Structural, Magnetic, and Electrical Properties and Magnetoresistance of Monovalent K-Substituted La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) Manganite

Abstract

The effects of K⁺ substitution at the Ba-site on the structural, magnetic, and electrical properties and magnetoresistance (MR) of La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) manganites prepared via the solid-state method were investigated. Rietveld refinement of X-ray diffraction data confirmed that both samples were crystallized in the rhombohedral structure with the R3$\overline{c}$ space group. In addition, the unit cell volume, V, and the average grain size also increased with K⁺ ions. Magnetization versus applied field (M–H) measurement was carried out, and the saturation magnetization (M_s) was found to increase from 1.81 ${\mu }_B$/f.u. (x = 0) to 4.11 ${\mu }_B$ /f.u. (x = 0.04), implying that K⁺ ions strengthened the ferromagnetic (FM) interaction. Furthermore, the metal–insulator transition temperature, T_MI, increased from 257 K (x = 0) to 271 K (x = 0.04). The observed behaviour may be related to the enhancement of double-exchange (DE) interaction due to the increase in Mn-O-Mn bond angle and electronic bandwidth (W), favouring the increasing rate of the e_g electron hopping process. The fitting of the electrical resistivity data in the metallic region describes the significance of residual resistivity, electron–electron and electron–magnon scattering processes to elucidate the electronic transport properties. Within the insulating region, variable range hopping (VRH) and small polaron hopping (SPH) models are proposed to describe the conduction mechanism.

1. Introduction

In recent years, increasing attention has been given to perovskite manganite with the general formula of La_1−xA_xMnO₃ (A = divalent elements such as Ca, Sr, and Ba) due to its unique feature of a phase diagram as a result of the complicated interplay between its charge, orbital ordering, spin, and lattice degrees of freedom [1]. The utmost traits of manganites include their multiferroic and magneto-transport properties, which can be tuned by external stimuli (such as magnetic field, pressure, X-rays radiation, injection of electrical current, etc.) or the substitution/addition of elements to improve the remarkable phenomena of magnetoresistance (MR) effect, where the electrical resistivity is reduced by orders of magnitude under the presence of an external magnetic field [2]. Manganites exhibiting high MR effects are extensively explored as they present opportunities for potential technological applications, such as magnetic sensors [3,4], non-volatile memory-based elements [5], and spintronic-based devices [6]. Interestingly, lanthanum perovskite manganite has also been recently proposed as a novel Dirac half-metal [7] material due to the large spin polarization and low energy consumption compared to conventional electronics. Pertaining to these remarkable findings, further LaMnO₃ studies are essential as they open the possibility of realizing colossal MR behaviours in manganite systems and emerging spintronic-based applications.

In perovskite manganites, the transport properties are mainly governed by Zener’s double-exchange (DE) mechanism involving the hopping of charge carriers from neighbouring Mn sites via O²⁻ ions in a Mn³⁺-O²⁻-Mn⁴⁺ network [8]. According to Zener’s model, e_g itinerant electrons can easily hop from the Mn³⁺ site to the Mn⁴⁺ site if the spins of manganese ions are aligned in parallel and are ferromagnetically coupled [8]. Further investigations have confirmed that the strength of the DE mechanism is also influenced by the lattice distortion effects due to the substitution of cations at the A-site, which induces changes in the average ionic radius (<r_A>), the Mn-O–Mn bond angle, as well as the Mn–O bond length [9,10,11]. These parameters are also regarded as controlling factors behind the occurrence of A-site disorder (${\sigma }^2$), tolerance factor (${\tau }_G$), electronic bandwidth (W), and many other phenomena in a perovskite system. According to previous studies, the introduction of a large A-site cation, such as Ag⁺ in La_0.7Ca_0.3−xAg_xMnO₃ [12] and K⁺ in La_1−xK_xMnO₃ [3], increases the bond angle and reduces the bond length. Consequently, the transfer of charge carriers between Mn ions via oxygen becomes easier and therefore facilitates the DE interaction. Furthermore, Zaidi et al. also presented the electrical behaviour study of La_0.67Pb_0.33−xNa_xMnO₃ [13] manganites_. The study attributed the decrease in T_MI to the reduction in <r_A> as a result of smaller Na⁺ substitution, which leads to the narrowing of the bandwidth (W) and in turn reduces the mobility of the e_g conduction electrons. In general, the influence of A-site substitution on transport properties depends on the size of doped ions and the degree of lattice distortion related to the modifications in bond length and bond angle.

