Structural, Optical, Charge-Transport, and Dielectric Properties of Double-Perovskite La2Co1−zFezMnO6 (z = 0, 0.2–1.0)

A series of double-perovskite La2Co1−zFezMnO6 (z = 0, 0.2–1.0) ceramics were synthesized using a well-established sol–gel method. The series of samples with a monoclinic phase and a P21/n symmetry were characterized by XRD, FTIR, conductivity, and capacitance measurement to extract charge-transport and dielectric characteristics at room temperature. The obtained IR spectra fitted well with the Lorentz oscillator model to calculate the damping factor, optical frequency, and oscillator strength and compared with the theory, which gave better agreement. The calculated activation energies from the Arrhenius plot supported the semiconducting nature of all samples. The temperature and frequency-dependent dielectric parameters, such as the real part (εr′), imaginary part (ε″) of the dielectric constant, dielectric loss (tanδ), and ac-conductivity (σac) were extracted. The dielectric constant (εr′, ε″) and dielectric loss (tanδ) were enhanced at a low frequency, while the ac-conductivity (σac) displayed higher values at higher frequencies. The enhancement in the dielectric parameters with increasing iron concentrations arose due to the higher surface volume fraction of iron (Fe3+) ions than the cobalt (Co3+) ions. The radius of the Fe3+ (0.645 Å) was relatively higher than the Co3+ ions (0.61 Å), significantly influenced by the grains and grain boundaries, and enhanced the barrier for charge mobility at the grain boundaries that play a vital role in space charge polarization.


Introduction
Over the last few decades, the field of multiferroics has attracted enormous attention from the scientific community owing to their unique physical properties, which hold great importance in practical applications [1][2][3]. Due to strong electron correlations in these materials, they show such effects as magnetism, ferroelectricity, multiferroic properties, or superconductivity. Several competing properties, such as magnetic order and electric polarization, often coexist [4][5][6]. In memory-storage devices, controlling magnetic components through electric manipulation will provide faster data storage and recovery. The merger of the magnetic order, switchable properties, and electric polarization on one data-storage device will increase the storage size by five orders of magnitude. Thus, the area currently required for 1 megabyte of information will be capable of carrying one terabyte. Understanding the physical effects behind these phenomena is essential to allow a systematic search for new composites with these combined properties.
The well-known ferroelectrics, including KNbO 3 , BaTiO 3 , BiFeO 3 , and PbTiO 3 , belong to first-class multiferroics. A few boracites with chemical formula M 3 B 7 O 13 X, with M and X representing the divalent metal and halogen, have tetrahedral phase ferroelectrics [5,7,8] Owing to the coexistence of ferromagnetism and electrical insulation within a single material, ferromagnetic materials with chemical formula RE 2 AMnO 6 , where RE and A correspond to the rare-earth element and transition metal element, respectively, attract a lot of interest. Of particular note, the double-perovskite La 2 CoMnO 6 in all forms, i.e., thin films,

Experiment
The series of double-perovskite La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) ceramics were synthesized by the well-established sol-gel process as described in [18] with all chemical purities higher than 99%. All samples were sintered for 6 h at 950 • C to obtain the required phase. We used an advanced diffractometer (Bruker, D8 DISCOVER, Billerica, MA, USA with Cu-Kα (λ = 1.5405 Å) radiation source) to verify the structure of ceramics, the phase was identified by Xpert HighScore software, and Rietveld refinement was carried out with the help of JANA 2006 software. Furthermore, using an Fourier-transform infrared (FTIR) spectrometer (Bruker, Vertex 80v, Billerica, MA, USA) close to normal incidence mode, infrared reflectivity (IR) spectra were determined. Diffuse reflectance spectra with a range of 30-7500 cm −1 were obtained from the spectrophotometer (PerkinElmer, Lambda 950 UV/VIS/NIR, Waltham, MA, USA) with a two-beam splitter detector. The electric and dielectric properties were tested with an LCR meter (GW Instek, 8101G, Taiwan, China) in a temperature range of 303-423 K and frequency range from 10 Hz-1 MHz. The chargetransport properties were performed using a Keithley meter (Keithley Instruments, 2400, Cleveland, OH, USA) at various temperatures.

