Nature of Magnetic Ordering in Cobalt‐Based Spinels

Abstract

In this chapter, the nature of magnetic ordering in cobalt‐based spinels Co3O4, Co2SnO4, Co2TiO4, and Co2MnO4 is reviewed, and some new results that have not been reported before are presented. A systematic comparative analysis of various results available in the literature is presented with a focus on how occupation of the different cations on the A‐ and B‐sites and their electronic states affect the magnetic properties. This chapter specifically focuses on the issues related to (i) surface and finite‐size effects in pure Co3O4, (ii) magnetic‐compensation effect, (iii) co‐existence of ferrimagnetism and spin‐glass‐like ordering, (iv) giant coercivity (HC) and exchange bias (HEB) below the glassy state, and (v) sign‐reversal behavior of HEB across the ferri/antiferromagnetic Néel temperature (TN) in Co2TiO4 and Co2SnO4. Finally, some results on the low‐temperature anomalous magnetic behavior of Co2MnO4 spinels are presented.

1. Introduction

The atomic arrangement of the spinel compounds is interpreted as a pseudo‐close‐packed arrangement of the oxygen anions with divalent cations occupying tetrahedral A‐sites and trivalent cations residing at the octahedral B‐sites of the cubic unit cell of space group Fd-3m(227). A spinel with the configuration (A²⁺) [B₂³⁺] O₄ is termed as “normal spinel” whereas the other possible configuration (B³⁺) [A²⁺B³⁺] O₄ is called “inverse spinel.” The continuum of possible atomic distribution between these two extremes is quantified by a parameter denoted as x (inversion parameter), which describes the fraction of “B” cations on tetrahedral sites. Thus, x = 0 for a normal spinel, 2/3 for a spinel with entirely arbitrary configuration, and 1 for a fully inverse spinel. Among many varieties of spinel compounds, ferrites and cobaltites are widely used in the high‐frequency electronic circuit components such as transformers, noise filters, and magnetic recording heads [1, 2]. The key property of these spinels is that at high frequencies (>1 MHz), their dielectric permittivity (ε) and magnetic permeability (μ) are much higher than those of metals with low loss‐factor (tanδ). These properties make them very advantageous for the development of magnetic components used in the power electronics industry. Also, the nanostructures of these spinels continue to receive large attention because of their potential applications in solid‐oxide fuel cells, Li‐ion batteries, thermistors, magnetic recording, microwave, and RF devices [1, 2].

In this review, we mainly focus on the nature of magnetic ordering of several insulating cobalt spinels of type Co₂MO₄ (where “M” is the tetravalent or trivalent metal cation such as Sn, Ti, Mn, Si, etc.) which are not yet well studied in literature. This review will primarily illustrate how the magnetic ordering changes when we substitute the above‐listed metal cations at the tetrahedral B‐sites. It is well known that the dilution of magnetic elements significantly disrupts the long‐range magnetic ordering and leads to more exotic properties like magnetic frustration, polarity reversal exchange bias, and reentrant spin‐glass state near the magnetic‐phase transition [3, 4]. The dilution essentially alters the super‐exchange interactions of J_AB, J_BB, and J_AA between the magnetic ions, which is the main source of anomalous magnetic behavior. In this review, we first start with the simplest case of antiferromagnetic (AFM) normal‐spinel Co₃O₄ with configuration (Co²⁺)[Co₂³⁺]O₄ and discuss the role of surface and finite‐size effects on antiferromagnetic (AFM) ordering. In the second section, we focus on the coexistence of ferrimagnetism and low‐temperature spin‐glass behavior of cobalt orthostannate (Co₂SnO₄) and cobalt orthotitanate (Co₂TiO₄). A detailed comparative analysis of some recent experimental results dealing with the temperature and frequency dependence of ac‐magnetic susceptibility is presented. In the subsequent section, some unusual magnetic properties of Co₂MnO₄ are discussed.

2. Magnetic properties of bulk versus nanocrystalline Co₃O₄

Cobalt forms two binary compounds with oxygen: CoO and Co₃O₄. While CoO has face‐centered cubic (NaCl‐type) structure, Co₃O₄ shows a normal‐spinel structure with a cubic close packed arrangement of oxygen ions and Co²⁺ and Co³⁺ ions occupying the tetrahedral “8a” and the octahedral “16d” sites, respectively [5]. The magnetic properties of Co₃O₄ were first investigated over 58 years ago; however, its magnetic behavior under reduced dimensions still attracted immense scientific interest mainly because of its distinctly different magnetic ordering under nanoscale as compared to its bulk counterpart [6]. Co₃O₄ can be synthesized in various nanostructural forms such as nanorods, nanosheets, and ordered nanoflowers with ultrafine porosity [7–10]. Such engineered nanostructures play vital roles as catalysts, gas sensors, magneto‐electronics, electrochromic devices, and high‐temperature solar selective absorbers [11–18]. At first glance, the normal‐spinel structure of Co₃O₄ may look similar to that of Fe₃O₄ (inverse spinel) but Co₃O₄ exhibits strikingly different magnetic ordering as compared to Fe₃O₄. In particular, Co₃O₄ does not exhibit ferrimagnetic ordering of the type observed in Fe₃O₄ because Co³⁺ ions on the octahedral B‐sites are in the low spin S = 0 state [5]. Instead, it exhibits antiferromagnetic ordering with each Co²⁺ ion at the A‐site having four neighboring Co²⁺ ions of opposite spins (with an effective magnetic moment of μ_eff ∼ 4.14 μ_B) [5]. Earlier studies by Roth reported that below the Néel temperature T_N ∼ 40 K, Co₃O₄ becomes antiferromagnetic in which the uncorrelated spins of the 8Co²⁺ (in 8(a), F.C. +000, $\frac{1}{4}\frac{1}{4}\frac{1}{4}$) cations in the paramagnetic state (space group O⁷_h—Fd‐3m) are split in the antiferromagnetic state (space group T²_d—F$\overline{4}$3m) into the two sublattices with oppositely directed spins of 4Co²⁺ ↑ (4(a), +000) and 4Co²⁺ ↓ (4(c) $\frac{1}{4}\frac{1}{4}\frac{1}{4}$) [5–7]. For T < T_N, the neutron diffraction studies did not show any evidence of a structural phase transition.

Figure 1.

(a) Temperature dependence of the dc‐magnetic susceptibility χ(T) for bulk Co₃O₄ under the zero‐field‐cooled (ZFC) and field‐cooled (FC) conditions. Here, T_p denotes the peak position in χ versus T plots. (b) Plots of (χ_pT) versus T (LHS scale) and d(χ_pT)/dT versus T plots (RHS scale) for the bulk Co₃O₄. Here, the paramagnetic susceptibility χ_p = χ–χ₀ with χ₀ = 3.06 × 10^‐6 emu/g Oe being the temperature‐independent contribution [6, 7].

