#
Grain-Size-Induced Collapse of Variable Range Hopping and Promotion of Ferromagnetism in Manganite La_{0.5}Ca_{0.5}MnO_{3}

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Structural Properties

#### 3.2. DC Resistivity

#### 3.3. Magnetization

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. AC Susceptibility

**Figure A1.**Magnetic ac susceptibility for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ ceramic samples: (

**a**) S4000; (

**b**) S400; and (

**c**) S40. Shown are the T-dependence of both the real ${\chi}^{\prime}$ (blue lines, left axis), and imaginary ${\chi}^{\u2033}$ (red lines, right axis) components, normalized to the mass of the sample. The cooling and warming curves differ only in the case of S4000.

**Figure A2.**Temperature dependence of the (

**a**) real ${\chi}^{\prime}$; and (

**b**) imaginary ${\chi}^{\u2033}$ part of the ac susceptibility at low T for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ ceramic sample (the same batch as S4000) at different frequencies. AC susceptibility depends only weakly on frequency.

## Appendix B. Grain Size Distribution

**Figure A3.**Grain size distribution in the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ ceramic samples: (

**a**) S4000; (

**b**) S400; and (

**c**) S40. The average grain size values $4100\pm 1400$, $400\pm 120$ and $43\pm 13\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ for S4000, S400 and S40, respectively, are indicated by the vertical black line.

## Appendix C. Fitting of Variable-Range-Hopping Mechanism

**Figure A4.**lnW vs. lnT for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ ceramic samples: (

**a**) S4000; (

**b**) S400; and (

**c**) S40, see text for details. Black dashed lines correspond to the slope expected for the 3D VRH ($p=1/4$) and agree with the measured data only for S4000 in the T-range 150–40 K.

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**Figure 1.**(

**a**) Conventional phase diagram of La${}_{1-x}$Ca${}_{x}$MnO${}_{3}$. PMI stands for a paramagnetic insulating phase, FMM for a ferromagnetic metallic, COI for a charge-ordered insulating, AFM for an antiferromagnetic, FMI for a ferromagnetic insulating and CAFMI for a canted antiferromagnetic insulating state. This figure is based on data from Ref. [9]. (

**b**) Splitting of the Mn 3d levels into ${e}_{g}$ and ${t}_{2g}$ orbitals by crystal field and Jahn–Teller distortion.

**Figure 2.**The surface morphology of the ceramic La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples. SEM images obtained from: (

**a**) S4000; (

**b**) S400; and (

**c**) S40 with the average grain size of $4100\pm 1400$, $400\pm 120$ and $43\pm 13\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$, respectively. The scale bar is indicated in each image.

**Figure 3.**X-ray scans of the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples for: (

**a**) S4000; (

**b**) S400; and (

**c**) S40. Red lines are the experimental data, blue lines are Rietveld fits, and green marks are the maximum positions. Corresponding residuals are shown below each X-ray spectrum. Several additional maximums for S400, approximately at ${34}^{\xb0}$, ${49}^{\xb0}$ and ${61}^{\xb0}$, not related to the ${P}_{nma}$ structure of La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ indicates the presence of small amounts of CaMnO${}_{3}$.

**Figure 4.**(

**a**) DC resistivity and (

**b**) its logarithmic derivative $\mathrm{dln}\rho /\mathrm{d}(1/T)$ as a function of temperature for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples: S4000 (red line), S400 (blue line) and S40 (green line). The resistivities are normalized to the room temperature values to emphasize the difference in the T-evolution for different samples. The arrows indicate cooling and warming. The $\mathrm{dln}\rho /\mathrm{d}(1/T)$ curve for S4000 is shown only in warming for clarity.

**Figure 5.**(

**a**) $\mathrm{log}\rho -{T}^{-1/4}$ plot of the normalized resistivity, suitable for the Mott 3D VRH mechanism, for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples: S4000 (red circles), S400 (blue line) and S40 (green line). Black dashed line is a fit to the 3D VRH $\rho \left(T\right)\propto \mathrm{exp}{({T}_{0}/T)}^{1/4}$. Only the warming curve is shown for S4000 for clarity. (

**b**) Schematic representation of the core–shell model in La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$, where the core of a grain is CO/AFM, and the shell is predominantly FM (see text).

