Microstructural Characterization and Magnetic, Dielectric, and Transport Properties of Hydrothermal La₂FeCrO₆ Double Perovskites

Abstract

Double perovskite La₂FeCrO₆ (LFCO) powders were synthesized via the hydrothermal method, which crystallized in an orthorhombic (Pnma) structure and exhibited a spherical morphology with an average particle size of 900 nm. Fourier transform infrared spectroscopy demonstrated the presence of fingerprints of vibrational modes of [FeO₆] and [CrO₆] octahedra in the powders. The XPS spectra revealed dual oxide states of Fe (Fe²⁺/Fe³⁺) and Cr (Cr³⁺/Cr⁴⁺) elements, and the oxygen element appeared as lattice oxygen and defect oxygen, respectively. The LFCO powders exhibited weak ferromagnetic behavior at 5 K with a Curie temperature of 200 K. Their saturation magnetization and coercive field were measured as 0.31 μ_B/f.u. and 8.0 kOe, respectively. The Griffiths phase was observed between 200 K and 223 K. A butterfly-like magnetoresistance (MR)–magnetic field (H) curve was observed in the LFCO ceramics at 5 K with an MR (5 K, 6 T) value of −4.07%. The temperature dependence of resistivity of the LFCO ceramics demonstrated their semiconducting nature. Electrical transport data were fitted by different conduction models. The dielectric behaviors of the LFCO ceramics exhibited a strong frequency dispersion, and a dielectric abnormality was observed around 260 K. That was ascribed to the jumping of electrons trapped at shallow levels created by oxygen vacancies. The dielectric loss showed relaxation behavior between 160 K and 260 K, which was attributed to the singly ionized oxygen vacancies.

1. Introduction

Perovskite transitional metal (TM) oxide nanostructures have gained a lot of attention due to their wide spectrum of intriguing properties [1,2,3] and promising technical applications [4]. Recently, nanostructures of double perovskites (DPs) A₂B′B″O₆ with hybrid 3d, 4d, or 5d TM ions at the B-site have attracted much attention because of their potential applications in the fields of microelectronic devices such as magnetic memories [5], magnetic tunnel junctions [6], and spin filtering devices [7]. In such a system, the interplay between the B′ (localized 3d-block TM ions) and B″ ions (delocalized 4d or 5d-block TM ions) provides more compositional flexibility to generate fascinating multifunctionalities [8,9]. Therefore, under the hybrid 3d- with 4d- or 5d-block TM ions at the B-site, highly spontaneous ordering can be achieved at the B-site because of their sizable differences in chemical valence states and the ionic radii between B′ and B″ TM ions. For example, structural ordered SrFeMoO₆ [5], Sr₂FeReO₆ [10], and Sr₂CrReO₆ [11] bulk DP oxides have been successfully synthesized. However, in the special case of two 3d-block TM ions positioned at the B′ and B″ sites, the synthesis of the 3d (B′)–3d (B″) ordered DP oxides is much more difficult owing to their similar ionic radii [12,13].

As a representative 3d–3d DP oxide, La₂FeCrO₆ (LFCO) has drawn considerable attention since it provides a model system for testing the propensity for ferromagnetism, given this combination and configuration of Cr³⁺ and Fe³⁺ ions, and the general infallibility of the Goodenough–Kanamori (GK) rules [14,15]. In the past decade, several works have been conducted in the LFCO oxide system from theoretical and experimental aspects. Ueda et al. [16] first realized the ferromagnetic (FM) ordering in the artificial LaCrO₃–LaFeO₃ superlattices, which were fabricated by laser molecular beam epitaxy via alternative depositions of Cr³⁺ and Fe³⁺ ions on the (111) plane. The magnetic coupling between Cr³⁺ and Fe³⁺ ions was confirmed to be FM in terms of superexchange [14,15]. Unfortunately, the saturated magnetization, M_S, was measured to be 3.0 μ_B/f.u., which is much lower than the theoretical one (M_S~7.0 μ_B/f.u.). The Rietveld refinements on the neutron diffraction data indicated that the LFCO compound crystallized in an orthorhombic perovkite structure (Pbnm) with a random positioning of the Fe and Cr cations at the B-sited sublattices, displaying antiferromagnetic (AFM) behavior at 265 K [17]. That is in agreement with the AFM coupling expected from the linear d³(Cr³⁺)–d³(Cr³⁺) and d⁵(Fe³⁺)–d⁵(Fe³⁺) superexchange (SE) interactions [18]. According to the GK rules, the FM coupling between the Fe³⁺ and Cr³⁺ ions in the LFCO DP oxide with a rock-salt ordering is expected due to the SE interaction via the Fe(d⁵)–O–Cr(d³) magnetic path [14,15]. Recently, theoretical calculations have demonstrated a ferrimagnetic (FiM) ground state in an ordered LFCO DP oxide with Cr³⁺ and Fe³⁺ ions coupled antiferromagnetically [19]. This conclusion was experimentally verified in the well-ordered epitaxial LFCO thin films with a B-site ordering degree as high as 90% [20], and the film had an M_S value of 2.0 μ_B/f.u. at 5 K, which was in accordance with the Pickett’s model [21], but the KG rules were broken. In the bulk LFCO samples, only the AFM behavior was observed, and the disappearance of the FM order was ascribed to the random positioning of the Fe³⁺ and Cr³⁺ ions at the B-sited sublattices [22]. As described above, different magnetic behaviors have been reported in the LFCO oxide artificial superlattice, thin films, and bulk counterparts, which are believed to be closely related to the diverse couplings of the Fe–O–Fe, Cr–O–Cr, and Fe–O–Cr bonds [16]. To date, although the structure and physical properties of the LFCO oxide system are widely studied, there is still much controversy in their magnetic behaviors because the magnetic properties of LFCO DP oxides are very susceptible to the formation of oxygen vacancies (${\mathrm{V}}_{\ddot{\mathrm{O}}}$) and anti-site defects (ASDs) and to the volatilization of La during their high-temperature synthesis, which favors the formation of different chemical oxide states of Fe and Cr TM ions. A competition between the AFM and FM interactions via the different magnetic paths can result in different magnetizations in the LFCO oxides [23]. Thus, in order to explore the physical mechanisms behind these anomalous magnetic behaviors in LFCO DP oxides, more systematic investigations are highly required. Recently, Sun et al. [24] theoretically investigated the structural and magnetic properties of R₂CrFeO₆ (R = rare earth elements) DP oxides. They found an antiparallel alignment of the spins of the Fe and Cr TM ions existing in the R₂CrFeO₆ oxides, and such AFM interaction resulted in Ferrimagnetism (FiM). Since the FiM state has the lowest energy in the monoclinic P2₁/n phase, it could be regarded as a ground state. They also found that by increasing the R radius, the energy difference between the P2₁/n and R-3 phases were reduced. The theoretical calculations indicate that the magnetic properties of R₂CrFeO₆ DP oxides are not only dependent upon the Cr–O–Fe bond angles and tilting angles but also upon the different constituents of the material [24]. An insulating ferrimagnet LFCO with antiparallelly aligned S = 3/2 (Cr³⁺) and S = 5/2 (Fe³⁺) ions is predicted through different first-principles approaches [19]. Similarly, the magnetic orders in the (LaFeO₃)_n–(LaCrO₃)_n superlattices (denoted as SL(n)) were investigated by Monte Carlo simulations, and the results were compared with those from the LaFe_0.5Cr_0.5O₃ bulk counterpart [25]. It is also noticed that both FM and FiM behaviors appeared in the SL(1) and SL(3), whereas two different AFM orders occurred in the SL(2) and SL(4), respectively. The present results not only match well with the experimental data in the SL(1) but also demonstrate some novel ordered phases in the SLs constructed with other periods. However, the magnetic transport properties of the LFCO oxide system have not been reported yet despite the fact that they play important roles in the applications of spintronic devices.