As shown in Figure 1, the recent magnetic studies by Dutta et al. [6] have reported a significant difference in the antiferromagnetic ordering temperature T_N ∼ 30 K of Co₃O₄ as compared to the earlier data (40 K); this new result however is in excellent agreement with T_N = 29.92 ± 0.03 K obtained by the heat capacity C_p versus T measurements reported by Khriplovich et al. [19]. It is well known that the peak in the magnetic susceptibility data of antiferromagnets usually occurs at a temperature few percent higher than T_N because the magnetic specific heat of a simple antiferromagnet (in particular, the singular behavior in the region of the transition) should be closely similar to the behavior of the function d(χ_pT)/dT [14]. Therefore, T_N is better defined by the peak in ∂(χT)/∂T versus T plot [20]. Figure 1 shows the temperature dependence of paramagnetic susceptibility χ_p(T) (LHS scale) and d(χ_pT)/dT versus T (RHS scale). For bulk Co₃O₄ the peak temperature value (30 K) in the d(χ_pT)/dT versus T plots is lower than T_N ≃ 40 K often quoted for Co₃O₄ [5–7, 9–10]. Thus, T_N = 30 K determined from two independent techniques (i.e., χ_p and C_p measurements) is consistent with each other and is the accurate characteristic value for bulk Co₃O₄. On the other hand, the nanoparticles of Co₃O₄ exhibit lower T_N values and reduced magnetic moment than the bulk value (30 K, 4.14 μ_B) which is a consequence of finite‐size and surface effects [6]. Salabas et al. first reported Co₃O₄ nanowires of diameter 8 nm and lengths of up to 100 nm by the nanocasting route and observed T_B ≃ 30 K and exchange bias (H_e) for T < T_B [10].

Figure 2.

Variation of the Néel temperature T_N of Co₃O₄ as a function of particle size (d). The data pertaining to solid‐square symbols are taken from Refs. [6–8] in which T_N is considered as a peak point in”d(χ_pT)/dT” versus”T” data. However, the data related to solid green circles are taken from Refs. [8, 9] in which T_N is considered as just the peak point in the “χ“versus”T” plot, not from the derivative plots. The scattered symbols are the raw data corresponding to T_N and the solid lines are the best fit to Eq. (1).

A plot of T_N values versus particle size d reported by several authors for various crystallite sizes of Co₃O₄ is shown in Figure 2. The lowest T_N value reported till now is about 15 ± 2 K for 4.34 nm size Co₃O₄ particles [8]. These nanoparticles were synthesized using biological containers of Listeria innocua Dps proteins and LDps as constraining vessels. Lin and Chen studied the magnetic properties of various sizes (diameter d = 16, 35, and 75 nm) of Co₃O₄ nanoparticles prepared by chemical methods using CoSO₄ and CoCl₂ as precursors [9]. These authors reported that the variation of T_N follows the finite‐size scaling relation:

$T_{\text{N}}\left(D\right)=T_{\text{N}}\left(\infty \right)\left[1-{\left(\frac{{\xi }_0}{d}\right)}^{\lambda }\right]$E1

for various sizes of Co₃O₄ nanoparticles. Accordingly, they obtained the shift exponent $\lambda$ = 1.1 ± 0.2 and the correlation length ξ_o = 2.8 ± 0.3 nm from the fitting analysis of T_N versus d (Figure 2). However, these authors considered T_N values as the direct peak temperature values from χ versus T instead of the peak point in d(χ_pT)/dT. Also, for the bulk grain sizes T_N (∞) = 40 K was considered instead of 30 K obtained from d(χ_pT)/dT analysis as discussed above. Therefore, we repeated the analysis but considering T_N values obtained from d(χ_pT)/dT versus T and the T_N (∞) = 30 K for various sizes of the Co₃O₄ nanoparticles obtained by sol‐gel process (these values were obtained from our earlier works [6, 7]). Accordingly, we obtained $\lambda$ = 1.201 ± 0.2 and the correlation length ξ_o = 2.423 ± 0.46 nm, which are slightly different from the earlier reported values $\lambda$ = 1.1 ± 0.2 and the correlation length ξ_o = 2.8 ± 0.3 nm [9]. Nonetheless, in both the cases T_N follows the finite‐size scaling relation Eq. (1).

Figure 3.

1/χ_p versus T plots for the bulk and nanocrystalline (∼17 nm) Co₃O₄ with χ₀ = 3.06 × 10^‐6 emu/g Oe (LHS scale). The solid lines represent linear fit to the Curie‐Weiss law: χ_p = C/(T + θ). On the RHS scale same figures are plotted except for χ₀ = 9 × 10^‐6 and 7.5 × 10^‐6 emu/g Oe for the bulk and Co₃O₄ nanoparticles of size ∼17 nm, respectively [6, 7].

For T > T_N, the data of χ versus T (Figure 3) are fitted to the modified Curie‐Weiss law χ_P = χ₀ + [C/(T + θ)] with C = Nμ²/3k_B, μ² = g² J(J + 1)μ_B², θ is the Curie‐Weiss temperature and χ₀ contains two contributions: the temperature‐independent orbital contribution mentioned earlier and the diamagnetic component χ_d = −3.3 × 10⁻⁷ emu/g Oe [6]. Usually χ₀ is estimated from the plot of χ versus 1/T in the limit of 1/T → 0 using the high‐temperature data. The value of χ₀ was estimated as 3.06 × 10⁻⁶ emu/g Oe for bulk Co₃O₄ using the inverse paramagnetic susceptibility (1/χ_P) versus temperature (T) data (shown in the left‐hand‐side scale of Figure 3) [5, 6]. A similar procedure for experimental data, shown in the right‐hand side scale of Figure 3, yields χ₀ = 9 × 10⁻⁶ and 7.5 × 10⁻⁶ emu/g Oe for the bulk and nanoparticles (size ∼17 nm) of Co₃O₄, respectively.