Apart from structural modification due to the replacement of different ionic radii, the substitution of monovalent ions (e.g., Li, Na, K, and Ag) is also interesting, owing to its ability to induce twice more Mn⁴⁺ ions with x substitution compared to divalent ions [9,10]. In this case, only a small doping x concentration is required to induce the optimum ratio of Mn³⁺/Mn⁴⁺ at 0.33, where the magnetotransport properties are the most prominent [14]. Potassium (K⁺) exhibits the largest ionic radius among other monovalent elements and is often substituted in La-based manganite compounds [6,15,16,17,18]. Interestingly, the substitution of K⁺ has been reported to enhance the DE process and the magnetic properties by increasing the value of metal–insulator transition temperature, T_MI, and Curie temperature, Tc, in several manganites, such as La_1−xK_xMnO₃ [3,17], La_0.7Ca_0.3−xK_xMnO₃ [19], and Pr_0.8Na_0.2−xK_xMnO₃ [20]. Furthermore, K⁺ has also been reported to destabilize the charge-ordering (CO) state in Pr_0.75Na_0.25−xK_xMnO₃ and drive the system towards ferromagnetic–metallic (FMM) behaviour [21]. Meanwhile, Jeddi et al. found that with the increase in the K⁺ fraction x in Nd_0.6Sr_0.4−xK_xMnO₃ [9], T_C and the T_MI decrease with a corresponding rise in the MR effect. Therefore, the substitution of K⁺ is expected to promote rich functionalities in the transport system of perovskite manganites.

In the present work, the effect of K⁺ substitution in La_0.7Ba_0.3−xK_xMnO₃ is elaborated to investigate its structural, magnetic, and electrical transport properties. La_1−xBa_xMnO₃ manganite is an extensively studied compound that exhibits an FM state at x > 0.25 above room temperature while also displaying a long range of FM ordering that is highly desired from the viewpoint of technological applications [1,22,23,24]. In this regard, La_0.7Ba_0.3MnO₃ manganite is selected as the parent compound as it exhibits T_C and T_MI close to room temperature [25,26]. To our knowledge, neither magnetic nor electrical properties have been investigated in K-substituted La_0.7Ba_0.3−xK_xMnO₃. Different theoretical models have been used to explain the charge conduction mechanism responsible for the temperature-dependent resistivity graph.

2. Results and Discussion

2.1. Structural Properties

2.1.1. X-ray Diffraction (XRD) Analysis

Figure 1 displays the room temperature X-ray diffraction (XRD) pattern along with the Rietveld refinement for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04). Both samples exhibit a single phase and are indexed in a rhombohedral structure (a = b $\ne$ c) with the R3$\overline{c}$ space group. The sharpness of intensity peaks indicates the formation of samples with high crystallinity [1,27], and the location of peaks was also consistent with the previous studies of La_0.7Ba_0.3MnO₃ [27,28]. A minor additional peak denoted by the (*) symbol in Figure 1 associated with the Mn₃O₄ phase was detected for sample x = 0.04 at angle 2θ = 36.1° using X’pert High Score Plus software (reference code: 00-024-0734). Furthermore, the small difference between the observed and calculated data (blue line) suggested a good fitting quality, which can also be seen by the small value of the goodness factor (χ² ~ 1), as shown in

Table 1. Lattice parameters, unit cell volume (V), and goodness of fit (χ²) obtained from Rietveld refinement for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04). ${\sigma }^2$, ${\tau }_G$, W, and T_MI represent the A-site disorder, tolerance factor, electronic bandwidth, and metal–insulator transition temperature, respectively.

Sample	x = 0	x = 0.04
Crystal structure	Rhombohedral
Space group	R3$\overline{c}$
a = b (Å)	5.5344 (18)	5.5405 (13)
c (Å)	13.527 (9)	13.524 (5)
V (Å)3	358.81 (28)	359.53 (18)
Average ionic radius, <rA>	1.295	1.299
Bond length, <Mn–O> Å	1.969 (6)	1.966 (7)
Bond angle, <Mn–O–Mn> (°)	166.6 (0)	168.7 (4)
χ2	1.70	2.14
Rwp (%)	6.87	7.59
Rp (%)	4.90	5.25
${\sigma }^2$(×10−2) (${\mathsf{\AA }}^2$)	1.3548	1.4932
${\tau }_G$	0.9523	0.9534
W	0.0927	0.0934
TMI (K)	257	271

.

