Tailoring dielectric permittivity of epitaxial Gd-doped CeO2−x films by ionic defects

Engineering materials with highly tunable physical properties in response to external stimuli is a cornerstone strategy for advancing energy technology. Among various approaches, engineering ionic defects and understanding their roles are essential in tailoring emergent material properties and functionalities. Here, we demonstrate an effective approach for creating and controlling ionic defects (oxygen vacancies) in epitaxial Gd-doped CeO2−x (CGO)(001) films grown on Nb:SrTiO3(001) single crystal. Our results exhibit a significant limitation in the formation of excess oxygen vacancies in the films during high-temperature film growth. However, we have discovered that managing the oxygen vacancies in the epitaxial CGO(001) films is feasible using a two-step film growth process. Subsequently, our findings show that manipulating excess oxygen vacancies is a key to the emergence of giant apparent dielectric permittivity (e.g. ε′ ≈ 106) in the epitaxial films under electrical field control. Overall, the strategy of tuning functional ionic defects in CGO and similar oxides is beneficial for various applications such as electromechanical, sensing, and energy storage applications.


Introduction
Tuning parameters in material engineering are often correlated.For example, the presence of charged point defects in the lattice of a material leads to significant internal strain due to atomic and/or electronic rearrangement, as well as changes in local or macroscopic symmetry [1][2][3][4][5].These defects have been extensively employed as a critical component to manipulate the properties of functional oxide materials for their device applications such as resistive random-access memories, conducting electrolytes, and solid-state fuel cells [2,[6][7][8].Thus, effective control of such dynamic ionic defects and understanding their role are crucial in functionalizing highly tunable oxide materials.
The introduction of aliovalent cation into fluorite oxides deliberately arrange anion vacancies, which causes deviations from the typical fluorite structure and gives rise to multiple intermediate elastic/electronic phases [9].For instance, when the Ce 4+ ions of CeO 2 are replaced with rare-earth 3+ cations (e.g.rare-earth element, R = La, Sm, Nd, Gd), it induces the creation of oxygen vacancies (V O ) as compensating defects to maintain charge neutrality.In these systems, it is believed that their electromechanical properties are strongly coupled with the rate-dependence of dynamic V O under electric field.Recently, exceptionally large electromechanical effects in the doped fluorites have been found, prominently appearing at low frequencies (f < 10 Hz) [1,10].Similar large electrostrictive effects were also found in nonlinear materials such as bismuth oxides (Bi 7 Nb 2−x Y x O 2−2/x ) and halide perovskites (MAPbI 3 ) [11,12].These materials with nonlinear properties under field excitation typically exhibit intrinsic relative permittivity denoted as ε r in the range of approximately 20-70, particularly at relatively high frequencies (>1 kHz).This is generally determined by applying low electric fields, typically a few kilovolts per centimeter or even lower.The permittivity values for many nonlinear oxides and similar systems can be found in [13].However, under sufficiently high electric fields, dynamic V O redistribution can significantly affect the apparent mechanical and electric displacement of the systems.This arises from macroscopic oxygen variations and charge rearrangement [1].The dielectric permittivity is a key property of electromechanical coupling where the polarization (P)-induced electrostrictive strain is expressed by x = QP 2 = M(P/ε) 2 , where Q and M are the polarization-induced and electric field-induced electrostrictive coefficient, respectively [14].Hence, it is essential to understand the correlation between the role of ionic defects and the corresponding relaxation behavior of apparent dielectric permittivity under electric fields.Subsequently, a recent study showed the potential for generating substantial piezoelectricity in centrosymmetric cubic Gd-doped CeO 2−x (CGO), which nominally forbids piezoelectricity.This achievement was made possible by utilizing the piezoelectric effect induced by a static electric field [1].The control of the V O dynamics within a polycrystalline CGO film was achieved by simultaneously applying static (E DC ) and driving alternating (E AC ) fields.The size of the induced piezoelectric effect of the CGO is remarkably large with apparent coefficients of ≈100 000 pm V −1 at low frequencies, while the relatively high-frequency piezoelectric coefficient is ≈100 pm V −1 at f = 1 kHz.In parallel, it was found that the apparent dielectric permittivity (ε = ∂P/∂E AC ) of the system shows a strong frequency dependence, dominated by the static field-forced migration (/hopping) of V O .These results directly indicate that the impact of dynamic V O under the applied field strongly governs both the dielectric and electromechanical properties of the system.
