Effect of Sr-doping on electronic and thermal properties of Pr2-xSrxFeCrO6 (0≤x≤1) oxide materials synthesized by using sol-gel technique

ABSTRACT The thermoelectric properties of a new type of Pr2-xSrxFeCrO6 double perovskite were investigated at higher temperature after its sol-gel synthesis. The XRD results validate the single-phase orthorhombic structure, and the crystallite sizes meet the morphological measurements. XPS examination, which formed the defect sites in these oxides, confirmed the varied oxidation states of constituents. The temperature-dependent electrical conductivity (σ) was poor in the original Pr2FeCrO6 composition, but after substituting Sr on the Pr-site, a significant rise in σ with two semiconductors and one metal transition was observed. A positive Seebeck coefficient confirmed the presence of a p-type charge carrier in the entire composition, and the charge transport mechanism was driven by the SPH model. Thermal conductivity increases in all doped samples, while it decreases in pristine compounds over the broad analyzed temperature range. Thermal expansion coefficient increased after doping in oxygen-deficient compound. The PrSrFeCrO6 compound had the maximum ZT (0.105), which was 3.9 times higher than that of the pristine compound.


Introduction
Because of the growing difficulty of meeting energy demands, renewable energy has been one of the most intensively researched areas in recent decades. The rising need for oil and electricity is a well-known fact that simply cannot be ignored. On a worldwide scale, more than 70% of energy produced is squandered in some way. Thermoelectric materials transform waste heat energy coming from the automotive exhausts, factory chimneys, and other sources into electrical energy, even without emitting greenhouse gases into the atmosphere [1]. To measure the performance of thermoelectric devices, a dimensionless figure of merit (ZT) is utilized, which is represented as, where, S, σ, T and κ tot or κ denote the Seebeck coefficients, electrical conductivity, absolute temperature (in K), and total thermal conductivity respectively. Additionally, thermal conductivity can be expressed mathematically as: κ ¼ κ e þ κ l where κ e is electronic thermal conductivity resulting from electrons and holes and κ l is lattice thermal conductivity resulting from phonon. The ZT value of a proposed thermoelectric material should be high. A combination of low thermal conductivity, high Seebeck coefficient, and high electrical conductivity should result in a higher ZT value. For reaching high ZT values, Slack [2] and Rowe [3] presented the "phonon glass and electron crystal" (PGEC) idea. The ideal thermoelectric material, according to this paradigm, should combine crystallike electrical capabilities with the thermal conductivity of amorphous materials, which is difficult to accomplish in a single molecule [4].
Additionally, the recent discovery of highly meritorious thermoelectric double perovskite oxides has rekindled interest in high-efficiency oxide materials (ZT). Double perovskite oxides have a lot of potential for thermoelectric applications due to their inherent low and high electrical transport capabilities. Their general formula is A 2 BB'O 6 , where A stands for alkaline earth elements and B for transition metals (B and B' are organized in octahedral position with six oxygen atoms in the form of BO 6 and B'O 6 ) [20]. Intriguingly, Pr 2 FeCrO 6 (PFCO) is also a recently identified novel double perovskite oxide. Its unusual physical properties, such as multiferroic [21], high T c and room temperature magneto-resistance [22], optical and magnetic properties [23], and magnetic and magnetocaloric properties [24], have recently caught the attention of researchers. On the other hand, no tests have ever been done on their thermoelectric properties. Therefore, we are presenting the Pr 2 FeCrO 6 double perovskite oxide's temperature-dependent thermoelectric properties. The A-and/or B-site cations can be partially substituted to change the degree of the perovskite structural distortion that affects the material properties. As an illustration, the majority of undoped perovskite materials have low oxide ion conductivity in open air. The formation of oxygen vacancies and subsequent increase in ionic conductivity occur when the A-site cation or B-site cation is partially substituted by acceptor cations. Lattice energy can, however, sometimes be lowered by arranging part or all of the ions over the available crystallographic sites when charges or the ionic radii of host-and dopant-cations differ sufficiently in certain circumstances. The researchers made it abundantly clear that by using the proper dopant, the physical, chemical, structural, magnetic, thermoelectric, and other properties could be tuned [5,16,20,[25][26][27][28][29][30][31]. In order to increase the electrical conductivity of thermoelectric materials, doping is also crucial [27][28][29]. This raises power factor and improves ZT. Notably, the high-temperature thermoelectric (TE) performances of rare earth-based materials doped with strontium (Sr) have shown promise recently, but their ZT values need to be increased before they can be taken into consideration for commercialization. The highest ZT of 0.05 was attained in Pr 1.8 Sr 0.2 CoFeO 6 at 673 K in a recent study by H. Wu et al. [20]. The present study hypothesizes that by altering the material chemistry and processing parameters, we can achieve even higher ZT values as compared to pristine Pr 2 FeCrO 6 .Hence, in present study, Sr-doped Pr-based family of Pr 2-x Sr x FeCrO 6 (PSFCO) (x = 0.0, 0.25, 0.50, 0.75, and 1.0) double perovskite has been synthesized by sol-gel technique and the effects of doping on temperaturedependent thermoelectric properties were examined in the temperature range of 300 K to 1400 K. It has been shown that the dimensionless ZT in PSFCO double perovskites can be improved by varying the concentration of A-site doping. These materials may be useful as multifunctional components in contemporary electronics, such as electrodes for batteries, electrolyzers or hydrogen fuel cells, solid electrolytes, thermoelectric devices, etc.

