhombohedral R3m structure. We confirmed that the electronic structures are essentially the same for the tetragonal (Fig. 4) and rhombohedral (Supplementary Fig. 7) cells. The data of the oxidized sample (Fig. 8a) could be traced by a single log-normal function, and the long-wavelength edge (LWE) is roughly estimated to be ≈660 nm (1.88 eV). This is in satisfactory agreement with the Fe3+-derived PV onsets (1.9–2.0 eV; see Fig. 6a, b). The PL spectra of the reduced sample (Fig. 8b) could not be reproduced by a single function but are are well fitted by a superposition of two log-normal functions. The LWEs are evaluated to be ≈660 nm (1.88 eV) and ≈570 nm (2.18 eV), which agree quantitatively with the Fe3+-derived second onset (1.9 eV) and the Fe2+-derived third onset (2.3 eV), respectively (Fig. 6d). The cathode-luminescence (CL) analysis of Fe (1%)-doped BaTiO3 ceramics (Supplementary Fig. 8) gave consistent results: LWEs of ≈680 nm (1.82 eV) for the oxidized ceramic and those of ≈630 nm (1.97 eV) and ≈530 nm (2.34 eV) for the reduced ceramic. Moreover, the integrated data of the reduced ceramic indicate a CL peak with a LWE of ≈860 nm (1.44 eV), which agrees with the Fe2+-derived first onset (<1.6 eV).

## Discussion

Our study demonstrates that the donor and acceptor states derived from iron ions activate the visible-light PV effect that is mediated through the successive Fe3+/Fe2+ redox cycles. Another promising dopant is Mn that has the valence states of 2 +, 3 +, and 4+ 57,58. As shown in Supplementary Figs. 9 and 10, the mixed valence state of Mn2+ and Mn3+ can activate sequential redox reactions that yield e’-h pairs. The Mn2+-VO•• associate creates a donor state at a depth of 1.3 eV from the CBM, thereby giving rise to the first PV onset. Considering the isolated Mn3+ having a acceptor state at 1.9 eV from the VBM, we expect that e’-h pairs can be generated at ≈1.9 eV, thereby leading to an intense PV response. The multiple gap states arising from Mn could produce the visible-light PV effect.

In summary, it is expected that our approach can be applied for photocatalysts of TiO2 and SrTiO3, as displayed in Supplementary Figs. 11 and 12, respectively. In these titanate systems, rigid TiO6 octahedra govern the electronic structure near the bandgap, and the overall features of the iron-derived gap states are essentially the same. Moreover, the multivalent cations are not the unique dopants for e’-h pair generation. We can adopt a co-doping strategy: donor and acceptor states arise from cations of different elements, where their defect concentrations are easily controllable. Our work offers a starting point for further investigation of successive redox-mediated functions and opens an unprecedented route to robust photoinduced effects based on gap-state engineering.

## Methods

### Sample preparation and measurements

An Fe (0.3%)-doped BaTiO3 bulk single-crystal was grown by a top-seeded solution growth method. After cutting the crystals, we annealed them in air at 1250 °C for 12 h in air for recovery from mechanical damage. To control the valence state of iron, the reduced sample was annealed at an oxygen partial pressure at 900 °C (Po2900 °C) of 10−23 atm for 100 h, which was followed by quenching to 150 °C, and was slowly cooled to room temperature. The oxidized sample was also prepared by annealing at 1200 °C at an Po21200 °C of 0.2 atm in a similar manner. A commercial single-crystal of BaTiO3 (NEOTRON CO., LTD.) was used as a reference. Platinum electrodes were fabricated on the sample surface by sputtering. As a poling treatment, we applied an electric field of 2 kV cm−1 during a slow cooling from 150 °C to room temperature through the Curie temperature (TC ≈ 130 °C). This process enables us to obtain the single-domain sample with Ps//[001]. Ceramics of Fe (1.0%)-doped BaTiO3 were prepared by solid-state reaction for luminescence measurements.

### Photo- and cathode-luminescence measurements

Photo-luminescence (PL) spectra of the single-crystal samples were measured at 10 K with a spectrometer (Horiba LabRam-HR PL) through a long-wavelength-pass filter (>325 nm). Data were collected with a He-Cd laser excitation at 325 nm (a spot size of ≈100 μm and a laser power of ≈1.7 mW). As the data at a wavelength (λ) above 800 nm were strongly influenced by an inherent optical property of the filter, we analyzed the PL intensity (IPL) in the λ range of 375–800 nm using the following log-normal function:

$$I_{{\mathrm{PL}}} = H \cdot {\mathrm{exp}}\left[ {\frac{{ - \ln 2}}{{\left( {\ln \rho } \right)^2}}\left\{ {\ln \left( {\frac{{\left( {\lambda - \lambda _0} \right)\left( {\rho ^2 - 1} \right)}}{{w\rho }}} \right) + 1} \right\}^2} \right]$$
(1)

where H denotes the peak height, λ0 the shift parameter, w the full-width at half maximum, and ρ the half-width ratio. Cathode-luminescence (CL) spectra were also collected at 298 K with a scanning electron microscopy system (JEOL JSM-7800F Prime). Unfortunately, the CL intensities of the single-crystal samples were below the detection limit. Instead, we measured CL data of Fe (1.0%)-doped BaTiO3 ceramics so that the output signal-to-noise ratio should be as high as possible. We analyzed the CL intensity (ICL) data using the log-normal function in the similar manner.

