Utilizing ferrorestorable polarization in energy-storage ceramic capacitors

Abstract

A self-powered system with a long lifetime would represent an opportunity in the development of a next-generation, standalone Internet of Things. Ceramic capacitors are promising candidates for energy storage components because of their stability and fast charge/discharge capabilities. However, even the energy density of state-of-the-art capacitors needs to be increased markedly for this application. Improving the breakdown electric field represents a potential solution, but operations at such high fields relying on unchanged dielectric permittivity sacrifice the lifetime of the capacitor to some degree. Here, we report ferrorestorable polarization engineering capable of more than doubling the effective permittivity. Our experiments and ab initio calculations demonstrate that a defect dipole composed of Cu³⁺ and oxygen vacancy in a prototypical ferroelectric BaTiO₃ ceramic is coupled with spontaneous polarization. The resultant ferrorestorable polarization delivers an extraordinarily large effective relative permittivity, beyond 7000, with a high energy efficiency up to 89%. Our work paves the way to realizing efficient ceramic capacitors for self-powered applications.

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 systems^3,4, safety technologies^5,6, and health care applications^7,8. 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 lifetimes^9,10.

To design self-powered systems, the energy density of ceramic capacitors must be markedly improved. Various polar materials, including paraelectrics^11,12,13, ferroelectrics^14,15,16, antiferroelectrics^17,18, and relaxors^19,20, 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 P_max is the maximum P induced by the maximum field (E_max), P_r is the remnant polarization, and P₀ is the initial P in the charge–discharge process¹⁰.

The U_rec 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 U_rec. A common approach is to form relaxors or solid solutions^21,22 that lead to a suppressed P_r and a higher η, whereas the P_max is reduced²³; a tradeoff exists between U_rec and η. Moreover, the performance is often evaluated from the polarization curves around a breakdown field (E_BD), i.e., E_max ~ E_BD, because the resultant U_rec 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 (ε_{r, eff}) expressed as ε₀ε_{r, eff} = 2U_rec/E_max², where ε₀ is the dielectric permittivity of vacuum. This is because ceramic capacitors need to be operated at fields much lower than E_BD to exploit their long lifetimes.

Fig. 1: Ferrorestorable polarization.

a Typical P-E loop of ferroelectrics (pristine). b Shifted P-E loop with an internal electric field (E_i) caused by the ground-state configuration of μ_def || P_s (controlled). The controlled sample has a large U_rec as a result of ΔP, which is termed ferrorestorable polarization. The interaction between μ_def and P_s stabilizes the downward polarization (P_down) at zero field, i.e., P₀ = P_down, because the P-E loop shifts to a positive field by the magnitude of E_i. E_i is defined as the average of E_c+ and E_c−, that is, $E_i = (E_{{{{\mathrm{c}}}} + } + E_{{{{\mathrm{c}}}} - })/2$, where E_c+ and E_c− are the electric fields at the extreme polarization switching currents in the positive and negative field sweeps, respectively.

We report that ferrorestorable polarization arising from an internal field (E_i) enhances ε_{r, eff}. In principle, E_i originates from an interaction between P_s (spontaneous polarization vector) and μ_def (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 E_i 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 μ_def composed of Cu³⁺ and oxygen vacancies (V_O^••). Our density functional theory (DFT) calculations reveal that V_O^•• is stabilized on the first nearest neighbor of Cu³⁺ on the Ti⁴⁺ site (forming μ_def) and that the ground-state configuration of μ_def || P_s leads to E_i. We found that our controlled sample, with a strong E_i, displays an extraordinarily large ε_{r, eff} of approximately 7,000 and an unexpectedly high η of 89%.

Electronic structure of Cu-doped BaTiO₃ with V _O ^••

Figure 2 shows the DFT results for the transition metal (TM)-doped cells with V_On^•• (V_On^••-TM), where On indicates the n^th nearest-neighbor oxygen with respect to TM and V_On^•• denotes an oxygen vacancy in site On (n = 1–21). Figure 2a exhibits the dependence of the total energy (E_total) on n, where the vertical axis is the E_total relative to that of n = 1. For the Cu cells, the E_total of n =1–3 is smaller than that of n = 4–21 irrespective of the valence states of Cu [except for n = 6 of V_On^••-Cu⁺ (Supplementary Fig. 1)]. This upward tendency indicates that the system is stabilized by V_O^•• trapping by Cu owing to an attractive interaction. Provided that V_O^•• 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 V_O^••; in other words, a CuO₅ pyramid is formed with μ_def (Fig. 2b–d).

