Comparison of Magnetic and Electrical Properties of Manganese-Doped Cobalt Ferrite Nanoparticles

Abstract

Cobalt ferrites have been regarded as one of the competitive candidates in diversified applications because of their tailored electrical and magnetic properties by tuning structural parameters with doping of transition and/or rare earth elements. To meet the ever-growing technological demands, varieties of new compositions of cobalt ferrites with diverse properties are continuously developed and produced through different synthetic routes across the globe and are made available to the consumers. In the recent past, three compositions of manganese-doped cobalt ferrite nanoparticles have been developed through a solid-state reaction route by using the planetary ball milling technique. In this chapter, a comparison on their novelty in respect of their magnetic and electrical properties is made to spot their possible applications. Besides, few suggestions have been put forwards for further investigations to meet the challenges in the days to come.

1. Introduction

Cobalt ferrite (CoFe₂O₄) is a ceramic metal oxide having an inverse spinel structure with the basic composition formula (B)^Tet [AB]^OctO₄. Its tetrahedral and octahedral sites are formed by oxygen ions in a face-centered cubic crystal structure. Fe³⁺ ions occupy the tetrahedral site, and Fe³⁺ and Co²⁺ ions occupy the octahedral site in equal proportionate. This cobalt ferrite exhibits the property of an electrical insulator and magnetic conductor/semiconductor similar to ferrite materials with a large number of diversified applications. An excellent ability to incorporate metals ions in its lattice attracts materials scientists to synthesize and optimizes them with new interesting properties. The manganese-doped cobalt ferrites mostly demonstrate magnetic, electrical, and catalytic properties, which are correlated to their structural properties. The doping of metallic ions having an ionic radius less than unity (<1 Å) changes the structural and morphological properties in the doped cobalt ferrites [1, 2, 3]. The cation distribution over the tetrahedral and octahedral sites governs tuning and/or tailoring the structural properties of the doped cobalt ferrites that in turn lead to change their magnetic, electrical, and catalytic behavior. This distribution of cations depends on various factors like the size of the dopant ion, charge possessed by the dopant ion, site preference of the dopant ion, synthesis technique, preparation conditions, heat treatments, and the extent of dopant. Besides, the particle size plays a significant role to change the structural, morphological, magnetic, electrical, and catalytic properties of doped cobalt ferrite nanoparticles [4, 5]. Over the years, cobalt ferrites doped with transition (TE) and rearearth (RE) elements have been synthesized by various techniques and optimized for multifarious applications. Recently, nanoparticles of cobalt ferrites doped with metallic ions have received renewed attention from the scientific community across the globe to optimize them in medical science, catalyst, environmental, and sensor applications [6, 7, 8, 9, 10, 11, 12]. By the time, a series of investigations on structural, electrical, and magnetic properties of manganese-doped cobalt ferrite nanoparticles were reported with composition formulas Co_1−xMn_xFe₂O₄, CoMn_xFe₂O_4, and Co_1+xMn_xFe_2−xO₄, where x refers to the concentration levels, $0.125\le x\le 0.5$. In the first composition, manganese (Mn) was substituted for cobalt (Co), and in the second composition, Mn was substituted for Iron (Fe). Both of them were stoichiometric compositions. In the third composition, Mn substituted for Fe, but the same concentration/amount of Co was added in this composition to make this system a non-stoichiometric one [3, 4, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The main purpose of this chapter is to consolidate and make a comparison of the obtained results to focus their possible applications in light of their electrical and magnetic properties and also suggest further investigations. In this chapter, significant structural, magnetic, and electrical properties of manganese-doped cobalt ferrite nanoparticles with stoichiometric and non-stoichiometric compositions have been discussed along with their possible applications considering their academic and research values.

1.1 Theoretical aspects

1.1.1 Solid-state reaction and its mechanism

Solid-state reaction simply refers to all solventless processes leading a solid reactant to a solid product [24]. It occurs within the rigid constraint environment of the crystal lattice. The solid-state reaction provides the extreme case for evaluating the effect of intermolecular forces on a reaction and their influence on reaction mechanism and direction. The confined environment of the reactant crystal lattice can control the kinetic features of a reaction and hence the nature of the products. This solid-state reaction can take place with minimum requirements of energy and atomic or molecular motions according to the topochemical principle, proposed by Cohen and Schmidt. The advantages of solid-state reactions are as follows: (i) the atom economic nature of the solid-state reactions and the limited formation of side products, (ii) no solvents are required in the reaction so no waste disposed of issue associated with the solvent comes out. As such products do not require vigorous purification, (iii) the constrained environment can lead to novel chemical reactions, and (iv) this reaction is a bit faster than others [25]. To understand the mechanism of the solid-state reaction, let us consider the thermal reaction of two crystals of compounds A and B, which are in intimate contact across one face as shown in Figure 1. When no melt is formed during the reaction, the reaction has to occur initially at the points of contact between A and B compounds and later by diffusion of the constituents through the product’s phase. Hence, there will be two stages in the solid-state reaction, namely:

Figure 1.

Schematic diagram of the mechanism of solid-state reaction. (i) The first stage of the reaction is the formation of nuclei of the product phase C at the interface between A and B. After nucleation of product C has occurred, a product layer is formed as shown in b. (ii) At this stage, there are two reaction interfaces, one between A and C and another between C and B. For further reaction to occur, counter-diffusion of ions from A and B must occur through the existing product layer C to the new reaction interfaces.

As the reaction progresses, the product layers become thicker, which result in progressively longer diffusion paths and slower reaction rates because the product layer between the reacting particles acts as barriers. So, the reaction is controlled by the lattice diffusion and the rate law has a parabolic form and is given by $\frac{\mathit{dx}}{\mathit{dt}}=\mathrm{\kappa }{x}^{-1}$, where x is the amount of reaction, which is equal to the thickness of the growing product layer. Ions are normally regarded as being trapped on their appropriate lattice sites, and it is difficult for them to move to the adjacent sites. Only at very high temperatures, the ions have sufficient energy to diffuse through the crystal lattice. As a thumb rule, two-third of the melting temperatures of one constituent is sufficient to achieve diffusion sufficiently and hence to enable the solid-state reaction. From this discussion, it is clear that the reaction between two solids may not occur even if thermodynamic considerations favor the product formation. So, three important factors influence the rate of reaction between solids as (i) the area of contact between reacting solids and hence their surface area, (ii) the rate of nucleation of the product phase, and (iii) the rate of diffusion through the product’s phase. However, apart from the problems arising from the nucleation and diffusion, this method suffers from several additional disadvantages such as: (i) the undesirable phase may be formed, (ii) the homogeneous distribution of dopants sometimes is difficult to achieve, (iii) there is a limited possibility to monitor the progress of the reaction, (iv) because of this difficulty, mixtures of reactants and products are frequently obtained, so separating the desired products from the mixture is generally difficult, and (v) in many systems, the reaction temperature cannot be raised as high as necessary for the reasonable reaction, because one or more components of the reacting mixture may be volatile. So, it is necessary to optimize critical parameters. A variety of techniques are there for achieving the products by this solid-state reaction. Despite some disadvantages like the irregular shape of the particle, the possibility to be infected by impurities, ball milling is a versatile and cost-effective technique owing to having control over the particle size to some extent by the milling time [26].