The refined lattice parameters, bond angle, and bond length are summarized in Table 1. It can be seen in Figure 2a that the structural parameters a and b increased from 5.5344 Å (x = 0) to 5.5405 Å (x = 0.04), while c slightly decreased from 13.527 Å (x = 0) to 13.524 Å (x = 0.04) with K⁺ substitution. In addition, the unit cell volume, V, increased as K⁺ was substituted from 358.81 (Å)³ (x = 0) to 359.53 (Å)³ (x = 0.04). The observed behaviour is justified due to the fact that larger K⁺ (1.55 Å) is substituted into smaller Ba²⁺ (1.47 Å) ions [16], which possibly induced the lattice expansion in both a and b directions, thereby leading to the increase in unit cell volume. Hence, the increase in average ionic radius at A-site, $<r_A>$ was expected (Table 1) and subsequently gave rise to the lattice-distortion effect, which can be characterized by the A-site disorder (${\sigma }^2$) parameter expressed using the following equation [16]:

$${\sigma }^2={\displaystyle \sum }{x}_i{r}_i^2-{<r_A>}^2$$

(1)

where $x_i$ and $r_i^2$ is the fractional occupancy of A-site ions and ionic radius. The increasing value of ${\sigma }^2$ from 1.3548 ${\mathsf{\AA }}^2$ (x = 0) to 1.4932 ${\mathsf{\AA }}^2$ (x = 0.04), as shown in Table 1, indicates that K⁺ increases the disorder and the lattice distortion due to the mismatch in the cation size.

Consequently, lattice distortion may also influence the stability of the perovskite manganite compound governed by Goldschmidt’s tolerance factor (${\tau }_G$) [29,30]. The tolerance factor was calculated using the followed expression [9,31]:

$${\tau }_G=\frac{r_{A }+r_B}{\sqrt{2} \left(r_B+r_o\right)}$$

(2)

where $r_A, r_B$, and $r_O$ represent the average ionic radius at A-, B-, and O- sites. The perovskite manganite compound is considered stable when the ${\tau }_G$ value lies between the range of 0.89 and 1.2 [32]. Here, it is noted that ${\tau }_G$ increased from 0.9523 (x = 0) to 0.9534 (x = 0.04), hence implying that the stability of the system increased. Upon K⁺ substitution, the Mn–O–Mn bond angle increased from 166.6° (x = 0) to 168.7° (x = 0.04), whereas the Mn–O bond length slightly decreased from 1.969 (x = 0) to 1.966 (x = 0.04). The changes in bond length, as well as bond angle, can further be used to determine the e_g electron bandwidth, W, according to the following relation [33]:

$$W=\frac{\cos 1/2 \left(\mathsf{\pi }-<{\mathrm{M}}_{\mathrm{n}}-\mathrm{O}-{\mathrm{M}}_{\mathrm{n}}>\right)}{{\left(<{\mathrm{M}}_{\mathrm{n}}-\mathrm{O}>\right)}^{3.5}}$$

(3)

The increasing trend of W from 0.0927 (x = 0) to 0.0934 (x = 0.04) suggested that K⁺ results in the widening of electron bandwidth and the overlapping between the Mn-3d and O-2p orbitals [34]. The increasing stability reflected by high ${\tau }_G$, as well as larger W as K⁺ was substituted may enhance the magneto-transport properties, which will be further discussed in the following section.

2.1.2. Scanning Electron Microscope

A scanning electron microscope (SEM) integrated with energy-dispersive X-ray (EDX) spectroscopy revealed more detailed insight into the grain morphology and the elemental distributions in the studied compounds. Figure 3a,b shows the SEM-EDX analysis for La_0.7Ba_0.3−xK_xMnO₃ (x= 0 and 0.04). The SEM micrographs revealed that the grains in both compounds are inhomogeneous and exhibit irregular shapes. In addition, the average grain sizes estimated via ImageJ software increased from 0.71 μm (x = 0) to 1.34 μm (x = 0.04). The observed increase in grain size is understood to be due to the substitution of larger K⁺, which is consistent with the increasing trend of unit cell volume in Table 1. Similar behaviour was also reported in another K-substituted compound [15] due to the substitution of a larger doped ion size. EDX analysis was carried out to determine the presence of elements in La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04). The EDX spectra displayed characteristic peaks of La, Ba, K, Mn, and O elements. No foreign peaks were detected, confirming that both samples contain all the expected chemical elements with no impurities. Moreover, the values of atomic composition (%) obtained from EDX quantitative analysis (

Table 2. Atomic composition (%) of each element in La_0.7Ba_0.3−xK_xMnO₃ for (x= 0 and 0.04) samples obtained from EDX quantitative analysis.