Structural Properties
The X-ray diffraction pattern confirmed the structure of the double-perovskite La 2 Co 1−z Fe z MnO 6 ceramics at room temperature. Further investigation confirmed that this type of compound has a monoclinic phase with a P2 1 /n symmetry at room temperature. With the help of the fitting software, XRD peaks are well indexed, indicating that all series of this compound have a monoclinic phase with a P2 1 /n space group with a slight increase in the lattice parameters. The maximum intensity peak is observed at 2θ ≈ 32 • , representing the principal peak of the P2 1 /n space group. All peaks are well fitted to the calculated pattern, indicating that the double-perovskite La 2 Co 1−z Fe z MnO 6 has a single phase that displays the purity of the ceramics shown in Figure 1.

Structural Properties
The X-ray diffraction pattern confirmed the structure of the double-perovski La2Co1−zFezMnO6 ceramics at room temperature. Further investigation confirmed that th type of compound has a monoclinic phase with a P21/n symmetry at room temperatur With the help of the fitting software, XRD peaks are well indexed, indicating that all seri of this compound have a monoclinic phase with a P21/n space group with a slight increa in the lattice parameters. The maximum intensity peak is observed at 2θ ≈ 32°, represen ing the principal peak of the P21/n space group. All peaks are well fitted to the calculate pattern, indicating that the double-perovskite La2Co1−zFezMnO6 has a single phase th displays the purity of the ceramics shown in Figure 1.
The Braggs peaks were slightly shifted towards the right with more iron substitutio because the ionic radius of the Fe +3 (0.645 Å ) ion is relatively larger than the Co +3 (0.61 Å ion. The larger ionic radius caused expansion or shrinkage of the structure and a sligh The X-ray diffraction scans of (a) La 2 CoMnO 6 (b) La 2 Co 0.8 Fe 0.2 MnO 6 (c) La 2 Co 0.6 Fe 0.4 MnO 6 (d) La 2 Co 0.4 Fe 0.6 MnO 6 (e) La 2 Co 0.2 Fe 0.8 MnO 6 (f) La 2 FeMnO 6 .
The Braggs peaks were slightly shifted towards the right with more iron substitution because the ionic radius of the Fe +3 (0.645 Å) ion is relatively larger than the Co +3 (0.61 Å) ion. The larger ionic radius caused expansion or shrinkage of the structure and a slight increase in the lattice parameters a, b, and c. Due to the variation in particle size, the surface-to-volume ratio also fluctuated. The obtained refined lattice-parameter values are listed in Table 1, which are in agreement with the parameters reported by others [19]. The plot of the lattice parameters versus the contents is shown in Figure 2. We found that the volume and grain size fluctuated with the increase in Fe +3 concentrations during refining the structural parameters. The average crystalline grain size of La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) estimated the (2 0 0) peak with the help of the Scherrer formula given by Equation (1), and the values are listed in Table 1 [12].
where k represents the geometry factor, and its value is considered as 0.9, θ is a diffracted Braggs angle, and β defines the full width at half maximum. The plot of the lattice parameters versus the contents is shown in Figure 2. We fou that the volume and grain size fluctuated with the increase in Fe +3 concentrations dur refining the structural parameters. The average crystalline grain size of La2Co1−zFezMn (z = 0, 0.2-1.0) estimated the (2 0 0) peak with the help of the Scherrer formula given Equation (1), and the values are listed in Table 1 [12].

= cos
( where k represents the geometry factor, and its value is considered as 0.9, θ is a diffrac Braggs angle, and β defines the full width at half maximum.