1.1.2 Basics on ferrite structure

The spinel is any of a class of minerals, whose name coined from the mineral spinel MgAl₂O₃. The general formula of the spinel is given by$A^{2+}{B}_2^{3+}{O}_4^{2-}$ when it is crystallized in the cubic (isometric) crystal systems. In it, the oxide ions (anions) are arranged in a cubic close-packed lattice. A and B are the cations bearing charges 2⁺ and 3⁺ to have it overall charge neutral. The cations A and B occupy some or all of the tetrahedral and octahedral sites in the lattice depending on the preferences of sites. The A cations are usually divalent and B cations trivalent. But other combinations are also possible. The anions are usually oxygen; when the other chalcogenides¹ constitute the anion sublattices, then they are referred to as a Thiospinel. It notable here that A and B can also be the same metal but with different valences. For example, magnetite, whose formula can be written as ${\mathit{Fe}}_3{O}_4$ or ${\mathit{Fe}}^{2+}{\mathit{Fe}}_2^{3+}{O}_4^{2-}$, is the most abundant member of the spinel group. The spinels are grouped in series by the B cations. The main spinel groups are shown in Table 1 with respective generic formula, where A cations are exhibited by X for convenience to denote its variable constituents.

Group	Cations		Anions	Formula	X = A cation	Remark
	A	B
Aluminum Spinel	X	Aluminum (Al)	Oxygen (O)	XAl2O4	X = Mg, Be, Zn, Fe, Mn
Iron Spinel	X	Iron (Fe)	Oxygen (O)	XFe2O4	X = Co, Cu, Mn, Mg, Fe, Ni, Ti, (Zn, Fe)	Ferrites
Chromium Spinel	X	Chromium (Cr)	Oxygen (O)	XCr2O4	X = Fe, Mg, Zn

Table 1.

Main spinel group with respective generic formula.

1. Franklinite is excluded from the given formula because it is represented as (Fe, Mn, Zn)(Fe, Mn)₂O₄.

2. The heterogeneity of spinel group members varies based on composition with ferrous and magnesium-based members varying greatly as in solid solution, which requires similarly sized cations. However, ferric and aluminum-based spinels are almost entirely homogeneous due to their large size difference.

They usually have fcc packing of anions. In the spinel structure, there are 64 tetrahedral and 32 octahedral sites. 1/8th tetrahedral and 1/2th octahedral sites are usually occupied in its unit cell. Accordingly, a spinel unit cell is made up of eight fcc cells of oxygen ions in the configuration 2 × 2 × 2. So it is a big structure consisting of 32 oxygen ions, 8 A ions, 16 B ions, and a total of 56 ions [3]. Thus, a spinel unit cell contains two types of sublattices as depicted in Figure 2. These two types of sublattices repeat alternately in a three-dimensional array to produce a spinel unit cell, which ultimately requires eight sublattices.

Figure 2.

Schematic diagram of the unit cell structure of spinel ferrite [4, 5].

The interstices available in an ideal close-packed structure of rigid oxygen anions can incorporate in the tetrahedral sites, only the metal ion with a radius r_tetra ≤ 0.30 Å and in octahedral sites, only ions with a radius r_oct ≤ 0.55 Å. To accommodate cations like Co²⁺, Cu²⁺, Mg²⁺, Ni^2+, and Zn²⁺, the lattice has to be expanded. The difference in the expansion of octahedral and tetrahedral sites is characterized by a parameter called oxygen parameter (u). In an ideal spinel, the tetrahedral and octahedral sites are enlarged in the same ratio, and accordingly, the distance between the tetrahedral is (0 0 0) and the oxygen site 3a /8 or 0.375 a where ‘a’ is the lattice constant [3, 6] and hence u_ideal = 3/8 0r p.375. The tetrahedral sites are usually too small for the metal ions. Accordingly, the incorporation of divalent metal ions in tetrahedral sites induces a larger expansion of the tetrahedral sites, leading to a large value for ‘u’ than the ideal value. The tetrahedral sites are expanded by an equal displacement of the four oxygen ions onwards, along the body diagonals of the cube, still occupying the corners of an expanded regular tetrahedron. The four oxygen ions of the octahedral sites are shifted in such a way that this oxygen octahedron shrinks by the same amount as the first expands. As such, a remarkable characteristic of spinel structure is that it can form an extremely wide variety of total solid solutions. This means that the composition of a given ferrite can be strongly modified, while the basic crystalline structure remains the same. Depending on how cations occupy the interstices, spinel structure can be categorized as (i) normal spinel structure (NSS), (ii) inverse spinel structure (ISS), and mixed spinel structure (MSS). The description of all these categories of spinel structure is summarized in Table 2.

Structure	Chemical formula	Sites		Remark
		Tetrahedral (th); Coordination = 4	Octahedral [Oh]; Coordination = 6
NSS	$\left(A^{2+}\right){\left[B^{3+}\right]}_2{O}_4^{2-}$	All A^2+ cations occupy	All B^3+ cations occupy
ISS	$\left(B\right)\left[\mathit{AB}\right]O_4$	½ B^3+ cations occupy	½ B^3+, and all A^2+ cations occupy
MSS	$\left(A_{1-\mathrm{\delta }}{B}_{\mathrm{\delta }}\right){\left[A_{\frac{\mathrm{\delta }}{2}}{B}_{1-\frac{\mathrm{\delta }}{2}}\right]}_2{O}_4$	‘δ’ is called the inversion degree. It can take value between 0 and 1, i.e., 0 < δ < 1. If δ = 0, then it becomes NSS, and if δ = 1, then it becomes ISS. So, ‘δ’ can be referred to as a tunable factor between NSS and ISS in MSS.

Table 2.

Summary description of various categories of spinel structure.

Note: cations in NSS: $A^{2+}-\frac{2}{4}$ and $B^{3+}-\frac{3}{6}$; oxygen valance: $\frac{2}{4}\times 1+\frac{3}{6}\times 3=2.$

Doped ferrites are fabricated classes of ferrites where in some impurity atoms are doped in their existing ferrous or ferric matrix. They may be magnetic or ceramic compounds depending upon their compositions and also be of spinel structure or perovskite structure. The possibilities of doping in producing doped ferrites along with the generic formula of such ferrites are in its brief discussion focusing on the spinel structure. In this case, it is possible to introduce impurity of the same valance either in A or B and even in both sites by maintaining wt% or at% of M and impurity (M’). Accordingly, the generic formula for doped spinel ferrites may be written as summarized in Table 3.