Sample	Atomic Composition (%)
	x = 0	x = 0.04
La	13.75	13.78
Ba	6.25	5.45
K	0	0.77
Mn	20.44	20.73
O	59.56	59.27
Total	100	100

) for all samples were almost equal to their respective stoichiometric values. This result suggests that the investigated samples were appropriately synthesized to form compounds with the desired composition.

2.2. Magnetic Properties

Figure 4 represents the graph of the temperature dependence of the real part of AC susceptibility, χ′, for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) carried out under an AC field of 2 Oe and frequency of 325 Hz. The temperature in which the transition of ferromagnetic to paramagnetic state (FM–PM) occurs, known as Curie temperature (T_C), was determined based on the minimum value of dχ′/dT versus T plot [30,35]. In the present study, no drop in χ′ was observed, implying that both samples possibly exhibit larger T_C above 300 K. This behaviour is coherent with the earlier studies of the parent compound, which reported the value of T_C at 340 K [26] and 348 K [25]. Furthermore, we also note that instead of the drop, a kink in χ’ is observed around 180 K in our crystals, similar to the previous study of (La_0.70Ba_0.3MnO₃)_1−x/(Al₂O₃)_x [36]. Additional measurement of the magnetization curve (M–H) was performed to verify the magnetic behaviour for both samples at 300 K using a vibrating sample magnetometer (VSM). The result is presented in Figure 5. The S-shape curve’s appearance for both samples indicates ferromagnetic behaviour [37]. The inset of Figure 5 shows an enlarged view in M–H hysteresis loops, signifying the FM ordering of the samples [38]. The magnetization increased sharply at a low magnetic field for both samples; nonetheless, the value of saturation magnetization, M_s, increased from 1.81 ${\mu }_B$/f.u. to 4.11 ${\mu }_B$/f.u. for x = 0 and x = 0.04, respectively (

Table 3. Magnetic parameters of La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) obtained from VSM measurement.

Sample	x = 0	x = 0.04
Ms (${\mu }_B$/f.u.)	1.81	4.11
Hc (G)	29.7	32.4

). The rise in the M_s value and electronic bandwidth, W, by K⁺ substitution indicates the increasing overlapping of Mn-3d orbital ions with the O-2p orbital, thereby leading to the increase in effective FM interaction between Mn ions [9,31]. Moreover, the increasing Mn–O–Mn bond angle (listed in Table 1) possibly contributed to stronger FM, as highlighted in a previous study of Nd_0.7−xLa_xSr_0.3MnO₃ [11]. Apart from that, the enhancement in magnetic properties can likewise arise from the enlargement of grain size with K⁺ substitution. According to Ibrahim et al. [39], compounds consisting of larger grain sizes with smaller grain boundaries exhibit a reduction in magnetic inhomogeneities, hence promoting the growth of the FM phase. On the other hand, the small value of coercivity, H_c, is associated with the magnetic domains, which can be rotated easily in the direction of a magnetic field [32]. The obtained values of saturation magnetization, M_s, and coercivity, H_c, are listed in Table 3.

2.3. Electrical Properties

Figure 6 displays the temperature dependence of resistivity (ρ) data for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) at 0 T. Interestingly, the metal–insulator transition temperature, T_MI increased from 257 K (x = 0) to 271 K (x = 0.04), while the peak value of resistivity decreased from 14.5 ohm.cm (x = 0) to 0.08 ohm.cm (x = 0.04). The reduction in ρ could be attributed to the rise in the Mn–O–Mn bond angle (°) and increased bandwidth, W, with increasing <r_A> (see Table 1). Based on Table 1, K⁺ substitution increased the Mn–O–Mn bond angle and reduced the Mn–O bond length. This observed behaviour may facilitate the transfer of e_g conduction electrons and enhance the delocalization of charge carriers. As a result, the electron mobility improves because the e_g hopping process from Mn³⁺ to Mn⁴⁺ in the Mn³⁺–O²⁻–Mn⁴⁺ double-exchange (DE) process is enhanced, leading to a reduction in resistivity. This is also evidenced by the increased bandwidth with K⁺ (Table 1). The increasing value of W signifies that the overlapping between 3d-manganese and 2p-oxygen orbitals increased, thus favouring the transfer interaction of e_g electrons between neighbouring Mn-sites [40]. Therefore, the strength of DE interaction in the La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) system becomes stronger and T_MI shifts to a higher temperature. Further, the reduction of grain boundaries effect may also contribute to the decrease in ρ. It can be seen that magnitude of ρ is lowered by several orders (10⁻² Ω·cm) with K⁺ substitution, which can be compared to the ceramic and single-crystal LCMO [41]. For granular systems, larger grain size may reduce the scattering at grain boundaries, which improves electron mobility and electrical conductivity [39,42]. Notably, the increase in grain size value and reduced grain boundaries for sample x = 0.04 indicates the rise in the conduction channel for e_g electrons’ transfer to move between grains, which may also aid the DE process. These factors together may contribute to the significant decrease in resistivity. Similar findings were also discussed by Siwach et al. [43] and Huang et al. [41]. The obtained results are in good accordance with those interpreted in previous works [39,42].

