## Introduction

Miniaturized energy storage has played an important role in the development of high-performance electronic devices, including those associated with the Internet of Things (IoTs)1,2. Capacitors with a high power density are expected to provide innovative advances for energy management systems3,4, safety technologies5,6, and health care applications7,8. A key challenge is the creation of a standalone energy storage system with a long lifetime. A system equipped with a harvester or a wireless power transfer enables the semipermanent operation of IoT products beyond the constraints of energy supply9. Ceramic capacitors are considered the leading storage components because of their robustness and extremely long lifetimes9,10.

To design self-powered systems, the energy density of ceramic capacitors must be markedly improved. Various polar materials, including paraelectrics11,12,13, ferroelectrics14,15,16, antiferroelectrics17,18, and relaxors19,20, have been investigated. The following two indices obtained from polarization (P)-electric field (E) properties have been widely used to assess the energy storage performance: the recoverable energy density $$U_{{{{\mathrm{rec}}}}} = \mathop {\int }\nolimits_{P_{{{\mathrm{r}}}}}^{P_{{{{\mathrm{max}}}}}} EdP$$ and the energy efficiency $$\eta = U_{{{{\mathrm{rec}}}}}/\mathop {\int }\nolimits_{P_0}^{P_{{{{\mathrm{max}}}}}} EdP$$, where Pmax is the maximum P induced by the maximum field (Emax), Pr is the remnant polarization, and P0 is the initial P in the charge–discharge process10.

The Urec of typical ferroelectrics corresponds to the shaded area in the P-E loop shown in Fig. 1a. A large polarization change $${{\Delta }}P = P_{{{{\mathrm{max}}}}} - P_{{{\mathrm{r}}}}$$ results in a large Urec. A common approach is to form relaxors or solid solutions21,22 that lead to a suppressed Pr and a higher η, whereas the Pmax is reduced23; a tradeoff exists between Urec and η. Moreover, the performance is often evaluated from the polarization curves around a breakdown field (EBD), i.e., Emax ~ EBD, because the resultant Urec becomes large24. Such operations relying on an unchanged dielectric permittivity, however, result in lifetime degradation to some extent. We therefore think that an appropriate indicator should be taken into account, that is, an effective relative permittivity (εr, eff) expressed as ε0εr, eff = 2Urec/Emax2, where ε0 is the dielectric permittivity of vacuum. This is because ceramic capacitors need to be operated at fields much lower than EBD to exploit their long lifetimes.

We report that ferrorestorable polarization arising from an internal field (Ei) enhances εr, eff. In principle, Ei originates from an interaction between Ps (spontaneous polarization vector) and μdef (defect dipole vector) [hereafter, Λ (bold) denotes the vector of its scalar Λ, i.e., Λ = |Λ|], which has been recognized as symmetry-conforming short-range ordering25. Compared with the pristine material (Fig. 1a), a controlled sample with Ei exhibits a shifted P–E loop with a markedly large ΔP, which is termed ferrorestorable polarization (Fig. 1b).

We selected a prototypical ferroelectric BaTiO3 ceramic as a model material and chose μdef composed of Cu3+ and oxygen vacancies (VO••). Our density functional theory (DFT) calculations reveal that VO•• is stabilized on the first nearest neighbor of Cu3+ on the Ti4+ site (forming μdef) and that the ground-state configuration of μdef || Ps leads to Ei. We found that our controlled sample, with a strong Ei, displays an extraordinarily large εr, eff of approximately 7,000 and an unexpectedly high η of 89%.

## Electronic structure of Cu-doped BaTiO3 with VO••

Figure 2 shows the DFT results for the transition metal (TM)-doped cells with VOn•• (VOn••-TM), where On indicates the nth nearest-neighbor oxygen with respect to TM and VOn•• denotes an oxygen vacancy in site On (n = 1–21). Figure 2a exhibits the dependence of the total energy (Etotal) on n, where the vertical axis is the Etotal relative to that of n = 1. For the Cu cells, the Etotal of n =1–3 is smaller than that of n = 4–21 irrespective of the valence states of Cu [except for n = 6 of VOn••-Cu+ (Supplementary Fig. 1)]. This upward tendency indicates that the system is stabilized by VO•• trapping by Cu owing to an attractive interaction. Provided that VO•• has a certain mobility at a moderate temperature (e.g., 80 °C) and the system is equilibrated (so-called ‘aging’), which is followed by cooling to room temperature, we can assume that Cu is associated with VO••; in other words, a CuO5 pyramid is formed with μdef (Fig. 2b–d).