Fig. 2: Interaction between Cu and V_O^••.

a Total energy (E_total) of V_On^••-Cu⁺, V_On^••-Cu²⁺, V_On^••-Cu³⁺ and V_On^••-V²⁺. In our DFT calculations, low-spin (LS) and high-spin (HS) states were adopted for V_On^••-Cu³⁺ with n = 1–3 and n ≥ 4, respectively, according to the ground-state arrangements (Supplementary Fig. 1a, b). b–d Schematics of possible configurations of P_s with μ_def in Cu³⁺. μ_{def, 1} is the most stable for Cu³⁺ because its E_total is 0.15–0.36 eV lower than those of μ_{def, 2} and μ_{def, 3}. For Cu²⁺, μ_{def, 2} is the majority configuration because μ_{def, 1} and μ_{def, 2} have almost the same E_total. For Cu⁺, μ_{def 1} and μ_{def, 3} coexist because the difference in E_total is quite small. e–g Total and partial density of states of V_O1^••-Cu³⁺ (LS), V_O3^••-Cu³⁺ (LS), and V_O4^••-Cu³⁺ (HS). The valence band (VB) and the conduction band (CB) are mainly composed of O-2p orbitals and Ti-3d orbitals, respectively. Red upward and blue downward arrows indicate the majority and minority spin bands, respectively. h Unoccupied antibonding d_x2−y2^* orbital in V_O1^••-Cu³⁺. i Partially occupied antibonding d_z2^* orbital in V_O4^••-Cu³⁺.

For the V²⁺ cells, the relative E_total decreases with increasing n, showing that a VO₆ octahedron is preserved; i.e., V_O^•• 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 V_O^•• and the negatively charged cations and suggest the importance of the electronic contribution.

The Cu cells with n = 1–3 have different μ_def configurations, μ_def || P_s (μ_{def, 1}), μ_def ⊥ P_s (μ_{def, 2}), and μ_def || −P_s (μ_{def, 3}), as displayed in Fig. 2b–d. μ_{def, 1} is the most stable for Cu³⁺ because its E_total is 0.15–0.36 eV lower than those of μ_{def, 2} and μ_{def, 3}. For Cu²⁺, μ_{def, 1} and μ_{def, 2} have almost the same E_total, which is lower than that of μ_{def, 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, μ_{def, 2} becomes the majority configuration. For Cu⁺, μ_{def 1} and μ_{def, 3} coexist because the difference in E_total is quite small and E_total of μ_{def, 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 μ_{def, 1} is expected for Cu³⁺.

To explore the origin of μ_def formation, we focused on the variation in the electronic structure of the Cu³⁺ cells. As shown in Supplementary Fig. 1, V_On^••-Cu³⁺ with n = 1–3 has a lower energy in the low spin (LS) state, while V_On^••-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 V_O1^••-Cu³⁺(LS), V_O3^••-Cu³⁺(LS), and V_O4^••-Cu³⁺(HS). The orbital hybridization between Cu-3d and the adjacent O-2p plays a crucial role in their energy levels and the resultant E_total. 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 V_O1^••-Cu³⁺ (Fig. 2e) and V_O3^••-Cu³⁺ (Fig. 2f), the electron-occupied states of d_z2, d_x2−y2, d_xy, d_yz, and d_zx are located near the bottom of the valence band (−5.5 eV to −3 eV). Moreover, the unoccupied d_x2−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 V_O4^••-Cu³⁺ cell, not only the unoccupied d_x2−y2^* state but also the partially occupied d_z2^* state is present above the VBM in the minority spin band (Fig. 2g); d_z2^* interacts with O1-2p and O3-2p (Fig. 2i). The partial occupation of the d_z2^* state inside the gap is the main reason why the V_O4^••-Cu³⁺ cell with the CuO₆ octahedron is higher in energy than that (n = 1–3) with the CuO₅ pyramid. The stabilization of V_O1^••-Cu³⁺ is consistent with the theory of symmetry-conforming short-range ordering, which describes reversible domain switching in acceptor-doped BaTiO₃ crystals well²⁵.

pO₂ ^{900 °C} dependence of the Cu valence

Figure 3 shows the pO₂^{900 °C} dependence of the effective magnetic moment (μ_eff) for the Cu (1.5%) samples estimated from the temperature dependence of the inverse magnetic susceptibility (1/χ) (Supplementary Fig. 2). The following four pO₂^{900 °C} regions appear: Region I, pO₂^{900 °C} ≤ 10⁻²⁵ atm, where μ_eff is negligibly small; Region II, 10⁻²⁵ atm < pO₂^{900 °C} ≤ 10⁻¹⁰ atm, where μ_eff sharply increases from zero to 1.7; Region III, 10⁻¹⁰ atm < pO₂^{900 °C} < 10⁻² atm, where μ_eff is almost constant; and Region IV, pO₂^{900 °C} ≥ 10⁻² atm, where μ_eff further increases. In Region I, the negligible μ_eff 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 μ_eff 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 pO₂^{900 °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 μ_eff can be explained by the oxidation of Cu²⁺ to Cu³⁺ (d⁸, ${{{\mathrm{Cu}}}}_{{{{\mathrm{Ti}}}}}^\prime$). At pO₂^{900 °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 pO₂^{900 °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).

