Crystal structure, bandgap, and the changes in the charge conduction mechanisms in ceramics are interrelated, and the underlying physics unifies all these different phenomena. The experimental and theoretical evaluation of the electronic properties of the solid solution of (1 − x)BaTiO_{3}–(x)LaFeO_{3} (x = 0, 0.015, 0.031, 0.062) is attempted in this work. Bandgap was observed to be tunable with La/Fe doping from 3.2 eV (x = 0) to 2.6 eV (x = 0.06), while the lattice disorder was found to increase. A theoretical assessment confirms a considerable shift of valence band maxima and conduction band minima with an introduction of additional defect states within the bandgap. Electron localization was also confirmed theoretically with doping. Such changes in the electronic properties were experimentally confirmed from dielectric/AC - conductivity/impedance spectroscopy studies. From different transportation models, hopping is a preferred mechanism in the less distorted BaTiO_{3}. However, a large polaron tunneling process can be justified for the doped samples at lower temperatures. Only at higher temperatures, a small polaron tunneling can be justified for the doped samples. The transportation is affected by the grain boundaries as much as the grains themselves. A complete analysis using Nyquist plots reveals the competing contributions of these regions to the transportation mechanism and is correlated to the disorder/distortions in the lattice in terms of the formation of oxygen vacancies.

## INTRODUCTION

Transition metal based oxide perovskites are widely used in optoelectronic devices due to their electronic structure, charge transport mechanisms, and defect states.^{1–3} The optical properties and electrical conduction mechanisms are correlated in optoelectronic materials.^{4–8} Optical memories, electro-optic devices, actuators, sensors, photovoltaics, photocatalysis, etc., are applications where bandgap tuning and associated changes in electrical and optical properties are essential to be explored in materials where there is the flexibility of modifications using various probes, including chemical modifications.^{9–12}

BaTiO_{3} (BTO) is a wide bandgap semiconductor and a well-known dielectric material with a bandgap of ∼3.2 eV.^{13} The bandgap can be tuned depending on the crystal structure and defects. The tetragonal structure of BTO has a higher bandgap than the cubic structure.^{14} This variation of hybridization of orbitals in the crystal structure is an important factor.^{15} Experimentally and theoretically, Fe-doped BTO^{16} and La-doped BTO^{17} were found to have a reduced bandgap. However, there needs to be more explanation of band structures. While some describe Fe-doped BTO leads to the modification of the Ti orbitals,^{18} others refer to the formation of Fe defect bands near the valence band.^{19,20}

(1 − x)BaTiO_{3}-(x)LaFeO_{3} (x = 0, 0.015, 0.031, 0.062) solid solution (BLFT) has been studied earlier for the structure-correlated multiferroic and magnetodielectric applications.^{21} Fe^{3+} is introduced at the Ti^{4+} site to induce magnetic properties in the ferroelectric BaTiO_{3} lattice. La^{3+} is substituted in Ba^{2+} sites to tune the electrical properties of BaTiO_{3}. Hence, to maintain charge neutrality of the lattice, La^{3+} substitution of Ba^{2+} seemed justified to maintain a ferroelectric component while introducing magnetism with Fe^{3+} substitution of Ti^{4+} in BaTiO_{3}. A pure tetragonal *P4mm* structure was observed for x ≤ 0.01, while for x > 0.01, a dual phase of tetragonal and cubic (*Pm-3m*) structure was observed. With increasing x, structural distortion increases. A mixed valence state of Ti^{3+}/Ti^{4+} and Fe^{2+}/Fe^{3+} were observed in these samples. Oxygen vacancies (O_{V}) increased with doping. For x > 0.01, an increase in the lossy nature of the samples was observed, hinting at a better transport property of these materials. Hence, a study of the electronic, dielectric, and transport properties of the doped materials become important. Knowing that these properties are correlated to the modifications in the electronic density of states (DOS), orbital shapes, etc., an attempt to correlate the theoretical and experimental results is of prime interest. The introduction of lattice disorder/distortions by introducing foreign elements and oxygen vacancies may lead to such modifications in the DOS and orbital shapes, thereby making this solid solution interesting.

The formation of lattice distortions often leads to a transformation of the transport mechanism from hopping to different types of tunneling mechanisms.^{22} The structural changes leading to changes in the electronic properties of these materials provide an ideal opportunity to investigate such possibilities. In this work, an attempt has been made to justify the experimental data with different transportation mechanisms, keeping in mind the grain and grain boundary contributions.

## METHODOLOGY

(1 − x)BaTiO_{3}–(x)LaFeO_{3} (x = 0, 0.015, 0.031, 0.062) are prepared through the sol-gel method as described in a previous report.^{21} The samples are named as BTO for x = 0, BLFT1 for x = 0.015, BLFT2 for x = 0.031, and BLFT3 for x = 0.062. The bandgap of all the samples was determined from the diffuse reflectance spectra (DRS) using a UV-vis-NIR Shimadzu 2600 spectrophotometer. The AC conductivity is studied using the dielectric measurements with the 10 mm pellets, which are sintered at 1350 °C for 4 h. Dielectric measurement has been done with silver electrodes on both sides. A Newton's 4th Ltd. dielectric spectrometer is used to measure with a signal strength of 1 V_{rms}. The data were obtained in the frequency range of 1 Hz–1 MHz from 323 to 563 K.

