R3c to P4bm phase transition assisted small to large polaron crossover in sodium bismuth titanate-based ferroelectrics

Sodium bismuth titanate (NBT) with the chemical formula (Na_0.5Bi_0.5TiO₃) is a promising lead-free perovskite oxide possessing interesting ferroelectric (remnant polarization P_r = 38 μC/cm², coercive field E_c in the range 50–60 kV/cm) and piezoelectric properties (d₃₃ ∼ 70 pC/N).¹ It reveals a rhombohedral (R3c) structure at room temperature. With the rise of temperature, a tetragonal (P4bm) phase starts to appear at a depolarization temperature (T_d ∼ 423 K).^2,3 The structural distortions generating ferroelectricity reduce with the advent of a P4bm phase, reducing the polarization.^4,5 A temperature-dependent structural study revealed a change in the Ti-O and covalent Bi–O bonds at T_d.^6,7 Hence, polarization and structural changes are correlated. The interaction of the charge carriers with the lattice is mostly affected by the changes in factors such as thermal energy and chemical potentials.^8,9 Understanding the phonons and how they change with such determining factors is one way to understand the underlying correlation between the lattice and the charge carriers. A high-temperature study of phonons revealed a significant change in the phonon modes related to the Ti–O/Bi–O bond vibrations at T_d.³ On the other hand, it is well known that structural changes have serious consequences on the vibrational properties of the component atoms.^3,10

In a distorted lattice, the distortion itself plays a vital role in modifying the nature of the charge carriers, creating fermionic quasiparticles called polarons.^11,12 These are formed in polarizable materials due to the coupling of excess electrons or holes with ionic vibrations at the distorted sites. The presence of the polaron reduces the energy of the lattice by an amount of energy known as the polaron energy (W_P).¹³ The changes in the e⁻-ph interactions lead to modifications in the polaron energy.¹⁴ Functional properties of perovskite ABO₃ oxides are affected by the confinement and correlation of the transition metal (TM) d-orbitals within the BO₆ octahedra. Hence, the absence of an oxygen atom is sensitive enough to confine the d-orbital environment, sometimes resulting in partial hybridization of TM d-orbitals with oxygen 2p orbitals.¹⁵ This results in various trap and recombination states in the electronic structure. Such modifications in the electronic structure due to oxygen vacancies (O_v) generally prompt polaron formation around the vacant site.¹⁶ Lattice distortions can happen irrespective of O_vs and can produce polarons. First-principles calculations show that mechanical strain can transform excess electrons from free carriers to localized polaronic states by modulating electron–phonon coupling.⁹ Moreover, in ATiO₃-based systems, the A-site cations and the phase transition modify the polaron energetics and dynamics.¹⁴ Any modification in the lattice due to an irregularity can modify the polarons. Such polaron dynamics from small-to-large polarons are prominently observed during the structural change from anatase to a rutile phase in TiO₂.¹⁷ 

In ferroelectric materials, the effect of spontaneous polarization on the structural changes and the related changes in the behavior of the polaron dominates the transport properties.¹¹ Frequency and temperature-dependent dielectric and loss measurements are crucial to understanding the polarization mechanisms, conduction processes, impurities, and phase transitions. Deep defect states near the Fermi level (often linked to structural irregularities) significantly influence AC conductivity.¹⁸ The AC conductivity spectra can be modeled in terms of Jonscher's power law (JPL): $σ a c= σ d c+A ω n$⁠. The exponent term “n” is frequency-dependent such that 0 ≤ n ≤ 1. It provides information about the degree of interaction between the mobile ions with the neighboring rigid lattice in ferroelectric ceramics.^19,20 These interactions can be either of translational type or localized in nature. The pre-exponential factor “A” defines the strength of the polarizability. The steady frequency-independent $σ a c$ region at low frequencies is often called a plateau region. In this frequency range, due to the lower AC-field energy (i.e., availability of a longer time), a long-range translational motion of polarons (σ_dc) can be expected, enabling successful hopping of the ions to the nearest available site.²¹ 

The distinction between small and large polaron formation is often discussed in the light of variation of frequency exponents (n) with temperature.^22–25 The temperature dependence of the frequency exponent “n” can be correlated to the different possible transport mechanisms:

• Correlated barrier hopping (CBH) in which “n” decreases continuously with temperature^26,27

• Overlapping large polaron tunneling (OLPT) in which “n” decreases first and attains a minimum and thereafter increases¹⁸ 

• Quantum mechanical tunneling (QMT) mechanism in which “n” is independent of temperature and generally has a value of 0.8²⁸ 

