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Ferrimagnetic Clusters as the Origin of Anomalous Curie–Weiss Behavior in ZnFe_{2}O_{4} Antiferromagnetic Susceptibility

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## Abstract

**:**

_{2}O

_{4}ferrite have revealed the positive character of its Curie–Weiss temperature, contradicting its observed antiferromagnetic behavior which is characterized by a well-defined susceptibility peak centered around the Neel temperature (10 K). Some approaches based on ab initio calculations and mixture of interactions have been attempted to explain this anomaly. This work shows how for very low values of the inversion parameter, the small percentage of Fe atoms located in tetrahedral sites gives rise to the appearance of ferrimagnetic clusters around them. Superparamagnetism of these clusters is the main cause of the anomalous Curie–Weiss behavior. This finding is supported experimentally from the thermal dependence of the inverse susceptibility and its evolution with the degree of inversion.

## 1. Introduction

_{2}O

_{4}) is a model system for the study of these competing interactions. These spinel ferrites have the formula MFe

_{2}O

_{4}, where M is usually one or more divalent or trivalent metals as long as positive charges are compensated for the neutrality of the unit cell. In the general case, the ionic distribution is mixed, and it can be represented by [M

_{1−δ}Fe

_{δ}]

^{A}[M

_{δ}Fe

_{2−δ}]

^{B}O

_{4}, where δ is the inversion parameter, which specifies the fraction of Fe

^{+3}ions in A sites. Accordingly, δ = 0 and 1 stand for the normal and inverse cases, respectively. The normal (Zn)

^{A}(Fe

_{2})

^{B}O

_{4}is antiferromagnetic but the exchange of few Zn-Fe cations between A and B sites gives place to (Zn

_{1−δ}Fe

_{δ})

^{A}[Zn

_{δ}Fe

_{2−δ}]

^{B}O

_{4}. Recent evidence of coexisting ferrimagnetic clusters inside of an antiferromagnetic matrix [8,9,10] suggests the use of this material as a model to correlate the increase of Zn–Fe exchange with an anomalous increasing positive Curie–Weiss temperature despite the antiferromagnetic character of the sample.

_{1-δ}Fe

_{δ})

^{A}[Zn

_{δ}Fe

_{2-δ}]

^{B}O

_{4},where A and B represent the tetrahedral and octahedral sites, respectively, and δ the inversion parameter. For δ = 0, the structure is a normal spinel and presents a paramagnetic behavior, with a transition to antiferromagnetic order near 10 K. Although the equilibrium cation distribution of bulk zinc ferrites can be assumed to be completely normal, it is very difficult to prevent any exchange between the Zn and Fe cations. Thus, the partially inverted ZnFe

_{2}O

_{4}(δ > 0) has been intensively studied to understand the relationship between cation distribution and magnetic properties [14].

## 2. Materials and Methods

_{2}O

_{3}in stoichiometric ratio, annealed at 1200 °C for 24 h. Subsequently, the sample was subjected to 50 h of mechanical grinding and annealed for 1 h at different temperatures. C samples were obtained by a mechanical milling of ZnO and α-Fe

_{2}O

_{3}at 1:1 molar ratio for 150 h and subjecting the samples to different annealing temperatures. Table 1 summarizes the synthesis procedures and the subsequent milling times and annealing temperatures and times.

## 3. Results

## 4. Discussion

_{B}>, of the ferromagnetic clusters. It is worth noting that T

_{B}only depends on the cluster volume and not on the volume of the particles. In fact, the changes of T

_{B}with the particle sizes show a tendency opposite to that expected for superparamagnetic particles. This is due to the inverse relation between particle size and average cluster volume. This relation can be understood by taking into account that the smaller the particle size, the larger the inversion degree and, consequently, the larger the average cluster volume, this last being the relevant volume for the superparamagnetic behavior.

^{−28}m

^{3}. Therefore, if we consider that there is only one jump per unit cell, the fraction these cells that have undergone an elementary inversion is 40% for a macroscopic inversion value δ = 0.05.

_{B}the blocking temperature, and V the cluster volume. As illustrated by Figure 1, the susceptibility reaches a maximum at temperatures between 10 and 100 K for all the samples. Therefore, if K = 1.3 × 10

^{5}Jm

^{−3}[22] and for instance, it is applied to sample B1 with T

_{B}= 19.9 K, the effective average volume of the ferrimagnetic clusters is in the order of 5 10

^{−26}m

^{3}, which corresponds to a volume of 100 unit cells, that represents single domains of 4 nm size.

_{B}= θ.

^{2}µ

_{B}. Since the maximum magnetic moment per cell corresponding to a local δ = 0.5 becomes 47 µ

_{B,}then a group of 2–3 unit cells with δ = 0.5 or a group of 4–6 unit cells with local δ = 0.25 could account for these observations.

_{B}> and $\theta $ = ${T}_{B}^{\mathrm{max}}$, is an index of the width of the clusters size distribution.