Figure 7a,b shows the resistivity data for both samples with the absence and the presence of the magnetic field. At 0.8 T, resistivity for both samples is found to be decreased. This observed behaviour indicates the enhancement of delocalization related to the increased rate of e_g itinerant hopping, alongside the improvement in charge carriers’ mobility in the DE process [44,45,46]. The suppression of resistivity under the applied field could be interpreted by analyzing the influence of the magnetic field on the alignment of magnetic spins [30]. It is known that when the magnetic field is applied, the spin disorder is suppressed, and Mn spins within the disordered region realign along the field direction. Accordingly, more spins are ferromagnetically coupled, and the transfer of itinerant electrons in Mn³⁺–O²⁻–Mn⁴⁺ is favoured, resulting in the improvement in electrical conductivity [17,21,44]. Nevertheless, the T_MI for both samples remained unchanged upon applying an external magnetic field, in contrast with the previous report of Nd_0.6Sr_0.4−xK_xMnO₃ [9], where T_MI slightly shifted to a higher temperature.

The temperature dependence of resistivity data can be divided into two regions: metallic (T < T_MI) and insulating region (T > T_MI), in which the nature of electrical conduction through the scattering process is more dominant in the former, while the hopping mechanism is more dominant in the latter [47]. In the metallic region (T < T_MI), the resistivity data for both samples at 0 T and 0.8 T were fitted according to the following equation [5,36,48]:

ρ = ρ₀ + ρ₂T² + ρ_4.5T^4.5

(4)

where ρ₀ is the residual resistivity due to scattering by impurities, defects, grain boundaries, or domain walls; ρ₂ arises from electron–electron scattering; and ρ_4.5 is a resistivity factor due to electron–magnon scattering processes [5,36,48,49]. The data fit well and are of high quality with the square value of the linear correlation coefficient, R²~99.9%. The related scattering parameters are listed in

Table 4. Best fit parameters for metallic region of La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) using Equation (4).

Sample, x	${\mathit{\rho }}_{\mathit{o}} (\mathsf{\Omega } \mathbf{cm})$		${\mathit{\rho }}_2$$(\mathsf{\Omega } \mathbf{cm}/{\mathbf{K}}^{-2})$		${\mathit{\rho }}_{4.5}$$(\mathsf{\Omega } \mathbf{cm}/{\mathbf{K}}^{-4.5})$
	H = 0 T	H = 0.8 T	H = 0 T	H = 0.8 T	H = 0 T	H = 0.8 T
0	12.30	11.89	4.92 × 10−6	3.06 × 10−6	3.01 × 10−11	2.65 × 10−11
0.04	0.03	0.02	1.19 × 10−6	1.08 × 10−6	−4.26 × 10−13	−4.13 × 10−13

. Parameter ${\rho }_o$ yielded the highest value compared to ${\rho }_2$ and ${\rho }_{4.5}$ for both samples, indicating that the residual resistivity due to the grain boundary is more significant in the conduction process in the metallic region. It is also noticed that the residual resistivity decreases by more than two orders of magnitude. As evidenced in the XRD and SEM-EDX results, both investigated samples are well-crystallized. Thus, the contribution of impurities may not be responsible for such a reduction. This could be due to the additional influence of sample porosity, in addition to the grain boundary effect [30]. In the present study, the decreasing porosity with K⁺ substitution may also increase the electron transport channels. As seen in Figure 3b, grains in the K⁺-substituted sample become more tightly packed, and hence the transfer of e_g electrons between grains makes it easier to enhance the DE process [50]. Consequently, a drastic reduction in ${\rho }_{\mathrm{o}}$ as well as resistivity in the K-substituted sample is observed (Figure 7a,b). On the other hand, the small value of ${\rho }_{4.5}$ could be attributed to the smaller spin inhomogeneity [30]. According to the increasing tolerance factor (Table 1), K⁺ increases the stability of the perovskite system and possibly causes a decline in spin fluctuations. As reported by Sangeetha et al., the carrier scattering by thermal spin fluctuations plays an important role in the resistivity drop below T_MI [51]. Since parameter ${\rho }_{4.5}$ represent the resistivity factor due to electron–magnon scattering, the substitution of non-magnetic K⁺ possibly reduced the spin wave effect as well as spin fluctuations [52]; as a result, the value of ${\rho }_{4.5}$ decreased. A similar finding related to the obtained $-{\rho }_{4.5}$ value was also reported in the study of La_0.7−xY_xBa_0.3Mn_1−xFe_xO₃ [53], La_0.7−xBi_xSr_0.3MnO₃ [52]_, and La_0.65−xBi_xCa_0.35MnO₃ [48]. Based on Table 4, parameters ${\rho }_o$ and ${\rho }_2$ are reduced with x due to the increase in grain size associated with the reduction in grain boundaries [30,42], which corresponded well to our SEM analysis result in Figure 3a,b.

