Study of small polaronic, variable hopping conduction and its exploration by impedance analysis in I - substituted CaCu 3 Ti 4 O 12 at O - ion site by fine-tuning the electrical properties, grains and grain boundaries

In this study, dielectric and impedance related investigation are carried out in solid-state synthesized iodine doped CaCu 3 Ti 4 O 12-x I x (x= 0, 0.005, 0.05 and 0.2) at anion site in the varying temperature (300-500 K) and frequency (20 Hz - 1 MHz). The detailed analysis of dielectric and scaled plot (Z″, M″) confirm the Maxwell-Wagner relaxations unlike Debye type. Broad relaxation peaks in spectroscopic plots and nearly straight lines in admittance (Y″ vs. Yʹ) and ε″ vs. εʹ reveals the occurrence of various relaxation processes with a narrow distribution of time constants. Mismatch in peak frequencies of Z″/ Z″ max and M″/ M″ max suggest that relaxation mechanism is dominated by short-range (localized) movement of charge carriers. The twinning values of E a, τ ,M″ (≈ 0.067 eV) and E a, τ , Z” (≈0.081 eV) indicate the involvement of same type of charge carriers in conduction and relaxation processes. Correlating Jonscher’s power law and overlapping peaks in the combined plot of M″, Z″ vs. log f indicates high loss, due to DC conduction through localized hopping of small polarons via some defect states through Mott’s VRH mechanism, among the highly concentrated density of states, N (E f ), confined in the very low-temperature zone and nearest-neighbor hopping (NNH) of polarons due to Arrhenius in high-temperature zone both reserved in low-frequency region (≤1 kHz) only.

through nano-sized grains at oxygen site. To ensure the changes in crystal structure of the CCTO ceramics lattice parameter of CaCu3Ti4O12-xIx (x= 0, 0.005, 0.05 and 0.2) was calculated using equation = √ℎ 2 + 2 + 2 . Small deviation in lattice parameter from the standard value (7.391 Å) for all compositions were noticed as shown in table 1. This, may be i). due to differences in the ionic radius of the host oxygen anion (0.152 nm) and substituent iodine anion (0.22 nm) or, ii) due to the presence of Ti 4 as Ti 3 leading to Jahn-Teller distortions in TiO6 octahedra. [20] In zoomed view of (220) peak (Fig. S1) shows the shifting of (220) peak at 34 o towards low angle side for all compositions indicating a change in lattice parameter due to lattice distortions owing to compressive residual stress of crystal structures. [13] Table 1. Tabulated value of grain size, lattice parameter, crystallinity, and particle size.
Grain size/crystallite size, D for the bulk pellet was calculated using Scherer equation, D = Kλ/βcosθ , where K=0.9 or 1 is the structure factor, λ=wavelength of Cu-Kα radiation (1.54 Å), β is the FWHM (full-width half maxima) in radians and is given by β=√ (βᵒ − β i ), where β° is the width observed from the XRD peak, β i is the width due to instrumental effects and θ is the peak position value in radians. Peak gets broader as the grain size gets smaller (Fig. S1) justifying Scherer formula as FWHM is inversely proportional to grain size

Particle size (μm) SEM
Density and porosity measurement were calculated through Archimedes' principle using equations S1 and S2 (supplementary information). Theoretical density was calculated from the lattice parameter and molecular weight of the sample. Some pores were observed at the triple points at grain boundaries (Fig. 3). Porosity decreases with increasing the value of x and it reveals that iodine improves densification.