The total energy and electronic DOS calculations are based on density functional theory as implemented in the Vienna a*b initio* software package (VASP).^{23,24} Projected augmented wave (PAW) pseudopotentials in the generalized gradient approximation (GGA) have been used to perform the calculations, which are applied to the Perdew–Burke–Ernzerhof (PBE) exchange-correlation function.^{25} The calculations were performed on a unit cell (5 atoms) of tetragonal BTO, and a plane wave energy cutoff of 800 eV was used. The energy convergence criterion was set to 10^{−6} eV, and an 8 × 8 × 8 K-point mesh was used. We have also calculated the electronic DOS and total energy for La- and Fe-doped BTO using a (2 × 2 × 4) supercell (80 atoms). To simulate the Fe doped system, we have replaced one Ti by Fe out of 16 available Ti sites (BaTi_{0.9375}Fe_{0.0625}O_{3}) and additionally replaced Ba by La to simulate Ba_{0.9375}La_{0.0625}Ti_{0.9375}Fe_{0.0625}O_{3}. The total energy and electronic DOS in all the systems were calculated in fully relaxed structures. We have used denser 8 × 8 × 4 k-grid points in the Brillouin zone for the electron localization function and electronic DOS.

## RESULTS AND DISCUSSION

### Electronic bandgap studies

_{g}, was measured using the UV-visible absorption spectrum. The optical bandgap was estimated by fitting the absorption coefficient in Tauc's relation given by (αhν)

^{n}= C(hν–Eg),

^{26}where C is the proportionality constant, h is the Planck constant, E

_{g}is the optical bandgap, and n is 2 for the direct bandgap.

^{27}Tauc's plot of all the samples is given in Fig. 1. The extrapolation of the straight portion of the curve to the energy axis will give the bandgap. The structural disorder and the defects in the lattice are associated with the Urbach energy (E

_{U}).

^{28}The spectral dependence of the absorption coefficient α in the spectral region corresponding to transitions involving the tails of the electronic density of states is described by the Urbach equation,

_{0}is a constant, E

_{U}is the energy that reflects structural disorder and defects of a semiconductor.

E_{g} constantly decreased from the UV region ∼3.25 eV (BTO) to the visible region ∼2.6 eV (BLFT3) (Fig. 1). The band structure of BTO is formed mainly due to the Ti_{3d} (CB) and O_{2p} (VB) orbital.^{29} Changes in the overlapping pattern of the Ti and O orbitals can lead to modifications in the energy gap. Modifications of electronic hybridization ultimately redefine the energy gap. The shift of Ti^{4+} ions with doping leads to distortion of the TiO_{6} octahedra, which results in changes in the strength of hybridization in the Ti–O bond through crystal field splitting.^{30} Panchal *et al*.,^{31} with the help of x-ray absorption, studied the difference in the strength of Ti–O hybridization in tetragonal and cubic phases. The enhanced extent of hybridization of e_{g} and t_{2g} orbitals and a smaller crystal field due to the reduction in tetragonal distortion and lifting of the TiO_{6} octahedral symmetry led to a reduction in the effective bandgap. E_{U} increases from 126 meV for pure BTO to 917 meV for BLFT3 as shown in Fig. 1. This increase confirms the structural disorder in BTO with La and Fe substitution.

In BTO, doping causes structural asymmetry due to the displacement of Ba and Ti atoms. This introduces atomic orbital destabilization and causes Jahn–Teller (J–T) effect.^{32} In pure BTO, Ti is supposed to be in the Ti^{4+} state. This Ti^{4+} ion does not have a d-electron and hence cannot participate in a Jahn–Teller distortion. On the other hand, Ti^{3+} has one 3d-electron which can participate in J–T distortion. Similarly, Fe^{3+} is not a J–T ion but Fe^{2+} exhibits J–T distortion. In the BLFT samples, both Ti^{3+} and Fe^{2+} ions were observed suggesting an increase of J–T distortion. Displacements in Ti–O/Fe–O and Ba–O/La–O bonds for the BLFT samples^{21} ensure J–T effect. The Jahn–Teller distortions modify the energy and the splitting. Hence, a lowering of the bandgap is observed along with the reduction of the energies of the band edges. There can also be an increase of intermediate states in the bandgap.^{33} It has been observed that Fe generates defect states near E_{F}^{34} and inside the CB. It was reported that dopant cations and oxygen vacancies can form defect clusters in doped lattices and charge transfer via defect clusters.^{35} This inhomogeneous charge distribution allows the trapping of electrons. Hence, an increase in carrier concentration can be expected. For the states near E_{F}, one can also expect an increase in conduction mediated through these defect states. From XPS studies, multiple defects could be observed in terms of O_{V} and multiple oxidation states of ions. Such defects can also contribute to localized electronic states inside the gap. Electrons can be trapped in these states, thereby increasing the carrier concentration.