The electromechanical properties of CGO have been reported for polycrystalline materials including particles, ceramics, and textured microscale films for their potential electromechanical applications such as actuators and sensors [1,11,13,15].These materials inherently possess microstructural characteristics such as varying grain sizes, shapes, orientations, and grain boundary density.These characteristics significantly influence ionic motion and transport within the system.Often, they lead to non-uniform or diminished dielectric and electromechanical properties, making it challenging to establish a direct correlation between field-induced V O dynamics and the resulting electrical polarization of the system.
To gain a clearer understanding of these correlated effects, utilizing single-crystalline CGO film materials is a straightforward approach in conjunction with optimizing epitaxial film growth parameters such as growth temperature and background gas pressure.Tuning the strain of epitaxial CGO films has been effectively evoked by varying the thickness of the films on distinct underlying materials/substrates and external dopant concentration [16][17][18].However, some of the research has shown that there is no significant change in the strain of CGO films (thickness > 50 nm) grown at high temperature (e.g.T > 500 • C) although the films were prepared under different oxygen background pressure [19,20].This indicates the limited control of V O contents in the employed film growth regime.To the best of our knowledge, there has been no direct investigation yet on the control of V O contents and the associated strain effect in epitaxial CGO films.This exploration is beyond the compensating effect of V O concentration in the CGO system, typically determined by Gd doping concentration.Therefore, understanding the growth characteristics of epitaxial CGO films is paramount to clarify the role of the compensating and excess V O contents for exploring and controlling the strain effect and correlated functional properties.
Here, we demonstrate the influence of growth conditions of epitaxial CGO(001) films on Nb-doped SrTiO 3 (001) single crystal with the aim of controlling ionic V O defects.Our results reveal a significant limitation to the oxygen reduction within the epitaxial CGO films during high-temperature growth (e.g.T = 700 • C) regardless of the oxygen background pressure.However, an effective control of Vo contents in epitaxially grown films is achievable by employing a two-step growth strategy-an initial growth of CGO buffer layer at high temperature (T = 700 • C), followed by a subsequent low-temperature (T = 400 • C) growth of CGO film.Subsequently, we show that manipulating the excess V O in CGO is a crucial factor in generating high dielectric permittivity under electric field control.

Film fabrication
Epitaxial CGO films were grown on TiO 2 -terminated Nb(0.5 at.%)-dopedSrTiO 3 (Nb:STO)(001) substrates [5 × 5 × 0.5 (z) mm 3 size] by using pulsed laser deposition (PLD) (a KrF excimer laser, λ = 248 nm).For film growth, a CGO (Gd: 20%) ceramic target was used and the target-to-substrate distance was fixed at 4.5 cm.All the films were grown by a laser energy fluence of approximately 1.5 J cm −2 with a repetition rate of 3 Hz.Two different growth processes were employed: (i) 700 • C for one-step growth and (ii) 700 • C for the CGO buffer layer and 400 • C for the main film in a two-step growth.The V O contents in the grown samples were controlled by adjusting oxygen partial pressure (P O = 1 × 10 −1 to 7 × 10 −6 mbar).After the film growth, all the film samples were cooled down to room temperature at the same P O used during growth.For verification work, SrRuO 3 (SRO) layers (∼15 nm-thick) grown at 700 • C in P O = 1 × 10 −1 mbar were employed prior to CGO film growth (see the supplementary information).