Generally, all scattering parameters were observed to be reduced with K⁺, evidencing that charge carriers experience less scattering and the rate of e_g transfer between Mn ions in the DE process increases. The magnetic field application (0.8 T) further minimizes the scattering of charge carriers due to the parallel spin alignment of e_g in the Mn–O–Mn network. Therefore, exchanges of interaction between Mn-sites increase and lead to the improvement in electrical conduction. Our result is consistent with the previous studies of electrical resistance in manganites under the influence of a magnetic field [44,45,46].

On the other hand, the conduction mechanism in the insulating region has been described through well-established hopping models: small polaron hopping (SPH) at T > θ_D/2 (θ_D is Debye’s temperature) and variable range hopping (VRH) at T < θ_D/2. SPH is usually considered in the high-temperature insulating region, where thermal energy is sufficient to assist the hopping of charge carriers to their nearest neighbours with high mobility [21,54]. Small polaron hopping is expressed using the following equation [25]:

ρ = ρ₀T exp (E_a/k_BT)

(5)

where E_a is the activation energy and k_B is Boltzmann’s constant. The values of E_a for both samples are determined by the slope of ln ($\rho$/T) versus 1000/T curves, as shown in Figure 8a,b and the values are listed in

Table 5. Fitting parameters of the experimental resistivity data for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) according to SPH and VRH models. E_a is the activation energy obtained in the SPH model; T_OM, R_h, E_h, and N(E_F) correspond to the Mott’s temperature, hopping distance, hopping energy, and charge carrier density of state (DOS) near the Fermi level in the VRH model, respectively.

Sample	x = 0		x = 0.04
	H = 0 T	H = 0.8 T	H = 0 T	H = 0.8 T
	SPH model
Ea (meV)	48.39	47.70	40.07	38.86
	VRH model
TOM × 106 (K)	1.291	1.267	0.646	0.496
Rh (Å)	13.93	13.86	11.71	10.97
Eh (meV)	49.46	49.22	41.59	38.94
N(EF) (×1020) (eV−1 cm−3)	1.774	1.810	3.550	4.620

. A reduction in E_a was observed from 48.39 meV (x = 0) to 40.07 meV (x = 0.04), implying that K⁺ enhanced the delocalization of e_g electrons due to the reduction in grain-boundary effects and the widening of the e_g electron bandwidth. As a result, the possibility of the conduction electron hopping to neighbouring sites was enhanced, thereby reducing the values of E_a. As the magnetic field is applied, the value of E_a for both samples is further decreased. This behaviour signifies the weakening of attraction between lattice and electrons in the form of polarons, hence assisting the conductivity of e_g carriers through the hopping process. The reduction in E_a for both samples at 0.8 T is in agreement with the trend of decreasing resistivity observed in Figure 7a,b.

Unlike the conduction mechanism described using the SPH model, the thermal energy is not large enough to allow electrons to hop to their nearest neighbours in a low-temperature insulating region. Therefore, it needs to hop further to find a site with a smaller potential difference [49,53]. In this case, variable range hopping (VRH) is more appropriate to illustrate the transport mechanism, which can be described by the following equation [55]:

$$\rho ={\rho }_{OM}\exp {\left(T_{OM}/T\right)}^{\frac{1}{4}}$$

(6)

where ${\rho }_{OM}$ is Mott’s residual resistivity and $T_{OM}$ is Mott’s temperature. $T_{OM}$ is obtained from the gradient of ln ($\rho$) versus T^1/4 plots (see Figure 9a,b), which can further be applied to determine the value of the charge carrier density of state (DOS) near the Fermi level N(E_f), hopping energy E_h, and hopping distance R_h [21,55]. Here, $\alpha$ and $k_B$ each represent the localization length at $\alpha$ = 4.5 Å and Boltzmann’s constant, respectively [55]. The linear fit using Equation (6) presents a straight line with R²~99.9%, satisfying the VRH mechanism.