Optical Properties
The IR reflectivity spectra of the series of double-perovskite La2Co1−zFezMnO6 (z 0.2-1.0) ceramics were measured using FTIR at room temperature, as shown in Figure  and 4. There were 24 Raman-active modes (12 Ag + 12 Bg) and 33 infrared reflectivity ( active modes (ГIR = 17 Au + 16 Bu) for the monoclinic phase according to the group the [13]. We conducted IR tests, and only 12 active phonon modes were observed in our sa ples instead of 33 phonons in La2Co1−zFezMnO6. We did not observe other active phon modes due to the polycrystalline nature of the material, in which grains have differ

Optical Properties
The IR reflectivity spectra of the series of double-perovskite La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) ceramics were measured using FTIR at room temperature, as shown in Figures 3 and 4. There were 24 Raman-active modes (12 A g + 12 B g ) and 33 infrared reflectivity (IR) active modes (Г IR = 17 A u + 16 B u ) for the monoclinic phase according to the group theory [13]. We conducted IR tests, and only 12 active phonon modes were observed in our samples instead of 33 phonons in La 2 Co 1−z Fe z MnO 6 . We did not observe other active phonon modes due to the polycrystalline nature of the material, in which grains have different orientations. However, our samples exhibited the monoclinic phase, but we observed less polar phonons modes than the theoretically predicted 33 polar phonon modes for this structure. For the searching of optical phonons mode, we refined our experimental results with the help of the Lorentz oscillator model. The equation of the dielectric function and IR via Fresnel's law are expressed in Equations (2) and (3).
where orientations. However, our samples exhibited the monoclinic phase, but we observed less polar phonons modes than the theoretically predicted 33 polar phonon modes for this structure. For the searching of optical phonons mode, we refined our experimental results with the help of the Lorentz oscillator model. The equation of the dielectric function and IR via Fresnel's law are expressed in Equations (2) and (3).
where ϒ j and Sj are the damping factor and the oscillator strength of the j-th optical pho-non, ε∞ displays the charge polarization, and represents the j-th optical transverse phonons [18,20]. The fitted parameters of optical transverse phonon values of La2Co1−zFezMnO6 (z = 0, 0.2-1.0) are listed in Table 2. The parameters of the phonon oscillator strength amplitude "S" are listed in Table 3. The damping factor "γ" with the width of the vibrational oscillation phonon mode is inserted in Table 4. In the low-frequency region, the observed IR spectra of La2Co1−zFezMnO6 display that the phonon modes are slightly shifting with the concentration of Fe +3 . Two reasons potentially caused the shifting of frequency below 200 cm −1 are mass substitution effects and the constant force described by the harmonic oscillator equation.
= √D/u (4) where D represents the constant force that depends on the bond length, bond angle, and lattice parameters. This constant force is highly sensitive to the bond length and lattice parameters. u represents the reduced mass of those ions, which contributes to the related phonon modes. Hence, the phonon frequency will be enhanced or suppressed due to the variation of constant lattice parameters. Due to the higher surface volume fraction, Fe 3+ ions' influence on the optical transverse mode causes an increase in the phonon frequency. j and S j are the damping fact j-th optical phonon, ε ∞ displays the charge polarization, and ω 2 TO(j) represents the j-th optical transverse phonons [18,20]. The fitted parameters of optical transverse phonon values of La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) are listed in Table 2. The parameters of the phonon oscillator strength amplitude "S" are listed in Table 3. The damping factor "γ" with the width of the vibrational oscillation phonon mode is inserted in Table 4. In the low-frequency region, the observed IR spectra of La 2 Co 1−z Fe z MnO 6 display that the phonon modes are slightly shifting with the concentration of Fe +3 . Two reasons potentially caused the shifting of frequency below 200 cm −1 are mass substitution effects and the constant force described by the harmonic oscillator equation.
where D represents the constant force that depends on the bond length, bond angle, and lattice parameters. This constant force is highly sensitive to the bond length and lattice parameters. u represents the reduced mass of those ions, which contributes to the related phonon modes. Hence, the phonon frequency will be enhanced or suppressed due to the variation of constant lattice parameters. Due to the higher surface volume fraction, Fe 3+ ions' influence on the optical transverse mode causes an increase in the phonon frequency.