Sample characterizations
Structural properties of the grown CGO films were investigated using an X-ray diffraction (XRD) analysis (Rigaku, SmartLab X-ray Diffractometer) at room temperature using Cu-Kα1 radiation (λ = 1.54056Å).Atomic force microscopy (AFM) was employed to study the surface topology of the Nb:STO and grown CGO films.AFM topology was analyzed using a Bruker Dimension Icon.ScanAsyst-Tapping mode was used with a ScanAsyst-Air tip (0.4 N m −1 ).The scanned size was 5 × 5 µm 2 with a 1 Hz scan rate.X-ray photoemission spectroscopy (XPS) measurements were performed under ultra-high vacuum (base pressure = 3 × 10 −11 mbar) using an ESCALAB Xi+ x-ray photoelectron spectrometer microprobe (Thermo Fisher Scientific, Ease Grinstead, UK) and a monochromatic Al Kα x-ray source (hv = 1486.6eV).The emitted photoelectrons from the surface of the samples were collected with a take-off angle of 90 • .The binding energy scale was calibrated to the C 1s core-level peak at the binding energy of 284.5 eV with an overall energy resolution of 470 meV.The XPS spectra were fitted using a Tougaard background subtraction and Voigt line shape.Compositional ratios of the elements (Ce, Gd, and O) in the films were determined after correcting for the inelastic mean free paths of the emitted photoelectrons via the TPP-2M formula [21] and by applying a Scofield photoionized cross-section [22].The impedance measurements were performed by using a Solartron 1255B Frequency Response Analyzer and a 1296 Dielectric Interface.The complex impedance (Z * ) of the film samples was measured by applying alternating voltages (V AC ) as where I is the measured current, Z ′ and Z ′ ′ are the real and imaginary parts of complex impedance, respectively.The complex permittivity (ε * ) of the samples can be derived from the relation: where ε ′ (ω) and ε ′ ′ (ω) are the real part and imaginary part of the permittivity, respectively, and C 0 is the vacuum capacitance.Both ε ′ (ω) and ε ′ ′ (ω) were extracted from the impedance in a conventional way as the following equations: where d is the thickness of CGO film, A is the electrode area, and ε 0 is the vacuum permittivity.
To fit the dielectric spectra of the CGO film samples, the two main dielectric relaxation processes across the frequency range are given as: The real part (ε ′ MW ) and imaginary part (ε ′ ′ MW ) of the complex Maxwell-Wagner (MW) functions can be expressed by: The real part (ε ′ U ) and imaginary part (ε ′ ′ U ) of the complex universal dielectric relaxation (UDR) functions can be described by a power law as:

Results and discussion
To investigate the effect of V O on the lattice strain in grown films, we prepared CGO film samples by PLD with different oxygen partial pressures, P O = 1 × 10 −1 mbar, 1 × 10 −2 , 5 × 10 −3 , 5 × 10 −4 , and 7 × 10 −6 mbar.Note that the P O = 7 × 10 −6 mbar was the base pressure at 700 • C for the PLD chamber used for film growth (see the Experimental section).All the CGO films were epitaxially grown on TiO 2 -terminated Nb:STO(001) single crystal at 700 • C and exhibited atomically flat surfaces (figure 1(a)).
The thickness of the grown films kept being similar in the range of ∼78 to ∼85 nm, where the films are fully relaxed without significant epitaxial constraint in the given growth condition (see figures 1(b) and (c)).Figure 1(d) shows XRD θ-2θ patterns of the CGO films as a function of P O , grown on Nb:STO(001).We observed no significant change in the CGO(002) peak positions and reciprocal space map (RSM) CGO(024) reflections of the grown films with different P O (figures 1(d) and (e)).This suggests no prominent variations in the out-of-plane strain (x z < +0.018%) of the films.This was calculated by using the c-lattice parameter (5.40 ± 0.01 Å) of the CGO film grown at 1 × 10 −1 mbar.These results indicate that changing the oxygen partial pressure in the chamber seems not to influence the films with the P O .To further confirm this, new films were grown by similar reduced conditions (e.g.T = 700 • C and P O = 5 × 10 −4 mbar) at different PLD chambers in different places (MLU, Germany).Similarly, the results show no significant change in the CGO(002) peak position of the films grown with other P O (figure 1(f)).These results were confirmed in different labs: (i) a CGO(001) film grown on Nb:STO(001) at 1 × 10 −1 mbar at Technical University of Denmark, (ii) a CGO(001) film grown on Nb:STO(001) at P O = 1 × 10 −1 mbar at Martin-Luther-Universität (MLU), and (iii) a CGO(001) film grown on Nb:STO(001) at P O = 5 × 10 −4 mbar at MLU.The change of P O during the high-temperature growth and cooling process does not cause any significant variation in the lattice constant, from which we can infer that the pressure in the given range does not impact on the V O concentration in the grown films, and that the deviation from V O concentration is small.It is well-documented that oxygen-deficient ceria and ceria-based materials have a propensity to readily react and store oxygen [23,24].Consequently, the ability to manipulate the content of oxygen vacancies in the doped films is restricted under the prevailing high-temperature film growth conditions that encourage oxidation activities.