It is well known that the origin of the antiferromagnetic ordering in transition metal oxides can be explained by the super‐exchange interaction between the magnetic elements via oxygen ion. In the present case, there are two possible paths for super‐exchange interaction between magnetic ions in Co₃O₄, i.e., Co²⁺ ions: (tetrahedral site) A – O (oxygen) – A (tetrahedral site) and A – O – B – O – A with the number of nearest‐neighbors z₁ = 4 and next‐nearest‐neighbors z₂ = 12, respectively. If the corresponding exchange constants are represented by J_1ex and J_2ex, the expressions for T_N and θ, using the molecular‐field theory, can be written as [5, 6]

$T_{\text{N}}=\frac{J(J+1)}{3k_{\text{B}}}\left(J_{1\text{ex}}{z}_1-J_{2\text{ex}}{z}_2\right)$E2

$\theta =\frac{J(J+1)}{3k_{\text{B}}}\left(J_{1\text{ex}}{z}_1+J_{2\text{ex}}{z}_2\right)$E3

In order to determine J_1ex and J_2ex, the magnitude of effective J(J + 1) for Co²⁺ is required. Since the Curie constant C is equivalent to Nμ²/3k_B with μ = g [ J(J + 1)]½μ_B where g is the Landé g‐factor and J is the total angular momentum. Using the magnitude of g = 2 and C from Figure 3, one can estimate the effective magnetic moment μ_eff = 4.27μ_B for bulk Co₃O₄ and μ = 4.09μ_B for Co₃O₄ nanoparticles of size ∼17 nm. The spin contribution to the above magnitudes of μ is 3.87μ_B for Co²⁺ with spin S = 3/2. Obviously, there is some additional contribution resulting from the partially restored orbital angular moment for the ⁴F_9/2 ground state of Co²⁺ [5, 6]. Using Eqs. (2) and (3) and the values of “θ,” “T_N,” and “μ” for the two cases yields J _1ex = 11.7 and J_2ex = 2.3 K for bulk, and J_1ex = 11.5 and J_2ex = 2.3 K for the Co₃O₄ nanoparticles (d ∼ 17 nm). Thus, both the exchange constants J _1ex and J_2ex correspond to antiferromagnetic coupling. From the magnitudes of C in Figure 3, the value of μ is obtained as 3.28 and 3.43 μ_B for bulk and Co₃O₄ nanoparticles (d ∼17 nm), respectively. These magnitudes of μ are lower than the spin contribution (3.87 μ_B) of Co²⁺ ion itself. Consequently, the magnitudes of “θ” in Figure 3 seem to be questionable. This may be due to the fact that the use of molecular‐field theory in determining the exchange constants has its own limitations since higher order spin correlations are neglected in this model [6].

Figure 4.

The magnetization (M) versus external applied field (H) plots recorded in standard five‐cycle hysteresis mode for bulk and nanoparticles (d ∼17 nm) of Co₃O₄ measured at 5 K in the lower field region of ±1 kOe. (a) Irreversibility observed for the direct and reverse field scans for bulk Co₃O₄, however, (b) a asymmetric shift in the hysteresis loop with enhanced coercivity can be clearly noticed in the case of nanoparticles (d ∼ 17 nm) of Co₃O₄ measured under field‐cooled protocol (FC) of H = 20 kOe [6, 7].

For a typical bulk antiferromagnet, below T_N, the magnetization is expected to vary linearly with applied external magnetic field H below the spin‐flop field. Therefore, the corresponding coercive field H_c and exchange‐bias field H_e must become zero. This was indeed observed in bulk Co₃O₄ (Figure 4a) [5, 6]. Conversely, for the Co₃O₄ nanoparticles (d ∼ 17 nm), the data at 5 K show a symmetric hysteresis loop with H_c = 200 Oe for the zero‐field‐cooled sample and asymmetric (shifted) hysteresis loop with H_c = 250 Oe and H_e = −350 Oe for the sample cooled in magnetic field H = 20 kOe from 300 to 5 K as shown in Figure 4b. Thus, cooling the sample in a magnetic field produces an exchange bias and leads to the enhancement of H_c as well. The temperature dependence of H_c and H_e for the nanoparticles of Co₃O₄ cooled under H = 20 kOe from 300 K to the measuring temperature is shown in Figure 5. Both H_c and H_e approach to zero above T_N. The inset of Figure 5 depicts the training effect, i.e., change in the magnitude of H_e for the sample cycled through several successive hysteresis loops (designated by “n” at 5 K) [6, 10]. A similar effect has been recently reported by Salabas et al. [10] in the Co₃O₄ nanowires of 8 nm diameter although the magnitudes of H_e and H_c in their case are somewhat smaller. The existence of the exchange bias suggests the presence of a ferromagnetic (shell)/antiferromagnetic (core) interface with FM‐like surface spins covering the core of the antiferromagnetically ordered spins in the nanoparticles of Co₃O₄. Salabas et al. reported that H_e falls by ~25% measured between the first and the second loops. The observation of the training effect and open loops of up to 55 kOe suggests that the surface spins are in an unstable spin‐glass‐like state [10]. Such a spin‐glass ordering results from the weaker exchange coupling experienced by the surface spins due to reduced coordination at the surface. These effects however disappear above T_N when the spins in the core become disordered. The observation of somewhat lower magnetic moment per Co²⁺ ion, smaller values of exchange constants J_1ex and J_2ex, and lower T_N was noticed for the nanoparticles of Co₃O₄ in relation to the bulk Co₃O₄. This could be due to the weak exchange coupling and reduced coordination of the surface spins.

Figure 5.

The temperature dependence of H_c and H_e for the nanoparticles of Co₃O₄ (d ∼ 17nm) measured from T = 5 to 40 K under field‐cooled (FC) mode at 20 kOe and at 5 K under zero‐field‐cooled (ZFC) condition [6]. One can clearly notice H_c → 0, H_e → 0 as T approaches T_N. The inset shows progressive decrease of the magnitude of H_e after successive scan (number of cycles”n”) at 5 K [6].

3. Co‐existence of ferrimagnetism and spin‐glass states in some inverted spinels

In 1975, Sherrington and Kirkpatrick (SK) first predicted the reentrant behavior of spinels using mean‐field approach for certain relative values of the temperature and exchange interaction [21, 22]. Later, Gabay and Toulouse extended the SK Ising model calculation to the vector spin glasses and showed that it is possible to have multiple phase transitions such as ferro/ferri/antiferromagnetic state

paramagnetic state

Mixed phase‐1

Mixed phase‐2 [22, 23]. The present section deals with such kind of systems in which the longitudinal ferrimagnetic ordering coexists with the transverse spin‐glass state below the Néel temperature T_N [23–28]. In this connection, we mainly focus on the magnetic ordering in two inverse‐spinel systems, namely (i) cobalt orthostannate (Co₂SnO₄) and (ii) cobalt orthotitanate (Co₂TiO₄) which exhibits the reentrant spin‐glass behavior [24–40]. At first glimpse, both systems Co₂SnO₄ and Co₂TiO₄ are expected to show similar magnetic properties because of the fact that nonmagnetic ions Sn and Ti play identical role on the global magnetic ordering of the system. But in reality, they exhibit markedly different magnetic structures below their ferrimagnetic Néel temperatures [24–40], which is discussed in detail below.