Preparation of the Pr 2-x Sr x FeCrO 6 double perovskite
An Sr-doped Pr 2-x Sr x FeCrO 6 with 0≤x ≤ 1, double perovskite was synthesized by using ceramic reactions. The nitrate precursors of Pr, Sr, Fe, and Cr were allowed to mix with the necessary stoichiometry of DI water and citric acid monohydrate (10% by weight). By stirring at 353 K for 4 hours with a pH near to 3, a homogeneous solution was produced using the sol-gel method. The resulting mixture was an extremely viscous residual gel, which was subsequently dried overnight in an oven at 373 K. After drying, a high-energy planetary ball mill (Make: Fretsch) with a maintained BPR (ball-to-powder ratio) of 10:1 was used for wet milling (using ethanol) in order to avoid agglomeration of the powder for 24 hours at 500 rpm, followed by conventional calcination at 1073 K for 6 hours. To achieve the desired nano-size of the calcined powders, the milling process was repeated at 500 rpm for 24 hours in a planetary micro mill (Fritsch, PULVERISETTE 7 premium line, Rhineland Palatinate, Germany). The milled calcined powders were bindered with PVA (poly vinyl alcohol) and then uniaxially pressed for 60 seconds at 600 MPa to form rectangular pellets. The pellets were then visually examined, and no surface cracks were observed. Ultimately, dense pellets of PSFCO ceramic samples were formed by sintering at 1473 K in open air for 6 hours.

Characterization of Pr 2-x Sr x FeCrO 6 double perovskite
The structural identity was determined using X-ray diffraction (XRD) on a Rigaku diffractometer with Cu-K radiation (λ = 1.5406 Å) at a scan rate of 1°/min and 0.02° step size. A Supra55 Zeiss field emission scanning electron microscope (FESEM) and a JEOL JEM-1400 transmission electron microscope (TEM) were used to examine the morphological characteristics of crushed pellets. X-ray photoelectron spectroscopy (XPS) using K-alpha radiation and monochromatic X-rays were used to investigate the elemental state of chemical in a synthesized Pr 2-x Sr x FeCrO 6 sample. The rectangularshaped sample was used to measure the temperaturedependent thermoelectric properties, like electrical conductivity (σ) and See-beck coefficient (S) with the help of ZEM-3 M10 apparatus (ULVAC-RICO Inc.). In order to minimize heat loss through radiation, the thermal diffusivity (α) was measured using a Hot Disk Thermal Constant Analyzer (ULVAC TC-7000) and the temperature-dependent specific heat constant (C P ) was investigated using the DSC technique. Finally, the thermal conductivity was calculated by using following relation: κ=αρC P ,where ρ is the density measured by Archimedes principle. The margins of uncertainty in high-temperature electrical conductivity, Seebeck coefficient, thermal conductivity, and figure of merit measurements were ± 5, ±5, ±5, and ± 2%, respectively.