• Non-overlapping small polaron tunneling (NSPT) model in which an increase in “n” is observed with the rise in temperature.^18,29

Studying polarons in ferroelectrics is important for optimizing charge transport and electrical properties.³⁰ Additionally, understanding polaron dynamics helps one to understand whether the stability and efficiency of ferroelectric-based applications such as energy harvesters and photovoltaics have been improved.^11,31 As NBT is an important lead-free ferroelectric material, understanding the origin and possibility of polaron formation is vital. Oxygen vacancies (O_v) are highly prone in NBT as Na and Bi are highly volatile at the A-site. Hence, while the ceramics are sintered at high temperatures, O_v centers are formed,^32,33 thereby facilitating the polaron formation near the O_v centers. The charge carriers are affected by such O_vs and depend on the structure. Hence, the study of such changes is extremely important.

This work attempts to understand the polaronic transport mechanism using experimental temperature-dependent AC conductivity measurement and ab initio studies to substantiate a probable model using O_v as a source of the polarons. The room-temperature R3c and high-temperature P4bm structures of parent NBT are explored in terms of structural distortions, charge density, and polaronic barriers to study the type of polarons associated with different structures and their change in polaron formation energies. The structure is also affected by factors like doping.^34–36 Such chemical modification modifies the electronic properties and related factors like piezoelectric, dielectric, and transport properties.³⁷ Hence, in this work, an extension is attempted to experimentally understand the effect of chemically modified lattice distortion on the polaronic nature in (1-x)NBT.xBCZT solid solution. Incorporation of multiple cations at A and B sites of NBT results in the reduction of T_d from 456 K (x = 0) to 352 K (x = 0.09) due to the increase in lattice disorder, where the jump of the phonon frequency with temperature around T_d for NBT-BCZT solid solutions was also studied.³⁸ This study discusses the effect of chemical substitution on energy barriers and conductivity.

(1−x) Na_0.5Bi_0.5TiO₃. (x) Ba_0.85Ca_0.15Ti_0.90Zr_0.10O₃ (x = 0, 0.03, 0.06, and 0.09) samples were prepared using a sol-gel synthesis route, which is mentioned in detail in this paper.³⁷ The A-site (Na/Bi) is substituted with Ba²⁺ and Ca²⁺, whereas the B-site (Ti) with Zr⁴⁺. Here, x = 0 has been mentioned as parent NBT and others as chemically modified NBT or NBT-BCZT. The AC conductivity measurements were performed using a broadband dielectric spectrometer (Newton's 4th Ltd. phase-sensitive multimeter) in the temperature range of 323–723 K and frequency range of 100 Hz to 1 MHz. The multimeter had a signal strength of 1 V_rms. The depolarization phenomenon has been studied by recording the current changes through the samples in capacitance mode due to a negligible voltage bias of 0.1 μV.^35,38 The voltage and the current were applied and recorded by a Keithley 2450 source meter. The temperature was ramped up at 1 K/min. The high-temperature phonon modes were studied from Raman spectroscopy using a Horiba-made LabRAM HR Evolution Raman spectrometer (spectral resolution 1 cm⁻¹) having He-Ne LASER of wavelength 633 nm in the temperature range of 323–473 K. The room temperature x-ray diffraction was done using a Bruker D2-Phaser, and the lattice parameters were calculated after doing Rietveld refinement in the Full-Prof suite.

Ab initio calculations were conducted employing the Projected Augmented Wave (PAW) basis set within the framework of VASP, ensuring precise determination of spin polarization.^39–41 The Perdew–Burke–Ernzerhof (PBE) functional, encompassing the generalized gradient approximation (GGA), accounted for the exchange-correlation term.⁴² Valence states were accurately configured: Na (3 s), Bi (6 s, 6p), Ti (3d, 4 s), and O (2 s, 2p). Due to computational limitations, each system was examined within a 2 × 2 × 2 supercell, where convergence was precisely assessed. E_cut was determined as 520 eV, harmonizing computational efficiency and accuracy. A Monkhorst–Pack k-point grid of 4 × 4 × 4 and 3 × 3 × 4 was tailored to R3c and P4bm phases of NBT, respectively, ensuring meticulous balance. Striving for precision, Hellmann–Feynman residual forces were closely reduced to less than 0.001 eV/Å during cell relaxations and defect (O_v) calculations. Energy convergence criteria were stringently set at 1E-07 eV for self-consistent field calculations. Incorporating self-interaction correction through the DFT + U formalism remained pivotal throughout the project. The choice of the +U parameter is significant and can be tailored to ensure agreement on calculated properties.^12,43 A rotationally invariant Hubbard U correction was adopted as proposed by Dudarev et al., with U_eff set at 4.46 eV for Ti-3d orbitals.⁴⁴ This parameter choice, rooted in empirical knowledge, had demonstrated efficacy in reproducing vacancy formation properties within the PbTiO₃/SrTiO₃ systems.¹⁴ 