In view of the above facts, in the present work, LFCO oxides were synthesized by the hydrothermal method. This process is characterized as a powerful method for the synthesis of perovskite oxide powders with controllable sizes and morphologies via the modifications of the hydrothermal processing parameters (e.g., precursor types, hydrothermal reaction temperature and time, pH value, and the type and concentration of the used mineralizers). The structural, magnetic, and dielectric properties of the hydrothermal LFCO DP oxides and their electrical and magnetic transport properties were comprehensively studied to better understand the relationships between the microstructure and physical properties of the LFCO oxides.

2. Materials and Methods

2.1. Synthesis of LFCO Powders

Hydrothermal process was used to synthesize the LFCO powders, which was carried out in a Teflon-lined stainless steel autoclave. The filling capacity of the autoclave was 80%. First, solutions of La(NO₃)₃·6H₂O, Fe(NO₃)₃·9H₂O, and Cr(NO₃)₃·9H₂O were prepared with 0.5 M concentration, and the molar ratio of La:Fe:Cr was kept as 2:1:1. For a typical synthesis of the LFCO powders, 10 mL suspension with Cr(NO₃)₃·9H₂O and Fe(NO₃)₃·9H₂O was made with the addition of 2 g KOH. Then, 10 mL La(NO₃)₃ solution was put into the above suspension, followed by the addition of another 8 g KOH (used as a mineralizer) under continuous stirring for 30 min at room temperature (RT) to form a mixed solution. The above mixed solution was transferred into hydrothermal autoclaves, which were kept at 433 K for 4 h and then cooled naturally to RT in air. Deionized water was used to wash the resulting powders, which were filtered and dried at 353 K for 12 h in an oven. Finally, the obtained brown powders were further post-annealed at 1473 K for 12 h in air in a tube furnace.

2.2. Microstructural Characterization and Physical Properties

All of the post-annealed LFCO powders were characterized by X-ray diffraction (XRD) at RT by using a SIEMENS D5000 diffractometer (Simens, Berlin, Germany) under Cu Kα radiation (λ = 1.54056 Å). XRD data were collected in a step-scan mode with 2θ in the range of 20° to 80°. The step size was 0.02°/s, and the collecting time for each step was 10 s. Structural parameters of the LFCO powders were extracted from Rietveld refinements on the powder XRD data by using the general structure analysis system (GSAS) software (GSAS-II) [26]. Fourier transform infrared (FTIR) spectrum of the LFCO powders was recorded in the wavenumber range of 400 cm⁻¹ to 1600 cm⁻¹ by using a PerkinElmer FT-IR 65 spectrometer. The morphology and chemical compositions of the tested samples were examined using a scanning electron microscope (SEM, FEI QUANTA 650, Hillsboro, OR, USA) with attached energy-dispersive X-ray spectroscopy (EDS). The EDS data were acquired in a mapping mode. X-ray photoelectron spectroscopy (XPS) was utilized to determine the chemical binding energies (BEs) and oxide states of the constituent elements in the LFCO powders. The La 3d, Fe 2p, Cr 2p, and O 1s XPS spectra were collected by using a PHI 5000 spectrometer (Versa Probe, ULVAC-PHI, Kanagawa, Japan) under Al Kα radiation at 1486.60 eV as X-ray source. All of the collected XPS spectra were calibrated by C 1s core-level positioned at a BE value of 284.60 eV. Dielectric measurements of the LFCO ceramics were carried out by using an Agilent impedance analyzer (Model, Agilent 4192 A, Agilent technologies, Santa Clara, CA, USA) operated in a frequency range of 10² Hz to 10⁶ Hz and a temperature range of 173–373 K controlled by a controller (DMS-2000, Partulab Technology, Wuhan, China). The D.C. magnetization (M) data were measured using a SQUID magnetometer (MPMS3, Quantum Design, San Diego, CA, USA). The M-T curves were recorded in the ZFC (zero-field cooling) and FC (field cooling) protocols under a magnetic field of 500 Oe over a temperature range of 2 K to 300 K. The M-H hysteresis loops were measured at 5 K and 300 K, respectively, within the magnetic field of ±6 T. All of the magnetic measurements were performed with the powder samples installed inside a Teflon capsule. A standard four-probe method was used to measure the resistivity (ρ) of the LFCO ceramics over a temperature range of 2 K to 800 K without applying a magnetic field. The magnetic field dependence of ρ for the LFCO ceramics was also measured from −6 to 6 T at 5 K and 300 K, respectively, from which the plot of magnetoresistance (MR) versus magnetic field (H) was extracted.