Microstructure charecterization
SEM images in Fig. 3 shows the typical micrographs of cuboid, and elongated rectangularshaped morphologies with particle size in range of 1 to 7μm. Segregation of Cu at grain boundary along with few secondary phases were confirmed via EDS analysis. Cu segregation led to the formation of the liquid phase at grain boundaries which assisted diffusion leading to grain growth. [13] As reported large grain size enhances the dielectric constants. [21] In this report largest grain size was noticed for CaCu3Ti4O11.8I0.2 (Fig. 3d). This lead to enhancement in dielectric constant from ≈1× 10 3 (undoped: CaCu3Ti4O12, Fig. 4a) to ≈ 1× 10 4 for doped composition (CaCu3Ti4O11.8I0.2; Fig. 5a) and it has also explained in accordance with equation (1) in the dielectric analysis of CCTO.  Maximum dielectric constant (≈2.97-0.29) × 10 3 and minimum loss (≈0.1-0.2) occurred for CaCu3Ti4O11.8I0.2 for almost entire temperature and frequency range (Table S2). In this case dielectric constant increased by 10 decades and loss decreased by 100 order at 20 Hz (Figs. 5a, 5b). This can be attributed to large grain size and formation of IBLC's. If it is assumed that, grain and grain boundary form a two-layer capacitor with a thickness (dg + dgb), where dg and dgb are thicknesses of grain and grain-boundary layer respectively. The effective dielectric constant for IBLC's is given by, ε s ' = ε ( + ) ---------(1), (where ε is the dielectric constant of the grain boundary layer, which even low, will give large, ε s ' for large ratio, ( + ), clearly showing its dependency on microstructure and hence maximum permittivity for large grains. [22] Prominent mechanisms that lead to enhanced dielectric constants are ferroelectricity, chargedensity wave (CDW) formation, hopping charge transport, the metal-insulator transition, and various kinds of interface effects. [23] Hopping conduction is the typical charge transport process of localized charge carriers. In electronic conductors, electrons/holes can localize due to disorder. In amorphous structure, the disorder may be due to doping (substitutional disorder) whereas in pure crystals it is due to slight deviations from stoichiometry or lattice imperfections. [23] The dielectric behavior can be judged in three regions.
Very low-frequency region, (20- which shifts exponentially to a higher frequency with temperature seems to be characteristics of Debye-type dipolar relaxation with thermally activated relaxation rate. [24] , [25]. This behaviour hints CCTO to be of non-ferroelectric nature due to absence of ferroelectric phase transition. But at the same time as the inflection point in εr vs. T and loss peak in tan δ vs. T are not at the same temperature (so Debye-type relaxation too is ruled out). Speculating now that CDC behavior may be due to the Maxwell-Wagner type extrinsic effect. At interfaces (metal to insulator contacts, intergrain boundaries) depletion layers are formed due to the formation of Schottky barriers forming BLC's yielding Maxwell-Wagner type relaxations. In general, if the work function of the metal electrode is higher than of electron semiconductor, then in the contact region of the semiconductor the electron concentration is curbed and a depletion layer appears which may lead to capacitive surface, forming BLC's giving rise to Maxwell-Wagner effect, which is also confirmed from humpy, semicircular arcs of Z″ vs. Zʹ, (Fig. 8a). [26] Now further increasing frequency but still in low-frequency zone ≤ 100 Hz due to inertial effect/ randomization, heavy ions/dipoles begin to lag behind the field rapidly decreasing the permittivity and enhancing the loss with increasing temperature. Furthermore by and large electronic polarization (10 14 -10 15 ) Hz, which is shifting of electron cloud with respect to nucleus, and ionic polarization (10 10 -10 13 ) Hz independent of temperature give rise to instantaneous saturation polarization. This do not make any contribution to loss due to their high frequency operating range in the measured frequency 20 Hz to 1 MHz and temperature 300-450 K. This clarifies reason for loss in all compositions as shown in Figs. 4d, 5d, S2d,

S3d.
Mid-frequency region, (100 Hz-1 kHz), ω < 1/ Owing to IBLC effect at grain boundaries (due to segregation of defects/ions or depletion region, porosity, grooves, Cu segregation, etc.) leads to dipolar polarization and enhances the dielectric constant but of lower magnitude and due to inertia and its randomization with temperature rise, the loss gets enhanced. [27] Correlating this with electrical microstructure as shown in Fig. 15d Very high-frequency region, (1 kHz -1 MHz), ω≫ 1/ dipoles, space charge do not follow such high-frequency field and won't respond at all so, εr= ε ∞ ≤ 10 2 is mostly due to instantaneous saturation polarization via electronic and ionic polarization, which remains almost independent of entire frequency and temperature range. So successively lowest magnitude εr and tan δ almost dormant with temperature were noticed. The underlying physics using Jonscher's power law revealed that it's the small polaron hopping/lagging within the grain, the reason for permittivity/loss as clarified in ac conductivity study.