For the V2+ cells, the relative Etotal decreases with increasing n, showing that a VO6 octahedron is preserved; i.e., VO•• is distanced from V2+ due to a repulsive interaction. The results of Fig. 2a cannot be explained solely by a Coulomb interaction between the positively charged VO•• and the negatively charged cations and suggest the importance of the electronic contribution.

The Cu cells with n = 1–3 have different μdef configurations, μdef || Ps (μdef, 1), μdef Ps (μdef, 2), and μdef || −Ps (μdef, 3), as displayed in Fig. 2b–d. μdef, 1 is the most stable for Cu3+ because its Etotal is 0.15–0.36 eV lower than those of μdef, 2 and μdef, 3. For Cu2+, μdef, 1 and μdef, 2 have almost the same Etotal, which is lower than that of μdef, 3 by 0.10 eV. Because the multiplicity of the O1 site is one while that of the O2 site is four in the Cu-doped BaTiO3 supercell, μdef, 2 becomes the majority configuration. For Cu+, μdef 1 and μdef, 3 coexist because the difference in Etotal is quite small and Etotal of μdef, 2 is higher than that of the other two by approximately 0.16 eV. These results indicate that the highest probability of the formation of μdef, 1 is expected for Cu3+.

To explore the origin of μdef formation, we focused on the variation in the electronic structure of the Cu3+ cells. As shown in Supplementary Fig. 1, VOn••-Cu3+ with n = 1–3 has a lower energy in the low spin (LS) state, while VOn••-Cu3+ with n ≥ 4 is stabilized in the high spin (HS) state. This is because the Cu3+ with n = 1–3 is positioned in a weaker ligand field of the CuO5 pyramid, whereas the Cu3+ with n ≥ 4 is located in a stronger ligand field of the CuO6 octahedron. Figure 2e–g show the density of states (DOS) of VO1••-Cu3+(LS), VO3••-Cu3+(LS), and VO4••-Cu3+(HS). The orbital hybridization between Cu-3d and the adjacent O-2p plays a crucial role in their energy levels and the resultant Etotal. Here, the d states with and without an asterisk denote the antibonding and bonding states, respectively. In the partial DOS of the Cu-3d of VO1••-Cu3+ (Fig. 2e) and VO3••-Cu3+ (Fig. 2f), the electron-occupied states of dz2, dx2−y2, dxy, dyz, and dzx are located near the bottom of the valence band (−5.5 eV to −3 eV). Moreover, the unoccupied dx2−y2* state appears just above the valence band maximum (VBM), as in Mn-doped BiFeO326, where this gap state arises primarily from an interaction with O2-2p (Fig. 2h). For the VO4••-Cu3+ cell, not only the unoccupied dx2−y2* state but also the partially occupied dz2* state is present above the VBM in the minority spin band (Fig. 2g); dz2* interacts with O1-2p and O3-2p (Fig. 2i). The partial occupation of the dz2* state inside the gap is the main reason why the VO4••-Cu3+ cell with the CuO6 octahedron is higher in energy than that (n = 1–3) with the CuO5 pyramid. The stabilization of VO1••-Cu3+ is consistent with the theory of symmetry-conforming short-range ordering, which describes reversible domain switching in acceptor-doped BaTiO3 crystals well25.