3. Results and Discussion

3.1. Microstructural Characterization

The XRD pattern of the LFCO powders measured at RT is shown in Figure 1. All of the XRD peaks can be indexed in an orthorhombic perovskite structure with a space group of Pnma (JCPDS file, No. 89-0478), indicating that the LFCO powders crystallize in an orthorhombic lattice symmetry. The profile of the Rietveld refinements on the powder XRD data and the allowed Bragg reflections are also demonstrated in Figure 1. Three reliability factors (R_p, R_wp, and χ²) were utilized to assess the fitting quality between the experimental XRD data and the theoretical ones, which were determined as R_wp = 6.03%, R_p = 5.98%, and χ² = 2.32.

The small value of χ² indicated a good fittingness. The refined structural parameters at RT are tabulated in Table S1, matching well with the data reported previously [27,28]. The Goldschmidt’s tolerance factor (t) of the LFCO powders was also calculated to be 0.961, indicating that an orthorhombic crystal structure was preferred for the present LFCO powder [29]. It is also noticed that any XRD peak representing the B-site cationic ordering does not appear in Figure 1, suggesting that the rock-salt type ordering of the Fe and Cr ions at B-sited sublattices is not realized due to their similar ionic radii and the same chemical valence states [30]. The Scherrer formula described by Equation (1) was used to determine the average crystallite size (D) of the LFCO powders [31]:

$$D=\frac{0.9\lambda }{\beta cos{\theta }_{\mathrm{B}}}$$

(1)

where λ is the X-ray wavelength of the Cu Kα radiation (λ = 1.54056 Å), β is the full width at half maximum (HWHM) of the (200)/(121) diffraction peak (in radians), and θ_B is the Bragg diffraction angle. The average crystallite size, D, was determined to be 41.0 nm.

The typical low-magnification SEM image of the LFCO powders is displayed in Figure 2a, and Figure 2b presents the histogram of the LFCO particle size distribution. It was noticed that the LFCO particles had a spherical morphology and their average particle size was determined to be 0.90 μm by fitting the particle size distribution in Figure 2b. Figure 2c displays an EDS spectrum acquired from the LFCO powders in a mapping mode, illustrating the EDS signals of the constituent elements, as expected. There are no other elements in the LFCO powders. The atomic molar ratio of La:Fe:Cr was determined to be 2:0.94:0.93 from the quantitative EDS data, which approached the nominal chemical compositions of the powders.

FTIR spectroscopy was used to identify the local structures of the hydrothermal LFCO powders. Figure 3 shows the FTIR spectrum obtained from the powders, where two absorption bands positioned at 437 cm⁻¹ and 603 cm⁻¹ are clearly observed. The weak absorption band located at 437 cm⁻¹ is attributed to the bending vibrations of the Fe–O bonds within the FeO₆ octahedron [32], while the intense absorption band positioned at 603 cm⁻¹ is a result of the stretching vibration of the Cr-O bonds within the CrO₆ octahedra [33]. Thus, the octahedral coordination of Fe and Cr ions with oxygen ions (O²⁻) in the powders is confirmed by the FTIR spectrum.