structure. For the searching of optical phonons mode, we refined our experiment with the help of the Lorentz oscillator model. The equation of the dielectric func IR via Fresnel's law are expressed in Equations (2) and (3).
where ϒj and Sj are the damping factor and the oscillator strength of the j-th opt non, ε∞ displays the charge polarization, and ( ) 2 represents the j-th optical tr phonons [18,20]. The fitted parameters of optical transverse phonon v La2Co1−zFezMnO6 (z = 0, 0.2-1.0) are listed in Table 2. The parameters of the phon lator strength amplitude "S" are listed in Table 3. The damping factor "γ" with t of the vibrational oscillation phonon mode is inserted in Table 4. In the low-fr region, the observed IR spectra of La2Co1−zFezMnO6 display that the phonon m slightly shifting with the concentration of Fe +3 . Two reasons potentially caused th of frequency below 200 cm −1 are mass substitution effects and the constant force d by the harmonic oscillator equation.

= √D/u
where D represents the constant force that depends on the bond length, bond an lattice parameters. This constant force is highly sensitive to the bond length an parameters. u represents the reduced mass of those ions, which contributes to th phonon modes. Hence, the phonon frequency will be enhanced or suppressed d variation of constant lattice parameters. Due to the higher surface volume frac ions' influence on the optical transverse mode causes an increase in the phonon fr    The intermediate phonon is observed above 200 cm −1 due to the motion of B ions against the vibrations of oxygen (MnO6). At higher frequencies (>500 cm −1 ), the phonon modes are attributed to the stretching of the octahedral (Fe/Mn)O6. In the low-frequency range 200 cm −1 , one phonon ωTO1 is noted at 127.10, 124.83, 130.83, 129.10, 128.75, and 127.40 cm −1 for (z = 0, 0.2-1.0), which shows the altering of phonon modes due to the reduced mass effect of Fe +3 . These parameters are confirmed from the harmonic oscillator relation corresponding to the force constant and reduced mass. In the low-frequency region, the phonons (ωTO4) modes are missing for the z = 0.8 and 1.0 contents, which may be shifting of phonon modes due to the reduced mass effect of magnetic metal ions (Fe +3 and Co +3 ). In the high-frequency region (400-700 cm −1 ), the phonon mode ωTO9 is missing for z = 1.0. Due to the stretching of phonon modes or octahedral distortions of MnO6 in the    The intermediate phonon is observed above 200 cm −1 due to the motion of B ions against the vibrations of oxygen (MnO 6 ). At higher frequencies (>500 cm −1 ), the phonon modes are attributed to the stretching of the octahedral (Fe/Mn)O 6 . In the low-frequency range ≤200 cm −1 , one phonon ω TO1 is noted at 127.10, 124.83, 130.83, 129.10, 128.75, and 127.40 cm −1 for (z = 0, 0.2-1.0), which shows the altering of phonon modes due to the reduced mass effect of Fe +3 . These parameters are confirmed from the harmonic oscillator relation corresponding to the force constant and reduced mass. In the low-frequency region, the phonons (ω TO4 ) modes are missing for the z = 0.8 and 1.0 contents, which may be shifting of phonon modes due to the reduced mass effect of magnetic metal ions (Fe +3 and Co +3 ). In the high-frequency region (400-700 cm −1 ), the phonon mode ω TO9 is missing for z = 1.0. Due to the stretching of phonon modes or octahedral distortions of MnO 6 in the higher frequency regions above (400-700 cm −1 ), the exploration of optical phonon modes is interesting. The structural distortion is produced due to the reduction of masses related to La, Fe, and Mn ions caused by the vibration of the oxygen atoms in comparison to the limitation of the harmonic oscillator. The doping of lighter or heavier ions in the La 2 Co 1−z Fe z MnO 6 compound causes expansion or shrinkage in the size of the whole crystal. Due to the shrinkage, the crystal's influence on the volume function ultimately gives an upsurge for the variation of the phonon mode frequencies.

