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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.
Department of Physics, Indian Institute of Technology, Assam, India
Department of Physics & Astronomy, West Virginia University, Morgantown, USA
*Address all correspondence to: email@example.com
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 JAB, JBB, and JAA 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 . 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 . 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 . 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 μeff ∼ 4.14 μB) . Earlier studies by Roth reported that below the Néel temperature TN ∼ 40 K, Co3O4 becomes antiferromagnetic in which the uncorrelated spins of the 8Co2+ (in 8(a), F.C. +000, ) cations in the paramagnetic state (space group O7h—Fd‐3m) are split in the antiferromagnetic state (space group T2d—F3m) into the two sublattices with oppositely directed spins of 4Co2+ ↑ (4(a), +000) and 4Co2+ ↓ (4(c) ) [5–7]. For T< TN, the neutron diffraction studies did not show any evidence of a structural phase transition.
As shown in Figure 1, the recent magnetic studies by Dutta et al.  have reported a significant difference in the antiferromagnetic ordering temperature TN ∼ 30 K of Co3O4 as compared to the earlier data (40 K); this new result however is in excellent agreement with TN = 29.92 ± 0.03 K obtained by the heat capacity Cp versus Tmeasurements reported by Khriplovich et al. . It is well known that the peak in the magnetic susceptibility data of antiferromagnets usually occurs at a temperature few percent higher than TN 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(χpT)/dT. Therefore, TN is better defined by the peak in ∂(χT)/∂Tversus Tplot . Figure 1 shows the temperature dependence of paramagnetic susceptibility χp(T) (LHS scale) and d(χpT)/dTversus T(RHS scale). For bulk Co3O4 the peak temperature value (30 K) in the d(χpT)/dTversus Tplots is lower than TN ≃ 40 K often quoted for Co3O4 [5–7, 9–10]. Thus, TN = 30 K determined from two independent techniques (i.e., χp and Cp 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 TN values and reduced magnetic moment than the bulk value (30 K, 4.14 μB) which is a consequence of finite‐size and surface effects . Salabas et al. first reported Co3O4 nanowires of diameter 8 nm and lengths of up to 100 nm by the nanocasting route and observed TB ≃ 30 K and exchange bias (He) for T< TB .
A plot of TN values versus particle size dreported by several authors for various crystallite sizes of Co3O4 is shown in Figure 2. The lowest TN value reported till now is about 15 ± 2 K for 4.34 nm size Co3O4 particles . 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 . These authors reported that the variation of TN follows the finite‐size scaling relation:
for various sizes of Co3O4 nanoparticles. Accordingly, they obtained the shift exponent = 1.1 ± 0.2 and the correlation length ξo = 2.8 ± 0.3 nm from the fitting analysis of TN versus d(Figure 2). However, these authors considered TN values as the direct peak temperature values from χ versus Tinstead of the peak point in d(χpT)/dT. Also, for the bulk grain sizes TN (∞) = 40 K was considered instead of 30 K obtained from d(χpT)/dTanalysis as discussed above. Therefore, we repeated the analysis but considering TN values obtained from d(χpT)/dTversus Tand the TN (∞) = 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 = 1.201 ± 0.2 and the correlation length ξo = 2.423 ± 0.46 nm, which are slightly different from the earlier reported values = 1.1 ± 0.2 and the correlation length ξo = 2.8 ± 0.3 nm . Nonetheless, in both the cases TN follows the finite‐size scaling relation Eq. (1).