## pO2900 °C dependence of the Cu valence

Figure 3 shows the pO2900 °C dependence of the effective magnetic moment (μeff) for the Cu (1.5%) samples estimated from the temperature dependence of the inverse magnetic susceptibility (1/χ) (Supplementary Fig. 2). The following four pO2900 °C regions appear: Region I, pO2900 °C ≤ 10−25 atm, where μeff is negligibly small; Region II, 10−25 atm < pO2900 °C ≤ 10−10 atm, where μeff sharply increases from zero to 1.7; Region III, 10−10 atm < pO2900 °C < 10−2 atm, where μeff is almost constant; and Region IV, pO2900 °C ≥ 10−2 atm, where μeff further increases. In Region I, the negligible μeff indicates that the dominant species is Cu+ (d10 electron configuration) with a spin-only magnetic moment ($$\mu _{{{{\mathrm{eff}}}}}^{{{{\mathrm{spin}}}}}$$) of zero. The increase in μeff to approximately 1.7 in Region II is associated with the oxidation of Cu+ to Cu2+ (d9, $$\mu _{{{{\mathrm{eff}}}}}^{{{{\mathrm{spin}}}}} = 1.73$$), which is accompanied by a color change from black to dark brown. The concentration of Cu2+ ($$\left[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^{\prime \prime }} \right]$$) is maximized at pO2900 °C ~ 10−10 atm, where the charge neutrality is expressed as $$\left[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^{\prime \prime }} \right] \approx [V_{{{\mathrm{O}}}}^{ \cdot \cdot }]$$. In Region III, the plateau of μeff can be explained by the oxidation of Cu2+ to Cu3+ (d8, $${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^\prime$$). At pO2900 °C ≈ 1 × 10−2 atm, Cu3+ is the majority species, and the charge neutrality is expressed as $$\left[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^\prime } \right] \approx 2[V_{{{\mathrm{O}}}}^{ \cdot \cdot }]$$, where the following two Cu3+ coordination environments arise: the Cu3+O6 octahedron in the HS state ($${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{oc}}}}}^\prime$$, $$\mu _{{{{\mathrm{eff}}}}}^{{{{\mathrm{spin}}}}} = 2.83$$) and the Cu3+O5 pyramid in the LS state ($${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{py}}}}}^\prime$$, $$\mu _{{{{\mathrm{eff}}}}}^{{{{\mathrm{spin}}}}} = 0$$) (Supplementary Fig. 3). In Region IV, an increase in pO2900 °C causes a steep rise in $$\mu _{{{{\mathrm{eff}}}}}$$ because of $$[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{oc}}}}}^\prime }] \gg [ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{py}}}}}^\prime }]$$, which is accompanied by a color change from light yellow to gray. We confirmed that impurity phases are not formed even after annealing under reducing atmospheres (Supplementary Fig. 4).

## Polarization properties

Figure 4a–c shows the P-E loops of the undoped samples (pO2900 °C = 0.2 atm). The Pr of the as-prepared sample with neither poling pretreatment nor aging is 13.1 μC/cm2 (Fig. 4a). The coercive field (Ec) is 0.4 kV/cm for Ec+ and −0.5 kV/cm for Ec−. The subsequent aging does not influence the properties with or without the poling pretreatment (Fig. 4b, c).

The Cu (1.5%) samples display diverse properties depending not only on the aging and poling pretreatment but also on the Cu valence (dependent on pO2900 °C). Figure 4d–f shows the P-E loops at pO2900 °C = 3 × 10−6 atm. The as-prepared sample presents a loop similar to the undoped sample, regardless of the Cu valence. The aged sample without poling exhibits an antiferroelectric-like slim loop (Fig. 4e). This indicates that the aging stabilizes a multidomain (MD) state with zero net polarization at E = 0 and that an application of E transforms to a quasi-single domain (SD) state followed by recovery of the original MD state after the field is turned off (Supplementary Fig. 10). This polarization behavior has been observed in single crystals25 and Mn-doped ceramics27. Moreover, the sample (pO2900 °C = 3 × 10−6 atm) with poling pretreatment followed by aging (hereafter denoted ‘controlled’) has a P-E loop shifted in the positive direction. It has an extraordinarily large Ei of 47.5 kV/cm, which is defined by the average of Ec+ = 50.1 kV/cm and Ec− = 45.0 kV/cm (Fig. 4f). Namely, a negative polarization state is stabilized at E = 0. As a result, polarization switching cannot be observed in the range of E < 0. Note that the resultant ΔP (ferrorestorable polarization) is as high as 43.8 μC/cm2 at 76 kV/cm. The aged samples (pO2900 °C ≥ 4.0 × 10−11 atm) without poling display an antiferroelectric-like pinched curve (Supplementary Fig. 5a, c, e), whereas those at pO2900 °C = 2.0 × 10−20 atm present a typical loop (Supplementary Fig. 5g).