This work similarly treats the R3c and P4bm systems to investigate oxygen vacancy's electronic and structural effects on polaron properties. Polaron formation energy (E_polaron) was calculated using Eq. (2) for each system. The total energy of the bulk supercell was calculated for three different scenarios for each structure: (i) a perfect 2 × 2 × 2 supercell without oxygen vacancy, (ii) a perfect (unrelaxed structure) 2 × 2 × 2 supercell with a single oxygen vacancy, and (iii) a distorted (relaxed structure) 2 × 2 × 2 supercell with a single oxygen vacancy. For the R3c structure, the total energy was calculated to be −530.824, −521.107, and −521.463 eV, while for the P4bm structure, these values were −528.213, −518.421, and −518.685 eV, respectively. The energies for the oxygen vacant cases for unrelaxed and relaxed cases include the self-trapping energy. The charge-neutral oxygen vacancy formation energies are 5.379 eV (R3c) and 5.189 eV (P4bm), calculated using Eq. (1). Post-processing of the simulated data was performed using VASPKIT, and VESTA was used to visualize spin density, charge density differences, and structural changes induced by the polaron.^47,48 The electronic structures of these polaron formations are analyzed using orbital-resolved density of states (DOS) calculations.

The electron localization function for two conditions is plotted for both R3c and P4bm relaxed structures: (1) without O_v and (2) with O_v (Fig. 1). The details of the atoms and their atomic positions for both the structures with and without O_v are tabulated in Table SI in the supplementary material. For the R3c and P4bm NBT structures, it can be observed that with the introduction of O_v, the Bi–O and Ti–O electronic clouds were distorted (Fig. 1). The effect of O_v on the Bi atoms and TiO₆ octahedra in both structures is shown in Fig. S1–S4 in the supplementary material.

Observing the changes in the electronic clouds, a detailed analysis of the localization is essential. Such localization can be correlated to the presence of polaron. Often, the associated atomic distortions in such materials are due to defects like O_v. Hence, a discussion is required for both phases in these lines to understand the possibility of the polaronic transport mechanism in NBT.

The density of states (DOS) is plotted for both with and without O_v scenarios for the R3c and P4bm phases (Fig. 2). For R3c, the Ti(3d) states majorly contribute to the conduction band, while the O(2p) state contributes to the valence band [Fig. 2(a)]. For the R3c NBT without O_v, the Fermi level coincided with the valence band maxima in agreement with the reported literature.^49–51 With the introduction of O_v, the Fermi level shifted to ∼1.2 eV above the valence band [Fig. 2(b)]. This is because a charge-neutral-O_v behaves as a double donor state. The O_v-induced structural distortion leads to the self-trapping states of those donor electrons. This self-trapping is the electron localization that creates states inside the bandgap around the Fermi energy. These localized states are mainly due to the contribution from the Bi(p), Ti(d), and O(p) states. However, the bandgap of ∼2.6 eV remained unchanged for both with and without O_v cases. It is to be noted that the energy gap between the Fermi level and the conduction band was reduced.

Similar to the R3c phase, Ti(3d) at the conduction band and O(2p) at the valence band contributed majorly to the bandgap for the P4bm phase [Fig. 2(c)]. The P4bm bandgap was estimated to be lesser, ∼2.4 eV, than the R3c phase, in agreement with the literature.⁵² In the P4bm phase, like the R3c phase, the Fermi level and the valence band maxima coincided for the NBT without O_v. With the incorporation of O_v, the Fermi level shifted ∼2.4 eV above the valence band [Fig. 2(d)]. Hence, the energy gap between the Fermi level and the conduction band is much lesser in the P4bm phase than in the R3c phase. As O_v is responsible for lattice distortion, which is further responsible for the generation of trapped states inside the bandgap and raises the Fermi level closer to the conduction band, thereby the energy gap between the trap states and the conduction band can be held analogous to an energy barrier (ΔE) required for the polarons to transport successfully. The ab initio calculations revealed this energy barrier (ΔE) to be larger (∼1.4 eV) for R3c [Fig. 2(b)] and much smaller (∼0.2 eV) for P4bm [Fig. 2(d)].