For the BTO, the absorption edge is sharp [Fig. 1(a)] which shows the negligible presence of defect states in the lattice and a very low value of E_{U}. In the case of doped samples, there is a gradual reduction in the sharpness of the absorption edge from BLFT1 to BLFT3. Such a reduction in the sharpness clearly indicates the presence of band tails due to strain from structural deformation, resulting in an increase in E_{U}. However, one should not rule out the presence of dopant-initiated states very close to the conduction band maximum (CBM) and valence band minimum (VBM). A theoretical assessment has been performed using DOS analysis in the next section to verify such effects.

A hump-like broad absorption peak is observed for BLFT samples [Fig. 1(b)]. These absorption edges correspond to the absorption due to intermediate states. As the defect states increase, doping leads to an increase in the intensity of this pre-absorption edge at a lower energy than the bandgap energy. DOS analysis is essential to understand such states responsible for the pre-edge absorption.

### Density of states analysis

To investigate experimentally obtained reduction in bandgap and additional absorption pre-edge, electronic total density of states (TDOS) and partial density of states (PDOS) in BTO and BLFT3 have been performed using the fully relaxed crystal structure, as shown in Fig. 2. For simplicity, E_{F} is shifted in both compounds to 0 eV. The electronic density of states shown is mainly dominated by Ba 5p, Ti 3d, O 2p, Fe 3d, and La 5d states. The partial density of states of Ba 5p orbitals near the Fermi level indicates Ba has a small electronic contribution to the conduction band; it acts as a charge balancer to balance the system charge.^{36} The contribution from La is not significant in BLFT3 compared to Ba in electronic DOS.

In the case of BTO, the valence band and conduction band edges near E_{F} show a large density of states and sharp edges compared to BLFT3. This correlates with the sharp absorption edge experimentally obtained in the UV–visible absorption spectra of BTO, revealing lower E_{U} but a tailing absorption edge for doped BLFT samples revealing higher E_{U}. The bandgap of both the samples were found to be of the order of ∼1.6 eV. The VBM of BTO was observed at 0.031 eV lower than E_{F} while that of the BLFT3 was observed at 0.799 eV lower than E_{F}, a shift to a lower energy by 0.768 eV. On the other hand, the CBM of BTO was observed at 1.507 eV above E_{F} while that of the BLFT3 was observed at 0.803 eV above E_{F}, a shift to a lower energy by 0.704 eV. Hence, both VBM and CBM display a shift toward lower energies for the BLFT3 sample as compared to BTO. This change in the shift of CBM and VBM leads to moving E_{F} closer to the CBM. A point to observe is that in the BTO, E_{F} is close to the VBM while it moves toward the middle of the gap with doping. The closeness of E_{F} to the VBM in the case of BTO is indicative of a p-type nature of the sample. On the other hand, the Fermi level rises with doping for BLFT3, indicating an increase in the negative carriers and a decrease in the positive carriers.

In the case of BTO, the CBM is contributed mainly by the Ti states, while the VBM is contributed by the O states. Hence, the bandgap is mainly associated with the hybridization of Ti–O bonds. In the BLFT3 sample, apart from the O states, the Fe states also contribute to VBM. On the other hand, the CBM is a contribution of not only the Ti-states but also the Fe-states. Hence, Fe seems to affect both the VBM and CBM. Hence, the presence of Fe in the lattice modifies the electronic charge distribution in such a manner that shifts the Fermi level to a higher position in the bandgap.

It is notable that in BTO, there are no defect states inside the bandgap. However, with the advent of La and Fe in the lattice, a considerable amount of defect states is observed near E_{F} of the BLFT3 sample. These states are contributed mainly by Fe, followed by O and nominally by Ti. In a p-type semiconductor, E_{F} is generally shifted toward the VBM due to the acceptor level near the VBM, while in a n-type semiconductor, E_{F} is generally shifted toward the CBM due to the donor level being near the CBM. Hence, the shift of E_{F} toward the middle of the gap may be a consequence of the presence of these defect states at the middle of the band in the BLTF3. The presence of both Ti^{3+}, Ti^{4+} and Fe^{2+}, Fe^{3+} ions in the BLFT samples was observed from XPS studies.^{21}. As a result, O_{V} was also found to increase, which changes the density of states. Such changes in the cationic valence state and oxygen content are expected to generate changes in the DOS. Such changes are hereby observed in terms of the new states inside the bandgap. Thus, the lowering of the CBM and VBM with respect to E_{F} is correlated with the changes in the cationic valence states.