The chemical valence states of the constituent elements (La, Fe, Cr, and O) in the LFCO powders were verified by the XPS spectra at RT. Figure 4a exhibits a wide scanning XPS spectrum for the LFCO powders over the scanning energy from 200 to 1000 eV, where the La 3d, Fe 2p, Cr 2p, and O 1s core-level XPS peaks are clearly observed. In addition, the C 1s core-level XPS peak is also observed at 284.60 eV, which is attributed to the conductive carbon tape employed in the XPS measurements. High-resolution XPS spectra of the La 3d, Fe 2p, Cr 2p, and O 1s core levels are demonstrated in Figure 4b–e, respectively. In Figure 4b, the La 3d core-level XPS spectrum is deconvoluted into two doublets positioned in the 831–840 eV and 850–858 eV regions, respectively. The former doublet corresponds to the La 3d_5/2 level, and the latter one corresponds to the La 3d_3/2 level. In addition, the satellite peaks of the La 3d_5/2 and La 3d_3/2 XPS peaks appeared at 838.43 eV and 855.32 eV, respectively. These two satellite peaks are ascribed to the electron transferring from the oxygen valence band to the empty La 4f level [34]. The difference (Δ) between the BE values of the La 3d_5/2 (834.47 eV) and La 3d_3/2 (851.35 eV) XPS peaks was measured to be 16.88 eV, which corresponds to the spin–orbit coupling (SOC) of the La element. The Δ value of the La element as well as the BE values of the La 3d_5/2 and La 3d_3/2 XPS peaks confirm the presence of La³⁺ ions in the LFCO powder sample [35]. Figure 4c depicts the Fe 2p_3/2 core-level XPS spectrum over the scanning energy range of 706–715 eV, where two characteristic peaks are observed at 710.15 eV and 711.40 eV, respectively. They can be assigned to Fe²⁺ and Fe³⁺ ions, respectively [36]. The percentage molar ratio of Fe²⁺ to Fe³⁺ species was extracted from the peak fitting of the Fe 2p_3/2 XPS spectrum, which was 18%:82%. Thus, the effective oxide state of the Fe ion was +2.82. Figure 4d displays the Cr 2p_3/2 XPS spectrum over the energy region of 574–582 eV. Similarly, the Cr 2p_3/2 XPS peak is also deconvoluted into two peaks located at BE values of 576.25 eV and 578.90 eV, respectively. They are assigned to the Cr³⁺ and Cr⁴⁺ species, respectively [37,38]. The percentage molar ratio of the Cr³⁺ to Cr⁴⁺ species was calculated as 74%:26%. Therefore, the effective oxide state of the Cr ions was +3.26. In Figure 4e, the O 1s XPS spectrum in the scanning energy region of 526–536 eV exhibits an asymmetric feature, indicating more than one kind of oxygen species in the powders. Previously, three kinds of oxygen species named as lattice oxygen (denoted as O_α), defect oxygen (represented as O_β), and surface adsorbed oxygen (designated as O_γ) were reported to appear in the energy regions of 529–530 eV, 530–532 eV, and 533–534 eV, respectively [39,40]. Here, only two types of oxygen species appear in the present O 1s XPS spectrum, which are lattice oxygen (O_α) with a BE value of 529.42 eV and defect oxygen (O_β) with a BE value of 531.70 eV, respectively. The percentage molar ratio of the O_α to O_β species was estimated to be 46%:54%. This result indicates a higher molar ratio of oxygen defects in the LFCO powders. The species, peak positions, and percentage molar ratios obtained from the peak fittings of the Fe 2p_3/2, Cr 2p_3/2, and O 1s XPS spectra are summarized in Table S2, where the effective chemical oxidation states of the Fe and Cr ions are also presented.

3.2. Magnetic Properties

Figure 5a displays the M-H hysteresis loops of the LFCO powders recorded at 5 K and 300 K, respectively. It is noticed that the M-H curve of the LFCO powders exhibits a linear-like behavior at 300 K with almost zero remnant magnetization and coercivity, indicating paramagnetic-like behavior, while at 5 K, an open hysteresis loop is clearly observed, which is saturated at a particular magnetic field, H_F = 40 kOe. Beyond 40 kOe, the magnetization increases linearly with the magnetic field, indicating AFM behavior in the LFCO powders. That is ascribed to the SE interactions via the Fe³⁺–O–Fe³⁺ and Cr³⁺–O–Cr³⁺ magnetic paths. In addition, the observed small open hysteresis loop indicates a weak ferromagnetism, which is a result of the double-exchange interactions via the Fe³⁺–O–Cr³⁺, Fe³⁺–O–Cr⁴⁺, Fe²⁺–O–Cr³⁺, and Fe²⁺–O–Cr⁴⁺ magnetic paths. The magnetic parameters such as the remanent magnetization (M_r) and coercive field (H_c), could be obtained from the M-H hysteresis loop recorded at 5 K, which were M_r = 0.23 emu/g (or 0.02 μ_B/f.u.) and H_c = 8.0 kOe, respectively. It is noticed that beyond the H_F, the M-H curve exhibits a linear relationship; thus, the measured M-H curve is composed of two parts: one is from the saturated hysteresis loop at H_F, and another one is from the AFM magnetization following a linear relationship with the magnetic field. Therefore, the M(H) at the high-field region can be expressed by Equation (2) [41]:

$$M\left(H\right)={\chi }_{AF}H+M_{\mathrm{S}}$$

(2)

where the ${\chi }_{AF}H$ part is obtained from the AFM magnetization and M_S is the saturation magnetization of the weak ferromagnetism. To evaluate the M_S value of the samples, the law of approach to saturation is utilized, according to Equation (3) [42]:

$$M=M_{\mathrm{S}}\left(1-\frac{A}{\sqrt{H}}-\frac{B}{H}\right)$$

(3)