**Figure 6.**Standard ZFC and FC magnetization curves in $H=100\phantom{\rule{3.33333pt}{0ex}}\mathrm{Oe}$ for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples: (

**a**) S4000; (

**b**) S400; and (

**c**) S40. In the case of S4000, the FCC and FCW curves do not overlap and are indicated by the blue and red lines, respectively. In the case of S400 and S40 the FCC and FCW curves are identical (not shown) which was confirmed on other samples. The ZFC curves in all three panels are shown by the black lines. The sharp peak in ZFC and sudden jump in FCC and FCW curves at $T\approx 40\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$ for S4000 are extrinsic and come from a tiny amount of Mn${}_{3}$O${}_{4}$ phase which often occurs during the sample synthesis [69,70,71].

**Figure 7.**Magnetic hysteresis loops for the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples at: (

**a**) 2 K; (

**b**) 100 K; and (

**c**) 200 K in the field range $-3<H<3\phantom{\rule{3.33333pt}{0ex}}\mathrm{kOe}$. All curves were measured in the FCC protocol. Magnetization is expressed in units of Bohr magneton ${\mathsf{\mu}}_{\mathrm{B}}$ per Mn atom. Red, blue and green symbols refer to the S4000, S400 and S40 sample, respectively. The full range of H is shown in insets. The low saturation values of M indicates that the FM fraction of La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ is $<5\%$, $\approx 35\%$ and $\approx 40\%$ for the S4000, S400 and S40 sample, respectively (see text).

**Table 1.**Structural properties of the La${}_{0.5}$Ca${}_{0.5}$MnO${}_{3}$ samples obtained from SEM imaging and Rietveld fits. In all samples, the crystallites are significantly smaller than the grains, indicating the presence of many crystallites within each grain.

Sample Label | Grain Size (nm) | Crystallite Size (nm) | a (Å) | b (Å) | c (Å) |
---|---|---|---|---|---|

S4000 | $4100\pm 1400$ | $301\pm 10$ | 5.4099(3) | 7.6240(4) | 5.4183(3) |

S400 | $400\pm 120$ | $28\pm 3$ | 5.464(2) | 7.787(3) | 5.504(2) |

S40 | $43\pm 13$ | $14\pm 2$ | 5.502(3) | 7.786(4) | 5.424(3) |

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**MDPI and ACS Style**

Novosel, N.; Rivas Góngora, D.; Jagličić, Z.; Tafra, E.; Basletić, M.; Hamzić, A.; Klaser, T.; Skoko, Ž.; Salamon, K.; Kavre Piltaver, I.;
et al. Grain-Size-Induced Collapse of Variable Range Hopping and Promotion of Ferromagnetism in Manganite La_{0.5}Ca_{0.5}MnO_{3}. *Crystals* **2022**, *12*, 724.
https://doi.org/10.3390/cryst12050724

**AMA Style**

Novosel N, Rivas Góngora D, Jagličić Z, Tafra E, Basletić M, Hamzić A, Klaser T, Skoko Ž, Salamon K, Kavre Piltaver I,
et al. Grain-Size-Induced Collapse of Variable Range Hopping and Promotion of Ferromagnetism in Manganite La_{0.5}Ca_{0.5}MnO_{3}. *Crystals*. 2022; 12(5):724.
https://doi.org/10.3390/cryst12050724

**Chicago/Turabian Style**

Novosel, Nikolina, David Rivas Góngora, Zvonko Jagličić, Emil Tafra, Mario Basletić, Amir Hamzić, Teodoro Klaser, Željko Skoko, Krešimir Salamon, Ivna Kavre Piltaver,
and et al. 2022. "Grain-Size-Induced Collapse of Variable Range Hopping and Promotion of Ferromagnetism in Manganite La_{0.5}Ca_{0.5}MnO_{3}" *Crystals* 12, no. 5: 724.
https://doi.org/10.3390/cryst12050724