The pre-edge absorption found experimentally from absorption spectroscopy [Fig. 1(b)] is a consequence of the mid-band defect states. Photo-excited electrons from the O-states in the VB to Ti-states in the CB will be significantly affected by the changes in the effective charge of the Ti atoms and the absence or presence of the O atoms. Such changes, along with the introduction of the Fe-states at both the VBM and CBM, are also partially contributors to the shifting of the E_{F}. Such effects also may be responsible for the loss of sharpness of the VBM and CBM edges. The observed defect states within the bandgap can trap the electrons and act as localized distortions or polarons in the lattice.^{37} These polaron formations and how the electron charge distribution between Ti and O varies with doping can be analyzed with electron localization function calculation detailed in the section on “Electron localization function.”

## ELECTRON LOCALIZATION FUNCTION

It is necessary to understand how the doping of La and Fe affects the BTO lattice in terms of electronic charge distribution, in other words, the hybridization of the atoms. A theoretical estimation can be obtained by comparing the electron localization plots of a pure and doped sample. In this study, such a comparison is made between the pure BTO and the maximum doped BLFT3 samples. We have shown electron density distribution in ab-planes, separately for the Ba–O planes [Figs. 3(a) and 3(b)] and the Ti–O planes [Figs. 3(c) and 3(d)]. The warmer red color depicts the high electronic charge density, and the colder blue color represents the absence of electrons. The covalent nature of the bonds is revealed from the contours of charge density. For the Ti–O planes, as expected, the charge distribution around each Ti and O atoms in BTO are identical. However, with doping, a distortion in charge density distribution appears in the vicinity of the dopant atom and modifies the surrounding lattice to some extent. Note that in the doped samples, the blue-colored regions were increased in size as compared to the pure BTO lattice, revealing the localization of electrons in the BLFT3 lattice. As a result, the continuity of the redness in between two oxygen atoms is reduced in the doped lattice. This hints at the localization of electrons or the formation of polarons in the lattice due to La and Fe doping. Similarly, in the Ba–O planes, variations are observed around the La sites. Note that the blue-colored regions have intensified between the Ba atoms due to doping, and the continuity of the redness has disappeared. This further strengthens the claim of localization of electrons leading to polaron formation in the lattice.

From the localization of electrons, it seems that a stronger hybridization between the Ti and O atoms is present in the BTO lattice than in the BLFT3 lattice. Similarly, the same can be inferred for the hybridization between the Ba and O atoms. This means that the bond strength of both the Ti–O and Ba–O bonds weakens with doping. The polaron formation is strong evidence of improvement in the conduction process. Hence, studying the conduction process becomes extremely necessary in this context, leading to an AC conductivity study.

### AC conductivity studies

AC conductivity (σ_{ac}) can be obtained from the formula σ_{ac }= 2πfε_{0}ε″, where f is the frequency (Hz), ε_{0} is the permittivity of vacuum, and ε″ is the imaginary part of dielectric permittivity. Frequency-dependent σ_{ac} (10 Hz–1 MHz) was estimated in the temperature range of 323–563 K [Fig. 4(a)] to understand the conduction mechanism of the BLFT samples. Three distinct regions appear in three different frequency ranges of the σ_{ac} data:

a low frequency regime (f < 10

^{3}Hz),an intermediate frequency regime (f ∼ 10

^{3}–10^{4}Hz),a higher frequency regime (f > 10

^{4}Hz).

The low-frequency regime reveals a plateau region and generally corresponds to the total DC conductivity of the grain and grain boundary.^{38} The intermediate frequency regime reveals the first dispersion region and is generally a characteristic of localized charges in the grains and trapped charges in the grain boundaries.^{38} The higher frequency regime reveals a second dispersion region and is representative of the presence of an increased mobility of the charge carriers.^{39} This change in the conductivity with frequency is correlated to the jump relaxation model (JRM) proposed by Funke.^{40} It depicts a transition from long-range hopping to short-range motion due to the short time available for carrier movement at higher frequencies. The successful hopping of charged species to nearby available sites contributes to the frequency-independent plateau at low frequencies. At high frequencies, conductivity is the result of two processes. (i) the correlated forward and backward motion of charge carriers, which results in unsuccessful hopping, and (ii) the motion of charge carriers to the new sites, leading to a relaxation process and thereby a successful hopping. The more dispersive nature at higher frequencies is due to the higher rate of successful to unsuccessful hopping.^{39} Hence, conductivity at different frequencies is explained using the model Jonscher's double power law: σ_{ac} = σ_{dc} + A*w*^{n }+ B*w*^{m}, where A, B are constants, n and m are frequency exponents. Long-range translational motion of charge carriers contributes to the dc conductivity σ_{dc}. Short-range translational motion contributes to the low-frequency dispersion region A*w*^{n} where 0 < n < 1. Re-orientational motion at high frequencies contributes to the third term, B*w*^{m}, where 0 < m < 2.^{40,41} The values of n and m are obtained by fitting the conductivity data [Fig. 4(b)].

Four models can define the conduction mechanism in this material.^{42}

Quantum mechanical tunneling (QMT),

Correlated barrier hopping (CBH),

Non-overlapping small polaron tunneling (NSPT),

Overlapping large polaron tunneling (OLPT).