To alleviate such an oxidation effect observed during high-temperature film growth, we employed a two-step film growth: (i) an initial growth of oxidized CGO buffer layer (approximately 10 nm thickness) at T = 700 • C and (ii) the subsequent 100 nm-thick film growth at T = 400 • C as schematically illustrated in the inset of figure 2(a).The assumption is that the low-temperature growth can minimize the oxidation during and after CGO film growth assuming that the oxidation in oxides typically becomes significant above ∼500 • C with the surrounding oxygen environment [25].Figure 2(a) exhibits the XRD θ-2θ patterns of the CGO films on TiO 2 -terminated Nb:STO(001) as a function of P O , using the two-step growth technique.The high-temperature CGO buffer (∼10 nm thick) was grown on Nb:STO(001) with a positive in-plane strain of ε x ≈ +1.8% and a negative out-plane strain of average ε z ≈ −0.5%, confirmed by RSM.A corresponding Poisson coefficient of ≈0.31 was measured for the elastic distortion of the epitaxial thin film, which is consistent with the reported values [26].The biaxial lattice strain occurs due to the lattice mismatch between CGO (a [100] = 5.425 Å) and Nb:STO (a [110] = 5.52 Å).
The low-temperature (400 • C) CGO films were grown by varying P O after the growth of a consistent high-temperature CGO buffer layer (T = 700 • C in P O = 0.1 mbar).The thickness of the films remained constant at about 100-120 nm.Notably, we observed that the oxidized CGO buffer layer effectively promoted the epitaxial growth of CGO(001) films even at the low temperature.The epitaxial relationship between the CGO and Nb:STO was found to be  2(a)).RSM further confirms that the in-plane lattice of the CGO film (approximately 120 nm-thick) grown in P O = 1 × 10 −1 mbar is still under epitaxial strain of ε x ≈ +0.5%.This further suggests that the low-temperature film growth prevents oxygen diffusion/reorganization in the film.In contrast, it was found that the in-plane parameter relaxes toward the bulk value (Gd 20%, a = 5.425 Å) [1] and the out-of-plane c-axis expands with ε z ≈ +0.72% when lowering the P O (figure 2(b)).Thus, the reduced CGO films are tetragonally distorted, verified by measuring their asymmetric {024} planes at four different φ angles (0 • , 90 • , 180 • , and 360 • ) in XRD.However, the CGO peak position no longer shifted with further increase in the reduction condition (P O ⩽ 5 × 10 −3 mbar), as shown in figure 2(d), probably reaching a limited V O concentration under the given growth condition.The observed out-of-plane lattice expansion could be attributed to the inclusion and/or preferential ordering/accumulation of V O along the (001) direction of the CGO(001) films, leading to a tetragonal distortion.The preferential out-of-plane lattice expansion was further confirmed by growing the same low-temperature CGO film layer (∼120 nm) on a CGO buffer (∼10 nm)/SRO (∼15 nm)/Nb:STO(001) (see supplementary figure S2).Similar tetragonal lattice distortion was observed in ceria, characterized by a cubic (space group: Fm-3m) to tetragonal (P4 2 /nmc) phase transition with approximately 1.3% volume expansion [27].Further, this phase transition is expected to be promoted and stabilized with a high V O concentration due to the associated stress, driven by the change in the host cation radii, i.e., Ce 4+ (97 pm) → Ce 3+ (114 pm).To verify the changes in the oxidation state of Ce, we performed XPS measurements for the CGO films, grown at P O = 1 × 10 −1 and 5 × 10 −3 mbar.Figure 2(e) illustrates the Ce 3d core level spectra for the films.The spectral fittings include the six peaks (v, v ′′ , v ′′′ , u, u ′′ , u ′′′ ) and four peaks (v 0 , v ′ , u 0 , u ′ ), corresponding to three pairs of spin-orbit doublets for the Ce 4+ valence state and two pairs of doublets for the Ce 3+ state, respectively [28].Clearly, we found that the surface of the film grown at P O = 5 × 10 −3 mbar possesses a higher Ce 3+ concentration [Ce 3+ /(Ce 3+ +Ce 4+ ) = 25.1 ± 1.2%], compared to the grown film at P O = 1 × 10 −1 mbar, which has a concentration of 11.3 ±1.3%.The corresponding surface oxygen deficiency, δ, of the Gd 0.2 (Ce 3+/4+ ) 0.8 O 2−δ films were determined to be ≈11.33%(P O = 1 × 10 −1 mbar) and ≈25.1% (P O = 5 × 10 −3 mbar), respectively, corresponding to a larger excess V O content (≈12.5%) in the grown film at the lower P O .Note that these XPS results only indicates a tendency for the V O formation in the film with P O as the probing depth is limited to the surface of the films.Hence, our results qualitatively suggest that the tetragonal CGO(001) films with the out-of-plane expansion are stabilized by a high V O concentration, together with the concentration (10%) induced by 20% trivalent Gd doping.