Phase identification in Pr 2-x Sr x FeCrO 6 double perovskite
Well-sintered, dense pellets of Pr 2-x Sr x FeCrO 6 with 0≤x ≤ 1 were used to investigate the crystal structure using X-ray diffraction at room temperature. The diffraction peaks in Figure 1 (a) are indexed to the search for perovskite structures, and the presence of many peaks suggests that the entire composition of the synthesized PSFCO family (x = 0 to 1) was crystalline in nature with an orthorhombic crystal structure. The single-phase solid solution with negligible PrCrO 2 (in x = 0) and SrFeO 2 (in x = 1) residue in entire compositions is further supported by the x-ray diffraction pattern. However, in the proposed investigation, the quantitative analysis through Rietveld refinement by using FullProf software [32] states that the amount of impurity phase in each synthesized sample was 1% (w/ w) and the refinement of final doped sample (x = 1) is shown in Figure 1 The controlled Sr substitution at the Pr-site in Pr 2- x Sr x FeCrO 6 revealed that it had no impact over the crystallographic structure of the host material, because both the pure and Sr-doped samples were crystallized in an orthorhombic structure with Pbnm space group (SFCO). In Table 1, all of the structural characteristics and unit cell parameters are listed. Because Sr 2+ has a larger ionic radius (1.18) than Pr +3 (0.99), the unit cell parameters in the PSFCO compound increase as the Sr content increases, as would be expected. Additionally, the average crystallite sizes for the current series of compositions were calculated using the Scherrer formula for the FWHM value of (112) reflection [23], and it was discovered that they increase with increasing dopant concentration, as shown in Table 1. The increase in crystallite size in PSFCO double perovskite can be attributed to a remarkable interplay of factors [3,17,[20][21][22][23][24][25][26][27][47][48][49][50][51][52]. The tunning in the crystallite size is a result of Sr doping, leading to the formation of solid solutions and enhanced grain growth. The lattice strain caused by the difference in ionic radii between Pr and Sr ions contributes to the remarkable transformation. As Sr concentration increases, a thorough solid solution forms, giving rise to larger crystallites. Also, the Sr dopants act as nucleation centers, fueling the mesmerizing dance of crystal development and catalyzing grain growth. The thermodynamics of the system undergo a dramatic change, enabling Ostwald ripening and the growth of larger crystallites. Interatomic spacing, on the other hand, primarily depends on lattice parameters determined by crystal structure and ion sizes, rather than oxygen content. Sr doping influences the lattice parameters and interatomic spacing, inducing slight movement of oxygen ions toward more charged cations. As a result, very little lattice distortion might appear in the samples, which would cause all of the diffraction peaks to move slightly, by about 0.03 to 0.04º, toward a lower diffraction angle. Additionally, the subsequent increase in diffraction intensity with increasing Sr-concentration can be attributed to growing particle size (as validated by morphological examination in Figure 2). Furthermore, a compound's stability/distortion status must be classified according to the tolerance factor (t) in order to be approximated with a reasonable degree of certainty [1]. Thus, the tolerance factor (t), introduced by Goldschmidt in 1926, can be used to indicate how much perovskite oxides deviate from their ideal structure. Equation (2) can be used to calculate the tolerance factor from the Goldschmidt relation:  where, the ionic radii of A, A', B, B' cations, and oxygen ions are r A , r A' , r B , r B' , r o respectively. The Pr 3+ (0.99 Å) and Sr 2+ (1.18 Å) ions at A-site had average radii of r A and r A' , respectively, while Fe 3+ (0.645 Å), Cr 3+ (0.615 Å) and O 2-(1.40 Å) ions r B, r B' and r o , respectively. Table 1 shows how the values of t increased with increasing Sr-content from x = 0 to 1 (for orthorhombic, t = 0.71 to 0.9). Because Pr 2-x Sr x FeCrO 6 has a lower tolerance factor, it was expected that the influence of short-range trivalent metal ions like Fe 3+ and Cr 3+ during Sr-doping would be mitigated by long-range Pr 3+ ions with negligible lattice strain (less than unity). The estimated tolerance factor confirms that the system must have significant lattice distortion.  Figure 2 (a-b) clearly indicates that the samples comprised agglomerates of crystallites/grains because of the increased rate of precipitation during synthesis [23]. It is also worth noting that the average particle size grew as Sr-doping concentration rose. The mean particle sizes were estimated by taking both agglomerated and non-agglomerated nanoparticles into consideration. By counting the number of particles on the SEM micrograph of Pr 2-

Morphologies of the particles
x Sr x FeCrO 6 for compositions x = 0 and 1, respectively, it was found that the mean particle size (inset Figure 2(a,b)) was in the range of 13.91 ± 3 nm to 31.64 ± 3 nm. Figure 2 (c-d) further displays the TEM images of both pure PFCO and PSFCO double perovskites. Similar to FESEM, TEM projected that non-spherical nanoparticles may share certain edges, and if there was a size discrepancy, it might be brought on by the existence of those particles that weren't quite spherical. The results from SEM micrographs were supported by the observation of a variation in grain size, ranging from 12.1 ± 5 nm to 28.3 ± 5 nm, as the Sr content was raised. The mean particle size derived by morphological investigations agrees well with the crystallite size reported by the XRD analysis, as shown in Table 1.