The variation of exponent n with temperature (T) reveals which type of conduction mechanism is happening in the material. A T-independent “n” corresponds to QMT. On the other hand, n increases with T for a NSPT model while it decreases with T for a CBH model. If n decreases with T, attains a minimum, and thereafter increases, then an OLPT model can be justified. Variation of the exponent n with T is plotted in Fig. 5(a).

In the above equations, n′ is 2 for bi-polaron and 1 for single polaron hopping mechanism, N is the density of the localized states, R_{ω} is the hopping distance, τ_{0} is the characteristic relaxation time, and W_{H} is the barrier height of the energy band needed to hop from one site to another. At room temperature W_{H }≫ *k _{B}*T (

*k*T ∼ 0.026 eV). Hence, $ n=1\u22126 k B T / W H $; which justifies the decreasing nature of n with increasing T. Note that $1\u2212 n=6 k B T / W H $, for uniformity for all models fitting has done for 6/(1 − n) vs T.

_{B}At room temperature if W_{H} ≫ k_{B}T, then one may write $ n=1+4 k B T / W H$, which justifies the increasing nature of n with increasing T. From the relation, W_{H} can be estimated by fitting the plots of 6/(1 − n) vs T.

_{H}called the polaron tunneling energy given by

_{p}is the polaron radius, where ε

_{P}is the effective dielectric constant. W

_{HO}is associated with charge transfer between the overlapping sites. The tunneling process takes place when the polaron is incapable of hopping over the potential barrier, W

_{H}. The AC conductivity can be defined as

_{F}) is the density of states at the Fermi level, α is the decay parameter for the localized wave function. It is the spatial decay parameter for the s-like wave function assumed to describe the localized state at each site.

^{42}However, the inter-site separation $ R \omega $ is not independent of temperature and frequency and can be expressed as:

_{HO}is assumed to be constant for all sites for all temperatures and frequencies. The equation of frequency exponent “n” is simplified as

_{H}. In the OLPT model, n is both temperature-dependent and frequency-dependent. At low temperatures, n decreases from unity with an increase in temperature. However, at high temperatures, the variation of n with temperature depends on the polaron radius. For large values of r

_{p}′, n continues to decrease with increasing temperature, eventually tending to the value of n predicted by the QMT model of non-polaron forming carriers. On the other hand, for small values of r

_{p}′, it exhibits a minimum value at a certain temperature and subsequently increases with increasing temperature in a similar fashion to the case of NSPT.

^{43,44}

In BTO, the exponent n decreases with temperature [Fig. 5(a)] throughout the temperature range, revealing a CBH model. The charge carriers can hop over the Coulomb barrier, separating the potential wells. The activation energy calculated from the conductivity plots (W_{HO} ∼ 0.212 eV) [Fig. S4 in the supplementary material] is nearly equal to the barrier height (W_{H} ∼ 0.268 eV) obtained from the fitting of the CBH model in BTO [Fig. 5(b)]. Hence, charge carriers prefer to hop above the potential barrier and follow CBH instead of tunneling through the OLPT mechanism. Moreover, the chances of polaron formation will be negligible in BTO, according to the above discussions related to defect formation and electron localization studies. In BLFT samples, the exponent n decreases with T at low temperatures and attains a minimum, and afterwards increases with temperature. This trend is the case for the OLPT model with small values of r_{p}′.^{43} The smaller nature of the polaron radius may also indicate a probable transition of a low temperature OLPT to high temperature NSPT transition, as temperature increases. The 6/(1 − n) vs T plot is fitted with OLPT at low temperature range (<503 K for BLFT1 and BLFT2, <483 K for BLFT3) and NSPT with a higher temperature range (>503 K for BLFT1 and BLFT2, >483 K for BLFT3) [Fig. 5(b)].^{43,45} An attempt was made to fit the entire temperature range with the OLPT model. However, the model did not respond well enough. For the lower temperature region of BLFT samples, the possibility of the CBH model in place of the OLPT model can be neglected, as for the doped samples, the W_{H} value obtained from the CBH model increases with doping [Fig. S5 in the supplementary material], and the conductivity increases with doping. This is not acceptable, and hence, the CBH model can be neglected, making the OLPT model suitable to explain the conduction mechanism for doped samples at lower temperatures.

The fitted plots of log σ_{dc} vs 1000/T to calculate W_{HO} is shown in Fig. S4 in the supplementary material. The parameters obtained after fitting 6/(1 − n)−T plots with the OLPT, CBH, NSPT models are tabulated in Table S6 in the supplementary material. A comparison of the W_{H} values of the CBH model with the OLPT and NSPT models reveals a much higher value of the W_{H} than the tunneling models for the doped samples. This implies that with La and Fe doping, the energy needed for the charge carriers to hop is much more than to tunnel. Hence, in these doped samples, polarons take part in the conduction process and reduce the energy required to travel between potential wells. This is reflected by the increased conductivity of the doped samples.