To investigate the effect of V O concentration on the electrical and dielectric properties of the CGO films grown at different P O , we performed impedance spectroscopy measurements.Sputtered Al (approximately 100 nm) was used as top electrode on all the film samples and the electrical measurements were performed in out-of-plane capacitor geometry as schematically illustrated in figure 3(a).To eliminate possible external effects (e.g. an interfacial effect between the CGO film and Nb:STO substrate or a dielectric response of substrate), all the side walls of Nb:STO substrates were electrically shorted to a bottom metal plate with conductive Ag paste.Moreover, no interfacial effects were found when compared two samples, CGO/Nb:STO and CGO/SRO/Nb:STO, showing almost identical electrical/dielectric responses (see the supplementary figure S3).The real part ε ′ (ω) and imaginary part ε ′ ′ (ω) of the permittivity were derived from the measured complex impedance [Z * (ω) = Z ′ (ω) + jZ ′ ′ (ω), where ω (= 2π f) is the angular frequency] in a conventional way.Details of the calculations and data fitting functions are given in the Experimental section.
Figure 3(b) shows the complex apparent dielectric permittivity of the CGO film samples grown at different P O (1 × 10 −1 , 1 × 10 −2 mbar, and 5 × 10 −3 mbar), measured by applying a constant alternating electric field (E AC = 3.4 kV cm −1 ) across a range of f from 100 mHz to 200 kHz.The results indicate a significant increase in the relative permittivity of the films when f → 0, while remaining high frequency permittivity (ε ′ ) of 25-28.Clearly, this feature is further obvious with the films grown at P O < 1 × 10 −1 mbar showing a strong frequency dependence with two prominent dielectric relaxation behaviors: (I) in the high frequency (100 Hz < f < 10 kHz) region, the ε ′ is around 70-100, which is certainly larger than the intrinsic high-frequency dielectric constant (ε ∞ ∼20-30) of CGO and similar oxides, and (II) in the low frequency regime (f < 100 Hz), the permittivity considerably increases (|ε r | > 10 4 ) following a power law [∝ ω − (1−s) , where s is a dimensionless exponent [29,30].Thus, two main contributions were considered to describe the observed dielectric response of the CGO films grown at relatively lower P O .The high-frequency dielectric response in the region (I) could reflect the conductivity difference between the relatively oxidized buffer and reduced film layer.These two elements lead to a Debye-like relaxation with a peak loss at ∼2-5 kHz, based on charge accumulation at the interface with different carrier relaxation times (τ ) in the two film layers, which stands for the releasing time of excess carriers across the interface [31].In this regime, the interface capacitance is larger than the capacitance of the film layers.Once the interfacial conductivity becomes comparable with or smaller than the conductivity of the film layer, the loss (ε ′ ′ ) increases toward lower frequencies as seen in figure 3(b).This is the so-called MW relaxation.In fact, this feature is absent in the film grown at P O = 1 × 10 −1 mbar as the conductivity of the two layers (buffer and film) are similar.On the other hand, we found an increase in both the ε ′ and ε ′ ′ of all the films when lowering f, but with different relaxation times.This is dissimilar to typical Debye-type dipolar or MW relaxation models, which lead to a ε ′ plateau and an increase in ε ′ ′ with a slope of ∝ ω −1 with decreasing f.If the exponent s < 1 (in equation (6.2)), it is an indication for the contribution of hopping conduction to the dielectric relaxation, which is associated with ionic migration within the CGO films, described by Jonscher's UDR [29].Thus, the conductivity of the CGO system can be described as σ (ω) = σ DC + σ U ω s , where σ DC is the dc conductivity and σ U is the hopping conductivity.Based on these, we employed complex dielectric functions for data fittings describing the two dominant relaxations, MW and UDR (see the Experimental section).Our fitting results indicate the largely increased ε ′ ′ and a faster relaxation toward higher frequencies due to the increased σ (ω) of the samples grown at lower P O .This implies the presence of a higher concentration of mobile V O species in the samples (P O < 1 × 10 −1 mbar), confirmed by XRD and XPS.