To establish the homogeneity of the double perovskite family Pr 2-x Sr x FeCrO 6 (0≤x ≤ 1), energy dispersive x-ray spectroscopy (EDS) was performed on the whole compositions. From Figure 3, the EDS examination of PSFCO (x = 1) sample confirmed the existence of the elemental constituents Pr, Sr, Fe, Cr, and O in the proper proportions. Moreover, certain unavoidable faint signals of C that were seen during the EDS investigation may have been caused by the use of conductive carbon tape in the FESEM study. As anticipated, no evidence of phase segregation could be seen in the Pr 2-x Sr x FeCrO 6 oxide samples, and measured atomic percentage (at %) of Pr, Sr, Fe, Cr, and O were in good agreement with theoretical values (inset of Figure 3-a).

X-Ray photoelectron spectroscopy and carrier concentration
X-ray photoelectron spectroscopy (XPS) was used to examine the chemical valence states of Pr, Sr, Fe, Cr, and O in the final doped PSFCO compound (at x = 1, i.e. PrSrFeCrO 6 ), as shown in Figure 4 (a-e). The XPS spectrum for Pr is shown in Figure 4(a), where the Pr 2p spectrum has two spin-orbit components with Pr 3d 5/2 at 935.4 eV and Pr 3d 3/2 at 954.2 eV and the obtained binding energy difference (△β E ), i.e. 18.8 eV was the spin orbit splitting energy (Gaussian fitting), suggesting high spin Pr. At the low-energy side of both The aforementioned XPS discovery also showed that the propensity of spin-orbit splitting in the 2p edges of Cr and Fe may be caused by either the final 2p 5 core hole state or by their various oxidation states. While Fe and Cr share a similar valence of + 3 in BO 6 and B'O 6 octahedra, respectively, these elements can alter their oxidation states by a super interchange of valence between the two octahedra. Because of this, Fe shows up as 2+ and Cr as 4+, and these two can begin their spin-orbit coupling as a result of unpaired cations with various oxidation states. Moreover, the quantity of unpaired electrons in each orbital can affect the values of the L-S coupling as well as the l and s quantum {∑l (l: azimuthal quantum number) and ∑s (s: spin quantum number)} [16]. Such an exciting finding could help to explain the Cr and Fe disorder in the Pr 2-x Sr x FeCrO 6 lattice. Given that Pr and Sr, unlike Cr and Fe, have stable valence states of Pr 3+ and Sr 2+ , Sr doping on Pr-sites is likely to enhance the charge carrier concentration (n p ) of Pr 2-x Sr x FeCrO 6 [20].
It's interesting to note that the carrier concentration (n p ) of Pr 2-x Sr x FeCrO 6 double perovskite was calculated using the following relation for all compositions (x = 0 to 1) at room temperature: n p = 1/R He , where R H is the measured Hall coefficient and e is the electron's charge in a forward and reverse magnetic field of around 10,500 G. As can be seen in Figure 4 (f), it was discovered that the n p increases from 1.07 × 10 14 to 1.84 × 10 20 cm −3 with increasing doping concentration. The impact of doping concentration on the behavior of n p (specific parameter) is evident. As the concentration increases, n p experiences accelerated growth. This is especially evident at higher doping concentrations, such as x = 1, where n p exhibits an exponential rise. This suggests a strong correlation between doping concentration and the significant amplification of n p , highlighting the influential role of higher dopant levels in shaping the observed trends. 0.0, 0.25, 0.50, 0.75, and 1.0
As shown in Figure 5 (a), electrical conductivity increases with increasing temperature and peaking at 942 K (Region-II)forpristine Pr 2 FeCrO 6 (PFCO) composition, whereas, two electrical conductivity peaks were observed with varying compositions from x = 0.25 to 1.0, indicating the semiconductor (dσ/dT > 0) to metallic (dσ/dT˂0) to semiconductor (dσ/dT > 0) transitions in all three regions (I, II, and III) respectively, which also signify the semiconducting behavior in PSFCO compounds at elevated temperature.The highest σ was recorded in PSFCO compounds was 6688 S/m, 11074 S/m, 15918 S/m, and 28,272 S/m for x = 0.25, 0.50, 0.75 and 1.0, respectively. At room temperature, the lowest σ (17.18 S/m) was observed in pristine PFCO but in entire PSFCO (x = 0.25 to 1.0) compound, the observed value of σ was in the order of 10 2 S/m. After substituting the Sr on A-site, the significant increase in semiconductor to metallic transition temperatures (T S→M ) was noticed along with increasing doping concentration (x = 0 to 1), as witnessed in Figure 5(a).