$$T_{OM}=18 / (k_{B}N\left(E_F\right){\alpha }^3)$$

(7)

$$R_h\left(T\right)=\left(3/8\right) \alpha {\left(T_{OM}/T\right)}^{\frac{1}{4}}$$

(8)

$$E_h\left(T\right)=\left(1/4\right)k_B T^{\frac{3}{4}} {T_{OM}}^{\frac{1}{4}}$$

(9)

The parameters related to the VRH model are listed in Table 5. It can be seen that the value of $T_{OM}$ decreases from 1.291 × 10⁶ K (x = 0) to 0.646 × 10⁶ K (x = 0.04) when K⁺ is substituted into the compound. Moreover, one can observe that the values of E_h and R_h decrease with K⁺ (Table 5), whereas the value of N(E_F) increases. Such an increase in N(E_F) reflects the increase in the density of states and the number of available hopping sites for the charge carriers. In the present study, it is suggested that K⁺ substitution enhanced the interconnectivity between the grains as the grain size rose. Consequently, the hopping process becomes easier as the e_g electron does not require large energy (E_h) to hop to a site beyond the nearest neighbour. The average hopping length is smaller than the average distance between the sites [53], and therefore, the hopping distance (R_h) decreased with K⁺ substitution. Based on Table 5, both R_h and E_h were found to be decreased in the presence of a magnetic field, indicating the enhancement in delocalization while increasing the charge carrier density, N(E_F), to enhance the hopping process and conductivity in the insulating region. These results agreed well with the reduction in resistivity observed in Figure 7a,b. Moreover, the obtained values of N(E_F) were in the range of DOS at the Fermi level reported previously in other manganite systems [27,56,57].

2.4. Magnetoresistance (MR)

Magnetoresistance (MR) is calculated by the following equation [9,58]:

$$MR (\%)=\frac{\rho \left(0, T\right)-\rho \left(H, T\right)}{\rho \left(0, T\right)}\times 100$$

(10)

where ρ(0) and ρ(H) are the resistivity under zero magnetic field (0 T) and applied magnetic field (0.8 T), respectively. Figure 10 illustrates the temperature dependence of MR (%) under H = 0.8 T and its value at three respective temperatures; 30 K, T_MI, and 300 K, which are tabulated in

Table 6. T MR (%) obtained for La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) at three respective temperatures: 30 K, T_MI, and 300 K.

Sample	MR (%)
	30 K	TMI	300 K
x = 0	3.25	2.75	3.01
x = 0.04	25.0	5.31	4.24

. MR has been previously discussed with respect to different contributions: intrinsic and extrinsic [28,42,44]. For the intrinsic MR effect, the spin ordering due to the suppression of spin fluctuations is responsible for explaining resistivity reduction under an applied magnetic field [13,28,42]. On the other hand, the extrinsic MR effect is explained based on spin-polarized tunnelling (SPT) related to the scattering of carriers at the grain boundaries. It is often discussed in polycrystalline granular systems [21,37,42,44]. As the electrons move between grains, they may encounter an insulating barrier consisting of disordered and scattered Mn spins, reducing connectivity among the grains. The application of a magnetic field may overcome the scattering effects [30,44]. As a result, the MR effect is observed at low temperatures. At 30 K, the MR (%) significantly increased from 3.25% (x = 0) to 25% (x = 0.04). As discussed above, this behaviour could be ascribed to the influence of the extrinsic MR effect, whereby the SPT across grain boundaries is more dominant [37,58,59]. Grain boundaries acted as barriers to charge transport at zero field and restricted the tunnelling process due to disordered Mn spins [37]. In this study, a small substitution of K⁺ possibly improved the spin alignment of the charge carrier at the grain boundaries region, and it was further enhanced with the application of a magnetic field. The realignment of disordered Mn spins in the grain boundary regions may reduce the magnetic disorder and facilitate the increase in intergrain SPT; thus, the MR effect is enhanced [30,58]. Meanwhile, the rise in MR (%) in the vicinity of T_MI from 2.75% (x = 0) to 5.31% (x =0.04) possibly arose from the increase in the DE interactions as a result of the parallel alignment of spin at Mn–O–Mn couplings under magnetic field [30,58]. The similar behaviour of increased MR at T_MI region due to enhanced DE has also been reported in La_0.65Ca_0.35−xLi_xMnO₃ [44] and La_0.5Nd_0.15Ca_0.25A_0.1MnO₃ (A = Ca, Li, Na, K) [60]. The highest MR (%) observed at 30 K for sample x = 0.04 indicates that the extrinsic MR effect related to spin-polarized tunnelling is more pronounced in this system. For a small concentration of K⁺ at x = 0.04, the enhancement in the MR (%) value observed at 300 K from 3.01 % (x = 0) to 4.24% (x = 0.04) in a moderately low applied field gives this system a future prospect for advancement in room-temperature technological applications.