Figures 3(a) and 3(b) display the calculated real-space spin polarization density of the relaxed 2 × 2 × 2 supercell, resulting from O_v within the R3c and P4bm phases of NBT, respectively. For both the R3c and P4bm phases, the top view of the (100) plane clearly illustrates the localization of electron density on Ti atoms near the oxygen vacancy. As previously discussed for both the R3c and P4bm phases, a charge-neutral oxygen vacancy acts as a double donor defect. For the R3c structure, these two electrons are localized at two neighboring Ti sites near the oxygen vacancy, with two different spin states. In contrast, for the P4bm structure, the spin density is spread mostly in the middle of the supercell with weak contributions at the peripheral atoms. These pictures reveal a comparatively extended localization for P4bm compared to the R3c phase [Figs. 3(b) and 3(c)]. Thus, from these calculations, it is exposed that the P4bm structure with O_v supports larger localization due to structural distortions, which will be discussed in the following paragraphs.

A good way to understand the modifications in the charge densities of the respective phases due to O_v is to calculate the charge density difference (Δρ) between the relaxed localized structures with and without O_v [Figs. 3(d) and 3(e)]. These plots are at the same isosurface level (0.06) to compare both phases effectively. The First Nearest Neighbor (1NN) atoms around O_v showed a prominent Δρ for the R3c phase compared to the weak Δρ for the Second Nearest Neighbor (2NN) atoms [Fig. 3(d)]. However, Δρ is equally high for both 1NN and 2NN atoms for the P4bm phase [Fig. 3(e)]. This proves that the P4bm phase has a larger structural and electronic distortion than the R3c phase, showing a transition from small to large distortion, i.e., from a small to a large polaron.

Apart from the changes in the DOS and charge densities, the introduction of O_v resulted in atomic displacements for both the R3c and P4bm phases, which were also localized around O_v. Such atomic displacements can be studied from the displacement vectors of atoms surrounding O_v (Fig. 4). The displacement vectors of the Na, Bi, and O-atoms [Fig. 4(a)] and the Ti and O-atoms [Fig. 4(b)] in the R3c (110) plane reveal a much larger displacement of the Bi atom than the O, Ti, and Na atoms in the Ov neighborhood. The displacement decreased for the 2NN compared to the 1NN atoms confirming a localized distortion around O_v.

For the P4bm phase, the displacements of Bi, Ti, O [Fig. 4(c)], Na, Ti, O [Fig. 4(d)], and Ti, O [Fig. 4(e)] atoms are visualized along (010), (100), and (110) planes, respectively, to enable appropriate visualization of the particular atoms. A striking difference was observed between the displacements of atoms in R3c and the P4bm structures (Table SII in the supplementary material). In the R3c phase, a considerable atomic displacement was observed for only the 1NN atoms, out of which Bi was much more displaced than the O, Ti, and Na atoms. In the P4bm phase, the O-atom displacement was comparable to that of the Bi-atom of the R3c phase. Moreover, this enhanced displacement was extended to the 2NN O-atoms as well. The Ti, Bi, and Na cationic displacements were lesser than the O-atoms. However, the cationic displacements were comparable to the O-displacements. Like R3c, the 1NN atomic displacements were larger than the 2NN atoms, but the magnitude of the 2NN atomic displacement in the case of P4bm was much higher than the R3c phase. Therefore, from this analysis, it can be inferred that the P4bm phase experiences a larger structural distortion in the locality of O_v than the R3c phase. This further supports the large polaron formation in the P4bm structure.

From the above discussions, it can be said that the structural and electronic cloud distortions are more extended over larger atomic sites for the P4bm phase than the R3c NBT lattice. The polaron formation energy due to O_v was calculated for the R3c and P4bm structures. It was estimated that the P4bm (−0.26 eV) phase possesses a relatively lower E_polaron than the R3c (−0.36 eV) phase of NBT. Also, the large polarons in P4bm NBT require a much lesser energy barrier (∼0.2 eV) than the small polarons in R3c NBT (∼1.4 eV) for conduction. This ensures that an R3c to P4bm structural transformation can change the polaronic nature from small to large with different energy barriers and polaron formation energy in NBT.

The ab initio studies concluded that the changes in the electronic cloud distortion, density of states, and structural distortion are related to the R3c to P4bm phase change. The barrier energy (ΔE) is higher for the R3c phase than the P4bm. The extensive structural, electronic cloud distortion, and charge density difference in P4bm confirmed the presence of a large polaron compared to the small polaron in the R3c phase of NBT. Hence, such a transformation in terms of conductivity needs to be explored experimentally using AC conductivity studies.