For T> TN, the data of χversus T(Figure 3) are fitted to the modified Curie‐Weiss law χP = χ0 + [C/(T + θ)] with C= Nμ2/3kB, μ2 = g2J(J+ 1)μB2, θis the Curie‐Weiss temperature and χ0 contains two contributions: the temperature‐independent orbital contribution mentioned earlier and the diamagnetic component χd= −3.3 × 10−7 emu/g Oe . Usually χ0is estimated from the plot of χversus 1/Tin 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/χP) 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 J1ex and J2ex, the expressions for TN and θ, using the molecular‐field theory, can be written as [5, 6]
In order to determine J1ex and J2ex, the magnitude of effective J(J+ 1) for Co2+ is required. Since the Curie constant Cis equivalent to Nμ2/3kB with μ= g[ J(J+ 1)]½μB where gis the Landé g‐factor and Jis the total angular momentum. Using the magnitude of g= 2 and Cfrom Figure 3, one can estimate the effective magnetic moment μeff = 4.27μB for bulk Co3O4 and μ= 4.09μB for Co3O4 nanoparticles of size ∼17 nm. The spin contribution to the above magnitudes of μis 3.87μB for Co2+ with spin S= 3/2. Obviously, there is some additional contribution resulting from the partially restored orbital angular moment for the 4F9/2 ground state of Co2+ [5, 6]. Using Eqs. (2) and (3) and the values of “θ,” “TN,” and “μ” for the two cases yields J1ex = 11.7 and J2ex = 2.3 K for bulk, and J1ex = 11.5 and J2ex = 2.3 K for the Co3O4 nanoparticles (d∼ 17 nm). Thus, both the exchange constants J1ex and J2ex correspond to antiferromagnetic coupling. From the magnitudes of Cin Figure 3, the value of μis obtained as 3.28 and 3.43 μB for bulk and Co3O4 nanoparticles (d∼17 nm), respectively. These magnitudes of μare lower than the spin contribution (3.87 μB) 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 .
For a typical bulk antiferromagnet, below TN, the magnetization is expected to vary linearly with applied external magnetic field Hbelow the spin‐flop field. Therefore, the corresponding coercive field Hc and exchange‐bias field He 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 Hc = 200 Oe for the zero‐field‐cooled sample and asymmetric (shifted) hysteresis loop with Hc = 250 Oe and He = −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 Hc as well. The temperature dependence of Hc and He for the nanoparticles of Co3O4 cooled under H= 20 kOe from 300 K to the measuring temperature is shown in Figure 5. Both Hc and He approach to zero above TN. The inset of Figure 5 depicts the training effect, i.e., change in the magnitude of He 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.  in the Co3O4 nanowires of 8 nm diameter although the magnitudes of He and Hc 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 He 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 . 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 TN when the spins in the core become disordered. The observation of somewhat lower magnetic moment per Co2+ ion, smaller values of exchange constants J1ex and J2ex, and lower TN 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.
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
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 TN [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 μB and the magnetic moment at octahedral B‐sites μ(B) is equal to 5.19 and 4.91 μB 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 TN ∼ 44 ± 2 K . They also suggested that Co2SnO4 should contain two equally populated sublattices that align collinearly and couple antiferromagnetically . 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 . They also reported very high values of coercive field HC > 50 kOe below TN . The reported magnetization value at 16 K per Co2+ ion was about 2.2 × 10−3 µB 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 ± 0.2µB was close to the standard value of 4.13 emu/moland 4.8 µB, 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 TN .
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 . 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 . In 1987, Srivastava et al. reported multiple peaks in the temperature dependence of ac‐magnetic susceptibility χac(T) for both Co2SnO4 and Co2TiO4 below their TN providing the evidence of Gabay and Tolouse mixed‐phase transitions [25, 28]. Nevertheless, some recent studies proved the existence of transverse spin‐glass state TSG (∼39 K) just below the TN (= 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 TSG (∼46 K) and TN (=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 χdc(T) for both Co2SnO4 and Co2TiO4 measured under ZFC and FC (H@100 Oe) conditions in Figure 6. These χdc(T) plots show typical characteristics of ferrimagnetic ordering with peaks across the Néel temperatures TN = 47 K (for Co2TiO4) and 39 K (for Co2SnO4). However, for T≤ 31.7 K an opposite trend in the χdc(T) values was noticed for Co2TiO4 with χdc ∼ 0 at magnetic‐compensation temperature TCMP = 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 χdc(T) approaches to zero. Consequently, χdc‐ZFC exhibits negative magnetization until TSG. It is expected that the different magnetic moments on the tetrahedral (A) and the octahedral (B) sites, and their different temperature dependence (i.e., μA(T), μB(T)) play a major role on the global magnetic ordering of both systems.