In brief, K⁺ substitution in the La_0.7Ba_0.3MnO₃ system enhanced the electrical conductivity and strengthened the magnetic interactions, which are suggested to be due to the improvement in microstructure, changes in bond angle, and increased hopping probability of the e_g conduction electrons. Since K⁺ contributes to the rise of the MR effect, even under a low applied magnetic field, we believe that the present work is of considerable significance for the application of magnetic field sensors.

3. Materials and Methods

Polycrystalline samples of La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) composition were synthesised via the solid-state method using La₂O₃, BaCO₃, Mn₂O₃, and K₂CO₃ as starting materials with 99.9% purity (Sigma-Aldrich). The powders were mixed and ground in an agate mortar for 1 h before the calcination process at 900 °C for 24 h. The processes of grinding and calcination were repeated twice to eliminate volatile foreign particles and carbon compounds in the samples [61]. After regrinding, the samples were pressed into pellets (13 mm diameter and 2.5 mm thickness) with an applied pressure of 5 tons using a hydraulic press, followed by a sintering process at 1100 °C for 24 h. The pellets were left to cool down to room temperature in the furnace before final grinding for 30 min for each sample. In addition, a slow cooling process at a rate of 1 °C/min was conducted after each calcination and sintering process to recover the oxygen lost at high temperature, thus maintaining the expected oxygen stoichiometry [37,62]. X-ray diffraction (XRD) was performed to examine the crystal structure of the sample using PANalytical model Xpert PRO with Cu Kα (λ = 1.5406 Å) radiation. The data were collected in the angle range of 2θ = 10°–80° with a step size of 0.017° and further analysed through the Rietveld refinement method using the General Structure Analysis System (GSAS) and Graphical User Interfaces (EXPGUI) software programs. The grain morphology and the elemental analysis of the studied compounds were obtained via a scanning electron microscope (LEO model 982 Gemini) integrated with energy-dispersive X-ray spectroscopy (EDX). The electrical properties and magnetoresistance measurements were conducted using the standard four-probe technique in Janis model CCS-900T/204 cryostat at 30 K–300 K under zero field and applied field of 0.8 Tesla. Alternating Current (AC) susceptibility measurement was performed using Lakeshore AC susceptometer. The DC magnetic measurement was carried out using a Vibrating Sample Magnetometer (VSM, Lakeshore Model 7400) at 300 K under the magnetic fields of ±14 kOe.

4. Conclusions

In summary, the effects of K⁺ substitution on the structural, magnetic, and electrical properties and magnetoresistance of La_0.7Ba_0.3−xK_xMnO₃ (x = 0 and 0.04) perovskite manganites were investigated. Structural analysis revealed that both compounds exhibit a single-phase rhombohedral structure with the R3$\overline{c}$ space group. SEM analysis results showed that K⁺ substitution leads to larger average grain sizes. Magnetization versus applied field (M–H) measurement demonstrated the strengthening of ferromagnetic (FM) interaction with K⁺ substitution. Furthermore, electrical measurement displayed a reduction in resistivity and the metal–insulator transition temperature, T_MI shifted to a higher temperature as K⁺ was introduced into the system. The observed behaviour is correlated to the increasing tolerance factor (${\tau }_G$) and bandwidth (W) with K⁺ substitution, which increases the stability of the perovskite system and consequently enhances the mobility of charge carriers in the double-exchange (DE) mechanism. The grain boundary effects, which decreased as grain size increased for the K-substituted sample, play a significant role in the electrical conductivity behaviour. Suppression of resistivity for both samples was observed with the application of an external magnetic field of 0.8 T due to spin-ordering and improvement in the conduction process. Both samples exhibit metallic behaviour in the low-temperature region, which is well-fitted with the equation ρ = ρ₀ + ρ₂T² + ρ_4.5T^4.5. Meanwhile, VRH and SPH models are used to elucidate the transport mechanism of both samples in the insulating region.