To further understand such relaxation behaviors, we performed the permittivity measurements for the two samples (P O = 1 × 10 −1 and 5 × 10 −3 mbar) at various temperatures (T = 298-333 K) (figure 3(c)).We found that the two samples show different relaxation behaviors: the film grown at P O = 1 × 10 −1 mbar only shows relaxation (II) over the measured f range due to the absence of MW effect, while the film grown at P O = 5 × 10 −3 mbar shows two types of relaxation (I) and (II).The transition time (t r ) for the characteristic dielectric relaxation of the samples was determined by linear extrapolation of the onsets to the relative plateaus.Interestingly, the t r shifts toward higher frequencies with a continuous increase in both the ε ′ and ε ′ ′ when increasing T (figure 3(d)), indicating that the ionic conduction also contributes an increase in the space charge polarization of the system.Figure 3(e) shows the Arrhenius plots of Ln(t r ) in 1000/T.The activation energy, E A (I), of the t r for the high frequency dielectric relaxation (I) was determined to be 0.40 ± 0.01 eV for P O = 5 × 10 −3 mbar.This could be predominantly associated with polarons hopping as an ionic conduction is limited at the high frequency regime [32].The E A (II) of the t r for the relaxation (II) was determined to be 0.50 ± 0.01 eV and 0.48 ± 0.01 eV for P O = 1 × 10 −1 mbar and 5 × 10 −3 mbar, respectively [1,33].The CGO films grown at lower P O possess the compensating oxygen vacancies (5% of oxygen sites due to the 20% Gd dopants on Ce sites) and additional oxygen vacancies (y), which stem from a reducing process step in the film growth (see figure 2) [34].The complete formula can be written in the Kröger-Vink notation as Ce × 0.8−2y Ce ′ 2y (Gd ′ Ce ) 0.2 Ce × 1.9−y (V •• O ) 0.1+y .The activation energy for the V O hopping is in a range of 0.45-0.70eV in undoped CeO 2−x , strongly dependent on environmental conditions and the microstructure [32,33,35].In doped ceria with 20% Gd, this value increases to 0.75 eV due to the trapping of V •• O near the Gd dopants [33].Hence, the excess V O (y) compensated by Ce 3+ can be more mobile than those associated with Gd, resulting in a large increase in the permittivity of the reduced CGO films with lower E A [1].Thus, when an electric field is applied to the sample at the low frequency regime, a macroscopic/long-range charge separation may appear as the V •• O and the small polarons (4f electron of Ce 3+ ) are diffusing along the field.Both have similar activation energies of approximately 0.5 eV.