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 . On the contrary, the Co2SnO4 shows the perfect tetravalent nature of stannous ions without any trivalent signatures of Co3+ [Co2+][Co2+Sn4+]O4 . 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 TN, for example, exhibiting the magnetic‐compensation behavior, sign reversible zero‐field exchange‐bias, and negative slopes in the Arrott plots (H/Mversus M2) [27, 34].
At high temperatures (for all T> TN), 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/χ0) – [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 = Ta =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 Ta. For both the systems, the effective magnetic moment μeff is determined from the formula C= Nμeff2/3kB. The experimentally observed magnetic moments at the B‐sites μ(B) = 5.19 μB 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 μB) and Ti3+ (1.73 μB), μ(B) = . Also, the analysis of the dc and ac susceptibilities combined with the weak anomalies observed in the Cp versus Tdata has shown the existence of a quasi‐long‐range ferrimagnetic state below TN ~ 47.8 K and a compensation temperature of TCMP ~ 32 K .
σo (Oe mol K emu−1)
The list of various parameters obtained from the Néel fit of χ−1 versus Tcurve recorded under zero‐field‐cooled condition.
The real and imaginary components of the temperature dependence of ac‐susceptibility data χac(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 (TP(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 Hac = 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 TN. For example, applying the Vogel‐Fulcher law (below equation) for interacting particles
and the best fits of the experimental data (the logarithmic variation of relaxation time”τ”as a function of 1/(TF ‐ T0) as shown in Figure 8a yields the following parameters: interparticle interaction strength T0 = 39.3 K and relaxation time constant τ0 = 7.3 × 10−8 s for Co2SnO4. Here, we define the freezing temperature for each frequency is TF, angular frequency ωas 2πf(ω=1/τ), kB is the Boltzmann constant, and Eais 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 TSG(note that the TP(f) data represent a relatively small temperature interval):
The least‐square fit using the power law of the data shown in Figure 7 is depicted in Figure 8b. Here, TSG is the freezing temperature, τ0 is related to the relaxation of the individual cluster magnetic moment, and zνis a critical exponent. The least‐square fit analysis for Co2SnO4 gives TSG = 39.6 K, τ0 = 1.4 × 10−15 s, and zν= 6.4. Since the value of zνobtained 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 TN ∼ 41 K [20, 27]. In these studies, the difference in T0 and TSG 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 . 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 (HDC) in a similar way as it occurs in a typical spin‐glass system perfectly following the linear behavior of H2/3 versus TP (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 TN [27, 31, 33]. Earlier studies by Hubsch et al. have shown unusual temperature dependence of coercive field HC(T) in polycrystalline Co2TiO4 sample where the M‐H loops were measured in 20‐kOe field at different temperatures below 60 K . It is well known that the discovery of exchange‐bias (HEB) 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 . Generally, HEB 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 HEB even under the zero‐field‐cooled samples of bulk Ni‐Mn‐In alloys and in bulk Mn2PtGa [48, 49]. The source of such unusual HEB 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 HC→0 anomalies across TCMP, TSG, and TN in the temperature‐dependent data of HC for Co2TiO4 samples . Slightly different results were reported by Nayak et al. in Ref. , where HC values drops monotonically on approaching TN. However, the behavior of temperature dependence of exchange‐bias field HEB(T) and remanent magnetization MR(T) in Ref.  closely resembles with the trend of HC(T) reported by Hubsch et. al. in polycrystalline Co2TiO4 samples. On the other hand, the temperature behavior of HEB(T), HC(T), and MR(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 HEB(T), HC(T), and MR(T) observed below TN. 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 HC and MR.
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(Cr0.95Fe0.05)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 , energy‐conversion devices , and as a catalyst for the reduction of nitrogen oxides . Moreover, Co2MnO4 nanocrystals have demonstrated outstanding catalytic properties for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) . 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 , 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 .
In addition to the novel catalytic properties, Co2MnO4 spinel exhibits intriguing magnetic properties. Lotgering first observed the existence of ferromagnetic ordering in Co2MnO4 spinels . Ríos et al. systematically studied the effect of Mn concentration on the magnetic properties of Co3‐xMnxO4 solid solutions prepared by spray pyrolysis . As we have discussed in Section 2 that Co3O4 has an antiferromagnetic order with TN = 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 xother than 0 (Co3O4) and 1 (Co2MnO4), both ferromagnetic and antiferromagnetic magnetic exchange interactions coexist in the system, which yields a ferrimagnetic ordering in the Co3‐xMnxO4 (0 < x< 1) solid solutions . Tamura performed pressure‐dependent study of Curie temperature (TC) and found that TC decreases with increase in pressure .