Degree of inversion (i)	Occupation of sites	Structure of spinel ferrites	Generic formula
0	A site by $\left(M_{1-x}-M_x^{\prime }\right)$ and B site by Fe	NSS	${\left(M_{1-x}-M_x^{\prime }\right)}^A{\left[{\mathit{Fe}}_2\right]}^B{O}_4$
1	A site by Fe and B site by $\left(M_{1-x}-M_x^{\prime }\right)$ and Fe	ISS	${\left(\mathit{Fe}\right)}^A{\left[\left(M_{1-x}-M_x^{\prime }\right)\mathit{Fe}\right]}^B{O}_4$
0.5	A site by ½ M and ½ Fe, B site by ½ $\left(M_{1-x}-M_x^{\prime }\right)$ and $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ Fe	MSS	${⟨{\left(M_{1-x}-M_x^{\prime }\right)}_{0.5}{\mathit{Fe}}_{0.5}⟩}^A{\left[{\left(M_{1-x}-M_x^{\prime }\right)}_{0.5}{\mathit{Fe}}_{1.5}\right]}^B{O}_4$

Table 3.

Summary description of doped spinel ferrites’ structure.

Note: 1. M’ represents the impurity of the same valance. 2. x represents the wt% or at%.

1.2 Comprehensive results and novelty of Mn-doped cobalt ferrite nanoparticles

1.2.1 Used technique for synthesis and characterization

The samples of three systems of Mn-doped CoFe₂O₄ ferrite nanoparticles were prepared through the above-explained solid-state reaction route by using the ball milling technique (Model). Laboratory graded oxide powders of cobalt (Co), manganese (Mn), and iron (Fe) were mixed in a mortar with a pestle for 2 h and then ball milled for 10 h. The ball-milled powders were then calcined at 750°C in a furnace for 1 h in the air atmosphere. The calcined powders were pelletized in the form of disc and toroid for electrical and magnetic measurements, respectively [17, 18, 22]. These toroid and disc-shaped samples were again sintered at 1050°C in the same furnace for 1 h in the air atmosphere. The Waynekerr impedance analyzer 6500B was used to measure electrical and magnetic properties over the frequency band 100 Hz–120 MHz. The calcined powders were used in the EMPYREAN (PAN analytical) XRD machine to record diffraction data for structural analysis and field emission scanning electron microscope (FESEM, JEOL, JSM-7600F) integrated with energy dispersive spectroscopy (EDS) for morphological and elemental analysis. The homemade vibrating sample magnetometer (VSM) was used to measure DC magnetization by using the calcined powders for the purpose [17, 22, 23, 24]. The formulas used for calculation of different parameters associated with structural, electric, and magnetic properties to analyze the investigated samples have been presented in tabular form as Appendix A.

1.2.2 Structural properties

The XRD patterns for all three compositions showed the sharp peaks and confirmed their crystallinity with a single-phase spinel structure. They were closely resembled and well-matched with the standards JCPDS card No. 22-1086 for CoFe₂O₄ [24]. Their FESEM micrographs (representative micrographs shown in Appendix B) showed little agglomerated particles with nearly spherical shapes and pores [17, 18, 22]. Their EDS spectrum also confirmed the presence of their compositional elements without any impurities. The Xpert pro-High Score Plus software was used to estimate their structural parameters from their strongest peaks correspond to miller plane (311), and their particle size was determined by using the Image-J software and found to be in the nanoscaled range [17, 18, 22]. All these structural and morphological parameters have been reproduced and presented in tabular form in Table 4 to demonstrate their tunability by the Mn content or concentration level (x) and to analyze the variation nature system to system:

Parameters	Co1−xMnxFe2O4				CoMnxFe2−xO4				Co1+xMnxFe2−xO4
	0.125	0.25	0.375	0.5	0.125	0.25	0.375	0.5	0.125	0.25	0.375	0.5
a (Å)	8.383	8.382	8.381	8.380	8.415	8.397	8.370	8.358	8.410	8.406	8.403	8.400
D (nm)	64.02	66.90	67.10	67.17	53.424	53.436	53.454	53.462	253.22	217.72	117.14	116.13
V (Å)^3	589.24	588.96	588.76	588.47	596.07	592.17	586.38	583.99	594.82	593.94	593.41	592.77
ρx (gm/cm^3)	5.104	5.107	5.109	5.111	4.750	4.779	4.824	4.842	3.351	3.355	3.358	3.362
LA (Å)	3.630	3.629	3.627	3.623	3.644	3.636	3.624	3.619	3.64153	3.63971	3.63863	3.63733
LB (Å)	2.964	2.963	2.962	2.960	2.975	2.969	2.959	2.955	2.97335	2.97187	2.97098	2.96992
A-O (Å)	1.817	1.815	1.813	1.811	1.822	1.818	1.812	1.809	1.82085	1.81994	1.81940	1.81875
B-O (Å)	2.097	2.095	2.092	2.089	2.103	2.099	2.092	2.089	2.10250	2.10145	2.10082	2.10007
Strain	0.00181	0.00170	0.00169	0.00168	0.00214	0.00213	0.00212	0.00211	0.14561	0.13036	0.11828	0.10833
PS (nm)	32	30	28	24	22	24	27	32	34	32	29	28

Table 4.

Structural parameters and average particle size of all three compositions [17, 18, 22].

The obtained lattice constant (a) was 8.380–8.383 Å for Co_1−xMn_xFe₂O₄ composition, 8.358–8.415 Å for CoMn_xFe_2−xO₄ composition, and 8.4–8.41 Å for Co_1+xMn_xFe_2−xO₄ composition is found almost in agreement with the literature values [25, 26, 27, 28]. The decreasing trend in lattice constant (a), cell volume (V), hopping lengths (L_A and L_B), bond lengths (A-O, and B-O), and strain were observed with the Mn content (x) for them. The increasing trend in the X-ray density with increasing Mn content (x) was observed in all three systems, which was found to maintain an agreed inverse relationship with the cell volume as illustrated in Table 1 [17, 18, 22]. The crystallite size (D_x) for the Co_1−xMn_xFe₂O₄ composition and CoMn_xFe_2−xO₄ composition was reported to increase with increasing Mn content (x) but to decrease for Co_1+xMn_xFe_2−xO₄ composition with much-enhanced value as compared to other two stoichiometric systems as illustrated in Table 1 [17, 18, 22]. This enhancement in crystallite size and the gradual decreasing trend was due to the concurrent contribution of Co²⁺ ions in this composition and the relatively smaller ionic radius of Mn²⁺ ions and was found to follow Vegard’s law as explained in the literature. The decreasing trend in particle size (PS) was noticed in Co_1−xMn_xFe₂O₄ and Co_1+xMn_xFe_2−xO₄ compositions, but its increasing trend for CoMn_xFe_2−xO₄ composition is evident in Table 4 [17, 18, 22]. Among these three systems, the crystallite size in Co_1+xMn_xFe_2−xO₄ composition is novel and significant due to the formation of granules by fudging particles together by the enhanced calcination temperature and concurrent addition of Co²⁺ with its relatively larger ionic radius (78 Å). This fact is led to changes in other physical properties of this composition and accordingly discussed in the subsequent sections to explore and/or exploit them in their possible applications.