Usually, the ferrimagnetic ordering in Co₂SnO₄ and Co₂TiO₄ arises from the unequal magnetic moments of Co²⁺ ions at the tetrahedral A‐sites and octahedral B‐sites [24–40]. The corresponding magnetic moment at the tetrahedral A‐sites μ(A) is equal to 3.87 μ_B and the magnetic moment at octahedral B‐sites μ(B) is equal to 5.19 and 4.91 μ_B for Co₂TiO₄ and Co₂SnO₄, respectively [27–33]. In 1976, Harmon et al. first reported the low‐temperature magnetic properties of polycrystalline Co₂SnO₄ system and showed the evidence for ferrimagnetic ordering with T_N ∼ 44 ± 2 K [24]. They also suggested that Co₂SnO₄ should contain two equally populated sublattices that align collinearly and couple antiferromagnetically [24]. On the basis of the Mössbauer spectroscopy results, these authors calculated the internal dipolar fields at the Sn sites from the two Co²⁺ sublattices to be > 80 kOe [24]. They also reported very high values of coercive field H_C > 50 kOe below T_N [24]. The reported magnetization value at 16 K per Co²⁺ ion was about 2.2 × 10⁻³ µ_B with zero magnetization value at 12 K. The Curie‐Weiss constant (C) = 4.3 ± 0.2 emu/mol, and the effective magnetic moment of the Co²⁺ ions = 5.0 ± 0.2 µ_B was close to the standard value of 4.13 emu/mol and 4.8 µ_B, respectively. A year later, Sagredo et al. reported that the zero‐field‐cooled and field‐cooled magnetization curves of single crystal Co₂SnO₄ exhibit strong irreversibility below T_N [39].

Sagredo et al. reported that thermoremanent magnetization, magnetic training effects, and spin‐glass phases present in this system are driven by the disordered‐spin configurations [39]. Accordingly, they speculated that the random distribution of Sn⁴⁺ ions on the B‐sites might break the octahedral symmetry of the crystal field and result in the frustrated magnetic behavior [39]. In 1987, Srivastava et al. reported multiple peaks in the temperature dependence of ac‐magnetic susceptibility χ_ac(T) for both Co₂SnO₄ and Co₂TiO₄ below their T_N providing the evidence of Gabay and Tolouse mixed‐phase transitions [25, 28]. Nevertheless, some recent studies proved the existence of transverse spin‐glass state T_SG (∼39 K) just below the T_N (= 41 K) in Co₂SnO₄ [27, 38, 40]. Similar type of results in Co₂TiO₄ was reported by Hubsch et al. and Srivastava et al. but with different T_SG (∼46 K) and T_N (=55 K) [25, 28, 31, 34].

In order to get a precise understanding of the magnetic properties of these systems, we have plotted the temperature dependence of dc‐magnetic susceptibility χ_dc(T) for both Co₂SnO₄ and Co₂TiO₄ measured under ZFC and FC (H@100 Oe) conditions in Figure 6. These χ_dc(T) plots show typical characteristics of ferrimagnetic ordering with peaks across the Néel temperatures T_N = 47 K (for Co₂TiO₄) and 39 K (for Co₂SnO₄). However, for T ≤ 31.7 K an opposite trend in the χ_dc(T) values was noticed for Co₂TiO₄ with χ_dc ∼ 0 at magnetic‐compensation temperature T_CMP = 31.7 K at which the two‐bulk sublattices magnetizations completely balances with each other [31–33]. Such compensation behavior in Co₂SnO₄ system is expected to appear at very low temperatures (T < 10 K) as χ_dc(T) approaches to zero. Consequently, χ_dc‐ZFC exhibits negative magnetization until T_SG. It is expected that the different magnetic moments on the tetrahedral (A) and the octahedral (B) sites, and their different temperature dependence (i.e., μ_A(T), μ_B(T)) play a major role on the global magnetic ordering of both systems.

Figure 6.

The temperature dependence of dc‐magnetic susceptibility χ_dc(T) measured under ZFC and FC (H@100 Oe) conditions for both Co₂TiO₄ (LHS) and Co₂SnO₄ (RHS). The dotted line shows the T_CMP.

Recent X‐ray photoelectron spectroscopic studies reveal that the crystal structure of Co₂TiO₄ consists of some fraction of trivalent cobalt and titanium ions at the octahedral sites, i.e., [Co²⁺][Co³⁺Ti³⁺]O₄ instead of [Co²⁺][Co²⁺Ti⁴⁺]O₄ [34]. On the contrary, the Co₂SnO₄ shows the perfect tetravalent nature of stannous ions without any trivalent signatures of Co³⁺ [Co²⁺][Co²⁺Sn⁴⁺]O₄ [27]. Such distinctly different electronic structure of the ions on the B‐sites of cobalt orthotitanate plays a significant role on the anomalous magnetic ordering below T_N, for example, exhibiting the magnetic‐compensation behavior, sign reversible zero‐field exchange‐bias, and negative slopes in the Arrott plots (H/M versus M²) [27, 34].

At high temperatures (for all T > T_N), the experimental data of inverse dc‐magnetic susceptibility (χ⁻¹) for both the systems Co₂TiO₄ and Co₂SnO₄ fit well with the modified Néel expression for ferrimagnets (1/χ) = (T/C) + (1/χ ₀) – [σ₀/(T − θ)]. Table 1 summarizes various fitting parameters obtained from the Néel expression for both Co₂SnO₄ and Co₂TiO₄. The fit for Co₂TiO₄ yields the following parameters: χ₀ = 41.92 × 10⁻³ emu/mol‐Oe, σ₀ = 31.55 mol‐Oe‐K/emu, C = 5.245 emu K/mol Oe, θ = 49.85 K. The ratio C/χ₀ = T_a =125.1 K represents the strength of the antiferromagnetic exchange coupling between the spins on the A‐ and B‐sites and is often termed as the asymptotic Curie temperature T_a. For both the systems, the effective magnetic moment μ_eff is determined from the formula C = Nμ_eff²/3k_B. The experimentally observed magnetic moments at the B‐sites μ(B) = 5.19 μ_B obtained from the temperature dependence of magnetization values are in line with the above‐discussed spectroscopic properties of Co₂TiO₄, i.e., the total moment μ(B) is perfectly matching with the contribution from the magnetic moments due to Co³⁺ (4.89 μ_B) and Ti³⁺ (1.73 μ_B), μ(B) = $\sqrt{{\left({\mu }_{{\text{Co}}^{+3}}\right)}^2+{\left({\mu }_{{\text{Ti}}^{+3}}\right)}^2}$ [34]. Also, the analysis of the dc and ac susceptibilities combined with the weak anomalies observed in the C_p versus T data has shown the existence of a quasi‐long‐range ferrimagnetic state below T_N ~ 47.8 K and a compensation temperature of T_CMP ~ 32 K [34].