An AC conductivity study was performed to experimentally evaluate the polaronic energy barrier and polaronic type with structural change. Previous temperature-dependent physical properties studies revealed an R3c to P4bm phase transition of parent NBT associated with a depolarization phenomenon at T_d.³⁸ The AC conductivity was calculated from the frequency-dependent relative permittivity (ɛ_r) and loss tangent (tan δ) of the material: $σ a c=ω ε 0 ε rtan⁡δ$⁠, where σ_ac is the AC conductivity, ɛ₀ is the dielectric permittivity in a vacuum, ɛ_r is the relative permittivity, tan δ is the loss tangent, and ω is the angular frequency.^21,53 The σ_ac spectra showed two different frequency regions at low temperatures (Fig. S5 in the supplementary material). The frequency dependence was slower or steady at lower frequencies while comparatively faster or frequency-dependent at higher frequencies. This characteristic frequency for each temperature for which the transition happens from a steady to a frequency-dependent nature increases with an increase in temperature. Hence, a steadier frequency-dependent σ_ac was observed at lower temperatures corresponding to dc conductivity, σ_dc. However, at higher frequencies, the conductivity showed dispersive behavior indicating contribution from the AC conductivity.

To explore the conductivity mechanism in the parent NBT material, Jonscher's Power Law (JPL) was utilized to fit the AC conductivity spectra.¹⁹ A JPL model was followed up to 403 K. Beyond 403 K, a Jonscher's Double Power Law (JDPL): $σ a c= σ d c+A ω n$ $+B ω m$ model was necessary⁵⁴ (Fig. S5 in the supplementary material). The pre-exponential factors define the strength of polarizability in two different frequency regions. The exponents “n” and “m” are such that 0 ≤ n ≤ 1 and 1 ≤ m ≤ 2 correspond to the low- and high-frequency regions, respectively.

The temperature dependence of the exponent “n” can shed light on the transport processes. The values of “n” were obtained from the power law fitting, which were found to be in the 0.88 ≤ n ≤ 1 range for the T < T_d range. For T < T_d, “n” increased with temperature [Fig. 5(a)]. The increasing temperature-dependent “n” trend confirms an NSPT behavior [Fig. 5(a)]. The polarons are localized enough that their distortions do not overlap. This gives rise to the formation of small polarons. In such cases, the electron undergoes tunneling between the sites closer to the Fermi level, thus leading to the NSPT model. The tunneling energy (W_H) depends on the polaron energy, which can be written as $W H= 1 2 W P$⁠. In the NSPT model, the exponent (n) can be written as $n = 1 + 4 k B T W H − k B T l n ( ω τ 0 )$⁠, where T is the absolute temperature, k_B is the Boltzmann constant, W_H is the maximum barrier height, τ₀ is the characteristic relaxation time, and ω is the angular frequency. For W_H ≫ k_BT, “n” can be written as $n = 1 + 4 k B T W H$⁠.^18,29

For T ≥ T_d, considering the low frequencies (n < 1), the exponent “n” was observed to follow the OLPT model for all the compositions (Fig. S5 in the supplementary material). Long proposed a model that projects the mechanism of overlapping distortions called large polarons between which tunneling is possible, called the OLPT model.¹⁸ The tunneling energy can be expressed as $W H = W H O ( 1 − r P R )$⁠, where r_P is the polaron radius, which is smaller than the variable site separation distance, R. W_HO is the activation energy associated with charge transfer between the overlapping sites that is assumed to be constant for all the sites, given by $W H O = e 2 4 ε P r P$ , where ɛ_P is the effective dielectric constant. W_HO is assumed constant for all sites, while the inter-site separation R is a random variable. The AC conductivity for the overlapping large polaron tunneling model is given by the relation $σ a c ( ω ) = [ π 4 e 2 ( k B T ) 2 { N ( E F ) } 2 12 ] × ω R ω 4 2 α k B T + W H O r P R ω 2$⁠, where k_B is the Boltzmann constant, T is the absolute temperature, N(E_F) is the density of states at the Fermi level, α is the decay parameter for the localized wave function, and Rω is the tunneling length at a frequency ω. The frequency exponent “n” is given as $n = 1 − 8 α R ω + ( 6 W H O r p R ω k B T ) { 2 α R ω + ( W H O r P R ω k B T ) } 2$⁠.