1.2.3 Electrical properties

Cobalt ferrites are considered to be composed of layers similar to any ferrite materials. They are grain and grain boundaries [29, 30]. The electrical properties of these materials are influenced by the dopants or impurities due to change in their structural and morphological properties. These electrical properties encompass the dielectric constant or permittivity and conductivity and/or resistivity. Both the conductivity and dielectric constant have a common origin according to Koop’s phenomenological theory in the manganese-doped cobalt ferrites similar to any other doped ferrite materials [17, 18, 22, 31]. The dielectric constant comes out from the polarization of the material. There are four types of polarization in the material depending on its mechanism, namely electronic polarization, atomic (orientation) polarization, ionic polarization, and space-charge (interfacial) polarization. The conductivity or resistivity arises from the mechanism of transportation of charge carriers in the material. Cobalt ferrites are the magnetic semiconductor in which Co²⁺ ions and Fe³⁺ ions are the p-type and n-type charge carriers. In the doped cobalt ferrites, interfacial polarization plays the dominant role in their dielectric behavior and electrical conductivity due to created heterogeneity by the incorporation of dopants. The frequency and temperature response of both dielectric constant and resistivity or conductivity are instrumental to explore or exploit the possible applications in diversified fields. The frequency response of the dielectric constant exhibits the ferrimagnetic nature of the materials in all three compositions as mentioned above [17, 18, 22, 31, 32]. The increasing trend in dielectric constant with the Mn content is observed for the Co_1−xMn_xFe₂O₄ at RT due to the decreased density of Co²⁺ ions for being replaced by Mn²⁺ ions in the B site [31]. But its unpredictable and irregular variations with Mn content for the compositions CoMn_xFe_2−xO₄ and Co_1+xMn_xFe_2−xO₄ are caused by the inhomogeneity of charge careers (p-type and n-type) across the grain boundaries due to dispersed particle size distribution as reported in the literature [22, 23, 24, 33]. The electrical modulus formalism was used to all the three compositions for the purpose as (i) to identify and understand the bulk properties, electrical conductivity, and relaxation time (ii) to differentiate the grain and grain boundary conduction process from the electrode polarization effect, and (iii) microstructural correlation in both the electrical and magnetic properties of the materials. The peaks were observed in the spectra of both the real and imaginary parts of the electrical modulus for the samples of Co_1+xMn_xFe_2−xO₄ composition. The frequencies corresponding to the peak in the dispersion of the real part of electrical modulus (M’) divide the spectra into two wings (wing-I and wing-II) corresponding to the long-range mobility and short-range mobility of charge carriers. The critical relaxation time constant as determined from the peaks of the dispersion of the real part of the electrical modulus sets the boundary below of which the n-type (Fe²⁺/Fe³⁺) and above it p-type (Co²⁺/Co³⁺) carriers play the dominant role in their conductivity [18]. The frequency corresponding to the observed peak in the absorption of the imaginary part of electrical modulus (M″) is termed as the characteristic frequency, f_max, that varies with the Mn content (x). The maximum magnitude of M″ at the characteristic frequency implies the increased eddy current loss as heat radiation, and it is possible to tune this loss at any desired value by changing the Mn content. As such, it is expected that this material may be suitable to be used in hyperthermia and medical-related research. The characteristic dielectric relaxation time constant determined from the characteristic frequency depends both on the concentration level and temperature. The observed critical temperature exists in the low-temperature regime and signifies the ferromagnetic-to-spin glass state transition [18, 22]. The normal behavior of both the real part and imaginary part of complex permittivity is observed with the increase in applied frequency for the samples of Co_1−xMn_xFe₂O₄ composition. The increasing trend in the real part of its complex permittivity in the lower frequencies is due to the decreased density of Co²⁺ ions in the B site. The minimum relaxation time constant as determined from the peaks in the spectra of the imaginary part of its complex electrical modulus shows the higher conductivity at a specific Mn content (x = 0.25) due to a higher hopping rate between F²⁺ to Fe³⁺ ions across the grain boundaries. The Nyquist plot of its complex electrical modulus identifies the dominance of carrier contributions in the conduction mechanism [18, 22]. An unpredictable variation in the value of a relative dielectric constant at Mn content (x) = 0.125 and 0.5 in the samples of CoMn_xFe_2−xO₄ composition originated from the inhomogeneity of charge careers (p-type and n-type) across the grain boundaries due to dispersed particle size distribution [17, 31]. Two semicircles appeared in the Nyquist plot of electrical modulus (M′ vs. M″) as designated by semicircle-I and semicircle-II for the samples of CoMn_xFe_2−xO₄ composition. The semicircle-I occurs in the low-frequency region and corresponds to grain-boundary contribution whereas the semicircle-II in the high-frequency region corresponding to the grain contribution [18]. The increasing trend in the estimated activation energy of the samples of Co_1+xMn_xFe_2−xO₄ composition with Mn content (x) at room temperature signifies the phase tuning effect from the ferrimagnetic-to-paramagnetic phase. The temperature response of AC resistivity in the low-temperature regime for the representative sample of Co_1+xMn_xFe_2−xO₄ composition showed the metallic or insulating behavior of the material. The increasing trend of activation energy with the manganese content signifies the ferrimagnetic-to-paramagnetic phase transition and thus tuning the samples of CoMn_xFe_2−xO₄ to behave as soft magnetic material [18]. The linear decreasing trend in its DC resistivity with the increase in temperature demonstrated the semiconducting behavior of the material above room temperature [17, 31]. The decreasing trend in AC resistivity of the samples of CoMn_xFe_2−xO₄ with the increase of temperature implies their semiconducting behavior of the material [18, 22]. The negative value of magnetoresistance implies the dominance of n-type charge carriers (Fe²⁺/ Fe³⁺) of both samples of Co_1−xMn_xFe₂O₄ and Co_1+xMn_xFe_2−xO₄ compositions in their hopping process. The complex impedance marks a single metallic band in the stoichiometric composition Co_1−xMn_xFe₂O₄ and a double metallic band in the non-stoichiometric composition Co_1+xMn_xFe_2−xO₄. This double metallic band may make this sample suitable to be used in the switching as well as actuator devices. Conversely, the higher conductivity in the single metallic band is expected to generate heat by the eddy current loss for the stoichiometric composition Co_1−xMn_xFe₂O₄, which may also make it suitable to be used in hyperthermia and medical science-related research [23, 24, 25, 26]. The measurement of conductance of both stoichiometric and non-stoichiometric compositions of Mn-doped cobalt ferrite nanoparticles at RT shows that “the magnitude of AC conductivity depends almost linearly with the Mn content and thus provides tunability by Mn content. This AC conductivity causes the eddy current loss while applied a varying magnetic field across the investigated sample and dissipate to heat the materials. The heat energy is expected to be selectable over the frequency band between two cutoff frequencies as determined by their 3dB points from their respective peaks and therefore make them suitable for use in temperature switching/sensing and/or thermoelectric devices. Besides the frequency exponent for both the compositions displayed that the material is a mixer of both ionic and Debye dipole-type crystals and with the increasing Mn content they tend towards more Debye dipole-type crystals” as reported in recent literature [33].