Figure 5a shows the Ei as a function of pO2900 °C for the controlled samples (their P-E loops are displayed in Supplementary Fig. 5b, d, f, h). The Ei becomes stronger with increasing pO2900 °C; the Ei starts to rise at approximately pO2900 °C = 1 × 10−10 atm and reaches 47.5 kV/cm at pO2900 °C = 3 × 10−6 atm. The pO2900 °C region where Ei rises coincides with Region III (Fig. 3), where the oxidation of Cu2+ to Cu3+ proceeds. These results indicate that controlling the Cu valence to Cu3+ is important for strengthening Ei. Considering that the Ei saturates at pO2900 °C = 0.2 atm in Region IV, we think that [$${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{py}}}}}^\prime$$] is more crucial than [$${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{oc}}}}}^\prime$$] in achieving a high Ei. We consider that the optimal pO2900 °C for enhancing Ei exists at approximately 0.2 atm because annealing at a higher pO2900 °C results in a low probability of μdef, 1 because of $$[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{oc}}}}}^\prime }] \gg [ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{py}}}}}^\prime }]$$. It is also notable that a $$[V_{{{\mathrm{O}}}}^{ \cdot \cdot }]$$ difference can be excluded from the major origin of the dependence of Ei on pO2900 °C because the samples annealed at a higher pO2900 °C exhibit a higher Ei despite a lower concentration of $$V_{{{\mathrm{O}}}}^{ \cdot \cdot }$$ required to form μdef.

A similar ferrorestorable polarization appears for the controlled samples with Cu (0.3%) (Supplementary Fig. 6), whereas this is not observed for the samples with V (0.3%), irrespective of pO2900 °C, as shown in Fig. 4g–i and Supplementary Fig. 7.

## Energy storage performance

By analogy with the relative permittivity $$\varepsilon _{{{\mathrm{r}}}}$$ for linear dielectrics with Urec = $$\varepsilon _0\varepsilon _{{{\mathrm{r}}}}E_{{{{\mathrm{max}}}}}^2/2$$, we define the effective relative permittivity εr, eff for nonlinear dielectrics with Urec = $$\varepsilon _0\varepsilon _{{{{\mathrm{r}}}},{{{\mathrm{eff}}}}}E_{{{{\mathrm{max}}}}}^2/2$$. Figure 5b shows the pO2900 °C dependence of εr, eff and η for the controlled samples with Cu (1.5%). The corresponding Urec is shown in Supplementary Fig. 8. With increasing pO2900 °C, εr, eff rises and is as high as 7,000. Moreover, η reaches 89% in the same pO2900 °C region.

Here, we discuss why an extraordinarily large εr, eff is achieved in the relatively high pO2900 °C region. Compared with the as-prepared material (Fig. 5c), the controlled sample has a strong Ei (μdef, 1 as the majority), providing a marked shift in the P-E loop (Fig. 5d). Our DFT calculations indicate that μdef, 1 is preferable for Cu3+ because its Etotal is 0.15–0.36 eV lower than those of μdef, 2 and μdef, 3. Given that an [VO••] and its random distribution equilibrated at pO2900 °C > 10−10 atm are frozen by successive quenching to room temperature, the charge neutrality is expressed as $$2\left[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^{\prime \prime }} \right] + [{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^\prime ] \approx 2[V_{{{\mathrm{O}}}}^{ \cdot \cdot }]$$; Cu3+ and Cu2+ coexist. During the thermal treatment at 200 °C (>TC), $$V_{{{\mathrm{O}}}}^{ \cdot \cdot }$$ migrates toward Cu3+ owing to an attractive interaction, and eventually, μdef (Cu3+O5 pyramid) is formed in the paraelectric cubic lattice (Supplementary Fig. 9a); i.e., $$[ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^\prime }] = [ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{py}}}}}^\prime }] + [ {{{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}},{{{\mathrm{oc}}}}}^\prime }]$$.