By the way, high loss lowers CCTO's marketing zone. A profound study has to be done to resolve this issue. The dielectric loss of dielectric material results from (1) distortional (related to electronic and ionic polarization which is negligible due to their very high resonance frequencies), (2) interfacial (movement or rotation of atom or molecules in an ac field) and (3) conduction loss attributed to dc electrical conductivity which represents the actual flow of charge through materials or hopping mechanism via oxygen vacancy at grain boundary due to leaky grain boundary (i.e., low Rgb).

Loss in very low and mid-frequency zone
The plateau in low frequency region shown in the Mʹ vs. log f (Fig.13c) indicated negligible electrode polarization, also validated from the high value of Ea τs-el ≈ 1.2 eV as shown in Fig. 15c (concludes that SBLC effect too for loss is ruled out here). The overlapped peaks (very low-frequency zone) in the combined normalized plots (Fig. 14 (Fig. 15a, shown for CaCu3Ti4O0.995I0.005) suppresses the BLC effect, therefore, permittivity decreases with T, so ultimately loss gets maximized in low-frequency zone with temperature rise due to dc conduction and due to incomplete reversal of ions/dipoles owing to their inertia upon ac field reversal. [29] , [4] High-frequency Zone As tanδ = ωCR g , so in the high-frequency field, the feeble response of ions/charges within grain makes εr low, furthermore Rg decreases very rapidly with increasing temperature in the high-frequency zone (Fig. 8a.) so do the loss and both are, almost independent of temperature and frequency. In Figs. 4c, 5c, S2c, S3c, εr vs. log f, dispersion of εr in the low-frequency region is seen. Generally, this behavior is found for electrically insulating materials i.e. dielectrics in which mechanism of conduction via hopping type is present. [30] , [31] With increasing frequency the degree of dispersion decreases and beyond kHz no dispersion was found, also confirmed from Figs. 4a, 5a, S2a, S3a, favoring the Jonscher's UDR power law, (iωτ) β with a negative exponent (β ˂ 1).This indicates hopping of small polarons, as β increases with T, (Fig. 6b) via defect states. [31] This was also supported also by overlapping peaks in M″ vs. log f ( Fig. 13b and S8b) in the frequency≈1 MHz (evidence of long-range conduction in the high-frequency zone) for CaCu3Ti4O12 and. CaCu3Ti4O0.995I0.005 respectively. The core physics behind these are clarified using Jonscher's power law (UDR) in the coming section ac conductivity study.

Effect of AC field on dielectric properties (AC conductivity study)
AC electrical measurements are utilized in identifying the dominant conduction process within the grain and grain boundary in BLC's, based on the formation of enormous micro capacitors of grain as electrode and grain boundary as the dielectric in CCTO. [32] Conduction originates due to electrons, ions, and polarons (via. defect states). AC conductivity was calculated using σac = ωε0ε″--(3), ε* = ε′+ ε″ = is the free space permittivity. Real ac conductivity, σtotal=σ1(T)+σ2(ω,T)=σdc + σac=σdc + , the frequency dependence of conductivity is given by Jonscher's power law which is a common feature of amorphous semiconductors and disordered systems. . [33] , [34] Constant A, the strength of polarizability has a unit of conductivity, and β (T, ω) is a dimensionless parameter, represents the degree of interaction between mobile ions and lattice and its variation with temperature sheds light on the suitable process for the conduction mechanism. For an electrically heterogeneous structure with grain, grain boundary and sample-electrode interface, low-frequency conductivity is due to sampleelectrode interface, mid frequency due to grain boundary and high frequency as the bulk effect is a typical of thermally assisted tunneling between localized states. [34] σ in the highfrequency region is described by UDR (Universal Dielectric Response) given by the equation, A(iωτ) β . [35] The variation of β with temperature can be taken as a criterion for conduction mechanism. [36] According to Jonscher's power law if β decreases with T it indicates hopping conduction mechanism. [37] If β increases with the temperature a small polaron is the predominant mechanism, whereas the large overlap polaron is characterized by a minimum followed by an increase of β with a further increase of the temperature. In cases when β is temperature independent a quantum mechanical tunneling is expected. According to Funke when β is equal to 1 then the hopping motion involves a translational motion with sudden hopping and when β >1 then the motion involves localized hopping in the neighborhood without leaving its position. [37] The frequency at which change in slope takes place is known as the hopping frequency of polarons, ωp which is temperature-dependent.