From the temperature-dependent variation of exponent “n,” it can be inferred that a transition happened at T_d (shown pink color) from a small polaron to a large polaron tunneling [Fig. 5(a)] (see Fig. S7 in the supplementary material for model fitting).

The polaron radius (r_p) distinguishes a small polaron from a large one. The r_p is dependent on the lattice parameters.⁵⁵ In the case of titanate-based perovskites, the hopping of charge carriers takes place from a Ti site to another Ti site mediated by O-atoms. For NBT, the Ti–Ti distance increases with an increase in temperature.⁷ Hence, it is speculated that the r_p will also increase. The ab initio calculations showed that the atomic displacement and charge density distortion are extended over larger atomic sites for the P4bm phase compared to the R3c phase. Hence, the polaron radius will increase at the T_d as the phase transforms to P4bm.

Barrier height (W_H) is an important parameter for understanding conduction behavior. T_d demarcates for the parent NBT (x = 0), introducing the P4bm phase for T > T_d from an R3c phase for T < T_d. For T < T_d, the barrier height (W_H) was found to be ∼1.60 eV for the small polaron (Table I). For T > T_d, the large polaron's barrier height (W_H) was 0.043 eV. On the other hand, the ab initio calculations revealed a higher ΔE for the R3c phase (∼1.40 eV) than the P4bm phase (∼0.20 eV). These values are in close approximation to the experimentally calculated barrier heights. Moreover, the P4bm phase has a larger structural and electronic distortion than the R3c phase, confirming a small-to-large polaron transition. These one-to-one correlations enable one to correlate the trap states and the energy gap to the polaronic barrier. It is to be noted that the energy gap (ΔE) and the barrier height (W_H) for the R3c phase are comparable but an order different for the P4bm phase. Such an ambiguity for the higher temperature region could possibly be due to thermally evolved defect concentrations. Moreover, the presence of a complex mixed phase of R3c and P4bm can be responsible for such behaviors.^2,4,6,56 Therefore, the above experimental and theoretical comparison correlates the role of defects to the formation of polarons and confirms a small to large polaron transformation at higher temperatures facilitated by a phase transition, significantly reducing the barrier energy.

It was confirmed from both computational and experimental studies that at T_d, parent NBT possess transition from a small polaronic to large polaronic type conduction due to the associated R3c to P4bm phase transition. A few chemically modified NBT materials were prepared to verify such findings by modifying the A site with Ba²⁺ and Ca²⁺ and the B site with Zr⁴⁺. The chemical formula can be written as Na_0.5(1−x)Bi_0.5(1−x)Ba_0.85xCa_0.15xTi_0.90(1−x)Zr_0.10xO₃, which is represented in a simpler way as (1−x)NBT.x(BCZT) (where x = 0.03, 0.06, and 0.09). At room temperature, the x = 0.03 and 0.06 ceramics show an R3c phase. The T_d of these samples are found to reduce with an increase in BCZT content from ∼456 K (x = 0) to 411 K (x = 0.03) to 381 K (x = 0.06). For the x = 0.09 composition, the T_d reduced to 352 K, close to the room temperature. Beyond T_d, the structure is a mixed phase of R3c and P4bm. Hence, it was evident that BCZT incorporation in the lattice reduced the T_d and introduced a mixed phase at a lower temperature.^37,38 This was previously reported with details on the modifications of the A–O and B–O bond lengths in the chemically modified structures. It was also discussed that the octahedral tilt decreased due to the A/B-site modifications (Table SII in the supplementary material), thereby modifying the ɛ_r and loss, influencing the conductivity in the modified NBT. Therefore, the AC conductivity spectra were analyzed, and the transport mechanism was further investigated experimentally to verify the small to large polaronic conduction at T_d for the BCZT-modified NBT ceramics.

Like the parent NBT, the conductivity spectra of (1-x)NBT.(x) BCZT were analyzed using power laws. A JPL model was followed for x = 0.03 and 0.06. For x = 0.03, the JPL model was used up to 403 K. Beyond 403 K, a JDPL model was used. For x = 0.06, JPL was used for T ≤ 383 K and JDPL for 403 K ≤ T ≤ 723 K (Fig. S3 in the supplementary material). On the other hand, the x = 0.09 sample could not be fitted with JPL and hence could only be fitted with the JDPL model for the entire temperature range of 323 K ≤ T ≤ 723 K.