As shown in Figure 5 (b), the positive Seebeck coefficient (S) suggested that only p-type charge carrier was present in Pr 2-x Sr x FeCrO 6 double perovskite. The maximum observed value of S was 438 µV/K at 380 K in PFCO compound, decreases after substituting the Sr on A-site in PFCO compound, followed by further decrease with increase in doping concentration, as expected from the drastic rise in electrical conductivity. The decrease in Seebeck coefficient with increasing temperature suggested the presence of screening effect in all the samples, and also this behaviour contradicts Loffe theory [34], and can be expressed as: where, S is the Seebeck coefficient at temperature T, S 0 is the Seebeck coefficient at a reference temperature, k B is the Boltzmann constant, e is the elementary charge, n o is the carrier concentration at temperature T 0 , n p is the p-type carrier concentrations at temperature T, γ is the scattering factor of the material. Moreover, the little gain in thermo-power (S) was noticed at that temperature, where the σ was decreased and vice-versa.
The effects of electrical conductivity and the Seebeck coefficient of Pr 2-x Sr x FeCrO 6 (0≤x ≤ 1) double perovskite were combined to compute the power factor (i.e. PF=S 2 σ), as illustrated in Figure 5 (c). Initially, the PF increases rapidly up to the specified temperatures, thereafter a decreasing trend was observed as expected, with further increase in temperature. Because of the low electrical conductivity, the PF of pristine PFCO compound was very low around 24.31 µWm −1 K −2 near 760 K, while, the highest PF was 117.03 µWm −1 K −2 observed at 900 K for PSFCO compound at x = 1.0, which was 4.81 times more as compared to PFCO compound. As a result, the power factor was notably dependent on substitution of Sr.
To calculate the thermal conductivity (κ) in Pr 2- x Sr x FeCrO 6 (0≤x ≤ 1) compounds, thermal diffusivity (D) was measured in the temperature ranging from 300 K to 1400 K, as illustrated in Figure 5 (d). The following formula was used to compute the thermal conductivity: κ=DρC p , where D is thermal diffusivity, C p is heat capacity, and ρ is sample density. In PFCO compound, κ decreases from 1.29 Wm −1 K −1 at 300 K to 0.545 Wm −1 K −1 at 1400 K, but after substituting Sr on Pr-site, the sudden drop in κ was observed in every PSFCO compound (x = 0.25 to 1.0). However, after increasing the doping concentration of Sr, the rise in κ was also noticed in entire temperature range, validating the outcomes of electronic thermal conductivity (κ e ), as shown in Figure 5 (e). The variation in thermal conductivity (κ) across the doped compounds exhibited diverse patterns, characterized by both significant decreases and increases. The changes were observed to be both substantial and subtle, with instances of sudden shifts and occasional dramatic jumps occurring with increasing temperature. The Wiedemann-Franz law is applicable for the computation of electronic thermal conductivity (κ e ), which correlates with the electronic contribution to thermal conductivity [35,36] and in this study, the validity of the Lorenz number was based on an assumption. Moreover, for the PFCO compound, κ e stays extremely low throughout the entire temperature range studied but dramatically rises when Sr was substituted on the Pr-site. The observed peak values of κ e in Pr 2-x Sr x FeCrO 6 (0≤x ≤ 1) compounds were 0.012 W m −1 K −1 , 0.077 W m −1 K −1 , 0.152 W m −1 K −1 , 0.217 W m −1 K −1 , and 0.306 W m −1 K −1 , for x = 0.0, 0.25, 0.50, 0.75, and 1.0, respectively. Further, the other measure portion of conductivity associated with the thermal and electronic thermal conductivities in these crystalline solids was due to the lattice vibrations, i.e. the lattice thermal conductivity (κ l ), which was in similar fashion of thermal conductivity as shown in Figure 5 (f). It can be seen in Figure 5 (f) that the PFCO compound shows drastic drop in κ l up to 0.541 Wm −1 K −1 at 1400 K, while after introducing Sr dopant, it decreases linearly up to the saturation (in region-II, of Figure 5 (a)) before reaching the maximum values at elevated temperature.