The bandgap in BaTiO_{3} is formed by Ti 3d (conduction band) and O 2p (valence band) orbitals. DFT calculations on BLFT3 reveal that the bandgap is due to the separation of the same Ti 3d and O 2p orbitals. However, the relative position of the Fermi energy with respect to the valence band and the conduction band was found to be modified. To add to these changes the E−k diagram reveals modifications in the Ti 3d and O 2p orbitals. Such changes are due to the presence of La and Fe in the lattice. A more visible effect is observed from the ELF studies. It was observed that the electron density is modified around the Ti and O sites. Further, a structural disorder in the lattice can also be observed from the modification of electron density near Ba and Ti ions. Hence, the lattice gets disordered due to the incorporation of La and Fe. Such changes in the band structure, i.e., the orbitals may be instrumental in reducing the requirement of thermal energy for a transition of a large polaron to a small polaron, i.e., an OLPT model to NSPT model. Hence, there is a gradual evolution of the BaTiO_{3} lattice with La and Fe addition that modifies the bonding orbitals of Ba, Ti and O. This clearly indicated the deformation of the lattice wave functions such that the potential wells were highly modified. Such modifications not only modified the experimental and theoretical bandgap but also influenced the transportation of the carriers. The defects thus formed take part in the short-range conduction process, as discussed in the AC conductivity results through tunneling. After understanding the nature of transport and its correlation to the distortion in the lattice, one should focus on the contributions from the grain and grain boundaries toward the conduction mechanism. Impedance and modulus formalism helps to study these contributions and the change from long-range to short-range conduction mechanism.

### Impedance and modulus analysis

The dielectric data can also be represented in the form of complex impedance Z* and complex modulus M*. Z* is related to complex permittivity ε* by the equation: ε* = 1/i*ω*c_{0}Z* where C_{0} is the empty cell capacitance and *ω* is the angular frequency. The complex number Z* can be expressed as Z* = Z′ + iZ″, where Z′ and Z″ are the real numbers belonging to the real and imaginary components of Z*. These are calculated from the capacitance (C) and resistance (R) of the material and can be defined by: Z′ = R/[1 + (*ω*RC)^{2}] and Z″ = *ω*R^{2}C/[1 + (*ω*RC)^{2}]. On the other hand, the electrical modulus, M* is related to ε* as M* = 1/ε*. Similar to Z*, the complex quantity M* can be expressed as M^{* }= M′ + iM″ = i*ω*C_{0}Z*. Hence, M′ = *ω*C_{0}Z″ and M″ = *ω*C_{0}Z′. Impedance formalism indicates the electric resistance of the sample and is used to evaluate the frequency of relaxation of the most resistive component of the sample.^{46} Compared to impedance, the modulus formalism^{47} helps to interpret the dielectric relaxation mechanism, since it neglects the contribution from the interfacial effects and capacitance due to electrode interface.^{48} Also, it highlights whether a process is a short-range charge displacement or can be attributed to long-range charge transport.

Impedance spectroscopy analysis is a useful tool to clarify the physical mechanism of dielectric behavior and grain/grain-boundary contribution to the impedance and modulus. The Nyquist plot of all samples (Fig. 6) for different temperatures shows distorted or depressed semicircles, indicating a deviation from the ideal Debye model which should be a perfect semicircle. The plots were fitted with an equivalent electrical circuit (Fig. 6) with an RC network corresponding to grain or bulk (R_{b}, C_{b}) and grain boundary (R_{gb}, C_{gb}). We have added a component common phase element (CPE) to incorporate the non-ideal behavior of these ceramics. The CPE impedance is given by $ Z CPE= 1 / Q ( i \omega ) p$, where *Q* and *p* are constants, and $0\u2264 p\u22641$. For p* = *1, the CPE is described as the ideal capacitor with Q = C; for p = 0 with Q = 1/R, it is an ideal resistor. The grain (R_{b} and C_{b}) and grain boundary (R_{gb} and C_{gb}) components obtained after fitting seem to vary with temperature. The temperature-dependent variation of all R_{b}, C_{b} R_{gb}, and C_{gb} are plotted (Fig. 7).

Both R_{b} and R_{gb} decrease with temperature for all samples [Figs. 7(a) and 7(b)]. A significant decrease of R_{b} with doping has been observed from BTO to BLFT3. From the ELF calculations, a strong localisation was observed with the introduction of La and Fe in the BLFT3 sample. These are indications of modified orbitals resulting in modified defects/disorder in the lattice. Such modified defects/disorders may enhance conduction involving modified polarons, thereby reducing R_{b} with increasing La/Fe content. However, not much variation in R_{gb} has been observed with doping at room temperature. Grain boundaries are formed due to non-continuation of a long-range crystalline order. Such a non-continuation can be a result of excessive strain in the lattice which limits the continuity. The strain in the lattice can be modified by foreign elements. The grain boundaries are generally thin and composed of a non-structured combination of cations and anions. The nature of the non-structured combination may also change due to the presence of foreign elements in the bulk/grain. Such changes are not considerably high at room temperature resulting in comparable R_{gb} for all samples. However, as temperature increases R_{gb} for the doped samples are comparable but are nominally higher than the BTO. Close to the phase transition temperature the BTO sample shows a certain increase of R_{gb} but is not observed in R** _{b}**. Such a change is not observed in the BLFT samples. The appearance of the anomaly of the R