## 5. Conclusions

^{3+}located in the A sites, which appear in samples with a very low degree of inversion. At temperatures higher than ${T}_{B}^{max}$, provided that the Curie temperature of the ferrimagnetic clusters is well above the measuring one, the inverse of the susceptibility approaches the typical thermal dependence $\chi =\frac{C}{T-\theta}$. The apparent Curie–Weis temperature, θ, is indeed a temperature corresponding to the blocking temperature distribution of the clusters, its particular position in the spectrum, close to ${T}_{B}^{max}$, depends on the shape and width of the distribution. In the case of a uniform distribution, θ can be considered to be ${T}_{B}^{max}$. In summary, in samples with very low inversion degree and blocking temperature well below the Curie one, the contribution of the superparamagnetic effect becomes dominant. The superparamagnetic apparent Curie–Weiss temperature, being a blocking one, is the temperature at which the magnetic relaxation time is similar to the measurement time; above this temperature, the behavior is superparamagnetic but below it, the system is ferromagnetic.

_{2}O

_{4}samples where there is not any trace of ferrimagnetic clusters. However, since this ideal case is very difficult to achieve, the previously reported anomalous sign for the Curie–Weiss temperature can be understood as a consequence of the superparamagnetism associated with the almost unavoidable presence of a few ferrimagnetic clusters.

## Supplementary Materials

_{_B}^max, 1/χ, (S2), tends towards a straight line with slope 1/C. As the average blocking temperature is 40K its difference with θ = 50 K is a measure of the half width of the distribution.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**Left panel**) shows ZFC-FC curves of zinc ferrites with different inversion degree (δ) and crystallite size (d). The (

**right panel**) shows the inverse of the dimensionless susceptibility and the corresponding linear fit with the Curie–Weiss law. The samples labels are identified in Table 2.

Sample Family | Synthesis Route Type | Sample | Thermo-Mechanic Treatment | |
---|---|---|---|---|

Milling (h) | Annealing (°C) | |||

A | Commercially supplied | A1 | - | 1100, 24 h |

A2 | 50 | |||

A3 | 50 | 400, 1 h | ||

A4 | 50 | 500, 1 h | ||

B | Ceramic | B1 | 1200, 24 h | |

B2 | 2 | |||

B3 | 10 | |||

B4 | 50 | |||

C | Mechano-Synthesis | C1 | 150 | |

C2 | 150 | 300, 1 h | ||

C3 | 150 | 400, 1 h | ||

C4 | 150 | 500, 1 h | ||

C5 | 150 | 600, 1 h |

**Table 2.**Inversion degree, saturation magnetization, apparent Curie–Weiss (θ), and average blocking temperature <T

_{B}> of the studied samples.

Samples | Inversion Parameter | Ms (5 K) | θ | <T_{B}> |
---|---|---|---|---|

(A/m × 10^{3}) | (K) | (K) | ||

A1 | 0.05(2) | 12.7(1) | 87(5) | 14(3) |

B1 | 0.10(2) | 80(1) | 185(5) | 19(3) |

C5 | 0.18(3) | 106(1) | 170(5) | 52(3) |

A4 | 0.21(4) | 168(1) | 162(5) | 88(5) |

B2 | 0.23(5) | 164(1) | 172(5) | |

C4 | 0.26(5) | 137(1) | 70(5) | |

A3 | 0.27(5) | 272(2) | 129(5) * | |

B3 | 0.41(5) | 340(3) | 235(5) * | |

C2 | 0.52(5) | 423(4) | 270(5) * | |

C1 | 0.56(6) | 388(4) | >300 | |

B4 | 0.59(6) | 409(4) | 263(5) |

_{B}).

Sample | Slope (1/C) Δ(1/X)/ΔT | C | m (µ_{B})(×10 ^{2}) |
---|---|---|---|

A1 | 133.7/213 | 1.59(2) | 5.2(1) |

B1 | 75/115 | 1.53(2) | 0.8(1) |

C5 | 15.3/130 | 8.5(1) | 3.3(1) |

A3 | 4.7/138 | 29(1) | 7.1(2) |

^{a}The standard deviations are in parenthesis.

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**MDPI and ACS Style**

Hernando, A.; Cobos, M.Á.; Jiménez, J.A.; Llorente, I.; García-Escorial, A.; de la Presa, P.
Ferrimagnetic Clusters as the Origin of Anomalous Curie–Weiss Behavior in ZnFe_{2}O_{4} Antiferromagnetic Susceptibility. *Materials* **2022**, *15*, 4789.
https://doi.org/10.3390/ma15144789

**AMA Style**

Hernando A, Cobos MÁ, Jiménez JA, Llorente I, García-Escorial A, de la Presa P.
Ferrimagnetic Clusters as the Origin of Anomalous Curie–Weiss Behavior in ZnFe_{2}O_{4} Antiferromagnetic Susceptibility. *Materials*. 2022; 15(14):4789.
https://doi.org/10.3390/ma15144789

**Chicago/Turabian Style**

Hernando, Antonio, Miguel Ángel Cobos, José Antonio Jiménez, Irene Llorente, Asunción García-Escorial, and Patricia de la Presa.
2022. "Ferrimagnetic Clusters as the Origin of Anomalous Curie–Weiss Behavior in ZnFe_{2}O_{4} Antiferromagnetic Susceptibility" *Materials* 15, no. 14: 4789.
https://doi.org/10.3390/ma15144789