1.2.4 Magnetic properties

The AC permeability and DC magnetization are considered to be the key parameters for understanding and explaining the magnetic behavior of the material. The magnetic modulus is similarly important to separate the local behavior of defects in the material from the effects of external agents like an air gap, stray effect, etc. It helps to understand the dynamic mechanism of permeability under the influence of the AC magnetic field. A pragmatic enhancement in the real part of permeability with the concentration levels, Mn(x) over the whole frequency band is noticed for the samples of Co_1+xMn_xFe_2−xO₄ composition [24]. Above room temperature, the increasing trend in the real part of permeability is caused by the dipolar orientations due to increased crystallite and/or grain size. The appearance of their relaxation peaks at around 537.5 K may be related to the spin resonance, and its fall from the peaks marks the ferromagnetic-to-paramagnetic phase transition and corresponds to the Curie temperature. Whereas the declining trend of the real part of permeability from the broadened peaks with the drop in temperature is occurred by the ceasing of dipolar orientations in the grains due to freezing effects. A crossover in the dispersion of the real part of magnetic modulus $M_m^{\prime }$ is noticed at a particular frequency around 58 kHz. Below this frequency $M_m^{\prime }$ increases but above it decreases with Mn content. The well-resolved peaks in the dispersion of the imaginary part of the magnetic modulus correspond to the resonance frequency and are found to decrease with the Mn content due to the observed decreasing trend in the crystallite size. The saturation magnetization and initial permeability are found to increase with the concentration levels x due to the antiferromagnetic effect of Mn²⁺ in the tetrahedral site according to Neel’s two sublattices models [24, 33, 34]. The anomalous variation in remnant magnetization and coercivity is observed due to the migration of Co²⁺ ions from the octahedral site to the tetrahedral site during phase formation in the solid-state reaction. The remnant ratio is less than unity (<<1), which marks the superparamagnetic behavior of the material. The grain boundaries interact and reduce the domain wall motion that results in lower permeability at higher sintering temperatures. Below room temperature (300 K), the magnetization increases with the drop in temperature, which occurs from the dominance of effective uniaxial anisotropy constant (K_ef) due to the additional presence of Co²⁺ ions in the octahedral (B) site because of the non-stoichiometric composition. The coercivity decreases with a further drop in temperature from a certain peak value is the signature of the possible superparamagnetic behavior of the material. The Weiss constant depends inversely with Mn content, which signifies the ferromagnetic nature of the material in the low-temperature regime [24]. A diamagnetic behavior of the materials is also marked over the frequency band 1 kHz–500 kHz for the samples of this composition due to negative values of the real part of permeability, which is the indication of the metamaterial according to Veslago through which electromagnetic wave cannot propagate and thus may make this material suitable to be used in the magnetic shielding operation. Besides, μ-negative (MNG) and double-negative (DNG) medias are identified over the frequency band 3 MHz–120 MHz at Mn(x) = 0.5 [1]. In the low-temperature regime, the dispersion of the AC permeability shows the transformation of the ferromagnetic phase to the spin-glass state at a certain temperature peak due to the frozen tendency of spins. This peak temperature may be termed as transition temperature, which is found to vary irregularly with the Mn content in the sample of composition CoMn_xFe_2−xO₄. The remnant ratio is found to be very less than unity, which marks its superparamagnetic behavior, and may this material be suitable to be used in the spintronics applications. The frequency response of both the real and imaginary parts of AC permeability demonstrates the normal behavior of the samples of Co_1−xMn_xFe₂O₄ compositions. A nonlinear increase in the real part of magnetic modulus ($M_m^{\prime }$) for the samples of this composition is observed with the increase of the applied frequency that signifies the contributions of both the wall motion (wall relaxation) and the spin rotations (rotational resonance) in its magnetization [18]. Afterward, $M_m^{\prime }$ is found to a slight linear increase to a single asymptotic value, which implies the ceasing of wall motions but the only presence of spin rotations. Both the increasing and decreasing trend up to and from well-resolved peaks is observed with the increase of frequency in the dispersion or absorption of $M_m^{"}$. The frequency that corresponds to the peak is known as the resonance frequency $f_{\mathit{res}}$ and follows the increasing trend in the crystallite or grain size. Besides, the magnitude of $M_m^{"}$ at the corresponding resonance frequency is also marked to increase with the increase in Mn content. This fact implies the more absorption of magnetic energy from the magnetic field that in turn leading to decrease spin rotations. This behavior of the material is expected to be suitable for electromagnetic suppression operations. The improvement of saturation magnetization is observed for the sample of Mn content (x = 0.5) as compared to that of un-doped cobalt ferrite nanoparticles (x = 0).

1.2.5 Comparison of electrical and magnetic properties

The applications of the synthesized materials in the real world mostly depend on the electric and magnetic properties and their tuning or alteration by the external agents like DC and/or AC electric and/or magnetic fields, electromagnetic field, the influence of temperatures both below and above room temperatures, environment, etc., thereon. A comparison is made based on the above discussion and presented in tabular form to explore the novelty of electrical and magnetic properties of manganese-doped cobalt ferrite nanoparticles of three different systems as synthesized with the composition formulas as Co_1+xMn_xFe_2−xO₄, Co_1−xMn_xFe₂O₄, and CoMn_xFe_2−xO₄ in Table 5:

Properties	Co1+xMnxFe2−xO4	Co1−xMnxFe2O4	CoMnxFe2−xO4
Dielectric constant	The decreasing trend in the real part of the dielectric constant exhibits the ferrimagnetic nature of the material. The peaks in the absorption of the imaginary part marked the predominant grain boundary contribution to the resistivity of the material [4, 17, 22]	The frequency response of both the real and imaginary parts of the dielectric constant exhibits normal behavior. The decreasing trend in the real part of the dielectric constant exhibits the ferrimagnetic nature of the material [22]	Both dielectric constant and D-factor show the normal behavior up to 2 kHz and above 2 kHz anomalous behavior. Grain boundaries are more responsive below 10 kHz, and conductive grains become more responsive above 10 kHz that ultimately increasing the conductivity of the material almost exponentially [18, 22, 28]
Electrical modulus	A boundary is marked between the long-range and short-range mobility of charge carriers. The critical relaxation time constant set a boundary below of which n-type and above it p-type charge carriers play a predominant role in their conductivity. The lowest value of activation energy at Mn(x) = 0.375 signifies faster hopping between Fe^3+ and Fe^2+ ions in the conductive grains and is associated with the deficiencies of oxygen ions in the B-site due to additional density of Co^2+ ions because of their; non-stoichiometry [18, 22]	An increasing trend in the real part of electrical modulus’ with the applied frequency is observed due to short-range mobility of charge carriers (p-type and n-type) because of the shorted hopping length (LB) in the B site. No boundary is observed between the short-range and long-range mobility of charge carriers. In the low-frequency regime, the higher values of the imaginary part of the electrical modulus indicate lower conductivity due to interfacial polarization in the grain boundaries. The Nyquist plots of complex electrical modulus separate the contribution of n-type and p-type ions in the hoping mechanism [22]	Two semicircles (semicircle-I and II) of non-ideal shape are found to occur in the Nyquist plot of electrical modulus that separates the grain boundary and grain contributions to the electrical conductivity of the material. Semicircle-I corresponds to the grain boundary, and semicircle-II corresponds to the grain contribution to the conductivity of the material [22, 23]
Resistivity/conductivity	The temperature response of AC resistivity in the low-temperature regime shows the metallic or insulating behavior of the material. The linear decreasing trend in the DC resistivity with the increase in temperature exhibits the semiconducting behavior of the material in the high-temperature regime. The higher rate of change of resistivity is marked with the applied magnetic field due to the combined effects of the faster hopping and higher mobility of n-type charge carriers across the grain boundaries [17, 22, 32]	The normal behavior of AC resistivity with the applied frequency is observed. The AC resistivity is found to decrease with Mn(x) below 0.01 MHz and above it independent of Mn(x). The value AC resistivity is large as compared to CoMnxFe2−xO4 due to the slower mobility of Co^2+ (p-type carriers) ions in the B site [22, 31]	The increasing trend of activation energy with the manganese content, as determined from the temperature-dependent AC conductivity curves, signifies ferrimagnetic-to-paramagnetic phase transition and thus tuning the sample to behave as soft magnetic material [18, 22, 33]
Conductance	Frequency non-responsive conductive band (FNRC) observed over the band 100 Hz–3 kHz, and frequency responsive conductive band (FRC) observed over the band 3 kHz–1 MHz [22, 31, 33]	Frequency non-responsive conductive band (FNRC) observed over the band 100 Hz–1 kHz, and frequency responsive conductive band (FRC) observed over the band 1 kHz–1 MHz [22, 32]	—
Impedance spectroscopy	The double metallic band is observed [20, 22, 26]	A single metallic band is observed [22, 29, 30]	—
Permeability	A pragmatic enhancement in the real part of permeability with Mn(x) over the 100 Hz–1 MHz.; μ-negative and double negative media identified over frequency band 3MH–120 MHz at Mn(x) = 0.5. A diamagnetic behavior is marked over the frequency band 1 kHz–500 kHz at Mn (x) = 0.125 [16, 22]	The frequency response of both the real and imaginary parts of AC permeability demonstrate normal behavior. The real part of permeability is non-responsive over frequency band 550 Hz–1 MHz. The decreasing trend in relaxation time with the Mn (x) implies the damping of spins in the high-frequency range [22]	The real part of AC permeability is found to increase up to 265 Hz (snapshot) and a slight decrease over a wide band of frequencies up to around 30 MHz. Afterward, its sudden fall demonstrates anomalous behavior. In the low-temperature regime, dispersion of AC permeability shows the transformation of the ferromagnetic phase to the spin-glass state at a certain peak temperature due to the frozen tendency of spins [18, 22]
Magnetic modulus	A cross-over frequency was observed at 58 kHz. Below this frequency real part of the magnetic modulus is inversely, and above it is directly proportional to Mn(x). In the low-temperature regime, a transition of ferromagnetic phase to spin glass state is marked at 131.59 K. The critical relaxation constant increases with Mn(x) that decreases, in turn, the damping of domain walls and results in both dipolar orientations and more spin rotations to increase the permeability of the material due to change in the metal chemistry. This implies that the porosity decreased due to the additional contribution of Co^2+ ions, which in turn removes pinning sites of the wall motion in the material [19, 22]	The resonance frequency as observed in the imaginary part of the magnetic modulus follows the increasing trend in the crystallite size. Besides, the magnitude of the imaginary part at the corresponding resonance frequency is also marked to increase with the increase in Mn (x). This fact implies the more absorption of magnetic energy from the magnetic field that in turn leading to decrease spin rotations. The relaxation time decreases with Mn (x) that signifies the increase in the damping mechanism of wall motion that causes a decrease in permeability of the material [17, 22]	The increasing trend of magnetic modulus up to around 248 kHz may be due to the collective effects of wall motion and spin rotation. Afterward, almost constancy is observed over the frequency band 248 kHz–10 MHz, which can be attributed to the spin-only rotation. The increasing trend in relaxation time with Mn(x) implies the predominance of dipolar orientations to contribute more to the permeability [18, 22]
Magnetization	Normal behavior is marked in the variation of saturation magnetization with Mn (x). In the low-temperature regime, a large saturation magnetization is observed that ranges from 120 to 240 emu g^−1 [22]	The improvement of saturation magnetization is observed over un-doped cobalt ferrite nanoparticles [22]	Both the saturation magnetization and initial permeability decrease with Mn(x) at RT. It exhibits normal magnetization behavior [18, 22]
Coercivity	The anomalous nature of coercivity exhibits mono-domain formation with the drop in temperature at Mn (x) 0.125 and 0.375 [22]	The anomalous nature in the variation of coercivity with Mn(x) is also marked [22, 27]	The increasing trend in coercivity shows the normal magnetic behavior of this material [17, 22]
Remnant; ratio	The remnant ratio was found to be below unity (<<1) and exhibiting possible superparamagnetic nature of the material [22]	Possible superparamagnetic behavior is observed due to the value of remnant ratio < <1 [17, 22, 27]	The remnant ratio is found to be very less than unity, which marks its superparamagnetic behavior [17, 22]
Curie; temperature	The Curie temperature is found to be tunable with Mn(x) over the range of 535 K–565 K and is found to agree with literature values [22]		The Curie temperature is marked to be controlled by the Mn(x) content, and thus, TC regulates magneto-mechanical hysteresis [22]
Magneto; resistance	The negative values of magnetoresistance showed the dominance of n-type charge carriers in its conduction mechanism [10, 22, 34]	The negative values of magnetoresistance showed the dominance of n-type charge carriers in their conduction mechanism [22]	—

Table 5.

Comparison of electric and magnetic properties of three systems of Mn-doped cobalt ferrite nanoparticles.

1.2.6 The novelty

From the comparison of manganese-doped cobalt ferrite nanoparticles in three systems as seen in Table 5, the followings are the significances or novelty in their properties:

1. All three compositions exhibit the ferrimagnetic nature of materials and almost in agreement with the cobalt ferrites. Only the lattice parameters and crystallite size are being influenced and modified by the antiferromagnetic effect of Mn²⁺ ions, and also, the cation distribution is deviated a bit from their idealistic situation according to their initial stoichiometry.

2. All the compositions show semiconducting behavior. But, the non-stoichiometric composition Co_1+xMn_xFe_2−xO₄ exhibits either insulating or metallic behavior and semiconducting behavior, which is its novel or unique property as compared to the other two compositions.