_{gb}at the transition temperature of BTO and the disappearance of the same in the doped samples hints at a more organized grain boundary for the BTO which is compromised with the advent of the dopants. This may result in further increase of R

_{gb}for the doped samples. The other possibility for the nominal increase of R

_{gb}at high temperatures may be correlated with the thermally activated decrease in the grain size in the case of doped samples. Chemical modification generated changes in the electronic orbitals may lead to modified lattice disorder. Such disorders are instrumental in decreasing R

_{b}and increasing R

_{gb}with composition.

The bulk capacitance C_{b} at lower temperatures varies from 2 to 10 nF for smaller values. However, at higher temperatures, the C_{b} reduces for BTO to 1–2 nF, while it increases to 40–70 nF for the doped samples [Figs. 7(c) and 7(d)]. This opposite trend between pure BTO and the BLFT samples hints at the contribution of defect clusters (clusters formed by the charged ions with O_{v}) in the doped samples with temperature. These defect clusters are absent in BTO. With temperature due to molecular vibration there is a tendency of misalignment of the inherent dipoles in BTO. However, when the defect dipoles contribute, the effect is different. With temperature, there may be an increase of these defect dipoles as the movement of ions and vacancies will increase. For the doped samples at a higher temperature, the C_{b} decreases with doping. This may be a consequence of the decreased resistance of the samples with an increase in doping concentration. On the other hand, C_{gb} consistently reduces with doping concentration throughout all temperatures. The decrease of the grain boundary capacitance with an increase in La and Fe content may be due to the lack of proper structure of the grain boundaries due to an increasing lack of oxygen.^{49} One important aspect of this study is that the C_{b} and C_{gb} values are comparable at low temperatures but are much higher (∼50 nF) for the doped samples at higher temperatures.

The comparison of the imaginary part of impedance and modulus is used to find out whether the conduction mechanism is long-range transport or short-range transport. The permittivity for a pure dielectric Debye response is described by the following equation:^{26}^{,} $ \epsilon \u2217= 1 /( i \omega \epsilon 0 Z \u2217) . d / a= \epsilon \u221e+( \epsilon S - \epsilon \u221e)/( 1 + i \omega \tau )= \epsilon \u221e$ [1 + (r − 1)/(1 + iωτ)] where the high frequency limit of permittivity is denoted by ε_{∞} and low frequency permittivity is denoted by $ \epsilon S$ and r = ε_{s}/ε_{∞}. The permittivity plot with frequency was plotted in a previous report.^{6}

Here our focus is on the imaginary part (Z″) of the impedance and modulus (M″) of the material in a temperature range of 323–563 K (Fig. 8). The variation of real part of the impedance (Z′) and modulus (M′) with frequency is shown in Figs. S1 and S2 in the supplementary material. The frequency dependence of Z″ and M″ reveals a maximum at a particular frequency. This maximum corresponds to the most probable relaxation mechanism happening within the material and is generally found to spread over a region due to imperfections in the lattice. In BLFT samples, more than one relaxation peaks are observed for Z″ and M″ which becomes more evident at higher temperatures. Note that for the BTO sample, only a single relaxation peak can be observed for all temperatures. The multiple peaks in Z″ and M″ correspond to the different relaxation mechanisms associated with difference in capacitance and resistance values in grain and grain boundary. It has already been discussed that the capacitance and resistance values from the grain and grain boundaries are different with composition and temperature, which can further lead to different relaxation mechanisms at different frequencies and hence result in multiple relaxation peaks in Z″ and M″ plots.

Depending on r, the imaginary parts for the different physical quantities (ε″, Z″, M″) may differ significantly. As the value of r increases, the separation between the maxima of Z″ and M″ peaks increase, indicating a non-Debye type relaxation. Whereas, for r = 1, the curve converges and corresponds to a Debye type relaxation. A non-Debye type relaxation is generally associated with a short-range motion of charge carriers confined to nearby potential wells.^{50} On the other hand, the Debye type relaxation corresponds to long-range motion of charge carriers. Note that for BTO sample (Fig. 9), the low temperature data (323 K) reveal a small disparity in the overlap of the *Z*″ and *M*″ peaks indicating the possibility of nominally non-Debye type relaxation. However, at higher temperatures (563 K), the overlap is more prominent indicating a better Debye type relaxation. This indicates a transition from a shorter long-range motion to a longer long-range motion. However, for the doped BLFT samples, the non-overlapping *Z*″ and *M*″ peaks indicate short-range motion of the carriers.