Discussion
Generally, the perovskite structures get distorted by the tiny ions, which also reduce the electronic conductivity of compounds and vice versa. When the ions differ in both valences and radii during aliovalent substitution on the A-site cations, a more complicated scenario occurs. The valence difference caused by the A-site must be compensated for in order to maintain electrical neutrality, either by a multivalent B-site cation (transition metal) or by producing vacancies on the oxygen sublattice. Here, the possibility of a positive charge carrier source for Pr 2-x Sr x FeCrO 6 compounds was raised by the presence of multivalent cations like Fe and Cr in the B site of these double perovskites. It's interesting to note that after replacing the formal charge (B-site Fe 3+ /Fe 4+ ) with both Cr 2+ and Cr 3+ , it was anticipated that the charge compensation mechanism might produce holes (h*) as the majority charge carriers.
In Pr 2-x Sr x FeCrO 6 compounds, the Kröger-Vink notation is very helpful for expressing defect reactions, These defect reactions show that the presence of Cr 2 + /Cr 3+ and Fe 3+ /Fe 4+ in this Pr 2-x Sr x FeCrO 6 system can donate holes, implying p-type behavior in these double perovskites, as corroborated by the positive S in Figure 5 (b). Moreover, as illustrated in Figures 2 and 3, lattice defects in Pr 2-x Sr x FeCrO 6 double perovskite oxides can interact with carriers to create tiny polarons with increased effective mass, m*, potentially preventing free movement of these polarons as carriers in typical degenerate semiconductors [1]. Yet, the additional holes caused by oxygen loss at high temperatures (as confirmed by iodometric titration test in Figure 6-a) combined with ionic vibrations result in the creation of surplus polarons, transforming the PSFCO combination into a highly polarizable p-type semiconducting material. On an interesting note, depending on the characteristics of the material and the partial pressure of oxygen in the air, the electrical conduction can be either n-type or p-type. As an oxygen deficiency is necessary to produce electrons, ntype electronic conductors are often unstable at high temperatures in air or other oxidizing atmospheres. Ptype electrical conductors, however, are often stable in air since it takes a lot of oxygen to produce holes [37]. The iodometric titration test (KIO3 − ) is depicted in Figure 6 (a). The image clearly shows that pristine PFCO and doped PSFCO samples were oxygen-deficient up to the specified temperature. Evidently, in doped PSFCO, the oxygen loss increases as temperature rises, whereas the pristine PFCO sample showed a slight change in deficient behavior near the transition point, which correlates well with the thermal expansion coefficient (TEC), as shown in Figure 6 (b) and conductivity plots along with increasing temperature. The thermal behavior of BO 6 octahedrons determines the extent of thermal expansion in several perovskites, which is connected with octahedral tilt, distortion, and bonding between the B-cation and the surrounding anions [38]. Moreover, TEC is affected by magnetic and electronic transitions, as well as chemical composition changes that occur with increasing temperature. Consequently, when temperature rises, the number of oxygen vacancies in transition metal oxides increases, resulting in a drop in the transition metal cation oxidation state and, as a result, a lengthening of the metal-oxygen link [37]. The doping of strontium causes oxygen deficiency as confirmed by iodometric titration test, hence, the TEC of PFCO is substantially lower than that of PSFCO, suggesting that the PSFCO compound can be mechanically more stable at higher operating temperature.
The quantity of porosity and various phase transition kinetics in the sintered powder sample might be related to the TEC values discovered using dilatometry measurements as shown in Figure 6 (b). The doping of Sr content increases typically result in favorable increases in average TEC values. Sr doping, however, favors electrical transport as confirmed by electrical conductivity in Figure 5 (a), by positively influencing the formation of oxygen vacancies in materials. The doping content across the studied temperature range is linked to the main/significant contribution to thermal expansion (chemical expansion). Also, the polarizability of PSFCO compounds rises with increasing operating temperature and percentage of Sr 2+ substitution. XPS analysis, on the other hand, suggested that, due to the stable valence state of Pr 3+ and Sr 2+ , doping of Sr on Pr-sites can also help in achieving enhanced charge carrier (n p ) [20], as shown in Figure 4 (f), and this effect of increased charge density can be related to increased electrical conductivity, as shown in Figure 5 (a).