The BCZT-modified NBTs also transitioned from the NSPT to the OLPT model, similar to the parent NBT case [Figs. 5(b)–5(d)]. This transition was reduced to a lower temperature as the BCZT content increased, revealing the dependence of T_d on the crossover of small to large polaronic-type conduction. It was observed that for large polarons, the r_p’ increased with BCZT substitution. This increase of r_p’ may be due to changes in the lattice due to A/B site modifications. For T < T_d, the W_H for the small polaron was found to decrease from 1.60 eV (x = 0, parent NBT) to 0.50 eV (x = 0.09) (Table I). Similarly, for T > T_d, W_H for the large polaron was also found to decrease from 0.043 eV (x = 0, parent NBT) to 0.018 eV (x = 0.09). It can be inferred that the chemical modifications in the parent NBT lattice reduced the barrier and enhanced the conductivity with increased BCZT content [Fig. 6(b)]. The reduction in W_H seemed to help the tunneling with BCZT incorporation. Hence, the lattice distortion created by the different dopant atoms probably enabled the lattice to interact with the charge carriers more effectively, enabling better transport through the tunneling process. This further confirmed that the structural transition enabled the small to large-polaronic type conductivity crossover at T_d for both NBT and NBT-BCZT solid solutions.

Poling is a necessary method in piezoceramics to study piezoelectricity. At room temperature, it was studied that the parent and modified samples show changes in the octahedral tilt, A–O, B–O bond lengths, and lattice parameters with poling (Table SIII in the supplementary material).³⁷ Hence, the poled systems are also studied to confirm the small to large polaronic-type conductivity in the parent and BCZT-modified NBT ceramics [Figs. 5(e)–5(h)].

For T < T_d, poling decreased W_H for small polarons in each composition. Also, similar to the unpoled cases, BCZT incorporation reduced the W_H from 1 eV (x = 0) to 0.31 eV (x = 0.06). The in-built electric field aligns the domains, reducing the arbitrary potential from multiple polarization directions. This seems instrumental in reducing the effective potential, allowing the small polarons to tunnel easily. The reduction of the barrier height with poling in the T > T_d temperature region for the large polarons was also observed to reduce from 0.03 eV (x = 0) to 0.011 eV (x = 0.09), thereby strengthening the claim. The above data revealed that the reduction in the W_H with poling for the large polaron in the T > T_d regime was lesser compared to the small polarons in the T < T_d regime. This may be due to the effect of depolarization at higher temperatures.

As observed from the AC conductivity studies, there is a huge decrement in W_H for the large polarons above T_d compared to the small polarons below T_d. To understand such variation, an attempt is made to visualize the effect of atomic vibrations on the polaronic transport in the NBT-based samples. For the T > T_d regime, the phonon energy for parent NBT related to the A–O (∼0.034 eV) and B–O (∼0.016 eV) vibrations is of the same order as the W_H of the large polaron (∼0.030 eV). On the other hand, W_H is much higher (∼1 eV) for small polarons than for the phonon energies [Fig. 6(a)]. Similar to the parent NBT, the BCZT-modified NBT also revealed the resemblance of large polaron W_H within the range of phonon vibration energy. This displayed a phonon-assisted large-polaronic transport at the T > T_d regime, resulting in much lesser W_H and higher conductivity. The lattice vibrations of the A–O and B–O bonds thereby play a critical role in constructing the shape, depth, electron localization, and energy landscape of the potential well, all of which significantly influence the charge carrier's mobility and transport properties within the material.

A large polaron generally forms over multiple sites; hence, its motion affects others, becoming coherent. The mobility of large polarons happens to be μ > 1 cm²/V s. On the other hand, the mobility of small polarons happens to be μ ≪ 1 cm²/V s, due to the incoherent charge transport of the small polarons’ extremely localized nature. Such a distinction of mobility can also be a potential tool in understanding the transition from a small-to-large polaronic type of transport.⁵⁷ In the case of AC conductivity, up to T < T_d, the conductivity variation is meagre compared to T > T_d [Fig. 6(b)]. This further validates the presence of large polarons, resulting in a huge decrement in the W_H for T > T_d compared to T < T_d, which strongly agrees with the ab initio study.