With increase in temperature, the frequency of Z″_{max}, M″_{max} shifts toward a higher frequency (Fig. 9). Hence, the relaxation process is thermally activated. A shift of the maxima toward higher frequencies at higher temperatures corresponds to lower time constants and, therefore, faster relaxation processes. Note that the separation between the *Z*″ and *M*″ peaks increases with doping. This indicates a shorter short-range motion of charge carriers with doping. Hence, a transition from the long-range to short-range mobility has been observed with doping.^{48} The above observations are exactly in line with the information on distortions, defects, and polaron formation obtained in the experimental and theoretical studies and the changes in the conduction phenomena.

## CONCLUSION

(1 − x)BaTiO_{3}-(x)LaFeO_{3} solid solution has revealed bandgap tuning from the UV region (3.25 eV in BTO) to the visible region (2.54 eV in BLFT3) with La/Fe doping, which can be correlated to an increase of lattice distortion and disorder from a previous study on these samples. Theoretical calculations reveal shifts in the valence band maxima and conduction band minima, along with the formation of defects states inside the bandgap with La/Fe doping. The electronic states are modified due to such defects states and correlated structural distortions/disorders which increase the Urbach energy. Such electronic modifications were justified by transportation measurement using dielectric properties, AC conductivity, and impedance spectroscopy. A theoretical assessment of the modifications was estimated using electron localization function calculations, revealing a more distorted lattice for the doped samples, suggesting localization of the electrons. An experimental verification of the localization was estimated using different transportation models which revealed the presence of large polaron hopping at the lower temperature regime and small polaron hopping at the high temperature regime for the La/Fe-modified BLFT samples. For the less distorted pure BTO lattice, a hopping mechanism was found to be a justified transportation method. With doping, the local structural deformations and the polaron formations lead the transportation toward a tunneling process. The defects induce bandgap tuning, and the associated polaron induced conduction mechanism makes these samples suitable for optoelectronic applications.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the real part of impedance and modulus plots (Figs. S1 and S2), simplification of the exponent n for the OLPT fitting (Explanation, Fig. S3), σac vs 1000/T plots (Fig. S4), CBH and OLPT fitting at the low temperature region of 6/(1 − n) vs T plots (Fig. S5). The values of WH, r P ′, R ω ′, WHO obtained from the fitted conduction models and conductivity plots are shown in Table S6.

## ACKNOWLEDGMENTS

P.M. and K.S.S. would like to acknowledge the Ministry of Education, Government of India, for the Prime Minister Research Fellowship (PMRF). P.M. acknowledges Professor Rupesh S. Devan and his research group for UV-visible spectroscopy measurement. Author S.C.B. would like to acknowledge DST INSPIRE for Providing fellowships (No. IF190617). The authors would like to acknowledge the Department of Science and Technology (DST), Government of India, for providing funds for (No. DST/TDT/AMT/2017/200).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no conflicts of interest.

### Author Contributions

**P. Maneesha:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Methodology (lead); Writing – original draft (lead). **Koyal Suman Samantaray:** Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting). **Suresh Chandra Baral:** Data curation (supporting); Writing – review & editing (supporting). **R. Mittal:** Investigation (supporting); Software (supporting). **Mayanak K. Gupta:** Data curation (equal); Investigation (supporting); Software (equal). **Somaditya Sen:** Resources (supporting); Supervision (lead); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## REFERENCES

*Optical Properties of Metal Oxide Nanostructures*

*Optical Properties of Metal Oxide Nanostructures*

*Emerging Applications of Low Dimensional Magnets*

*Fundamentals of Low Dimensional Magnets*

*Fundamentals of Low Dimensional Magnets*

*Emerging Applications of Low Dimensional Magnets*

_{2}NiMnO

_{6}double perovskites for various applications: Challenges and opportunities

_{2}NiMnO

_{6}double perovskite

_{3}+ doped CuO powders under low power visible light and natural sunlight

_{3}for optoelectronic devices applications

_{3}using the Hubbard U correction

^{3+}doping on the physical properties of BaTiO

_{3}

_{3}doped with lanthanum (La): Insight from DFT calculation

_{3}: DFT based calculation

_{3}.(x) LaFeO

_{3}solid solution

*Ab initio*molecular dynamics for liquid metals

*ab initio*total-energy calculations using a plane-wave basis set

_{3}by transition metals co-doping for visible-light photoelectrical applications: A first-principles study

_{3}

_{3}

^{2+/3+}-transition in BaTiO

_{3}and SrTiO

_{3}single crystals

_{3}” (last accessed Nov 23, 2023).

_{0.75}Pb

_{0.25}Ti

_{1−x}Zr

_{x}O

_{3}ceramics

_{1−x}Co

_{x}O

_{3}(0 ≤ x ≤ 0.1)

_{0.1}Ti

_{0.9}O

_{3}and BaZr

_{0.2}Ti

_{0.8}O

_{3}thin films

_{3}Se

_{5}

_{2}WO

_{4}

**27**, 378–385 (

_{2}Co

_{2}Fe

_{12}O

_{22}(Co

_{2}Y) doped with Bi

_{2}O

_{3}

_{3}