In the cooling process, a phase transition from the paraelectric phase to the ferroelectric phase occurs. In this as-prepared sample, an MD structure with zero net polarization is formed (Supplementary Fig. 9b). Because the $$V_{{{\mathrm{O}}}}^{ \cdot \cdot }$$ distribution above TC is frozen at room temperature, the probabilities of μdef, 1, μdef, 2, and μdef, 3 seem to be 1/6, 2/3, and 1/6, respectively; no correlation between the configurations of μdef-Ps exists in each ferroelectric domain.

In the sample with Cu3+ as the majority, the poling pretreatment by applying negative fields results in a transformation from the MD state with P = 0 to a quasi-SD state with Pdown (Fig. 5e). Successive aging at 80 °C (<TC) promotes a rearrangement of $$V_{{{\mathrm{O}}}}^{ \cdot \cdot }$$ to attain the ground-state defect structure (Fig. 2a); $$V_{{{\mathrm{O}}}}^{ \cdot \cdot }$$ is stabilized on the O1 site, which increases μdef, 1 and decreases μdef, 2 and μdef, 3. As aging proceeds, μdef, 1 becomes the majority, which corresponds to State 1 in Fig. 5d, f. In other words, the SD structure with Pdown is stabilized by μdef, 1, which leads to a certain Ei.

When an upward Emax is applied to the controlled sample, the polarization is switched from Pdown to Pmax, which is accompanied by a change from μdef, 1 to μdef, 3. During the subsequent field decrease, the polarization returns to the initial Pdown in the SD state, where ΔP (= PmaxPdown) corresponds to ferrorestorable polarization. Given that a negative field is applied to the sample with Pdown, the Pdown state remains unchanged regardless of the field strength, and thereby, the linear polarization appears in the negative field region, as shown in Fig. 4f.

## Discussion

The εr, eff of our controlled samples along with those of the previously reported capacitors are plotted in Fig. 6. In principle, a tradeoff exists between εr, eff and η except for a few examples of antiferroelectrics. In contrast, our controlled sample (pO2900 °C = 3 × 10−6 atm), denoted “b”, possesses both a large εr, eff (7,000) and a high η (89%). This εr, eff is over twice as large as that of typical BaTiO3-based ceramics28,29,30 This excellent performance is due to the large ferrorestorable polarization (ΔP) and the small hysteresis loop of the P-E curve that arises from the strong Ei introduced without sacrificing the polar nature. We consider that the ferrorestorable polarization can be utilized in BaTiO3 doped with other TM acceptors, such as Mn27 and Fe31, that can trap VO••. Our approach using the interaction between μdef and Ps is effective in breaking the tradeoff between Urec and η. Since a fabrication process of BaTiO3-based multilayered ceramic capacitors (MLCCs) has been established, we can readily adapt our material design to energy-storage MLCCs. Moreover, it is expected that employing Bi-based ferroelectrics with a larger Ps can further enhance εr, eff.

Our experimental and theoretical investigations demonstrate that a built-in internal field arising from defect-polarization interactions delivers excellent energy storage performance in ferroelectrics. This method is applicable to relaxor ferroelectrics32,33, antiferroelectrics17,34, and ferrielectrics35,36. Our findings will pave the way for energy storage capacitors utilizing ferrorestorable polarization in self-powered systems.