Electrical Properties
By adopting the two probe methods, we measured the electrical properties of the double-perovskite La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) ceramics in the temperature range of 303-423 K. From the I-V measurement, we observed a linear response with an increase in the current in all temperature ranges that obeyed Ohm's law. Furthermore, the current was enhanced with the specific voltage V o = 15V with temperature, displaying the semiconductor behavior of this ceramic. The I-V curves at various temperatures for La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) are shown in Figure 5. It can be observed that the current increases significantly when adding Fe doping, indicating an enhancement of the electrical behavior of the doped samples. This increment behavior may result from creation of polaron generated from the distorted lattice structure by the substitution of Co with Fe [19]. Temperature-dependent resistivity was calculated using the equation: According to the adiabatic nearest-neighbor hopping over small polarons (Holstein polaron) model (ANHSP), the decline in resistivity with the enhancement in temperature is due to the thermally triggered charges corresponding to the conduction mechanism of hopping [21][22][23], and the activation energy of the samples was calculated based on the formula expressed in Equation (6), with the help of the Arrhenius plot of Lnρ versus 1/kβ T. According to the adiabatic nearest-neighbor hopping over small polarons (Holstein polaron) model (ANHSP), the decline in resistivity with the enhancement in temperature is due to the thermally triggered charges corresponding to the conduction mechanism of hopping [21][22][23], and the activation energy of the samples was calculated based on the formula expressed in Equation (6), with the help of the Arrhenius plot of Lnρ versus 1/k β T.
where β is the Boltzmann constant, E a is the activation energy, ρ represents the resistivity, and T is the absolute temperature. The activation energy of the samples increases with the mole percentage of Fe +3 due to the large atomic radii compared to the Co 3+ . Usually, the activation energy (E a ) of the polaronic conduction of holes and electrons is close to 0.2 eV, but in the case of oxides, ionic conductors have E a greater than 0.90 eV. The calculated activation energies from the Arrhenius plot support the semiconducting nature of all samples and the negative correlation between the resistivity and the iron concentration.
The polaronic hopping conduction in this temperature range (303-423 K) is explained by the hopping polaron model [24][25][26][27][28], which explains the bandgap conduction mechanism. Usually, the low-temperature resistivity curve is explained by the electron-phonon scattering or impurity scattering mechanisms by changing the slope of ρ (T) around a charge carrier's localization temperature [29]. In the high-temperature regime, small polaron hopping appears to be effective by the linear curve fit, while variable range hopping (VRH) conduction is dominated for the low-temperature regime. The temperature-dependent resistivity (left side) and the Arrhenius law (inset of right side) for La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) is shown in Figure 6. The insets represent the concentration-dependent resistivity and the extracted activation energy from the Arrhenius law. It can be seen that the activation energy increases with Fe doping and reaches a maximum with a 0.5 Fe doping ratio, then suppression may be due to the agglomeration and segregation of Fe nano-additives at grain boundaries.
Materials 2022, 15, x FOR PEER REVIEW 9 of of Fe +3 due to the large atomic radii compared to the Co 3+ . Usually, the activation energy (Ea) of th polaronic conduction of holes and electrons is close to 0.2 eV, but in the case of oxides, ionic co ductors have Ea greater than 0.90 eV. The calculated activation energies from the Arrhenius pl support the semiconducting nature of all samples and the negative correlation between the resisti ity and the iron concentration. The polaronic hopping conduction in this temperature range (303 423 K) is explained by the hopping polaron model [24][25][26][27][28], which explains the bandgap conductio mechanism. Usually, the low-temperature resistivity curve is explained by the electron-phono scattering or impurity scattering mechanisms by changing the slope of ρ (T) around a charge ca rier's localization temperature [29]. In the high-temperature regime, small polaron hopping appea to be effective by the linear curve fit, while variable range hopping (VRH) conduction is dominate for the low-temperature regime. The temperature-dependent resistivity (left side) and the A rhenius law (inset of right side) for La2Co1−zFezMnO6 (z = 0, 0.2-1.0) is shown in Figure 6. Th insets represent the concentration-dependent resistivity and the extracted activation energy fro the Arrhenius law. It can be seen that the activation energy increases with Fe doping and reaches maximum with a 0.5 Fe doping ratio, then suppression may be due to the agglomeration and segr gation of Fe nano-additives at grain boundaries.