Thus ac conductivity follows Almond West relation, where ωp is the hopping frequency and β is the Jonscher's constant. [38] , [39] As ωp =σdc according to . [37] The high-frequency variation of σac was found to obey Jonscher's power law, σac =A(ω) β ,------------ (8), where ω is the angular frequency of the ac field. [40] The plot of ln σtotal vs. ln ω (Fig. 6a) shows frequencyindependent ac conductivity at very low frequency ascribed as dc conduction (σdc) and dispersion of ac conductivity at higher frequencies is seen with the change of slope at a frequency known as hopping frequency (ωp) which is shifted to higher frequency side with an increase of temperature for all samples. By nonlinear fitting ln σtotal vs. ln ω at various temperatures σdc, A, the strength of polarizability and β can be determined. The plot of σdc, β, and A vs. T in the Fig. 6b shows an increase of σdc and β with temperature. The increase of β with T clues small polaronic conduction as the dominant conduction mechanism. Here β is greater than 1 (here β≈5. 44) indicates localized hopping motion of small polarons without changing neighborhood position. [37] Also, coherence length, L (radius of polaron), called the size of polarization cloud, Lcoherence=D 1/3 was calculated which was less than the lattice parameter for all compositions again indicating small polaron hopping conduction. [41] Thus strongly favors the result got from power law as shown in Fig. 6f for various compositions. The Plot of σac vs.1000/T obeys Arrhenius law (Fig. 6d). Its linear fit shows E aσac which were found to decrease with increasing frequency as shown in inset Fig. 6e. This signifies an increase in ac conductivity with frequency, contributes minute loss in high-frequency zone. [34] Thus justifying Figs. 4b, 4d, 5b, 5d, S2b, S2d, S3b, S3d. Fig. 6a shows, the rapid increase of the high frequency ac conductivity for higher temperatures as compared to low T's. This strongly favors that ac conductivity is due to polaronic conduction which is both temperature and frequencydependent. Due to lagging of polaronic conduction loss occurs with increasing temperature and frequency which justifies the Figs. 4b, 4d, 5b, 5d, S2b, S2d, S3b, S3d. Polaronic relaxation means existence of UDR in high dielectric constant materials. [34] , [42] The plot of A vs. T in Fig. 6b shows a rapid decrease of strength of polarization ≈ 330 K then shows almost becomes temperature independent behaviour in high T zone. This explains the reason for the decrease in εr with temperature rise and further its temperature independency in hightemperature zone and rationalizes (Figs. 4a, 5a, S2a, S3a) very well. The ac conductivity data has been used to calculate the density of states, N (Ef) at Fermi level using the relation σac , where is the electronic charge, f0 is the photon frequency, is the localized wave function. Assuming f0=1013 Hz and =1010 m -1 at various operating frequencies and temperatures. [37] The plot of N (Ef) vs. ln ω at various temperatures shows that the density of states increases with increasing temperature in the low-frequency zone and decreases exponentially in the high-frequency zone for all temperatures. So high N (Ef) in low-frequency zone enables hopping between pairs of sites thereby increase of loss in the low-frequency zone with temperature is seen. On similar note drastic decrease of density of states in the high-frequency zone decreases loss.