3. The electrical modulus itself manifests the novelty of all three compositions of manganese-doped cobalt ferrite nanoparticles wherein short-range and long-range mobility of charge carriers are distinctly identified for the composition Co_1+xMn_xFe_2−xO₄. The dielectric relaxation is of single relaxation non-Debye type for Co_1+xMn_xFe_2−xO₄ and Co_1−xMn_xFe₂O₄ compositions, whereas it is a double relaxation type for CoMn_xFe_2−xO₄ composition.

4. Impedance spectroscopy shows the double metallic band for the Co_1+xMn_xFe_2−xO₄ composition, whereas a single metallic band is observed in Co_1−xMn_xFe₂O₄. These are unique for these two systems, and therefore, it may be regarded as the novel property.

5. A diamagnetic behavior manifests μ-negative (MNG) and double-negative (DNG) media in Co_1+xMn_xFe_2−xO₄, which are absent in the other two compositions and therefore unique or novel property.

6. The anomalous magnetic behavior in respect of coercivity and remnant magnetization is unique for the samples of Co_1+xMn_xFe_2−xO₄ compositions below the room temperature as compared to the other two compositions.

7. The transformation of the ferromagnetic phase to the spin-glass state is another important novel property of Co_1+xMn_xFe_2−xO₄ and CoMn_xFe_2−xO₄ compositions.

8. The magnetic modulus manifests that the dependence of permeability on manganese constant that reverses at a particular frequency termed as crossover frequency, which is unique for Co_1+xMn_xFe_2−xO₄ composition.

9. The decreasing trend in the resonance frequency as determined from the well-resolved peaks of the imaginary part of the magnetic modulus with increasing Mn(x) for Co_1+xMn_xFe_2−xO₄ follows the decreasing trend of its crystallite size that manifest the correlation between permeability and crystallite or grain size and therefore novel behavior of the material.

10. The increasing trend in resonance frequency with Mn(x) for the Co_1−xMn_xFe₂O₄ follows the increasing trend of its crystallite size and shows the novel behavior.

11. The tunability of the Curie temperature by manganese content in CoMn_xFe_2−xO₄ facilitates control over the mganetomechanical hysteresis of the material.

1.3 Possible applications

The novel behavior of manganese-doped cobalt ferrite nanoparticles in three compositions as discussed above may make them suitable to be used in the following possible applications:

1. The semiconducting nature of the material may make them suitable to be used in thermoelectric devices.

2. The decreasing trend in the coercivity of Co_1+xMn_xFe_2−xO₄ composition may make this material suitable to be used as a ferrite core in the high-frequency inductor, transformer, and various microelectronic devices

3. The appearance of peaks in the dispersion of the imaginary part of the dielectric constant shows the grain boundary contribution to the resistivity in Co_1+xMnxFe_2−xO₄ which in turn corresponds to the thermal energy dissipation. This behavior of dielectric constant may make this material suitable for use in hyperthermia or related research in medical science.

4. The equality in the real part and imaginary parts of complex permittivity at resonance suggests this material of Co_1+xMnxFe_2−xO₄ composition may be used in high-frequency switching devices.

5. The enhanced negative magnetoresistance and its variation nature make the materials of Co_1+xMnxFe_2−xO₄ composition be used to identify and control the electron transport properties in the sensor and spintronics applications.

6. The varied nature of temperature-dependent resistivity may make the material of Co_1+xMnxFe_2−xO₄ composition suitable to be used in the high-temperature sensor devices.

7. The dispersion of the real part of permeability is found almost independent over the frequency band 550 Hz–1 MHz, and the imaginary part shows non-responsive over frequency band 1.5 kHz–1 MHz and makes this material of Co_1−xMn_xFe₂O₄ composition a good candidate to be used in high-frequency applications.

8. The increasing nature of magnetization with the temperature at constant concentration levels x provides the ability to control and probe the magnetization of the material and thus make the system suitable to be used in the field of sensor application.

9. The metallic frequency band as observed at x = 0.5 of all three systems in their anomalous zone above 1 MHz may make the material suitable to be used in the radio frequency detectors and fast switching devices.

10. This negative values of μ′ are the indication of a possible metamaterial that is capable of distorting or absorbing the electromagnetic waves, which makes this material suitable to be used in electromagnetic suppression or shielding operation. Conversely, the constancy in μ′ and 𝜀′ over a frequency band 500 kHz–100 MHz is an indication of potential technological material for patch/dielectric antenna.

11. The increasing nature of magnetization with the temperature at constant concentration levels x provides the ability to control and probe the magnetization of the material and thus make the system suitable to be used in the field of sensor applications.

12. The decreasing rate of resistivity with the applied magnetic field at room temperature may make this sample suitable to be used in the field of position and biosensors.

13. The relative enhancement in saturation magnetization and initial permeability in Co_1−xMn_xFe₂O₄ and CoMn_xFe_2−xO₄ as compared to that in un-doped cobalt ferrite nanoparticles reveals the use of Mn in fine-tuning the new magnetic materials for technological applications in microelectronic devices.

14. The tunability of the Curie temperature of the material of composition CoMn_xFe_2−xO₄ by the Mn(x) content may make this sample suitable to be used in the magneto-mechanical sensors.

2. Conclusion

From the above discussion, it is interesting to note that, creating defects intentionally in a methodic way, the physical properties will be changed, which is the crux route to tune or tailor the properties in the solids. Intensive investigations, therefore, were carried out over the years, and their outcomes have put in practical applications in diversified fields. Manganese-doped cobalt ferrite irrespective of its bulk and nano-dimension forms has already been utilized in applications starting from the electronic industry to biomedical research, diagnostic and labeling, and targeted delivery of medicines. In this chapter, some novel approach like electrical modulus, magnetic modulus, and impedance spectroscopy has been used to identify the change and new properties of manganese-doped cobalt ferrite nanoparticles and highlighted some possible applications that include hyperthermia and medical-related research, magnetic field shielding/suppression operation, etc. The tunability of AC conductivity of both stoichiometric and non-stoichiometric compositions of Mn-doped cobalt ferrite nanoparticles is suggested to be suitable for use in thermoelectric devices. However, there is still room for further investigations more rigorously, a few of them are suggested here for the researchers, scientists, engineers, and technologists to open up their thoughts across the globe:

1. The photocatalytic conversion of visible solar energy to generate e⁻/h⁺ in turn may produce reactive oxygen species through the redox process for the degradation of pollutants in water.

2. In a quest of exploring new alternatives of RE permanent magnets by investigating non-stoichiometric manganese-doped cobalt ferrites and compositing them to nickel to enhance magnetism and coercivity.

3. An investigation of this material is required to enhance the dye degradation capability.

4. Improvement of the efficiency and savings of energy in renewable energy devices using cobalt ferrite nanoparticles with non-stoichiometric composition.

Acknowledgments

The authors are thankful to the International Science Programs (ISP), Uppsala University, Sweden, for financial and technical support, also to the Department of Physics, Bangladesh University of Engineering and Technology, and the Department of Physics, Military Institute of Science and Technology for experimental support.