Fig. 3: pO₂^{900 °C} dependence of μ_eff for the Cu (1.5%)-doped samples.

μ_eff is estimated from magnetic susceptibility measurements (Supplementary Fig. 2). Sample images are also shown. The spin configurations with their $\mu _{{{{\mathrm{eff}}}}}^{{{{\mathrm{spin}}}}}$ values of Cu⁺, Cu²⁺, Cu³⁺ (LS), and Cu³⁺ (HS) are displayed in Supplementary Fig. 3.

Polarization properties

Figure 4a–c shows the P-E loops of the undoped samples (pO₂^900 °C = 0.2 atm). The P_r of the as-prepared sample with neither poling pretreatment nor aging is 13.1 μC/cm² (Fig. 4a). The coercive field (E_c) is 0.4 kV/cm for E_c+ and −0.5 kV/cm for E_c−. The subsequent aging does not influence the properties with or without the poling pretreatment (Fig. 4b, c).

Fig. 4: Polarization properties.

a–c Undoped samples. d–f, Cu (1.5%)-doped samples at pO₂^900 °C = 3 × 10⁻⁶ atm. g–h V (0.3%)-doped samples at pO₂^900 °C = 3 × 10⁻¹⁶ atm. a, d, g, As-prepared. b, e, h, Aged without poling pretreatment. c, f, i, Aged with poling pretreatment (controlled).

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 pO₂^{900 °C}). Figure 4d–f shows the P-E loops at pO₂^900 °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 (pO₂^{900 °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 E_i of 47.5 kV/cm, which is defined by the average of E_c+ = 50.1 kV/cm and E_c− = 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 (pO₂^{900 °C} ≥ 4.0 × 10⁻¹¹ atm) without poling display an antiferroelectric-like pinched curve (Supplementary Fig. 5a, c, e), whereas those at pO₂^900 °C = 2.0 × 10⁻²⁰ atm present a typical loop (Supplementary Fig. 5g).

Figure 5a shows the E_i as a function of pO₂^900 °C for the controlled samples (their P-E loops are displayed in Supplementary Fig. 5b, d, f, h). The E_i becomes stronger with increasing pO₂^{900 °C}; the E_i starts to rise at approximately pO₂^{900 °C} = 1 × 10⁻¹⁰ atm and reaches 47.5 kV/cm at pO₂^{900 °C} = 3 × 10⁻⁶ atm. The pO₂^{900 °C} region where E_i 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 E_i. Considering that the E_i saturates at pO₂^{900 °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 E_i. We consider that the optimal pO₂^{900 °C} for enhancing E_i exists at approximately 0.2 atm because annealing at a higher pO₂^{900 °C} results in a low probability of μ_{def, 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 E_i on pO₂^900 °C because the samples annealed at a higher pO₂^{900 °C} exhibit a higher E_i despite a lower concentration of $V_{{{\mathrm{O}}}}^{ \cdot \cdot }$ required to form μ_def.

Fig. 5: Impact and origin of ferrorestorable polarization.

a, b pO₂^{900 °C} dependence of E_i and effective relative permittivity (ε_{r, eff}) and energy efficiency (η) of the controlled samples with Cu (1.5%). c, d Unipolar polarization curves of the as-prepared and the control samples at pO₂^900 °C = 3 × 10⁻⁶ atm. Schematics of the polarization and μ_def configurations of the sample subjected to the poling pretreatment (e) and the controlled sample (f) at zero field. g Configuration of the controlled sample with E = E_max. State 1 of (f) and State 2 of (g) correspond to those in (d).

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 pO₂^900 °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 U_rec = $\varepsilon _0\varepsilon _{{{\mathrm{r}}}}E_{{{{\mathrm{max}}}}}^2/2$, we define the effective relative permittivity ε_{r, eff} for nonlinear dielectrics with U_rec = $\varepsilon _0\varepsilon _{{{{\mathrm{r}}}},{{{\mathrm{eff}}}}}E_{{{{\mathrm{max}}}}}^2/2$. Figure 5b shows the pO₂^{900 °C} dependence of ε_{r, eff} and η for the controlled samples with Cu (1.5%). The corresponding U_rec is shown in Supplementary Fig. 8. With increasing pO₂^{900 °C}, ε_{r, eff} rises and is as high as 7,000. Moreover, η reaches 89% in the same pO₂^{900 °C} region.