AC conductivity in the high-temperature zone can also be explained based on oxygen loss at high temperature, according to Oo=½O2+ ·· +2e ʹ , where Oo, O2 , ·· and e ʹ represent oxygen at oxygen site in the lattice, oxygen gas, a doubly positively oxygen vacancy and negatively charged electron respectively following with Kroger-Vink notation of defects. The electron released in the process makes the material semiconducting. [14] In CCTO, Ti 3 and Ti 4 form Ea is the activation energy for hopping, kB is the Boltzmann constant and T is the absolute temperature. [34] Mott first pointed out that at a low temperature most frequent hopping process is not due to nearest neighbor and Mott's variable range hopping conduction sets in. [34] The VRH  , --- (11) and Hopping Range, ] / , --- (12), where W is the Hopping energy, R is Hopping Range of polarons respectively. From equations (10), (11) and (12) One obtains, W=0.25k B T 3/4 1/4 --- (13).
The density of states, N(Ef)=3.36 × 10 30 eV/m 3 was calculated using equation (9) by taking ξ=a=0.37 nm, the distance between neighboring Ti ions of CCTO. [34] It is clear now, dc conductivity was due to VRH of polarons in the very low-temperature zone, the hopping range of which decreases with temperature but hopping energy increases with temperature  Table S3.

Impedance Analysis A ceramic piece consists of small crystallites called grains joined
randomly with strained and mismatch bonds. Properties of grain boundary differ from grains.
For device applications, the electrode is required. Thus, an electronic component may now be treated as a grain-grain boundary-electrode system. Properties of these are distinct and highly sensitive to processing variables as heating-cooling rates, atmosphere, starting ingredients, etc. Interplaying with these desired properties can be achieved. High permittivity in BLCs unlike ferroelectrics or relaxors (which shows high permittivity due to polarization) may be associated with sample geometry, thin layer effects associated with grain, grain-boundary and sample-electrode interface governed by the equation C= ʹ , (14) where ε ʹ is the overall permittivity of dielectric regions, A, the cross-sectional area and d its thickness. Equation  Table S1 shows the wide range of phenomenon which can be studied ranging from low-frequency electrochemical processes to high-frequency effects associated with dielectric relaxations and ion hopping between. [44] The various phenomena are characterized by the difficulty and/or the frequency of their occurrence. Thus, lattice vibrations occur at the infrared frequency (10 12  inversely with the capacitive contribution of that part. The value of the resistance can be deduced using ω * RC=1, where ω * is the angular frequency at the peak of the arc and C is the capacitance of that contribution. [14] The relaxation time, for one charge transfer process/polarization process is given by =RC= Second mode of analysis (ii) Impedance or Modulus Spectroscopy process. In immittance analysis, Z"and M" vs. log f is plotted. If the relaxation time (τ=RC) for the different process are widely different, then one observes a peak corresponding to each process. The peak position gives the relaxation frequency. Height of the peak in Z" vs. log f gives 2 at which loss is maximum. Z'' vs. log f highlights the process with maximum resistance (grain boundary or electrode specimen interface). Height of the peak in M"vs. log f is given by where Co is the capacitance of empty cell with air as dielectric and C is the capacitance with dielectric. Since these immittance functions have inverse relations, they can be used simultaneously to focus on certain information that is being suppressed in other representations. Complex plane plots are convenient to extract the capacitance and dc resistance of various contributions thereby dc conductivity can be measured. Whereas spectroscopic plots give the relaxation frequencies. Impedance and modulus plots give complementary information. Impedance plots highlight the process with maximum resistance (grain boundary or electrode specimen interface), while the modulus plot highlights the process with the least capacitance (grain or bulk contribution). If the capacitance is in the range≈ pf then the contribution is due to bulk (grain) bulk and if ≈ nf then capacitance is due to grain boundary. Cgb > Cg indicates well-sintered ceramics with narrow grain boundary. [46]  indicates that parallel CPE to be included in the model. [46] CPE represents the distribution in the properties. CPE is a mathematical realization of the system in which the phase angle between the applied ac voltage and the resulting current remains independent of frequency [47] , [48] There are other views for possible sources of CPE, like CPE indicates inhomogeneity in electrode or formation of electrical double layer capacitor, but majority favors that CPE behavior arises due to the presence of distribution in time constants in the sample-electrode. [46] , [45] This makes ubiquitous use of CPE in impedance spectroscopy.