where A and B are constants, which are related to the micro-stress and magneto-crystalline anisotropy of the samples, respectively. Figure 5b displays the M(H) plotted as a function of $\frac{1}{\sqrt{H}}$, from which the M_S of the LFCO powders can be extracted as $\frac{1}{\sqrt{H}}$ approaching zero (or H→∞). Thus, the extracted M_S value of the LFCO powders from Equation (3) was 3.64 emu/g (or 0.31 μ_B/f.u.) at 5 K. This value was higher than the previously reported M_S values for the hydrothermal LaFe_0.5Cr_0.5O₃ powders (M_S = 0.21 μ_B/f.u.) [43], LFCO nanoparticles (M_S = 0.046 μ_B/f.u.) synthesized via the citrate auto-combustion technique [44], and LaFe_0.5Cr_0.5O₃ ceramics (M_S = 0.04 μ_B/f.u.) prepared by the solid-state reaction method [17]. However, the present M_S value was still much smaller than the theoretical value (M_S = 4.0 μ_B/f.u.) predicted for the LaFe_0.5Cr_0.5O₃ compound with an atomic order of Fe³⁺(d⁵)-O-Cr³⁺(d³) under high-spin states [45] or the M_S = 2.0 μ_B/f.u. reported for the LFCO thin films with a B-site ordering degree of ~90% [20]. Based on the theoretical calculations of the local spin density, Pickett et al. reported that the FiM ground state in the LCFO compound with an M_S of 2 μ_B/f.u. exhibited much more stability than that in the FM ones with an M_S of ∼7.0 μ_B/f.u. [21]. It is known that the magnetic properties of the LFCO samples are influenced not only by the magnetic coupling manners between the Fe³⁺ and Cr³⁺ ions at the B-sited sublattices (e.g., long range FM or AFM order), but also by the ordering degree of the Fe³⁺ and Cr³⁺ ions at B sublattices (e.g., fully ordering, partial ordering, or complete disordering) due to their different d-orbitals and occupied states (e.g., t_2g or e_g orbitals; empty, half-filled, or fully filled orbital states) and different magnetic moments. The present XRD data reveal the almost random positioning of the Fe³⁺ and Cr³⁺ ions at B-sited sublattices, which destroys the long-range alignments of magnetic moments. That is the reason why the experimental M_S value is much smaller than the theoretical M_S value (2.14 μ_B/f.u.) of the LFCO powders predicted based on an intuitive ionic model. The concentration of ASDs in the LFCO powders was evaluated to be 42.8%, and the corresponding B-site ordering degree ($\eta$) in the LFCO powders was determined to be 14.4%. The details for the calculations of the ASD concentration and B-site ordering degree are described in Supplementary Materials (Section S1). The temperature dependence of the M_ZFC and M_FC curves is shown in Figure 5c. The two M-T curves demonstrate a typical ferrimagnetic (FiM) transition around 200 K, which is a field-independent magnetic phase transition at T_C = 200 K, as confirmed by the derivative dM/dT curves for both the M_ZFC and M_FC, and it is plotted as an inset of Figure 5c. Such a transition temperature is smaller than that reported for the bulk samples (T_C = 265 K) [17]. The difference (ΔT_C = 65 K) can be ascribed to the finite size effect and/or surface strain effect of nanopowders [46]. Upon cooling, the M_ZFC curve varied very smoothly, and the paramagnetic (PM)-to-FiM transition behavior was very weak, and a much broad phase transition peak appeared. The blocking temperature (T_B) associated with this broad phase transition can be determined to be 113 K from the minimum position of the first derivative dM_ZFC/dT curve (see inset in Figure 5c). The low-temperature magnetic transition from the FiM to AFM phases appeared around T_N = 10 K. A large irreversibility (ΔM, defined as (M_FC–M_ZFC)/M_FC) was observed between the ZFC and FC curves, and the ΔM value reached ~71% at 50 K. That was ascribed to the fast increase in the predominant alignments of the spin orientations under 500 Oe. It was also noticed that the M_ZFC and M_FC curves did not merge together at T_C, and the irreversibility was still maintained well above T_C. That indicated that the true paramagnetic behavior was not realized immediately above T_C. It was noticed that the magnetic moments (M) of the LFCO powders were much smaller (in the range of 0.01–0.05 emu/g) within the investigated temperature range (2–300 K), which is attributed to the almost random positioning of the Fe³⁺ and Cr³⁺ ions at the B-sited sublattices and/or oxygen vacancies (${\mathrm{V}}_{\ddot{\mathrm{O}}}$). Figure 5d shows the temperature-dependent inverse of the susceptibility (χ⁻¹), where χ⁻¹ varies linearly with the temperature in the range of 225–280 K, indicating PM behavior. Thus, χ⁻¹(T) follows the Curie–Weiss (CW) law, as expressed by Equation (4):

χ⁻¹(T) = (T − θ_p)/C

(4)

where C denotes the CW parameter and θ_P represents the PM Curie temperature, which are tabulated in Table S1. The negative θ_P (θ_P = −441 K) suggests the predominant AFM interaction in the LFCO powders, which accords well with the AFM behavior reflected by the M-H loop at 5 K. As the temperature further decreases below 223 K, χ⁻¹ decreases smoothly in a quasi-linear manner at an extended temperature range. From the CW parameter, C, the effective magnetic moments (μ_eff) in the PM phase can be calculated as 6.64 μ_B/f.u. (@500 Oe), and the theoretical magnetic moment, μ_cal (per formula unit), of the La₂${\mathrm{F}\mathrm{e}}_{0.18}^{2+}{\mathrm{F}\mathrm{e}}_{0.82}^{3+}{\mathrm{C}\mathrm{r}}_{0.74}^{3+}{\mathrm{C}\mathrm{r}}_{0.26}^{4+}$O₆ can be calculated as 6.80 μ_B/f.u. Details of the μ_eff and μ_cal calculations are described in Supplementary Materials (Section S2). It is noticed that the μ_cal of the LFCO powders is slightly larger than the μ_eff under a magnetic field of 500 Oe. This is attributed to the weak magnetic coupling between the Fe and Cr ions because of the high concentration of ASD content in the LFCO powders. The calculated μ_eff and μ_cal for the LFCO powders are tabulated in Table S1. In Figure 5d, the χ⁻¹ − T curve exhibits a severe downturn deviation from the CW law below the temperature of 223 K (which is denoted as the Griffiths temperature, $T_C^G=223\mathrm{K}$), indicating the existence of a Griffiths phase (GP) in the LFCO powders. The GP phase lies between the magnetically ordered state and the completely disordered paramagnetic high-temperature regime [47]. In the GP region, χ⁻¹ − T holds a power law described by Equation (5) [48,49]:

$${\chi }^{-1}\left(T\right) \propto (T-T_C^R{)}^{1-\gamma }$$

(5)

where γ is an exponent, indicating the strength of GP, and $T_C^R$ is a random critical temperature. The $T_C^R$ can be obtained from γ = 0 in the CW regime [50] and is equivalent to the θ_P. Thus, γ = 0 represents the PM state, and γ = 1 represents a GP phase. A nonlinear fitting curve of χ⁻¹ in the temperature range between T_C and $T_C^G$ is presented in Figure 5d, from which the γ value is determined as 0.268, and the corresponding value of $T_C^R$ is 146.2 K. The small γ value indicates the weak Griffiths singularity in the LFCO powders.

3.3. Dielectric Properties

At RT, the dielectric properties of the LFCO ceramics measured as a function of the frequency is illustrated in Figure 6a. It is observed that both the dielectric constant (ε_r) and dielectric loss (tanδ) fall continuously as the frequency increases, displaying a dielectric behavior with a strong frequency dispersion. Rapid decreases in ε_r and tanδ at low frequencies can be contributed to the onset of several polarization mechanisms (e.g., space charge and dipolar, ionic, and electronic polarizations). Under the applied electric field, electrons can hop across the grains and grain boundaries of the LFCO ceramics. Charge carriers can pile up at the grain boundaries due to their higher resistance, leading to a space charge polarization [50]. The dipolar polarization and interfacial polarization make significant contributions to the larger ε_r at low frequencies. In order to make these charge carriers move along grain boundaries, much energy is needed, leading to a high value of tanδ. At higher frequencies, only the electronic and ionic polarization mechanisms are capable of reacting to the applied electric field, leading to smaller values of ε_r and tanδ.

In addition, the charge carriers gathered at the grain boundaries are also reduced at higher frequencies. Thus, these confined charge carriers are scattered, resulting in smaller ε_r and tanδ values. Such dielectric dispersion can be well explained based on the Maxwell–Wagner relaxation model [51].

Under a series of frequencies, the ε_r and tanδ values of the LFCO ceramics were measured as a function of temperature, and the results are displayed in Figure 6b and Figure 6c, respectively. As shown in Figure 6b, the ε_r grows slowly in the low-temperature region below 200 K, whereas beyond 200 K, the ε_r value increases fast and reaches a maximum peak at a temperature of around 260 K, and then it decays following the CW law. In addition, a collapse of the ε_r peak value takes place as the frequency is increased, but the peak positions do not shift towards the high-temperature direction. That is much clearly observed in the ε_r–T curves measured at low frequencies (1 kHz–40 kHz). This dielectric abnormality can be attributed to the jumping of electrons that are weakly bound to ${\mathrm{V}}_{\ddot{\mathrm{O}}} \mathrm{o}\mathrm{r} {\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$ vacancies. These trapped electrons can be excited into the conduction band of the LFCO ceramics under thermal excitation at ~260 K. The thermal energy (E = k_BT) at 260 K is 0.0224 eV, close to the activation energy (E_a = 0.02 eV) of the jumping electrons. As the applied electric field frequency increases, the trapped electrons are unable to catch up with the field, leading to a smaller contribution to the ε_r. Thus, a collapse of the ε_r value with an increasing frequency appears, as shown in Figure 6b. However, at higher frequencies (e.g., 400, 800, and 1000 kHz), the maximum peak of ε_r around 260 K disappeared; instead, a diffused phase transition occurred, which was mainly due to the almost random positioning of Fe and Cr cations at the B-sited sublattices in the LFCO ceramics with a high content of ASDs. The compositional fluctuations at the B-site can result in microscopic heterogeneity in the LFCO compound, leading to statistics of local Curie points around the average one. In addition, at higher frequencies, the contributions to the ε_r peak from the dipolar and interfacial polarizations become much less significant because the two polarization mechanisms are no longer operative to respond to the applied electric field. In Figure 6c, tanδ has a much smaller value in the low-temperature region below 260 K, and beyond 260 K, a fast increase in tanδ appears, especially for those measured at low frequencies. The tanδ–T curves in the local temperature between 180 K and 260 K is shown as an inset in Figure 6c, where the relaxation peaks for tanδ move towards q higher temperature with an increasing frequency from 1 kHz to 10 kHz, and the peak values also collapse simultaneously. This dielectric anomaly exhibiting the typical characteristics of dielectrics are associated with the oxygen vacancies (${\mathrm{V}}_{\ddot{\mathrm{O}}}$) and the related defect dipoles [52]. In the LFCO compound with a high content of ASDs, the ASDs can appear in the manner of the Fe-on-Cr site (Fe_Cr) or Cr-on-Fe site (Cr_Fe), respectively. Due to their different chemical oxide states, ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{\prime }$ (Fe²⁺ on Cr³⁺ site or Fe³⁺ on Cr⁴⁺ site) and ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{″}$ (Fe²⁺ on Cr⁴⁺ site) defects are readily formed at B-sites. The ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{\prime }$ and ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{″}$defects can attract the ${\mathrm{V}}_{\ddot{\mathrm{O}}}$ to form defect dipoles, namely ${{(\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{\prime }-{\mathrm{V}}_{\ddot{\mathrm{O}}})}^{·}$or ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{″}-{\mathrm{V}}_{\ddot{\mathrm{O}}}.$ These defect dipoles can change their orientations with respect to the jumping of O²⁻ into a vacant oxygen site of the oxygen octahedron or due to the hopping of electrons (among Fe²⁺/Fe³⁺ and/or Cr³⁺/Cr⁴⁺ pairs) that are weakly bound to ${\mathrm{V}}_{\ddot{\mathrm{O}}} \mathrm{o}\mathrm{r}$ ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$. At a low temperature below 200 K, the ${\mathrm{V}}_{\ddot{\mathrm{O}}}$ defects, ${{(\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{\prime }-{\mathrm{V}}_{\ddot{\mathrm{O}}})}^{¨}$or ${\mathrm{F}\mathrm{e}}_{\mathrm{C}\mathrm{r}}^{″}-{\mathrm{V}}_{\ddot{\mathrm{O}}}$defect dipoles, and the free electrons jumping between the Fe²⁺/Fe³⁺ and/or Cr³⁺/Cr⁴⁺ pairs, are bound at their respective defect sites. Thus, they make a small contribution to the ε_r and tanδ values at low temperatures [53]. However, a fast increase in the ε_r value appeared around 260 K at low frequencies, which was attributed to the ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}} \mathrm{o}\mathrm{r}$ ${\mathrm{V}}_{\ddot{\mathrm{O}}}$ defects in the LFCO ceramics, which were capable of moving across the whole sample, leading to the onsets of several polarization mechanisms (e.g., space charge and interfacial and/or defect dipolar polarizations) and affecting the dielectric relaxation behavior of the entire system [54]. In addition, the thermal excitation of electrons that are weakly bound to ${\mathrm{V}}_{\ddot{\mathrm{O}}} \mathrm{o}\mathrm{r} {\mathrm{V}}_{\ddot{\mathrm{O}}}$ vacancies and their hopping between Fe²⁺/Fe³⁺ and/or Cr³⁺/Cr⁴⁺ pairs also contribute to the rapid increase in the ε_r value around 260 K at low frequencies. Beyond 260 K, the ε_r value decays following the CW law. As shown in Figure 6c, the sharp increase in tanδ at temperatures beyond 260 K can be ascribed to the rapid increase in the electrical conduction of the LFCO ceramics. To determine the E_a value associated with the dielectric relaxation behavior exhibited by tanδ between 160 K and 260 K, a plot of Lnω versus 1000/T is used, as described by Equation (6):