We consider that luminescence profiles depend not only on the energy levels of defect states but also on the density-of-states of the relevant bands59,60. It is reasonable to estimate the energy levels of gap states with respect to the valence band maximum (VBM) or the conduction band minimum (CBM) from the long-wavelength edge (LWE) rather than the peak wavelength of the PL/CL luminescence profiles. However, the spectra exhibit an inherent tail on the long-wavelength side, which prevents us from determining the edge unambiguously. Instead, we evaluate the LWE from the linear slope (dashed lines in Fig. 8 and Supplementary Fig. 8) of the log-normal distribution.

### Photovoltaic analysis

The current density Ji is the component along the i direction arising from the bulk-PV effect and is expressed by the following equation using a third-rank bulk PV tensor βijk:11,12,13

$$J_i = I_{{\mathrm{opt}}}\beta _{ijk}{\mathrm{e}}_j{\mathrm{e}}_k$$
(2)

where Iopt denotes the light intensity and ej and ek are the components of the unit vectors along the j and k directions, respectively. We adopt the standard 3 × 6 matrix notation: βijkβ with λ = 1, 2, …, 6, as used in the representation of the piezoelectric tensor. As the direction of Iopt is i = 1, that of Ps is i = 3, and the light-polarization (Θ) is defined as the angle between the polarization plane of light and the measurement direction of J (Fig. 7b), the photocurrent densities in 4 mm point group22 are expressed by

$$J_{1} = 0\left( {//I_{{\mathrm{opt}}}} \right) \,\qquad \qquad \quad$$
(3)
$$J_{2} = I_{{\mathrm{opt}}}\beta _{15}{\mathrm{sin2}}\varTheta ( \bot P_{\mathrm{s}}) \, \,\, \, \quad$$
(4)
$$J_{3} = I_{{\mathrm{opt}}}\left[ {\left( {\beta _{33} + \beta _{31}} \right){\mathrm{/}}2 + \left( {\beta _{33} - \beta _{31}} \right){\mathrm{/}}\left.{2 {\mathrm{cos2}}{\mathrm{\varTheta }}} \right)} \right]\left( {{\mathrm{//}}P_{\mathrm{s}}} \right)\,$$
(5)

We measured the current density (J)–bias voltage (V) characteristics under light with an intensity of 5.6 W cm−2 using a laser module with a wavelength of 405 nm (a photon energy hv of 3.1 eV). The light-polarization was controlled by a half-wavelength plate and a polarizer. The details are described in the preceding paper22. We investigated the normalized short-circuit current density Jsc Iopt−1 as a function of hv. Monochromatic light from a Xe lamp passing through a MgF2-coated diffraction grating was irradiated to the sample using an optical fiber. The wavelength resolution of our optical system was estimated to be 20 nm. Through the PV measurements, we excluded a transient current due to the capacitance and resistance of the samples together with a pyroelectric current caused by the photothermal61 effect. In determining the open-circuit voltage (Voc), we extrapolated the linear JV characteristics in a limited range owing to our current amplifier. All the PV measurements were performed at 25 °C.

### Defect concentration calculations

The defect concentrations in an equilibrium state were calculated on the basis of the thermodynamic data set for BaTiO3, the details of which are described in Supplementary Information. The calculations were conducted by a generalized-reduced-gradient non-linear least-squares method. We made the following assumptions: the lattice is equilibrated at 900 °C or 1200 °C (the annealing temperature) at a specific Po2, and then the concentration of oxygen vacancy (VO••) is quenched (fixed) at room temperature. We take into account the electronic reactions, such as the redox of Fe3+/Fe2+ and the thermally activated electron-hole pair generation of nulle’ + h in the whole temperature range.

### Density-functional theory (DFT) calculations

DFT calculations were conducted using the generalized gradient approximation62 with a plane-wave basis set. We used the projector-augmented wave method63 as implemented in the Vienna ab initio simulation package (VASP)64. We employed the Perdew–Burke–Ernzerhof gradient-corrected exchange-correlation functional revised for solids (PBEsol)65 and a plane-wave cut-off energy of 520 eV. Within the simplified generalized gradient approximation (GGA) + U approach66, we added on-site Coulomb interaction parameters of UJ of 2 eV to Fe-3d and Mn-3d throughout our calculations. We created TM-substituted cells (TM = Fe and Mn) with 3 × 3 × 3 from the optimized BaTiO3 cell with tetragonal P4mm symmetry, leading to the tetragonal (T) Ba27Ti27O81 structure. As displayed in Supplementary Fig. 2, there exists twenty-one types of oxygen atoms with different site symmetry. We therefore performed calculations for all the supercells with VOn••, where VOn•• denotes the oxygen vacancy on the n-th nearest-neighbor site with respect to Fe ions. Structural relaxation was performed for the atoms with less than 0.6 nm distance from the original position of VOn••. The valence states of Fe atoms were controlled by adjusting the total number of electrons. We employed the Γ-centered 3 × 3 × 3 k-point mesh for the structural relaxations and the 5 × 5 × 5 k-point mesh for density-of-states and band structure calculations. We added UJ of 8 eV to Ti-3d in addition to UJ of 2 eV for Fe-3d and Mn-3d for the electronic structure calculations.