Dielectric Properties
The dielectric properties of double-perovskite La2Co1−zFezMnO6 (z = 0, 0.2-1.0) com posites were measured by conductance (G) and capacitance (C) with an LCR meter. Th real part ( ′ ) and imaginary part ( ′′ ) of the dielectric constant, dielectric loss (tanδ), an ac-conductivity ( ac) were studied in the frequency range of 10 Hz-1 MHz between 30 and 423 K. The real part of the dielectric constant ( ′ ) displays the capacitive energy sto age in the material when the electric field is applied, as shown in Equation (7).
where εo represents the permittivity of the free space, A is the area, and d is the thickne

Dielectric Properties
The dielectric properties of double-perovskite La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) composites were measured by conductance (G) and capacitance (C) with an LCR meter. The real part ( ε r ) and imaginary part (ε ) of the dielectric constant, dielectric loss (tanδ), and ac-conductivity (σ ac ) were studied in the frequency range of 10 Hz-1 MHz between 303 and 423 K. The real part of the dielectric constant ( ε r ) displays the capacitive energy storage in the material when the electric field is applied, as shown in Equation (7).
where ε o represents the permittivity of the free space, A is the area, and d is the thickness of the pellet. According to the Wagner dielectric response theory, the grain boundaries are nonconducting layers, while the grains are conducting due to the hopping process. The charges move to the poorly conducting layer from the good one with an applied electric field [30,31]. The real part (Log ε r ) of the dielectric constant versus frequency at various temperatures for La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0) is shown in Figure 7a-f. The inset of Figure 7a-f represents the temperature-dependent real part of each plot. The values of the real part (Log ε r ) increase with the increase in temperature and Fe concentration due to the relaxation of electric dipoles, but at a higher frequency, they are found to be flat, coinciding with each other close to zero. This reveals that the relaxation procedure ceases at a higher frequency due to the short time scale involved. The sharp fall in real part (Log ε r ) at higher frequency originates from dielectric relaxation. The gradual decline in the real part of the dielectric constant (Log ε r ) in the low-frequency region can be ascribed to space charge polarization due to blocked charge carriers at physical barriers, such as grain boundaries. At low frequencies, grain boundaries or interfacial polarization play a vital role in dominating the dielectric properties of the material. At maximum values of the real part ( ε r ) at low frequency, the charges begin to accumulate at grain boundaries. The net polarization is changed due to the dominance of the space charge polarization [32]. The increase in the dielectric parameters (Log ε r ) with increasing z-contents may arise due to the surface volume fraction of Fe 3+ ions compared to the cobalt Co 3+ ions. The radius of the Fe 3+ is (0.645 Å) in a high-spin state is relatively higher than the Co 3+ ions (0.61 Å) in a low-spin state [14,33,34], which enhances the barrier for charge mobility at the grain boundaries and plays an important role in the space charge polarization. The contribution of net dipoles becomes slow at higher frequencies, and only electronic polarization remains active due to the very small net polarization in comparison to low-frequency polarization. Similarly, at 10 Hz and 423 K, the peak values of the real part (Log ε r ) are 10.60, 10.61 11.47, 10.99, 10.91, and 10.01 with concentrations (z = 0, 0.2-1.0). It is found that the magnitude of the dielectric constant (Log ε r ) increases with temperature and Fe concentrations (z = 0, 0.2-1.0). Due to increase in temperature, space charges are released, which participate in polarization. Grain and grain boundaries are also increased in size due to an increase in temperature.