$$\omega ={\omega }_0\exp \left(\frac{-E_{\mathrm{a}}}{k_{\mathrm{B}}T}\right)$$

(6)

where ω represents the angular frequency corresponding to the tanδ peak value, ω₀ denotes the characteristic relaxation angular frequency at infinite temperature, and k_B and T have their normal meanings, as described in the textbook. When fitting the Lnω vs. 1000/T curve with the Arrhenius law, a linear behavior between Lnω and 1/T was observed, as shown in Figure 6d, from which the E_a value was evaluated to be 0.586 eV. It was reported that the E_a value for the singly ionized ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$ in perovskite oxides is about 0.3–0.5 eV, while the value is ~1.0 eV for the doubly ionized ${\mathrm{V}}_{\ddot{\mathrm{O}}}$ vacancies [55]. The present E_a value was 0.586 eV, close to that reported for the ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}},$ implying that this dielectric relaxation is associated with the movement of singly ionized ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$.

3.4. Electrical and Magnetic Transport Properties

Figure 7a shows the resistivity Lnρ(T) of the LFCO ceramics measured with respect to the temperature without applying magnetic fields, where a continuous decrease in Lnρ(T) within the investigated temperature range (2–800 K) is verified by the negative $\frac{\mathrm{d}\mathrm{L}\mathrm{n}\rho }{\mathrm{d}T}$. That reveals the semiconducting nature of the LFCO ceramics. Notably, around 20 K, the plot of dLnρ/dT against T exhibits a valley, indicating a steep reduction in the resistivity Lnρ(T) taking place around 20 K. This phenomenon could be ascribed to the phase transition from antiferromagnetic to ferrimagnetic phases as the temperature is increased. Such a phase transition could make the spin states fluctuate around 20 K, leading to a sharp increase in the charge carrier concentrations. Correspondingly, a steep decrease in the resistivity Lnρ(T) appeared. To clarify the conduction mechanisms of the LFCO ceramic samples over 2–800 K, different conduction models such as Mott’s variable range hopping (VRH) model [56], the thermal activation model [57], and the small polaron hopping (SPH) model [58] are used to fit the resistivity data. In the high-temperature region of 350–683 K, the VRH model described by Equation (7) [56] was used to fit the resistivity data.

$$\rho ={\rho }_0\exp {(\frac{T_{\mathrm{o}}}{T})}^{\frac{1}{4}}$$

(7)

where ${\rho }_0$ is a resistivity pre-factor, and T_o = 18/k_BN(E_F)η³, denoting the Mott characteristic temperature. Here k_B, N(E_F), and η stand for the Boltzmann’s constant, the density of electronic states at the Fermi level (E_F), and the electron localization length, respectively. Figure 7b shows a linear fitting of the plot of the lnρ(T) vs. (1000/T)^1/4 curve in the temperature between 350 K and 683 K, which indicates that the electrical transport behavior of the LFCO ceramics in this temperature region is controlled by Mott’s VRH mechanism. The linear fitting of the lnρ(T) vs. (1000/T)^1/4 curve yields a T_o value of 5989.7 K, and the corresponding N(E_F) value of 4.42 × 10²⁴ eV⁻¹ cm⁻³, where the η value for the small polarons was chosen as 1.99 Å at the scale of the average bong length of the <Fe-O> or the <Cr-O> bond. The VRH transport is typically observed in a system with a random potential. Such random potential is provided by the random positioning of Fe and Cr ions at B-sited sublattices in the present samples, as confirmed by the above XRD patterns and magnetic data. A slight deviation from the linear curve below 350 K in Figure 7b may be related to the magnetic scattering from the localized moments of Fe/Cr ions. In the temperature region between 275 K and 350 K, the plot of Lnρ versus the inverse temperature (1000/T) shown in Figure 7c exhibits almost perfect linear behavior. Such kind of T-dependent resistivity is usually observed in semiconductors owing to thermal activation, which is described as by Equation (8) [57]:

$$\rho ={\rho }_O\exp (\frac{E_{\mathrm{a}}}{k_{\mathrm{B}}T})$$

(8)