Our observations suggest that the ionic conductivity of the system is key for tailoring the permittivity of the CGO system.We conducted the impedance measurements for the film sample (P O = 5 × 10 −3 mbar), excited by combined electric fields, simultaneous applications of a constant driving AC field (E AC = 3.4 keV) under different static fields (E DC = 0.04, 0.07, and 0.18 MV cm −1 ) as schematically illustrated in figure 4(a).Firstly, we examined the σ (ω) of the CGO film sample over a frequency range of 100 mHz-1 MHz (figure 4(b)).When the E AC was applied to the sample, it shows an ionic relaxation behavior at f < 5 kHz, expressed by σ ∝ ω 0.82 [1].The exponent range of 0 < s < 1 represents dispersive ionic diffusion, while at f > 5 kHz a universal limiting ac-conductivity was found with σ ∝ ω [36].Importantly, a large increase in the σ (ω) was found with s → 0 when simultaneously increasing the E DC .This indicates the occurrence of frequency-independent dc-conductivity, indicating greatly enhanced long-range ionic diffusion or hopping process [37].The static field-enhanced ionic conductivity is further signified with the level of excess V O when comparing the two samples (P O = 1 × 10 −1 and 5 × 10 −3 mbar) as shown in figure 4(c).Figure 4(d) exhibits the frequency dependent dielectric permittivity of the CGO film (P O = 5 × 10 −3 mbar) as a function of E DC .Notably, as the E DC increases (up to 0.18 MV cm −1 ), we observed a large increase in both the ε ′ (≈10 6 ) and ε ′ ′ (≈10 7 ) of the film at low frequencies (f < 1 Hz).Moreover, higher E DC strength effectively shifts the t r toward high frequencies (f > 1 kHz).These observations are consistent with our previous result [1].The enhanced V O hopping/conduction in the system is associated with lowered activation energies when the static fields are applied.Consequently, our results show that the dielectric permittivity of the single crystalline CGO films strongly depends on the activation energy of V O hopping and its frequency dependence under static field applications, which can be effectively tuned by incorporating excess V O .However, we found that effective ionic conduction for the emergence of large permittivity in single crystalline CGO films can be driven by a much smaller E DC strength (e.g.0.18 MV cm −1 ) than that required for polycrystalline films (e.g. 1 MV cm −1 ).This could be due to higher excess V O concentration and facilitated hopping by the absence of microstructural or other external effects (e.g., grain boundaries).

Summary
In conclusion, we demonstrate an effective way to control the V O content in epitaxial CGO(001) films grown on Nb:STO(001).Our findings show that oxygen reduction in the epitaxial CGO films is limited during high-temperature growth.In contrast, the two-step growth effectively induces excess V O in the single crystalline CGO(001) films, stabilizing tetragonal CGO films with a preferential out-of-plane lattice expansion.Subsequently, we found that controlling excess V O contents is essential for the emergence of giant apparent dielectric permittivity (e.g.ε ′ ≈ 10 6 ) of the system, which can be effectively tuned by simultaneous static electric field applications, potentially enabling the generation of substantial electromechanical coupling and piezoelectric effects.Therefore, this work provides crucial insights for designing fluorites or similar oxides using ionic defects for use in energy-efficient applications such as electromechanical and energy storage devices.
[110] CGO ||[100] STO and [001] CGO ||[001] STO as schematically shown in figure 2(c) and supplementary figure S1.Moreover, clear shifts in the CGO film peaks toward lower 2θ angles when the P O decreased to 5 × 10 −3 mbar, revealing the c-lattice elongation of the CGO films with lower P O (figure

Figure 3 .
Figure 3. Dielectric permittivity of the CGO film samples as a function of PO.(a) A schematic for the electrical measurement of the CGO films on Nb:STO(001), performed on an out-of-plane capacitor geometry.(b) The εr of the CGO films as a function of f, grown at PO = 1 × 10 −1 , 1 × 10 −2 , and 5 × 10 −3 mbar, measured by applying EAC = 3.4 kV cm −1 .(c) Variations in the εr of the CGO films, grown at PO = 1 × 10 −1 (left panel) and 5 × 10 −3 mbar (right panel), measured by EAC = 3.4 kV cm −1 at various T (298-333 K).The transient times, tr, for the characteristic dielectric relaxation in the CGO films were determined by linear extrapolation of the onsets to the plateaus.(d) The corresponding Cole-Cole plots of the films at variable T. Two characteristic dielectric relaxations (region-I and -II) of the CGO film (PO = 5 × 10 −3 mbar) appear when lowering the measurement f, while the dielectric relaxation of the CGO film (PO = 1 × 10 −1 mbar) is dominated in the low f regime (region-II).(e) Arrhenius plots of Ln(1/tr) in 1000/T for deriving the EA for the characteristic dielectric relaxation of the CGO films.