Appendix: used formulas

The formulas used for calculation of various parameters for analysis of structural, magnetic, and electrical properties along with software are listed in Table A.1.

Parameters	Formula	Remarks
Structural
Lattice constant	$d=\frac{a}{\sqrt{h^2+k^2+l^2}}$	Using High Score Plus software
Crystallite size	$0.9\mathrm{\lambda }/\beta cos\theta$	Debye–Scherrer formula
Tetrahedral hoping length	$\left(L\right)=\frac{a\sqrt{3}}{4}\mathring{\mathrm{A}}$	Stanley’s equation
Octahedral hoping length	$\left[L\right]=\frac{a\sqrt{2}}{4}\mathring{\mathrm{A}}$
Tetrahedral bond length	$A-O=\left(u-\frac{1}{4}\right)a\sqrt{3}\mathring{\mathrm{A}}$
Octahedral bond length	$B-O=\left(\frac{5}{8}-u\right)a\mathring{\mathrm{A}}$
X-ray density	${\rho }_x=\mathit{ZM}/N{a}^3$
Porosity	$P\%=\left(1-\frac{{\rho }_x}{\rho }\right)\times 100$
Strain	$\beta cos\theta =\frac{k\mathrm{\lambda }}{D}+4ϵsin\theta$	Williamson–Hall equation (plot)
Magnetic
Magnetization	$\left(V_{\mathit{\text{measd}}}\times \mathit{Cac}-\mathit{\text{Const}}\right)/M_{\mathit{\text{measrd}}}$	Using VSM
Coercivity	$\left(H_{C1}+H_{c2}\right)/2$	Using hysteresis curves
Remnant magnetization	$\left(M_{C1}+M_{c2}\right)/2$
Curie temperature	Extrapolated intercept value	Using M–T curves
Anisotropy constant
Initial permeability	The slope value at initial magnetization	Using M–H curves
Susceptibility	$M/{\mu }_{0}H$
Electrical
DC resistivity	${\rho }_{\mathit{dc}}=\mathit{RA}/l$	Using the 4-probe method
DC conductivity	${\sigma }_{\mathit{dc}}=1/{\rho }_{\mathit{dc}}$
AC resistivity	${\rho }_{\mathit{dc}}=\mathit{RA}/l$	Using Wynekerr impedance analyzer
AC conductivity	${\sigma }_{\mathit{dc}}=1/{\rho }_{\mathit{dc}}$
Magnetoresistance	$\mathit{MR}\%=\frac{{\rho }_H-{\rho }_0}{{\rho }_0}\times 100$	Using the 4-probe method
Electromagnetic
Complex permittivity	$\epsilon ={\epsilon }^{\prime }-{\mathit{j\epsilon }}^{\prime \prime }$	Measurement of real and imaginary parts by impedance analyzer
Complex permeability	$\mu ={\mu }^{\prime }-{\mathit{j\mu }}^{\prime \prime }$
Impedance	$Z=\sqrt{R^2+X^2}$
Characteristics impedance	$Z_0=\sqrt{{\mu }^{\prime }/{\epsilon }^{\prime }}$
Miniaturization factor	$n=\sqrt{{\mu }^{\prime }\times {\epsilon }^{\prime }}$
Dielectric constant	${\epsilon }_r=\frac{C}{C_0}$	Measurement of capacitance by impedance analyzer
Dielectric loss	$tan\delta ={\epsilon }^{\prime \prime }/{\epsilon }^{\prime }$	Measurement of real and imaginary parts by impedance analyzer
Magnetic loss	$tan\delta ={\mu }^{\prime \prime }/{\mu }^{\prime }$
Eddy current loss	$W_{\mathit{ed}.\mathit{\text{loss}}}=\omega {\mu }_r{\mu }_0{H}^{2}Vsin\delta$	Usual meanings
Currie constant, C	The slope of $\frac{1}{\mathrm{\chi }}-T$ plot	Usual meanings
Weise constant, $\theta$	The intercept at the temperature axis
Electric modulus	$M^{\prime }\left(\omega \right)=\frac{{\epsilon }^{\prime }\left(\omega \right)}{{\epsilon }^{\prime }{\left(\omega \right)}^2+\epsilon "{\left(\omega \right)}^2}$; $M^{"}\left(\omega \right)=\frac{{\epsilon }^{"}\left(\omega \right)}{\epsilon \prime {\left(\omega \right)}^2+\epsilon "{\left(\omega \right)}^2}$	Real part; Imaginary part
Magnetic modulus	$M_m^{\prime }\left(\omega \right)=\frac{{\mu }^{\prime }\left(\omega \right)}{{\mu }^{\prime }{\left(\omega \right)}^2+\mu "{\left(\omega \right)}^2}$; ${M^{"}}_m\left(\omega \right)=\frac{{\mu }^{"}\left(\omega \right)}{\mu \prime {\left(\omega \right)}^2+\mu "{\left(\omega \right)}^2}$	Real part; Imaginary part

Table A.1.

The formula/procedure used for calculation of different parameters used in this thesis paper [18, 19, 22].

Abbreviations

Co	symbol for cobalt
Mn	symbol for manganese
Fe	SYMBOL of iron
RT	Room temperature
x	signifies Mn concentration/concentration levels at weight percentage
Å	Angstrom (unit of dimension) = 10^−10 m
K	Kelvin scale of temperature
AC	alternating current
nm	nanometer (unit of dimension) = 10^−9 m
emu g^−1	electromagnetic unit per gram (unit of magnetization
XRD	X-ray diffraction
FESEM	field emission electron microscopy
M (H)	magnetization as a function of magnetic field
MS	saturation magnetization
MR	remnant magnetization
HC	coercivity
${\mu }_i$	initial permeability
nB	magnetic moment per formula unit in Bohr Magneton
${\mu }^{\prime }$	real part of AC permeability
${\mu }^{"}$	imaginary part of permeability
χ	susceptibility
$tan\delta$	loss factor (corresponds to both dielectric and magnetic loss)
${\epsilon }^{\prime }$	real part of permittivity
${\epsilon }^{"}$	imaginary part of permittivity
${\epsilon }_r$	relative dielectric constant
$C$	capacitance
Z	impedance
X	reactance
R	resistance
${\rho }_{\mathit{ac}}$	AC resistivity
${\sigma }_{\mathit{ac}}$	AC conductivity
fo	resonance frequency
τ	relaxation time constant
fc	critical frequency
fmax	maximum characteristic frequency
M’	real part of electric modulus
M”	imaginary part of electric modulus
Mm’	real part of magnetic modulus
Mm”	imaginary part of magnetic modulus
$∆M\left(T\right)$	change of magnetization as a function of temperature
$∆T$	change in temperature
$∆H$	change in magnetic field
$∆S$	change in entropy
$∆S_m$	change in magnetic entropy
$∆S_{\mathit{lat}}$	change in entropy due to lattice vibrations
$∆S_{\mathit{\text{elec}}}$	change in entropy due to electronic contribution