Analysis of grain and grain boundary behavior in CaCu 3 Ti 4 O 12 synthesized via solidstate route and sintered at 1050 o C for 12 hours
In general, for =1 and for widely separated time constants, i.e., vs.  vs. Over in (Fig. 8a), the decreasing diameter of the semicircular arcs of non-zero intercept with increasing temperature indicates a decrease in Rg, Rgb, Rs-el (high, mid, and low-frequency response), which reduces the barrier to the motion of charge carriers increasing electrical transport and increasing ac conductivity thereby contributing loss with frequency rise in their respective frequency zone, also verified by ln σac vs. ln ω plot, (Figs. S4a -S6a) from where it was found that activation energy for ac conductivity decreases with the increase of frequency enhancing conductivity which is the cause of loss in the high-frequency regime. In the plot Z″ vs. Zʹ, Fig. 8a, the center of the fitted semicircle lies below the abscissa indicating a departure from pure Debye type and confirming the Jonscher's universal power law. The inclination of a semicircle with respect to real axis with increasing temperature indicates that departure from Debye nature was partly thermally assisted. [49] , [50] In Figs. 8b and S7b, Z″ vs. log f, shows broad and asymmetrical peak around ( ″ ) which gets broader and shifts to high-frequency side with temperature rise, the peak broadening suggests spread of relaxation times i.e. existence of various temperature-dependent electrical relaxation phenomenon in the material. Also, peak shifting points that it is a thermally activated process (non-Debye type).
Again the merging of Z″ values at high frequency may be due to accumulation of charge carriers at interfaces (space charge) in the material. The most probable relaxation time τm= 1 ω m obeys Arrhenius law, ω m =ω 0 −E a /k B T , where ω 0 is the pre-exponential factor, kB is the Boltzmann constant, give the activation energy(E aωZ″ ≈ 0.087 eV) for most probable jump frequency. It's attention-grabbing that E aωZ″ (Fig. 8c) represents dielectric relaxation (for localized conduction). [25] The relaxation process may be due to immobile charges/electrons at low temperatures and defects at high temperatures. [47] At high-temperature zone the relaxation time, τ is less (means faster electrical relaxation process via defects).In Fig. 8d, Zʹ vs. log f, Zʹ (resistance) of the sample decreases with a sudden dip (due to sudden decrease in DOS, N (Ef)) with an increase of temperature in low-frequency zone exhibiting (NTCR) type behavior analogous to semiconductors and merge with very low magnitude plateau in higher frequency zone, for all temperatures which are due to release of space charge by decreasing potential barrier with increasing temperature. Thus enhancing ac conductivity with an increase of temperature at high frequencies, which again very well explains loss behavior in the high-frequency zone. [51] A similar explanation holds for CaCu3Ti4O0.995I0.005.   (Fig. 10 c) and ε″ vs. εʹ (Fig. 10d) result in nearly straight line indicative of more polarization processes with a narrow distribution of time constants. [46] , [45] The behavior of CCTO and CaCu3Ti4O0.995I0.005 are in a quite close resemblance.