## Methods

### Preparation of Cu-doped and V-doped BaTiO3 ceramics

Cu-doped and V-doped BaTiO3 powders were prepared by a solid-state reaction. BaCO3 (99.99%), TiO2 (99.99%), CuO (99.9%) and V2O5 (>99%) powders were mixed by ball milling. The mixtures of the raw materials were calcined at 850–1000 °C for 5–10 h. The calcined powders were crushed by ball milling again and pelletized into disks (10 mm diameter) at 120 MPa by uniaxial pressing. The disks were pressed isostatically at 150 MPa for 1 h and then sintered at 1200–1400 °C for 5–10 h. The sintered ceramics were cut and polished; the resulting sample size was 8 mm × 4 mm × 0.3 mm. The samples were annealed again at 1200 °C for 24 h to heal the microcracks introduced during the preparation process. The oxygen vacancy concentration ([VO••]) and valence state of Cu were controlled by annealing at 900 °C for 12 h at an oxygen partial pressure (pO2900 °C) ranging from 4 × 10−20 atm to 0.2 atm followed by quenching to room temperature (25 °C) in a short time (1–3 s). A sample annealed at a pO2900 °C of 1 × 102 atm was also prepared by hot isostatic pressing.

### Poling and aging treatments

Some samples were heated at 200 °C (>Curie temperature TC of 120–135 °C) for 10 min in air to homogenize the distribution of VO••. After cooling to 25 °C, an external electric field of −40 kV/cm was applied for 5 s to achieve a negatively poled state, which is termed ‘poling pretreatment’. The poled sample was aged at 80 °C (<TC) for 24 h (aging) to accelerate the diffusion of VO•• in the presence of Ps and then cooled to 25 °C. Before and after aging, polarization measurements were performed at 25 °C and 1 Hz (or 100 Hz) for the samples with and without the poling pretreatment.

### Magnetic susceptibility measurements

To determine the valence state of Cu, the magnetic susceptibility χ of the 1.5% Cu-doped samples equilibrated at various pO2900 °Cs followed by quenching to room temperature was measured with a superconducting quantum interference device (Quantum Design Ltd., MPMS-XL). The size of the sample was 3 × 4 × 5 mm3, and the measurements were performed at a magnetic field of 1 T in the temperature range of 50–300 K.

### Ab initio calculations

Density functional theory (DFT) calculations for tetragonal BaTiO3 with VO•• (TM = Cu+, Cu2+, Cu3+, and V2+) were performed to find a stable site of VO•• according to the literature35. DFT calculations were carried out with the generalized gradient approximation (GGA+U)37 using a plane wave basis set as implemented in the Vienna ab initio simulation package (VASP)38. We used projector-augmented wave potentials39 with valence-electron configurations of 5s25p66s2 for Ba, 3p63d24s2 for Ti, 3d104s1 for Cu, 3d34s2 for V, and 2s22p4 for O. A plane-wave cutoff energy of 520 eV was adopted, and all calculations were conducted until the total energy converged to less than 10−6 eV. To investigate the electronic states, a supercell of BaTiO3 was constructed by the following procedure. First, a BaTiO3 lattice in space group P4mm was structurally optimized until the Hellmann-Feynman force on each atom was smaller than 0.1 eV/nm. A Monkhorst-Pack k-mesh of 5 × 5 × 5 centered at the Γ point was used for structural optimization for lattice parameters and fractional coordinates. Next, a supercell of 3 × 3 × 3 (Ba27Ti27O81) was constructed using the optimized unit cell. One Ti atom in the supercell was replaced by one TM to obtain the supercell (Ba27Ti26TMO81). The valence state of the dopants was controlled by changing the total number of electrons. For geometry optimization of the supercell, a simplified local spin density approximation +U approach40 was adopted as a correction for localized and strongly correlated electrons within on-site Coulomb terms of UJ of 2 eV for both Cu-3d and V-3d and 0 eV for Ti-3d. Structural optimization of the supercell was performed using a k-mesh of 3 × 3 × 3 centered at the Γ point.

In the next step, calculations of the VO••-containing cells (Ba27Ti26TMO80) were conducted to investigate the energetically favorable site of VO••. From the optimized Ba27Ti26TMO81 supercell, one O atom was removed from a specific O site. Structural optimization of the fractional coordinates was performed for all atoms and a fixed cell size in the same manner as that for the defect-free supercell. The oxygen atom on the nth nearest-neighbor O site with respect to TM is defined as “On”, and the oxygen vacancy created by removing On is expressed as “VOn••”. The VOn••-containing cell is expressed as “Von••- TM”. For the calculations of DOS and band structures, the UJ of Ti-3d is set to 8 eV in a manner similar to that in the literature31.