In the same temperature and frequency range, the imaginary part (Log ε " ) is also calculated by using Equation (8). By applying an AC field, the imaginary part of the dielectric displays the energy dissipated within the material [35]. The variation in the imaginary part versus frequency at selected temperatures from 303 to 423 K with different concentrations of z = 0, 0.2-1.0 composites is shown in Figure 8a-f. The estimated values of the imaginary part (Log ε " ) versus temperature are represented in the inset of each graph with z = 0, 0.2-1.0. The imaginary part displays higher values at low frequency and is found to be constant at high frequency. According to the Maxwell-Wagner model and Koop's theory, the dielectric medium is assumed to be made of well-conducting grains separated by grain boundaries.   As the imaginary part (Log ε") depends on the conductance, none of the electrons in the dielectric ceramic are bound. These free electrons play a vital role in conductance. This current loss may be due to the migration of free electrons or any other energy consumption, whereas only ion vibration losses contribute at high frequencies [36,9]. The imaginary part of the dielectric constant (Log ε") shows no response at a higher frequency due to a short time period. Most carriers cannot follow the external electric field, and the material response becomes very small. Grain boundaries are more effective at low frequency, Figure 8. (a-f) Imaginary part of dielectric (Log ε " ) versus Log f graph for La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0). Inset of (a-f) represents the temperature-dependent imaginary part of each plot.
As the imaginary part (Log ε " ) depends on the conductance, none of the electrons in the dielectric ceramic are bound. These free electrons play a vital role in conductance. This current loss may be due to the migration of free electrons or any other energy consumption, whereas only ion vibration losses contribute at high frequencies [9,36]. The imaginary part of the dielectric constant (Log ε " ) shows no response at a higher frequency due to a short time period. Most carriers cannot follow the external electric field, and the material response becomes very small. Grain boundaries are more effective at low frequency, due to which charges hop, resulting in a higher dielectric constant, and the carriers can follow the frequency of externally applied ac field.
The dependence of the dielectric loss or tanδ on the frequency at different temperatures for the double perovskites is shown in Figure 9(a-f). Tangent loss is generally referred to as a dissipation factor and calculated from Equation (9), and it is significant to the mechanisms of dielectric relaxation and ac conduction.
Materials 2022, 15, x FOR PEER REVIEW 13 due to which charges hop, resulting in a higher dielectric constant, and the carrier follow the frequency of externally applied ac field. The dependence of the dielectric loss or tanδ on the frequency at different tem tures for the double perovskites is shown in Figure 9(a-f). Tangent loss is general ferred to as a dissipation factor and calculated from Equation (9), and it is significa the mechanisms of dielectric relaxation and ac conduction.
The dissipation of energy (tanδ) shows maximum values with the temperatu crease at low frequencies. The tanδ has a higher value compared to the energy stor low frequencies. The tanδ decreased with increasing frequency due to the attri charges to the field [37]. The variations in tanδ were observed with frequency for dif concentrations of iron. The dielectric loss increased with increasing the concentration temperature variation is also shown in each inset of Figure 10a-f. The energy losses be attributed to the collision, vibration, and other charges' phenomena. Moreover, th is related to the period difference between relaxation and the applied field. Low relax time results from the instantaneous polarization process while large relaxation time sociated with the delayed polarization process. When the applied field has the sam riod of relaxation, the loss contribution is at its maximum. Losses turn smaller whe relaxation time greatly differs between the period of the applied field. The loss bec small if the relaxation time is relatively larger than the applied field frequency. I situation, the polarization mechanism is much slower than the reversal of the field, w cannot make ions follow the applied field. Similarly, if the relaxation time is much sm than the applied field frequency, the polarization process can easily follow the ap field with no lag, creating a small loss. The dissipation of energy (tanδ) shows maximum values with the temperature increase at low frequencies. The tanδ has a higher value compared to the energy stored at low frequencies. The tanδ decreased with increasing frequency due to the attributed charges to the field [37]. The variations in tanδ were observed with frequency for different concentrations of iron. The dielectric loss increased with increasing the concentration, and temperature variation is also shown in each inset of Figure 10a-f. The energy losses may be attributed to the collision, vibration, and other charges' phenomena. Moreover, the loss is related to the period difference between relaxation and the applied field. Low relaxation time results from the instantaneous polarization process while large relaxation time is associated with the delayed polarization process. When the applied field has the same period of relaxation, the loss contribution is at its maximum. Losses turn smaller when the relaxation time greatly differs between the period of the applied field. The loss becomes small if the relaxation time is relatively larger than the applied field frequency. In this situation, the polarization mechanism is much slower than the reversal of the field, which cannot make ions follow the applied field. Similarly, if the relaxation time is much smaller than the applied field frequency, the polarization process can easily follow the applied field with no lag, creating a small loss.