System	C (emu K/mol/Oe)	χo (emu/mol/Oe)	σo (Oe mol K emu^−1)	θ (K)	μeff (μB)	μ (A) (μB)	μ(B) (μB)
Co2TiO4	5.245	0.0419	31.55	49.85	6.5	3.87	5.19
		NAA	NAB	NBB	JAA	JAB	JBB
		17.319	35.700	12.720	3.25 kB	4.47 kB	3.18 kB
Co2SnO4	4.889	0.0436	102.370	39.5	6.25	3.87	4.91
		NAA	NAB	NBB	JAA	JAB	JBB
		21.564	33.201	10.68	4.05 kB	5.26 kB	4.28 kB

Table 1.

The list of various parameters obtained from the Néel fit of χ⁻¹ versus T curve recorded under zero‐field‐cooled condition.

The real and imaginary components of the temperature dependence of ac‐susceptibility data χ_ac(T) (= χ′(T) + i χ″(T)) recorded at different frequencies for both the polycrystalline samples Co₂SnO₄ and Co₂TiO₄ show the dispersion in their peak positions (T_P(f)) similar to the compounds exhibiting spin‐glass‐like ordering [27, 33]. Figure 8 shows the χ′(T) and χ″(T) data of Co₂SnO₄ and Co₂TiO₄ recorded at different measuring frequencies ranging from 0.17 to 1202 Hz with peak‐to‐peak field amplitude H_ac = 4 Oe under zero dc‐bias field. It is clear from these figures that the peaks seen in both cases show pronounced frequency dependence, which suggests the dynamical features analogous to that of observed in spin‐glass systems. A detailed analysis of such frequency dependence of χ′(T) and χ″(T) using two scaling laws described below provides the evidence for spin‐glass‐like characteristics below T_N. For example, applying the Vogel‐Fulcher law (below equation) for interacting particles

$\tau ={\tau }_0\exp \left(\frac{E_a}{k_{\text{B}}\left(T-T_0\right)}\right)$E4

and the best fits of the experimental data (the logarithmic variation of relaxation time”τ” as a function of 1/(T_F ‐ T₀) as shown in Figure 8a yields the following parameters: interparticle interaction strength T₀ = 39.3 K and relaxation time constant τ ₀ = 7.3 × 10⁻⁸ s for Co₂SnO₄. Here, we define the freezing temperature for each frequency is T_F, angular frequency ω as 2π f (ω =1/τ), k_B is the Boltzmann constant, and E_a is an activation energy parameter. Such large value of τ₀ indicates the presence of interacting magnetic spin clusters of significant sizes in the polycrystalline Co₂SnO₄ system. The origin of such spin clusters may arise from a short‐range magnetic order occurring due to the competition between ferrimagnetism and magnetic frustration. Another characteristic feature that the spin‐glass systems follows is the power law (Eq. (5)) of critical slowing down in a spin‐glass phase transition at T_SG (note that the T_P(f) data represent a relatively small temperature interval):

$\tau ={\tau }_0{\left(\frac{T}{T_g}-1\right)}^{-z\upsilon }$E5

Figure 7(a).

The temperature dependence of the ac‐magnetic susceptibilities (i) real χ′(T) component and (ii) imaginary χ″(T) component of bulk polycrystalline Co₂SnO₄ system recorded at various frequencies under warming condition using dynamic magnetic field of amplitude h_ac = 4 Oe and zero static magnetic field H_dc = 0. The lines connecting the data points are visual guides [27].

Figure 7(b).

The temperature dependence of the ac‐magnetic susceptibilities (iii) real χ′(T) component and (iv) imaginary χ″(T) component of bulk polycrystalline Co₂TiO₄ system recorded at various frequencies under warming condition using dynamic magnetic‐field of amplitude h_ac = 4 Oe and zero static magnetic field H_dc = 0. The lines connecting the data points are visual guides [33].

Figure 8.

The best fit of the relaxation times to the (a) Vogel‐Fulcher law and the (b) power law for the spinels Co₂SnO₄ and Co₂TiO₄.

The least‐square fit using the power law of the data shown in Figure 7 is depicted in Figure 8b. Here, T_SG is the freezing temperature, τ₀ is related to the relaxation of the individual cluster magnetic moment, and zν is a critical exponent. The least‐square fit analysis for Co₂SnO₄ gives T_SG = 39.6 K, τ₀ = 1.4 × 10⁻¹⁵ s, and zν = 6.4. Since the value of zν obtained in the present case lies well within the range (6–8) of a typical spin‐glass systems; thus, one can conclude that Co₂SnO₄ exhibits spin‐glass‐like phase transition across 39 K just below the T_N ∼ 41 K [20, 27]. In these studies, the difference in T₀ and T_SG is very small (∼0.3 K) suggesting the close resemblance between the current Co₂SnO₄ system and the compounds exhibiting spin‐glass‐like transition. However, the situation for Co₂TiO₄ is bit different; in particular, the best fit to the Vogel‐Fulcher law yields T₀ = 45.8 K and τ₀ = 3.2 × 10⁻¹⁶ s and the power law yielded fairly unphysical values of the fitting parameters: for example, τ₀ ~ 10⁻³³ s with zν > 16, indicating the lack of spin‐glass‐like phase transition [33]. Although, the magnitude and shift of the ac‐susceptibility values both χ′(T) and χ″(T) strongly suppressed in the presence of dc‐magnetic field (H_DC) in a similar way as it occurs in a typical spin‐glass system perfectly following the linear behavior of H^2/3 versus T_P (AT‐line analysis). Under such tricky situation, it is very difficult to conclude that Co₂TiO₄ is a perfect spin‐glass or not (of course one can call it as a pseudo‐spin‐glass system). Nevertheless, the ac‐magnetic susceptibility data and its analysis suggested that the both Co₂SnO₄ and Co₂TiO₄ systems consist of interacting magnetic clusters close to a spin‐glass state.