The linear fitting gives out the E_a value of 11.43 meV, which is comparable to the dopant levels in normal semiconductors. In the temperatures below 275 K, the plot of the lnρ vs. 1000/T curve becomes more deviated from a linear fitting. Instead, in the temperature between 15 K and 77 K, two linear fittings can be found for lnρ/T vs. (1000/T), as illustrated in Figure 7d. That indicates that the electrical conduction is governed by the SPH mechanism in this temperature range, as described by Equation (9) [58]:

$$\rho ={\rho }_{O}T\exp (\frac{E_{\mathrm{a}}}{k_{\mathrm{B}}T})$$

(9)

The linear fitting of the ln(ρ/T) vs. (1000/T) curve shown in Figure 7d yields the E_a values for the small polarons hopping, which are 4.85 meV and 2.20 meV, respectively, in the two temperature regions, I (30–77 K) and II (15–30 K).

Magnetoresistance (MR) is described as a fractional change in the electrical resistance of the material when subjected to a magnetic field (H) at temperature (T), which is defined by Equation (10) [59]:

$$\mathrm{M}\mathrm{R}\left(T,H\right)=\left[\rho \left(T,H\right)-\rho \left(T,H_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}\right)\right]/\rho (T,H_{\mathrm{peak}})$$

(10)

where H_peak is the magnetic field at which the resistivity ρ reaches the maximum value. The MR dependent upon the field for the LFCO ceramics was measured at 5 K and 300 K, which is demonstrated in Figure 7e and Figure 7f, respectively. Negative MR behavior was observed at 5 K and 300 K. It is noticed that in Figure 7e, the MR-H plot at 5 K exhibits a typical butterfly-like shape, and the MR (5 K, 6 T) is measured to be −4.07%. Such MR value is higher than that reported for the semiconducting DP oxides such as Sr₂CrReO₆ ceramics with MR (4.2 K, 7 T) = −3.0% [60], Sr₂CrHfO₆ ceramics with MR (2 K, 7 T) = −2.73% [61], and Sr₂Fe_0.5Hf_1.5O_6−δ ceramics with MR (2 K, 7 T) = −2.05% [62]. However, the butterfly-like MR-H curve disappeared at 300 K in Figure 7f, which was attributed to the absence of M-H hysteresis during the magnetic domain rotation under the excitation of the magnetic field. In addition, the MR value varied rapidly in a small magnetic field region but slowly in the high magnetic fields, exhibiting a similar response of the steep magnetization. That indicates that such negative MR behavior attributed to the magneto-tunneling effect in the LFCO ceramics.

4. Conclusions

In summary, we successfully synthesized the LFCO DP powders via the hydrothermal process and investigated their structural, magnetic, dielectric, and electrical/magnetic transport properties. The Rietveld refinement on the powder XRD pattern revealed that the LFCO powders crystallized in an orthorhombic distorted perovskite structure with the Pnma space group. The SEM images demonstrated the spherical morphology of the LFCO powders with an average particle size of 0.90 μm. The EDS data gave out the atomic ratio of La:Fe:Cr equal to 2:0.94:0.93. FTIR spectroscopy demonstrated the structural units of the [FeO₆] and [CrO₆] octahedra in the powders based on their fingerprints of vibrational modes. The XPS spectra verified the dual chemical valence states of the Fe (Fe²⁺/Fe³⁺) and Cr (Cr³⁺/Cr⁴⁺) elements as well as two kinds of oxygen species, namely lattice oxygen and defect oxygen. The LFCO powders exhibited weak ferromagnetic behavior at 5 K with M_S = 0.31 μ_B/f.u. and H_c = 8.0 kOe, respectively. The T_C was determined to be 200 K, and a strong irreversibility between the M_ZFC and M_FC curves appeared around 295 K, which was attributed to the fast increase in the predominant alignments of the magnetic clusters under an external magnetic field. A Griffiths phase appeared in the temperature range between 200 K and 223 K. The LFCO ceramics exhibited butterfly-like MR-H behavior at 5 K, and the MR (5 K, 6 T) value was measured to be −4.07% due to the magneto-tunneling effect. The temperature-dependent resistivity data for the LFCO ceramics exhibited semiconducting behavior, and the resistivity data were well fitted by different conduction models in different temperature regions. The LFCO ceramics displayed dielectric behavior with a strong frequency dispersion, and the dielectric abnormality observed around 260 K was attributed to the jumping of electrons that were trapped in a shallow level created by oxygen vacancies. The activation energy associated with dielectric relaxation behavior, which was exhibited by the dielectric loss between 160 K and 260 K, was determined to be 0.586 eV, approaching the data reported for a singly ionized ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$. The movements of singly ionized ${\mathrm{V}}_{\stackrel{·}{\mathrm{O}}}$vacancies are responsible to such relaxor dielectric behavior.