It is probable that charge carriers are concentrated at defect regions in this current investigation (Cr 0 B ,Cr 00 B ; Fe=Fe 0 B ). As a consequence, it appears that small polaron Pr 2-x Sr x FeCrO 6 system. The electrical conductivity induced by SPH conduction may be explained using the following Equation [39][40][41][42]: where, σ o stands for a constant, K B stands for the Boltzmann constant, E hop stands for the activation energy of tiny polaron hopping conduction, and T stands for temperature.
In both semiconductor areas, the small polaron hopping conduction process is depicted in Figure 7 as a plot between ln(σT) Vs 1/K B T. The linearly fitted curve between ln(σT) Vs 1/K B T in the complete Pr 2- x Sr x FeCrO 6 system uses a small polaron hopping conduction mechanism with temperatures spanning from 300 K to 1400 K, as illustrated in Figure 7. The SPH conduction mechanism was seen in both the semiconductor zones in every composition (except x = 0.0, pure PFCO), and data was also included in Table 2. It's interesting to note that, as shown in Table 2, for all compositions of Pr 2-x Sr x FeCrO 6 double perovskite, the high-temperature semiconductor phase requires less activation energy than the low temperature semiconductor phase. Interestingly, the activation energy for hoping was found to decrease as Sr-content increased, as indicated in Figure 7 and Table 2. The Heikes formula, as modified by Chaikin and Beni [43,44], is a popular method for illustrating the link between S and n p : where, e stands for the electronic charge and c for the fractional concentration of small polarons generated by the multivalent cations in the system. Equation 6 was used to determine the fractional polaron concentration. The PFCO (x = 0) compound had the lowest value, while PSFCO had the greatest value (x = 1), and this pattern of the "c" values was consistent with the electrical conductivity plot. As the dopant concentration grows (from x = 0 to 1), the polaron concentration, on the other hand, climbs dramatically. This discovery seems to coincides nicely with enhanced n p , as revealed by Hall measurement, which is depicted in Figure 4 (e). During the tested temperature, the Seebeck coefficient and power factor were likewise in good accord with these charge transfer mechanisms. The thermally stimulated holes that may occupy the conduction band as temperature rises is another explanation for the quick increase in PF with increased Srcontent in addition to the findings discussed above. Despite this, at higher temperatures, PF declines after reaching peak values, which is thought to be caused by the lower effective carrier density at higher temperature.
After doping, the increase in thermal conductivity of PSFCO compounds, in comparison to pristine PFCO compounds, can be attributed to the combined effects of lattice vibration and electronic dispersion effects, as illustrated in Figure 5 (f). These effects contribute significantly to the efficient transfer of heat within the PSFCO compounds, resulting in their higher thermal conductivity across a wide temperature range. Interestingly, after doping, the rise in lattice thermal conductivity (κl) with temperature suggests a tuned rate of phonon scattering at higher temperatures, i.e. the rate of phonon scattering can be tuned by adjusting the p-type carrier concentration, which suited well to our explanation of electrical conductivity in Figure 5 (a). In fact, a decrease in the p-type carrier concentration does not directly lead to a decrease in the rate of phonon scattering but the presence of p-type carriers, such as holes, can actually enhance phonon scattering and decrease thermal conductivity in some materials [45]. Therefore, a decrease in p-type carrier concentration would result in reduced scattering of phonons and potentially higher thermal conductivity. This was either due to the structural variations or the presence of impurities in the materials. Surprisingly, the slight increase in κ l with temperature was accompanied by a decrease in electrical conductivity. This suggests that p-type carrier concentration decreased due to increased diffuse scattering, indicating that thermal conductivity rises with temperature and eventually reaches a plateau before phonon-phonon scattering takes over at higher temperatures.