Another interesting feature of both Co₂SnO₄ and Co₂TiO₄ is that they show asymmetry in M‐H hysteresis loops unveiling giant coercivities and bipolar exchange bias under both ZFC and FC cases below their T_N [27, 31, 33]. Earlier studies by Hubsch et al. have shown unusual temperature dependence of coercive field H_C(T) in polycrystalline Co₂TiO₄ sample where the M‐H loops were measured in 20‐kOe field at different temperatures below 60 K [31]. It is well known that the discovery of exchange‐bias (H_EB) effect in the structurally single‐phase materials with mixed magnetic phases has recently gained tremendous attention because of its technological applications in the development of Read/Write heads of the magnetic recording devices [41]. Generally, H_EB has been experimentally observed only in the systems cooled in the presence of external magnetic field (FC mode) from above the Néel temperature or spin‐glass freezing point. Such systems usually comprise of variety of interfaces such as ferromagnetic (FM)‐antiferromagnetic (AFM), FM‐SG, FM‐ferrimagnetic, AFM‐ferrimagnetic, and AFM‐SG [41–49]. However, few recent papers have reported significant H_EB even under the zero‐field‐cooled samples of bulk Ni‐Mn‐In alloys and in bulk Mn₂PtGa [48, 49]. The source of such unusual H_EB under zero‐field‐cooled sample was attributed to the presence of complex magnetic interfaces such as ferrimagnetic/spin‐glass or AFM/spin‐glass phases [48–51]. Some recent reports have suggested that large exchange anisotropy can originate from the exchange interaction between the compensated host and ferromagnetic clusters [48–51]. Strikingly, Hubsch et al. observed the H_C→0 anomalies across T_CMP, T_SG, and T_N in the temperature‐dependent data of H_C for Co₂TiO₄ samples [31]. Slightly different results were reported by Nayak et al. in Ref. [33], where H_C values drops monotonically on approaching T_N. However, the behavior of temperature dependence of exchange‐bias field H_EB(T) and remanent magnetization M_R(T) in Ref. [33] closely resembles with the trend of H_C(T) reported by Hubsch et. al. in polycrystalline Co₂TiO₄ samples. On the other hand, the temperature behavior of H_EB(T), H_C(T), and M_R(T) in Co₂SnO₄ is way different from that of Co₂TiO₄ though they are isostructural with each other. It is likely that the different magnitudes and different temperature dependences of the moments on the Co²⁺ ions on the “A”‐ and “B”‐sites in Co₂SnO₄ are responsible for the anomalous behavior in the H_EB(T), H_C(T), and M_R(T) observed below T_N. Moreover, in Co₂SnO₄ below about 15 K, the data suggest that there is nearly complete effective balance of the antiferromagnetically coupled Co²⁺ moments at the”A”‐ and”B”‐sites leading to negligible values of H_C and M_R.

Results from neutron diffraction in Co₂TiO₄ suggested the presence of canted‐spins, likely resulting from magnetic frustration caused by the presence of nonmagnetic Ti⁴⁺ ions on the”B”‐sites. A similar canting of the spins might be present in Co₂SnO₄ although neutron diffraction studies are needed to verify this suggestion [27, 31, 33]. Other Co‐based spinel compounds that display the reversal in the orientation of the magnetic moments along with negative magnetization due to the magnetic‐compensation phenomena are CoCr₂O₄ and Co(Cr_0.95Fe_0.05)₂O₄ [3, 52, 53].

4. Magnetic properties of bulk and nanocrystalline Co₂MnO₄

Among various Co₂XO₄ (X = Mn, Ni, Co, Zn, etc.) spinels, Co₂MnO₄ has retained a unique place. In particular, Mn‐ and Co‐based spinel oxides have gained considerable interest in the recent past due to their numerous applications in the Li‐ion batteries [54, 55], sensors [56–58], thermistors [59], energy‐conversion devices [60], and as a catalyst for the reduction of nitrogen oxides [61]. Moreover, Co₂MnO₄ nanocrystals have demonstrated outstanding catalytic properties for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) [60]. ORR and OER are the essential reactions in the electrochemistry‐based energy‐storage and energy‐conversion devices. Co₂MnO₄ has shown superior catalytic activities compared to the commercial 30 wt% platinum supported on carbon black (Pt/C). Due to the special surface morphology [62], Co₂MnO₄ spinel is a very promising pseudo‐capacitor material [63, 64]. Co₂MnO₄ has also demonstrated potential applications for protective coating on ferrite stainless steel interconnects in solid‐oxide fuel cells (SOFCs) [65, 66]. Furthermore, colossal magnetoresistance (CMR) has been observed in the Mn‐ and Co‐based spinel oxides [67].

In addition to the novel catalytic properties, Co₂MnO₄ spinel exhibits intriguing magnetic properties. Lotgering first observed the existence of ferromagnetic ordering in Co₂MnO₄ spinels [68]. Ríos et al. systematically studied the effect of Mn concentration on the magnetic properties of Co_3‐xMn_xO₄ solid solutions prepared by spray pyrolysis [69]. As we have discussed in Section 2 that Co₃O₄ has an antiferromagnetic order with T_N = 30 K. When we replace Co in Co₃O₄ by Mn cation, large ferromagnetic ordering appears, which ultimately dominates the antiferromagnetic ordering at x = 1. Therefore, pure Co₂MnO₄ shows ferromagnetic behavior and this has been confirmed by detailed magnetic measurements [70, 71]. However, for value of x other than 0 (Co₃O₄) and 1 (Co₂MnO₄), both ferromagnetic and antiferromagnetic magnetic exchange interactions coexist in the system, which yields a ferrimagnetic ordering in the Co_3‐xMn_xO₄ (0 < x < 1) solid solutions [72]. Tamura performed pressure‐dependent study of Curie temperature (T_C) and found that T_C decreases with increase in pressure [71].