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 Co3‐xMnxO4 (0 < x< 1) have been reported in literature [59, 70–84]. From the magnetic measurements, Wickham and Croft proposed the following cationic distribution: for solid solutions of Co3‐xMnxO4 systems obtained after the thermal decomposition of co‐precipitated manganese and cobalt salts . Later studies by Blasse suggested a different cationic distribution: . Based on the physicochemical properties, Ríos et al. proposed a more complicated cationic distribution: 0.06]O4 for Co2MnO4 powder samples prepared by thermal decomposition of nitrate salts . From the neutron diffraction and magnetic measurements, Boucher et al. reported the cationic distribution similar to the one reported by Wickham and Croft [72, 75]. Yamamoto et al.  performed neutron diffraction measurements on Co2MnO4 oxides prepared by chemical methods at low temperatures, and reported Mn1‐nCon[MnxCo2‐n]O4 as atomic distribution. Here, nis the inversion parameter of the inverse‐spinel structure. Gautier et al. suggested two different possible cationic distributions: and 0.05]O4 [77, 78]. On the contrary, the electrical conductivity measurements suggest a different cationic distribution: , and or [59, 79] for Co2MnO4 spinel. Aoki studied the phase diagram and cationic distribution of various compositions of manganese and cobalt mixed spinel oxides . 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].
In 2008, Bazuev et al. investigated the effect of oxygen stoichiometry on the magnetic properties of Co2MnO4+δ . Accordingly, two oxygen‐rich compositions, (i) Co2MnO4.62 and (ii) Co2MnO4.275, were prepared by the thermal decomposition of presynthesized Co and Mn binary oxalates (Mn1/3Co2/3C2O4· 2H2O). Above studies also report existence of anomalous behavior in the magnetic properties of Co2MnO4.275 spinel at low temperatures and high magnetocrystalline anisotropy in Co2MnO4.62. Bazuev et al. also noticed that TN of Co2MnO4+δ is highly sensitive to the oxygen stoichiometry, imperfections in the cationic sublattice and variation in the Mn oxidation states. The imperfect Co2MnO4+δ inherits ferrimagnetic ordering that arises due to the antiferromagnetic exchange between Co2+ () and Mn4+ ()) 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 . XANES spectra of both Co2MnO4.62 and Co2MnO4.275 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 . Bazuev at al. further proposed following cationic distributions for Co2MnO4.275 and Co2MnO4.62 compositions: and , respectively. Here, CoIII 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  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< HC), 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.  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 . 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 (dcr) of a single‐domain wall in Co2MnO4 and obtained that dcr = 39 nm . Therefore, all particles with diameter d< dcr can be considered as a single‐domain particle. Since particles with d< dcr show normal hysteresis while particles with d> dcr 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. . 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 peakat the onset of ferromagnetic ordering (TC = T1 = 176.4 K) and a broad cusp centered at T2 (<TC).
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 (TSG) = 162.6 K for T1 (119.85 K for T2). Since the values of zνfor 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 TC ∼ 176.4 K . 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 . 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 (twt) dependence of magnetization relaxation (aging) at 50 K in both MZFC and MFC was noticed in these Co2MnO4 nanoparticles, which further supports the presence of the spin‐glass behavior observed in this system.
The high‐temperature inverse magnetic susceptibility (1/χZFC versus T) data of Co2MnO4 nanoparticles (Figure 12) fit well with the Néel's expression for ferrimagnets:
The fit (red line in Figure 12) yields: C= 0.0349 emu K/g Oe,= 4.5 × 10-5 emu/g Oe, = 5.76 × 104 g Oe K/emu, = 169.8 K, and the asymptotic Curie temperature Ta = C/= 775.6 K [90, 91]. Ta gives us information about the strength of antiferromagnetic exchange coupling between Mn3+ and Mn4+ at octahedral sites. Moreover, the effective magnetic moment μeff calculated using expression: turnouts to be .
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.