The Debye approximation may also be used to explain the center for the decrease in κ l with temperature [5]: where, x=hω/k B T, are the dimensionless parameters, h stands for the Planck constant, ω stands for the angular frequency, θ D stands for the Debye temperature, k B stands for the Boltzmann constant, τ P stands for the relaxation rate, and expressed the average phonon velocity. Nevertheless, the relaxation rate depends on scattering at grain boundaries, point defects, and phonon-phonon umklapp. Conduction may have decreased over the whole temperature range as a result of the Umklapp and point defect phonon scattering processes, which indicates substantial structural instability in contrast to mass impact [46]. Intriguingly, increasing Sr-content in PSFCO results in distortion (confirmed by tolerance factor in Table 1), larger grains (based on morphological study in Figure 2), and enhancement in bond length (confirmed by XRD analysis in Table 1), which may be the mechanism behind scattering. In addition to the aforementioned, we believe that the lower size of the phonon mean free path in contrast to the crystal size was what led to the thermal conductivity saturation before to reaching high temperatures. The distinction suggests that not all phonons can achieve comparable effectiveness, which lowers the mean free path for entropy transfer [16,17,19,[42][43][44][47][48][49][50]. The conclusion that all Pr 2- x Sr x FeCrO 6 ceramic compositions have intrinsically poor thermal conductivity, nearly less than 70-80% of thermoelectric oxides, can thus be drawn [51]. In particular, the PFCO compound exhibits extremely low thermal conductivity at high temperatures, indicating the glass-like properties. Finally, the ZT was computed and presented in Figure 8 based on the observed electrical characteristics at various temperatures. Lower S and higher σ provide a remarkably high ZT, which was further enhanced by selective Sr-doping in Pr 2-x Sr x FeCrO 6 double perovskites from x = 0 to 1. In the Pr 2-x Sr x FeCrO 6 double perovskite, the Sr content has a considerable influence on the temperature dependence of ZT. The ZT of samples with low Sr content obviously rises linearly, but samples with high Sr content (x ≥ 0.5) exhibit an exponential rise in ZT, which reaches a maximum of 0.105 at 942 K for x = 1.0 and was around 3.9 times greater than the pure compound (x = 0.0). It was observed that, throughout the entire composition (x = 0 to 1), the ZT values increased due to the substitution of Sr on the Pr-site up to a certain temperature, then decreased with further increase in temperature, and varied slightly depending on the transport properties and lattice reduction [52-55, , 56]. Unexpectedly, this ZT in Pr 2- x Sr x FeCrO 6 double perovskite value was greater than the already researched p-type Pr 2 CoFeO 6 (ZT = 0.05 at 673 K) [20], and Ca 2 (Zn x Fe 2-x )O 5 (ZT = 0.002 at 1073 K) 52,53], earlier published values. This demonstrates how carefully choosing the dopants may significantly increase the ZT of Pr 2-x Sr x FeCrO 6 double perovskite.

Conclusion
The thermoelectric potential of Sr-doped Pr 2 FeCrO 6 double perovskite is reported in this paper. By using a sol-gel technique, a disordered single-phase double perovskite Pr 2-x Sr x FeCrO 6 (x = 0.0, 0.25, 0.50, 0.75, and 1.0) material was synthesized, followed by calcination and traditional sintering. The every nano-sized crystalline composition holds orthorhombic crystal structure with Pbnm space group, as confirmed by Rietveld refinement of XRD data in Pr 2-x Sr x FeCrO 6 double perovskite. The average particle/grain size values determined by morphological analysis were consistent with the crystallite size values observed from the XRD analysis. The EDS analysis confirms that there exists a homogeneous distribution of elements throughout the compositions in Pr 2-x Sr x FeCrO 6 material. The XPS results exhibited the uniform distribution of elements/cations with presence of Pr 3+ , Sr 2+ , Fe 4+ /Fe 3+ , and Cr 2+ /Cr 3+ ions on the surface of Pr 2-x Sr x FeCrO 6 double perovskite. During temperature dependent conductivity measurement, the entire Sr-doped PSFCO (x = 0.25 to 1) samples showed an intermediate metallic phase (dσ/dT˂0) transition between the two semiconductor (dσ/dT > 0) phases. The p-type conductivity was maintained in all the samples, as confirmed by the positive value of Seebeck coefficient. Interestingly, in these double perovskites, a small polaron hopping conduction mechanism was observed. Due to the formation of defects, Sr 2+ doping on Pr 3+ sites suggested the significant improvement in the carrier density and, as a result, electrical transport performance and thermoelectric power factor tuned, while Sr-doping also significantly lowers the activation energy barrier for SPH, benefiting from the high electrical transport properties. The doping causes oxygen deficiency and structural modifications in compounds that increase the TEC and may result into the formation of mechanically stable compound for high temperature operation conditions. In conjunction with the reduction of inherent low κ l caused by sol-gel synthesis induced nanograins and Sr-doping induced lattice distortions alongwith point defects, the highest figure of merit (ZT = 0.105) was calculted for PrSrFeCrO 6 at 942 K. Due to higher ZT and enhanced TEC values in PSFCO compound, and further by employing multi-strategies, like, band engineering, optimizing carrier density, energy filtering, etc., this composition can be a good choice as a catalyst or cathode or anode in the area of energy generation, conversion and storage application.