Pure Co₂MnO₄ possess a cubic inverse‐spinel structure: (B³⁺)[A²⁺B³⁺]O₄ (shown in Figure 9). In an ideal case, the octahedral [B] sites are occupied by divalent cations together with half of the trivalent cations and the rest of the trivalent cations occupy the tetrahedral (A) sites in the spinel structure. However, due to the presence of different oxidation states of Mn and Co cations at (A) and [B] sites, the actual cationic distribution of Co₂MnO₄ is very complex and it has been a topic of considerable debate [59, 70–84]. Different sample preparation conditions also play an important role in the cationic distribution of Co₂MnO₄. On the basis of X‐ray diffraction, electrical conductivity, magnetic, physiochemical, and neutron diffraction measurements, several different cationic distributions for nonstoichiometric Co_3‐xMn_xO₄ (0 < x < 1) have been reported in literature [59, 70–84]. From the magnetic measurements, Wickham and Croft proposed the following cationic distribution: ${\text{Co}}^{2+}[{\text{Co}}_{2-x}^{3+}{\text{Mn}}_x^{3+}]{\text{O}}_4(0<x<2)$ for solid solutions of Co_3‐xMn_xO₄ systems obtained after the thermal decomposition of co‐precipitated manganese and cobalt salts [72]. Later studies by Blasse suggested a different cationic distribution: ${\text{Co}}^{2+}[{\text{Co}}_{2-x}^{2+}{\text{Mn}}_x^{4+}]{\text{O}}_4$ [73]. Based on the physicochemical properties, Ríos et al. proposed a more complicated cationic distribution: ${\text{Co}}_{0.88}^{2+}{\text{Mn}}_{0.12}^{2+}[{\text{Co}}_{0.87}^{3+}{\text{Co}}_{0.22}^{2+}{\text{Mn}}_{0.09}^{3+}{\text{Mn}}_{0.76}^{4+}$$■$_0.06]O₄ for Co₂MnO₄ powder samples prepared by thermal decomposition of nitrate salts [74]. From the neutron diffraction and magnetic measurements, Boucher et al. reported the cationic distribution ${\text{Co}}^{2+}[{\text{Co}}_{2-x}^{3+}{\text{Mn}}_x^{3+}]{\text{O}}_4$ similar to the one reported by Wickham and Croft [72, 75]. Yamamoto et al. [76] performed neutron diffraction measurements on Co₂MnO₄ oxides prepared by chemical methods at low temperatures, and reported Mn_1‐nCo_n[Mn_xCo_2‐n]O₄ as atomic distribution. Here, n is the inversion parameter of the inverse‐spinel structure. Gautier et al. suggested two different possible cationic distributions: ${\text{Co}}^{\text{2+}}[{\text{Co}}^{\text{3+}}{\text{Mn}}_{0.35}^{2+}{\text{Mn}}_{0.29}^{3+}{\text{Mn}}_{0.36}^{4+}]{\text{O}}_4$ and ${\text{Co}}^{\text{2+}}[C{o}_{0.95}^{3+}{\text{Mn}}_{0.015}^{2+}{\text{Mn}}_{0.50}^{3+}{\text{Mn}}_{0.485}^{4+}$$■$_0.05]O₄ [77, 78]. On the contrary, the electrical conductivity measurements suggest a different cationic distribution: ${\text{Co}}^{\text{2+}}[C{o}_x^{2+}{\text{Co}}_{2(1-x)}^{3+}{\text{Mn}}_x^{4+}]{\text{O}}_4$, and ${\text{Co}}^{\text{2+}}[{\text{Co}}^{\text{2+}}{\text{Mn}}^{\text{4+}}]{\text{O}}_4$ or ${\text{Co}}^{\text{3+}}[{\text{Co}}^{\text{3+}}{\text{Mn}}^{\text{2+}}]{\text{O}}_4$ [59, 79] for Co₂MnO₄ spinel. Aoki studied the phase diagram and cationic distribution of various compositions of manganese and cobalt mixed spinel oxides [80]. He further investigated the effect of temperature and Mn concentration on the structure of manganese‐cobalt spinel oxide systems. Control over morphology, crystallite site, grain size, and specific surface area of Co₂MnO₄ powders can be achieved by thermal decomposition of precursors in a controlled atmosphere [81, 82].

Figure 9.

Cubic inverse‐spinel structure of Co₂MnO₄ in Fd‐3m (227) space group.

In 2008, Bazuev et al. investigated the effect of oxygen stoichiometry on the magnetic properties of Co₂MnO_4+δ [83]. Accordingly, two oxygen‐rich compositions, (i) Co₂MnO_4.62 and (ii) Co₂MnO_4.275, were prepared by the thermal decomposition of presynthesized Co and Mn binary oxalates (Mn_1/3Co_2/3C₂O₄· 2H₂O). Above studies also report existence of anomalous behavior in the magnetic properties of Co₂MnO_4.275 spinel at low temperatures and high magnetocrystalline anisotropy in Co₂MnO_4.62. Bazuev et al. also noticed that T_N of Co₂MnO_4+δ is highly sensitive to the oxygen stoichiometry, imperfections in the cationic sublattice and variation in the Mn oxidation states. The imperfect Co₂MnO_4+δ inherits ferrimagnetic ordering that arises due to the antiferromagnetic exchange between Co²⁺ ($e_g^4{t}_{2g}^3$) and Mn⁴⁺ ($t_{2g}^3{e}_g^0$)) cations located at the tetrahedral and octahedral sites, respectively. To further investigate the electronic states of Mn and Co cations in the Co₂MnO₄ lattice, Bazuev at al. employed the X‐ray absorption near‐edge spectroscopy (XANES) to probe the electronic states of the absorbing atoms and their local neighborhood [83]. XANES spectra of both Co₂MnO_4.62 and Co₂MnO_4.275 compositions revealed that Co is present in both Co²⁺ and Co³⁺ oxidation states while Mn is present as Mn⁴⁺ and Mn³⁺. Additionally, it was found that Mn is located in a higher symmetry octahedral crystal field environment [83]. Bazuev at al. further proposed following cationic distributions for Co₂MnO_4.275 and Co₂MnO_4.62 compositions: ${\text{Co}}_{0.936}^{2+}[{\text{Co}}_{0.936}^{\text{III}}{\text{Mn}}_{0.421}^{3+}{\text{Mn}}_{0.515}^{4+}]{\text{O}}_4$ and ${\text{Co}}_{0.66}^{2+}[{\text{Mn}}_{0.866}^{4+}{\text{Co}}_{1.072}^{\text{III}}]{\text{O}}_4$, respectively. Here, Co^III is a low‐spin cation while Co²⁺ and Mn³⁺ are high‐spin cations.