The potential well produced by ion relaxation about a static point charge is given by $V e q L R(r)=− ( 1 ε ∞ − 1 ε 0 ) e 2 | r |$⁠.^13,57 The parameter $ε 0/ ε ∞$ is a determining factor for the polaron to be small or large.^31,57–59 A short-range electron–phonon interaction in a covalent-type material has been found to have $ε 0/ ε ∞→1$ while a long-range electron–phonon interaction typically found in electronic materials has $ε 0/ ε ∞>2$⁠. The latter type has often been reported to have a large polaron formation, while the first type supports the formation of small polarons. In the absence of $ε 0$ (static dielectric constant) and $ε ∞$ (optical dielectric constant), a fair approximation can be accepted for the ratio to be $ε 100 Hz/ ε 1 MHz$⁠, where it is assumed that $ε$ reduces with frequency. With this point of view, $ε 0/ ε ∞$ will be greater than $ε 100 Hz/ ε 1 MHz$⁠. However, the trend of this ratio with temperature further confirmed the nature of the polaron involved in different temperature regions. The fraction $ε 100 Hz/ ε 1 MHz$ decreased initially in the range T < T_d and thereafter increased for T > T_d for all the compositions [Fig. 6(c)]. The above discussion is based on the fact that the small polaron to large polaron transformation at T_d is further validated by such observation.

In ferroelectric-like materials, an intense local electric field can induce collective polarization and local phase transitions, forming ferroelectric large polarons with polar nanodomains that order around the charge. This results in significant screening of the Coulomb potential and potential barriers. We understand that these ferroelectric large polarons are complex due to the need for theoretical models that go beyond the harmonic approximation to account for collective polarization and soft phonons.⁶⁰ An attempt to understand the limitations of the lattice in holding on to a particular structure below a certain temperature is the uniqueness of this study, which connects the nature of the polaron to such internal/external changes.

Temperature-dependent AC conductivity measurements on the parent NBT system reveal a small to large polaronic-type conduction transformation at T_d, associated with a structural phase transition from R3c to P4bm. The conductivity increased drastically in the temperature region T > T_d compared to T < T_d. This phenomenon was found to be correlated to the decreased barrier energy (W_H). A theoretical calculation using ab initio studies revealed the small to large polaronic transition in NBT due to the R3c to P4bm transition. As polaron formation is associated with structural distortions due to strain and chemical modifications, an attempt was made to estimate the changes in the density of states due to the presence of an oxygen vacancy. Trap states were observed to form in the bandgap, corresponding to an energy gap of ∼1.4 eV for R3c and ∼0.2 eV for P4bm. On the other hand, the experimental polaronic energy barrier (W_H) was found to be ∼1.6 eV for T < T_d and ∼0.043 eV for T > T_d. A resemblance between the energy gap and W_H in the T < T_d region, where only the R3c phase is dominant, indicates a strong correlation between the oxygen vacancy-induced distortion and small polaron formation. For T > T_d, W_H reduced drastically with the advent of the P4bm phase, supported by the smaller energy gap calculated from ab initio studies. This enhanced the conductivity in NBT for T > T_d. Charge density and atomic displacement analysis revealed higher distortions in P4bm, with lower polaron formation energy (−0.26 eV) than R3c (−0.36 eV). The phonon energy was observed to be in the same order as the W_H of the large polaron beyond T_d but was much lower for the small polaron below T_d. This indicated a phonon-assisted large-polaronic transport (for T > T_d). The role of structural distortion due to chemical modification was studied for NBT-BCZT solid solutions. The small-to-large polaronic crossover happened at a lower T_d, confirming the contribution of structural transition. The BCZT incorporation reduced barrier height from ∼1.6 eV (x = 0) to ∼0.5 eV (x = 0.09) for small polarons and ∼0.04 eV (x = 0) to ∼0.01 eV (x = 0.09) for large polarons, which consequently increased the conductivity.

See the supplementary material for the details of the calculated atomic positions and bond lengths with and without oxygen vacancy in the R3c and P4bm phases of NBT. The AC conductivity and OLPT model fitting are also detailed. It contains figures and tables that present the fitting parameters and show the fitting for all the compositions. The experimental lattice parameters and variations in A-O/B-O bond lengths for parent and modified compositions are tabulated.

K.S.S. and M.P. acknowledge the Ministry of Education, India, for providing the Prime Minister Research Fellowship (PMRF). D.S. thanks the University Grant Commission of India for financial assistance. S.S. is grateful to the Department of Science and Technology (DST), Government of India, for providing funding through Grant No. DST/TDT/AMT/2017/200.

The authors have no conflicts to disclose.

Koyal Suman Samantaray: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Resources (lead); Visualization (lead); Writing – original draft (lead). Dilip Sasmal: Data curation (supporting); Formal analysis (equal); Methodology (equal); Software (lead); Writing – original draft (supporting). P. Maneesha: Formal analysis (supporting); Writing – review & editing (supporting). Somaditya Sen: Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.