Here, we discuss why an extraordinarily large ε_{r, eff} is achieved in the relatively high pO₂^{900 °C} region. Compared with the as-prepared material (Fig. 5c), the controlled sample has a strong E_i (μ_{def, 1} as the majority), providing a marked shift in the P-E loop (Fig. 5d). Our DFT calculations indicate that μ_{def, 1} is preferable for Cu³⁺ because its E_total is 0.15–0.36 eV lower than those of μ_{def, 2} and μ_{def, 3}. Given that an [V_O^••] and its random distribution equilibrated at pO₂^900 °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 (>T_C), $V_{{{\mathrm{O}}}}^{ \cdot \cdot }$ migrates toward Cu³⁺ owing to an attractive interaction, and eventually, μ_def (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 T_C is frozen at room temperature, the probabilities of μ_{def, 1}, μ_{def, 2}, and μ_{def, 3} seem to be 1/6, 2/3, and 1/6, respectively; no correlation between the configurations of μ_def-P_s 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 P_down (Fig. 5e). Successive aging at 80 °C (<T_C) 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 μ_{def, 1} and decreases μ_{def, 2} and μ_{def, 3}. As aging proceeds, μ_{def, 1} becomes the majority, which corresponds to State 1 in Fig. 5d, f. In other words, the SD structure with P_down is stabilized by μ_{def, 1}, which leads to a certain E_i.

When an upward E_max is applied to the controlled sample, the polarization is switched from P_down to P_max, which is accompanied by a change from μ_{def, 1} to μ_{def, 3}. During the subsequent field decrease, the polarization returns to the initial P_down in the SD state, where ΔP (= P_max − P_down) corresponds to ferrorestorable polarization. Given that a negative field is applied to the sample with P_down, the P_down 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 ε_{r, eff} of our controlled samples along with those of the previously reported capacitors are plotted in Fig. 6. In principle, a tradeoff exists between ε_{r, eff} and η except for a few examples of antiferroelectrics. In contrast, our controlled sample (pO₂^900 °C = 3 × 10⁻⁶ atm), denoted “b”, possesses both a large ε_{r, eff} (7,000) and a high η (89%). This ε_{r, eff} is over twice as large as that of typical BaTiO₃-based ceramics^28,29,30 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 E_i 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 V_O^••. Our approach using the interaction between μ_def and P_s is effective in breaking the tradeoff between U_rec 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 P_s can further enhance ε_{r, eff}.

Fig. 6: Energy storage performance of ceramic capacitors.

The horizontal and vertical axes indicate the effective relative permittivity (ε_{r, eff}) and energy efficiency (η), respectively. The red rectangles indicate the data of the Cu (1.5%) samples in this study; a Aged sample at pO₂^900 °C = 3.0 × 10⁻⁶ atm without the poling pretreatment; (b) controlled sample at pO₂^900 °C = 3.0 × 10⁻⁶ atm; (c) controlled sample at pO₂^900 °C = 1.6 × 10⁻⁷ atm; (d) controlled sample at pO₂^900 °C = 0.2 atm. The green triangles denote the previous studies on thin films and the blue circles those on bulk ceramics. Most of the points are located in the shaded area because of the tradeoff between U_rec and η, except for a few examples of antiferroelectric ceramics. The numbers assigned to each plot correspond to the reference numbers in the Supplementary information. The details are listed in Supplementary Table 1.

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 ferroelectrics^32,33, antiferroelectrics^17,34, and ferrielectrics^35,36. 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 ([V_O^••]) and valence state of Cu were controlled by annealing at 900 °C for 12 h at an oxygen partial pressure (pO₂^{900 °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 pO₂^{900 °C} of 1 × 10²atm was also prepared by hot isostatic pressing.

Poling and aging treatments

Some samples were heated at 200 °C (>Curie temperature T_C of 120–135 °C) for 10 min in air to homogenize the distribution of V_O^••. 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 (<T_C) for 24 h (aging) to accelerate the diffusion of V_O^•• in the presence of P_s 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 pO₂^900 °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 V_O^•• (TM = Cu⁺, Cu²⁺, Cu³⁺, and V²⁺) were performed to find a stable site of V_O^•• 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 U – J 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 V_O^••-containing cells (Ba₂₇Ti₂₆TMO₈₀) were conducted to investigate the energetically favorable site of V_O^••. 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 n^th nearest-neighbor O site with respect to TM is defined as “On”, and the oxygen vacancy created by removing On is expressed as “V_On^••”. The V_On^••-containing cell is expressed as “Vo_n^••- TM”. For the calculations of DOS and band structures, the U – J of Ti-3d is set to 8 eV in a manner similar to that in the literature³¹.