AC conductivity is shown in the inset of each plot in Figure 11a-f. The values of σac also increased with the temperature because charge carriers can more easily hop the barrier at high temperatures compared to low temperatures [38]. The other reason is that large ionic radii Fe +3 (0.645 Å ) in a high-spin state with relatively higher surface effects than the Co 3+ ion (0.61 Å ) in a low-spin state pile up the free carriers at the grain boundaries, where these carriers have enough energy to jump over the barriers at high frequencies. Figure 11 (a-f) Ac conductivity (σac) versus Log f graph for La2Co1−zFezMnO6 (z = 0, 0.2-1.0). Inset of (a-f) represents the temperature-dependent Ac conductivity part of each plot. Figure 11. (a-f) Ac conductivity (σ ac ) versus Log f graph for La 2 Co 1−z Fe z MnO 6 (z = 0, 0.2-1.0). Inset of (a-f) represents the temperature-dependent Ac conductivity part of each plot.
The σ ac values remained almost constant for all samples at low frequency and showed an abrupt increase at high frequency. At low frequency, the hopping of charges across the grain boundaries is minimal and increases with frequency. The temperature-dependent AC conductivity is shown in the inset of each plot in Figure 11a-f. The values of σ ac also increased with the temperature because charge carriers can more easily hop the barrier at high temperatures compared to low temperatures [38]. The other reason is that large ionic radii Fe +3 (0.645 Å) in a high-spin state with relatively higher surface effects than the Co 3+ ion (0.61 Å) in a low-spin state pile up the free carriers at the grain boundaries, where these carriers have enough energy to jump over the barriers at high frequencies.

Conclusions
A series of double-perovskite La 2 Co 1−z Fe z MnO 6 (z = 0., 0.2-1.0) composites were synthesized successfully by the sol-gel technique. The structural properties were investigated by XRD and confirmed that the synthesized composites possess a monoclinic phase with a P2 1 /n symmetry at room temperature. The obtained IR spectra were fitted well by the Lorentz oscillator model to calculate the damping factor, optical frequency, and oscillator strength and compared with the theory, which gave better agreement. Moreover, I-V measured the curve at room temperature following Ohm's law. The calculated activation energies from the Arrhenius plot support the semiconducting nature of all samples. The resistivity decreases with increased concentration. Furthermore, the frequency and temperature-dependent dielectric constant ( ε r , ε ) and dielectric loss (tanδ) were enhanced at low frequencies while the ac-conductivity (σ ac ) displayed the maximum value at high frequencies. The increase in the dielectric parameters with increasing z-contents may arise due to the surface volume fraction of Fe 3+ ions compared to the cobalt Co 3+ ions. The radius of the Fe 3+ is (0.645 Å) is relatively higher than the Co 3+ ions (0.61 Å), which can be significantly influenced by the grains and grain boundaries, enhancing the barrier for charge mobility at the grain boundaries that play an important role in the space charge polarization.