Scaling Behavior
The Nyquist plot (Z″ vs. Zʹ) is useful for materials which possess one or more widely separated relaxation process of comparable magnitudes and obey Debye law. where ωm corresponds to the peak value of Z″ and M″ vs. log f plot. The co-occurrence of all curves/peaks in scaled impedance and electric modulus master curve clearly indicates that the relaxation mechanism is nearly temperature independent. Also, the FWHM observed in both plots is greater than 1.14 decades indicating relaxation to be temperature independent and non-exponential in nature. [49] The shift of Z″ and M″ peaks with frequency in their spectroscopic plots Figs. 8b and S7b and Figs. 12b and S8b indicates relaxation to be non -Debye type (i.e. poly-dispersive). These Modulus analysis for CaCu3Ti4O12 In Fig. 12a, M″ vs. Mʹ plot, one semicircle appears with the decreased diameter and high M″ value, with increase in temperature. This can be related to decrease in the contribution of grain-boundary resistance to the modulus total resistance with increase of temperature. [37] , [49] Information regarding electrical transport and polarization process can be collected via the electric modulus plot. In Mʹ vs. logω (Fig. 12c ),very low magnitude plateau in the low-frequency zone appears for all temperatures signifying negligible or absence of electrode polarization. [53] , [30] Further a sigmoidal increase in the value of Mʹ, tending towards M ∞ , the asymptotic value in high-frequency zone showing dispersion signifies conduction due to short-range mobility of charge carriers (via ions). [49] If instead of dispersion, M ∞ had been saturated at higher temperatures in higher frequency zone then the conduction would have been through the long-range movement of charge carriers. [49] M″ (ω) exhibits a slightly asymmetric peak, (M″max.) at which relaxation occurs, shifts to a higher frequency with increasing temperature (Fig. 12b). The most probable relaxation time, τ m follows Arrhenius law, gives activation energy, E aωM″ =0.067 eV. It's noteworthy that activation energy, E aωZ″ represents localized conduction (i.e. dielectric relaxation) and that of M″( E aωM″ ) represents nonlocalized conduction (i.e. long-range conductivity). [28] The twinning values of E aωM ″ = 0.067 eV and E aωZ″ = 0.081 eV suggests that conduction and relaxation processes occur by the same type of charge carriers. [28] It is notable in M″ (ω) ( Fig. 12b) that, considering parameters, M″max. the shift in fpeak suggests the variation in resistance, whereas the only variation in M″max. with no change in fpeak suggests a change in both resistance and capacitance. [28] However, Fig. 12b shows variations in both parameters of M″max. and fpeak indicating the variation in capacitance. The thin layer associated with grain and grain boundary and sample-electrode interface strongly support BLCs. Furthermore, both Z″ (ω) (Figs. 8b and S7b ) and M″(ω) (Figs. 12b and S8b) shows an overlapping in the highfrequency range (≈1 MHz) for all temperature range investigated, being characteristics of long-range conductivity process which is the reason for the loss in the high-frequency range [52] and similar explanation appears for CaCu3Ti0.995O0.005, (Fig. S7). The combined plot of Z″ and M″ vs. log ω for CaCu3Ti4O12 and CaCu3Ti11.995O0,005 shown in Fig. 13 provides information whether its localized/short range, ( i.e .defect relaxation) or nonlocalized (i.e. ionic or electronic conductivity) that governs the relaxation process. [53] The mismatch in the peak frequency in M″ and Z″ indicates relaxation process is dominated by the short-range movement of charge carriers and deviates from ideal-Debye like behavior, while their overlap indicates the dominance of long range movement of charge carriers. [28] In Fig. 13 plot Z″/Z″max. and M″/ M″max. vs. log ω, the mismatch in the peak frequencies for CaCu3Ti4O12 and CaCu3Ti11.995O0,005 suggests non-Debye like behavior and simultaneous occurrence of both long-range and localized relaxation with the short-range movement of charge carriers dominating. The long-range is dominated at a low frequency, which is known as dc conductivity in the absence of interfacial polarization. Localized electron is mobilized with the aid of lattice oscillation via hopping (Fig. 13). In addition the low magnitude of activation energy,(Figs.8c and S6c), E aωZ″ =0.088 eV (for CCTO) and E aωZ″ =0.089 eV (for CaCu3Ti11.995O0,005), favors the charge transport via small polaron hopping mechanism in high-frequency zone. [53] For one of the composition, CaCu3Ti4O0.995I0.005 it was found that Rg, R gb, R s-el decreases with temperature, (Fig. 14a).The time constants, τs-el>τgb>τg, (Fig.14c and Table 2) obeys Arrhenius format and show NTCR behavior with the activation energies much larger as compared to polarons. So clearly indicating it's the polarons hopping and lagging due to their low activation energies, managing the electrical behavior of CaCu3Ti4O12-xIx dominantly. Supervision, Writing-review and editing. MKG: Writing-review and editing.

Data availability
All data generated or analyzed during this study are included in this submission [and its supplementary information files].

Conflict of interest
The authors declare that they have no conflicts of interest.