Co₂MnO₄ spinels have gained peculiar interest of researchers due to their unusual magnetic hysteresis behavior at low temperatures. Joy and Date [70] first observed the unusual magnetic hysteresis behavior in Co₂MnO₄ nanoparticles below 120 K. By means of the magnetic hysteresis loop measurements at low temperatures, they realized that the initial magnetization curve (virgin curve) lies outside the main hysteresis loop at 120 K. However, for T > 120 K, they observed normal hysteresis behavior in Co₂MnO₄. This unusual behavior of hysteresis loop at low temperatures can be associated with the irreversible domain wall motion. At low magnetic fields (H < H_C), domain walls experience substantial resistance during their motion with increasing magnetic field. Similar behavior has been observed for some other alloys [85–87]. Such irreversible movement of domain walls was ascribed to the rearrangement of valence electrons at low temperatures. Generally, in a system with mixed oxidation states of Mn at high temperatures, short‐range diffusion of Co ions, Co³⁺ ⇆ Co²⁺ associated with Mn³⁺ ⇆ Mn⁴⁺, gets activated at low temperatures. This may cause change in the local ordering of the ions at the octahedral sites. Consequently, the resistance for domain wall motion increases and this slows down the motion of domain walls. Soon after Joy and Date, Borges et al. [84] confirmed that the unusual hysteresis of Co₂MnO₄ compound indeed arises due to the irreversible domain wall motion. Borges et al. prepared various different size nanoparticles of Co₂MnO₄ by Pechini method and performed the magnetic hysteresis measurements at low temperatures [84]. They observed that the samples below a critical diameter (d < 39 nm) exhibit normal hysteresis behavior, while the bulk grain size samples (d ∼200 nm) show unusual hysteresis behavior at low temperatures. Using the spherical particle model, Borges et al. further calculated the critical diameter (d_cr) of a single‐domain wall in Co₂MnO₄ and obtained that d_cr = 39 nm [88]. Therefore, all particles with diameter d< d_cr can be considered as a single‐domain particle. Since particles with d < d_cr show normal hysteresis while particles with d > d_cr show unusual hysteresis behavior, one can conclude that the unusual hysteresis behavior is indeed due to the irreversible motion of the domain walls.

The dynamic magnetic properties of Co₂MnO₄ nanoparticles of average diameter 28 nm were reported by Thota et al. [89]. A detailed study of dc‐ and ac‐magnetic susceptibility measurements of these nanoparticles reveals the low temperature spin‐glass‐like characteristics together with the memory and aging effects (Figure 10). Figure 10 shows the temperature dependence of real and imaginary part of the ac‐susceptibility (χ′(T) and χ″(T)) of Co₂MnO₄ nanoparticles recorded at different values of frequencies (f) between 1 Hz and 1.48 kHz at a peak‐to‐peak amplitude of 1 Oe. Both χ′(T) and χ″(T) exhibit a sharp peak at the onset of ferromagnetic ordering (T_C = T₁ = 176.4 K) and a broad cusp centered at T₂ (<T_C).

The temperature at which both χ′(T) and χ″(T) attain the maximum value shifts toward high‐temperature side as the frequency increases from 1 Hz and 1.48 kHz similar to spin‐glass behavior. Such frequency dependence of χ′(T) and χ″(T) follows the Vogel‐Fulcher law (Eq. (4) and insets of Figures 10a(ii) and b(ii)) and power law of critical slowing down (Eq. (5) and insets of Figures 10a(i) and b(i)). Least‐square fit to these equation yields the following parameters (insets of Figure 10): interparticle interaction strength ( T₀) = 162 K for T₁ (118 K for T₂) and relaxation time constant τ₀ = 6.18 × 10⁻¹⁵ s for T₁ (4.4 × 10^-15 s for T₂), critical exponent (zν) = 6.01 for T₁ (7.14 for T₂), and spin‐glass transition temperature (T_SG) = 162.6 K for T₁ (119.85 K for T₂). Since the values of zν for both the peaks T₁ and T₂ of Co₂MnO₄ nanoparticles lie well within the range (6–8) of a typical spin‐glass systems, one can conclude that Co₂MnO₄ exhibits spin‐glass‐like phase transition across 162.6 K just below the T_C ∼ 176.4 K [89]. Figure 11a shows the magnetization relaxation of Co₂MnO₄ nanoparticles under ZFC and FC protocols with a temperature quench to 70 K at H = 250 Oe [89]. The magnetization relaxation during the third cycle appears as a continuation of first cycle (Figures 11b and c). Relaxation of ZFC magnetization with temperature quenching confirms the existence of memory effects in Co₂MnO₄ nanoparticles. A noticeable wait‐time (t_wt) dependence of magnetization relaxation (aging) at 50 K in both M_ZFC and M_FC was noticed in these Co₂MnO₄ nanoparticles, which further supports the presence of the spin‐glass behavior observed in this system.

Figure 10.

The real and imaginary components of the ac‐magnetic susceptibilities (χ′(T) and χ″(T)) of Co₂MnO₄ nanoparticles measured at various frequencies. The insets a(i) and b(i) represent the Vogel‐Fulcher law whereas the insets a(ii) and b(ii) represent the power law of critical slowing down for both the peaks T₁ and T₂.

Figure 11.

Magnetization relaxation M(t) under ZFC and FC protocols with a temperature quench to 70 K at H = 250 Oe. The continuation of the first and third relaxation process during (b) ZFC and (c) FC cycles.

The high‐temperature inverse magnetic susceptibility (1/χ_ZFC versus T) data of Co₂MnO₄ nanoparticles (Figure 12) fit well with the Néel's expression for ferrimagnets:

$\frac{1}{\chi }=\frac{T}{C}+\frac{1}{{\chi }_0}-\left[\frac{{\sigma }_0}{T-\theta }\right]$E6

Figure 12.

The scattered data represent the high‐temperature inverse magnetic susceptibility (1/χ_ZFC versus T) of Co₂MnO₄ nanoparticles and the solid line represents the best fit to the Néel's expression for ferrimagnets Eq (6).

The fit (red line in Figure 12) yields: C = 0.0349 emu K/g Oe, ${\chi }_0$ = 4.5 × 10^-5 emu/g Oe, ${\sigma }_0$ = 5.76 × 10⁴ g Oe K/emu, $\theta$ = 169.8 K, and the asymptotic Curie temperature T_a = C/${\chi }_0$ = 775.6 K [90, 91]. T_a gives us information about the strength of antiferromagnetic exchange coupling between Mn³⁺ and Mn⁴⁺ at octahedral sites. Moreover, the effective magnetic moment μ_eff calculated using expression: $C=\frac{N{\mu }_{\text{eff}}^2}{3k_{\text{B}}}$ turnouts to be $8.13 {\text{μ}}_{\text{B}}$.

5. Concluding remarks

In this review, magnetic properties of bulk and nanoparticles of the Co‐based spinels Co₃O₄, Co₂SnO₄, Co₂TiO₄, and Co₂MnO₄ have been summarized. The fact that the observed magnetic properties of these spinels are so different is shown to result from the different occupation of the cations on the A‐ and B‐sites and their different electronic states at these sites. The richness of the properties of spinels thus results from these differences.

Acknowledgments

The authors thank Prof. M. S. Seehra for his suggestions and guidance in organizing this chapter. Sobhit Singh acknowledges the support from the West Virginia University Libraries under OAAF program.